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## Material Information- Title:
- The Synthesis of minimum phase transfer functions by zero sharing
- Creator:
- Walter, William Austin, 1937-
- Publication Date:
- 1964
- Copyright Date:
- 1964
- Language:
- English
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- x, 142 leaves. : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Circles ( jstor )
Degrees of polynomials ( jstor ) Electrical impedance ( jstor ) Mathematical optima ( jstor ) Polynomials ( jstor ) RC circuits ( jstor ) Realizability ( jstor ) RLC circuits ( jstor ) Transfer functions ( jstor ) Zero ( jstor ) Dissertations, Academic -- Electrical Engineering -- UF Electric networks ( lcsh ) Electrical Engineering thesis Ph. D - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis -- University of Florida.
- Bibliography:
- Bibliography: leaves 140-141.
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- Also available on World Wide Web
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- Manuscript copy.
- General Note:
- Vita.
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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13977417 ( OCLC ) ADA5078 ( NOTIS )
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THE SYNTHESIS OF MINIMUM PHASE TRANSFER FUNCTIONS BY ZERO SHARING By WILLIAM AUSTIN WALTER A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA December, 1964 ACKNOWLEDGMENTS The author would like to express his gratitude to the members of his supervisory committee and to Dr. W. H. Chen for his supervision and guidance. The author also wishes to thank the College of Engineering for granting the funds which made it possible for him to study for the degree of Doctor of Philosophy. TABLE OF CONTENTS ACKNOWLEDGMENTS . . . . . . . . . LIST OF FIGURES . . . . . . . . . ABSTRACT . . . . . . . . . . Chapter I. INTRODUCTION . . . . . . . . 1.1 The Basic Zero Sharing Procedure . . II. CONDITIONS OF PHYSICAL REALIZABILITY . . 2.1 Conditions of Physical Realizability of the Transfer Functions, T(s), With Ladder Networks . . ..... 2.2 Conditions of Physical Realizability For the Parameters z21 and z22 or -y21 and y22 With Ladder Network . . III. TRANSFER FUNCTIONS WHICH ARE POSITIVE REAL . . . . . . . . . . IV. TRANSFER FUNCTIONS WITH REAL NEGATIVE TRANSMISSION ZEROS . . . . . . 4.1 RC or RL Transfer Functions Which Are Positive Real . . . . . 4.2 A Special Class of RC and RL Transfer Functions . . . . . 4.3 Synthesis of T1(s) With RC Networks . . . . . . . . 4.4 Synthesis of the Network N' as an RC Ladder Network . . . . . 4.5 Illustrations of the Synthesis of ZT(s), -A(s), or -A*(s) With RC Networks . . . . . . . . 4.6 Synthesis of T2(s) With RC Networks . . . . . . . . Page ii v viii . 1 S6 S. 10 . 11 S. 12 S. 14 S. 18 S. 18 S. 20 S. 21 S. 29 S. 36 S- 45 TABLE OF CONTENTS--Continued Chapter Page 4.7 Synthesis of the Network N" . . ... 53 V. TRANSFER FUNCTIONS WITH PURELY IMAGINARY TRANSMISSION ZEROS . . . .. 54 5.1 Limitations of the Zero Sharing Technique in the Synthesis of LC Networks . . . . .... .. 54 5.2 Synthesis of A*(s) or G*(s) With LC Networks . . . . . . . .. 59 VI. TRANSFER FUNCTIONS WITH COMPLEX CONJUGATE AND NEGATIVE REAL TRANSMISSION ZEROS .... 71 6.1 Transfer Functions Whose Denominator Polynomial Is of Second Degree . . .. 72 6.2 The Derivation of Regions In Which A Positive Real Function, F(s) May Have Poles ............ . 85 6.3 Transfer Functions Whose Denominator Polynomial Is of Third Degree . . .. 95 6.4 Synthesis of the Networks N' Or N" As RC, RL, or RLC Ladders . . . .. 109 6.5 Illustrations of the Synthesis of RLC Networks . . . . . . ... 123 VII. CONCLUSIONS . . . . . . . . .. 136 LIST OF REFERENCES . . . . . . . .. 140 BIOGRAPHICAL SKETCH . . . . . . . ... 142 LIST OF FIGURES Figure 1.1. A general four-terminal network . . . 1.2. Network arrangement for a prescribed transfer function of the set Tl(s) . . 1.3. Network configuration for prescribed transfer function of the set T2(s) . . 4.1. L network configuration . . . . . 4.2. Standard ladder configuration for prescribed z21 and z22 . . . . . 4.3. Generalized basic sections (a) A type 1 section (b) A type 2 section . . . 4.4. Basic RC sections (a) Type 1 RC section (b) Type 2 RC section . . . . . 4.5. Ladder network for ZT(s) as given in (4-29) . . . . . . . 4.6. Network prediction for N' . . . . 4.7. Ladder network for A(s) as given in (4-38) with K/H 3 . . . . . 4.8. Ladder network for A(s) as given in (4-38) with K/H 15/7 . . . . 4.9. L network configuration . . . . . 6.1. The construction of a region in which poles of F(s) may be selected . . . 6.2. A region in which poles of F(s) may be selected, for F(s) as given in (6-29) . 6.3. A typical C path with arrows indicating its negative direction . . Page . 1 8 8 S 27 S. 31 S. 31 S. 32 S. 38 S 42 S. 43 S. 46 S. 51 . 82 LIST OF FIGURES--Continued Figure Page 6.4. A group of C paths on which poles of F(s) may be placed . . . . ... 93 6.5. Regions of the s plane in which condition (6-71) is satisfied . . ... 102 6.6. Regions of the s plane in which condition (6-71) is satisfied . . ... 103 6.7. Generalized basic sections (a) A type 3 section (b) A type 4 section . . ... 110 6.8. Type 1 RLC section . . . . . ... 111 6.9. Type 2 RLC section . . . . . ... 114 6.10. Type 3 basic section with a shunt resistance R(h) . . . . . . . 117 6.11. Type 4 basic section with a series resistance R(h) . . . . . . . 121 b 6.12. First network realization for Y (s) as given in (6-133) . . . . . 124 6.13. Regions in which the poles of -y21 may be chosen, for YT(s) as given in (6-133) . . 125 6.14. Second network realization for YT(s) as given in (6-133) . . . . . . . 126 6.15. Permissible pole locations of z21 for ZT (s) as given in (6-139) . . . . .. 127 6.16. First network realization for ZT(s) as given in (6-139) . . . . . .. 128 6.17. Second network realization for ZT(s) as given in (6-139) . . . . . .. 129 LIST OF FIGURES--Continued Figure Page 6.18. Permissible pole locations of z21 for A(s) as given in (6-149) . . . .. 130 6.19. Network realization for A(s) as given in (6-149) . . . . ... 132 6.20. Network realization for A(s) as given in (6-157) . . . . ... 135 Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE SYNTHESIS OF MINIMUM PHASE TRANSFER FUNCTIONS BY ZERO SHARING By William Austin Walter December, 1964 Chairman: Dr. W. H. Chen Major Department: Electrical Engineering A new philosophy is presented for synthesizing ladder networks from a prescribed minimum phase transfer function, T(s). Synthesis procedures are developed in which a sharing of transmission zeros is effected between the net- work parameters, z21 and z22 (or -y21 and y22), with the result that the synthesis problem is divided into two parts, a two-terminal realization and a four-terminal realization. This division usually reduces the labor involved in the synthesis and often requires a smaller number of elements in the final network than are required by other methods. A wide range of problems exists in which the zero sharing technique may be advantageously applied. Methods are described and illustrated for the synthesis of RC, RL, viii LC, and RLC networks. For the synthesis of two-element- kind networks, from transfer functions whose transmission zeros lie on the negative portion of the a axis or on the ju axis, the zero sharing approach is straightforward, regardless of the complexity of the transfer function, i.e., regardless of the degree of the numerator or denom- inator polynomial. In the synthesis of transfer functions which have complex as well as other types of transmission zeros, the zero sharing approach is applicable to those transfer functions whose denominator polynomial is of rather low degree in the complex frequency variable s. A central portion of the zero sharing approach is concerned with the choice of the parameters z21 and z22 (or -y21 and y22) from a prescribed transfer function, T(s) = P(s)/Q(s), of the response/excitation type. It is shown that this selection is equivalent to removing certain of the poles of 1/T(s) = Q(s)/P(s) and replacing them with new poles in such a way that the modified function, Q(s)/P'(s), has the following three properties: 1) Q(s)/P'(s) is positive real. 2) Q(s)/P'(s) may be broken into a sum of two parts, each of which is positive real. One part must have for its poles, all of the poles which were added in forming Q(s)/P'(s). This part must not have any of the poles of the original function, Q(s)/P(s). The second part must have as its poles only the poles of the original function, Q(s)/P(s). 