UFDC Home  myUFDC Home  Help 



Full Text  
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, SeonKyung 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 THIRDORDER INTERCEPT POINT ................................................10 3.1 Definition of 1 dB Gain Compression and Thirdorder 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 CommonSource 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 IP1dB Estimation from Twotone 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 Onetone Data................................... .................75 6.4 IP1dB Estimation from the Twotone 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 NMOS AND PMOS TRANSISTOR................................113 B VOLTERRAKERNELS OF A COMMONSOURCE 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 11 The 1 dB gain compression point and IIP3 of various circuits..............................3. 41 The sum m ary of fitting results ....... .......................................... ....................34 42 Frequency effect on the fitting algorithm ............................................................. 39 43 F hitting R results ........................................................................50 51 The summary of the measurement results of commercial amplifiers ....................58 52 The summary of the estimated IIP3 of commercial amplifiers.............................61 53 The measurement data and calculated IIP3 of a commercial amplifier....................63 54 The summary of the application results of the IP1dB estimation algorithm.............67 61 The summary of the measurement results of commercial PAs.............................75 62 The summary of fitting results (amplifier A) ............................... ...... ...............79 63 The summary of fitting results (amplifier A) ............................... ...... ...............83 64 The summary of fitting results of the first model (amplifier B) ..............................83 65 The summary of fitting results according to fitting region (amplifier B) ................86 66 The summary of fitting results of the first model (amplifier C).............................90 67 The summary of fitting results (amplifier C) ............. ........................................96 68 The summary of fitting results of the first model (amplifier D) ............................102 69 The summary of fitting results (Amplifier D)...................................................... 102 610 The comparison between two fitting models ............................... ................102 611 The summary of the application results of the IP1dB estimation algorithm...........104 A1 BSIM 3 model of nM OSFET...................................................... ..................113 A2 BSIM3 model of pMOSFET............ ... .... ......... ............... 114 LIST OF FIGURES Figure p 21 The output current of an ideal class C amplifier for a sinewave input .................6 22 The configuration of a singletone test........ .......................................... 9 23 The configuration of a twotone test. .............................................. ............... 9 31 Definition of 1dB gain compression point.....................................10 32 Intermodulation in a nonlinear system ...... ................. ...............11 33 Definition of thirdorder intercept point..... ................................12 34 The definition of Intercept points in onetone test.............................. ............... 16 41 A schematic of a commonsource amplifier .................................... ............... 25 42 DC characteristic of a commonsource amplifier. ................................................26 43 The AC simulation of a commonsource amplifier.......................................27 44 The results of onetone sim ulation ...................................................... ............... 28 45 The results of a twotone simulation......................... ............................ ............ ..29 46 The difference between amplitudes at two frequencies .......................................29 47 Extracted nonlinear coefficients ......................... .................................... 31 48 Standard errors of nonlinear coefficients ..................................... .................32 49 The sum of squares of the residual................................ ..................... ............... 33 410 Gain curves with 0.1% and 2% random noise ............................... ............... .35 411 Calculated thirdorder intercept point with 0.1 percent added random noise ..........36 412 Calculated thirdorder intercept point with 2.0 percent added random noise ..........37 413 The influence of added random error on the fitting results................................ 37 414 A n equivalent circuit ...................................... ............... .... ....... 41 415 A schematic of a commonsource amplifier with an active load ...........................43 416 DC characteristics of a commonsource amplifier with an active load....................44 417 The AC simulation of a commonsource amplifier with an active load ..................45 418 The results of onetone simulation ..................................................................... 46 419 The results of twotone simulation..................................................................... 46 420 Extracted nonlinear coefficient K ........................................ ....................... 47 421 Extracted nonlinear coefficient K 3 ........................................ ....................... 48 422 Extracted nonlinear coefficient K 5 ........................................ ....................... 49 423 Standard errors of nonlinear coefficients ..................................... .................49 51 ER A 1 am plifier in a test board........................................... .......................... 52 52 The spectrum from the signal source ............................................ ............... 