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RF Circuit Nonlinearity Characterization and Modeling for Embedded Test


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RF CIRCUIT NONLINEARIT Y CHARACTERIZATION AND MODELING FOR EMBEDDED TEST By CHOONGEOL CHO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by CHOONGEOL CHO

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This document is dedicated to the graduate students of the University of Florida.

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iv ACKNOWLEDGMENTS I would like to express my sincere appreci ation to my advisor, Professor William R. Eisenstadt, whose encouragement, guida nce, and support throughout my work have been invaluable, I also would like to thank Pr ofessors Robert M. Fox, John G. Harris, and Oscar D. Crisalle for their interest in this work and their guidance as the thesis committee members. I thank Motorola Company for financial support. I also thank Bob Stengel and Enrique Ferrer for their dedication to my res earch. In addition, I thank all of the friends who made my years at the University of Flor ida such an enjoyable chapter of my life. I am grateful to my parents and pare nts in law for their unceasing love and dedication. Finally, I thank my wife, Seon-Kyung Kim, whose endless love and encouragement were most valuable to me. Most importantly, I w ould like to thank God for guiding me everyday.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES...........................................................................................................ix ABSTRACT.....................................................................................................................xi v CHAPTER 1 INTRODUCTION........................................................................................................1 1.1 Motivation...............................................................................................................1 1.2 Research Goals.......................................................................................................2 1.3 Overview of Dissertation........................................................................................3 2 BACKGROUND..........................................................................................................5 2.1 Classifications of Distortions..................................................................................5 2.2 Taylors Series Expansion......................................................................................7 2.3 Measurement of Nonlinear System........................................................................8 3 THE RELATIONSHIP BETWEEN THE 1 dB GAIN COMPRESSION POINT AND THE THIRD-ORDER INTERCEPT POINT...................................................10 3.1 Definition of 1 dB Gain Compression and Third-order Intercept Point...............10 3.2 Classical Approach to Model IIP3........................................................................12 3.3 New Approach to Model Gain Compression Curve.............................................15 3.4 Fitting Polynomials Data by Using Linear Regression Theory............................20 3.5 Summary...............................................................................................................23 4 SIMULATION...........................................................................................................24 4.1 A MOSFET Common-Source Amplifier..............................................................24 4.2 Measurement Error Consideration........................................................................35 4.3 Frequency Effect on the Fitting Algorithm..........................................................38 4.4 Load Effect on the Fitting Algorithm...................................................................41 4.5 Summary...............................................................................................................51

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vi 5 COMMERCIAL RF WIDEBAND AMPLIFIER......................................................52 5.1 Nonlinearity Test..................................................................................................52 5.2 IIP3 Prediction from the Gain Compression Curve..............................................58 5.3 The Application of the Proposed Algorithm at High Frequency..........................62 5.4 IP1-dB Estimation from Two-tone Data.................................................................63 5.5 Summary...............................................................................................................68 6 POWER AMPLIFIERS..............................................................................................69 6.1 Linear and Nonlinear Power Amplifiers...............................................................69 6.2 Measurement of Commercial Power Amplifiers..................................................70 6.3 IIP3 Estimation from the One-tone Data...............................................................75 6.4 IP1-dB Estimation from the Two-tone Data.........................................................103 6.5 Summary.............................................................................................................104 7 SUMMARY AND SUGGESTI ONS FOR FUTURE WORK.................................105 7.1 Summary.............................................................................................................105 7.2 Suggestions for Future Work..............................................................................107 7.2.1 Nonlinearity of a Mixer............................................................................108 7.2.2 Modeling a Mixer Embedded Test...........................................................111 APPENDIX A BSIM3 MODEL OF N-MOS AND P-MOS TRANSISTOR...................................113 B VOLTERRA-KERNELS OF A COMMON-SOURCE AMPLIFIER.....................115 C MATLAB PROGRAM FOR FITTING ALGORITHM..........................................118 LIST OF REFERENCES.................................................................................................124 BIOGRAPHICAL SKETCH...........................................................................................126

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vii LIST OF TABLES Table page 1-1 The 1 dB gain compression point and IIP3 of various circuits...................................3 4-1 The summary of fitting results.................................................................................34 4-2 Frequency effect on the fitting algorithm.................................................................39 4-3 Fitting Results..........................................................................................................50 5-1 The summary of the measurement results of commercial amplifiers......................58 5-2 The summary of the estimated IIP3 of commercial amplifiers.................................61 5-3 The measurement data and calculated IIP3 of a commercial amplifier....................63 5-4 The summary of the application results of the IP1-dB estimation algorithm.............67 6-1 The summary of the measurement results of commercial PAs................................75 6-2 The summary of fitting results (amplifier A)..........................................................79 6-3 The summary of fitting results (amplifier A)..........................................................83 6-4 The summary of fitting results of the first model (amplifier B)...............................83 6-5 The summary of fitting results according to fitting region (amplifier B)................86 6-6 The summary of fitting results of the first model (amplifier C)...............................90 6-7 The summary of fitting results (amplifier C)..........................................................96 6-8 The summary of fitting results of the first model (amplifier D)............................102 6-9 The summary of fitting results (Amplifier D).......................................................102 6-10 The comparison between two fitting models.........................................................102 6-11 The summary of the application results of the IP1-dB estimation algorithm...........104 A-1 BSIM3 model of n-MOSFET.................................................................................113

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viii A-2 BSIM3 model of p-MOSFET.................................................................................114

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ix LIST OF FIGURES Figure page 2-1 The output current of an ideal cl ass C amplifier for a sine-wave input.....................6 2-2 The configuration of a single-tone test.......................................................................9 2-3 The configuration of a two-tone test..........................................................................9 3-1 Definition of 1-dB gain compression point..............................................................10 3-2 Intermodulation in a nonlinear system.....................................................................11 3-3 Definition of third-order intercept point...................................................................12 3-4 The definition of Interc ept points in one-tone test...................................................16 4-1 A schematic of a common-source amplifier............................................................25 4-2 DC characteristic of a common-source amplifier....................................................26 4-3 The AC simulation of a common-source amplifier..................................................27 4-4 The results of one-tone simulation...........................................................................28 4-5 The results of a two-tone simulation........................................................................29 4-6 The difference between amp litudes at two frequencies...........................................29 4-7 Extracted nonlinear coefficients...............................................................................31 4-8 Standard errors of nonlinear coefficients................................................................32 4-9 The sum of square s of the residual...........................................................................33 4-10 Gain curves with 0.1% and 2% random noise.........................................................35 4-11 Calculated third-order intercept poi nt with 0.1 percent added random noise..........36 4-12 Calculated third-order intercept poi nt with 2.0 percent added random noise..........37 4-13 The influence of added random error on the fitting results......................................37

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x 4-14 An equivalent circuit................................................................................................41 4-15 A schematic of a common-source amplifier with an active load.............................43 4-16 DC characteristics of a common-s ource amplifier with an active load....................44 4-17 The AC simulation of a common-s ource amplifier with an active load..................45 4-18 The results of one-tone simulation...........................................................................46 4-19 The results of two-tone simulation...........................................................................46 4-20 Extracted nonlinear coefficient K1...........................................................................47 4-21 Extracted nonlinear coefficient K3...........................................................................48 4-22 Extracted nonlinear coefficient K5...........................................................................49 4-23 Standard errors of nonlinear coefficients................................................................49 5-1 ERA1 amplifier in a test board.................................................................................52 5-2 The spectrum from the signal source.......................................................................53 5-3 One-tone test scheme...............................................................................................54 5-4 The measurement data of one-tone test (ERA1 amplifier)......................................54 5-5 Two-tone test scheme...............................................................................................55 5-6 The measurement data of tw o-tone test (ERA1 amplifier)......................................55 5-7 The measurement data of one-tone test (ERA2 amplifier)......................................56 5-8 The measurement data of tw o-tone test (ERA2 amplifier)......................................56 5-9 The measurement data of one-tone test (ERA3 amplifier)......................................57 5-10 The measurement data of tw o-tone test (ERA3 amplifier)......................................57 5-11 One-tone data and extraction from ERA1 device at 100 MHz................................59 5-12 A flow chart for estimation of IIP3 from one-tone measurement............................60 5-13 One-tone data and extraction from ERA2 device at 100 MHz................................61 5-14 One-tone data and extraction from ERA3 device at 100 MHz................................62 5-15 One-tone data and extracti on from ERA2 device at 2.4 GHz..................................63

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xi 6-1 The measurement data of one-tone test of the amplifier A......................................71 6-2 The measurement data of twotone test of the amplifier A......................................71 6-3 The measurement data of one-tone test of the amplifier B......................................72 6-4 The measurement data of twotone test of the amplifier B......................................72 6-5 The measurement data of one-tone test of the amplifier C......................................73 6-6 The measurement data of twotone test of the amplifier C......................................73 6-7 The measurement data of one-tone test of the amplifier D......................................74 6-8 The measurement data of twotone test of the amplifier D......................................74 6-9 The value of coefficient K1 (amplifier A)................................................................77 6-10 The value of coefficient K3 (amplifier A)................................................................77 6-11 Standard errors of K1 and K3 (amplifier A)..............................................................78 6-12 Sum of squares of the residuals (amplifier A)..........................................................78 6-13 The value of coefficient K1 (amplifier A)................................................................80 6-14 The value of coefficient K3 (amplifier A)................................................................80 6-15 The value of coefficient K5 (amplifier A)................................................................81 6-16 Standard errors of K1 and K3 (amplifier A)..............................................................81 6-17 Standard Error of K5 (Amplifier A)..........................................................................82 6-18 Sum of squares of the residuals (Amplifier A)........................................................82 6-19 The value of coefficient K1 (amplifier B)................................................................84 6-20 The value of coefficient K3 (amplifier B)................................................................84 6-21 Standard errors of K1 and K3 (amplifier B)..............................................................85 6-22 Sum of squares of the residuals (amplifier B)..........................................................85 6-23 The value of coefficient K1 (Amplifier B)...............................................................87 6-24 The value of Coefficient K3 (Amplifier B)..............................................................87 6-25 The value of Coefficient K5 (Amplifier B)..............................................................88

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xii 6-26 Standard errors of K1 and K3 (amplifier B)..............................................................88 6-27 Standard error of K5 (amplifier B)...........................................................................89 6-28 Sum of squares of the residuals (amplifier B)..........................................................89 6-29 The value of coefficient K1 (amplifier C)................................................................91 6-30 The value of coefficient K3 (amplifier C)................................................................91 6-31 Standard errors of K1 and K3 (amplifier C)..............................................................92 6-32 Sum of squares of the residuals (amplifier C)..........................................................92 6-33 The value of coefficient K1 (amplifier C)................................................................93 6-34 The value of coefficient K3 (amplifier C)................................................................93 6-35 The value of coefficient K5 (amplifier C)................................................................94 6-36 Standard errors of K1 and K3 (amplifier C)..............................................................94 6-37 Standard error of K5 (amplifier C)...........................................................................95 6-38 Sum of squares of the residuals (amplifier C)..........................................................95 6-39 The value of coefficient K1 (amplifier D)................................................................97 6-40 The value of coefficient K3 (amplifier D)................................................................97 6-41 Standard errors of K1 and K3 (amplifier D)..............................................................98 6-42 Sum of squares of the residuals (amplifier D)..........................................................98 6-43 The value of coefficient K1 (amplifier D)................................................................99 6-44 The value of coefficient K3 (amplifier D)................................................................99 6-45 The value of coefficient K5 (amplifier D)..............................................................100 6-46 Standard errors of K1 and K3 (amplifier D)............................................................100 6-47 Standard error of K5 (amplifier D).........................................................................101 6-48 Sum of squares of the residuals (amplifier D)........................................................101 7-1 A schematic of a gilbert cell mixer........................................................................108 7-2 The result of one-tone test with different LO powers............................................109

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xiii 7-3 The analysis of the gain curve at -15 dBm LO power...........................................109 7-4 A cascode structure................................................................................................110 7-5 Simple embedded system with a mixer..................................................................111 B-1 An equivalent circuit of a common-source amplifier.............................................116

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xiv Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy RF CIRCUIT NONLINEARIT Y CHARACTERIZATION AND MODELING FOR EMBEDDED TEST By Choongeol Cho December 2005 Chair: William R. Eisenstadt Major Department: Electrical and Computer Engineering This dissertation presents a fitting algorithm useful fo r characterizing nonlinearities of RF circuits, and is specifically designed to estimate the third-order intercept point ( IP3) by extracting the nonlinear coefficients from the one-tone measurement. And the dissertation proposes a method to predic t the 1 dB gain compression point ( P1-dB) from a two-tone measurement. The fitting algorithm is valuable for reducing production IC test time ans cost. A new relationship between the 1 dB gain compression point and the thirdorder intercept point has been derived. It follows that the difference between IP1-dB and IIP3 is not fixed, and the discrepancy is e xplained by the new proposed equation which includes the relevant nonlinear coefficients. The new fitting algorithm has been verified through application to the simulation of a common-source amplifier. The best fitting range has been identified through the minimiza tion of the standard er rors of the nonlinear coefficients and of the sum of squares of th e residuals. A robust al gorithm to predict IIP3 has been developed for wideband RF amplifiers an application that is of particular

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xv interest. The proposed fitting algorithm was successfully verified in experiments done on commercial RF power amplifiers. The estimated IIP3 values obtained from one-tone measurement data was close to the experime ntly measured values. The method proposed to predict IP1-dB from a two-tone measurement was also applied successfully to the same commercial RF power amplifiers. A simple embedded test using a direct conve rsion mixer can be realized to estimate the nonlinear characteristics of an amplifier based upon the estimate IIP3 from the onetone data. In this thesis, the nonlinear characte ristics of a mixer is researched and a mixer embedded test technique is suggested. The e ffects of the mismatches and phase offset will be researched for mixer test in the future. The methods developed in this thesis are useful tools in the context for typical RF/Mixed-signal production test. The advantage is that these methods avoid the difficulty of two-tone measurements or remove one-tone measurement test step. By developing the relationship between the 1 dB gain compression point and the third-order intercept point, a simpler embedded test model can be adopted avoiding the cost a nd time of a two-tone measurement.

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1 CHAPTER 1 INTRODUCTION 1.1 Motivation In the near future, RF microwave circu its will be embedded in highly integrated “Systems-on-a-Chip” (SoCs). These RF SoCs will need to be debugged in the design phase and will require expensive automated te st equipment (ATE) with microwave test capability when tested in production. RF/mixed-s ignal portions of a SoC must be verified with high-frequency parametric tests. Current ly, the ATE performs a production test on package parts with the assist ance of an expensive and elab orate device interface board (DIB) or load board. Alternative methods of onchip RF test should be explored to lower test cost [Eis01]. Another merit of the embedded test is to minimize test time. Current measurement is performed in the last stage of production. A parameter test of an RF circuit can be executed using the embedded circuit be fore packaging and even sorting. The 1 dB gain compression point and the th ird-order intercept point are important nonlinear parameters of the RF/mixed-signa l circuit and provide good verification of a circuit or device’s lin earity and dynamic range. The para meters can be connected to adjacent channel power ratio and error vector magnitude (EVM) in amplifiers and must be kept under control. Gain compression is a relatively simple microwave measurement since it requires a variable power single t one source and an output power detector. IIP3 characterization is more complicated and mo re costly since two se parate tones closely spaced in frequency must be generated and applied to the circuit under test (CUT) and

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2 the CUT’s fundamental and third order distor tion term power must be measured. Thus, measurement of IIP3 requires high Q filters to select first and third order distortion frequencies in the detector circuit [Eis02]. By developing an accurate relationship between gain compression and IIP3, the production testing of the manufactured IC can be greatly simplified. Although the accuracy of this approach may not be as great as direct IIP3 measurement, it has great appeal in test cost reduction and may be sufficient for production IC test. 1.2 Research Goals The first goal in this research is to de rive the relationship between 1 dB gain compression point and third-order intercept point. The published difference between 1 dB gain compression point and IIP3 is roughly 10 dB; this relation ship is derived using firstorder and third-order nonlinear coefficients of transistor amplifier circuits. This calculation assumes that higher-o rder nonlinear coefficients do not affect the 1 dB gain compression. The simulation using 0.25 m MOSFETs, 0.4 m MOSFETs and Si-Ge BJTs models in Table 1-1 shows that this cl assical relationship does not work in simple transistor circuits. The difference be tween 1 dB gain compression and IIP3 shows a variation of 8 dB to 13.7 dB. All the circuits (common-sour ce amplifier, differential amplifier with resistive load, and commonemitter amplifier except for a differential amplifier with an active load) s how that the difference between IIP3 and IP1-dB is no longer constant. To simplify the embedded te st, a more reliable relationship between these two parameters is required. The second goal is to verify the relationshi p between two parameters that is derived in the first goal. This dissertation examines the general types of amplifiers,

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3 Table 1-1 The 1 dB gain compression point and IIP3 of various circuits Model Circuit IP1-dB* (dBm) IIP3** (dBm) IIP3 -IP1-dB (dB) Common-Source Amplifier -2.2 10.0 12.2 Differential Amplifier (Resistive Load) 3.0 16.0 13.0 TSMC 0.25m MOSFET Differential Amplifier (Active Load) -13.0 -5.0 8.0 Common-Source Amplifier -1.75 10.25 12.0 Si-Ge IBM6HP MOSFET Differential Amplifier (Resistive Load) 2.65 15.25 12.6 Si-Ge IBM6HP BJT Common-Source Amplifier -20.25 -6.5 13.75 *One-tone test : Source frequency = 100 MHz **Two-tone test : frequencies = 100 MHz, 120 MHz commercial wideband RF amplifiers and comm ercial RF power amplifiers. In this dissertation, the fitti ng approach is developed to estimate IIP3 from a gain compression curve using higher-order nonlinear coefficients. Finally, a simple embedded test using a mixer is considered and suggested to measure IIP3 and IP1-dB. Through developing the re lationship between IIP3 and IP1-dB, the embedded test for IIP3 requires only a one-tone source. 1.3 Overview of Dissertation This Ph.D. dissertation consists of seven chapters. An overview of the research is given in this current chapter (Chapter 1), including the motivation, research goals, and the scope of this work. Chapter 2 reviews some background knowledge on this research. Basic concepts of both nonlinear system s and nonlinear analysis are described. In chapter 3, a new relationship between the 1 dB gain compression point and the third-order intercept point is derived. Fi rst, this relationship between IIP3 and IP1-dB is

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4 reviewed in classical prior analysis. The new relationship is derived by nonlinear analysis on the gain compression curve. The fitting algorithm to estimate IIP3 from a one-tone measurement and the calculation method to predict IP1-dB from a two-tone measurement are developed. The linear regression theory requ ired for the fitting algorithm is reviewed and modified for the applicati on of the devised algorithm. In chapter 4, the proposed fitting algorithm is verified through the application of the algorithm to the simulation of a commonsource amplifier. The best fitting range is chosen through the standard erro rs of nonlinear coefficients and the sum of squares of the residuals. The effects of meas urement errors at high fre quency are researched. Through the Voletrra series analysis, the lo ad effect on the algorithm is studied. In chapter 5, a robust algorithm to pred ict IIP3 is developed for wideband RF amplifiers. The IP1-dB prediction from two-tone measur ement has been applied to these wideband amplifiers. Through several steps of simple calculation using the third-order intercept point and the gain compre ssion at the fundamental frequency, IP1-dB is estimated under 1 dB error. In chapter 6, the fitting algorithm is a pplied to commercial RF power amplifiers. Through the inspection of the standard errors of fitting parameters, the best fitting range is chosen for the extraction of nonlinear coeffi cients used for the calculation of the thirdorder intercept point. The chosen fitting ra nge is confirmed by the quantity, sum of squares of the residuals. Another method to predict IP1-dB from a two-tone measurement is applied to the commerci al RF power amplifiers. Lastly, in chapter 7, the prim ary contributions of this dissertation are summarized and future work is suggested.

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5 CHAPTER 2 BACKGROUND 2.1 Classifications of Distortions All physical components and devices are in trinsically nonlinear. Nevertheless, the most circuit and system theory deal almost ex clusively with linear analysis. The reason is because linear systems are characterized in term s of linear algebraic, differential, integral, and difference equations that are relatively eas y to solve, most nonlinear systems can be adequately approximated by equivalent linear systems for suitably small inputs, and closed-form analytical solutions of non linear equations are not normally possible. However, linear models are incapable of explaining important nonlinear phenomena [Wei80]. This section review s types of distortions and nonlinearities fo r understanding nonlinear characterist ics of a system. Distortion actually refers to the distortion of a voltage or current waveform as it is displayed versus time [San99] Any difference between the sh ape of the output waveform and that of the input waveform is called distortion except for scaling a waveform in amplitude. In a circuit, the type of distortion is classified as one of two classes. First, linear distortion is caused by the application of a linear circuit w ith frequency-varying amplitude or phase characteristics. For exampl e, when a square-wave input is applied to a high-pass filter, the output waveform unde rgoes linear distortion. Second, nonlinear distortion is caused by nonlinea r transfer function characte ristics. For example, the application of a large sinusoidal wavefo rm to the exponential transfer function

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6 characteristic of a bipolar tr ansistor based amplifier can ca use a sharpening of one hump of the waveform and flattening of the other one. Nonlinear distortion is classified fine ly in two categories : weak and hard distortion. In the case of weak distortion, the harmonics gr adually shrink as the signal amplitude becomes smaller. However, th e harmonics are never zero. The harmonic amplitudes can easily be calculated from a Tayl or series expansion around the quiescent or operating point. In weakly nonlinear distor tion, the Volterra series can be used for estimating the nonlinear behavior of a circuit. Hard distortion, on the other hand, can be seen in Class AB, B, and C amplifiers. In these cases, a part of the sinusoidal wavefo rm is simply cut off, leaving two sharp corners. These corners generate a large num ber of high-frequency harmonics. They are the sources of hard distorti on. Hard distortion harmonics suddenly disappear when the amplitude of the sinusoidal waveform falls below the threshold, i. e., the edge of the transfer characteristic. The class C amplifier is considered below as an example of a circuit with hard distortion. Figure 2-1 show s the output current of an ideal class C Figure 2-1 The output current of an ideal class C amplif ier for a sine-wave input amplifier. The output current amplitude at a fundamental frequency is in the form of a nonlinear function which s hown in equation (2-1).

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7 2 sin 2 2o fundi i (2-1) where 2 is a conduction angle and is a nonl inear function of output amplitude io. 2.2 Taylor’s Series Expansion Let be ) ( x f continuous on a real interval I containing 0x ( and x ), and let ) () (x fn exist at x and ) () 1 ( nf be continuous for all I Then we have the following Taylor series expansion: ) ( ) )( ( 1 ... ) )( ( ' 3 2 1 1 ) )( ( ' 2 1 1 ) )( ( 1 1 ) ( ) (1 0 0 ) ( 3 0 0 2 0 0 0 0 0x R x x x f n x x x f x x x f x x x f x f x fn n n (2-2) where ) (1x Rn is called the remainder term. Then, Taylor's theorem provides that there exists some between x and 0x such that 1 0 ) 1 ( 1) ( )! 1 ( ) ( ) ( n n nx x n f x R (2-3) In particular, if M fn ) 1 (in I then 1 0 1)! 1 ( ) ( n nx x n M x R (2-4) which is normally small when x is close to 0x [Ros98]. For a nonlinear conductance, the current through the element, ) ( t iout, is a nonlinear function f of the controlling voltage, ) ( t vCONTR. This function can be expanded into a power series around the quiescent point) (CONTR OUTV f I [Wam98]. )) ( ( )) ( ( ) ( t v V f t v f t icontr CONTR CONTR OUT

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8 ) ( ) ( )) ( ( 1 ) (1t v v t v f k V fk contr V v k k k CONTRCONTR (2-5) Nonlinear coefficients are defined as follows CONTRV v k k nv t v f k K ) ( )) ( ( 1 (2-6) The expression of the AC current through the conductance is in equation (2-7). ... ) ( ) ( ) ( ) (3 3 2 2 1 t v K t v K t v K t icontr contr contr out (2-7) 2.3 Measurement of Nonlinear System A single-tone test is used for the me asurement of harmonic distortion, gain compression/expansion, large-signal impe dances and root-locus analysis. The configuration of a single-tone test can be seen in Figur e 2-2. When the input power sweeps a wide range, the output po wer at the same frequency as the input is measured in the sweep range. The cable and other interc onnection components that transfer power should be calibrated since these components have power loss. For the analysis of intermodulation, cro ss-modulation and desensitization, the twotone test is used. Figure 2-3 shows the confi guration of a two-tone harmonic test. In this test, the power combiner is used for comb ining two powers at di fferent frequencies. Through this test, a third-order in tercept point is determined. The next chapter develops the relationship between 1 dB gain compression point and third-order intercept point. Taylor series analysis above is essential for performing that analysis.

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9 Figure 2-2 The configuration of a single-t one test. ERA is a commercial amplifier. Figure 2-3 The configuration of a two-tone test. ERA is a commercial amplifier.