3) The poles which are selected for the modified function must not coincide with the zeros of Q(s). The construction of the modified function Q(s)/P'(s) is easily accomplished by inspection for those transfer functions, whose transmission zeros lie on the negative portion of the a axis or on the jw axis, and whose realiz- ation is to result in a two-element-kind network. For those transfer functions which have complex as well as other types of transmission zeros, the selection of poles for the modified function, Q(s)/P'(s), is restricted to certain regions of the s plane. The location and approx- imate shape of these regions are determined by a procedure similar to that used in the design of control systems by the root locus method. CHAPTER I INTRODUCTION Many practical problems encountered in the design of electrical networks are concerned with the transfer of a signal from one pair of network terminals to a second pair. In the four-terminal network shown in Fig. 1.1, terminals 1-1' are considered as the input, and transmission occurs from this terminal pair to the output terminals 2-2'. 1 Il 12 2 E1 t2R2 1' 2' Figure 1.1 A general four-terminal network For a given load the transfer properties of the network may be expressed in terms of a ratio of input voltage or cur- rent, E or I to output voltage or current, E or I . Expressed in the frequency domain, this description has the form of a ratio of two polynomials in the complex frequency variable, s o + ju, and is termed a transfer function. The synthesis problem considered here is one of finding a ladder network which possesses the properties that are spec- ified in terms of such a transfer function. The resulting 1 network is to consist of lumped, linear, bilateral, and pas- sive elements, and is to have no mutual inductances. Reali- zation procedures will be developed for networks consisting of RC, RL, LC, and RLC elements. The network specifications will be given in terms of one of six transfer functions of the response/excitation type. For a current input at terminals 1-1' of Fig. 1.1, three of these transfer functions are defined as (1-1) E2 12 I2 Z (s) = A(s) = and A*(s) - T 1l 1 1 R2/oo R2O R2=O T (s) will be used when referring to the set, Z (s), -A(s), or -A*(s). For a voltage input at terminals 1-1' of Fig. 1.1, three transfer functions are defined as (1-2) 12 E2 E2 Y(s) = G(s) - and G*(s) = - SE E E R2O R200 R2-00 T2(s) will be used to denote -YT(s), G(s), or G*(s). For convenience T(s) will be used when referring to the entire group of transfer functions, members of either T1(s) or T2(s). From (1-1) it is seen that 12 (E2/R2) A(s) -ZT(s)/R2 (1-3) I1 Ii2 If A(s) is specified for a network which is terminated in a finite non-zero load, an equivalent Z (s) may be easily T found from (1-3). It is sufficient, therefore, to consider only the synthesis of Z (s). It will be found advantageous, however, to consider the synthesis of the limiting case, A*(s), individually. Similarly, it is necessary to consider only YT(s) and the special case, G*(s). The transfer specifications in terms of T(s) will be a ratio of two polynomials in s having the form P(s) sn + an-lsn- +---+ als + a0 (1-4) T(s) = H = H Q(s) sm + b +--+ bs + b m-1 1 0 Assuming that P(s) and Q(s) have no common factors, the finite zeros of T(s) occur at the zeros of the numerator polynomial, P(s), and additional zeros may occur at infinity for m > n. The zeros of T(s) are termed "transmission zeros," since at these frequencies no transmission occurs through the network. Due to the assumed configuration of the resulting net- work, only those transfer functions, T(s), whose finite transmission zeros do not occur in the right-hand half of the s plane will be considered. Such transfer functions are commonly referred to as minimum phase functions. T(s) must possess a number of additional properties in order to be realizable in a ladder network made up of lumped, linear elements. Discussion of these properties, however, will be postponed until Chapter II. The problem of realizing a ladder network, whose trans- fer function is specified, is commonly handled by first finding a pair of network parameters, and from these param- eters, realizing a network. Use is made of the open circuit parameters zll, z12, z21, and z22 in the synthesis of the transfer functions, T (s). It may easily be shown that z21 Z (s) -RA(s) (1-5) 1 + z22/R2 Use is made of the short circuit parameters yll' y12' y21' and y22 in the synthesis of the set of transfer functions, T2(s). It may be shown that -Y21 -Y (s) G(s)/R2 (1-6) 1 + y22 R2 In selecting the network parameters, the ratio specified in (1-4) is inserted in (1-5) or (1-6) for the appropriate transfer function. From the expression which results, z2 and z22 (or -y21 and y22) are each selected as a ratio of two polynomials in s. In general a great many possible choices exist, and to ascertain which of these choices result in a realizable pair of parameters requires a com- plete description of the conditions of physical realizabil- ity for the parameters of a ladder network. Such a description is postponed until Chapter II. To form a basis for choosing the network parameters, attention is turned to the transmission zeros of T(s) and the behavior of the parameters at these zeros. From (1-5) (or (1-6)) it is evident that at a zero of T1(s) one of the following two conditions must hold: 1) z21 (or y21) must have a zero which is not a zero of the denominator, 1 + z22/R2 (or 1 + y22 R2)* 2) z22 (or Y22) must have a pole which is not a pole of z21 (or Y21). Two standard methods for choosing the parameters represent the widely differing approaches which one may take in making this choice [1l.* In the first approach z21 and z22 are chosen to have the same denominators, which rules out possibility 2. All transmission zeros are then the zeros of z21. The second approach assigns a constant value to z21 with the result that all transmission zeros are produced as private poles of z22. This method is restricted in its use to only those transfer functions which are posi- tive real. However, for this special case it yields a very simple solution which requires only two-terminal techniques in the synthesis. The zero sharing method is primarily concerned with a new philosophy which may be used in selecting network *Brackets denote entries in the List of References. parameters from the given transfer function. An orderly procedure is developed for utilizing the simplifying proper- ties of the second approach above while extending its application to all physically realizable minimum phase transfer functions. 1.1 The Basic Zero Sharing Procedure Consider the network arrangement of Fig. 1.1 terminated in a normalized load, R2 = 1. For such a termination (1-5) and (1-6), when solved for z22 and y22, have the form 0(s) z22 21 ( 1 (1-7) 22 21 HP(s) and Q(s) 2= y 1 (1-8) 22 21 HP(s) respectively. For the special open circuit and short circuit cases, A*(s) and G*(s), z22 and y22 have the form Q(s) 22 21 HP(s) (1-9) and Q(s) Y22 1 Ip(s) (1-10) respectively. The basic zero sharing procedure may be summarized in the following steps: 1) Determine whether or not Q(s)/P(s) is positive real. In the event that the positive real property is satisfied, 21 (or -y21) may be set equal to a constant, K, and step 2 may be omitted. 2) In the event that Q(s)/P(s) is not positive real, choose z21 KN(s)/D(s) (or -y21 KN(s)/D(s)) in such a way that the product, z21Q(s)/P(s) (or -y210(s)/P(s)) is posi- tive real and has no zeros on the jo axis. (In the synthe- sis of A*(s) or G*(s), however, zeros on the ju axis are permitted.) The degree of the polynomial N(s) and that of D(s) are to be as low as possible and still meet the desired conditions on the product z21Q(s)/P(s) (or -y21Q(s)/P(s)). N(s) is chosen to have certain of the zeros of P(s), as will be described later in detail. If N(s) has some but not all of the zeros of P(s) then the product z21Q(s)/P(s) (or -y21Q(s)/P(s)) must in general satisfy certain restrictions on the residues at each of its poles in addition to being positive real. It is this case which represents a sharing of transmission zeros and is of primary interest. 