53 53 O netone test scheme e ....................................................................... ..................54 54 The measurement data of onetone test (ERA1 amplifier) ....................................54 55 T w oton e test sch em e .................................................................... .....................55 56 The measurement data of twotone test (ERA1 amplifier) .............................55 57 The measurement data of onetone test (ERA2 amplifier) ....................................56 58 The measurement data of twotone test (ERA2 amplifier) .............. ...............56 59 The measurement data of onetone test (ERA3 amplifier) ....................................57 510 The measurement data of twotone test (ERA3 amplifier) ....................................57 511 Onetone data and extraction from ERA1 device at 100 MHz .............................59 512 A flow chart for estimation of IIP3 from onetone measurement..........................60 513 Onetone data and extraction from ERA2 device at 100 MHz .............................61 514 Onetone data and extraction from ERA3 device at 100 MHz .............................62 515 Onetone data and extraction from ERA2 device at 2.4 GHz..............................63 61 The measurement data of onetone test of the amplifier A.................. .............71 62 The measurement data of twotone test of the amplifier A.................. ............71 63 The measurement data of onetone test of the amplifier B ................. ..............72 64 The measurement data of twotone test of the amplifier B ................. ..............72 65 The measurement data of onetone test of the amplifier C ................. ..............73 66 The measurement data of twotone test of the amplifier C................. ..............73 67 The measurement data of onetone test of the amplifier D................. ...............74 68 The measurement data of twotone test of the amplifier D................. .......... 74 69 The value of coefficient K 1 (amplifier A) ..................................... .................77 610 The value of coefficient K3 (amplifier A) ..................................... .................77 611 Standard errors of K1 and K3 (amplifier A) .......................................................78 612 Sum of squares of the residuals (amplifier A) ......................................................78 613 The value of coefficient K 1 (amplifier A) ..................................... .................80 614 The value of coefficient K 3 (am plifier A ) ........................................ .....................80 615 The value of coefficient K5 (amplifier A) ..................................... .................81 616 Standard errors ofK 1 and K3 (amplifier A)................................... .................81 617 Standard Error of K5 (Amplifier A)................................ .............................. 82 618 Sum of squares of the residuals (Amplifier A) ...................................................82 619 The value of coefficient K 1 (amplifier B) ..................................... .................84 620 The value of coefficient K3 (amplifier B) ..................................... .................84 621 Standard errors of K1 and K3 (amplifier B).........................................................85 622 Sum of squares of the residuals (amplifier B) ...................................................... 85 623 The value of coefficient K1 (Amplifier B) .................................... ............... 87 624 The value of Coefficient K3 (Amplifier B) ................................... .................87 625 The value of Coefficient K5 (Amplifier B) ................................... .................88 626 Standard errors of K1 and K3 (amplifier B)..........................................................88 627 Standard error of K (am plifier B)...................................... ......................... 89 628 Sum of squares of the residuals (amplifier B) ...................................................... 89 629 The value of coefficient K1 (amplifier C) ..................................... .................91 630 The value of coefficient K3 (amplifier C) ..................................... .................91 631 Standard errors of K1 and K3 (amplifier C)..........................................................92 632 Sum of squares of the residuals (amplifier C) ......................................................92 633 The value of coefficient K1 (amplifier C) ..................................... .................93 634 The value of coefficient K3 (amplifier C) ..................................... .................93 635 The value of coefficient K5 (amplifier C) ..................................... .................94 636 Standard errors of K1 and K3 (amplifier C).........................................................94 637 Standard error of K5 (am plifier C) ................................... ............. ........ ....... 95 638 Sum of squares of the residuals (amplifier C) ......................................................95 639 The value of coefficient K1 (amplifier D) ..................................... .................97 640 The value of coefficient K3 (amplifier D) ..................................... .................97 641 Standard errors of K1 and K3 (amplifier D).........................................................98 642 Sum of squares of the residuals (amplifier D) ......................................................98 643 The value of coefficient K1 (amplifier D) ..................................... .................99 644 The value of coefficient K3 (amplifier D) ..................................... .................99 645 The value of coefficient K5 (amplifier D) .................................... ............... 100 646 Standard errors of K1 and K3 (amplifier D)........................................................ 100 647 Standard error of K5 (amplifier D) ................... ...... .............. 101 648 Sum of squares of the residuals (amplifier D)....................................................... 101 71 A schematic of a gilbert cell mixer ............................................................ 108 72 The result of onetone test with different LO powers ................. ................109 73 The analysis of the gain curve at 15 dBm LO power ............... .............. .....109 74 A cascode structure .................. ...................................... .. ........ .. 