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10 CHAPTER 3 THE RELATIONSHIP BETWEEN THE 1 dB GAIN COMPRESSION POINT AND THE THIRD-ORDER INTERCEPT POINT 3.1 Definition of 1 dB Gain Compressi on and Third-order Intercept Point The constant small-signal gain of a circuit is usually obtained with the assumption that the harmonics are negligible. However, as the signal amplitude increases, the gain begins to vary with input power. In most circ uits of interest, the output at high power is a compressive or saturating function of the i nput. In analog, RF and microwave circuits, these effects are quantified by the 1-dB gain compression point [Raz98]. Figure 3-1 shows the definition of the 1 dB gain compression point. The real gain curve, C is plotted on a log-log scale as a function of the input power level. The output level falls below its ideal value since the compression of the real gain curve is caused by the Figure 3-1 Definition of 1dB gain compression point

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11 Figure 3-2 Intermodulation in a nonlinear system nonlinear transfer characteristics of the circuit. Point A1-dB is defined as the input signal level in which the difference between the ideal linear gain curve B and the real gain curve C is 1 dB. Another important nonlinear characteristic is the intermodulation distortion in a two-tone test. When two signals with different frequencies, 1 and 2 are applied to a nonlinear system, the large signal output e xhibits some components that are not harmonics of the input frequencies. Of pa rticular interest are the third-order intermodulation products at ) 2 (2 1 and ) 2 (1 2 as illustrated in Figure 3-2. In this figure, two large signals at the left are inputs into an amplifier in the center. The output is shown on the right of the figure as the two fundamental signals plus the intermodulation frequencies ) 2 (2 1 and ) 2 (1 2 The corruption of signals due to thir d-order intermodulation of two nearby interferers is so common and so critical that a performance metric has been defined to characterize this behavior. The third-order intercept points IIP3 and OIP3 are used for characterizing this effect. These terms are de fined at the intersecti on of two lines shown in Figure 3-3. The first line has a slope of one on the log-log plot (20 log (amplitude at

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12 Figure 3-3 Definition of third-order intercept point w1)) and represents the input and output pow er of the fundamental frequency. The second line represents the growth of the ) 2 (2 1 intermodulation harmonic with input power, it has a slope of three. OIP3 is the output power at the intercept point and IIP3 is the input power at the intercept point, IP3. 3.2 Classical Approach to Model IIP3 In a nonlinear system without memory such as an amplifier at low frequency, the output can be modeled by a power series of the input in section 2.2. If the input of the nonlinear system is ) (t x, the output ) (t y of this system is as follows, ... ) ( ) ( ) ( ) (3 3 2 2 1 t x K t x K t x K t y (3-1) where iK is nonlinear coefficients of this syst em. This example is explained in equation (2-6) in section 2.2. The classical analysis of the nonlinear system uses the assumption that the fourth-order and hi gher-order terms in equation (3 -1) are negligible. In the classical analysis, the nonlinear system is modeled as follows [Raz98][Gon97], 3 3 2 2 1) ( ) ( ) ( ) ( t x K t x K t x K t y (3-2) 3OIP3IIP20log(Ain)) log( 201w at Amplitude ) 2 log( 202 1w w at Amplitude 3OIP3IIP20log(Ain)) log( 201w at Amplitude ) 2 log( 202 1w w at Amplitude

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13 If a sinusoidal input with a fundamental fre quency is applied to this nonlinear system ) cos( ) (t A t x (3-3) then the output of this system is represented by using the equation (3-2), ) 3 cos( 4 3 ) 2 cos( 2 ) cos( 4 3 2 ) (3 3 2 2 3 3 1 2 2t A K t A K t A K A K A K t y (3-4) At the fundamental frequency the gain is defi ned as a function of the input signal amplitude, 4 3 ) (3 3 1A K A K at Gain (3-5) If 3K has the opposite sign of 1K, the gain is a decreasi ng function of the input amplitude. The 1-dB gain compression point is defined in Figure 3-1, the equati on at this point is dB A K A K A KdB dB dB1 log 20 4 3 log 201 1 3 1 3 1 1 (3-6) where dBA1 is the input amplitude at the 1 dB gain compression point. The solution of equation (3-6) is 3 1 1145 0 K K AdB (3-7) In summary, the input amplitude referred to the 1 dB gain compre ssion point is found using only two nonlinear coefficients 1K and 3K

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14 The classical analysis is considered in a two-tone test. The input signal in the twotone test is composed of two signals w ith the same input amplitude and different frequencies. ) cos( ) cos( ) (2 1t A t A t x (3-8) When this input signal is applied to the non linear system represented by the equation (32), the output is, ... 2 cos 4 3 2 cos 4 3 ) cos( 4 9 ) cos( 4 9 ) (1 2 3 3 2 1 3 3 1 3 3 1 1 3 3 1 t A K t A K t A K A K t A K A K t y (3-9) The third-order intercept point is defined in Figure 3-3. At this point, the output amplitude at a fundamental frequency is the sa me as that at an in termodulation frequency. The input signal level satisfying the above condition is represented by, 4 33 3 3 3 1 IP IPA K A K (3-10) where 3 IPA is the input amplitude at the third-orde r intercept point. The solution of above equation (3-10) is 3 1 33 4 K K AIP (3-11) The input signal level at the third-order intercept point is al so found using two nonlinear coefficients 1K and 3K From equation (3-7) a nd equation (3-10), the relationship between the input signal levels at the 1 dB gain compression point and the third-order intercept point can be derived,

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15 dB A AIP dB6 9 3 / 4 145 03 1 (3-12) The classical analysis shows th at the relationship between tw o nonlinear characteristics is represented by equation (3-12). 3.3 New Approach to Model Gain Compression Curve For simplicity, the analysis is limited to memoryless, time-invariant nonlinear systems. Prior classical analysis limits the output to the th ird-order nonlinearity coefficient. For the more exact analysis, the relaxation of this limita tion is required. The nonlinear system in this analysis is represented by, 5 5 4 4 3 3 2 2 1) ( ) ( ) ( ) ( ) ( ) ( t x K t x K t x K t x K t x K t y (3-13) If a sinusoidal input such as equation (3-2) is applied to this system, then the output amplitudes at each odd-frequency are as follows, ) cos( 8 5 4 3 ) ; (5 5 3 3 1t A K A K A K t y (3-14) ) 3 cos( 16 5 4 1 ) 3 ; (5 5 3 3t A K A K t y (3-15) ) 5 cos( 16 1 ) 5 ; (5 5t A K t y (3-16) If a low input signal level is considered in equation (3-14), the following condition is satisfied, 5 5 3 3 18 5 4 3 A K A K A K (3-17) then the output amplitude at the fundamental frequency is A K1. From equation (3-15), the output amplitude at frequency 3 is 3 34 1 A K if

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16 Figure 3-4 The definition of Intercept points in one-tone test. A= 20 log (K1A), B=20 log (K3A3/4) and C=20 log (K5A5/16) 5 5 3 316 5 4 1 A K A K (3-18) As stated in the previous section, it is possible to define the intercept points shown in Figure 3-4 in one-t one test. In Figure 3-4, Li ne A represents the output amplitude at the fundamental frequency and has a slope of one in the log-log scale graph. This line is extrapolated from linear smallsignal area in equation (3-14) from equation (3-17). Line B is the output amplitude at tr iple fundamental freque ncy and has a slope of three. This line is also extrapolated from equation (3-15) under the condition of equation (3-18). Line C with a slope of five is the output amplitude at frequency 5 and is drawn from equation (3-16). The inter cept points between th ree lines in this figure are denoted by IP15, IP13 and IP35. At the point IP13, which is the intercepti on point between Line A, and Line B, the input signal level 13A can be found from,

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17 3 13 3 13 14 1 A K A K (3-19) where the value of Line A is the same as that of Line B at the input level 13A From this equation (3-19), the input si gnal level can be represented by using two nonlinear coefficients 1K and 3K such as, 2 1 3 1 132 K K A (3-20) At the intercept point IP15 between Line A and Line C, the input signal level 15A is found by solving the following equation, 4 15 5 15 116 1 A K A K (3-21) 4 1 5 1 152 K K A (3-22) The signal level 35A which is the input value at the intercept point IP35, can be described by, 4 35 5 3 35 316 1 4 1 A K A K (3-23) 2 1 5 3 352 K K A (3-24) The relationship between the input signal levels at the in tercept points can be derived from the equation (3-20) (3-22) and (3-24), 2 15 35 13A A A (3-25) The equation (3-25) can be expresse d differently by using logarithm,

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18 ) log( 20 2 ) log( 20 ) log( 2015 35 13A A A (3-26) In addition, this relations hip can be designated by, 15 35 132 IP IP IP (3-27) where ijIP is the input-referred power at the interception point ijIP. The 1 dB gain compression is considered in this approach. The input signal level dBA1 at this point can be represented by the equation dB A K A K A K A KdB dB dB dB1 log 20 8 5 4 3 log 201 1 5 1 5 3 1 3 1 1 (3-28) From the above equation, one can see that fi fth order nonlinear coefficient makes the gain compression change from equation (3-6) in the previous section. Generally, gain compression arises when K3 has the opposite sign of K1. From equation (3 -28), the gain compression curve is affected by K5. The simple equation is derived from equation (328). 0 109 0 4 3 8 52 1 1 3 4 1 1 5 dB dBA K K A K K (3-29) Using intercept points as defined abov e, this equation is represented ( K1>0, K3<0 and K5<0) by, 0 109 0 3 102 13 1 4 15 1 A A A AdB dB (3-30) If K5 is positive then, 0 109 0 3 102 13 1 4 15 1 A A A AdB dB (3-31)

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19 Therefore, if A13, A15 and A35 nonlinear coefficients are known, it is possible to evaluate dBA1. Equation (3-8) takes the form of the input signal in a two-t one test. If this input is applied to the system descri bed by equation (3-13), the out put has a more complicated form than that in the one-tone test. The output at fundamental freque ncy is described by, ) cos( 4 25 4 9 ) ; (5 5 3 3 1t A K A K A K t yi i (3-32) where i represents the fundamental frequenc y which is one of two frequencies 1 and 2 The amplitude of the output, ) ; (it y in equation (3-32) is different from that of the output ) ; ( t y in the equation (3-14) of the one-tone test. This difference is called as the term desensitization [Wam98]. This term is originated from communication circuits in which a weak signal is affected by an adjacent strong unwanted signal through a nonlinear transfer characteristic. The output amplitude at the intermodulation frequency ) 2 (2 1 is represented by the following equation, ) ) 2 cos(( 8 25 4 3 ) 2 ; (2 1 5 5 3 3 2 1t A K A K t y (3-33) The third-order intercept point is defined in the small-signal area of a gain plot. The input signal level at the third-order intercept point is the same as that in the classical analysis. Equation (3-11) shows the input signal level at th is intercept point. In this analysis, it is possible to estimate IIP3 and 1 dB gain compression points from the knowledge of the nonlinear coefficients since IIP3 and 1 dB gain compression point are represented by functions of nonlin ear coefficients. The gain compression curve in the result of a one-tone test is a functi on of the nonlinear coefficients represented by

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20 equation (3-14). It is possibl e to extract the nonlinear co efficients from the gain compression curve by applying a fitting method that will be explained in the next section. If the nonlinear coefficients are extracted fr om the gain compression curve in a one-tone test, it is possible to estimate IIP3 without the two-tone test. In the results of the two-tone test, the gain curve at each frequencies 1 and 2 is a function of the nonlinear coefficients described by e quation (3-32). Even though the equation (3-32) includes the desensitization factor, this f actor is also a function of th e nonlinear coefficients. It is possible to extract the nonlinear coefficients from these gain curves by using a numerical fitting method. In addition, the IIP3 measurement contains the nonlinear coefficients effects. It is possible to predict 1 dB gain compression points and estimate the gain curve without the one-tone test by usi ng the nonlinear coefficients ex tracted from two-tone test results. 3.4 Fitting Polynomials Data by Using Linear Regression Theory If a model has n sets of observations and is fitted with a series of linear parameters b0, b1, b2,…, bm by the method of least squares, the model is represented by the following matrix form [Dra95], E b X Y (3-34) where Y X b and E are 1 n,m n ,1 m and 1 n matrices respectively. In this equation (3-34), Y is a measured result, X is an input data, b is a set of linear parameters and E represents the error between a fitting model and a real model. The above linear regression model has thr ee basic assumptions. First, the average of the errors in the obser vations is zero and the va riance of these errors is 2. Second, iE and jE are uncorrelated if i is different from j. Third, the error follows the normal

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21 random variable distribution. These three basi c assumptions are included intrinsically or extrinsically in the statistical appr oach of linear regression analysis. The sum of squares of deviations or errors is, b X Y b X Y E E E SST T i ) ( (3-35) To find the sum of square of errors the least quantity, b can be found as follows, Y X X X bT T 1 (3-36) After finding b using above equation (3-36), the variance of b is, 2 1 X X b VT (3-37) where the diagonal terms of the above varian ce represent the variance of the parameter ib and the off-diagonal terms stand for the covariance of the pair ib and jb. The variance of errors, 2, is calculated through the analysis of th e residual. The residual is defined as the difference between a fitting model and a real model. The sum of square of the residuals is represented by the equation (3-35) The mean square of the residual is defined as the sum of square of the residuals divide d by the degree of freedom of the residual. If the number of the observations is n and the de gree of freedom of the regression parameter is m, the degree of freedom of the residual is n-m. The mean square of the residual is used as the estimate of the variance of errors, 2. The mean square of the residual is, m n Y X b Y Y m n b X Y b X Y MST T T T E 12 (3-38) In this research, the fitting model for th e gain curve is the odd-order polynomial equation, 5 5 3 3 1i i i ix K x K x K y (3-39)

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22 where iy is the estimate value of the fitting model. The residual is, i i iy y E (3-40) The matrices in the equation (3-34) can be formed as follows, ...2 1y y Y (3-41) ... ... ...5 2 3 2 2 5 1 3 1 1x x x x x x X (3-42) 5 3 1K K K b (3-43) ...2 1E E E (3-44) The degree of freedom of the parameter b is three. If the number of the data points is n, the degree of freedom of the residual is n-3. Using the degree of freedom of the residual, the variance of the errors can be calculated in equation (3-37). The standard error of the parameters is defined as the square root of the variance of the parameter. The standard errors of the parameters or nonlin ear coefficients can be calculated using the equation (3-37). The sum of square of the re sidual also can be cal culated using equation (3-35). These two analyses, the standard erro rs of the nonlinear coefficients and the sum of square of the residual are used in de termining the fitting range in this research.

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23 3.5 Summary In this chapter, the new relationship be tween the 1 dB gain compression point and the third-order intercept point has been de rived. First, this relationship between IIP3 and IP1-dB was reviewed in classical analysis The difference between two nonlinear characteristics was 9.6 dB and constant. The cl assical analysis included only third-order nonlinear coefficients. The new relationship was derived by expandi ng nonlinear analysis on the gain compression curve up to the fifthorder nonlinear coeffi cients. The difference between IP1-dB and IIP3 is not fixed and is explained by the equation including nonlinear coefficients. The fitting algorithm to estimate IIP3 from one-tone measurement and the calculation method to predict IP1-dB from two-tone measurement are devised. The linear regression theory required for the fitting al gorithm has been review ed and modified for the application of the algorithm.

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24 CHAPTER 4 SIMULATION 4.1 A MOSFET Common-Source Amplifier The modeling approach developed in a previ ous chapter is applied to the simulation of a weakly nonlinear system. A common sour ce amplifier is considered as a weakly nonlinear system. There are two methods that can estimate 1 dB gain compression point and IIP3 from one-tone test. The first method us es the ratio of nonlinear coefficients. These ratios are found from the harmonic powe r intercept points, which are explained in Chapter 3. For determining gain compression, measurement of the overall device or amplifier power is needed but for IIP3 estimation, the measurement of the third and fifth harmonic frequency magnitudes is require d. The second method to estimate IIP3 is fitting the gain curve at the fundamental freque ncy for the extraction of the nonlinear coefficients. The gain curve is fitted via a Matlab program develope d in this research. The second IIP3 determination technique is more useful than the first since the second technique does not need the third and fifth harmonic frequency amplitude coefficients to the measured. To demonstrate the fitting technique, a common-source amplifier with TSMC 0.25 m n-MOSFET is considered. Figure 4-1 sh ows the schematic for a common-source amplifier. In this amplifier design, supply voltage VDD is 3.3 V and the load resistor RD is 10 K The size of the transistor M1 is a minimum size (W/L=1.18 m/0.25 m) and BSIM3 model of this transistor is listed in the Appendix A. The gate bias voltage is set to produce the weakly nonlinear be havior in the amplif ier. This bias point was found by an

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25 Figure 4-1 A schematic of a common-source amplifier analysis of the DC characteristic of this amplifier using the Agilent ADS2002 software. The I-V characteristic curve and transfer charact eristic curve are shown in Figure 4-2. For a weakly nonlinear simulation, a gate bias is chosen near the center poi nt that is shown as point P in both Figure 4-2.A and Figure 4-2.B. At the gate bias, 1.1 V, an AC simulation was performed. Figure 4-3 shows the result of AC simulation. When the frequency increases above 1 GHz, the voltage gain decreas es significantly. This indicates that the parasitic capacitance of the n-MOSFET needs to be considered in the gain calculation and cannot be ignored above 1 GHz. As a result, the fitting algorithm in the frequencydomain has some errors related to the transi stor parasitic capacitances (gate and drain) since the power series does not include th e phase information caused by these parasitic capacitors.

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26 A B Figure 4-2 DC characteristic of a common-so urce amplifier. A) shows the drain current and B) shows the drain voltage.

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27 Figure 4-3 The AC simulation of a common-source amplifier At the same gate bias as the AC simulation a one-tone test and a two-tone test were simulated using the harmonic-balance simulator in the ADS2002 software. Figure 4-4 and Figure 4-5 show the simulation results of one-tone and two-tone vo ltage gain transfer function described in Chapter 2. Voltage gain curves are made by observing the amplifier output amplitude or power while sweeping the input voltage or power at a fixed frequency. In Figure 4-4, the amplitude of the input voltage is swept from 0.01 V to 2 V at the fundamental frequency of 100 MHz. Curv e A is the voltage gain curve which is the output amplitude measured at the drain node of M1 in Figure 4-1 of the common-source amplifier. Line B represents the ideal gain curve under the assumption that there are no harmonics at any amplifier power level. GC denotes the 1 dB compression point that shows an 1 dB difference between Curve A and Line B. At the voltage amplitude gain

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28 Figure 4-4 The results of one-tone simulation compression of 1 dB, the input voltage is 0.52 V in Figure 4-4. The applied frequencies in the two-tone test are 100 MHz and 110 MH z. In Figure 4-5, Curve A and B represent the amplifier output signal amplitudes at 100 MHz and at 90 MHz separately. Curve A is the fundamental frequency of 100 MHz, Curve B indicates the amplitude at intermodulation frequency of 90 MHz (2100 MHz – 110 MHz). Line C and D indicate the ideal harmonic response of the amplifier if no other harmonics are present. A real amplifier introduces an increasing number of harmonic components as gain compression is increased. The point of intersection of the two ideal harmonic lines, denoted by point TOI, indicates the intermodulation intercep t point that has a value of 2.05V. This intermodulation intercept point has the same de finition as that of the third-order-intercept point in a 50 system such as an RF system. In two-tone test, the amplitudes at two

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29 Figure 4-5 The results of a two-tone simulation Figure 4-6 The difference between amplitudes at two frequencies. A: Input frequency (100 MHz, 110 MHz) B: Intermod ulation frequency (90 MHz, 120 MHz)

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30 frequencies and two intermodulation frequencies should be considered. These amplitudes are compared in Figure 4-6. In this figur e, Curve A shows the difference between the amplitudes at two fundamental frequencie s (100 MHz and 110 MHz). Curve B is the difference between the amplitudes at two inte rmodulation frequencies (90 MHz and 120 MHz). Up to 0.2 V, the difference of amp litudes at two intermodul ation frequencies is under 0.001 dB. In the case of two fundamen tal frequencies, the difference is under 0.001 dB up to 0.34 V. Since the third-order interc ept point is defined in the small-signal area explained in Chapter 3, the third-order inte rcept point between tw o curves at 110 MHz and 120 MHz is the same as that between 2 curves at 100 MHz and 90 MHz. In view of the CMOS amplifier simulations, the difference between 1 dB gain compression point and third order intercept point is 12 dB. This outcome differs with 10 dB that is shown in the classical nonlinea r calculations for amplifiers [Raz98]. Nonlinear coefficients can be derived from the least-square polynomial fitting of the gain compression curve A of Figure 4-4. The polynomial model used in this fitting is as the power series that follows, 5 5 3 3 18 5 4 3 x K x K x K y (4-1) where Ki is a nonlinear coefficient representing th e output signal amplitude at harmonic i. According to linear regression theory, th e standard error can be analyzed on the individual coefficient, Ki, fitting outcomes. In addition, th e difference between the fitting model and the actual gain curve produces a fitting residual, a curve fitting error. Therefore, the best fitting range can be se lected by analyzing the fitting error and standard error of each fitting coefficient. Wh en the input amplitude decreases, the output amplitude includes more information of lower-order harmonics in the polynomials.

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31 Figure 4-7 Extracted nonlinear coefficients Selecting a range with a small input voltage should be the first co nsideration during the fitting process. The fitting algorithm extracts the nonlinear coefficients from the lowestorder harmonic to 5th-order harmonic or the highest-order harmonic. A small area is chosen in the small input signal region of the ga in curve data as the first fitting range. In this fitting range, each of the nonlinear coe fficients, the standard error of each nonlinear coefficients and residuals are extracted in the fitting process by using the polynomial model, equation (4-1). After finding the information in this range, the fitting process continues in a wider range than the first fitt ing range. The respective coefficients and the change of the fitting range are shown in Figure 4-7. The first fitting range in the gain compression curve is from 0.01 V to 0.3 V in a small input amplitude area. In Figure 4-7, K1 at the point that x-axis is 0.3 V represents the value of the first-order nonlinear coefficient extracted from this first fitting region. In addition, K3 and K5 indicate the

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32 third-order and the fifth-order nonlinear coe fficients correspondingly. A horizontal axis defines the width of the fitting range in Figur e 4-7 since the starting point of the fitting range is fixed at 0.01 V to include the small input amplitudes. The x-axis represents the end point of the fitting range in volts. This gr aph shows us that indi vidual coefficients K1, K3 and K5 vary based on the widths of the fitting ra nge. It is important to define the best fitting range to choose the extracted nonlinear coefficients as shown in Figure 4-7 since the nonlinear coefficients vary with the width of the data fitting range. Figure 4-8 Standard errors of nonlinear coefficients The standard errors of the separate coeffi cients in their fitting ranges are calculated by using the following equations that were presented in chapter 3. ii ib V K e s ) .( (4-2)

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33 where ) .( .iK e s is the standard error of the iK coefficient and iib V is the diagonal term of the variance of the parameters or nonlinear coefficients which are calculated by equation (3-28). Figure 4-8 shows the standard error of nonlinear coefficients according to the width of the fitting range. In this graph, ) .( .iK e s points to the sta ndard error of the nonlinear coefficient Ki. The lowest values for the standard errors of K1, K3 and K5 are shown by Point A in Figure 4-8. This point is about 0.5 V that is near 1 dB gain compression point, 0.52 V. The sum of square of the residual, ) (iE SS between the fitting model and the actual gain curve can be found using equation (4-3), Y X b Y Y E E E SST T T T i ) ( (4-3) where the calculation is the same as equation (3-26). This quantity shows the total error Figure 4-9 The sum of s quares of the residual

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34 caused by the difference between the fitting model and the actual model. Figure 4-9 shows the sum of square of the residual,) (iE SS. In this graph, the total error quantity is significantly lower under the 0.6 V fitting ra nge. Point A represents the lowest value found in Figure 4-8. From this graph, it is s hown that the width of the fitting range from 0.01 V to point A of Figure 4-8 has small ) (iE SS. Therefore, the fitting range that has the minimum standard error and the small tota l residuals can be de fined through Figure 48 and Figure 4-9. The fitting outcome for this set of prospective ra nges is summarized in Table 4-1. At the thirdorder-intercept point IIP3, the input voltage is 2.21 V or 6.9 dB input voltage with the resulting harmonic coe fficients. This measured input value shows about 0.7 dB difference in comparison with the simulated value of 6.24 dB. This error may be created by the phase information that is missing from the fitting model and is caused by errors in the calculation of the pow er measurements with no phase information. In general, Volterra series are used to model the case of weak ly nonlinear behavior. However, the one tone measurement is not sufficient to implement both phaseinformation detection and phase analysis that are required for Volterra-series parameters. In spite of these errors, 1 dB or less difference for th e third-order-intercept point simulation and measurement is a good result. Table 4-1 The summary of fitting results Parameters Values Coefficient K1 3.05 Coefficient K3 -0.83 Coefficient K5 -3.67 Calculated V(IIP3) 2.21V(6.9 dB) Simulated V(IIP3) 2.05V(6.24 dB)

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35 Figure 4-10 Gain curves with 0.1% and 2% random noise 4.2 Measurement Error Consideration Measurement error needs to be considered in the actual data measurement with the algorithm presented in Section 4.1. The effect of additional random noise upon the simulated data is analyzed in this section fo r the determination of the measurement error. In addition, the error boundaries are studied in order to pr oduce satisfactory results with the proposed algorithm. To analyze the effect of random noise, random noise is created by using random number generation in Matlab software and is added to the each one-tone data point used in the previous section. i i i in P y y z %) ( (4-4)

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36 Figure 4-11 Calculated thirdorder intercept point with 0.1 percent added random noise where iy is the voltage output amplitude of the individual data point, in is the real number that is chosen randomly from -1 to 1, P is the percentage of added random noise, and iz is a new data point that includes the output amplitude and random noise. Figure 410 shows two gain curves with 0.1 % and 2 % additional random noise. Curve A is the gain curve with 0.1 % additional random noi se and Curve B represents the gain curve including 2% additional random noise. In this section, the fitti ng range from 0.01V to 0.52 V that was defined in Section 4.1 is us ed for fitting the gain curve data including random noise. The gain curve data with 0.1 % added random noise is fitted in the fixed fitting range. The same procedure is repeated one thousand times averaging in the gain data with different 0.1 % random noise. The calculated third-order intercept points over one thousand samples are shown in Figure 411. The average value over one thousand

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37 Figure 4-12 Calculated thirdorder intercept point with 2.0 percent added random noise Figure 4-13 The influence of added random error on the fitting results

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38 samples is 2.216 V and the standard deviati on of these samples is 0.0149 V. Figure 4-12 shows the calculated third-or der intercept point with 2 pe rcent added random noise over one thousand samples. In this graph, the distribution of thirdorder intercept point calculated from the gain data added 2% random noise is wider than that added 0.1 % random data. The average value of these samp les is 2.3023 V and the standard deviation is 0.3692 V. Fitting results on the each additi onal random error amount and the associated standard deviation are shown in the Figure 4-13. In this graph, added random error increases by 0.1% increments from 0.1% to 2.0%. As the additional amount of random error increases, standard devi ation grows. Two dotted lines A and B show 0.5 dB as an acceptable range of third-order intercept point error for the center value of 6.9 dB. Each bar represents the standard deviation for one thousand samples. If the fitting results from random errors are limited within 0.5 dB the additional random error amounts are within 0.8%. In other words, if measurement error is in 0. 8% of the correct measurement value, the applied result through the fitting algorithm makes an IIP3 estimation with 0.5 dB accuracy possible. However, if more than 1% random errors are added to this data, it becomes difficult to predict the IIP3 from a single measuremen t of the gain curve and produce an accurate value. Data averaging with multiple data point measurements to reduce average error must be used. 4.3 Frequency Effect on the Fitting Algorithm What kind nonlinearity effects can hi gher frequencies produce? Frequency becomes higher for recent wireless circuits. In addition, the high frequency region of operation needs to be measured to analyze the nonlinearity of RF components. The extraction algorithm is applied to frequencies up to 30 GHz in the circuits used in the previous sections. The results are shown in the Table 4-2. According to Table 4-2, the

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39 difference between the extracted third-order intercept point and simulated value grows when the frequency increases from 100 MHz to 5 GHz. There is an evident difference between the simulation and the power series model used for the fitting algorithm. In order to analyze that difference, the gain comp ression curve is represented by Volterra-series components. Appendix B shows Volterra-kerne ls of a common-source amplifier. When the input is ) cos(1t V Vin (4-5) the output amplitude at the fundame ntal frequency is as follows, ...} ) exp( ) , , ( H 8 5 ) exp( ) , ( H 4 3 ) exp( ) ( H Re{ ) ; (1 1 1 1 1 1 5 5 1 1 1 1 3 3 1 1 1 1 t j V t j V t j V t Vout (4-6) Table 4-2 Frequency effect on the fitting algorithm Frequency V1-dB(dB) VIIP3(dB) Estimated VIIP3(dB) Difference (dB) 100 MHz 0.52 V (-5.68 dB) 2.05 V (6.24 dB) 2.21 V (6.89 dB) 0.65 dB 900 MHz 0.52 V (-5.68 dB) 2.05 V (6.24 dB) 2.23 V (6.97 dB) 0.73 dB 2 GHz 0.53 V (-5.68 dB) 2.06 V (6.28 dB) 2.29 V (7.20 dB) 0.92 dB 5 GHz 0.57 V (-5.51 dB) 2.07 V (6.32 dB) 2.43 V (7.71 dB) 1.39 dB 10 GHz 0.61 V (-4.88 dB) 2.04 V (6.19 dB) 2.21 V (7.00 dB) 0.70 dB 20 GHz 0.64 V (-4.29 dB) 2.05 V (6.24 dB) 2.24 V (6.24 dB) 0.77 dB 30 GHz 0.67 V (-3.88 dB) 2.12 V (6.53 dB) 2.41 V (7.64 dB) 1.11 dB

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40 At low frequency, the parasiti c capacitance between gate and drain, Cgd, can be ignored. Using the calculated Volterra-series kernel in Appendix B, the output amplitude is, ...} ) exp( 8 5 ) exp( 4 3 ) exp( Re{ ) ; (1 1 5 5 1 1 3 3 1 1 1 1 t j C j G V K t j C j G V K t j C j G V K t VL L L L L L out (4-7) From equation (4-7), each coefficient has same phase if gdC is disregarded and dbC is a linear parasitic capacitor. At low frequency, Volterra-series coefficients are similar to Power-series coefficients. The output amplit ude at the fundamental frequency is in equation (4-8). 5 5 3 3 1 1' 8 5 4 3 ) ( V K V K V K Vout (4-8) where 2 / 1 2 2 1 2' L L i iC G K K. At high frequency, gdC should be regarded in the calc ulation of Volterra-kernels. If gdC is included in Volterra-kernels, the output amplitude is ...} ) exp( 8 5 ) exp( 4 3 ) exp( ) ( Re{ ) ; (1 1 5 5 1 1 3 3 1 1 1 1 1 t j C j G V K t j C j G V K t j C j G V C j K t VL L L L L L gd out (4-9) Apart from disregarded gdC in the previous analysis, each coefficient has different phase. Due to the phase discrepancy, each coeffi cient needs to be described by a complex number. Therefore, another effect on the freque ncy exists when the result of one-tone test is modeled as a Volterra series. According to Table 4-2, the estimat ed deviation value by 5 GHz increases up to 1.4 dB. After that, the deviation does not increase with frequency.