3) z22 as given in (1-7) or (1-9), or y22 as given in (1-8) or (1-10), is next placed in the form z22 z' + Z (1-11) 22 22 s y y' + Y 22 22 p (1-12) respectively. Here z' (or y' ) is to have the same poles as z21 22 22 21 (or -y ) and both z' (or y' ) and Z (or Y ) are positive 21 22 22 s P real. The decomposition indicated in (1-11) or (1-12) per- mits a network realization of the form shown in Fig. 1.2 or Fig. 1.3, respectively. With i Zs parameters 1 z2 = z21 E2 R2 z' z -Z 22 22 s Fig. 1.2 Network arrangement for a prescribed transfer function of the set Tl(s) Fig. 1.3 Network configuration for a prescribed transfer function of the set T (s) 2^ 9 4) The synthesis is completed by realizing the network N' of Fig. 1.2 (or the network N" of Fig. 1.3) from its associated parameters z' and z' (or y' and y') in a 21 22 21 22 ladder configuration. The synthesis method which will be discussed was recently introduced by Chen [1]. This method provides a well organized and unified approach to synthe- sizing transfer functions of RC, RL, LC, and RLC ladder networks. CHAPTER II CONDITIONS OF PHYSICAL REALIZABILITY The six transfer functions, T(s), defined in (1-1) and (1-2) must satisfy a number of conditions in order that they represent the transfer properties of a physical network which is made up of lumped, linear, bilateral, and passive elements arranged in a ladder configuration. Similarly, the open circuit and short circuit network parameters must meet certain conditions in order that they be the associated parameters of a physical ladder network. From the previous discussion of the basic zero sharing process and from (1-5) and (1-6), it is seen that the synthesis of a given transfer function reduces to the problem of synthesizing a pair of network parameters, z21 and z22, (or -y21 and y22.) For this reason, those conditions on the parameters will be con- sidered which are particularly applicable to the pair of parameters z and z22 (or -y21 and y 22) The combined 21 22 21 22 realizability conditions on both T(s) and the parameters will play an important role in the development of the zero sharing method. First, these conditions aid in establishing a criterion for selecting the network parameters from T(s), and second, they are instrumental in determining permissible decompositions of the parameters z22 or y22 to allow the sharing of transmission zeros. Only a summary of the realizability conditions will be given here. For proofs of these conditions, see the references [1, 2]. 2.1 Conditions of Physical Realizability of the Transfer Functions, T(s), With Ladder Networks [1] The six transfer functions, T(s), must satisfy the following conditions: 1) T(s) is representable as a ratio of two polynomials in s with real coefficients. 2) The poles of T(s) if not on the ju axis are in the left-hand half of the s plane, and those on the jw axis are simple. (Of the six transfer functions, T(s), only A*(s) and G*(s) may have poles on the ju axis.) 3) For the expressions ZT(s)/R2, -A(s), -R2YT(s), G(s), -A*(s), and G*(s) the following conditions must be satisfied: (a) All coefficients in the numerators and denom- inators of these expressions must be nonnegative. (b) The numerator coefficients for each of these expressions must be no greater than the corresponding denominator coefficients. 4) The residues of the transfer functions A*(s) or G*(s) at any of their imaginary poles, s ju, are imaginary. 5) The transfer functions A*(s) and G*(s) do not have a pole at either s 0 or s -oo. These conditions must hold for a transfer function if it is to be realizable in a ladder network having no mutual inductances. Also, those cases of node bridging which pro- duce transmission zeros in the right-hand half of the s plane will not be considered. Therefore, T(s) will be a minimum phase function, and, as a sixth condition, the zeros of T(s) must be either in the left-hand half of the s plane or on the ju axis. Additional restrictions are placed on the transfer functions when the network is to consist only of two kinds of elements. RC, RL, and LC network realizations will be considered separately, and the additional conditions which apply will be considered along with each individual synthesis procedure. 2.2 Conditions of Physical Realizability For the Parameters z21 and z22 or -y21 and y22 With Ladder Networks [1] The following conditions which the network parameters must satisfy, are particular applicable to the pairs of parameters z21 and z22 or y21 and y22: 1) z22 (or y22) and zll (or y11) must be physically realizable driving point functions. 2) The poles of z21 (or y21), z22 (or y22), and zl (or y11), if not on the ju axis, are in the left-hand half of the s plane, and those on the ju axis are simple. 3) In general, the poles of z21 (or y21) are also the poles of z22 (or y22) and zll (or y11), but z22 or zll (Y22 or y11) may have poles in addition to those of z21 (or Y21). The poles of z21 (or y21) on the jw axis must always be poles of z22 (or y22) and zll (or yll). 4) All coefficients in the numerators and denominators of z21 (or -y21), z22 (or y22), and Z11 (or yll) are nonneg- ative. The numerator coefficients of z21 (or -y21) are no greater than the corresponding numerator coefficients of z2 (or y22) or zll (or y11) where z21, z22, and z11 (or -Y21' Y22 and y11) are placed in a form having the same denominators. 5) If z22 (or y22) or zll (or yll) is a LC driving point function, and, therefore, a ratio of odd and even polynomials in s, then z21 (or -y21) must also be a ratio of odd and even polynomials. CHAPTER III TRANSFER FUNCTIONS WHICH ARE POSITIVE REAL In the realization of a transfer function, T(s), in ladder form, it is desirable to first determine whether or not T(s) is positive real in addition to being physically realizable as a transfer function. In the event that T(s) does satisfy the positive real condition, a simple real- ization may be effected using only two-terminal techniques. In the interest of completeness, this case will now be discussed. For any of the six transfer functions under considera- tion, if T(s) is positive real then Q(s)/P(s), considered as an impedance, is a non-minimum-resistive function. This follows for ZT(s), A(s), Y (s), and G(s) since these func- tions are not permitted to have poles on the ju axis as discussed in Chapter II. Although A*(s) and G*(s) may have poles on the ju axis, the residues at such poles are imagi- nary as the conditions of physical realizability dictate. Imaginary residues at these poles,of course, rule out the possibility of the function being positive real. One may therefore conclude that if T(s) is positive real then O(s)/P(s) considered as an impedance is not a minimum resistive function. In the following discussion a synthesis procedure is given for realizing Z (s), A(s), and A*(s). A completely dual procedure may be followed for YT(s), G(s), and G*(s). Synthesis may be carried out by first placing Q(s)/P(s) in the form Q(s)/P(s) R + Z'(s) (3-1) where Z'(s) is a minimum resistance driving point impedance function. When Q(s)/P(s) is an RC or RL driving point function, or when all transmission zeros lie on the ju axis, the decomposition of O(s)/P(s) is easily effected by partial fractioning. If O(s)/P(s) is a general RLC driving point function, standard methods for the series removal of a purely resistive element may be used. With Q(s)/P(s) reduced as in (3-1), z22 has the form z21 z (R + Z' (s)) -1 (3-2) 22 H when Z (s) or A(s) is specified, and the network is to be T terminated in a normalized resistive load, R2 = 1. z22 has 2 z22 the form z21 z = --(R + Z'(s)) (3-3) 22 H for A*(s). 221 may now be chosen as a constant, K, and with the proper choice of values for the constants, K and H, z22 will be a physically realizable driving point function. In the synthesis of ZT(s) or A(s) it is evident from a standpoint of physical realizability that KR/H must be greater than one. When it is required that the network be synthesized within a constant multiplier it is convenient in many cases to choose a value of K Q 1 and assure that the necessary condition H KR (3-4) K + 1 is satisfied. When H is specified, an appropriate value of K is easily determined from (3-4). It is seen in (3-4) that the greatest possible upper bound on H is R. In the reali- zation of A*(s) it is only necessary to assure that H R (3-5) in (3-3). For ZT(s), A(s), or A*(s) the resulting network has the form of Fig. 1.2 in which the network N' consists of a single shunt resistance of K ohms and possibly a series resistance. When equality occurs in (3-4) or (3-5) the realization will not require the series resistive element. For Y (s), G(s), or G*(s) the resulting network has the form of Fig. 1.3. Here the network N" consists of a single 17 series resistance of 1/K ohms and possibly a shunt resist- ance. Equality in (3-4) or (3-5) eliminates the shunt resistive element. CHAPTER IV TRANSFER FUNCTIONS WITH REAL NEGATIVE TRANSMISSION ZEROS The cases which will now be considered are those in which the resulting network is to consist of only resistive and capacitive elements, or it is to consist only of resis- tive elements, and inductive elements with no mutual coupling. The six transfer functions, T(s), of networks which are composed of these kinds of elements, arranged in a ladder configuration with no-node bridging, will possess the following two properties in addition to the properties of physical realizability stated in Chapter II: 1) The transmission zeros must lie on the negative por- tion of the a axis including the points a 0 and o -co. 2) The poles of T(s) must be simple and lie on the neg- ative portion of the a axis excluding the points a 0 and 0 = 00. 