110 75 Simple embedded system with a mixer..... .......... .................... ..................111 Bl An equivalent circuit of a commonsource 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 thirdorder intercept point ( IP3) by extracting the nonlinear coefficients from the onetone measurement. And the dissertation proposes a method to predict the 1 dB gain compression point (P1dB) from a twotone 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 IP1dB 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 commonsource 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 onetone measurement data was close to the experimently measured values. The method proposed to predict IP1dB from a twotone 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/Mixedsignal production test. The advantage is that these methods avoid the difficulty of twotone measurements or remove onetone measurement test step. By developing the relationship between the 1 dB gain compression point and the thirdorder intercept point, a simpler embedded test model can be adopted avoiding the cost and time of a twotone measurement. CHAPTER 1 INTRODUCTION 1.1 Motivation In the near future, RF microwave circuits will be embedded in highly integrated "SystemsonaChip" (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/mixedsignal portions of a SoC must be verified with highfrequency 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 onchip 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 thirdorder intercept point are important nonlinear parameters of the RF/mixedsignal 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 thirdorder 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 thirdorder nonlinear coefficients of transistor amplifier circuits. This calculation assumes that higherorder nonlinear coefficients do not affect the 1 dB gain compression. The simulation using 0.25 [tm MOSFETs, 0.4 [tm MOSFETs and SiGe BJTs models in Table 11 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 (commonsource amplifier, differential amplifier with resistive load, and commonemitter amplifier except for a differential amplifier with an active load) show that the difference between IIP3 and IP1dB 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 11 The 1 Model TSMC 0.25[ m MOSFET SiGe IBM6HP MOSFET dB gain compression point and IIP3 of various circuits Ci t IPidB* IIP3** Circui(dBm) (dBm) CommonSource Amplifier 2.2 10.0 Differential Amplifier 3.0 16.0 3.0 16.0 (Resistive Load) Differential Amplifier13.0 5.0 (Active Load) CommonSource Amplifier 1.75 10.25 Differential Amplifier 2.65 15.25 (Resistive Load)15.25 (Resistive Load) SiGe IBM6HP CommonSource Amplifier 20.25 6.5 13.75 BJT *Onetone test: Source frequency = 100 MHz **Twotone 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 higherorder nonlinear coefficients. Finally, a simple embedded test using a mixer is considered and suggested to measure IIP3 and IP1dB. Through developing the relationship between IIP3 and IP1dB, the embedded test for IIP3 requires only a onetone 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 thirdorder intercept point is derived. First, this relationship between IIP3 and IP1dB is IIP3 IP1dB (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 onetone measurement and the calculation method to predict IP1dB from a twotone 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 commonsource 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 IP1dB prediction from twotone measurement has been applied to these wideband amplifiers. Through several steps of simple calculation using the thirdorder intercept point and the gain compression at the fundamental frequency, IP1dB 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 IP1dB from a twotone 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 closedform 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 frequencyvarying amplitude or phase characteristics. For example, when a squarewave input is applied to a highpass 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 highfrequency 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 21 shows the output current of an ideal class C 0 20 Figure 21 The output current of an ideal class C amplifier for a sinewave input amplifier. The output current amplitude at a fundamental frequency is in the form of a nonlinear function which shown in equation (21). ind = (20 sin 20) (21) 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+... (22) 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+ (23) (n + 1)! In particular, if f(n+ R n1 (x) < x xx "+ (24) (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) (25) k=1 k! (Dv)" k 1 k = VCONTR Nonlinear coefficients are defined as follows 1 af(v(t)) (26) K = (26) (k! ( Iv=CONTR The expression of the AC current through the conductance is in equation (27). iotn (t) = K Von (t) + K2onr (t) + Kv o(t)+... (27) 2.3 Measurement of Nonlinear System A singletone test is used for the measurement of harmonic distortion, gain compression/expansion, largesignal impedances and rootlocus analysis. The configuration of a singletone test can be seen in Figure 22. 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, crossmodulation and desensitization, the two tone test is used. Figure 23 shows the configuration of a twotone harmonic test. In this test, the power combiner is used for combining two powers at different frequencies. Through this test, a thirdorder intercept point is determined. The next chapter develops the relationship between 1 dB gain compression point and thirdorder intercept point. Taylor series analysis above is essential for performing that analysis. Figure 22 The configuration of a singletone test. ERA is a commercial amplifier. 'Power Combiner Figure 23 The configuration of a twotone test. ERA is a commercial amplifier. CHAPTER 3 THE RELATIONSHIP BETWEEN THE 1 dB GAIN COMPRESSION POINT AND THE THIRDORDER INTERCEPT POINT 3.1 Definition of 1 dB Gain Compression and Thirdorder Intercept Point The constant smallsignal 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 1dB gain compression point [Raz98]. Figure 31 shows the definition of the 1 dB gain compression point. The real gain curve, C is plotted on a loglog 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 31 Definition of 1dB gain compression point co co _o10_ : (91 (92 ,/ (y ^ (2cl c(,) (2co, ci) Figure 32 Intermodulation in a nonlinear system nonlinear transfer characteristics of the circuit. Point A1dB 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 twotone 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 thirdorder intermodulation products at (20), )2) and (202 0,), as illustrated in Figure 32. 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 thirdorder intermodulation of two nearby interferers is so common and so critical that a performance metric has been defined to characterize this behavior. The thirdorder intercept points IIP3 and OIP3 are used for characterizing this effect. These terms are defined at the intersection of two lines shown in Figure 33. The first line has a slope of one on the loglog plot (20 log (amplitude at A 20log(Amplitude at w,) OIP3  20 log(Amplitude at 2w w2) IIP3 20log(Ain) Figure 33 Definition of thirdorder 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 +... (31) where K, is nonlinear coefficients of this system. This example is explained in equation (26) in section 2.2. The classical analysis of the nonlinear system uses the assumption that the fourthorder and higherorder terms in equation (31) 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 (32) If a sinusoidal input with a fundamental frequency is applied to this nonlinear system x(t) = A cos(ot) (33) then the output of this system is represented by using the equation (32), KA2 3KA3 KA2 y(t) =  + KA + 3K3 cos(Ot) + KA2 cos(2ot) 2 4 2 (34) 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 (35) 4 If K3 has the opposite sign of K,, the gain is a decreasing function of the input amplitude. The 1dB gain compression point is defined in Figure 31, the equation at this point is 20 log KA 3K3 AdB3 = 20logK,1AI 1dB (36) 4 where A1, is the input amplitude at the 1 dB gain compression point. The solution of equation (36) is AdB = 0.145 (37) 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 twotone 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) (38) 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 (39) 3KA3 ,, 3K3A3'} ,, ^ + r3A3 2 cos((2w, O2 )t)+ 3K3A 2 cos((2co2 1 )t)+' 4 4 The thirdorder intercept point is defined in Figure 33. 