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41 Frequency has a greater effect on phase than amplitude at more than 5 GHz. In addition, the gain of the amplifier reduces drastically Though the deviation of the estimate grows the increased frequency, the deviation is lim ited by within 1.5 dB at any frequency. 4.4 Load Effect on the Fitting Algorithm The nonlinear characteristics of an amplifier is affected by the load of the amplifier. In this section, the effect of loads on the non linear characteristic is studied and the effect on the fitting algorithm is researched. Two type s of loads are considered for affecting the nonlinear characteristics. A passive load is composed of passive components such as resistors, capacitors and induc tors. An active load is a curr ent mirror that is made by an active device such as transistor. First, th e passive load effects on a single transistor amplifier are studied. The equiva lent circuit of a general single transistor amplifier is shown in Figure 4-14. In this figure, ZL represents the passive load, Zin is an input impedance, V1 and V2 are an input node and an output node respectively, i represents the input-voltage-controlled-current source. The pa ssive components used in the passive load Figure 4-14 An equivalent circuit

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42 are assumed to be linear components to simplify calculation. Under a given bias condition, the voltage-controlled-current source can be described by Taylor series at the quiescent point in equati on (2-6) in chapter 2. ... v v v3 3 2 2 1 K K K i (4-10) The passive load is a func tion of s in the s-domain, ) ( s Z ZL L (4-11) Volterra-kernel at the out put node in the equivalent circuit can be described by ) ( ) ,..., ( H1 2 1 i n n L i i is Z K s s s (4-12) The nonlinear characteristic of an amplifier is determined by the above Volterra-kernels. The output amplitude at a fundamental frequency in one-tone test of th is circuit is found by combining Volterra-kernels. If the input is ) cos(1t V Vin (4-13) then the output at fundamental frequency 1 is ...} ) exp( ) , , ( H 8 5 ) exp( ) , ( H 4 3 ) exp( ) ( H Re{ ) ; (1 1 1 1 1 1 5 5 1 1 1 1 3 3 1 1 1 1 t j V t j V t j V t Vout (4-14) The load impedance used in odd-order Volterra-kernels is 1 1 1j s si n i (4-15) ) ( ) (1 1j Z s ZL i n n L (4-16) Therefore, the odd-order Volterra-kernel at the fundamental frequency is ) ( H1 j Z KL i i (4-17)

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43 The passive load is used to the change of each odd-order Volterra-kernel. This change includes frequency effects. In the model used in this analysis, the change of Volterrakernel yields only frequency eff ects in the passive load. The f itting algorithm is not affect by the passive load except for the change caused by the operating frequency. The effect of the operating frequency on the amplifier wa s researched in the previous section. The effect of the active load on the non linear characteristic of a single transistor amplifier is studied. Figure 4-15 shows an exam ple of a single transistor amplifier with an active load. BSIM3 models of PMOS and NMOS FETs are listed in Appendix A. The current mirror consisted of two PMOS devices, M2 and M3, which act as an active load and the sizes of these two devices are th e minimum width and length ( W/L=1.18 m / 0.25 m). In this schematic, supply bias voltage VDD is 3.3 V, R1 is 10 k V1 is the gate Figure 4-15 A schematic of a common-s ource amplifier with an active load

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44 A B Figure 4-16 DC Characteristics of a common-source amplifier with an active load. A) is current and B) is voltage.

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45 Figure 4-17 The AC simulation of a comm on-source amplifier with an active load bias voltage of the current mirror. The I-V ch aracteristic curve and transfer characteristic curve are shown in Figure 4-16. In Figure 4-16.A, Iref is the current of the resistor R1 whose value is 151.2 A. I1 is the drain current of the transistor M1. In Figure 4-16.B, V1 is the gate bias of transistors M2 and M3 which construct current mirror, the value of V1 is 1.51 V. Vo is the drain voltage of transistor M1. For the simulation of AC, one-tone and two-tone test, the gate bias vol tage is chosen at the point VG in which all MOS transistors are in saturation mode. The result of AC simulation of this amplifier is shown in Figure 4-17. The 3-dB bandwidth of this amplifier is about 1 GHz. For applying the fitting algorithm to the common-source amplifier with active load, one-tone and two-tone tests are simulated in this schematic at the gate bias voltage VG. The result of the one-tone test

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46 Figure 4-18 The results of one-tone simulation Figure 4-19 The results of two-tone simulation

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47 is shown in Figure 4-18. In this test, the am plitude of the input voltage is swept from 0.001 V to 1 V at the fundamental frequency 100 MHz. Curve A is the gain curve and Line B represents the ideal gain curve unde r the assumption that there are no harmonics. The 1 dB gain compression point is denoted by GC in this graph. The value of input-referred 1 dB compression point is 0.08V Figure 4-19 shows the result of two-tone simulation. The applied frequencies in th e two-tone test are 100 MHz and 110 MHz. Curve A and B represent the output voltage s at 100 MHz and at 90 MHz separately. Curve A is the fundamental frequency of 100 MHz, Curve B indicates the amplitude at intermodulation frequency of 90 MHz. Li ne C and D indicate the ideal harmonic response. The point of intersection of the tw o ideal harmonic lines, denoted by point TOI, indicates the third order interc eption point that has a value of 0.25V. The fitting algorithm that extracts nonlinear coefficients fr om a gain compression curve is explained previously in Section 4-1. The same algorithm is applied to this single transistor amplifier Figure 4-20 Extracted no nlinear coefficient K1

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48 with an active load. Figure 4-20 shows the value of coefficient K1 though this fitting algorithm. In this figure, the value of K1 is extracted by the fitting method through the fitting range from 0.001 V to the end point of the fitting range. The vo ltage gain of this amplifier is greater than that of a common source amplifier in Section 4-1 since the value of K1 in this figure is greater than that of K1 in Figure 4-7. The value of K3 is shown in Figure 4-21. The negative sign of K3 causes the gain compression on Figure 4-21 Extracted n onlinear coefficient K3 the gain curve since the sign of K3 is opposite to the sign of K1. The extracted value of K5 is shown in Figure 4-22. The value of this co efficient decreases rapidly when the width of the fitting range increases. Through these thr ee graphs, the extracted values of nonlinear coefficients change according to the width of the fitting range. This result is the same as that in the previous section. Figure 423 shows the standard errors of nonlinear coefficients according to the width of the fitting range. In this graph, Ki represents the

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49 Figure 4-22 Extracted no nlinear coefficient K5 Figure 4-23 Standard errors of nonlinear coefficients

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50 standard error of the nonlinear coefficient Ki, curve K1 and K3 follows the left y-axis and curve K5 is represented by the right y-axis. The lo west values for the standard errors of each coefficients are shown by Point A in Figure 4-23. This point is about 0.14 V. This is greater than the 1 dB gain compression point, 0.08 V. The best fitting range of this circuit is different from that of a common-source am plifier discussed in S ection 4-1. Figure 4-24 shows the sum of square of the residual. In this graph, the total error quantity is significantly lower at point A in the standard error graph. Therefore, the fitting range that has the minimum standard error and the small total residuals can be defined through these two graphs. The fitting outcome for this set of prospective ranges is summarized in Table 4-3. At the 3rd order intercept point, the input voltage is 0.238 V and -12.48 dB. This value shows about a 0.44 dB difference in comparison with the simulated value -12.04 dB. The estimated third-order intercept point is close to the simulated third-order intercept point. The applied fitting algorithm works well in this example. The equivalent circuit of an amplifie r with an active load can be m odeled by the passive component at the quiescent point. The Volterra analysis researched in the previous part of this section is applied to this amplifier with an active load by replacing th e active load with the passive components. The fitting algorithm is not affect by the active load except for the change caused by the operating frequency. Table 4-3 Fitting results Parameters Values Coefficient K1 11.63 Coefficient K3 -274.7 Coefficient K5 3957 Calculated V(IIP3) 0.238 V (-12.48 dB) Simulated V(IIP3) 0.25 V (-12.04 dB)

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51 4.5 Summary In this chapter, the proposed fitting algorithm has been verified through the application of the algorithm to the simula tion of a common-source amplifier. The best fitting range has been chosen through the standa rd errors of nonlinear coefficients and the sum of squares of the residua ls. The effect of measurement error was explored by adding random noise to the simulation data. Fre quency effects on the fitting algorithm was examined up to 30 GHz. The IIP3 estimation error is less than 2 dB up to 30 GHz. Through the Volterra series analysis, the effect of loads on the algor ithm was investigated and there were no differences except fo r the changes in operating frequency.

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52 CHAPTER 5 COMMERCIAL RF WIDEBAND AMPLIFIER 5.1 Nonlinearity Test Commercial amplifiers and test boards (ERA-series, MiniCircuit, Co.) shown in Figure 5-1 were used for the measurement of onetone and two-tone test s. It is difficult to extract the nonlinear power coefficients for th ese ERA amplifiers due to noise sources in the measurement system. One of the noise sour ces is the signal ge nerator. Even though this signal generator is designed to give a pure signal source with elaborate internal units, it still has harmonics in the frequency-domain. A low pass filter is used to decrease the harmonics in the signal generator. Figure 5-2 shows the spectrum in the signal source with and without a low pass filter. In Figur e 5-2.A, Curve A repr esents the power at fundamental frequency, Curve B corresponds to the power at second harmonic frequency, and Curve C stands for the power at third harmon ic frequency. In this graph, the power at second harmonic and third harmonic frequencies exist in the signal generator. Figure 52(b) shows that the harmonic components de crease when the low pass filter is used. Figure 5-1 ERA1 amplifier in a test board

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53 A B Figure 5-2 The spectrum from the signal so urce. A) shows the sp ectrum without a low pass filter and B) shows the sp ectrum with a low pass filter. The ERA1 amplifier is measured in a one-t one test. The test scheme is shown in Figure 5-3. The fundamental fr equency used for this test is 100 MHz. Figure 5-4 shows the result of the one-tone measurement. In this graph, the input referred 1 dB compression point, IP1-dB is 1.7 dBm and is denoted by GC. In this figure, Curve A represents the measured compression curve and Li ne B is the ideal linear curve. Here, the gain compression curve at the fundamental fre quency, Curve A is used for the extraction of nonlinear coefficients. The application of a fitting method on this curve will be

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54 Figure 5-3 One-tone test scheme Figure 5-4 The measurement data of one-tone test (ERA1 amplifier) explained in next section. Figure 5-5 shows th e test scheme for the two-tone test. Source frequencies in this two-tone test are 100 MH z and 120 MHz. The result of the two-tone test is shown in Figure 5-6. In this figure, Curve A and B represent the amplifier output signal amplitudes at 100 MHz and at 80 MHz se parately. Curve A represents the output power at fundamental frequency of 100 MHz, Curve B indicates the output power at

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55 intermodulation frequency of 80 MHz (2100 MHz – 120 MHz). The point of intersection of the two dotted lines, IP3 indicates the third-order intercept point. The input-referred third-orde r intercept point, IIP3 is 16.3 dBm. GC denotes 1 dB gain compression point that has a value of -2.5 dB m. The 1 dB compression point in two-tone test is explained in section 3.3. Figure 5-5 Two-tone test scheme Figure 5-6 The measurement data of two-tone test (ERA1 amplifier)

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56 ERA2 and ERA3 amplifiers are also test ed in the same bias condition as ERA1 amplifier to verify the proposed algorithm to predict third-order intercept point using extraction of nonlinear coefficients from th e gain compression curve. The one-tone test data of ERA2 is shown in Figure 5-7. Input-referred 1 dB gain compression point denoted by GC is -0.8 dBm. Figure 5-8 shows two-tone data of ERA2 Like Figure 5-6, Figure 5-7 The measurement data of one-tone test (ERA2 amplifier) Figure 5-8 The measurement data of two-tone test (ERA2 amplifier)

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57 Figure 5-9 The measurement data of one-tone test (ERA3 amplifier) Figure 5-10 The measurement data of two-tone test (ERA3 amplifier)

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58 Table 5-1 The summary of the measurement results of commercial amplifiers Device IP1-dB* IP1-dB,2** IIP3** IIP3-IP1-dB ERA1 1.7 -2.5 16.3 14.6 ERA2 -0.8 -5 13.8 14.6 ERA3 -8 -13 4.5 12.5 *One-tone test : Source frequency = 100 MHz **Two-tone test : frequencies = 100 MHz, 120 MHz Curve A and Curve B represent the output power at 100 MHz and at 80 MHz separately. The input-referred third-order in tercept point is 13.8 dBm. The input-referred 1 dB gain compression point denoted by GC is -5 dBm. The 1 dB gain compression point of the ERA3 amplifier is -8 dBm denoted by GC in Figur e 5-9. The result of the two-tone test is shown in Figure 5-10. The 1 dB compressi on point is -13 dBm and the input-referred third-order intercept point is 4.5 dBm in Figure 5-10. The measurement data of these amplifiers are summarized in Table 5-1. In this table, the differe nce between 1 dB gain compression point and third-order intercep t point is not cons tant. Through real measurements, the relationship between two non linear characteristics can be constructed. 5.2 IIP3 Prediction from the Gain Compression Curve The application of the technique to manuf acturing test LNA measurements required substantial modification. IIP3 extraction from LNA gain simu lation data has the benefit of many decibel places of accuracy (high S/N) and an ideal (no loss) test system. Real measurements from spectrum analyzers can e xhibit roughly 1% accuracy in the data and substantial power loss due to cable s and fixtures in the test se tup. Even after using power magnitude calibration techniques on spectrum an alyzer data, signifi cant uncertainty can exist due to phase errors. In addition, it was found that to ex tract properly IIP3 parameters the data has to be measured on one power range of the spectrum analyzer.

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59 Using multiple spectrum analyzer power ranges introduced offset errors in the measured data. In summary, the simulated LNA data had remarkably high S/N across the entire data set while the measured LNA gain data had a mediocre S/N at high-power which decreased as power decreased. Given the situation, a global extracti on of the LNA power series expansion coefficients was not stable with small change s in the data set. Adding a few more data points at the high-power range woul d create large changes in the K3 and K5 extracted coefficients. To counteract this problem, which could be seen in ATE systems doing manufacturing test, a new parameter extrac tion methodology had to be created. After experimenting with many ways of perfor ming this parameter extraction, a regional parameter extraction methodology was devised. Figure 5-11 One-tone data and extr action from ERA1 device at 100 MHz

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60 A new robust measurement extraction algor ithm is developed for one-tone gain data as graphed in Figure 5-11 for ERA1 amp lifier. In this new algorithm, the nonlinear power coefficients are extracted regionally. To help understanding how to interpret this graph, the procedure is explai ned step by step in Figure 5-12. First, the entire one-tone Figure 5-12 A flow chart for estimati on of IIP3 from one-tone measurement power compression curve is measured as show n by curve B in Figure 5-11. Line B is a straight-line that can be fitted to the lowpower amplifier data. From this line B, the K1 factor is determined. The effects of the K1 factor are subtracted fr om the original gain compression curve A (slope 1/1). Curve C shows the remaining terms on the gain compression curve. From this curve C, it is easy to see that the K3 extraction region

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61 below point P of region R has ve ry high noise. Instead of K3, the K5 factor is extracted from the compression region R of Figure 5-11. This is easily verified because the slope of line D is 5/1. Fortunately, it is not necessary to know the slope of the K3 factor, since it is, idealy, 3/1. To determine the value K3, one need to know where it intercepts line D and that is at point P. Point P is at the intercept between the K3 line and the K5 line. Point P is also the highest S/N point in the measured K3 factor data. The calculated IIP3 from this point is 16.82 dBm is very close to the measured IIP3, 16.3 dBm. Figure 5-13 shows the application of the algorithm to ERA2 amplifier. The explanation of this graph is the same Figure 5-13 One-tone data and extr action from ERA2 device at 100 MHz Table 5-2 The summary of the estimated IIP3 of commercial amplifiers Device Measured IIP3 Estimated IIP3 ERA1 16.3 16.82 ERA2 13.8 12.81 ERA3 4.5 3.71

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62 Figure 5-14 One-tone data and extr action from ERA3 device at 100 MHz as that of Figure 5-11. The same analysis is applied to ERA3 amplifier in Figure 5-14. Table 5-2 shows the measured ch aracteristics and estimated IIP3 of the ERA devices (ERA1, ERA2 and ERA3). From the table, the difference between the measured IIP3 and the estimated IIP3 is less than 2 dB in all cases and le ss than or equal to 0.41 dB for most cases. Through these experiments a method fo r predicting IIP3 using a one-tone LNA gain measurement was developed. 5.3 The Application of the Propos ed Algorithm at High Frequency The ERA2 amplifier is retested at a rela tively high frequenc y 2.4 GHz in Figure 515. Fig 5-15 shows the new robust extraction al gorithm at high frequency. The result in Table 5-3 shows that this al gorithm is working in microwave frequencies. Through these experiments a method for predicting IIP3 using a one-tone LNA gain measurement was developed.

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63 Table 5-3 The measurement data and calculated IIP3 of a commercial amplifier Device IP1-dB* IIP3** IIP3*** ERA2 -0.4 12.1 12.08 *One-tone test : Source frequency = 2.4 GHz **Two-tone test : frequencies = 2.4 GHz, 2.475 GHz *** Calculated from the extracted nonlinear coefficients Figure 5-15 One-tone data and extr action from ERA2 device at 2.4 GHz 5.4 IP1-dB Estimation from Two-tone Data The algorithm in section 5.2 estimates IIP3 from the one-tone data. In this section, the estimation of the 1 dB gain compression poin t is investigated on the basis of the twotone measurement data. Let the input amplitude at the 1 dB compression point be A1-dB. From equation (3-24), The 1 dB compression point in the one-tone test is found in equation (3-24).

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64 0 109 0 4 3 8 52 1 1 3 4 1 1 5 dB dBA K K A K K (5-1) Two ratios, 1 5K K and 1 3K K are needed to solve above e quation (5-1). These ratios are found in the two-tone measurement data From equation (3-9), the ratio 1 3K K is determined using following equation, 2 3 1 31 3 4IPA K K (5-2) where 3 IPA is the input amplitude at the third-orde r intercept point. The sign of this ratio should be decided by inspecting the gain curv e at the fundamental frequency in the twotone measurement. For example, the sign of 1 3K K is negative in Figure 5-6 since Curve A shows the compression in the gain curve. Th e 1 dB gain compression point in the twotone test is explained by equation (5-3), 0 109 0 4 9 4 252 2 1 1 3 4 2 1 1 5 dB dBA K K A K K (5-3) where 2 1 dBA is the input amplitude at the 1 dB compression point of the two-tone test that is denoted by GC in Figure 56. From above equation, the ratio 1 5K K can be found by 109 0 4 9 25 42 2 1 1 3 4 2 1 1 5 dB dBA K K A K K (5-4)

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65 Inserting two ratios 1 5K K and 1 3K K to equation (5-1), th e input-referred 1 dB gain compression point, IP1-dB can be calculated. This method is applied to ERA amplifiers for verification. First, the ERA1 amplifier is considered as the application example of this algorithm. IIP3 of ERA1 amplifier is 16.3 dBm in Ta ble 5-1. The input amplitude at this point is 065 2 1020 / ) 10 ( 33 IIP IPA (5-5) where the unit of 3 IPAis volt (V) and the input resistor and the output resistor loads are 50 The absolute value of the ratio, 1 3K K can be found in equation (5-2). 3146 0 065 2 3 42 1 3 K K (5-6) The sign of above ratio 1 3K K is negative since the gain at the fundamental frequency compresses in Figure 5-6. The ratio 1 3K K is 3146 01 3 K K (5-7) From table 5-1, the IP1-dB,2 is -2.5 dBm. The input amplitude at this 1 dB compression point in two-tone is 2371 0 1020 / ) 10 ( 2 12 1 dBIP dBA (5-8)

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66 Using the value of 2 1 dBA and the ratio 1 3K K, the ratio 1 5K K is found in equation (5-4). The calculated value of 1 5K K is -3.5140. The 1 dB compression point in the one-tone test is now found by solving equation (5-1). The solution of this equation is 0.4192. The calculated dBIP 1 is 2.45 dBm and is very close to the measured value, 1.7 dBm. Through the application of the algorithm, two ratios 1 5K K and 1 3K K are found and 1 dB compression point is estimated. The differe nce between the measured value and the estimated value of 1 dB compression point is less than 1 dB. The data of the ERA2 amplifier is used for another example to verify this algorithm. First, the input amplitude at third-order intercept point is 5488 13IPA (5-9) since IIP3 of ERA2 amplifier is 13.8 dBm in Tabl e 5-1. From this value, the absolute value of the ratio 1 3K K is calculated. The sign of this ratio is negative since the gain curve compresses in Figure 5-8. The calculated value of 1 3K K is -0.5588. From 2 1 dBIP in Table 5-1, the calculated value of 2 1 dBA is 0.1778. Using the value of 2 1 dBAand the ratio 1 3K K, the calculated value of the ratio 1 5K K is -11.1224. Fina lly, the calculated value of dBA 1 from equation (5-1) is 0.5143. The estimated value of the 1 dB compression point in one-tone te st is -0.05 dBm. Compared to the measured value of 1 dB compression point, -0.8 dBm, the estimation error is less than 1 dB.

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67 Finally, the data of the ERA3 amplifier is used for the estimation of 1 dB compression point from two-tone data. In this amplifier, the input amplitude at thirdorder intercept point is 5309 03IPA (5-10) since IIP3 of ERA3 amplifier is 4.5 dBm in Tabl e 5-1. The absolute value of the ratio 1 3K K is calculated by using equation (5-2). Th e sign of this ratio need to be negative since the gain curve compresses in Figure 5-10. 7308 41 3 K K (5-11) From table 5-1, the IP1-dB,2 is -13 dBm. The input amplitude at this 1 dB compression point in two-tone data is 0708 02 1 dBA (5-12) Using the value of 2 1 dBAand the ratio 1 3K K, the calculated value of the ratio 1 5K K is 5 3541 5 K K (5-13) From equation (5-1), the calculated value of dBA 1 is 0.1248. The estimated value of the 1 dB compression point in one-tone test is -8.08 dBm and is very close to the measured value, -8 dBm. Table 5-4 summarize the results of these applications of three amplifiers. Table 5-4 The summary of the application results of the IP1-dB estimation algorithm Device Measured IP1-dB Estimated IIP3 ERA1 1.7 2.45 ERA2 -0.8 -0.05 ERA3 -8 -8.08

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68 In this table, the difference between the measured IP1-dB and the estimated IP1-dB is less than 1 dB in all cases and less than or e qual to 0.75 dB for most cases. Through these experiments a method for estimating IP1-dB using a two-tone LNA gain measurement was developed. 5.5 Summary In this chapter, a robus t algorithm to predict IIP3 has been developed for the wideband RF amplifier. Given the noisy measur ement situation, a global extraction of the LNA power series expansion coefficients was not stable with small changes in the data set. Adding a few more data points at the high-power range would create large changes in the K3 and K5 extracted coefficients. To counteract this problem, which would be seen in ATE systems doing manufacturing test, a new parameter extraction methodology had to be created. After experimenting with many ways of performing this parameter extraction, a regional parameter extr action methodology was devised. The IP1-dB prediction from two-tone measurement has been applied to these wideband amplifiers. Through several steps of simple calculation us ing the third-order in tercept point and the gain compression at the fundamental frequency, IP1-dB has been estimated within less than 1 dB error.

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69 CHAPTER 6 POWER AMPLIFIERS 6.1 Linear and Nonlinear Power Amplifiers The final stage which gives signal to an antenna is apower amplifier. The characteristics of this power amplifier is specified by the communication scheme. Linearity and efficiency among these characteri stics are important factors for determining power amplifiers in their circuit application. Nonlinear power amplif iers are preferred in constant envelope modulations and Linear power amplifiers are used in amplitude modulations and /4-QPSK in digital modulations. The issue of nonlinearity is directly re lated to the spectral regrowth in a communication system transmission. The sta ndards for various communication protocals define the nonlinearity as adjacent channe l power ratio (ACPR) or adjacent channel power (ACP). In the early stag e of designing RF components, IIP3 measured in two-tone test is used for the measure of nonlinearity instead of ACPR or ACP. IIP3 is easily measured and reported the nonlinearity estimation even though IIP3 is not exactly the same as ACPR or ACP. Linear power amplifier operation is classi fied as “class A” operation. An ideal linear amplifier doesn’t have nonlinear terms. But in actual case the active devices used for the amplifiers do produce harmonics. In cl ass A amplifier, nonlinear behavior is made be a weakly nonlinear behavior The definition of this w eakly nonlinear behavior is explained in chapter 2. This behavior can be analyzed by using the power series or

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70 Volterra series. In this chapter, the power amplifiers ar e investigated for their nonlinear characteristics. 6.2 Measurement of Commercial Power Amplifiers Four commercial RF power amplifiers are measured in one-tone and two-tone tests. To distinguish these amplifiers, amplifiers are named alphabetically. First, the result of one-tone test on the amplifier A is shown in Figure 6-1. The fundamental frequency used for this test is 2.45 GHz. In this figure, Curve A represents the measured output power and the dotted Line B is the ideal linear curve. The 1 dB gain compression point is denoted by GC in this figure. The input -referred 1 dB gain compression point, IP1-dB is 11 dBm. Figure 6-2 shows the result of a two-tone test. The two input frequencies are 2.45 GHz and 2.46 GHz. Curve A represents th e amplifier output signal power at 2.45 GHz and dotted Line B is the ideal linear curve extrapolated from Curve A. Curve C indicates the output power at intermodulation frequency of 2.44 GHz (22.45 GHz – 2.46 GHz). The dotted Line D has slope of 3 and is extrapolated from the low input power region of Curve C. The point of intersection of the two dotted line s, TOI indicates the third-order intercept point. The input-re ferred third-order intercept point, IIP3 is -4.4 dBm. GC2 denotes 1 dB gain compression point in two-tone test whic h has a value of -16 dBm. Figure 6-3 and Figure 6-4 show the re sults of nonlinear tests on amplifier B. The test condition of this amplifie r is the same as that of the amplifier A. In Figure 6-3, the input-referred 1 dB gain comp ression point is -8 dBm. IIP3 is 0 dBm and IP1-dB,2 denoted by GC2, is -13 dBm. The test results of the amplifier C are shown in Figure 6-5 and Figure 6-6. The same test conditi on as that of the amplifier A is applied to this test of the amplifier C. IP1-dB denoted by GC, is -6 dBm from the result of one-tone test in Figure 6-5. IP1-dB,2 is -10 dBm and IIP3 is 3 dBm in Figure 6-6. Figure 6-7 shows the results

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71 Figure 6-1 The measurement data of one-tone test of the amplifier A Figure 6-2 The measurement data of two-tone test of the amplifier A

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72 Figure 6-3 The measurement data of one-tone test of the amplifier B Figure 6-4 The measurement data of two-tone test of the amplifier B

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73 Figure 6-5 The measurement data of one-tone test of the amplifier C Figure 6-6 The measurement data of two-tone test of the amplifier C

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74 Figure 6-7 The measurement data of one-tone test of the amplifier D Figure 6-8 The measurement data of two-tone test of the amplifier D

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75 Table 6-1 The summary of the measurement results of commercial PAs Device Name IP1-dB* IP1-dB,2** IIP3** IIP3-IP1-dB Amplifier A -11 dBm -16 dBm -4.4 dBm 6.6 dB Amplifier B -8 dBm -13 dBm 0 dBm 8 dB Amplifier C -6 dBm -10 dBm 3 dBm 9 dB Amplifier D*** -8 dBm -12.5 dBm -1 dBm 7 dB *One-tone test : Source frequency = 2.45 GHz **Two-tone test : frequencies = 2.45 GHz, 2.46 GHz *** Amplifier D is tested at 5.2 GHz (and 5.21 GHz). of the one-tone test of the amplifier D. The input frequency in this one-tone test is 5.2 GHz. IP1-dB is -8 dBm in this figure. The applied two frequencies in two-tone test are 5.2 GHz and 5.21 GHz. IIP3 is -1 dBm and IP1-dB,2 is -12.5 dBm. The measurement data of these amplifiers are summarized in Table 6-1. In this table, the difference between 1 dB gain compression point and third-orde r intercept point is not constant. 6.3 IIP3 Estimation from the One-tone Data A robust algorithm which analyzes the measurement data with error is used for the wideband RF amplifier. In RF power amplifier, another algorithm is required to examine the nonlinear characteristics sin ce the nonlinear behavior of pow er amplifier is somewhat different from that of wideband RF amplifier. A fitting method which extracts the nonlin ear coefficients simultaneously from one-tone data and is used in the analysis of simulation data in chapter 4 is useful in the nonlinear analysis of RF power amplifiers. The fitting range is adjusted through linear regression analysis. The increase of fitti ng range reduces the standard error of each coefficient but the residual, the difference between real model and fitting model, increases since the high amplifier input power includes energy in higher-order nonlinear factors than that of fitting model. Both the analysis of the residual and that of standard

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76 errors of each coefficients help to define the appropriate range fo r the fitting model. Third-order intercept point can be estimate d using the extracted nonlinear coefficients after fitting this range. In this section, tw o fitting models are applied to the data of commercial power amplifiers. One of two fitting model is 3 3 14 3 x K x K y (6-1) Equation (6-1) is the simplest form for the explanation of the nonlinear gain curve. The other model applied to the fitting gain curve is 5 5 3 3 18 5 4 3 x K x K x K y (6-2) The difference between two fitting models is whether or not to include the fifthorder nonlinear coefficient in the fitting model. Through th e application of these two models to the data of commercial PAs, th e effect of adding fifth-order nonlinear coefficient is investigated. The first fitting model, equation (6-1) is applied to the one-tone data of the amplifier A shown in Figure 6-1. Figure 6-9 shows the value of coefficient K1 from the fitting results. In this graph, the x-axis represents the end point of fitting range from the starting point, -16 dBm. When the fitting range changes, the fitted result changes also. Figure 6-10 shows the value of coefficient K3. The best fitting range should be chosen using the standard errors of each coefficien ts and the sum of squares of the residuals which are explained in Chapter 4. The standa rd errors are defined in equation (4-2). Figure 6-11 shows the standard errors of coeffi cients K1 and K3. In this graph, Point A indicates the smallest value in both the standa rd errors of K1 and K3. From the standard error of the coefficients, the fitting range (16 dBm, -9 dBm) is chosen. The sum of