4.1 RC or RL Transfer Functions Which Are Positive Real As indicated in the discussion of the general procedure to be followed, it is desirable to first determine whether or not Q(s)/P(s) is positive real. Any one of the transfer functions Z (s), -A(s), and -A*(s) for RC networks or -YT(s), G(s), and G*(s) for RL networks is positive real if and only if its reciprocal, Q(s)/P(s), is of the form 18 Q (s) = R(s) P(s) (s p1)(s - (s z )(s - 1 p2)-----(s p) z )-----(s z ) 2 n where the p. and z. are real with 0 > P > Z1 > p1 > z2 > P2 ---- > Zn > Pn (4-1) R(s) = 1 s PO n(s) s Any one of the transfer functions -YT(s), G(s), and G*(s) for RC networks or ZT(s), -A(s), and A*(s) for RL networks is positive real if and only if its reciprocal, O(s)/P(s), is of the form Q(s) (s pl)(s p2)----(s pn) R(s) P(s) (s zl)(s z2)----(s zn) where the p and z are real with (4-2) 0 p > pl > >2 > z2 ---- > n > n 0> > PO > z > > z2 > zn > Pn f for R(s) 1 or R(s) (s p ), respectively. If T(s) satisfies the conditions of (4-1) or (4-2) a realization may be easily effected by following the procedure given in Chapter III. If, however, the resulting network configuration of Fig. 1.2 with a rather complex Zs and with N' consisting of a single shunt resistor, or the network of Fig. 1.3 in which N" consists of a single series resistor is undesirable, various degrees of zero sharing may be achieved by use of the methods which will be discussed in Section 4.3. 4.2 A Special Class of RC and RL Transfer Functions [3] In addition to those transfer functions which are pos- itive real, there exists another class which may be synthe- sized using only two-terminal techniques. For RC or RL ' networks, this class consists of those transfer functions, T(s), whose reciprocal, Q(s)/P(s), has the following properties: 1) Q(s)/P(s) is of the form Q(s) (s pl)(s p2)----(s ) (43) P(s) (s z1)(s z2)-----(s n) where n < m. 2) The zeros, pi, of Q(s) must be simple and must lie on the negative portion of the a axis excluding the points a = 0 and a -oo. 3) The zeros, zj, of P(s) must have no greater multi- plicity than two and must lie on the negative portion of the o axis including the points a = 0 and o = -co. 4) If one begins with the critical frequency of Q(s)/P(s) nearest the origin and divides the entire group of critical frequencies into adjacent pairs (counting a double pole as two adjacent poles), then each pair of critical frequencies must consist of a pole and a zero. The realization of transfer functions which satisfy these conditions may be carried out by using the method which is developed in Section 4.3. An example of the syn- thesis of such a transfer function is given in Section 4.5. 4.3 Synthesis of T1(s) With RC Networks Attention is now turned to the problem of synthesizing an RC network for which Z (s), -A(s), or -A*(s) is speci- fied. As mentioned previously, T1(s) is used to represent this set. The only restriction is that T1(s) be physically realizable with resistive and capacitive elements as described at the beginning of the present chapter. The transfer impedance or current-ratio function, T1(s), is given as a ratio of two polynomials in the form p(s) T () H(s) 1 Q(s) (s zl)(s z2)-----(s zn) H(s p)( ) - (s pl)(s p2)-----(p pm) with poles and sion for T1(s) solved for z22 zeros on the -o axis and m L n. The expres- in terms of the open circuit parameters when is Q(s) z22 21 HP(s)- 1 N(s) Q(s) z K/H 1 22 D(s) P(s) 0(s) =K/H ) 1 P'(s) for T (s) ) ZT(s) or ation, R2 = 1, or T1(s) = -A(s) and a normalized termin- Q(s) 22 = z21 (s) K P(s) (4-6) K/ (s) P K/H p'(s) for Tl(s) A*(s). As suggested in the outline of the general procedure to be followed, choosing the open circuit parameters z21 and z22 is the initial step in the network (4-4) (4-5) realization. z21 should be chosen as a rational function of the form N(s) K(s-zl)(s-z2)----(s-zn-q+k) 21 K (s) (s-r )(s-y2)----(s-y ) k(47) in which both poles and zeros are real. The transmission zeros zl, z2,----zn-q+k are chosen according to a set of rules which will be given and are not necessarily the first n-q+k zeros of T1(s) given in (4-4). For ZT(s) or A(s), q = m, and for A*(s), q may take on either of two values, q = m or q m + 1, as will become evident in the following discussion. The choice of poles and zeros of z21 is dictated by the following considerations: 1) The zeros of z21 are selected from the set of trans- mission zeros of the given transfer function as indicated in (4-7). 2) The poles and zeros of z21 must lie on the -a axis including the point a 0. Poles of z21, however, are not permitted at s oo. 3) The poles of z21 must not coincide with the zeros of Q(s). 4) The poles and zeros of z21 must be chosen in such a way that the product z21Q(s)/P(s) is an RC driving point function of the form Q(s) N(s) Q(s) Q(s) z21 P = K 1 T D(s) P(s) P'(s) (4-8) (s-pl)(s-P2)--(s-pm) K (s-l)(s-y2)--(s-yk) (s-k+l)--(s-zq) in which the denominator polynomial may or may not have a simple zero at s = 0. The set of transmission zeros, zk+l, z k+2---- z consists of those finite transmission zeros k+2' q which do not belong to the set, zl, z2,---, zq+k. The poles and zeros of z21Q(s)/P(s) must alternate on the -a axis with a pole as the leading singularity. The possible values of q, q = m for ZT(s) or A(s) and q = m or q m + 1 for A*(s), become evident when (4-8) is substituted in (4-5) and (4-6). 5) In order to take full advantage of the simplifica- tion which is possible using this technique one should assign as few poles and zeros to z21 as possible and still satisfy condition 4 above. Although such a choice of z2 is not essential, it does assure that two-terminal tech- niques will be used as much as possible throughout the remainder of the synthesis. With z21 chosen according to the considerations above, one next places the product z210(s)/P(s) in partial- fractioned form. z22 may then be represented as (4-9) [ al ak ak+l am z2 = K/H ( + -- +-+ --) + (-- +-+ ) - 2( + S-y s-z S-Z SS-1 -k k+l m for ZT(s) or A(s), and (4-10) [ al ak ak+l aq SK/H a + -- +--+ --) + (-- +--+ 22 S- I S-Tk S-Zk+l S-Zq for A*(s). In (4-10), a0 = 1 for q m, or a0 = 0 for q m + 1. At this point it is necessary to determine a value for K/H in (4-9) or (4-10), and it is well to consider briefly the significance of this choice. The constant H determines the overall magnitude of the voltage or current response for a given voltage or current excitation. The constant K was introduced as a scale factor of z21 in order to provide flexibility in synthesizing the networks N' and N" in Fig. 1.2 and Fig. 1.3 respectively. This flexibility is desirable since the network N' or N" is to be synthesized by one of the well known synthesis techniques for ladder networks which is directed toward synthesizing a given trans- fer function within a multiplicative constant. In a particular problem, the constant H may be speci- fied or it may simply be required that the synthesis be carried out to within a constant multiplier. When H is specified or when a high value of H is desired, the permis- sible range of values for H is of interest. An upper limit is placed on H by the conditions of physical realizability for transfer functions as stated in property number 4 of Chapter II. Additional limits are placed on H by the topo- logical form of the network realization and considerable attention has been given to determining these limits for specific network configurations [4]. Two sets of limits on K and H exist after the