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 (310) where AIP3 is the input amplitude at the thirdorder intercept point. The solution of above equation (310) is AI3 4K (311) 3K3 The input signal level at the thirdorder intercept point is also found using two nonlinear coefficients K, and K3. From equation (37) and equation (310), the relationship between the input signal levels at the 1 dB gain compression point and the thirdorder intercept point can be derived, .4 9.6dB (312) A_1 3 V4/3 The classical analysis shows that the relationship between two nonlinear characteristics is represented by equation (312). 3.3 New Approach to Model Gain Compression Curve For simplicity, the analysis is limited to memoryless, timeinvariant nonlinear systems. Prior classical analysis limits the output to the thirdorder 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 (313) If a sinusoidal input such as equation (32) is applied to this system, then the output amplitudes at each oddfrequency are as follows, y(t;))= KA+K3A3 +5 KA cos(ot) (314) 4 16 y(t;5w) = K5A 5cos(5wt) (316) 16 If a low input signal level is considered in equation (314), the following condition is satisfied, KA >> 3 KA3 +5 +KA5 (317) 4 8 then the output amplitude at the fundamental frequency is KA. From equation (315), the output amplitude at frequency 3o is K A3 if, 4 IPl S/ 35 B/ C Input Power Figure 34 The definition of Intercept points in onetone test. A= 20 log (KIA), B=20 log (K3A3/4) and C=20 log (KsA5/16) K3A3 >> 5KA (318) 4 16 As stated in the previous section, it is possible to define the intercept points shown in Figure 34 in onetone test. In Figure 34, Line A represents the output amplitude at the fundamental frequency and has a slope of one in the loglog scale graph. This line is extrapolated from linear smallsignal area in equation (314) from equation (317). Line B is the output amplitude at triple fundamental frequency and has a slope of three. This line is also extrapolated from equation (315) under the condition of equation (318). Line C with a slope of five is the output amplitude at frequency 5o and is drawn from equation (316). 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 (319) where the value of Line A is the same as that of Line B at the input level A13. From this equation (319), the input signal level can be represented by using two nonlinear coefficients K1 and K3 such as, 2 A13 = 2 (320) 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 (321) 16 K4 A5 = 2K (322) Ks5 The signal level A35, which is the input value at the intercept point IP35, can be described by, K3A35 = K5A354 (323) 4 16 K2 A35 = 32 (324) K5 The relationship between the input signal levels at the intercept points can be derived from the equation (320), (322) and (324), A3A35 = A52 (325) The equation (325) can be expressed differently by using logarithm, 20 log(A13) + 20 log(A35) = 2 20 log( A1) (326) In addition, this relationship can be designated by, IP13 + IP3 = 2 x IP1, (327) where IP, is the inputreferred 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 (328) 4 8 From the above equation, one can see that fifth order nonlinear coefficient makes the gain compression change from equation (36) in the previous section. Generally, gain compression arises when K3 has the opposite sign of K1. From equation (328), 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 (329) 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 (330) A 15 ) A 13 If Ks is positive then, 1 \iM 3 Ad 0.109 = 0 (331) A 15 A3 1 Therefore, if A13, A15 and A35 nonlinear coefficients are known, it is possible to evaluate AldB * Equation (38) takes the form of the input signal in a twotone test. If this input is applied to the system described by equation (313), the output has a more complicated form than that in the onetone test. The output at fundamental frequency is described by, y(t;0,)= KA+9 K3A3 +2 KA cos(C0t) (332) where 0, represents the fundamental frequency which is one of two frequencies 0, and 92,. The amplitude of the output, y(t; ,), ) in equation (332) is different from that of the output y(t; o) in the equation (314) of the onetone 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 +25KA cos((20, 2)t) (333) The thirdorder intercept point is defined in the smallsignal area of a gain plot. The input signal level at the thirdorder intercept point is the same as that in the classical analysis. Equation (311) 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 onetone test is a function of the nonlinear coefficients represented by equation (314). 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 onetone test, it is possible to estimate IIP3 without the twotone test. In the results of the twotone test, the gain curve at each frequencies oi and o2i is a function of the nonlinear coefficients described by equation (332). Even though the equation (332) 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 onetone test by using the nonlinear coefficients extracted from twotone 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 (334) where Y, X, b and E are nx 1l,nx m,mxl and nx 1 matrices respectively. In this equation (334), 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 *rYXeb) (335) To find the sum of square of errors the least quantity, b can be found as follows, b = *( X *(X *Y) (336) After finding b using above equation (336), the variance of b is, V (b)= (T X *a2 (337) where the diagonal terms of the above variance represent the variance of the parameter b, and the offdiagonal 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 (335). 