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77 Figure 6-9 The value of coefficient K1 (amplifier A) Figure 6-10 The value of coefficient K3 (amplifier A)

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78 Figure 6-11 Standard errors of K1 and K3 (amplifier A) Figure 6-12. Sum of squares of the residuals (Amplifier A)

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79 Table 6-2 The summary of fitting results (amplifier A) Parameters Values Coefficient K1 36.20 s.e.(K1) 0.15 Coefficient K3 -1026 s.e.(K3) 22 Calculated IIP3 -3.27 dBm squares of the residual in this fitting range is also small in Figure 6-12. The fitting results are summarized in Table 6-2. The calculated IIP3 is -3.27 dBm in this table. The difference between the estimated IIP3 and the measured IIP3 is 1.13 dB since the measured IIP3 is -4.4 dBm in Table 6-1. The second fitting model, equation (6-2) is applied the same data of the amplifier A. Figure 6-13, 6-14 and 6-15 show the values of K1, K3 and K5 respectively. From these graphs, the values of nonlinear coefficients will be chosen after the best fitting range is decided. From Figure 6-16 and 6-17, Point A denotes the end point of the best fitting range which satisfies the condition that the standard errors of nonlinear coefficients become small simultaneously. The best fitting range in this fitting process is (-16 dBm, -5 dBm). In this range, the sum of squares of th e residuals is small in Figure 6-18. The result of the fitting application is summarized in Table 6-3. In this table, the calculated value, -4.31 dBm is very close to the measur ed value, -4.4 dBm. The first fitting model has less standard errors of each coefficient than the second fitting model. The data of the amplifier B in Figure 6-3 is used for the IIP3 prediction from onetone measurement. Using fitting model, equation (6-1), the values of K1 and K3 are shown in Figure 6-19 and Figure 6-20. Figure 621 shows the standard errors of nonlinear

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80 Figure 6-13 The value of coefficient K1 (amplifier A) Figure 6-14 The value of coefficient K3 (amplifier A)

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81 Figure 6-15 The value of coefficient K5 (amplifier A) Figure 6-16 Standard errors of K1 and K3 (amplifier A)

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82 Figure 6-17 Standard error of K5 (amplifier A) Figure 6-18 Sum of squares of the residuals (amplifier A)

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83 Table 6-3 The summary of fitting results (amplifier A) Parameters Values Coefficient K1 36.96 s.e.(K1) 0.14 Coefficient K3 -1329 s.e.(K3) 29 Coefficient K5 22276 s.e.(K5) 1126 Calculated IIP3 -4.31 dBm Table 6-4 The summary of fitting results of the first model (amplifier B) Parameters Values Coefficient K1 34.50 s.e.(K1) 0.11 Coefficient K3 -404 s.e.(K3) 7 Calculated IIP3 0.43 dBm coefficients. In this figure, point A is the best region of this fitting. In the application of the first fitting model, the best fitting range is from -17 dBm to -5 dBm. In the graph of sum of squares of the residuals, Figure 6-22, th e value of sum of squares of the residual is under 0.1. In this best fitting range, the fitti ng result is summarized in Table 6-4. Using extracted nonlinear coefficients, the calculated IIP3 is 0.43 dBm. Compared to the measured IIP3, 0 dBm, the estimated IIP3 is close to the measured value. When the second fitting model is applied to the amplifie r B, the values of K1, K3 and K5 extracted from one-tone gain curve in Figure 6-3 are drawn in Figure 6-23, 6-24 and 6-25 separately. The standard errors of K1 and K3 are shown in Figure 6-26 and the standard

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84 Figure 6-19 The value of coefficient K1 (amplifier B) Figure 6-20 The value of coefficient K3 (amplifier B)

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85 Figure 6-21 Standard errors of K1 and K3 (amplifier B) Figure 6-22 Sum of squares of the residuals (amplifier B)

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86 error of K5 is shown in Figure 6-27. It is di fficult to find the end point of fitting range which gives small values simultaneously in th e standard errors of nonlinear coefficients in Figure 6-26 and 6-27. To choose the best fitting range, three re gions denoted by point A, point B and point C in these figures are comp ared in the standard errors with extracted nonlinear coefficients in Table 6-5. From this table, the region C has the smallest standard errors of three nonlinear coefficients and is chosen for the best fitting region. In this region C, the sum of squares of the re siduals are not big in Figure 6-28. Estimated IIP3 is -0.28 dBm calculated from the fitting re sult of fitting region (-17 dBm, 1 dBm) is near around the measured IIP3, 0 dBm. The one-tone data of the amplifier C is the third example for the application of fitting algorithm which estimates the third-order intercept point using nonlinear coefficients extracted from the one-tone data of the amplifier. The applied fitting algorithm to this example is the same as that of the previous examples. First, the fitting model represented by equation (6-1) is appl ied to the gain compression curve of the amplifier C. The values of nonlinear coefficients K1 and K3 from fitting results are shown in Figure 6-29 and 6-30. The best fitting range is chosen in the standard errors of Table 6-5 The summary of fitting results according to fitting region (amplifier B) Parameters Region A (-8 dBm) Region B (-2 dBm) Region C (-1 dBm) Coefficient K1 32.85 33.97 33.77 s.e.(K1) 0.07 0.14 0.15 Coefficient K3 -173 -505 -481 s.e.(K3) 22 12 10 Coefficient K5 -13790 3471 3003 s.e.(K5) 1352 198 135 Calculated IIP3 4.05 dBm -0.48 dBm -0.28 dBm

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87 Figure 6-23 The value of coefficient K1 (amplifier B) Figure 6-24 The value of coefficient K3 (amplifier B)

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88 Figure 6-25 The value of coefficient K5 (amplifier B) Figure 6-26 Standard errors of K1 and K3 (amplifier B)

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89 Figure 6-27 Standard error of K5 (amplifier B) Figure 6-28 Sum of squares of the residuals (amplifier B)

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90 Table 6-6 The summary of fitting results of the first model (amplifier C) Parameters Values Coefficient K1 24.50 s.e.(K1) 0.19 Coefficient K3 -196 s.e.(K3) 8 Calculated IIP3 2.2 dBm K1 and K3 in Figure 6-31. Point A indicates the best fitting range from -14 dBm to -3 dBm. To confirm this range in the aspects of the residuals, the su m of squares of the residuals is checked in Figure 6-32. The value of sum of squares of the residuals is small in this fitting range. The fitting results ar e listed in Table 6-6. The calculated IIP3 is 2.2 dBm. Figure 6-33, 6-34 and 6-35 shows the values of K1, K3 and K5 when the second fitting model is applied to the one-tone data of the amplifier C in Figure 6-5. To determine the values of the nonlinear coefficien ts, the best fitting range is chosen in the standard error graphs of Fi gure 6-36 and 6-37. In Figure 6-36 and 6-37, the standard errors of K1, K3 and K5 decrease when the fitting width increases. The best fitting range in this example is from -14 dBm to 0 dBm. Th e sum of squares of the residuals in Figure 6-38 is also small in this fitting range. Ta ble 6-7 summarizes the fitting result in the fitting range from -14 dBm to 0 dBm. The calculated value of IIP3 is 1.16 dBm. The measured IIP3 of the amplifier C is 3 dBm in Table 6-1. The fitting result of the first model is better than that of the second model in this example. The one-tone data of the amplifier D in Fi gure 6-7 is used as the final example of the fitting algorithm. In this example, the two fitting models are applied to this gain curve in Figure 6-7. The input freque ncy of the one-tone data is 5. 2 GHz and is different from

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91 Figure 6-29 The value of coefficient K1 (amplifier C) Figure 6-30 The value of coefficient K3 (amplifier C)

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92 Figure 6-31 Standard errors of K1 and K3 (amplifier C) Figure 6-32 Sum of squares of the residuals (amplifier C)

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93 Figure 6-33 The value of coefficient K1 (amplifier C) Figure 6-34 The value of coefficient K3 (amplifier C)

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94 Figure 6-35 The value of coefficient K5 (amplifier C) Figure 6-36 Standard errors of K1 and K3 (amplifier C)

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95 Figure 6-37 Standard error of K5 (amplifier C) Figure 6-38 Sum of squares of the residuals (amplifier C)

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96 Table 6-7 The summary of fitting results (amplifier C) Parameters Values Coefficient K1 24.94 s.e.(K1) 0.18 Coefficient K3 -255 s.e.(K3) 10 Coefficient K5 1297 s.e.(K5) 99 Calculated IIP3 1.16 dBm the previous three examples which is measur ed at 2.45 GHz. The application algorithm is the same as that applied to the previous examples. Since the applied fitting algorithm to this example is the same as that to the previous, the analysis of the fitting result is also the same. Figure 6-39 shows the value of K1 according to the fitti ng range. The value of K3 can be found simultaneously with the value of K1 during fitting process. Figure 6-40 shows the value of K3. Figure 6-41 shows the standard e rrors of nonlinear coefficients K1 and K3. From this graph, the best fitting range is chosen by point A which indicates small values of the standard errors of K1 and K3. The range from -13 dBm to -6 dBm is chosen for the best fitting range. The sum of squares of the residual in this range is small in Figure 6-42. The fitting re sults are listed in Table 6-8. The applic ation of the second fitting model to the one-tone gain curve of the amplifier D is summarized in Table 6-9. The fitting range is chosen from -13 dBm to -3 dBm. Figure 6-43, 6-44 and 6-45 show the extracted values of K1, K3 and K5 respectively. Figure 6-46 shows the standard errors of K1 and K3. Point A is defined as the best fitting range which is from -13 dBm to -3 dBm. Table 6-10 shows the summary of th e application of two fitting model to commercial PAs. Both fitting models give good results for the estimated IIP3 compared

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97 Figure 6-39 The value of coefficient K1 (amplifier D) Figure 6-40 The value of coefficient K3 (amplifier D)

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98 Figure 6-41 Standard errors of K1 and K3 (amplifier D) Figure 6-42 Sum of squares of the residuals (amplifier D)

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99 Figure 6-43 The value of coefficient K1 (amplifier D) Figure 6-44 The value of coefficient K3 (amplifier D)

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100 Figure 6-45 The value of coefficient K5 (amplifier D) Figure 6-46 Standard errors of K1 and K3 (amplifier D)

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101 Figure 6-47 Standard error of K5 (amplifier D) Figure 6-48 Sum of squares of the residuals (amplifier D)

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102 Table 6-8 The summary of fitting results of the first model (amplifier D) Parameters Values Coefficient K1 17.08 s.e.(K1) 0.06 Coefficient K3 -253 s.e.(K3) 5 Calculated IIP3 -0.47 dBm Table 6-9 The summary of fitting results (amplifier D) Parameters Values Coefficient K1 17.54 s.e.(K1) 0.12 Coefficient K3 -342 s.e.(K3) 12 Coefficient K5 3218 s.e.(K5) 242 Calculated IIP3 -1.65 dBm Table 6-10 The comparison between two fitting models Devices IIP3 Fitting Model 1 Fitting Model 2 Amplifier A -4.4 dBm -3.27 dBm -4.31 dBm Amplifier B 0 dBm 0.43 dBm -0.28 dBm Amplifier C 3 dBm 2.2 dBm 1.16 dBm Amplifier D -1 dBm -0.47 dBm -1.65 dBm to the measured IIP3. The fitting range for the fitting mode l 1, equation (6-1), is less than that for the fitting model 2, equation (6-2). These results are sufficient to choose the fitting model 1 for the IIP3 estimation from the one-tone data of power amplifiers.

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103 6.4 IP1-dB Estimation from the Two-tone Data In addition, the estimation algorithm of the 1 dB gain compression point from twotone data is applied to these power amplifiers This algorithm is explained in section 5.4. The examples are shown also in section 5.4. The amplifier A is considered as the firs t application example of this algorithm. IIP3 of the amplifier A is -4.4 dBm in Table 61. The input amplitude at this point is 190 0 1020 / ) 10 ( 33 IIP IPA (6-3) where the unit of 3 IPAis volt (V). The absolute value of the ratio, 1 3K K can be found in equation (6-3). 723 36 190 0 3 42 1 3 K K (6-4) The sign of above ratio 1 3K K is negative since the gain at the fundamental frequency compresses in Figure 6-2. From Table 6-1, the IP1-dB,2 is -16 dBm. The input amplitude at this 1 dB compression point in the two-tone data is 050 0 1020 / ) 10 ( 2 12 1 dBIP dBA (6-5) Using the value of 2 1 dBA and the ratio 1 3K K, the ratio 1 5K K is found in equation (5-4). The calculated value of 1 5K K is 2499 109 0 4 9 25 42 2 1 1 3 4 2 1 1 5 dB dBA K K A K K (6-6) The 1 dB compression point in one-tone test is now found by solving equation (5-1). The

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104 Table 6-11 The summary of th e application results of the IP1-dB estimation algorithm Device Measured IP1-dB Estimated IIP3 Amplifier A -11 dBm -12.2 dBm Amplifier B -8 dBm -8.46 dBm Amplifier C -6 dBm -5.47 dBm Amplifier D -8 dBm -8.8 dBm solution of this equation is 0.077. The calculated dBIP 1 is -12.2 dBm. The difference between the estimated dBIP 1 and the measured dBIP 1 is 1.2 dB. The same procedure is applied to three other amplifiers. The results of these applications are listed in Table 6-11. 6.5 Summary In this chapter, the fitting algorithm which had been developed in chapter 4 has been verified through commercial RF power am plifiers. The inspection of the standard errors of fitting parameters has given the best fitting range for the extraction of nonlinear coefficients used for the calculation of the third-order intercept point. Two fitting models was applied to the data of four power amplif iers. The standard errors analyzed by linear regression analysis showed the accuracy of fitting algorithm. The chosen fitting range had to be confirmed by the quantity of sum of squares of the residuals. The estimated IIP3 from one-tone measurement data was close to the measured IIP3. The other method to predict IP1-dB from two-tone measurement was applied successfully to these commercial RF power amplifiers.

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105 CHAPTER 7 SUMMARY AND SUGGESTIONS FOR FUTURE WORK 7.1 Summary In this dissertation, nonlinear characteris tics of amplifiers we re discussed, and a fitting algorithm which estimates the third-order intercept point by extracting the nonlinear coefficients from the one-tone meas urement was proposed and verified, and a method to predict the 1 dB gain compression point from the two-tone measurement was proposed and applied to some examples. Comm ercial wideband RF amplifiers and RF power amplifiers are tested and used for the verification of the proposed algorithm. A typical RF/Mixed-signal produc tion test may use this algor ithm or method to avoid the difficulty of the two-tone measurement or to remove the one-tone measurement test step. The major contributions for this thesis work are summarized as follows. In chapter 3, a new relationship between the 1 dB gain compression point and the third-order intercept point has been derived. First, this relationship between the thirdorder intercept point and the 1 dB gain compression point was reviewed in classical analysis. The difference between two nonlinear characteristics was 9.6 dB and constant. The classical analysis included only a th ird-order nonlinear coefficient. The new relationship has derived by expanding nonlinea r analysis on the gain compression curve up to the fifth-order nonlinear coe fficients. The difference between IP1-dB and IIP3 is not fixed and is explained by the equation including nonlinear coefficients. The fitting algorithm to estimate IIP3 from one-tone measurement and the calculation method to

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106 predict IP1-dB from two-tone measurement are devised. The linear regression theory required for the fitting algorithm has been review ed and modified to apply the algorithm. In chapter 4, the proposed fitting algorithm has been verified through the simulation of a common-source amplifier. The best fitting range has been chosen through the standard errors of nonlinear coefficients and the sum of squares of the residuals. The effect of measurement error was explored by adding random noise to the simulation data. Frequency effect on the fitting algorithm was examined up to 30 GHz. The IIP3 estimation error is less than 2 dB up to 30 GHz. Through the Volterra series analysis, the load effect on the algorithm was investigat ed and was not affected by the active load except for the change caused by the operating frequency. In chapter 5, the robust algorithm to pr edict IIP3 has been developed for the wideband RF amplifier. Given the situation, a global extraction of the LNA power series expansion coefficients was not stable with sm all changes in the data set. Adding a few more data points at the high-power range would create large changes in the K3 and K5 extracted coefficients. To counteract this pr oblem, which would be seen in ATE systems doing manufacturing test, a new parameter ex traction methodology had to be created. After experimenting with many ways of performing this para meter extraction, a regional parameter extraction methodology was devised. The IP1-dB prediction from two-tone measurement has been applied to these wide band amplifiers. Through several steps of simple calculation using the third-order inte rcept point and the ga in compression at the fundamental frequency, IP1-dB has been estimated well under 1 dB error. In chapter 6, the fitting algorithm that had been developed in chapter 4 has been verified through commercial RF power amplifie rs. The inspection of the standard errors

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107 of fitting parameters has given the best fitting range for the extraction of nonlinear coefficients used for the calculation of the third-order intercept point. Two fitting models were applied to the data of f our power amplifiers. The standard errors analyzed by linear regression analysis showed the accuracy of th e fitting algorithm. The chosen fitting range had to be confirmed by the quantity of sum of squares of the residuals. The estimated IIP3 from one-tone measurement data was close to the measured IIP3. The other method to predict IP1-dB from two-tone measurement was applied successfully to these commercial RF power amplifiers. In conclusion, a fitting algorithm to estimate IIP3 from the one-tone test and a modified robust algorithm for the wideband RF amplifiers has been developed and verified through simulation and measuremen t of wideband RF amplifiers and the commercial RF power amplifiers. A us eful calculation method to predict IP1-dB from twotone measurement has been developed because of nonlinear analysis and has been established in the application of wideband RF amplifiers a nd RF power amplifiers. These methods developed in this thesis are useful to typical RF/Mixed-s ignal production tests. By developing the relationship between the 1 dB gain compression point and the thirdorder intercept point, a simple embedded test model used avoiding the difficulty and test cost of two-tone measurement. 7.2 Suggestions for Future Work A simple embedded test using a direct conve rsion mixer can be re alized to measure the nonlinear characteristics of an amplifier. In this chapte r, the nonlinear characteristics of a mixer are reported and the mi xer embedded test is suggested.

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108 Figure 7-1 A schematic of a gilbert cell mixer 7.2.1 Nonlinearity of a Mixer A mixer performs a frequency translation by multiplying two signals. A downconversion mixer employed in the receive pa th has two distinctly different inputs, called the RF port and the LO port. The RF port senses the signal to be downconverted and the LO port is the periodic waveform fr om the receiver local oscillator. In the receiver, the signal amplified by the LNA is applied to the RF port of the mixer [Raz98]. If a mixer operates with a differential LO signal and a si ngle-ended RF signal, it is called as a single-balanced mixer. If a mi xer accommodates both differential LO and RF inputs, it is called as a double balanced mixer and has the form of a Gilbert cell. To study of nonlinear characteristics of a mixer, a BJT Gilbert cell mixer is used. The schematic of the Gilbert cell mixer is shown in Figure 71. The one-tone test is performed with different LO powers. The gain of mixers must be carefully defined to avoid confusion. The power conversion gain of the mixer is defined as the ratio of the power of

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109 Figure 7-2 The result of one-tone test with different LO powers Figure 7-3 The analysis of the gain curve at -15 dBm LO power

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110 Figure 7-4 A cascode structure the IF signal to the power of the RF signal. Th e result of the one-tone test can be seen in Figure 7-2. In this test, the IF output power shows a distinct behavior different from that of the amplifier. The gain curve of the mixe r is assumed to be the sum of two different gain curves. Figure 7-3 shows the assumption that the gain curve at -15 dBm of LO power is the combination of the curve A and th e curve B. To verify this assumption, the hand analysis of a cascode stru cture with two input signals is given in Figure 7-4. The result of the hand analysis is ... ...} 20 20 4 10 20 4 20 4 10 4 4 { ...} 20 12 12 6 6 2 5 12 3 6 2 4 6 2 3 2 2 {5 1 3 2 5 4 1 5 5 4 4 1 2 2 5 3 1 5 4 4 3 1 2 1 5 3 4 2 1 2 1 5 2 4 1 1 4 3 2 5 1 2 3 1 5 2 2 1 4 3 2 1 4 3 2 3 4 1 3 5 2 4 1 4 1 5 2 2 1 4 2 2 3 3 1 3 4 2 3 1 3 1 4 2 1 3 3 2 2 1 2 1 3 2 2 1 1 2 2 s s s s s s s s s s s s outV V V V V V V V V V V V i (7-1) Equation (7-1) shows that the IF output is com posed of two gain curves. One of them is linear to LO amplitude and the other is linear to triple times of LO amplitude. From this

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111 Figure 7-5 Simple embedded system with a mixer analysis, it is possible to assume that the cascode structure with two input signals make the two compression curves. In the future, the nonlinear characteristic with multiple inputs and multiple nonlinear systems need to be investigated in the future. 7.2.2 Modeling a Mixer Embedded Test Simple embedded test schemes for device unde r test can be seen in Figure 7-5. Through developing the re lationship between IIP3 and IP1-dB, embedded test for IIP3 requires only the one-tone test. For embedde d test, new algorithm that predicts IIP3 without the two-tone test was developed and verified in amplifiers. Through the one-tone test using this simple embedded scheme, nonlinear characteristics can be measured. The direct downconversion of the mixer in the receiver can be adopted in the suggested simple test scheme. One of the pr oblems of the direct dow nconversion receiver architecture is the envelope distortion due to the even -order nonlinearities [Kiv01]. Several mismatch factors cause even order intermodulation distorti on in a Gilbert cell type mixer. The calibration technique fo r the dc offset caused by even-order nonlinearities is intr oduced for the direct downconve rsion mixer [Hot04]. For the

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112 realization of the mixer embedded test, the e ffect of the mismatches and phase offset need to be investigated in the future

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113 APPENDIX A BSIM3 MODEL OF N-MOS AND P-MOS TRANSISTOR In Chapter 4, the used transistor is TSMC 0.25um MOSFET. The BSIM3 model of n-MOSFET is listed in Table A-1 and the BS IM3 model of p-MOSFET is summarized in Table A-2. Table A-1 BSIM3 model of n-MOSFET LEVEL = 49 VERSION = 3.1 TNOM = 27 TOX = 5.7E-9 +XJ = 1E-7 NCH = 2.3549E17 VTH0 = 0.4122189 +K1 = 0.4555302 K2 = 5.728531E-3 K3 = 1E-3 +K3B = 2.9233592 W0 = 1.816641E-7 NLX = 2.10175E-7 +DVT0W = 0 DVT1W = 0 DVT2W = 0 +DVT0 = 0.4608679 DVT1 = 0.5163149 DVT2 = -0.5 +U0 = 318.8665069 UA = -9.32302E-10 UB = 2.162432E-18 +UC = 3.868128E-11 VSAT = 1.458598E5 A0 = 1.626434 +AGS = 0.3003966 B0 = -4.103132E-7 B1 = 5E-6 +KETA = -4.297464E-4 A1 = 0 A2 = 0.4274964 +RDSW = 116 PRWG = 0.5 PRWB = -0.2 +WR = 1 WINT = 0 LINT = 1.056485E-8 +XL = 3E-8 XW = -4E-8 DWG = -1.833849E-8 +DWB = 2.423085E-9 VOFF = -0.1085255 NFACTOR = 1.6188811 +CIT = 0 CDSC = 2.4E-4 CDSCD = 0 +CDSCB = 0 ETA 0 = 4.244913E-3 ETAB = 5.001973E-4 +DSUB = 0.0551518 PCLM = 1.9563343 PDIBLC1 = 1 +PDIBLC2 = 7.485749E-3 PDIBLCB = -0.0349745 DROUT = 0.8869937 +PSCBE1 = 7.999904E10 PSCBE2 = 5E-10 PVAG = 0 +DELTA = 0.01 RSH = 4.4 MOBMOD = 1 +PRT = 0 UTE = -1.5 KT1 = -0.11 KT1L = 0 +KT2 = 0.022 UA1 = 4.31E-9 UB1 = -7.61E-18 UC 1 = -5.6E-11 +AT = 3.3E4 WL = 0 WLN = 1 WW = 0 WWN = 1 +WWL = 0 LL = 0 LLN = 1 LW = 0 +LWN = 1 LWL = 0 CAPMOD = 2 XPART = 0.5 +CGDO = 6.08E-10 CGSO = 6.08E-10 CGBO = 1E-12 +CJ = 1.758361E-3 PB = 0.99 MJ = 0.4595413 +CJSW = 3.984847E-10 PBSW = 0.99 MJSW = 0.3488353 +CJSWG = 3.29E-10 PBSWG = 0.99 MJSWG = 0.3488353 +CF = 0 PVTH0 = -0.01 PRDSW = -10 +PK2 = 2.330213E-3 WKETA = 7.327459E-3 LKETA =-6.323718E-3

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114 Table A-2 BSIM3 model of p-MOSFET LEVEL = 49 +VERSION = 3.1 TNOM = 27 TOX = 5.6E-9 +XJ = 1E-7 NCH = 4.1589E17 VTH0 = -0.5572867 +K1 = 0.611414 K2 = 4.87747E-3 K3 = 0 +K3B = 10.9112779 W0 = 1E-6 NLX = 1E-9 +DVT0W = 0 DVT1W = 0 DVT2W = 0 +DVT0 = 2.3635216 DVT1 = 0.8955611 DVT2 = -0.2135943 +U0 = 113.0309435 UA = 1.42794E-9 UB = 1E-21 +UC = -9.95235E-11 VSAT = 2E5 A0 = 0.9647086 +AGS = 0.2047206 B0 = 1.359601E-6 B1 = 5E-6 +KETA = 9.05781E-3 A1 = 3.192059E-3 A2 = 0.3 +RDSW = 1.107652E3 PRWG = 0.1255776 PRWB = -0.2785571 +WR = 1 WINT = 0 LINT = 3.560404E-8 +XL = 3E-8 XW = -4E-8 DWG = -3.109864E-8 +DWB = 3.826099E-9 VOFF = -0.1311284 NFACTOR = 1.0570063 +CIT = 0 CDSC = 2.4E-4 CDSCD = 0 +CDSCB = 0 ETA0 = 0.677192 ETAB = -0.4495342 +DSUB = 1.23934 PCLM = 1.2278829 PDIBLC1 = 4.941705E-3 +PDIBLC2 = 1.946865E-5 PDIBLCB = -1E-3 DROUT = 0.0556797 +PSCBE1 = 1.766372E10 PSCBE2 = 1.433404E-9 PVAG = 0 +DELTA = 0.01 RSH = 3.5 MOBMOD = 1 +PRT = 0 UTE = -1.5 KT1 = -0.11 +KT1L = 0 KT2 = 0.022 UA1 = 4.31E-9 +UB1 = -7.61E-18 UC1 = -5.6E-11 AT = 3.3E4 +WL = 0 WLN = 1 WW = 0 +WWN = 1 WWL = 0 LL = 0 +LLN = 1 LW = 0 LWN = 1 +LWL = 0 CAPMOD = 2 XPART = 0.5 +CGDO = 6.73E-10 CGSO = 6.73E-10 CGBO = 1E-12 +CJ = 1.883162E-3 PB = 0.9888002 MJ = 0.46754 +CJSW = 3.120022E-10 PBSW = 0.5590324 MJSW = 0.2692798 +CJSWG = 2.5E-10 PBSW G = 0.5590324 MJSWG = 0.2692798 +CF = 0 PVTH0 = 7.130423E-3 PRDSW = 14.8296171 +PK2 = 3.224494E-3 WKETA = 0.0282003 LKETA = -7.320113E-3

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115 APPENDIX B VOLTERRA-KERNELS OF A CO MMON-SOURCE AMPLIFIER Volterra-kernels are calculat ed in this section for a common-source amplifier. The equivalent circuit for this cal culation is shown in Figure B1. In this calculation, all parasitic capacitors are assumed as linear capacitors for simplicity. Here, the nonlinear source is transconductance only. Cgd is only c onsidered at high frequency. Nonlinear transconductance is repr esented by the bellows, ... v v v v v5 gs 5 4 gs 4 3 gs 3 2 gs 2 gs 1 K K K K K i (B-1) To calculate first-order Volterra-kernel, Kirchoff’s current law is applied at both node 1 and node 2 in Figure B-1. in 1V V (B-2) 0 V ) ( V K2 1 1 L LsC G (B-3) Above two equations can be comb ined in one matrix equation: 0 V V V K 0 1in 2 1 1 L LsC G (B-4) It is clear that V1 and V2 reduce to the transfer functi ons of the voltages at node 1 and node 2. These transfer functions are denoted by H1,1(s) and H1,2(s). The first subscript represents the order of transf er function and the second subs cript indicates the number of nodes. The matrix equation can be formed in terms of these transfer functions by replacing voltages to these transfer functions or Volterra-kernels.