zero sharing step and will now be considered. The first set is for the special case in which the synthesis is to be car- ried out entirely on a two-terminal basis. Here, O(s)/P(s) must satisfy the conditions of section 4.2. After the application of the zero sharing techniques just discussed, a pair of open circuit parameters z21 and z22 will result in which z21 is positive real. z21 may be represented in partial fractioned form as S B1 2 Bk I L K + --- + -- +---+ (4-11) s 1 s 2 s rk where SO = 1 if q n or B0 0 if q n. For the reali- zation of the network N' in an L network as shown in Fig. 4.1 in which Z1 and Z2 are RC driving point impedances, it is both necessary and sufficient that K/H 1 > KB, -- 0' (4-12) and that a /Bi > H for i 1, 2,-----, k, (4-13) for ZT(s) or A(s) and a resistive termination, R2 = 1. This realization for a specified A*(s) requires that ai/Bi > H for i = 0, 1, 2,----, k. (4-14) r- ---------- -------------------- --------------0 I I ZI O II 11 Network N' Fig. 4.1 L network configuration When synthesizing ZT(s) or A(s) one may easily select a suitable value of H from (4-13). This value of H will determine an acceptable range of K as indicated in (4-12). K may then be chosen within this range on the basis of producing desirable element values in the normalized network. If the value of I is of secondary importance, one may first make a desirable choice of K and accept further restric- tions on H. When synthesizing A*(s), H may be selected within the limits set by (4-14). The value of K is not restricted and may simply be chosen in such a way as to produce desirable element values in the normalized network. The second set of limits on K and H is for the case in which the network N' is to be synthesized in a general ladder configuration of resistive and capacitive elements. For ZT(s) or A(s), the open circuit parameters for the network N' are z' z (4-15) 21 21 and r al a2 ak ] z' K/ 1 + --- + -- +---+ -- -1 (4-16) 22 S-r1l s-Y2 s-k If each parameter is placed in the form of a ratio of two polynomials with the numerator in expanded form, property 4 for the network parameters may then be applied to determine limits on H and K. It is generally desirable when synthe- sizing N' in a general ladder configuration to select a value for the ratio K/H but to leave the particular values of K and H unspecified. The value of K/H may be selected in such a way as to control the spread of element values in the resulting network. For realizability it is necessary, but not sufficient, that the condition K/H > 1 (4-17) be satisfied. Equality may occur in (4-17) only when m > n. The manner in which K and H may be determined may be best explained with the illustrations which will be given in section 4.5. In synthesizing A*(s) the role of K/H is not critical and any convenient value of K/H may be selected. With the ratio K/H determined one continues the synthe- sis by realizing Z as an RC driving point impedance. From (4-9) or (4-10) Zs is given by Z K/H +- -------+ -- (4-18) s s z s z Zk+1 m-z Finally, the network N' is synthesized either by standard two-terminal techniques for RC networks or by the methods to be discussed in section 4.4. 4.4 Synthesis of the Network N' as an RC Ladder Network In this section a procedure will be described for syn- thesizing the network N', whose parameters are z' and z' 21 22 As explained earlier, the driving point impedance z22 is obtained by removing the series impedance Z from z22 and z' z The method which will be described was intro- 21 21 duced by Chen (1]. Although it produces the same results as zero-shifting methods, it has two distinct advantages. First, it is more straightforward in its application than the zero-shifting methods, and, second, it may be extended to the realization of transfer functions which have complex transmission zeros [1]. This extension will be described in Chapter VI. The underlying principles which will be considered now are applicable to LC, RC, RL, and certain RLC synthesis problems. The basic approach to synthesizing N' consists of the following two steps. First, a network configuration is predicted which consists of a number of basic transmission- zero-producing sections. Second, z' is synthesized as a 22 driving point function having the predicted configuration. The synthesis of z' is carried out section by section in 22 such a way as to assure that each section produces the desired transmission zero. Associated with the network N' is the pair of open- circuit parameters z21 and z'2. Due to the application of the zero sharing technique of section 4.3, z' and z' have 21 22 identical denominators. Therefore, all transmission zeros of the network N' are the zeros of z'1, and may be easily 2i\ determined by inspection. These transmission zeros will be produced by either shunt or series branches of the standard ladder configuration of Fig. 4.2. This configu- ration which is the assumed form of N' will have transmis- sion zeros which are produced either by a pole of a series impedance Zk (k 0) or by a pole of a shunt admittance Yi. In general, each finite nonzero pole of Zk(k / 0) or iY produces a transmission zero. However, several poles of Fig. 4.2 Standard ladder configuration for prescribed z21 and z22 Zk or Y at s = 0 or s = co may contribute to a simple transmission zero at s = 0 or s -oo respectively. The ladder structure of Fig. 4.2 is considered here to be made up of a predicted configuration of certain building blocks. These building blocks or generalized basic sections are shown in Fig. 4.3. Each of the two types of basic sections consists of a "principal branch" which is shown as a shaded box and an "auxiliary branch" which appears as a clear box. In general, the principal branch is used to produce either one, or a complex conjugate pair of finite, nonzero value transmission zeros. The auxiliary branch contributes to a transmission zero at s = 0 or s 00 , or it may make no contribution to a transmission zero. Zb /Zb o -o o- o (a) (b) Fig. 4.3 Generalized basic sections (a) A type 1 section (b) A type 2 section Attention is now turned to the problem of synthesizing N' as a ladder network which is made up of only resistive and capacitive elements. The first step is to predict a possible ladder configuration which has the form shown in Fig. 4.2 and consists of basic RC sections and possibly some additional RC branches. There are two basic RC sections, type 1 and type 2, as depicted in Fig. 4.4. The subscript (k) Rb Rb I -_- y(k) r- s-zTLk a R(k)I Sk Ci a L G(k) I- ---- Ga Zk+l Y* Zk Z* Y k k k+1 k k (a) (b) Fig. 4.4 Basic RC sections (a) Type 1 RC section (b) Type 2 RC section or superscript, k, indicates that this is the kth section in the ladder configuration with the sections numbered consecu- tively from right to left. In each basic section of Fig. 4.4 a transmission zero is produced at s = z In Fig. 4.4 (a), (k)k if Rk) is set equal to zero, the principal branch of this type 1 RC section contributes to a transmission zero at s o. In Fig. 4.4 (b), if R(k) is set equal to infinity, b the principal branch of this type 2 RC section contributes to a transmission zero at s 0. Transmission zeros may there- fore be produced at any point on the negative a axis, including s 0 and s -00 by the use of type 1 and type 2 RC sections. The prediction step is accomplished as follows: 1) Examine z21 to determine the transmission zeros of N'. 2) Arrange type 1 and/or type 2 RC sections (and possi- bly some additional RC branches) in the ladder configuration of Figure 4.2. (Note that a shunt element appears at the extreme left, and that ZO may or may not be present as indi- cated by its presence in dotted lines.) 3) In the arrangement described in 2, the principal branch of each RC section is assigned one transmission zero which it is to produce. 