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 nm. The mean square of the residual is used as the estimate of the variance of errors, a2. The mean square of the residual is, I\ 1 b XY MSE = 2 = YX ) X/ ) 1 X( Y (338) nm nm In this research, the fitting model for the gain curve is the oddorder polynomial equation, y = K x, +K3 x, +K, K x, (339) where y, is the estimate value of the fitting model. The residual is, E, = y, y, (340) The matrices in the equation (334) can be formed as follows, y1 Y= y2 (341) 3 5 x 1 x 1 x X= x x2 x25 (342) K, b= K3 (343) K5 El E= E2 (344) 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 n3. Using the degree of freedom of the residual, the variance of the errors can be calculated in equation (337). 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 (337). The sum of square of the residual also can be calculated using equation (335). 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 thirdorder intercept point has been derived. First, this relationship between IIP3 and IP1dB was reviewed in classical analysis. The difference between two nonlinear characteristics was 9.6 dB and constant. The classical analysis included only thirdorder nonlinear coefficients. The new relationship was derived by expanding nonlinear analysis on the gain compression curve up to the fifthorder nonlinear coefficients. The difference between IP1dB and IIP3 is not fixed and is explained by the equation including nonlinear coefficients. The fitting algorithm to estimate IIP3 from onetone measurement and the calculation method to predict IP1dB from twotone 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 CommonSource 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 onetone 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 commonsource amplifier with TSMC 0.25 [m nMOSFET is considered. Figure 41 shows the schematic for a commonsource 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 41 A schematic of a commonsource amplifier analysis of the DC characteristic of this amplifier using the Agilent ADS2002 software. The IV characteristic curve and transfer characteristic curve are shown in Figure 42. For a weakly nonlinear simulation, a gate bias is chosen near the center point that is shown as point P in both Figure 42.A and Figure 42.B. At the gate bias, 1.1 V, an AC simulation was performed. Figure 43 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 nMOSFET 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 42 DC characteristic of a commonsource 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 43 The AC simulation of a commonsource amplifier At the same gate bias as the AC simulation, a onetone test and a twotone test were simulated using the harmonicbalance simulator in the ADS2002 software. Figure 44 and Figure 45 show the simulation results of onetone and twotone 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 44, 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 41 of the commonsource 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 44 The results of onetone simulation compression of 1 dB, the input voltage is 0.52 V in Figure 44. The applied frequencies in the twotone test are 100 MHz and 110 MHz. In Figure 45, 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 thirdorderintercept point in a 50 Q system such as an RF system. In twotone test, the amplitudes at two 100 120 140 i 0.01 0.1 1 Vin(V) Figure 45 The results of a twotone 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 46 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 46. 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 thirdorder intercept point is defined in the smallsignal area explained in Chapter 3, the thirdorder 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 leastsquare polynomial fitting of the gain compression curve A of Figure 44. The polynomial model used in this fitting is as the power series that follows, 3 5 y = K,*x+K x3 +K5x5 (41) 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 lowerorder 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 47 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 5thorder harmonic or the highestorder 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 (41). 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 47. 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 47, KI at the point that xaxis is 0.3 V represents the value of the firstorder nonlinear coefficient extracted from this first fitting region. In addition, K3 and K5 indicate the thirdorder and the fifthorder nonlinear coefficients correspondingly. A horizontal axis defines the width of the fitting range in Figure 47 since the starting point of the fitting range is fixed at 0.01 V to include the small input amplitudes. The xaxis 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 47 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 48 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 (42) 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 (328). Figure 48 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 48. 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 (43), T  TT  SS(E,)=E E=YYYb X Y (43) where the calculation is the same as equation (326). 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 49 The sum of squares of the residual caused by the difference between the fitting model and the actual model. Figure 49 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 48. From this graph, it is shown that the width of the fitting range from 0.01 V to point A of Figure 48 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 49. The fitting outcome for this set of prospective ranges is summarized in Table 41. At the thirdorderintercept 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 Volterraseries parameters. In spite of these errors, 1 dB or less difference for the thirdorderintercept point simulation and measurement is a good result. Table 41 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 410 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 onetone data point used in the previous section. z, = y, + y, x (P%)x n, (44) .......... 2 ,4         i   "_ 2.3 2.2 0 200 400 600 800 1000 Samples Figure 411 Calculated thirdorder 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 thirdorder intercept points over one thousand samples are shown in Figure 411. The average value over one thousand o U I 2.1 0) 0 200 400 600 800 1000 Samples Figure 411 Calculated thirdorder 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 thirdorder intercept points over one thousand samples are shown in Figure 411. The average value over one thousand 4 I o "c 0 C 0 I0 200 400 600 800 1000 Samples Figure 412 Calculated thirdorder intercept point with 2.0 percent added random noise Added Random Error(%) Figure 413 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 412 shows the calculated thirdorder intercept point with 2 percent 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 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 413. 