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116 Figure B-1. An equivalent circ uit of a common-source amplifier 0 1 ) ( H ) ( H K 0 11,2 1,1 1s s sC GL L (B-5) The solution of above matrix equation is as follows; 1 ) ( H1,1s (B-6) L LsC G K s 1 1,2) ( H (B-7) After finding the first kern el, the matrix for the second kernel can be made. 2 2 1,1 1 1,1 2 2 1 2,2 2 1 2,1 1K 0 ) ( )H ( H K 0 ) ( H ) ( H K 0 1 s s s s s s sC GL L (B-8) The second Volterra-kernel can be f ound by solving above matrix equation. 0 ) ( H2 1 2,1s s (B-9) L LC s s G K s s) ( ) ( H2 1 2 2 1 2,2 (B-10)

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117 Higher Volterra-kernels can be found by re peating the same process done before. Volterra-kernels have been calculated up to 5th order. 0 ) , ( H3 2 1 3,1s s s (B-11) L LC s s s G K s s s) ( ) , ( H3 2 1 3 3 2 1 3,2 (B-12) 0 ) , ( H4 3 2 1 4,1s s s s (B-13) L LC s s s s G K s s s s) ( ) , ( H4 3 2 1 4 4 3 2 1 4,2 (B-14) 0 ) , , ( H5 4 3 2 1 5,1s s s s s (B-15) L LC s s s s s G K s s s s s) ( ) , , ( H5 4 3 2 1 5 5 4 3 2 1 5,2 (B-16) At high frequency, Cgd should be consid ered in the Volterra-kernel. Like low frequency analysis, the matrix for the firs t Volterra-kernel can be formed by similar process including Cgd. 0 1 ) ( H ) ( H ) ( K 0 11,2 1,1 1s s C C s G sCgd L L gd (B-17) The first Volterra-kernel at high frequency is not the same as that at low frequency. 1 ) ( H1,1s (B-18) ) ( ) ( ) ( H1 1,2 gd L L gdC C s G sC K s (B-19) From second-order kernel, the Volterra-kernels are similar to those at low frequency only if CL is replaced by (CL+Cgd).

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118 APPENDIX C MATLAB PROGRAM FOR FITTING ALGORITHM The MATLAB program for the fitting algorithm is listed as bellows, % Extracting K1,K3,K5 coefficients % device : Power amplifier % frequency = 2.45GHz, one-tone data close all clear all load PA.txt -ASCII; Pin=PA(:,1)'; Pin=Pin(:); Pout=PA(:,2)'; Pout=Pout(:); Vin=10.^((Pin-10)./20); % Data converting : power(dBm) to voltage(V) Vout=10.^((Pout-10)./20); % Extracting b1, b3, b5 ( y=b1*x+b3x^3+b5x^5+error ) Fit_start=5; % Define the first fitting range Fit_end=length(Vin); for Fit_n = Fit_start:Fit_end % Sweeping the range for fitting clear xx % local parameter xx and yy for for-loop clear yy xx=Vin(1:Fit_n); % Define fitting data (voltage)

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119 yy=Vout(1:Fit_n); n=length(xx); Pinx(Fit_n-Fit_start+1)=Pin(Fit_n); % Define fitting data in dBm unit % Polynomial regression model : y=b0+b1*x+error % Vector form : estimate of Vector b clear X % Construct X X(:,1) = ones(n,1); X(:,1) = xx.*X(:,1); X(:,2) = xx.^2.*X(:,1); X(:,3) = xx.^2.*X(:,2); C=inv(X'*X); bb=C*(X'*yy); % estimation of v ector bb = [(X' X)^-1]*(X'Y) b1=bb(1); b3=bb(2); b5=bb(3); Coeff_b1(Fit_n-Fit_start+1)=b1; Coeff_b3(Fit_n-Fit_start+1)=b3; Coeff_b5(Fit_n-Fit_start+1)=b5; % Analysis of Varianc % 1. degrees of freedom % 2. Sum of Squares % 3. Mean Squares= SS/d.f. % 4. F test

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120 % 5. Standard Error and t-distribution and confidence interval Total_df(Fit_n-Fit_start+1)=n; % Degrees of freedom Tcorrected_df(Fit_n-Fit_star t+1)=Total_df(Fit_nFit_start+1)-1; Regression_df(Fit_nFit_start+1)=length(bb)-1; Residual_df(Fit_n-Fit_start+ 1)=Tcorrected_df(Fit_n-Fit_start+1)Regression_df(Fit_n-Fit_start+1); Mx=mean(xx); % Mean Value My=mean(yy); Regression_SS(Fit_n-Fit_start+1)= bb'*X'*yy-n*My^2; % Sum of squares T_corrected_SS(Fit_n-Fit_start+1)=yy'*yy-n*My^2; Residual_SS(Fit_n-Fit_start+1) =T_corrected_SS(F it_n-Fit_start+1)Regression_SS(Fit_n-Fit_start+1); MSR(Fit_n-Fit_start+1)=Regression_ SS(Fit_n-Fit_start+1)/Regression_df(Fit_nFit_start+1); % Mean Squares MSE(Fit_n-Fit_start+1)=Residual_ SS(Fit_n-Fit_start+1)/Residual_df(Fit_nFit_start+1); F0(Fit_n-Fit_start+1)=MSR(Fit_n-Fit_st art+1)/MSE(Fit_n-Fit_start+1); % F-test F_1percent(Fit_n-Fit_star t+1)=finv(0.99,Regression_df(Fit_nFit_start+1),Residual_df(Fit_n-Fit_start+1)); disp('_________________________________________________________________'); dispftest={'F-test ','calcu lated F0=',num2str(F0),' <=> ','F(0.01,n1,n2)=',num2str(F_1percent)};

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121 disp(dispftest); disp(' ( > ) means H0:b1=b3=b5=0 is rejected '); %The coefficient of multiple determination RR(Fit_n-Fit_start+1)=Regression_SS( Fit_n-Fit_start+1)/T_corrected_SS(Fit_nFit_start+1); disp('______________________________________________________________ __'); dispRR=['The coefficient of multiple determination R^2:',num2str(RR) ]; disp(dispRR); Cov_b=MSE(Fit_n-Fit_start+1).*C; % Covariance matrix of Vector b : Cov(Vector b) =sig^2(X'X)^-1, Matrix C=(X'X)^-1 disp('_____________________________________________________________ ___'); disp('Covariance of estimation beta'); disp(Cov_b); se_b1(Fit_n-Fit_start+1)=sqrt(Cov_b(1, 1)); % Standard error of beta se(bi)=sqrt(Cjj*sig^2) se_b3(Fit_n-Fit_start+1)=sqrt(Cov_b(2, 2)); se_b5(Fit_n-Fit_st art+1)=sqrt(Cov_b(3,3)); tvalue=tinv(0.975,Residual_df); % Confidence interval using t distribution low_b1=b1-tvalue*se_b1; high_b1=b1+tvalue*se_b1;

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122 disp('_______________________'); disp('95% Confidence interval'); disp_b1=[num2str(low_b1),'<< b1 <<',num2str(high_b1)]; disp(disp_b1); % Nonlinear Coefficients K5=(8/5).*Coeff_b5; K3=(4/3).*Coeff_b3; K1=Coeff_b1; SEK5=(8/5).*se_b5; SEK3=(4/3).*se_b3; SEK1=se_b1; % Third-order Intercept Point AIP3=sqrt(abs(4.*K1./(3.*K3))); IIP3=20.*log10(AIP3)+10; Calculated_IIP3=IIP3(Index_reg) WriteMatrix(:,1)=Pinx'; WriteMatrix(:,2)=K1'; WriteMatrix(:,3)=SEK1'; WriteMatrix(:,4)=K3'; WriteMatrix(:,5)=SEK3'; WriteMatrix(:,6)=K5'; WriteMatrix(:,7)=SEK5'; WriteMatrix(:,8)=Residual_SS';

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123 WriteMatrix(:,9)=IIP3'; dlmwrite('p7analysis.txt',WriteMatrix,'\t');

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124 LIST OF REFERENCES [Dra98] N. R. Drapper and H. Smith, Applied Regression Analysis, 3rd ed., New York :John Wiley & Sons, 1998. [Eis01] W. R. Eisenstadt and S. Choi “Programmable Embedded IF Source For Wireless Test,” 1st Workshop on Test of Wireless Circuits and Systems, Atlantic, NJ, November 1-2, 2001. [Eis02] W. R. Eisenstadt, C. Cho, R. Ste ngel, and E. Ferrer, “Third-order intercept Point(IIP3) from Gain Compressi on Curve,” 2nd Workshop on Test of Wireless Circuits and Systems, Baltimore, MD, October 10-11,2002. [Fag02] C. Fager, J.C. Pedro, N. B. Carv alho, and H. Zirath, “Prediction of IMD in LDMOS Transistor Amplifiers usin g a New Large-Signal Model,” IEEE Trans. On Microwave Theory and Techniques, vol. 50, pp. 2834-2842,2002. [Fon98] K. L. Fong and R. G. Meyer “Hi gh-Frequency Nonlinearity Analysis of Common-Emitter and Differential-Pair Transconductance Stages”, IEEE Journal on Solid-State Circuits, vol. 33, No. 4, pp. 548-555, April 1998 [Gon97] G. Gonzalez, Microwave Transistor Amplifers, 2nd ed., Upper Saddle River : Prentice Hall, 1997. [Gra93] P. R. Gray and R. G. Meyer, Analysis and Design of Analog Integrated Circuits, 3rd ed., New York :John Wiley & Sons, 1993. [Hot04] M. Hotti, J. Ryynanen, K. Kiveka s, and K. Halonen, “An IIP2 Calibration technique for direct conversion,” ISCAS 2004, pp. IV-257-IV-260, 2004. [Kan03] S. Kang, B. Choi, and B. Kim, “Linearity Analysis of CMOS for RF Application,” IEEE Trans actions on Microwave Theo ry and Techniques, vol. 51, No.3, pp. 972-977, March 2003. [Kiv01] K. Kivekas, A. Par ssinen, and K. Halonen, “Cha racterization of IIP2 amd DCOFFSETS in Transconductance Mixers,” IEEE Transactions on Circuits and systems-II: Analog and Digital Sign al Processing, vol. 48, pp. 1028-1038, Nov. 2002. [Maa88] S. A. Maas, Nonlinear Microwave Circuits, Norwood :Artech House, 1988.

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125 [Mon01] D.C. Montgomery, E. A. Peck, and G. G. Vining, Introduction to Linear Regression Theory, 3rd ed., New York :John Wiley & Sons, 2001. [Mcc99] J. H. McClellan, R. W. Schafer and M. A. Yoder, DSP First, Upper Saddle River :Prentice Hall, 1999. [Ped91] D. O. Pederson and K. Mayaram, Analog Integrated Circuits for Communication: principles, simulation and design, Boston :Kluwer Academic Publishers, 1991. [Raz98] B. Razavi, RF Microelectronics, Upper Saddle River :Prentice-Hall, 1998 [Ros98] D. A. Ross, Master Math :Calculus, Franklin Lakes: CAREER PRESS, 1998. [San73] W. M. C. Sansen and R. G. Meyer, “Distortion in Bipolar Transistor VariableGain Amplifiers,” IEEE Journal on Solid -State Circuits, vol. 8, pp. 275-282, August 1973. [San99] W. Sansen, “Distortion in Elementary Transistor circuits ,” IEEE Transactions on Circuits and systems, vol. 46, pp. 315-325, March 1999. [Vuo03] J. Vuolevi and T. Rahkonen, Distortion in RF Amplifers, Norwood :Artech House, 2003. [Wam98] P. Wambacq and W. Sansen, Distortion Analysis of Analog Integrated Circuits, Boston :Kluwer Academ ic Publishers, 1998. [Wei80] D. D. Weiner and J. E. Spina, Sinusoidal Analysis a nd Modeling of Weakly Nonlinear Circuits, Distortion Analysis of Analog Integrated Circuits, New York :Van Nostrand Reinhold Company, 1980.

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126 BIOGRAPHICAL SKETCH Choongeol Cho was born in Korea, in 1969. He received a BS degree and a MS degree in physics from Yonsei University, South Korea, in 1991 and 1993, respectively. During his MS program, he had majored in solid-state physics for II-VI compound heterojunction structures. He worked as a process engineer for si x years at Samsung Semiconductor research lab. During this time, he joined a proce ss team that developed 256M DRAM and 1G DRAM. Since June 2001, he has been work ing on his Ph.D. degree in Radio Frequency Test Research Group at the University of Fl orida as a graduate research assistant. His research interests are nonlinear anal ysis of analog/RF mixed circuits, RF receiver design, and embedded test. He is part icularly interested in the improvement of linearity of RF components.


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Title: RF Circuit Nonlinearity Characterization and Modeling for Embedded Test
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Copyright Date: 2008

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RF CIRCUIT NONLINEARITY CHARACTERIZATION AND
MODELING FOR EMBEDDED TEST















By

CHOONGEOL CHO


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

CHOONGEOL CHO

































This document is dedicated to the graduate students of the University of Florida.
















ACKNOWLEDGMENTS

I would like to express my sincere appreciation to my advisor, Professor William

R. Eisenstadt, whose encouragement, guidance, and support throughout my work have

been invaluable, I also would like to thank Professors Robert M. Fox, John G. Harris, and

Oscar D. Crisalle for their interest in this work and their guidance as the thesis committee

members.

I thank Motorola Company for financial support. I also thank Bob Stengel and

Enrique Ferrer for their dedication to my research. In addition, I thank all of the friends

who made my years at the University of Florida such an enjoyable chapter of my life.

I am grateful to my parents and parents in law for their unceasing love and

dedication. Finally, I thank my wife, Seon-Kyung Kim, whose endless love and

encouragement were most valuable to me. Most importantly, I would like to thank God

for guiding me everyday.
















TABLE OF CONTENTS



A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES .............................................. .. ............... .............. vii

LIST OF FIGURES ......... ............................... ........ ............ ix

ABSTRACT .............. .......................................... xiv

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

1.1 M otiv action ............................................. 1
1.2 R research G oals ............................................................... 2
1.3 Overview of Dissertation..................... ........ ............................. 3

2 B A CK G R O U N D .................................... ...................... ........ .......... .... ....

2.1 Classifications of Distortions...................... ..... ............. ............... 5
2 .2 T aylor's Series E expansion ......................................................................... ...... 7
2.3 M easurement of Nonlinear System ................................................... ............... 8

3 THE RELATIONSHIP BETWEEN THE 1 dB GAIN COMPRESSION POINT
AND THE THIRD-ORDER INTERCEPT POINT ................................................10

3.1 Definition of 1 dB Gain Compression and Third-order Intercept Point ..............10
3.2 Classical Approach to M odel IIP3 ................................ .. ........................ 12
3.3 New Approach to Model Gain Compression Curve...........................................15
3.4 Fitting Polynomials Data by Using Linear Regression Theory............................20
3 .5 Su m m ary ................................23...........................

4 SIM U L A TIO N ............................................... .. .. .. .. ...... ........... 24

4.1 A MOSFET Common-Source Amplifier............................. ...............24
4.2 Measurement Error Consideration....................................................... 35
4.3 Frequency Effect on the Fitting Algorithm .................................. ............... 38
4.4 Load Effect on the Fitting Algorithm ....................................... ............... 41
4 .5 S u m m a ry ................................................... ................ ................ 5 1









5 COMMERCIAL RF WIDEBAND AMPLIFIER ............................................... 52

5.1 N onlinearity T est ...... .. .. ... .................................... .. ................ .. 52
5.2 IIP3 Prediction from the Gain Compression Curve ...........................................58
5.3 The Application of the Proposed Algorithm at High Frequency..........................62
5.4 IP1-dB Estimation from Two-tone Data ...................................... ............... 63
5 .5 S u m m a ry ...................................................................................................6 8

6 PO W ER A M PLIFIER S ........................................... .................. ............... 69

6.1 Linear and Nonlinear Power Amplifiers.................................... ............... 69
6.2 Measurement of Commercial Power Amplifiers...............................................70
6.3 IIP3 Estim ation from the One-tone Data................................... .................75
6.4 IP1-dB Estimation from the Two-tone Data ............................... ................103
6 .5 S u m m ary ............................... ......... ...... ................ ................ 10 4

7 SUMMARY AND SUGGESTIONS FOR FUTURE WORK............................105

7 .1 Su m m ary .................. ...................................... .......... ................ 10 5
7.2 Suggestions for Future W ork......................................... ......................... 107
7.2.1 N onlinearity of a M ixer................................ ......................... ........ 108
7.2.2 Modeling a Mixer Embedded Test.....................................................111

APPENDIX

A BSIM3 MODEL OF N-MOS AND P-MOS TRANSISTOR................................113

B VOLTERRA-KERNELS OF A COMMON-SOURCE AMPLIFIER.....................115

C MATLAB PROGRAM FOR FITTING ALGORITHM ..........................................118

L IST O F R E F E R E N C E S ...................................................................... ..................... 124

BIOGRAPHICAL SKETCH ............................................................. ............... 126
















LIST OF TABLES


Table page

1-1 The 1 dB gain compression point and IIP3 of various circuits..............................3.

4-1 The sum m ary of fitting results ....... .......................................... ....................34

4-2 Frequency effect on the fitting algorithm ............................................................. 39

4-3 F hitting R results ........................................................................50

5-1 The summary of the measurement results of commercial amplifiers ....................58

5-2 The summary of the estimated IIP3 of commercial amplifiers.............................61

5-3 The measurement data and calculated IIP3 of a commercial amplifier....................63

5-4 The summary of the application results of the IP1-dB estimation algorithm.............67

6-1 The summary of the measurement results of commercial PAs.............................75

6-2 The summary of fitting results (amplifier A) ............................... ...... ...............79

6-3 The summary of fitting results (amplifier A) ............................... ...... ...............83

6-4 The summary of fitting results of the first model (amplifier B) ..............................83

6-5 The summary of fitting results according to fitting region (amplifier B) ................86

6-6 The summary of fitting results of the first model (amplifier C).............................90

6-7 The summary of fitting results (amplifier C) ............. ........................................96

6-8 The summary of fitting results of the first model (amplifier D) ............................102

6-9 The summary of fitting results (Amplifier D)...................................................... 102

6-10 The comparison between two fitting models ............................... ................102

6-11 The summary of the application results of the IP1-dB estimation algorithm...........104

A-1 BSIM 3 model of n-M OSFET...................................................... ..................113









A-2 BSIM3 model of p-MOSFET............ ... .... ......... ............... 114
















LIST OF FIGURES


Figure p

2-1 The output current of an ideal class C amplifier for a sine-wave input .................6

2-2 The configuration of a single-tone test........ .......................................... 9

2-3 The configuration of a two-tone test. .............................................. ............... 9

3-1 Definition of 1-dB gain compression point.....................................10

3-2 Intermodulation in a nonlinear system ...... ................. ...............11

3-3 Definition of third-order intercept point..... ................................12

3-4 The definition of Intercept points in one-tone test.............................. ............... 16

4-1 A schematic of a common-source amplifier .................................... ............... 25

4-2 DC characteristic of a common-source amplifier. ................................................26

4-3 The AC simulation of a common-source amplifier.......................................27

4-4 The results of one-tone sim ulation ...................................................... ............... 28

4-5 The results of a two-tone simulation......................... ............................ ............ ..29

4-6 The difference between amplitudes at two frequencies .......................................29

4-7 Extracted nonlinear coefficients ......................... .................................... 31

4-8 Standard errors of nonlinear coefficients ..................................... .................32

4-9 The sum of squares of the residual................................ ..................... ............... 33

4-10 Gain curves with 0.1% and 2% random noise ............................... ............... .35

4-11 Calculated third-order intercept point with 0.1 percent added random noise ..........36

4-12 Calculated third-order intercept point with 2.0 percent added random noise ..........37

4-13 The influence of added random error on the fitting results................................ 37









4-14 A n equivalent circuit ...................................... ............... .... ....... 41

4-15 A schematic of a common-source amplifier with an active load ...........................43

4-16 DC characteristics of a common-source amplifier with an active load....................44

4-17 The AC simulation of a common-source amplifier with an active load ..................45

4-18 The results of one-tone simulation ..................................................................... 46

4-19 The results of two-tone simulation..................................................................... 46

4-20 Extracted nonlinear coefficient K ........................................ ....................... 47

4-21 Extracted nonlinear coefficient K 3 ........................................ ....................... 48

4-22 Extracted nonlinear coefficient K 5 ........................................ ....................... 49

4-23 Standard errors of nonlinear coefficients ..................................... .................49

5-1 ER A 1 am plifier in a test board........................................... .......................... 52

5-2 The spectrum from the signal source ............................................ ............... 53

5-3 O ne-tone test scheme e ....................................................................... ..................54

5-4 The measurement data of one-tone test (ERA1 amplifier) ....................................54

5-5 T w o-ton e test sch em e .................................................................... .....................55

5-6 The measurement data of two-tone test (ERA1 amplifier) .............................55

5-7 The measurement data of one-tone test (ERA2 amplifier) ....................................56

5-8 The measurement data of two-tone test (ERA2 amplifier) .............. ...............56

5-9 The measurement data of one-tone test (ERA3 amplifier) ....................................57

5-10 The measurement data of two-tone test (ERA3 amplifier) ....................................57

5-11 One-tone data and extraction from ERA1 device at 100 MHz .............................59

5-12 A flow chart for estimation of IIP3 from one-tone measurement..........................60

5-13 One-tone data and extraction from ERA2 device at 100 MHz .............................61

5-14 One-tone data and extraction from ERA3 device at 100 MHz .............................62

5-15 One-tone data and extraction from ERA2 device at 2.4 GHz..............................63









6-1 The measurement data of one-tone test of the amplifier A.................. .............71

6-2 The measurement data of two-tone test of the amplifier A.................. ............71

6-3 The measurement data of one-tone test of the amplifier B ................. ..............72

6-4 The measurement data of two-tone test of the amplifier B ................. ..............72

6-5 The measurement data of one-tone test of the amplifier C ................. ..............73

6-6 The measurement data of two-tone test of the amplifier C................. ..............73

6-7 The measurement data of one-tone test of the amplifier D................. ...............74

6-8 The measurement data of two-tone test of the amplifier D................. .......... 74

6-9 The value of coefficient K 1 (amplifier A) ..................................... .................77

6-10 The value of coefficient K3 (amplifier A) ..................................... .................77

6-11 Standard errors of K1 and K3 (amplifier A) .......................................................78

6-12 Sum of squares of the residuals (amplifier A) ......................................................78

6-13 The value of coefficient K 1 (amplifier A) ..................................... .................80

6-14 The value of coefficient K 3 (am plifier A ) ........................................ .....................80

6-15 The value of coefficient K5 (amplifier A) ..................................... .................81

6-16 Standard errors ofK 1 and K3 (amplifier A)................................... .................81

6-17 Standard Error of K5 (Amplifier A)................................ .............................. 82

6-18 Sum of squares of the residuals (Amplifier A) ...................................................82

6-19 The value of coefficient K 1 (amplifier B) ..................................... .................84

6-20 The value of coefficient K3 (amplifier B) ..................................... .................84

6-21 Standard errors of K1 and K3 (amplifier B).........................................................85

6-22 Sum of squares of the residuals (amplifier B) ...................................................... 85

6-23 The value of coefficient K1 (Amplifier B) .................................... ............... 87

6-24 The value of Coefficient K3 (Amplifier B) ................................... .................87

6-25 The value of Coefficient K5 (Amplifier B) ................................... .................88









6-26 Standard errors of K1 and K3 (amplifier B)..........................................................88

6-27 Standard error of K (am plifier B)...................................... ......................... 89

6-28 Sum of squares of the residuals (amplifier B) ...................................................... 89

6-29 The value of coefficient K1 (amplifier C) ..................................... .................91

6-30 The value of coefficient K3 (amplifier C) ..................................... .................91

6-31 Standard errors of K1 and K3 (amplifier C)..........................................................92

6-32 Sum of squares of the residuals (amplifier C) ......................................................92

6-33 The value of coefficient K1 (amplifier C) ..................................... .................93

6-34 The value of coefficient K3 (amplifier C) ..................................... .................93

6-35 The value of coefficient K5 (amplifier C) ..................................... .................94

6-36 Standard errors of K1 and K3 (amplifier C).........................................................94

6-37 Standard error of K5 (am plifier C) ................................... ............. ........ ....... 95

6-38 Sum of squares of the residuals (amplifier C) ......................................................95

6-39 The value of coefficient K1 (amplifier D) ..................................... .................97

6-40 The value of coefficient K3 (amplifier D) ..................................... .................97

6-41 Standard errors of K1 and K3 (amplifier D).........................................................98

6-42 Sum of squares of the residuals (amplifier D) ......................................................98

6-43 The value of coefficient K1 (amplifier D) ..................................... .................99

6-44 The value of coefficient K3 (amplifier D) ..................................... .................99

6-45 The value of coefficient K5 (amplifier D) .................................... ............... 100

6-46 Standard errors of K1 and K3 (amplifier D)........................................................ 100

6-47 Standard error of K5 (amplifier D) ................... ...... .............. 101

6-48 Sum of squares of the residuals (amplifier D)....................................................... 101

7-1 A schematic of a gilbert cell mixer ............................................................ 108

7-2 The result of one-tone test with different LO powers ................. ................109









7-3 The analysis of the gain curve at -15 dBm LO power ............... .............. .....109

7-4 A cascode structure .................. ...................................... .. ........ .. 110

7-5 Simple embedded system with a mixer..... .......... .................... ..................111

B-l An equivalent circuit of a common-source amplifier .......................................... 116















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

RF CIRCUIT NONLINEARITY CHARACTERIZATION AND
MODELING FOR EMBEDDED TEST

By

Choongeol Cho

December 2005

Chair: William R. Eisenstadt
Major Department: Electrical and Computer Engineering

This dissertation presents a fitting algorithm useful for characterizing nonlinearities

of RF circuits, and is specifically designed to estimate the third-order intercept point (

IP3) by extracting the nonlinear coefficients from the one-tone measurement. And the

dissertation proposes a method to predict the 1 dB gain compression point (P1-dB) from a

two-tone measurement. The fitting algorithm is valuable for reducing production IC test

time ans cost. A new relationship between the 1 dB gain compression point and the third-

order intercept point has been derived. It follows that the difference between IP1-dB and

IIP3 is not fixed, and the discrepancy is explained by the new proposed equation which

includes the relevant nonlinear coefficients. The new fitting algorithm has been verified

through application to the simulation of a common-source amplifier. The best fitting

range has been identified through the minimization of the standard errors of the nonlinear

coefficients and of the sum of squares of the residuals. A robust algorithm to predict IIP3

has been developed for wideband RF amplifiers, an application that is of particular









interest. The proposed fitting algorithm was successfully verified in experiments done on

commercial RF power amplifiers. The estimated IIP3 values obtained from one-tone

measurement data was close to the experimently measured values. The method proposed

to predict IP1-dB from a two-tone measurement was also applied successfully to the same

commercial RF power amplifiers.

A simple embedded test using a direct conversion mixer can be realized to estimate

the nonlinear characteristics of an amplifier based upon the estimate IIP3 from the one-

tone data. In this thesis, the nonlinear characteristics of a mixer is researched and a mixer

embedded test technique is suggested. The effects of the mismatches and phase offset

will be researched for mixer test in the future.

The methods developed in this thesis are useful tools in the context for typical

RF/Mixed-signal production test. The advantage is that these methods avoid the difficulty

of two-tone measurements or remove one-tone measurement test step. By developing the

relationship between the 1 dB gain compression point and the third-order intercept point,

a simpler embedded test model can be adopted avoiding the cost and time of a two-tone

measurement.














CHAPTER 1
INTRODUCTION

1.1 Motivation

In the near future, RF microwave circuits will be embedded in highly integrated

"Systems-on-a-Chip" (SoCs). These RF SoCs will need to be debugged in the design

phase and will require expensive automated test equipment (ATE) with microwave test

capability when tested in production. RF/mixed-signal portions of a SoC must be verified

with high-frequency parametric tests. Currently, the ATE performs a production test on

package parts with the assistance of an expensive and elaborate device interface board

(DIB) or load board. Alternative methods of on-chip RF test should be explored to lower

test cost [EisOl].

Another merit of the embedded test is to minimize test time. Current measurement

is performed in the last stage of production. A parameter test of an RF circuit can be

executed using the embedded circuit before packaging and even sorting.