4) The auxiliary branch of each RC section makes no contribution to a transmission zero. A number of predicted configurations are possible for a given set of transmission zeros. In this particular situa- tion two of the possible predictions are (a) a configuration consisting exclusively of type 2 RC sections with a shunt resistive branch at the far left, and (b) a configuration consisting exclusively of type 1 RC sections. It should be understood that some predictions will be realizable, while others may be nonrealizable. The appearance of negative elements will indicate a nonrealizable prediction. A simple rearrangement of the order of the transmission zeros will frequently bring a nonrealizable situation into a realizable form. The synthesis of the predicted network follows an orderly procedure in which one basic section is realized at a time. One begins by synthesizing the first section at the extreme right (k 1). For this section the impedance 1 Z z' or admittance Y1 is known, and, as a result 22 of prediction, a particular transmission zero has been assigned to this section. This is sufficient information to determine the elements of the first section. Upon removing the elements of this first section from the imped- ance Z1 or admittance Y1, one obtains the impedance or admittance looking into the next section, Z2 or Y2, respec- tively. This process is repeated for the entire predicted configuration. The technique for synthesizing a Type 1 RC section is as follows: 1) Evaluate R(k) where b Rb Zk(s) (4-19) s = z k 2) The admittance Y*k looking into the remaining por- tion of the ladder after R(k) is removed is b Y* = k (Z R(k) (4-20) k b 3) Y* is of the form k Mks Y* = --- + Y + 1 (4-21) k s z k k where [(s zk) Y| M = ---- Y* k k s S = Zk The elements of Y are then a (k) 1 (k) Mk R -andC (4-22) a Mk a -zk 4) The impedance looking into the next basic section is Zk+1 -(4-23) Y* Mks k s zk The technique for synthesizing a Type 2 RC section is (k) 1) Evaluate G(k), where a (k) (s) Ga v= (4-24) s z 2) The impedance Z* looking into the remaining portion k of the ladder after G (k) is removed is a Z* (4-25) k (Y G(k)) K a 3) Z* is of the form k Nk Z* = ---- + Z k s k + 1 , where (4-26) Nk (s zk) Z s zk The elements of Zb are then N 1 (k) k C and R (4-27) b Nk b -Zk 4) The admittance looking into the next basic section is I Yk + 1 ) Z Nk (4-28) S Z k 4.5 Illustrations of the Synthesis of ZT(s), -A(s), or -A*(s) With RC Networks For the first illustration a transfer function which belongs to the class described in section 4.2 will be considered. Let it be required to synthesize (s + 2)(s + 3) Z (s) H (s + 2)(s + (4-29) T (s + 1)(s + 4) in a network with a normalized load R2 1. In accordance with the procedure given in section 4.3, z21 is chosen as s + 2 z 2 K-- 21 s (4-30) Then SK/H F (s + l)(s + 4) 22 K/+H ) 22 | s(s + 3) (4-31) Upon partial fractioning z21Q(s)/P(s), z22 becomes (4-32) The limits on K and H are set by (4-11) and (4-12) as K/H 1 2 K (4-33) 2/3 > H (4-34) is chosen equal to its limiting value, that is H 2/3, H IK> = 2 -1 -H (4-35) K may be chosen equal to its limiting value, 2, in which case z22 and z21 become If H then [ 4/3 2/3 22 /H 1 + --- + -13] 22 1 s s+3 (4-36) 2 2 2 + 4/s + - 122 s + 3 z 2 + 4/s. 21 When the private pole of z22 is removed, the parameters of the network N' are z' = 2 + 4/s 22 and (4-37) z' = 2 + 4/s . 21 The resulting network for Z (s) as given in (4-29) is shown in Fig. 4.5. R-2/3 R2-1 Fig. 4.5 Ladder network for ZT(s) as given in (4-29) For the second illustration a transfer function will be considered whose realization requires that the network N' have the form of a ladder with more than one section. Let it be required to synthesize P(s) (s+2)(s+6)(s+7)(s+9)(s+10) -A(s) H H (4-38) Q(s) (s+l)(s+3)(s+5)(s+8)(s+12) in a network which is terminated in a normalized load, R 1. 2 As explained in section 4.3, one possible choice of z21 is 21 K (s + 7)(s + 9) (4-39) s(s + 4) For this choice of z21, z22 becomes 22 = K/n [(s+l)(s+3)(s+5)(s+8)(s+12) (4-40) 22 1 s(s+2)(s+4)(s+6)(s+10) The product, z21 O(s)/P(s), is next partial fractioned, which places z22 in the form [ 3 45/32 1 15/16 21/32 z22 K/H 1+-+---+-+---+ -1 (4-41) 22 s s+2 s+4 s+6 s+lOJ At this point it is necessary to select a value of K/IH. Property 4 for the network parameters, given in Chapter II, states that the numerator coefficients of z'2 must be greater than or equal to the corresponding numerator coefficients of z' Application of this property places the following 21 condition on H. H 5 12/63 (4-42) For this value of H, K/H must satisfy the condition K/H > 63/51 .(4-43) In selecting a desirable K/H within this wide range of permissible values, the following three facts are helpful: 1) z2 is given by z' K/H + 3 + -1 (4-44) 22 s s+4 As K/H decreases from infinity to its lower limit, the zeros of z' move to the left along the -a axis from their ini- 22 tial positions at the zeros of f(s), where 3 1 f(s) = 1 + + (4-45) s s+4 The zero movement is covered in detail in reference [5]. 2) To prevent the occurrence of a large spread in ele- ment values in the normalized network one should avoid selecting a value of K/H which places a zero of z' very near 22 to, but not touching, the transmission zero which is to be produced by the first section at the right of N'. 3) This shift in the zero locations of z' is identical 22 to the shift which occurs when synthesizing a type 1 RC section. With these facts in mind the synthesis is continued by simply selecting K/H = 2, the next larger integer value above the limiting value of 63/51, for H 12/63. As the first step in synthesizing the network N', a network config- uration is predicted as shown in Fig. 4.6. The realization of the first basic section at the right of N' is begun by evaluating Zl(s) -7 z22(s) -7 or Z(s) Ss -7s = -7 s = -9 z22(s) s Negative values are found for both s = -7 and s -9 which indicates that a negative resistance must appear in the position of R1l) in Fig. 4.6. Therefore, for K/H = 2, the network prediction of Fig. 4.6 is nonrealiz- able. According to remarks 1 and 2 above, a different choice of K/H may make the prediction of Fig. 4.6 realiz- able. (See reference [5]). A second choice, K/H = 3, eliminates the problem just encountered. For this choice, the open circuit parameters of N' are 2 9 3 2s + 20s + 36 (4 z' = 2 + + (4-46) 22 s s + 4 s(s + 4) and (s + 7)(s + 9)47) z' = =K (4-47) 21 21 s(s + 4) -----Section 2 (2) Rb S---Section 1--- I I i R^ bl) ] b Y* Z Y* 2 2 1 Fig. 4.6 Network prediction for N' For the network prediction of Fig. 4.6, the synthesis of N' is carried out as follows: 2 2s2 + 20s + 36 Z1 z2 + (4-48 1 22 s(s +4) ) R() 2/5 b 8 x 13 2 15 R() 4 x 13 a 25 c() 25 a 4 x 13x9 [s +5/2 (2) 12 x 13 b 5 x 7 R(2) 4 x 13 a 3 x 7 C(2) 3 a 4 x 13 The impedance Zs, whose position in the network is shown in Fig. 1.2, is given by Z 9/320 15+ 0 7 (4-49) s s + 2 s + 6 s + 10 The realization of Z may be carried out by any of the s standard synthesis procedures for RC driving point functions. The resulting network is shown in Fig. 4.7 I R(2 4x13 ,,(I 4x13 { R (2) 3 C(1) 25 a 4x13 a 4x13x9 Network N' Fig. 4.7 Ladder network for A(s) as given in (4-38) with K/H = 3 The situation frequently occurs in which K/H can be chosen so that z 2 will possess a zero at one of the trans- mission zeros which are to be produced by the network N'. This choice of K/H eliminates the need for the auxiliary branch in the first basic section at the right of network N'. The possibility of choosing K/H on this basis depends on the value of the expression within the first parentheses of (4-9). To use this criterion in choosing K/H, it is neces- sary that the inequality 0 < + + -- ---+ -- < 1 (4-50) s 1 s 2 s J be satisfied at one or more of the transmission zeros assigned to z21. Some flexibility in satisfying this condi- tion does exist, since a great many choices of N(s)/D(s) are possible, each of which meet the restrictions set by condi- tions 1 through 5 of section 4.3. However, it is difficult to select N(s)/D(s) on this basis. If (4-50) is satisfied, K/H may be chosen as (4-51) [ a, a2 ak -1 K/H + -- + -- + +-----+ - s YI s r2 s k s 1 where z1 is the transmission zero which is to be produced by the right most section of N'. The remainder of the synthe- sis is then carried out as described in section 4.4. The application of the above criterion in choosing K/H will now be illustrated for the transfer function given in (4-38). The expression indicated in (4-50) is first eval- uated at each of the transmission zeros to be produced by z21. From (4-39) and (4-41) F 3+ 1 5 1 + + 4- -2 (4-52) s s + 4 2-1 1 1 Is -7 + 3 + ] (4-53) s + 4 15 s = -9 are evaluated. For K/H chosen as K/H = 15/7 (4-54) z22 becomes (4-55) 15 x 45 15 x 15 45/7 7 x 32 15/7 7 x 16 15/32 z 8/7 + + + + + --- 22 s s + 2 s+4 s + 6 s + 10 With the series removal of the impedance 's from z22, z22 becomes 45/7 15/7 z' 8/7 + 45/7+ 15/7 (4-56) 22 s s+4 A network is then predicted having the form given in Fig. 4.6. The realization, however, will not require the resis- tive element R1). The synthesis procedure given in equations (4-19) through (4-23) is next carried out. The resulting network is shown in Fig. 4.8. 