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 thirdorder 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 42. According to Table 42, the difference between the extracted thirdorder 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 Volterraseries components. Appendix B shows Volterrakemels of a commonsource amplifier. When the input is V, = V cos(iclt) (45) 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 (46) Table 42 Frequency effect on the fitting algorithm Estimated Difference Frequency VidB(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 Volterraseries 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 ( (47) 5 K,V5 85G K5 WC exp(joOt) +...} 8 GL + jo,CL From equation (47), each coefficient has same phase if Cgd is disregarded and C, is a linear parasitic capacitor. At low frequency, Volterraseries coefficients are similar to Powerseries coefficients. The output amplitude at the fundamental frequency is in equation (48). 3 5 Vou, (o) = KI'*V + K3'.V3 +5 KsV'.5 (48) 4 8 where K'= K, (GL2 +1 2CL 2)2 At high frequency, Cgd should be regarded in the calculation of Volterrakernels. If Cgd is included in Volterrakernels, the output amplitude is (K1 jO1Cgd)V 3 K3V3 Vo, (t; ,) = Re{ exp(jai t) exp(joit) GL + jo),CL 4 G, + j1),C, (49) 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 onetone test is modeled as a Volterra series. According to Table 42, 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 414. 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 inputvoltagecontrolledcurrent source. The passive components used in the passive load Vi V2 Vin S Ziv ZL Figure 414 An equivalent circuit are assumed to be linear components to simplify calculation. Under a given bias condition, the voltagecontrolledcurrent source can be described by Taylor series at the quiescent point in equation (26) in chapter 2. i= Kv+K2V2 + K3V3 +... (410) The passive load is a function of s in the sdomain, ZL =ZL,(s) (411) Volterrakernel at the output node in the equivalent circuit can be described by H, (sI,S2,...S,)= KZ, ( s") (412) n=1 The nonlinear characteristic of an amplifier is determined by the above Volterrakemels. The output amplitude at a fundamental frequency in onetone test of this circuit is found by combining Volterrakernels. If the input is V= V cos(iot) (413) then the output at fundamental frequency o, is Vo,, (t; oi,) = Re{VH, (oi,) exp(jColt) + 3V3H3 (01, I1 1,ao) exp(jao),t) 5 (414) +5 V'H5 (0, C91, 1,m,,m0,)exp(jot) +...} 8 The load impedance used in oddorder Volterrakemels is s,=s= jco, (415) n=1 ZL ( s)= ZL (jC)1 (416) n=1 Therefore, the oddorder Volterrakernel at the fundamental frequency is H, = K, ZL(j) (417) The passive load is used to the change of each oddorder Volterrakernel. 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 415 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 IM2 V SVout 'iref  M R wth M1 Sin Figure 415 A schematic of a commonsource amplifier with an active load 44 180.0 ii i 160.0 140.0 120.0 100.0  80.0 II 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 416 DC Characteristics of a commonsource 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 417 The AC simulation of a commonsource amplifier with an active load bias voltage of the current mirror. The IV characteristic curve and transfer characteristic curve are shown in Figure 416. In Figure 416.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 416.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, onetone and twotone 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 417. The 3dB bandwidth of this amplifier is about 1 GHz. For applying the fitting algorithm to the commonsource amplifier with active load, onetone and twotone tests are simulated in this schematic at the gate bias voltage VG. The result of the onetone test 0.01 0.1 1 Vin (V) Figure 418 The results of onetone simulation 60 40 20 0 20 40 60 80 100 120 140 0.01 0.1 Vin (V) Figure 419 The results of twotone simulation is shown in Figure 418. 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 inputreferred 1 dB compression point is 0.08V. Figure 419 shows the result of twotone simulation. The applied frequencies in the twotone 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 41. 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 420 Extracted nonlinear coefficient Ki 48 with an active load. Figure 420 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 41 since the value of Ki in this figure is greater than that of Ki in Figure 47. The value of K3 is shown in Figure 421. The negative sign of K3 causes the gain compression on 100 120 140 160 180 200 220 240 260 280 300 0.00 Figure 421 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 422. 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 423 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 422 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 423 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 yaxis and curve K5 is represented by the right yaxis. The lowest values for the standard errors of each coefficients are shown by Point A in Figure 423. 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 commonsource amplifier discussed in Section 41. Figure 424 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 43. 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 thirdorder intercept point is close to the simulated thirdorder 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 43 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 commonsource 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 (ERAseries, MiniCircuit, Co.) shown in Figure 51 were used for the measurement of onetone and twotone 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 frequencydomain. A low pass filter is used to decrease the harmonics in the signal generator. Figure 52 shows the spectrum in the signal source with and without a low pass filter. In Figure 52.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 51 ERA1 amplifier in a test board 53 10 20 A 30 40 0 ( 0 50 v' 60 S70 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 SVo 60 100 18 16 14 12 10 8 6 4 2 0 2 Input Power (dBm) B Figure 52 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 onetone test. The test scheme is shown in Figure 53. The fundamental frequency used for this test is 100 MHz. Figure 54 shows the result of the onetone measurement. In this graph, the input referred 1 dB compression point, IP1dB 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 53 Onetone test scheme 20  25 20 15 10 5 0 5 10 Input Power (d Bm) Figure 54 The measurement data of onetone test (ERA1 amplifier) explained in next section. Figure 55 shows the test scheme for the twotone test. Source frequencies in this twotone test are 100 MHz and 120 MHz. The result of the twotone test is shown in Figure 56. 