The 1 dB gain compression point and the third-order intercept point are important

nonlinear parameters of the RF/mixed-signal circuit and provide good verification of a

circuit or device's linearity and dynamic range. The parameters can be connected to

adjacent channel power ratio and error vector magnitude (EVM) in amplifiers and must

be kept under control. Gain compression is a relatively simple microwave measurement

since it requires a variable power single tone source and an output power detector. IIP3

characterization is more complicated and more costly since two separate tones closely

spaced in frequency must be generated and applied to the circuit under test (CUT) and









the CUT's fundamental and third order distortion term power must be measured. Thus,

measurement of IP3 requires high Q filters to select first and third order distortion

frequencies in the detector circuit [Eis02].

By developing an accurate relationship between gain compression and IIP3, the

production testing of the manufactured IC can be greatly simplified. Although the

accuracy of this approach may not be as great as direct IIP3 measurement, it has great

appeal in test cost reduction and may be sufficient for production IC test.

1.2 Research Goals

The first goal in this research is to derive the relationship between 1 dB gain

compression point and third-order intercept point. The published difference between 1 dB

gain compression point and IIP3 is roughly 10 dB; this relationship is derived using first-

order and third-order nonlinear coefficients of transistor amplifier circuits. This

calculation assumes that higher-order nonlinear coefficients do not affect the 1 dB gain

compression. The simulation using 0.25 [tm MOSFETs, 0.4 [tm MOSFETs and Si-Ge

BJTs models in Table 1-1 shows that this classical relationship does not work in simple

transistor circuits. The difference between 1 dB gain compression and IIP3 shows a

variation of 8 dB to 13.7 dB. All the circuits (common-source amplifier, differential

amplifier with resistive load, and common-emitter amplifier except for a differential

amplifier with an active load) show that the difference between IIP3 and IP1-dB is no

longer constant. To simplify the embedded test, a more reliable relationship between

these two parameters is required.

The second goal is to verify the relationship between two parameters that is derived

in the first goal. This dissertation examines the general types of amplifiers,









Table 1-1 The 1
Model
TSMC 0.25[ m
MOSFET






Si-Ge IBM6HP
MOSFET


dB gain compression point and IIP3 of various circuits
Ci t IPi-dB* IIP3**
Circui(dBm) (dBm)
Common-Source Amplifier -2.2 10.0
Differential Amplifier 3.0 16.0
3.0 16.0
(Resistive Load)

Differential Amplifier13.0 -5.0
(Active Load)

Common-Source Amplifier -1.75 10.25

Differential Amplifier 2.65 15.25
(Resistive Load)15.25
(Resistive Load)


Si-Ge IBM6HP
Common-Source Amplifier -20.25 -6.5 13.75
BJT
*One-tone test: Source frequency = 100 MHz
**Two-tone test : frequencies = 100 MHz, 120 MHz

commercial wideband RF amplifiers and commercial RF power amplifiers. In this

dissertation, the fitting approach is developed to estimate IIP3 from a gain compression

curve using higher-order nonlinear coefficients.

Finally, a simple embedded test using a mixer is considered and suggested to

measure IIP3 and IP1-dB. Through developing the relationship between IIP3 and IP1-dB, the

embedded test for IIP3 requires only a one-tone source.

1.3 Overview of Dissertation

This Ph.D. dissertation consists of seven chapters. An overview of the research is

given in this current chapter (Chapter 1), including the motivation, research goals, and the

scope of this work. Chapter 2 reviews some background knowledge on this research.

Basic concepts of both nonlinear systems and nonlinear analysis are described.

In chapter 3, a new relationship between the 1 dB gain compression point and the

third-order intercept point is derived. First, this relationship between IIP3 and IP1-dB is


IIP3 -IP1-dB
(dB)
12.2
13.0


8.0


12.0


12.6









reviewed in classical prior analysis. The new relationship is derived by nonlinear analysis

on the gain compression curve. The fitting algorithm to estimate IIP3 from a one-tone

measurement and the calculation method to predict IP1-dB from a two-tone measurement

are developed. The linear regression theory required for the fitting algorithm is reviewed

and modified for the application of the devised algorithm.

In chapter 4, the proposed fitting algorithm is verified through the application of

the algorithm to the simulation of a common-source amplifier. The best fitting range is

chosen through the standard errors of nonlinear coefficients and the sum of squares of the

residuals. The effects of measurement errors at high frequency are researched. Through

the Voletrra series analysis, the load effect on the algorithm is studied.

In chapter 5, a robust algorithm to predict IIP3 is developed for wideband RF

amplifiers. The IP1-dB prediction from two-tone measurement has been applied to these

wideband amplifiers. Through several steps of simple calculation using the third-order

intercept point and the gain compression at the fundamental frequency, IP1-dB is estimated

under 1 dB error.

In chapter 6, the fitting algorithm is applied to commercial RF power amplifiers.

Through the inspection of the standard errors of fitting parameters, the best fitting range

is chosen for the extraction of nonlinear coefficients used for the calculation of the third-

order intercept point. The chosen fitting range is confirmed by the quantity, sum of

squares of the residuals. Another method to predict IP1-dB from a two-tone measurement

is applied to the commercial RF power amplifiers.

Lastly, in chapter 7, the primary contributions of this dissertation are summarized

and future work is suggested.














CHAPTER 2
BACKGROUND

2.1 Classifications of Distortions

All physical components and devices are intrinsically nonlinear. Nevertheless, the

most circuit and system theory deal almost exclusively with linear analysis. The reason is

because linear systems are characterized in terms of linear algebraic, differential, integral,

and difference equations that are relatively easy to solve, most nonlinear systems can be

adequately approximated by equivalent linear systems for suitably small inputs, and

closed-form analytical solutions of nonlinear equations are not normally possible.

However, linear models are incapable of explaining important nonlinear phenomena

[Wei80]. This section reviews types of distortions and nonlinearities for understanding

nonlinear characteristics of a system.

Distortion actually refers to the distortion of a voltage or current waveform as it is

displayed versus time [San99]. Any difference between the shape of the output waveform

and that of the input waveform is called distortion except for scaling a waveform in

amplitude. In a circuit, the type of distortion is classified as one of two classes. First,

linear distortion is caused by the application of a linear circuit with frequency-varying

amplitude or phase characteristics. For example, when a square-wave input is applied to a

high-pass filter, the output waveform undergoes linear distortion. Second, nonlinear

distortion is caused by nonlinear transfer function characteristics. For example, the

application of a large sinusoidal waveform to the exponential transfer function









characteristic of a bipolar transistor based amplifier can cause a sharpening of one hump

of the waveform and flattening of the other one.

Nonlinear distortion is classified finely in two categories : weak and hard

distortion. In the case of weak distortion, the harmonics gradually shrink as the signal

amplitude becomes smaller. However, the harmonics are never zero. The harmonic

amplitudes can easily be calculated from a Taylor series expansion around the quiescent

or operating point. In weakly nonlinear distortion, the Volterra series can be used for

estimating the nonlinear behavior of a circuit.

Hard distortion, on the other hand, can be seen in Class AB, B, and C amplifiers. In

these cases, a part of the sinusoidal waveform is simply cut off, leaving two sharp

corners. These corners generate a large number of high-frequency harmonics. They are

the sources of hard distortion. Hard distortion harmonics suddenly disappear when the

amplitude of the sinusoidal waveform falls below the threshold, i.e., the edge of the

transfer characteristic. The class C amplifier is considered below as an example of a

circuit with hard distortion. Figure 2-1 shows the output current of an ideal class C






0




20
Figure 2-1 The output current of an ideal class C amplifier for a sine-wave input

amplifier. The output current amplitude at a fundamental frequency is in the form of a

nonlinear function which shown in equation (2-1).









ind = (20 sin 20) (2-1)
27i

where 20 is a conduction angle and is a nonlinear function of output amplitude io.

2.2 Taylor's Series Expansion

Let be f(x) continuous on a real interval I containing x, ( and x), and let

f'"'(x) exist at x and f(" ")() be continuous for all < e I. Then we have the following

Taylor series expansion:


f(x) = f(x,) + -f'(x,)(x- x,) + f"(x)(x- xo)2
1 1.2

+ f .'(x,0)(x x,)3+... (2-2)
1.2.3
+ f (x,)(x- x0)" + R,, (x)
n!

where R,, (x) is called the remainder term. Then, Taylor's theorem provides that there

exists some between x and x, such that

f(n +l) ()
R,+ (x) (x x )n+ (2-3)
(n + 1)!

In particular, if f(n+

R n1 (x) < x xx "+ (2-4)
(n + 1)!

which is normally small when x is close to x, [Ros98].

For a nonlinear conductance, the current through the element, io, (t), is a

nonlinear function f of the controlling voltage, VCONR (t). This function can be

expanded into a power series around the quiescent point Iou = f(VcoNTR) [Wam98].

ioT (t) = f (VONTR (t)) = f(VCONTR + Vcon (t))










+ O1 k f(v(t)) k(25
Sf(VcONT ) E (a) .c (t) (2-5)
k=1 k! (Dv)"
k 1 k = VCONTR

Nonlinear coefficients are defined as follows

1 af(v(t)) (2-6)
K = (2-6)
(k! ( Iv=CONTR

The expression of the AC current through the conductance is in equation (2-7).

iotn (t) = K Von (t) + K2onr (t) + Kv o(t)+... (2-7)

2.3 Measurement of Nonlinear System

A single-tone test is used for the measurement of harmonic distortion, gain

compression/expansion, large-signal impedances and root-locus analysis. The

configuration of a single-tone test can be seen in Figure 2-2. When the input power

sweeps a wide range, the output power at the same frequency as the input is measured in

the sweep range. The cable and other interconnection components that transfer power

should be calibrated since these components have power loss.

For the analysis of intermodulation, cross-modulation and desensitization, the two-

tone test is used. Figure 2-3 shows the configuration of a two-tone harmonic test. In this

test, the power combiner is used for combining two powers at different frequencies.

Through this test, a third-order intercept point is determined.

The next chapter develops the relationship between 1 dB gain compression point

and third-order intercept point. Taylor series analysis above is essential for performing

that analysis.




















Figure 2-2 The configuration of a single-tone test. ERA is a commercial amplifier.


'Power Combiner

Figure 2-3 The configuration of a two-tone test. ERA is a commercial amplifier.













CHAPTER 3
THE RELATIONSHIP BETWEEN THE 1 dB GAIN COMPRESSION POINT AND
THE THIRD-ORDER INTERCEPT POINT

3.1 Definition of 1 dB Gain Compression and Third-order Intercept Point

The constant small-signal gain of a circuit is usually obtained with the assumption

that the harmonics are negligible. However, as the signal amplitude increases, the gain

begins to vary with input power. In most circuits of interest, the output at high power is a

compressive or saturating function of the input. In analog, RF and microwave circuits,

these effects are quantified by the 1-dB gain compression point [Raz98]. Figure 3-1

shows the definition of the 1 dB gain compression point. The real gain curve, C is

plotted on a log-log scale as a function of the input power level. The output level falls

below its ideal value since the compression of the real gain curve is caused by the


B


20log0 (A,,,,)
1dB










AldB 201og10(A,)

Figure 3-1 Definition of 1-dB gain compression point

















co co _o10_ :
(91 (92 ,/ (y ^

(2cl c(,) (2co, ci)

Figure 3-2 Intermodulation in a nonlinear system

nonlinear transfer characteristics of the circuit. Point A1-dB is defined as the input signal

level in which the difference between the ideal linear gain curve B and the real gain curve

Cis 1 dB.

Another important nonlinear characteristic is the intermodulation distortion in a

two-tone test. When two signals with different frequencies, i)1 and cm2, are applied to a

nonlinear system, the large signal output exhibits some components that are not

harmonics of the input frequencies. Of particular interest are the third-order

intermodulation products at (20), )2) and (202 0,), as illustrated in Figure 3-2. In

this figure, two large signals at the left are inputs into an amplifier in the center. The

output is shown on the right of the figure as the two fundamental signals plus the

intermodulation frequencies (2o0i 02) and (202, i,).

The corruption of signals due to third-order intermodulation of two nearby

interferers is so common and so critical that a performance metric has been defined to

characterize this behavior. The third-order intercept points IIP3 and OIP3 are used for

characterizing this effect. These terms are defined at the intersection of two lines shown

in Figure 3-3. The first line has a slope of one on the log-log plot (20 log (amplitude at









A 20log(Amplitude at w,)
OIP3 -


20 log(Amplitude at 2w w2)





IIP3 20log(Ain)
Figure 3-3 Definition of third-order intercept point


wl)) and represents the input and output power of the fundamental frequency. The

second line represents the growth of the (2o, 02) intermodulation harmonic with input

power, it has a slope of three. OIP3 is the output power at the intercept point and IIP3 is

the input power at the intercept point, IP3.

3.2 Classical Approach to Model IIP3

In a nonlinear system without memory such as an amplifier at low frequency, the

output can be modeled by a power series of the input in section 2.2. If the input of the

nonlinear system is x(t), the output y(t) of this system is as follows,

y(t)= K,x(t)+ K2(t2 +K3(t)3 +... (3-1)

where K, is nonlinear coefficients of this system. This example is explained in equation

(2-6) in section 2.2. The classical analysis of the nonlinear system uses the assumption

that the fourth-order and higher-order terms in equation (3-1) are negligible. In the

classical analysis, the nonlinear system is modeled as follows [Raz98][Gon97],

y(t) = K,x(t) + K2x(t)2 + Kx(t)3 (3-2)









If a sinusoidal input with a fundamental frequency is applied to this nonlinear system

x(t) = A cos(ot) (3-3)

then the output of this system is represented by using the equation (3-2),

KA2 3KA3 KA2
y(t) = -- + KA + 3K3 cos(Ot) + KA2 cos(2ot)
2 4 2 (3-4)
3KA3
+ cos(3ot)
4

At the fundamental frequency o), the gain is defined as a function of the input signal

amplitude,

3KA3
Gain(at )) = KA + 3K (3-5)
4

If K3 has the opposite sign of K,, the gain is a decreasing function of the input

amplitude.

The 1-dB gain compression point is defined in Figure 3-1, the equation at this point is


20 log KA 3K3 AdB3 = 20logK,1AI 1dB (3-6)
4

where A1, is the input amplitude at the 1 dB gain compression point. The solution of

equation (3-6) is


AdB = 0.145- (3-7)
V 3

In summary, the input amplitude referred to the 1 dB gain compression point is found

using only two nonlinear coefficients K, and K,.








The classical analysis is considered in a two-tone test. The input signal in the two-

tone test is composed of two signals with the same input amplitude and different

frequencies.

x(t) = Acos(it) + A cos(i2t) (3-8)

When this input signal is applied to the nonlinear system represented by the equation (3-

2), the output is,


y(t) = KIA + 93 cos(o)t) + K ,A + 9K33 cos(o)it)
4 4
(3-9)
3KA3 ,, 3K3A3'} ,, ^
+ r3A3 2 cos((2w, -O2 )t)+ 3K3A 2 cos((2co2 -1 )t)+'
4 4

The third-order intercept point is defined in Figure 3-3. At this point, the output

amplitude at a fundamental frequency is the same as that at an intermodulation frequency.

The input signal level satisfying the above condition is represented by,


KIAP3 3K3A (3-10)


where AIP3 is the input amplitude at the third-order intercept point. The solution of above

equation (3-10) is

AI3 4K (3-11)
3K3

The input signal level at the third-order intercept point is also found using two

nonlinear coefficients K, and K3. From equation (3-7) and equation (3-10), the

relationship between the input signal levels at the 1 dB gain compression point and the

third-order intercept point can be derived,









.4 -9.6dB (3-12)
A_1 3 V-4/3

The classical analysis shows that the relationship between two nonlinear characteristics is

represented by equation (3-12).

3.3 New Approach to Model Gain Compression Curve

For simplicity, the analysis is limited to memoryless, time-invariant nonlinear

systems. Prior classical analysis limits the output to the third-order nonlinearity

coefficient. For the more exact analysis, the relaxation of this limitation is required. The

nonlinear system in this analysis is represented by,

y(t)= Kx(t) + K,x(t)2 + K3x(t)3 + K4x(t)4 + Kx(t)5 (3-13)

If a sinusoidal input such as equation (3-2) is applied to this system, then the output

amplitudes at each odd-frequency are as follows,


y(t;))= KA+-K3A3 +5 KA cos(ot) (3-14)



4 16


y(t;5w) = K5A 5cos(5wt) (3-16)
16

If a low input signal level is considered in equation (3-14), the following condition

is satisfied,

KA >> 3 KA3 +5 +KA5 (3-17)
4 8

then the output amplitude at the fundamental frequency is KA. From equation (3-15),


the output amplitude at frequency 3o is -K A3 if,
4












IPl





S/ 35

B/
C

Input Power

Figure 3-4 The definition of Intercept points in one-tone test. A= 20 log (KIA),
B=20 log (K3A3/4) and C=20 log (KsA5/16)

K3A3 >> -5KA (3-18)
4 16

As stated in the previous section, it is possible to define the intercept points

shown in Figure 3-4 in one-tone test. In Figure 3-4, Line A represents the output

amplitude at the fundamental frequency and has a slope of one in the log-log scale graph.

This line is extrapolated from linear small-signal area in equation (3-14) from equation

(3-17). Line B is the output amplitude at triple fundamental frequency and has a slope of

three. This line is also extrapolated from equation (3-15) under the condition of equation

(3-18). Line C with a slope of five is the output amplitude at frequency 5o and is drawn

from equation (3-16). The intercept points between three lines in this figure are denoted

by IP15, IP13 and IP35. At the point IP13, which is the interception point between Line A,

and Line B, the input signal level A13 can be found from,










4
KA13 = K3A133 (3-19)


where the value of Line A is the same as that of Line B at the input level A13. From this

equation (3-19), the input signal level can be represented by using two nonlinear

coefficients K1 and K3 such as,


2
A13 = 2 (3-20)
K3

At the intercept point IP15 between Line A and Line C, the input signal level A1, is found

by solving the following equation,


KAA = IKAz4 (3-21)
16


K4
A5 = 2K (3-22)
Ks5

The signal level A35, which is the input value at the intercept point IP35, can be described

by,

-K3A35 = K5A354 (3-23)
4 16


K2
A35 = 32 (3-24)
K5

The relationship between the input signal levels at the intercept points can be derived

from the equation (3-20), (3-22) and (3-24),

A3A35 = A52 (3-25)

The equation (3-25) can be expressed differently by using logarithm,









20 log(A13) + 20 log(A35) = 2 20 log( A1) (3-26)

In addition, this relationship can be designated by,

IP13 + IP3 = 2 x IP1, (3-27)

where IP, is the input-referred power at the interception point IPI.

The 1 dB gain compression is considered in this approach. The input signal level

AdB at this point can be represented by the equation


20logK1Ad + K3A 3 +5 K, A =201ogKAn -1dB (3-28)
4 8

From the above equation, one can see that fifth order nonlinear coefficient makes the gain

compression change from equation (3-6) in the previous section. Generally, gain

compression arises when K3 has the opposite sign of K1. From equation (3-28), the gain

compression curve is affected by Ks. The simple equation is derived from equation (3-

28).

K +4- A 1 ,2 +0.109= 0 (3-29)
8 K, lf 4 J K, l

Using intercept points as defined above, this equation is represented (Ki>0, K3<0 and

Ks<0) by,


10 A) +3 A1 -0.109 = 0 (3-30)
A 15 ) A 13

If Ks is positive then,


1 \iM -3 Ad -0.109 = 0 (3-31)
A 15 A3 1









Therefore, if A13, A15 and A35 nonlinear coefficients are known, it is possible to evaluate

AldB *

Equation (3-8) takes the form of the input signal in a two-tone test. If this input is

applied to the system described by equation (3-13), the output has a more complicated

form than that in the one-tone test. The output at fundamental frequency is described by,


y(t;0,)= KA+-9 K3A3 +2 KA cos(C0t) (3-32)


where 0, represents the fundamental frequency which is one of two frequencies 0, and

92,. The amplitude of the output, y(t; ,), ) in equation (3-32) is different from that of the

output y(t; o) in the equation (3-14) of the one-tone test. This difference is called as the

term desensitization [Wam98]. This term is originated from communication circuits in

which a weak signal is affected by an adjacent strong unwanted signal through a

nonlinear transfer characteristic. The output amplitude at the intermodulation frequency

(20), 02) is represented by the following equation,


y(t;201 -0 )= IK3A3 +2-5KA cos((20, -2)t) (3-33)


The third-order intercept point is defined in the small-signal area of a gain plot. The input

signal level at the third-order intercept point is the same as that in the classical analysis.

Equation (3-11) shows the input signal level at this intercept point.

In this analysis, it is possible to estimate IIP3 and 1 dB gain compression points

from the knowledge of the nonlinear coefficients since IIP3 and 1 dB gain compression

point are represented by functions of nonlinear coefficients. The gain compression curve

in the result of a one-tone test is a function of the nonlinear coefficients represented by









equation (3-14). It is possible to extract the nonlinear coefficients from the gain

compression curve by applying a fitting method that will be explained in the next section.

If the nonlinear coefficients are extracted from the gain compression curve in a one-tone

test, it is possible to estimate IIP3 without the two-tone test. In the results of the two-tone

test, the gain curve at each frequencies oi and o2i is a function of the nonlinear

coefficients described by equation (3-32). Even though the equation (3-32) includes the

desensitization factor, this factor is also a function of the nonlinear coefficients. It is

possible to extract the nonlinear coefficients from these gain curves by using a numerical

fitting method. In addition, the IIP3 measurement contains the nonlinear coefficients

effects. It is possible to predict 1 dB gain compression points and estimate the gain curve

without the one-tone test by using the nonlinear coefficients extracted from two-tone test

results.

3.4 Fitting Polynomials Data by Using Linear Regression Theory

If a model has n sets of observations and is fitted with a series of linear parameters

bo, bi, b2,..., bm, by the method of least squares, the model is represented by the

following matrix form [Dra95],

Y= X +b+E (3-34)

where Y, X, b and E are nx 1l,nx m,mxl and nx 1 matrices respectively. In this

equation (3-34), Yis a measured result, X is an input data, b is a set of linear

parameters and E represents the error between a fitting model and a real model.

The above linear regression model has three basic assumptions. First, the average

of the errors in the observations is zero and the variance of these errors is -2. Second, EI

and E, are uncorrelated if i is different from j. Third, the error follows the normal








random variable distribution. These three basic assumptions are included intrinsically or

extrinsically in the statistical approach of linear regression analysis.

The sum of squares of deviations or errors is,

SS(E,)=E *~rE= (--X br *rY-Xeb) (3-35)

To find the sum of square of errors the least quantity, b can be found as follows,

b = *( X *(X *Y) (3-36)

After finding b using above equation (3-36), the variance of b is,

V (b)= (T X *a2 (3-37)

where the diagonal terms of the above variance represent the variance of the parameter

b, and the off-diagonal terms stand for the covariance of the pair b, and b The variance

of errors, "2, is calculated through the analysis of the residual. The residual is defined as

the difference between a fitting model and a real model. The sum of square of the

residuals is represented by the equation (3-35). The mean square of the residual is defined

as the sum of square of the residuals divided by the degree of freedom of the residual. If

the number of the observations is n and the degree of freedom of the regression parameter

is m, the degree of freedom of the residual is n-m. The mean square of the residual is

used as the estimate of the variance of errors, a-2. The mean square of the residual is,
I-\ 1 --b XY
MSE = 2 = Y-X ) X/- ) 1 X( Y (3-38)
n-m n-m

In this research, the fitting model for the gain curve is the odd-order polynomial

equation,

y = K x, +K3 x, +K, K x, (3-39)










where y, is the estimate value of the fitting model. The residual is,


E, = y, -y, (3-40)

The matrices in the equation (3-34) can be formed as follows,

y1
Y= y2 (3-41)



3 5
x 1 x 1 x
X= x x2 x25 (3-42)



K,
b= K3 (3-43)

K5

El
E= E2 (3-44)



The degree of freedom of the parameter b is three. If the number of the data points

is n, the degree of freedom of the residual is n-3. Using the degree of freedom of the

residual, the variance of the errors can be calculated in equation (3-37). The standard

error of the parameters is defined as the square root of the variance of the parameter. The

standard errors of the parameters or nonlinear coefficients can be calculated using the

equation (3-37). The sum of square of the residual also can be calculated using equation

(3-35). These two analyses, the standard errors of the nonlinear coefficients and the sum

of square of the residual are used in determining the fitting range in this research.









3.5 Summary

In this chapter, the new relationship between the 1 dB gain compression point and

the third-order intercept point has been derived. First, this relationship between IIP3 and

IP1-dB was reviewed in classical analysis. The difference between two nonlinear

characteristics was 9.6 dB and constant. The classical analysis included only third-order

nonlinear coefficients. The new relationship was derived by expanding nonlinear analysis

on the gain compression curve up to the fifth-order nonlinear coefficients. The difference

between IP1-dB and IIP3 is not fixed and is explained by the equation including nonlinear

coefficients. The fitting algorithm to estimate IIP3 from one-tone measurement and the

calculation method to predict IP1-dB from two-tone measurement are devised. The linear

regression theory required for the fitting algorithm has been reviewed and modified for

the application of the algorithm.














CHAPTER 4
SIMULATION

4.1 A MOSFET Common-Source Amplifier

The modeling approach developed in a previous chapter is applied to the simulation

of a weakly nonlinear system. A common source amplifier is considered as a weakly

nonlinear system. There are two methods that can estimate 1 dB gain compression point

and IIP3 from one-tone test. The first method uses the ratio of nonlinear coefficients.

These ratios are found from the harmonic power intercept points, which are explained in

Chapter 3. For determining gain compression, measurement of the overall device or

amplifier power is needed but for IIP3 estimation, the measurement of the third and fifth

harmonic frequency magnitudes is required. The second method to estimate IIP3 is fitting

the gain curve at the fundamental frequency for the extraction of the nonlinear

coefficients. The gain curve is fitted via a Matlab program developed in this research.

The second IIP3 determination technique is more useful than the first since the second

technique does not need the third and fifth harmonic frequency amplitude coefficients to

the measured.

To demonstrate the fitting technique, a common-source amplifier with TSMC 0.25

[m n-MOSFET is considered. Figure 4-1 shows the schematic for a common-source

amplifier. In this amplifier design, supply voltage VDD is 3.3 V and the load resistor RD is

10 KQ. The size of the transistor Ml is a minimum size (W/L=1.18 [m/0.25 tm) and

BSIM3 model of this transistor is listed in the Appendix A. The gate bias voltage is set to

produce the weakly nonlinear behavior in the amplifier. This bias point was found by an










VDD



RD


0 out


V, in-- M1








Figure 4-1 A schematic of a common-source amplifier


analysis of the DC characteristic of this amplifier using the Agilent ADS2002 software.

The I-V characteristic curve and transfer characteristic curve are shown in Figure 4-2. For

a weakly nonlinear simulation, a gate bias is chosen near the center point that is shown as

point P in both Figure 4-2.A and Figure 4-2.B. At the gate bias, 1.1 V, an AC simulation

was performed. Figure 4-3 shows the result of AC simulation. When the frequency

increases above 1 GHz, the voltage gain decreases significantly. This indicates that the

parasitic capacitance of the n-MOSFET needs to be considered in the gain calculation

and cannot be ignored above 1 GHz. As a result, the fitting algorithm in the frequency-

domain has some errors related to the transistor parasitic capacitances (gate and drain)

since the power series does not include the phase information caused by these parasitic

capacitors.




































0.5 1.0 1.5 2.0 2.5 3.0 3.5

Gate Bias Voltage, VG(V)


0.5 1.0 1.5 2.0 2.5 3.0 3.5


Gate Bias Voltage, VG(V)

Figure 4-2 DC characteristic of a common-source amplifier.
and B) shows the drain voltage.


B
A) shows the drain current


350.0


300.0


250.0


o 200.0
4-.
C
S150.0


- 100.0
0
Q


50.0


0.0
0.0


3.5


3.0


2.5


> 2.0

3)
0 1.5
C

C' 1.0
0


0.0 L-
0.0










12 ..... .... ....