4.6 Synthesis of T2(s) With RC Networks A synthesis procedure will now be developed for the realization of a network for which Y (s), G(s), or G*(s) is specified. The only restriction on T2(s) is that it must 2t (2) 12x13 b 7x7 Network N' Fig. 4.8 Ladder network for A(s) as given in (4-38) with K/H 15/7 be a physically realizable transfer function of an RC net- work, as described at the beginning of this chapter. The procedure for synthesizing T2(s) is the same as that which was developed for Tl(s) in section 4.3 except for variations in the permissible pole and zero locations in choosing the network parameters. The transfer admittance or voltage- ratio function is given in the form P(s) T (s) H (s) 2 Q(s) (4-57) H(s zl)(s z2)----.(s zn) (s Pl)(s p2)---(s p) (s pl)(S p2)-....(s -p) with poles and zeros on the negative a axis and m r n. From the expression for T2(s) in terms of the short circuit parameters, y22 is Q(s) Y22 -Y21 HP(s) N(s) O(s) K/H N(s) (s) 1 (4-58) D(s) P(s) 0(s) = K/H ) -1 P'(s) for T2(s) = YT(s) or T2(s) = G(s) and a normalized termi- nation, R2 = 1, or Q(s) 22 = -Y21 HP(s) (4-59) 0(s) K/H p'(s) for T2(s) = G*(s). -y21 is chosen as N(s) (s-) (s-z2)----(S-zn-q+k) -Y21 = K K (4-60) 21 D(s) (s-r )(s-2 )---(s-r ) where q m or q = m 1. Equation (4-60) indicates that the number of transmission zeros assigned to -Y21 is n-q+k. They are not necessarily the first n-q+k zeros of T2(s) as given in (4-57). The poles and zeros of y21 are chosen on the basis of the following considerations: 1) The zeros of y21 must be selected from the set of transmission zeros of the given transfer function. Although they will not be discussed here, certain cases do arise in which the choice of other zeros is advantageous. 2) The zeros of y21 may lie anywhere on the negative a axis, including s 0 and s = oo. The poles of y21 must lie also on the negative a axis including the point s oo but excluding the point s 0. 3) The poles of y21 must not coincide with the zeros of Q(s). 4) The poles and zeros of y21 must be chosen in such a way that the product -y21Q(s)/P(s) is an RC driving point admittance function of the form (4-61) Q(s) (s-Pl)(s-p2)--(s-Pr) P(s) (s-z )(s-)2)--(s-)k)( s-zk+1)--(-z in which q = m or q = m 1. The set of transmission zeros, zk+l, zk+2,---z consists of those finite transmission zeros which do not belong to the set z1, z2,---zn-q+k. The poles and zeros of -y21 (s)/P(s) must alternate on the negative a axis with a zero as the leading singularity. This of course excludes the possibility of y210(s)/P(s) having a pole at s 0. 5) One should assign as few poles and zeros as possible to y21 and still satisfy conditions 1 through 4 above. As before, various degrees of zero sharing between y21 and y22 may be produced by assigning more transmission zeros to y21 than the minimum required number. After choosing y21 in accordance with the above consid- erations one next places the product -y21Q(s)/P(s) in partial fractioned form. y22 may then be represented as i ails aks 2 K/H (a s + a + s +----+ ) + 2 -1 0 s r- s Tk (4-62) ak+ls as+) -1 s z s Zq k+l q for Y (s) or G(s) and Y22 K/H (a_ ls + a als +---+ ) + s r s k (4-63) ( k+ls as s Z s Z k+l ] for G*(s). In (4-62) and (4-63), al 1 for q m 1, or a-1 0 for q m. At this point it is necessary to assign a value to K/IH. As in section 4.3, two different sets of limits on K and H will be considered. The first is for the more restrictive case, in which the synthesis of N' is to be carried out entirely on a two-terminal basis. Here, Q(s)/P(s) must satisfy the conditions of section 4.2. After the application of the zero sharing techniques above, a pair of open circuit parameters, y21 and y22, will result in which y21 is an RC driving point admittance. y21 may be represented as - K B s + B + --- + + ---- (4-64) 21 -1 0 s r s r where -I 1I for n = q + 1, or 8_ = 0 for n I q +1. For the realization of the network N" in an L network as shown in Fig. 4.9, in which Z1 and Z2 are RC driving point imped- ances, it is both necessary and sufficient that (K/H)a0 1 2 K,0 (4-65) and that a. H > H for i = -1, 1, 2,----, k (4-66) i for YT(s) or G(s) with a resistive termination, R2 1. Such a realization for a specified G*(s) requires that "i -> H for i = -1, 0, 1, 2----, k. (4-67) When synthesizing Y(s) or ), may be selected in When synthesizing YT(s) or G(s), H may be selected in r----1- ----I--- Z2 Z1 0 -------------- Network 'N Fig. 4.9 L network configuration accordance with (4-66). A rearrangement of (4-65), K/H > 1 (4-68) a0 H8O shows the dependence of K/H on this choice of H. If the form of the network N" is restricted to that shown in Fig. 4.9, the maximum value of H for this configuration may be realized by equating H to its upper limit and then choosing K/H according to (4-68). A second alternative is to choose K/H in such a way as to reduce the spread in element values in the resulting network. Examination of (4-62) reveals very quickly the proper choice of K/H to reduce this spread. When G*(s) is to be synthesized, H may be selected within the limits set by (4-67). The value of K is not restricted and may be chosen so as to produce convenient element values in the normalized network. The second set of limits on K and H is applicable when N" is to be synthesized in a general ladder configuration composed of resistive and capacitive elements. Property 4 for the network parameters may be applied to y"1 and y2 to 21 a y'i2 to determine the limits on H and K. Here y" and y" are given 21 22 by y" = y (4-69) 21 21 and (4-70) Y[ als a2s aks y" = K/H a s + a + ---- + ---- +--+ -- . 22 -1 a0 s I s 2 s yk When synthesizing YT(s) or G(s), it is usually desirable to choose a value for K/H but to leave the individual values of K and H unspecified. The value of K/H may be selected in such a way as to control the spread of element values in the resulting network. In the synthesis of G*(s), the value of K/H is not critical and any convenient value may be chosen. After a suitable value of K/H has been selected, the synthesis is continued by realizing Y as an RC driving point admittance, where Sak+lS aqs Y K/H +---+ --- (4-71) I- k+1 q The remainder of the procedure consists of synthesizing the network N" either by standard RC two-terminal techniques or by the methods to be discussed in section 4.7. 4.7 Synthesis of the Network N" The method of synthesizing the network N" is essen- tially the same as that described in section 4.4 for the network N'. A modification of the predicted ladder config- uration is necessary due to the fact that the short circuit parameters y21 and y22 are prescribed for N" whereas the open circuit parameters were prescribed for the network N'. This modification is shown in Fig. 4.10. The predicted configuration again consists of type 1 and/or type 2 RC sections (and possibly some additional RC branches), however, in this case a series element appears at the extreme left. Except for this modification, the synthesis Zn, Zn-2 Z3 Yn-1 Y2 0 Y Lr- 0 Fig. 4.10 Standard ladder configuration for prescribed y21 and y22 procedure for the realization of N" is identical to that given in section 4.4. CHAPTER V TRANSFER FUNCTIONS WITH PURELY IMAGINARY TRANSMISSION ZEROS The general technique of sharing transmission zeros between z21 and z22 (or -y21 and y22) is based on the fact that one may replace certain poles of Q(s)/P(s) with poles which make the modified function, Q(s)/P'(s)= [N(s)/D(s)] x [Q(s)/P(s)], positive real. The success of this technique depends on the freedom which exists in making this replace- ment. It was seen in Chapter IV that for the RC and RL cases, poles were replaced in such a way that alternation on the negative a axis would exist between zeros and poles of the modified function. Also there were certain requirements at s 0 and s oo Since these were the only restric- tions on the replacement of poles, considerable freedom was permitted. In contrast to this freedom which exists for the RC and RL cases and that which will be found for RLC networks, the restrictions are so great in the LC case that the zero sharing technique is of no value in synthesizing ZT(s), A(s), YT(s), or G(s). 