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 thirdorder intercept point. The inputreferred thirdorder 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 twotone test is explained in section 3.3. Figure 55 Twotone 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 twotone test (ERA1 amplifier) Figure 56 /I 56 ERA2 and ERA3 amplifiers are also tested in the same bias condition as ERAl amplifier to verify the proposed algorithm to predict thirdorder intercept point using extraction of nonlinear coefficients from the gain compression curve. The onetone test data of ERA2 is shown in Figure 57. Inputreferred 1 dB gain compression point denoted by GC is 0.8 dBm. Figure 58 shows twotone data of ERA2. Like Figure 56, 30 25 B 20  0 10 C 0 GC 5 20 15 10 5 0 5 10 Input Power (dBm) Figure 57 The measurement data of onetone 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 58 The measurement data of twotone test (ERA2 amplifier) 35 30 25 20 15 10 5 0 5 Input Power (dBm) Figure 59 The measurement data of onetone 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 510 The measurement data of twotone test (ERA3 amplifier) BG GC \ GC Table 51 The summary of the measurement results of commercial amplifiers Device IP1dB* IP1dB,2** IIP3** IIP3IP1dB ERA1 1.7 2.5 16.3 14.6 ERA2 0.8 5 13.8 14.6 ERA3 8 13 4.5 12.5 *Onetone test : Source frequency = 100 MHz **Twotone test : frequencies = 100 MHz, 120 MHz Curve A and Curve B represent the output power at 100 MHz and at 80 MHz separately. The inputreferred thirdorder intercept point is 13.8 dBm. The inputreferred 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 59. The result of the twotone test is shown in Figure 510. The 1 dB compression point is 13 dBm and the inputreferred thirdorder intercept point is 4.5 dBm in Figure 510. The measurement data of these amplifiers are summarized in Table 51. In this table, the difference between 1 dB gain compression point and thirdorder 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 highpower 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 highpower 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 511 Onetone data and extraction from ERAl device at 100 MHz A new robust measurement extraction algorithm is developed for onetone gain data as graphed in Figure 511 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 512. First, the entire onetone Figure 512 A flow chart for estimation of IIP3 from onetone measurement power compression curve is measured as shown by curve B in Figure 511. Line B is a straightline 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 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 511. 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 513 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 513 Onetone data and extraction from ERA2 device at 100 MHz Table 52 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 514 Onetone data and extraction from ERA3 device at 100 MHz as that of Figure 511. The same analysis is applied to ERA3 amplifier in Figure 514. Table 52 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 onetone 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 515 shows the new robust extraction algorithm at high frequency. The result in Table 53 shows that this algorithm is working in microwave frequencies. Through these experiments a method for predicting IIP3 using a onetone LNA gain measurement was developed. Table 53 The measurement data and calculated IIP3 of a commercial amplifier Device IP1dB* IIP3** IIP3*** ERA2 0.4 12.1 12.08 *Onetone test: Source frequency = 2.4 GHz **Twotone 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 515 Onetone data and extraction from ERA2 device at 2.4 GHz 5.4 IP1dB Estimation from Twotone Data The algorithm in section 5.2 estimates IIP3 from the onetone 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 AidB. From equation (324), The 1 dB compression point in the onetone test is found in equation (324). "K5Ad 4+ K3 ,12dB +0.109=0 (51) 8 K, 4 K l Two ratios, KK5 ~and K31 are needed to solve above equation (51). These ratios are found in the twotone measurement data. From equation (39), the ratio K3 is determined using following equation, K3 4 1 K=x4 (52) KI 3 AI3 2 where AIP3 is the input amplitude at the thirdorder 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 56 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 (53), 25 K5 4 9 K3 4 K i,2 4 K Al d&,2 +0.109=0 (53) where A1 d,2 is the input amplitude at the 1 dB compression point of the twotone test that is denoted by GC in Figure 56. From above equation, the ratio 5 can be found K can be found by = 4 Al 4 j AldB, 2+0.109 (54) _d'I1rB,2 9 4 3 211 1 0 109 (5 Inserting two ratios L5 and K to equation (51), the inputreferred 1 dB gain compression point, IP1dB 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 51. The input amplitude at this point is A,3 10 310) 20 = 2.065 (55) 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 (52). K1 3K 1 4 3= x 2.065 = 0.3146 (56) K, 3 K The sign of above ratio 3 is negative since the gain at the fundamental frequency K1 compresses in Figure 56. The ratio K3 is K = 0.3146 (57) Ki ) From table 51, the IP1dB,2 is 2.5 dBm. The input amplitude at this 1 dB compression point in twotone is A1dB, = 10 (p ,')20 = 0.2371 (58) K K Using the value of A, d2 and the ratio the ratio a is found in equation (54) K K, The calculated value of K5 is 3.5140. The 1 dB compression point in the onetone lKi ) test is now found by solving equation (51). The solution of this equation is 0.4192. The calculated IP1 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 thirdorder intercept point is Ai3 =1.5488 (59) since IIP3 of ERA2 amplifier is 13.8 dBm in Table 51. 