11

10




n 7

6


< 4

3

2
1 -


10 107 108 10' 1010
Frequency (Hz)
Figure 4-3 The AC simulation of a common-source amplifier


At the same gate bias as the AC simulation, a one-tone test and a two-tone test were

simulated using the harmonic-balance simulator in the ADS2002 software. Figure 4-4

and Figure 4-5 show the simulation results of one-tone and two-tone voltage gain transfer

function described in Chapter 2. Voltage gain curves are made by observing the amplifier

output amplitude or power while sweeping the input voltage or power at a fixed

frequency. In Figure 4-4, the amplitude of the input voltage is swept from 0.01 V to 2 V

at the fundamental frequency of 100 MHz. Curve A is the voltage gain curve which is the

output amplitude measured at the drain node of Ml in Figure 4-1 of the common-source

amplifier. Line B represents the ideal gain curve under the assumption that there are no

harmonics at any amplifier power level. GC denotes the 1 dB compression point that

shows an 1 dB difference between Curve A and Line B. At the voltage amplitude gain











20

15 B

10

5



-5
0 GC
> -10 A
00
-15

-20

-25 Gain Curve

-30 Linear Gain Curve

-35
0.01 0.1 1
Vin(V)

Figure 4-4 The results of one-tone simulation


compression of 1 dB, the input voltage is 0.52 V in Figure 4-4. The applied frequencies

in the two-tone test are 100 MHz and 110 MHz. In Figure 4-5, Curve A and B represent

the amplifier output signal amplitudes at 100 MHz and at 90 MHz separately. Curve A is

the fundamental frequency of 100 MHz, Curve B indicates the amplitude at

intermodulation frequency of 90 MHz (2x 100 MHz 110 MHz). Line C and D indicate

the ideal harmonic response of the amplifier if no other harmonics are present. A real

amplifier introduces an increasing number of harmonic components as gain compression

is increased. The point of intersection of the two ideal harmonic lines, denoted by point

TOI, indicates the intermodulation intercept point that has a value of 2.05V. This

intermodulation intercept point has the same definition as that of the third-order-intercept

point in a 50 Q system such as an RF system. In two-tone test, the amplitudes at two































-100

-120

-140 i
0.01 0.1 1

Vin(V)

Figure 4-5 The results of a two-tone simulation


1.0

0.8

0.6

0.4

S0.2
70
o 0.0
0)
a -0.2

-0.4

-0.6

-0.8

-1.0


0.1 1
Vin(V)


Figure 4-6 The difference between amplitudes at two frequencies. A: Input frequency
(100 MHz, 110 MHz) B: Intermodulation frequency (90 MHz, 120 MHz)









frequencies and two intermodulation frequencies should be considered. These amplitudes

are compared in Figure 4-6. In this figure, Curve A shows the difference between the

amplitudes at two fundamental frequencies (100 MHz and 110 MHz). Curve B is the

difference between the amplitudes at two intermodulation frequencies (90 MHz and 120

MHz). Up to 0.2 V, the difference of amplitudes at two intermodulation frequencies is

under 0.001 dB. In the case of two fundamental frequencies, the difference is under 0.001

dB up to 0.34 V. Since the third-order intercept point is defined in the small-signal area

explained in Chapter 3, the third-order intercept point between two curves at 110 MHz

and 120 MHz is the same as that between 2 curves at 100 MHz and 90 MHz. In view of

the CMOS amplifier simulations, the difference between 1 dB gain compression point

and third order intercept point is 12 dB. This outcome differs with 10 dB that is shown in

the classical nonlinear calculations for amplifiers [Raz98].

Nonlinear coefficients can be derived from the least-square polynomial fitting of

the gain compression curve A of Figure 4-4. The polynomial model used in this fitting is

as the power series that follows,

3 5
y = K,*x+-K x3 +-K5x5 (4-1)
4 8

where K, is a nonlinear coefficient representing the output signal amplitude at harmonic

i. According to linear regression theory, the standard error can be analyzed on the

individual coefficient, K,, fitting outcomes. In addition, the difference between the fitting

model and the actual gain curve produces a fitting residual, a curve fitting error.

Therefore, the best fitting range can be selected by analyzing the fitting error and

standard error of each fitting coefficient. When the input amplitude decreases, the output

amplitude includes more information of lower-order harmonics in the polynomials.












4-

3







O
0 -2
SK

-3 -


-4 -

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
The End Point of Fitting Range

Figure 4-7 Extracted nonlinear coefficients


Selecting a range with a small input voltage should be the first consideration during the

fitting process. The fitting algorithm extracts the nonlinear coefficients from the lowest-

order harmonic to 5th-order harmonic or the highest-order harmonic. A small area is

chosen in the small input signal region of the gain curve data as the first fitting range. In

this fitting range, each of the nonlinear coefficients, the standard error of each nonlinear

coefficients and residuals are extracted in the fitting process by using the polynomial

model, equation (4-1). After finding the information in this range, the fitting process

continues in a wider range than the first fitting range. The respective coefficients and the

change of the fitting range are shown in Figure 4-7. The first fitting range in the gain

compression curve is from 0.01 V to 0.3 V in a small input amplitude area. In Figure 4-7,

KI at the point that x-axis is 0.3 V represents the value of the first-order nonlinear

coefficient extracted from this first fitting region. In addition, K3 and K5 indicate the









third-order and the fifth-order nonlinear coefficients correspondingly. A horizontal axis

defines the width of the fitting range in Figure 4-7 since the starting point of the fitting

range is fixed at 0.01 V to include the small input amplitudes. The x-axis represents the

end point of the fitting range in volts. This graph shows us that individual coefficients K1,

K3 and K5 vary based on the widths of the fitting range. It is important to define the best

fitting range to choose the extracted nonlinear coefficients as shown in Figure 4-7 since

the nonlinear coefficients vary with the width of the data fitting range.


0.11
0.10
0.09
S0.08 .e.()
A
) 0.07
O
O 0.06
0 0.05
2 s.e.(K)
-. 0.04
LI
S0.03
S0.02 s.e.(K ),
c 0.01 -
CU

C0 0.00
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

The End Point of Fitting Range (in Volts)

Figure 4-8 Standard errors of nonlinear coefficients


The standard errors of the separate coefficients in their fitting ranges are calculated

by using the following equations that were presented in chapter 3.

s.e.(K,) = Jj (4-2)










where s.e.(K,) is the standard error of the K, coefficient and Vb), is the diagonal term

of the variance of the parameters or nonlinear coefficients which are calculated by

equation (3-28). Figure 4-8 shows the standard error of nonlinear coefficients according

to the width of the fitting range. In this graph, s.e.(K,) points to the standard error of the

nonlinear coefficient K,. The lowest values for the standard errors of K1, K3 and K5 are

shown by Point A in Figure 4-8. This point is about 0.5 V that is near 1 dB gain

compression point, 0.52 V. The sum of square of the residual, SS(E,) between the fitting

model and the actual gain curve can be found using equation (4-3),

-T -- --T-T -
SS(E,)=E E=YYY-b X Y (4-3)

where the calculation is the same as equation (3-26). This quantity shows the total error



0.012


0.010


o- 0.008
O
4-
S0.006
E

S0.004

A
0.002


0.000

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

The End Point of Fitting Range
Figure 4-9 The sum of squares of the residual









caused by the difference between the fitting model and the actual model. Figure 4-9

shows the sum of square of the residual, SS(E,) In this graph, the total error quantity is

significantly lower under the 0.6 V fitting range. Point A represents the lowest value

found in Figure 4-8. From this graph, it is shown that the width of the fitting range from

0.01 V to point A of Figure 4-8 has small SS(E,). Therefore, the fitting range that has

the minimum standard error and the small total residuals can be defined through Figure 4-

8 and Figure 4-9. The fitting outcome for this set of prospective ranges is summarized in

Table 4-1. At the third-order-intercept point IIP3, the input voltage is 2.21 V or 6.9 dB

input voltage with the resulting harmonic coefficients. This measured input value shows

about 0.7 dB difference in comparison with the simulated value of 6.24 dB. This error

may be created by the phase information that is missing from the fitting model and is

caused by errors in the calculation of the power measurements with no phase information.

In general, Volterra series are used to model the case of weakly nonlinear behavior.

However, the one tone measurement is not sufficient to implement both phase-

information detection and phase analysis that are required for Volterra-series parameters.

In spite of these errors, 1 dB or less difference for the third-order-intercept point

simulation and measurement is a good result.

Table 4-1 The summary of fitting results
Parameters Values

Coefficient K1 3.05
Coefficient K3 -0.83
Coefficient K5 -3.67
Calculated V(IIP3) 2.21V(6.9 dB)
Simulated V(IIP3) 2.05V(6.24 dB)













2.0 A



1.5





0
> 0 -



0.5



0.0


0.0 0.5 1.0 1.5 2.0 2.5 3.0

Vin(V)

Figure 4-10 Gain curves with 0.1% and 2% random noise


4.2 Measurement Error Consideration

Measurement error needs to be considered in the actual data measurement with the

algorithm presented in Section 4.1. The effect of additional random noise upon the

simulated data is analyzed in this section for the determination of the measurement error.

In addition, the error boundaries are studied in order to produce satisfactory results with

the proposed algorithm.

To analyze the effect of random noise, random noise is created by using random

number generation in Matlab software and is added to the each one-tone data point used

in the previous section.

z, = y, + y, x (P%)x n, (4-4)


..........












2 ,4 -- -- -- -- -- -- -- -- i -- --



"_ 2.3



2.2








0 200 400 600 800 1000

Samples

Figure 4-11 Calculated third-order intercept point with 0.1 percent added random noise


where y, is the voltage output amplitude of the individual data point, n, is the real

number that is chosen randomly from -1 to 1, P is the percentage of added random noise,

and z, is a new data point that includes the output amplitude and random noise. Figure 4-

10 shows two gain curves with 0.1 % and 2 % additional random noise. Curve A is the

gain curve with 0.1 % additional random noise and Curve B represents the gain curve

including 2% additional random noise. In this section, the fitting range from 0.01V to

0.52 V that was defined in Section 4.1 is used for fitting the gain curve data including

random noise. The gain curve data with 0.1 % added random noise is fitted in the fixed

fitting range. The same procedure is repeated one thousand times averaging in the gain

data with different 0.1 % random noise. The calculated third-order intercept points over

one thousand samples are shown in Figure 4-11. The average value over one thousand
o U
I-
2.1

0)

0 200 400 600 800 1000
Samples

Figure 4-11 Calculated third-order intercept point with 0.1 percent added random noise


where y, is the voltage output amplitude of the individual data point, n, is the real

number that is chosen randomly from -1 to 1, P is the percentage of added random noise,

and z, is a new data point that includes the output amplitude and random noise. Figure 4-

10 shows two gain curves with 0.1 % and 2 % additional random noise. Curve A is the

gain curve with 0.1 % additional random noise and Curve B represents the gain curve

including 2% additional random noise. In this section, the fitting range from 0.01 V to

0.52 V that was defined in Section 4.1 is used for fitting the gain curve data including

random noise. The gain curve data with 0.1 % added random noise is fitted in the fixed

fitting range. The same procedure is repeated one thousand times averaging in the gain

data with different 0.1 % random noise. The calculated third-order intercept points over

one thousand samples are shown in Figure 4-11. The average value over one thousand













4









I-
o

"c





0-
C-
0
I0


200 400 600 800


1000


Samples


Figure 4-12 Calculated third-order intercept point with 2.0 percent added random noise


Added Random Error(%)


Figure 4-13 The influence of added random error on the fitting results


A 11



IT II II I I




.- ..-.....- .-- .. ....-.

B

*I ..... I I ..... I ..









samples is 2.216 V and the standard deviation of these samples is 0.0149 V. Figure 4-12

shows the calculated third-order intercept point with 2 percent added random noise over

one thousand samples. In this graph, the distribution of third-order intercept point

calculated from the gain data added 2% random noise is wider than that added 0.1 %

random data. The average value of these samples is 2.3023 V and the standard deviation

is 0.3692 V. Fitting results on the each additional random error amount and the associated

standard deviation are shown in the Figure 4-13. In this graph, added random error

increases by 0.1% increments from 0.1% to 2.0%. As the additional amount of random

error increases, standard deviation grows. Two dotted lines A and B show 0.5 dB as an

acceptable range of third-order intercept point error for the center value of 6.9 dB. Each

bar represents the standard deviation for one thousand samples. If the fitting results from

random errors are limited within + 0.5 dB, the additional random error amounts are

within 0.8%. In other words, if measurement error is in 0.8% of the correct measurement

value, the applied result through the fitting algorithm makes an IIP3 estimation with +0.5

dB accuracy possible. However, if more than 1% random errors are added to this data, it

becomes difficult to predict the IIP3 from a single measurement of the gain curve and

produce an accurate value. Data averaging with multiple data point measurements to

reduce average error must be used.

4.3 Frequency Effect on the Fitting Algorithm

What kind nonlinearity effects can higher frequencies produce? Frequency

becomes higher for recent wireless circuits. In addition, the high frequency region of

operation needs to be measured to analyze the nonlinearity of RF components. The

extraction algorithm is applied to frequencies up to 30 GHz in the circuits used in the

previous sections. The results are shown in the Table 4-2. According to Table 4-2, the









difference between the extracted third-order intercept point and simulated value grows

when the frequency increases from 100 MHz to 5 GHz. There is an evident difference

between the simulation and the power series model used for the fitting algorithm. In order

to analyze that difference, the gain compression curve is represented by Volterra-series

components. Appendix B shows Volterra-kemels of a common-source amplifier. When

the input is

V, = V cos(iclt) (4-5)

the output amplitude at the fundamental frequency is as follows,


3

5
Vo,,t (Q; ci8) = Re{VH (ai) exp(joclt) + 3 VH (01, oi ,-0i) exp(jo)lt)

+ 5 V'H5 (mi, o,0, o ,-i j,-mi) exp(jolt) +...}
8


(4-6)


Table 4-2 Frequency effect on the fitting algorithm

Estimated Difference
Frequency Vi-dB(dB) VIIP3(dB) Etim(dB) (d
VIIP3(dB) (dB)

100 z 0.52 V 2.05 V 2.21 V
100 MHz 0.65 dB
(-5.68 dB) (6.24 dB) (6.89 dB)

900 z 0.52 V 2.05 V 2.23 V
900 MHz 0.73 dB
(-5.68 dB) (6.24 dB) (6.97 dB)

0.53 V 2.06 V 2.29 V
2 GHz 0.92 dB
(-5.68 dB) (6.28 dB) (7.20 dB)
0.57 V 2.07 V 2.43 V
5 GHz 1.39 dB
(-5.51 dB) (6.32 dB) (7.71 dB)
0.61 V 2.04 V 2.21 V
10 GHz 0.70 dB
(-4.88 dB) (6.19 dB) (7.00 dB)

0.64 V 2.05 V 2.24 V
20 GHz 0.77 dB
(-4.29 dB) (6.24 dB) (6.24 dB)

SG 0.67 V 2.12 V 2.41 V
30 GHz 3 (71.11 dB
(-3.88 dB) (6.53 dB) (7.64 dB)









At low frequency, the parasitic capacitance between gate and drain, Cgd, can be ignored.

Using the calculated Volterra-series kernel in Appendix B, the output amplitude is,

K1V 3 K3V3
Vo.,(t;1) = Re{ --exp(jCoit) 3 K3 exp(jCo0t)
GL + jolCL 4 GL + jOCL (
(4-7)
5 K,V5
85G K5 WC exp(joOt) +...}
8 GL + jo,CL

From equation (4-7), each coefficient has same phase if Cgd is disregarded and C, is a

linear parasitic capacitor. At low frequency, Volterra-series coefficients are similar to

Power-series coefficients. The output amplitude at the fundamental frequency is in

equation (4-8).

3 5
Vou, (o) = KI'*V +- K3'.V3 +5 KsV'.5 (4-8)
4 8

where K'= -K, (GL2 +1 2CL 2)2

At high frequency, Cgd should be regarded in the calculation of Volterra-kernels. If

Cgd is included in Volterra-kernels, the output amplitude is

(K1 jO1Cgd)V 3 K3V3
Vo, (t; ,) = Re{ exp(jai t) exp(joit)
GL + jo),CL 4 G, + j1),C,
(4-9)
5 KV5
exp(joi1t) +...}
8 GL + jo,01C

Apart from disregarded Cgd in the previous analysis, each coefficient has different phase.

Due to the phase discrepancy, each coefficient needs to be described by a complex

number. Therefore, another effect on the frequency exists when the result of one-tone test

is modeled as a Volterra series. According to Table 4-2, the estimated deviation value by

5 GHz increases up to 1.4 dB. After that, the deviation does not increase with frequency.









Frequency has a greater effect on phase than amplitude at more than 5 GHz. In addition,

the gain of the amplifier reduces drastically Though the deviation of the estimate grows

the increased frequency, the deviation is limited by within 1.5 dB at any frequency.

4.4 Load Effect on the Fitting Algorithm

The nonlinear characteristics of an amplifier is affected by the load of the amplifier.

In this section, the effect of loads on the nonlinear characteristic is studied and the effect

on the fitting algorithm is researched. Two types of loads are considered for affecting the

nonlinear characteristics. A passive load is composed of passive components such as

resistors, capacitors and inductors. An active load is a current mirror that is made by an

active device such as transistor. First, the passive load effects on a single transistor

amplifier are studied. The equivalent circuit of a general single transistor amplifier is

shown in Figure 4-14. In this figure, ZL represents the passive load, Zin is an input

impedance, V1 and V2 are an input node and an output node respectively, i represents the

input-voltage-controlled-current source. The passive components used in the passive load



Vi V2








Vin


S Ziv ZL



Figure 4-14 An equivalent circuit









are assumed to be linear components to simplify calculation. Under a given bias

condition, the voltage-controlled-current source can be described by Taylor series at the

quiescent point in equation (2-6) in chapter 2.

i= Kv+K2V2 + K3V3 +... (4-10)

The passive load is a function of s in the s-domain,

ZL =ZL,(s) (4-11)

Volterra-kernel at the output node in the equivalent circuit can be described by


H, (sI,S2,...S,)= -KZ, (- s") (4-12)
n=1

The nonlinear characteristic of an amplifier is determined by the above Volterra-kemels.

The output amplitude at a fundamental frequency in one-tone test of this circuit is found

by combining Volterra-kernels. If the input is

V= V cos(iot) (4-13)

then the output at fundamental frequency o, is


Vo,, (t; oi,) = Re{VH, (oi,) exp(jColt) + 3-V3H3 (01, I1 1,-ao) exp(jao),t)
5 (4-14)
+-5 V'H5 (0, C91, 1,-m,,-m0,)exp(jot) +...}
8

The load impedance used in odd-order Volterra-kemels is


s,=s= jco, (4-15)
n=1


ZL ( s)= ZL (jC)1 (4-16)
n=1

Therefore, the odd-order Volterra-kernel at the fundamental frequency is

H, = -K, ZL(j) (4-17)








The passive load is used to the change of each odd-order Volterra-kernel. This change
includes frequency effects. In the model used in this analysis, the change of Volterra-
kernel yields only frequency effects in the passive load. The fitting algorithm is not affect
by the passive load except for the change caused by the operating frequency. The effect
of the operating frequency on the amplifier was researched in the previous section.
The effect of the active load on the nonlinear characteristic of a single transistor
amplifier is studied. Figure 4-15 shows an example of a single transistor amplifier with
an active load. BSIM3 models of PMOS and NMOS FETs are listed in Appendix A. The
current mirror consisted of two PMOS devices, M2 and M3, which act as an active load
and the sizes of these two devices are the minimum width and length ( W/L=1.18 [tm /
0.25 tm). In this schematic, supply bias voltage VDD is 3.3 V, R1 is 10 kM. V1 is the gate


VDD



V1

M3 V I--M2

V
SVout


'iref -- M
R wth M1
Sin



Figure 4-15 A schematic of a common-source amplifier with an active load







44



180.0 ii i

160.0

140.0

120.0

100.0 -

80.0
I-I

60.0

40.0

20.0

0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0

Gate Bias Voltage (V)
A

3.5 I


3.0 V


2.5

V
2.0 V1
> V
1.5



1.0


0.5


0.0 i I I I
0.0 0.5 1.0 1.5 2.0 2.5 3.0

Gate Bias Voltage (V)
B
Figure 4-16 DC Characteristics of a common-source amplifier with an active load. A) is
current and B) is voltage.










25 I


20 -


15





E \



0 -



-5. I . I . . I
108 109 1010
Frequency (Hz)

Figure 4-17 The AC simulation of a common-source amplifier with an active load


bias voltage of the current mirror. The I-V characteristic curve and transfer characteristic

curve are shown in Figure 4-16. In Figure 4-16.A, Iref is the current of the resistor R1

whose value is 151.2 pA. I1 is the drain current of the transistor M1. In Figure 4-16.B, Vi

is the gate bias of transistors M2 and M3 which construct current mirror, the value of Vi

is 1.51 V. Vo is the drain voltage of transistor Mi. For the simulation of AC, one-tone and

two-tone test, the gate bias voltage is chosen at the point VG in which all MOS transistors

are in saturation mode. The result of AC simulation of this amplifier is shown in Figure

4-17. The 3-dB bandwidth of this amplifier is about 1 GHz. For applying the fitting

algorithm to the common-source amplifier with active load, one-tone and two-tone tests

are simulated in this schematic at the gate bias voltage VG. The result of the one-tone test



































0.01 0.1 1


Vin (V)

Figure 4-18 The results of one-tone simulation


60

40

20

0

-20

-40

-60

-80

-100

-120

-140


0.01 0.1

Vin (V)

Figure 4-19 The results of two-tone simulation










is shown in Figure 4-18. In this test, the amplitude of the input voltage is swept from

0.001 V to 1 V at the fundamental frequency 100 MHz. Curve A is the gain curve and

Line B represents the ideal gain curve under the assumption that there are no harmonics.

The 1 dB gain compression point is denoted by GC in this graph. The value of

input-referred 1 dB compression point is 0.08V. Figure 4-19 shows the result of two-tone

simulation. The applied frequencies in the two-tone test are 100 MHz and 110 MHz.

Curve A and B represent the output voltages at 100 MHz and at 90 MHz separately.

Curve A is the fundamental frequency of 100 MHz, Curve B indicates the amplitude at

intermodulation frequency of 90 MHz. Line C and D indicate the ideal harmonic

response. The point of intersection of the two ideal harmonic lines, denoted by point TOI,

indicates the third order interception point that has a value of 0.25V. The fitting algorithm

that extracts nonlinear coefficients from a gain compression curve is explained

previously in Section 4-1. The same algorithm is applied to this single transistor amplifier

12.0

11.8

11.6

11.4
4-
Z 11.2

S 11.0
0
"5
10.8

> 10.6

10.4

10.2
10.0 I I I I, I
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
The End Point of Fitting Range

Figure 4-20 Extracted nonlinear coefficient Ki







48


with an active load. Figure 4-20 shows the value of coefficient K1 though this fitting

algorithm. In this figure, the value of K1 is extracted by the fitting method through the

fitting range from 0.001 V to the end point of the fitting range. The voltage gain of this

amplifier is greater than that of a common source amplifier in Section 4-1 since the value

of Ki in this figure is greater than that of Ki in Figure 4-7. The value of K3 is shown in

Figure 4-21. The negative sign of K3 causes the gain compression on


-100

-120

-140

-160

-180

-200

-220

-240

-260

-280

-300
0.00


Figure 4-21


0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
The End Point of Fitting Range

Extracted nonlinear coefficient K3


the gain curve since the sign of K3 is opposite to the sign of K1. The extracted value of K5

is shown in Figure 4-22. The value of this coefficient decreases rapidly when the width of

the fitting range increases. Through these three graphs, the extracted values of nonlinear

coefficients change according to the width of the fitting range. This result is the same as

that in the previous section. Figure 4-23 shows the standard errors of nonlinear

coefficients according to the width of the fitting range. In this graph, K, represents the












9000-


8000


7000


6000


5000


4000


3000


2000


1000

0-


0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

The End Point of Fitting Range

Figure 4-22 Extracted nonlinear coefficient K5


0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

The End Point of Fitting Region


Figure 4-23 Standard errors of nonlinear coefficients


90


80


70
CO
60
CO




30
40 0
-h

30


20


0.45


I,~ ~ ~ I* 1 1 1'1 1























I II II I I


K
K 5
5







A






K
- I-----------


I 1 n









standard error of the nonlinear coefficient K,, curve K1 and K3 follows the left y-axis and

curve K5 is represented by the right y-axis. The lowest values for the standard errors of

each coefficients are shown by Point A in Figure 4-23. This point is about 0.14 V. This is

greater than the 1 dB gain compression point, 0.08 V. The best fitting range of this circuit

is different from that of a common-source amplifier discussed in Section 4-1. Figure 4-24

shows the sum of square of the residual. In this graph, the total error quantity is

significantly lower at point A in the standard error graph. Therefore, the fitting range that

has the minimum standard error and the small total residuals can be defined through these

two graphs. The fitting outcome for this set of prospective ranges is summarized in Table

4-3. At the 3rd order intercept point, the input voltage is 0.238 V and -12.48 dB. This

value shows about a 0.44 dB difference in comparison with the simulated value -12.04

dB. The estimated third-order intercept point is close to the simulated third-order

intercept point. The applied fitting algorithm works well in this example. The equivalent

circuit of an amplifier with an active load can be modeled by the passive component at

the quiescent point. The Volterra analysis researched in the previous part of this section is

applied to this amplifier with an active load by replacing the active load with the passive

components. The fitting algorithm is not affect by the active load except for the change

caused by the operating frequency.

Table 4-3 Fitting results
Parameters Values

Coefficient K1 11.63
Coefficient K3 -274.7
Coefficient K5 3957
Calculated V(IIP3) 0.238 V (-12.48 dB)
Simulated V(IIP3) 0.25 V (-12.04 dB)









4.5 Summary

In this chapter, the proposed fitting algorithm has been verified through the

application of the algorithm to the simulation of a common-source amplifier. The best

fitting range has been chosen through the standard errors of nonlinear coefficients and the

sum of squares of the residuals. The effect of measurement error was explored by adding

random noise to the simulation data. Frequency effects on the fitting algorithm was

examined up to 30 GHz. The IIP3 estimation error is less than 2 dB up to 30 GHz.

Through the Volterra series analysis, the effect of loads on the algorithm was investigated

and there were no differences except for the changes in operating frequency.














CHAPTER 5
COMMERCIAL RF WIDEBAND AMPLIFIER

5.1 Nonlinearity Test

Commercial amplifiers and test boards (ERA-series, MiniCircuit, Co.) shown in

Figure 5-1 were used for the measurement of one-tone and two-tone tests. It is difficult to

extract the nonlinear power coefficients for these ERA amplifiers due to noise sources in

the measurement system. One of the noise sources is the signal generator. Even though

this signal generator is designed to give a pure signal source with elaborate internal units,

it still has harmonics in the frequency-domain. A low pass filter is used to decrease the

harmonics in the signal generator. Figure 5-2 shows the spectrum in the signal source

with and without a low pass filter. In Figure 5-2.A, Curve A represents the power at

fundamental frequency, Curve B corresponds to the power at second harmonic frequency,

and Curve C stands for the power at third harmonic frequency. In this graph, the power at

second harmonic and third harmonic frequencies exist in the signal generator. Figure 5-

2(b) shows that the harmonic components decrease when the low pass filter is used.


Figure 5-1 ERA1 amplifier in a test board








53





10


-20 A
-30

-40
0
(-
-0 -50

v' -60

S-70 C

-80

-90 I I I I I I
-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2
Input Power (dBm)
A






20 A
m

0-
40

S-Vo
-60






100
18 -16 -14 -12 -10 -8 -6 -4 -2 0 2
Input Power (dBm) B

Figure 5-2 The spectrum from the signal source. A) shows the spectrum without a low
pass filter and B) shows the spectrum with a low pass filter.


The ERA1 amplifier is measured in a one-tone test. The test scheme is shown in


Figure 5-3. The fundamental frequency used for this test is 100 MHz. Figure 5-4 shows


the result of the one-tone measurement. In this graph, the input referred 1 dB


compression point, IP1-dB is 1.7 dBm and is denoted by GC. In this figure, Curve A


represents the measured compression curve and Line B is the ideal linear curve. Here, the


gain compression curve at the fundamental frequency, Curve A is used for the extraction


of nonlinear coefficients. The application of a fitting method on this curve will be













20dB attenuator


Figure 5-3 One-tone test scheme


20 -


-25 -20 -15 -10 -5 0 5 10
Input Power (d Bm)

Figure 5-4 The measurement data of one-tone test (ERA1 amplifier)



explained in next section. Figure 5-5 shows the test scheme for the two-tone test. Source

frequencies in this two-tone test are 100 MHz and 120 MHz. The result of the two-tone

test is shown in Figure 5-6. In this figure, Curve A and B represent the amplifier output

signal amplitudes at 100 MHz and at 80 MHz separately. Curve A represents the output

power at fundamental frequency of 100 MHz, Curve B indicates the output power at


B \


Test board


I . I . I


I .... I... I


I ... I. ... I






55


intermodulation frequency of 80 MHz (2x100 MHz 120 MHz). The point of

intersection of the two dotted lines, IP3 indicates the third-order intercept point. The

input-referred third-order intercept point, IIP3 is 16.3 dBm. GC denotes 1 dB gain

compression point that has a value of -2.5 dBm. The 1 dB compression point in two-tone

test is explained in section 3.3.