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 58. The calculated value of K3 is 0.5588. From IP dB,2 lKi in Table 51, 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 K5 is 11.1224. Finally, the calculated value of Al d from equation (51) is 0.5143. The estimated value of the 1 dB compression point in onetone 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 twotone data. In this amplifier, the input amplitude at third order intercept point is A,3 = 0.5309 (510) since IIP3 of ERA3 amplifier is 4.5 dBm in Table 51. The absolute value of the ratio K3 is calculated by using equation (52). The sign of this ratio need to be negative Ki1 since the gain curve compresses in Figure 510. K3 =4.7308 (511) From table 51, the IP1dB,2 is 13 dBm. The input amplitude at this 1 dB compression point in twotone data is AldB,2 = 0.0708 (512) Using the value of A_ ,2 and the ratio K3, the calculated value of the ratio 5 is K= 354.5 (513) From equation (51), the calculated value of A1 d is 0.1248. The estimated value of the 1 dB compression point in onetone test is 8.08 dBm and is very close to the measured value, 8 dBm. Table 54 summarize the results of these applications of three amplifiers. Table 54 The summary of the application results of the IP1dB estimation algorithm Device Measured IP1dB Estimated IIP3 ERA1 1.7 2.45 ERA2 0.8 0.05 ERA3 8 8.08 In this table, the difference between the measured IP1dB and the estimated IP1dB 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 IP1dB using a twotone 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 highpower 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 IP1dB prediction from twotone measurement has been applied to these wideband amplifiers. Through several steps of simple calculation using the thirdorder intercept point and the gain compression at the fundamental frequency, IP1dB 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/4QPSK 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 twotone 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 onetone and twotone tests. To distinguish these amplifiers, amplifiers are named alphabetically. First, the result of onetone test on the amplifier A is shown in Figure 61. 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 inputreferred 1 dB gain compression point, IP1dB is  11 dBm. Figure 62 shows the result of a twotone 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 thirdorder intercept point. The inputreferred thirdorder intercept point, IIP3 is 4.4 dBm. GC2 denotes 1 dB gain compression point in twotone test which has a value of16 dBm. Figure 63 and Figure 64 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 63, the inputreferred 1 dB gain compression point is 8 dBm. IIP3 is 0 dBm and IPidB,2, denoted by GC2, is 13 dBm. The test results of the amplifier C are shown in Figure 65 and Figure 66. The same test condition as that of the amplifier A is applied to this test of the amplifier C. IP1dB, denoted by GC, is 6 dBm from the result of onetone test in Figure 65. IP1dB,2 is 10 dBm and IIP3 is 3 dBm in Figure 66. Figure 67 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 61 The measurement data of onetone 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 62 The measurement data of twotone 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 63 The measurement data of onetone 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 64 The measurement data of twotone 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 65 The measurement data of onetone 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 66 The measurement data of twotone 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 67 The measurement data of onetone 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 68 The measurement data of twotone 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 61 The summary of the measurement results of commercial PAs Device Name IP1dB* IP1dB,2** IIP3** IIP3IP1dB 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 *Onetone test: Source frequency = 2.45 GHz **Twotone test : frequencies = 2.45 GHz, 2.46 GHz *** Amplifier D is tested at 5.2 GHz (and 5.21 GHz). of the onetone test of the amplifier D. The input frequency in this onetone test is 5.2 GHz. IP1dB is 8 dBm in this figure. The applied two frequencies in twotone test are 5.2 GHz and 5.21 GHz. IIP3 is 1 dBm and IP1dB,2 is 12.5 dBm. The measurement data of these amplifiers are summarized in Table 61. In this table, the difference between 1 dB gain compression point and thirdorder intercept point is not constant. 6.3 IIP3 Estimation from the Onetone 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 onetone 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 higherorder 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. Thirdorder 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 (61) 4 Equation (61) 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 +5Kx5 (62) 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 fifthorder nonlinear coefficient is investigated. The first fitting model, equation (61) is applied to the onetone data of the amplifier A shown in Figure 61. Figure 69 shows the value of coefficient K1 from the fitting results. In this graph, the xaxis represents the end point of fitting range from the starting point, 16 dBm. When the fitting range changes, the fitted result changes also. Figure 610 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 (42). Figure 611 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 mm 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 610 The value of coefficient K3 (amplifier A) Figure 69 200 300 400 500 600 700 800 900 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 611 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 612. Sum of squares of the residuals (Amplifier A) IIIIIII _ Table 62 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 612. The fitting results are summarized in Table 62. 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 61. The second fitting model, equation (62) is applied the same data of the amplifier A. Figure 613, 614 and 615 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 616 and 617, 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 618. The result of the fitting application is summarized in Table 63. 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 63 is used for the IIP3 prediction from one tone measurement. Using fitting model, equation (61), the values of K and K3 are shown in Figure 619 and Figure 620. Figure 621 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 614 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 613 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 615 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 616 Standard errors of Ki and K3 (amplifier A) 80  0 "o 60 50 LU 40 O 20 20 10000 8000 6000 4000 2000 Figure 617 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 618 Sum of squares of the residuals (amplifier A) A I I I I I I I I I I I I II ~ 0 Table 63 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 64 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 622, 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 64. 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 onetone gain curve in Figure 63 are drawn in Figure 623, 624 and 625 separately. The standard errors of Ki and K3 are shown in Figure 626 and the standard 33 I 32 2 30  12 10 8 6 4 2 0 The End Point of Fitting Range Figure 619 The value of coefficient K1 (amplifier B) 250 400 450 L 12 10 8 6 4 2 The End Point of Fitting Range Figure 620 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 621 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 622 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 . .* ............ 