Figure 5-5 Two-tone test scheme


I I I I '


I I I I
IP3 --

GC










lip


-20 -15 -10 -5 0 5 10 15
Input Power (dBm)
The measurement data of two-tone test (ERA1 amplifier)


Figure 5-6


/I








56



ERA2 and ERA3 amplifiers are also tested in the same bias condition as ERAl


amplifier to verify the proposed algorithm to predict third-order intercept point using


extraction of nonlinear coefficients from the gain compression curve. The one-tone test


data of ERA2 is shown in Figure 5-7. Input-referred 1 dB gain compression point


denoted by GC is -0.8 dBm. Figure 5-8 shows two-tone data of ERA2. Like Figure 5-6,


30

25 B


20 -




0 10
C-


0 GC



-5

-20 -15 -10 -5 0 5 10
Input Power (dBm)

Figure 5-7 The measurement data of one-tone test (ERA2 amplifier)


30 IP

20

10

E 0 ,
O GC

10
0 -20 -

-30

0 -40

-50 -iP

-60 -

-70 . .
-20 -15 -10 -5 0 5 10 15 20
Input Power (dBm)

Figure 5-8 The measurement data of two-tone test (ERA2 amplifier)


































-35 -30 -25 -20 -15 -10 -5 0 5
Input Power (dBm)


Figure 5-9 The measurement data of one-tone test (ERA3 amplifier)


50

40
IP

20 .-

10 10.. ,"


10 -
S_4o B
CO -20 B

-30
0 -40

-50 IIP

-60


-30 -25 -20 -15 -10 -5 0 5 10
Input Power (dBm)



Figure 5-10 The measurement data of two-tone test (ERA3 amplifier)


BG







GC \

GC









Table 5-1 The summary of the measurement results of commercial amplifiers

Device IP1-dB* IP1-dB,2** IIP3** IIP3-IP1-dB

ERA1 1.7 -2.5 16.3 14.6

ERA2 -0.8 -5 13.8 14.6

ERA3 -8 -13 4.5 12.5
*One-tone test : Source frequency = 100 MHz
**Two-tone test : frequencies = 100 MHz, 120 MHz

Curve A and Curve B represent the output power at 100 MHz and at 80 MHz separately.

The input-referred third-order intercept point is 13.8 dBm. The input-referred 1 dB gain

compression point denoted by GC is -5 dBm. The 1 dB gain compression point of the

ERA3 amplifier is -8 dBm denoted by GC in Figure 5-9. The result of the two-tone test is

shown in Figure 5-10. The 1 dB compression point is -13 dBm and the input-referred

third-order intercept point is 4.5 dBm in Figure 5-10. The measurement data of these

amplifiers are summarized in Table 5-1. In this table, the difference between 1 dB gain

compression point and third-order intercept point is not constant. Through real

measurements, the relationship between two nonlinear characteristics can be constructed.

5.2 IIP3 Prediction from the Gain Compression Curve

The application of the technique to manufacturing test LNA measurements required

substantial modification. IIP3 extraction from LNA gain simulation data has the benefit

of many decibel places of accuracy (high S/N) and an ideal (no loss) test system. Real

measurements from spectrum analyzers can exhibit roughly 1% accuracy in the data and

substantial power loss due to cables and fixtures in the test setup. Even after using power

magnitude calibration techniques on spectrum analyzer data, significant uncertainty can

exist due to phase errors. In addition, it was found that to extract properly IIP3

parameters the data has to be measured on one power range of the spectrum analyzer.









Using multiple spectrum analyzer power ranges introduced offset errors in the measured

data. In summary, the simulated LNA data had remarkably high S/N across the entire

data set while the measured LNA gain data had a mediocre S/N at high-power which

decreased as power decreased.

Given the situation, a global extraction of the LNA power series expansion

coefficients was not stable with small changes in the data set. Adding a few more data

points at the high-power range would create large changes in the K3 and K5 extracted

coefficients. To counteract this problem, which could be seen in ATE systems doing

manufacturing test, a new parameter extraction methodology had to be created. After

experimenting with many ways of performing this parameter extraction, a regional

parameter extraction methodology was devised.



20 B




0 AA






-30 -D

-40


-20 -15 -10 -5 0 5 10

Pin(dBm)
Figure 5-11 One-tone data and extraction from ERAl device at 100 MHz









A new robust measurement extraction algorithm is developed for one-tone gain

data as graphed in Figure 5-11 for ERA1 amplifier. In this new algorithm, the nonlinear

power coefficients are extracted regionally. To help understanding how to interpret this

graph, the procedure is explained step by step in Figure 5-12. First, the entire one-tone


Figure 5-12 A flow chart for estimation of IIP3 from one-tone measurement


power compression curve is measured as shown by curve B in Figure 5-11. Line B is a

straight-line that can be fitted to the low-power amplifier data. From this line B, the K1

factor is determined. The effects of the K1 factor are subtracted from the original gain

compression curve A (slope 1/1). Curve C shows the remaining terms on the gain

compression curve. From this curve C, it is easy to see that the K3 extraction region









below point P of region R has very high noise. Instead of K3, the K5 factor is extracted

from the compression region R of Figure 5-11. This is easily verified because the slope of

line D is 5/1. Fortunately, it is not necessary to know the slope of the K3 factor, since it is,

idealy, 3/1. To determine the value K3, one need to know where it intercepts line D and

that is at point P. Point P is at the intercept between the K3 line and the K5 line. Point P is

also the highest S/N point in the measured K3 factor data. The calculated IIP3 from this

point is 16.82 dBm is very close to the measured IIP3, 16.3 dBm. Figure 5-13 shows the

application of the algorithm to ERA2 amplifier. The explanation of this graph is the same


20 A

10

01

0m -10

-20

-30
R
-40 -
D
I I I I I I
-20 -15 -10 -5 0 5 10

Pin(dBm)
Figure 5-13 One-tone data and extraction from ERA2 device at 100 MHz


Table 5-2 The summary of the estimated IIP3 of commercial amplifiers
Device Measured IIP3 Estimated IIP3
ERA1 16.3 16.82
ERA2 13.8 12.81

ERA3 4.5 3.71
















0

E
m -10 P .

o -20

-30 R

-40

-25 -20 -15 -10 -5 0 5
Pin(dBm)
Figure 5-14 One-tone data and extraction from ERA3 device at 100 MHz


as that of Figure 5-11. The same analysis is applied to ERA3 amplifier in Figure 5-14.

Table 5-2 shows the measured characteristics and estimated IIP3 of the ERA devices

(ERA1, ERA2 and ERA3). From the table, the difference between the measured IIP3 and

the estimated IIP3 is less than 2 dB in all cases and less than or equal to 0.41 dB for most

cases. Through these experiments a method for predicting IIP3 using a one-tone LNA

gain measurement was developed.

5.3 The Application of the Proposed Algorithm at High Frequency

The ERA2 amplifier is retested at a relatively high frequency 2.4 GHz in Figure 5-

15. Fig 5-15 shows the new robust extraction algorithm at high frequency. The result in

Table 5-3 shows that this algorithm is working in microwave frequencies. Through these

experiments a method for predicting IIP3 using a one-tone LNA gain measurement was

developed.









Table 5-3 The measurement data and calculated IIP3 of a commercial amplifier
Device IP1-dB* IIP3** IIP3***

ERA2 -0.4 12.1 12.08

*One-tone test: Source frequency = 2.4 GHz
**Two-tone test : frequencies = 2.4 GHz, 2.475 GHz
*** Calculated from the extracted nonlinear coefficients




20 B A

10

0 -

m -10
R P

0 -20

-30 D
-40 -


-20 -15 -10 -5 0 5 10

Pin(dBm)
Figure 5-15 One-tone data and extraction from ERA2 device at 2.4 GHz


5.4 IP1-dB Estimation from Two-tone Data

The algorithm in section 5.2 estimates IIP3 from the one-tone data. In this section,

the estimation of the 1 dB gain compression point is investigated on the basis of the two-

tone measurement data. Let the input amplitude at the 1 dB compression point be Ai-dB.

From equation (3-24), The 1 dB compression point in the one-tone test is found in

equation (3-24).









"K5-Ad 4+ K3 ,12dB +0.109=0 (5-1)
8 K, 4 K l


Two ratios, KK5 ~and K31 are needed to solve above equation (5-1). These ratios are


found in the two-tone measurement data. From equation (3-9), the ratio K3 is


determined using following equation,

K3 4 1
K=-x4 (5-2)
KI 3 AI3 2

where AIP3 is the input amplitude at the third-order intercept point. The sign of this ratio

should be decided by inspecting the gain curve at the fundamental frequency in the two-

K
tone measurement. For example, the sign of -3 is negative in Figure 5-6 since Curve A
K1

shows the compression in the gain curve. The 1 dB gain compression point in the two-

tone test is explained by equation (5-3),

25 K5 4 9 K3
4 K i-,2 4 K Al -d&,2 +0.109=0 (5-3)

where A1 d,2 is the input amplitude at the 1 dB compression point of the two-tone test


that is denoted by GC in Figure 5-6. From above equation, the ratio -5 can be found
K can be found

by
= 4 Al 4 j AldB, 2+0.109 (5-4)
_d--'I1rB,2- 9 4 3 -211 1 0 109 (5









Inserting two ratios L5 and K to equation (5-1), the input-referred 1 dB gain


compression point, IP1-dB can be calculated. This method is applied to ERA amplifiers for

verification.

First, the ERA1 amplifier is considered as the application example of this

algorithm. IIP3 of ERAl amplifier is 16.3 dBm in Table 5-1. The input amplitude at this

point is

A,3 10 3-10) 20 = 2.065 (5-5)

where the unit of Ap3 is volt (V) and the input resistor and the output resistor loads are


50 Q. The absolute value of the ratio, K3 can be found in equation (5-2).
K1

3K 1 4
3= x 2.065 = 0.3146 (5-6)
K, 3

K
The sign of above ratio -3 is negative since the gain at the fundamental frequency
K1


compresses in Figure 5-6. The ratio -K3 is
K


= -0.3146 (5-7)
Ki )

From table 5-1, the IP1-dB,2 is -2.5 dBm. The input amplitude at this 1 dB compression

point in two-tone is


A1dB, = 10 (p -,')20 = 0.2371


(5-8)








K K
Using the value of A, d2 and the ratio the ratio a is found in equation (5-4)
K K,

The calculated value of K5 is -3.5140. The 1 dB compression point in the one-tone
lKi )

test is now found by solving equation (5-1). The solution of this equation is 0.4192. The

calculated IP-1 d is 2.45 dBm and is very close to the measured value, 1.7 dBm. Through


the application of the algorithm, two ratios K- and KK3 are found and 1 dB


compression point is estimated. The difference between the measured value and the

estimated value of 1 dB compression point is less than 1 dB.

The data of the ERA2 amplifier is used for another example to verify this

algorithm. First, the input amplitude at third-order intercept point is

Ai3 =1.5488 (5-9)

since IIP3 of ERA2 amplifier is 13.8 dBm in Table 5-1. From this value, the absolute

value of the ratio K3 is calculated. The sign of this ratio is negative since the gain


curve compresses in Figure 5-8. The calculated value of K3 is -0.5588. From IP dB,2
lKi

in Table 5-1, the calculated value of Al B,2 is 0.1778. Using the value of A1 dB,2 and the

K K
ratio K 3, the calculated value of the ratio K-5 is -11.1224. Finally, the calculated


value of Al- d from equation (5-1) is 0.5143. The estimated value of the 1 dB

compression point in one-tone test is -0.05 dBm. Compared to the measured value of 1

dB compression point, -0.8 dBm, the estimation error is less than 1 dB.








Finally, the data of the ERA3 amplifier is used for the estimation of 1 dB

compression point from two-tone data. In this amplifier, the input amplitude at third-

order intercept point is

A,3 = 0.5309 (5-10)

since IIP3 of ERA3 amplifier is 4.5 dBm in Table 5-1. The absolute value of the ratio

K-3- is calculated by using equation (5-2). The sign of this ratio need to be negative
Ki1

since the gain curve compresses in Figure 5-10.


K3 =-4.7308 (5-11)

From table 5-1, the IP1-dB,2 is -13 dBm. The input amplitude at this 1 dB compression

point in two-tone data is

AldB,2 = 0.0708 (5-12)


Using the value of A_ -,2 and the ratio K3, the calculated value of the ratio -5 is


K-= -354.5 (5-13)


From equation (5-1), the calculated value of A-1 d is 0.1248. The estimated value of the

1 dB compression point in one-tone test is -8.08 dBm and is very close to the measured

value, -8 dBm. Table 5-4 summarize the results of these applications of three amplifiers.

Table 5-4 The summary of the application results of the IP1-dB estimation algorithm
Device Measured IP1-dB Estimated IIP3
ERA1 1.7 2.45
ERA2 -0.8 -0.05
ERA3 -8 -8.08









In this table, the difference between the measured IP1-dB and the estimated IP1-dB is less

than 1 dB in all cases and less than or equal to 0.75 dB for most cases. Through these

experiments a method for estimating IP1-dB using a two-tone LNA gain measurement was

developed.

5.5 Summary

In this chapter, a robust algorithm to predict IIP3 has been developed for the

wideband RF amplifier. Given the noisy measurement situation, a global extraction of the

LNA power series expansion coefficients was not stable with small changes in the data

set. Adding a few more data points at the high-power range would create large changes

in the K3 and K5 extracted coefficients. To counteract this problem, which would be seen

in ATE systems doing manufacturing test, a new parameter extraction methodology had

to be created. After experimenting with many ways of performing this parameter

extraction, a regional parameter extraction methodology was devised. The IP1-dB

prediction from two-tone measurement has been applied to these wideband amplifiers.

Through several steps of simple calculation using the third-order intercept point and the

gain compression at the fundamental frequency, IP1-dB has been estimated within less

than 1 dB error.














CHAPTER 6
POWER AMPLIFIERS

6.1 Linear and Nonlinear Power Amplifiers

The final stage which gives signal to an antenna is power amplifier. The

characteristics of this power amplifier is specified by the communication scheme.

Linearity and efficiency among these characteristics are important factors for determining

power amplifiers in their circuit application. Nonlinear power amplifiers are preferred in

constant envelope modulations and Linear power amplifiers are used in amplitude

modulations and 7t/4-QPSK in digital modulations.

The issue of nonlinearity is directly related to the spectral regrowth in a

communication system transmission. The standards for various communication protocals

define the nonlinearity as adjacent channel power ratio (ACPR) or adjacent channel

power (ACP). In the early stage of designing RF components, IIP3 measured in two-tone

test is used for the measure of nonlinearity instead of ACPR or ACP. IIP3 is easily

measured and reported the nonlinearity estimation even though IIP3 is not exactly the

same as ACPR or ACP.

Linear power amplifier operation is classified as "class A" operation. An ideal

linear amplifier doesn't have nonlinear terms. But in actual case the active devices used

for the amplifiers do produce harmonics. In class A amplifier, nonlinear behavior is made

be a weakly nonlinear behavior. The definition of this weakly nonlinear behavior is

explained in chapter 2. This behavior can be analyzed by using the power series or









Volterra series. In this chapter, the power amplifiers are investigated for their nonlinear

characteristics.

6.2 Measurement of Commercial Power Amplifiers

Four commercial RF power amplifiers are measured in one-tone and two-tone tests.

To distinguish these amplifiers, amplifiers are named alphabetically. First, the result of

one-tone test on the amplifier A is shown in Figure 6-1. The fundamental frequency used

for this test is 2.45 GHz. In this figure, Curve A represents the measured output power

and the dotted Line B is the ideal linear curve. The 1 dB gain compression point is

denoted by GC in this figure. The input-referred 1 dB gain compression point, IP1-dB is -

11 dBm. Figure 6-2 shows the result of a two-tone test. The two input frequencies are

2.45 GHz and 2.46 GHz. Curve A represents the amplifier output signal power at 2.45

GHz and dotted Line B is the ideal linear curve extrapolated from Curve A. Curve C

indicates the output power at intermodulation frequency of 2.44 GHz (2x2.45 GHz 2.46

GHz). The dotted Line D has slope of 3 and is extrapolated from the low input power

region of Curve C. The point of intersection of the two dotted lines, TOI indicates the

third-order intercept point. The input-referred third-order intercept point, IIP3 is -4.4

dBm. GC2 denotes 1 dB gain compression point in two-tone test which has a value of-16

dBm. Figure 6-3 and Figure 6-4 show the results of nonlinear tests on amplifier B. The

test condition of this amplifier is the same as that of the amplifier A. In Figure 6-3, the

input-referred 1 dB gain compression point is -8 dBm. IIP3 is 0 dBm and IPi-dB,2, denoted

by GC2, is -13 dBm. The test results of the amplifier C are shown in Figure 6-5 and

Figure 6-6. The same test condition as that of the amplifier A is applied to this test of the

amplifier C. IP1-dB, denoted by GC, is -6 dBm from the result of one-tone test in Figure

6-5. IP1-dB,2 is -10 dBm and IIP3 is 3 dBm in Figure 6-6. Figure 6-7 shows the results












28


26


24


22
E
m

20
0-

18


16


1 4 I I I I I I I I I
-16 -14 -12 -10 -8 -6 -4 -2 0

Pin (dBm)

Figure 6-1 The measurement data of one-tone test of the amplifier A


30 i
TOI -;
25

20
II I'" /

15 --"





-22 -20 -18 -16 -14 -12 -10 -6 -







Pin (-Bs m)
-1

/ ".lip
-20

-2 5 ,
-22 -20 -18 -16 -14 -12 -10 -8 -6 -4

Pin (dBm)

Figure 6-2 The measurement data of two-tone test of the amplifier A


B.







72



28

B
26

24 I

22

E20
"n
S18 GC
0
16 -

14

12

10 I I I I I I, ,
-18 -16 -14 -12 -10 -8 -6 -4 -2 0
Pin (dBm)
Figure 6-3 The measurement data of one-tone test of the amplifier B




40
TOI -
30
30 B ~ -B

20

10
D
m 0
o 0GC2

CL

-20

-30 -I3


-40 .
-24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4

Pin (dBm)

Figure 6-4 The measurement data of two-tone test of the amplifier B























E


o
20

0- 18


16


14

12


-16 -14 -12 -10 -8 -6 -4 -2 0 2

Pin (dBm)

Figure 6-5 The measurement data of one-tone test of the amplifier C




40

TOI -------- ,
30 -'


20


10 A -

SGC2 D
m 0
G6c
o -10
0CL

-20


-30 IIP


-40


-20 -18 -16 -14 -12 -10


18 -6 -4 -2 0 2
.8 -6 -4 -2 0 2 4


Pin (dBm)


Figure 6-6 The measurement data of two-tone test of the amplifier C


B


IIIIIIIIIIIIIIII


I I I I













24


22


20



E18



-n
' 16
0T
o

14



12


10


-14 -12 -10 -8 -6 -4 -2 0

Pin (dBm)

Figure 6-7 The measurement data of one-tone test of the amplifier D


m
-10
21-

a. -20


-30


-40


I I I


3


-24 -22 -20 -18 -16 -14 -12 -10


-8 -6 -4 -2 0


Pin (dBm)


Figure 6-8 The measurement data of two-tone test of the amplifier D


B


GC


I I -


I I I I II I I I I I I I I I I I I


IIIIII,I









Table 6-1 The summary of the measurement results of commercial PAs
Device Name IP1-dB* IP1-dB,2** IIP3** IIP3-IP1-dB
Amplifier A -11 dBm -16 dBm -4.4 dBm 6.6 dB
Amplifier B -8 dBm -13 dBm 0 dBm 8 dB
Amplifier C -6 dBm -10 dBm 3 dBm 9 dB
Amplifier D*** -8 dBm -12.5 dBm -1 dBm 7 dB
*One-tone test: Source frequency = 2.45 GHz
**Two-tone test : frequencies = 2.45 GHz, 2.46 GHz
*** Amplifier D is tested at 5.2 GHz (and 5.21 GHz).

of the one-tone test of the amplifier D. The input frequency in this one-tone test is 5.2

GHz. IP1-dB is -8 dBm in this figure. The applied two frequencies in two-tone test are

5.2 GHz and 5.21 GHz. IIP3 is -1 dBm and IP1-dB,2 is -12.5 dBm. The measurement data

of these amplifiers are summarized in Table 6-1. In this table, the difference between 1

dB gain compression point and third-order intercept point is not constant.

6.3 IIP3 Estimation from the One-tone Data

A robust algorithm which analyzes the measurement data with error is used for the

wideband RF amplifier. In RF power amplifier, another algorithm is required to examine

the nonlinear characteristics since the nonlinear behavior of power amplifier is somewhat

different from that of wideband RF amplifier.

A fitting method which extracts the nonlinear coefficients simultaneously from

one-tone data and is used in the analysis of simulation data in chapter 4 is useful in the

nonlinear analysis ofRF power amplifiers. The fitting range is adjusted through linear

regression analysis. The increase of fitting range reduces the standard error of each

coefficient but the residual, the difference between real model and fitting model,

increases since the high amplifier input power includes energy in higher-order nonlinear

factors than that of fitting model. Both the analysis of the residual and that of standard









errors of each coefficients help to define the appropriate range for the fitting model.

Third-order intercept point can be estimated using the extracted nonlinear coefficients

after fitting this range. In this section, two fitting models are applied to the data of

commercial power amplifiers. One of two fitting model is


y = K,x+-K33 (6-1)
4

Equation (6-1) is the simplest form for the explanation of the nonlinear gain curve.

The other model applied to the fitting gain curve is


y= Kx +-K3 +5-Kx5 (6-2)
4 8

The difference between two fitting models is whether or not to include the fifth-

order nonlinear coefficient in the fitting model. Through the application of these two

models to the data of commercial PAs, the effect of adding fifth-order nonlinear

coefficient is investigated.

The first fitting model, equation (6-1) is applied to the one-tone data of the

amplifier A shown in Figure 6-1. Figure 6-9 shows the value of coefficient K1 from the

fitting results. In this graph, the x-axis represents the end point of fitting range from the

starting point, -16 dBm. When the fitting range changes, the fitted result changes also.

Figure 6-10 shows the value of coefficient K3. The best fitting range should be chosen

using the standard errors of each coefficients and the sum of squares of the residuals

which are explained in Chapter 4. The standard errors are defined in equation (4-2).

Figure 6-11 shows the standard errors of coefficients K1 and K3. In this graph, Point A

indicates the smallest value in both the standard errors of K1 and K3. From the standard

error of the coefficients, the fitting range (-16 dBm, -9 dBm) is chosen. The sum of















m--m --m
II l I I I
























-12 -10 -8 -6 -4 -2 0

The End Point of Fitting Range


) The value of coefficient K1 (amplifier A)




I I II I I I

~- ^ -



/

*- /'-




*- /*-





--.


l- l


-12 -10 -8 -6 -4 -2 0

The End Point of Fitting Range


Figure 6-10 The value of coefficient K3 (amplifier A)


Figure 6-9


-200

-300

-400

-500

-600

-700

-800

-900

-1000

-1100

-1200
















- s.e.(K,)
Ss.e.(K,)


"~f1S* ---A


PE




p







S~
0*


-12 -10 -8 -6 -4

The End Point of Fitting Range



Figure 6-11 Standard errors ofK1 and K3 (amplifier A)


-2 0


2.5



'5 2.0
n,
o /

I,-


1.0
0*


C)
4-
o 0.5

E

0 -
I I I I I I I I

-12 -10 -8 -6 -4 -2 0

The End Point of Fitting Range



Figure 6-12. Sum of squares of the residuals (Amplifier A)


IIIIIII


_









Table 6-2 The summary of fitting results (amplifier A)
Parameters Values

Coefficient K1 36.20
s.e.(Ki) 0.15

Coefficient K3 -1026
s.e.(K3) 22

Calculated IIP3 -3.27 dBm


squares of the residual in this fitting range is also small in Figure 6-12. The fitting results

are summarized in Table 6-2. The calculated IIP3 is -3.27 dBm in this table. The

difference between the estimated IIP3 and the measured IIP3 is 1.13 dB since the

measured IIP3 is -4.4 dBm in Table 6-1.

The second fitting model, equation (6-2) is applied the same data of the amplifier

A. Figure 6-13, 6-14 and 6-15 show the values of K1, K3 and K5 respectively. From these

graphs, the values of nonlinear coefficients will be chosen after the best fitting range is

decided. From Figure 6-16 and 6-17, Point A denotes the end point of the best fitting

range which satisfies the condition that the standard errors of nonlinear coefficients

become small simultaneously. The best fitting range in this fitting process is (-16 dBm, -5

dBm). In this range, the sum of squares of the residuals is small in Figure 6-18. The

result of the fitting application is summarized in Table 6-3. In this table, the calculated

value, -4.31 dBm is very close to the measured value, -4.4 dBm. The first fitting model

has less standard errors of each coefficient than the second fitting model.

The data of the amplifier B in Figure 6-3 is used for the IIP3 prediction from one-

tone measurement. Using fitting model, equation (6-1), the values of K and K3 are

shown in Figure 6-19 and Figure 6-20. Figure 6-21 shows the standard errors of nonlinear




















-800 -


-1100 [-


-12 -10 -8 -6 -4 -2 0

The End Point of Fitting Range



The value of coefficient K1 (amplifier A)































-12 -10 -8 -6 -4 -2 0

The End Point of Fitting Range
The End Point of Fitting Range


Figure 6-14 The value of coefficient K3 (amplifier A)


* I I I I
U



U



U



U



U


U


* I I I I


-1200


-130C


-140C






Figure 6-13






37


)

I













30000
25000
20000
15000
10000
5000
0
-5000
-10000
-15000
-20000
-25000
-30000
-35000
-40000
-45000
-50000
-55000


I I I I I I I


-U

-.\-
- U U
- U


U


I I I I I I I I
-12 -10 -8 -6 -4 -2 0

The End Point of Fitting Range



Figure 6-15 The value of coefficient K5 (amplifier A)




14


Ss.e.(K,)



E 1.0
a) ,





0 0. -


0.4






0.0
-12 -10 -8 -6 -4 -2 0

The End Point of Fitting Range


Figure 6-16 Standard errors of Ki and K3 (amplifier A)


80 -
0
"o



60
50

LU

40 O
20



20













10000


8000




6000




4000


2000


Figure 6-17








06

-v
",

0.4

-
2o
to
0


02
Co
'S

E
CD 00


-12 -10 -8 -6 -4 -2 0

The End Point of Fitting Range



Standard error of K5 (amplifier A)


-12 -10 -8 -6 -4 -2 0

The End Point of Fitting Range



Figure 6-18 Sum of squares of the residuals (amplifier A)


A



I I I I I I I


I I I I I II







-~--- -0












Table 6-3 The summary of fitting results (amplifier A)
Parameters Values

Coefficient K1 36.96

s.e.(K1) 0.14

Coefficient K3 -1329

s.e.(K3) 29

Coefficient K5 22276

s.e.(K5) 1126

Calculated IIP3 -4.31 dBm


Table 6-4 The summary of fitting results of the first model (amplifier B)
Parameters Values

Coefficient Ki 34.50

s.e.(K1) 0.11

Coefficient K3 -404

s.e.(K3) 7

Calculated IIP3 0.43 dBm


coefficients. In this figure, point A is the best region of this fitting. In the application of

the first fitting model, the best fitting range is from -17 dBm to -5 dBm. In the graph of

sum of squares of the residuals, Figure 6-22, the value of sum of squares of the residual is

under 0.1. In this best fitting range, the fitting result is summarized in Table 6-4. Using

extracted nonlinear coefficients, the calculated IIP3 is 0.43 dBm. Compared to the

measured IIP3, 0 dBm, the estimated IIP3 is close to the measured value. When the

second fitting model is applied to the amplifier B, the values of K, K3 and K5 extracted

from one-tone gain curve in Figure 6-3 are drawn in Figure 6-23, 6-24 and 6-25

separately. The standard errors of Ki and K3 are shown in Figure 6-26 and the standard



















33 I-


32 2-


30 -


-12 -10 -8 -6 -4 -2 0

The End Point of Fitting Range

Figure 6-19 The value of coefficient K1 (amplifier B)


-250


-400


-450 L


-12 -10 -8 -6 -4 -2

The End Point of Fitting Range


Figure 6-20 The value of coefficient K3 (amplifier B)


I I I I

U -

'-U






Uj


I I I I I
U



U



U U

U

U



U

U
U
-U


''''`''''"'


'''''''''"'















s.e.(K )
s.e.(K1)













- *

OS


S *
0
5 o

S>
0x


-12 -10 -8 -6 -4 -2 0

The End Point of Fitting Range

Figure 6-21 Standard errors of K1 and K3 (amplifier B)


0.8 L-


-12 -10 -8 -6 -4 -2 0

The End Point of Fitting Range


Figure 6-22 Sum of squares of the residuals (amplifier B)


0.8


0.6


0.4


0.2


25




20 r



0
o
15

0




w
-a
10 -
c
(0


5


. .--*


............