5.1 Limitations of the Zero Sharing Technique in the Synthesis of LC Networks It will now be shown that the zero sharing approach cannot be applied advantageously to the synthesis of Z (s), A(s), YT(s), or G(s) with LC networks. Consider Q(s)/P(s) whose reciprocal is ZT(s), or -A(s), or -YT(s), or G(s) as defined in (1-1) and (1-2). From the conditions of physical realizability for these transfer functions, it is known that the zeros of Q(s)/P(s) are restricted to the left-hand half of the s plane excluding the ju axis. The poles of Q(s)/P(s) which, of course, represent the zeros of trans- mission, are to lie on the jw axis. The permissible pole locations on the ju axis for the modified function, Q(s)/P'(s), are fixed by the restriction that the residues of the function at each pole must be real. In the event that an odd number of finite poles are to be placed on the jw axis, the denominator will be of the form P'(s) s(s jl )(s + jX )(s jX2)(s + j%2)-- (s jXm)(s + jAm). (5-1) The angle associated with the residue at any one of the poles, jXi, is arg ki arg Q(jli) -arg (s) (5-2) s = ji The contribution to arg k from the poles is i 56 ar P(s) i(-X + Xi)(-X + X2)--(j2Xi)-- arg (s-JXi) i 2 2 S jXi (-Xi + X)2 (5-3) Oor 180 For ki to be real, it is then necessary that arg Q(jXi) = O'or 180 (5-4) Q(s) is now represented as n O(s) =Zaksk (5-5) k-0 where a 1. With s Re Q(s) becomes n Q(s) a RejkO (5-6) k-0 In order to meet the condition given in (5-4), the imaginary part of Q(s) is set equal to zero n 0 akRk sin kO (5-7) k=0 The only points of interest are those on the ju axis, for which (5-7) is satisfied. Therefore, let 9 r/2 in (5-7) which gives n 0 = akRk sin k n/2 (5-8) k-1 k odd When n, the number of finite zeros, is odd (5-9) 0 Rn(-l)(n-l)/2 + an-2Rn-2(-)(n-3)/2 +--+ aIR which has exactly n solutions. When n is even (5-10) 0 = a Rn-l(_-)(n-2)/2 + a Rn-3(-1)(n-4)/2+--+ a R n-I n-3 1 which has n-I solutions. From (5-9), (5-10) and the pre- ceding development it is seen that if an odd number of poles are to be placed on the jw axis in such a way as to make Q(s)/P'(s) positive real, the permissible pole locations are restricted to a set of n points when n is odd, or to a set of n-1 points when n is even. A similar development, when O(s)/P'(s) is to have an even number of finite poles, shows that arg P'(jyi) = 90" (5-11) In this case the real part of Q(s) is set equal to zero, which gives n 0 = akRk sin k t/2 (5-12) k-0 for 6 n/2. When n, the number of finite zeros is odd 0 nlRn-l ()(n-1)/2 a n-3 (l)(n-3)/2 0 a R (-1) + a R (-1) +---- n-1 n-3 (5-13) + a2R (-1) + a0 which has n-1 solutions. When n is even (5-14) 0 Rn(-l)n/2 + a Rn-2(-)(n-2)/2 +----+ a R2(-1) + a0 n-2 2 which has n solutions. If an even number of poles are to be placed on the ju axis so that O(s)/P'(s) is positive real, the permissible pole locations are restricted to n-1 points for n odd, or n points for n even. These severe restrictions are further strengthened by the fact that if z22 is a ratio of odd to even or even to odd polynomials, then z21 must also be a ratio of odd to even or even to odd polynomials. This condition stipulates that when P(s) is even, P'(s) must be odd, or that when P(s) is odd P'(s) must be even. In effect one is forced to choose precisely the same set of parameters as those which are chosen in standard LC synthesis methods (1, 5]. That is, z21 and z22 will have the same set of poles unless cancel- lation occurs. If such a cancellation is possible, it will be obvious in the application of the standard method for choosing z21 and z22 in the LC case. 5.2 Synthesis of A*(s) or G*(s) With LC Networks Although the zero sharing technique cannot be advanta- geously applied to the synthesis of ZT(s), A(s), YT(s), or G(s) with LC networks, the cases involving A*(s) and G*(s) can be handled very effectively using this method. The only restrictions placed on A*(s) and G*(s) are that they be physically realizable with inductive and capacitive ele- ments. In addition to the conditions of physical realiza- bility given in Chapter II, this requires that A*(s) or G*(s) have the following two properties: 1) The transmission zeros must lie on the ju axis including the points s 0 and s = 00 . 2) The poles of A*(s) or G*(s) must be simple and lie on the ju axis excluding the points s 0 and s = co . Before the zero sharing technique as it applies to the synthesis of LC networks is considered, it should be pointed out that a special class of LC transfer functions, similar to that for the RC and RL cases, exists in which the entire synthesis may be carried out using only two-terminal tech- niques. This class consists of those transfer functions A*(s) or G*(s), which are physically realizable with LC net- works, and whose reciprocal, Q(s)/P(s), has the additional two properties: 1) The zeros of P(s) must have no greater multiplicity than two. 2) If one begins with the critical frequency nearest the origin and divides the group of critical frequencies on the positive jw axis into adjacent pairs (counting a double pole of Q(s)/P(s) as two adjacent poles), then each pair of critical frequencies must consist of a pole and a zero. The synthesis of this class of transfer functions differs from the more general case only in the realization of the networks N' or N" shown in Fig. 2.1 and Fig. 2.2. A method will now be developed for synthesizing A*(s) or G*(s). Here, the only restrictions are those of physical realizability for LC networks as stated at the beginning of this section. The transfer current-ratio function, A*(s), or transfer voltage-ratio function, G*(s), is given in the form [-A*(s) or G*(s)] -HP(s)/Q(s) S J (5-15) 2 2 2 2 2 2 (s +-l)(s +-2)---(s +Xn) = -H 2 2 2 2 2 2 (s +2l)(S +22)---(S +2m) where poles and zeros lie on the jw axis and m n. The expression for A*(s) in terms of the open circuit parameters, or for G*(s) in terms of the short circuit parameters, may be solved for z22 or y22 respectively. This gives z22 z21 Q(s) (5-16) HP(s) Y22 = -y21Q(s)/HP(s) (5-17) It is convenient to choose z21 or -Y21 as z or -y H [21 or H D(s) (5-18) H/s(2 +2) ,-(+Xn-q+k) s 2 2 2 2 (--- 2 2,V - (s +) 2)--(s which forms the initial step in the synthesis procedure. Here, the poles and zeros of z21 (or -y21) lie on the jw axis. The transmission zeros, jX1, JX'---J n-q+k are not necessarily the first n-q+k zeros of -A*(s) or G*(s) as given in (5-15). The following considerations govern the choice of poles and zeros for z21 (or -y21): 1) The zeros of z21 (or -y21) should be selected from the set of transmission zeros of the given transfer function A*(s) (or G*(s)) as indicated in (5-18). 2) The poles and zeros of z21 (or -y21) must lie on the ju axis including the points s 0 and s oo . 3) The poles of z21 (or -y21) must not coincide with the zeros of Q(s). 4) The poles and zeros of z21 (or -y21) must be chosen in such a way that the product z21Q(s)/HP(s) (or -y21 Q(s)/HP(s) ) is an LC driving point function of the form 21 Q(s) or -_ Q(s)] 2 HP(s) 2HP(s)J (5-19) (s 2 s2 2)-(s 2 1/ 2 2 2 2 2 2 2 2 (s +v )--(s +v )(s +X1 )--(s +1 ) 1 k k+1 q in which the set of transmission zeros j k+l' J k+ 2' jXq consists of those finite transmission zeros which do not belong to the set +jXL, jX2,--- n-q+k. Acceptable values for q are q m and q m 1. The poles and zeros of z21 Q(s)/HP(s) (or y2l Q(s)/IIP(s) ) must alternate on the ju axis. 5) One should assign as few poles and zeros to z21 (or -Y21) as possible and still satisfy condition 4 above. With z21 (or -y21) chosen according to the above condi- tions one next places the product z21 Q(s)/HP(s) (or -y21 Q(s)/HP(s) ) in partial fractioned form. z22 (or y22) may then be represented as "a als aks [z22 or 22] (a s + + -- ---+ ) 22 22 1 2 2 2 2 s s +v s +V 1 k (5-20) ak+ls aqS +( +---+ ) 2 2 2 2 s + s + X k+l q where a 1 1 for q m 1 or a 0 for q m. At this point, Zs as shown in Fig. 2.1 or Y as shown in Fig. 2.2 P may be removed from z22 or Y22, respectively. The standard LC realization for Z or Y may then be performed. The remaining portion of the realization, the synthesis of the network N' or N", follows one of two courses. First, if A*(s) or G*(s) belongs to the special class of LC trans- fer functions mentioned earlier, then N' or N" may be syn- thesized in an L network of the form shown in Fig. 4.1 or Fig. 4.9 respectively. Only two-terminal LC synthesis tech- niques are required for this realization. Let z21 (or -y21) as given in (5-18) be represented in partial fractioned form as BO 31s Bks [z21 or -y21] Bis + + ---2 + (5-21) s s +v s +v 1 k Then the restrictions on the gain, H, for the L network configuration may be stated as a i/Bi 2 H for i = -1, 0, 1, 2,-----k. (5-22) The second course which may be taken is that of synthesizing N' or N" in a general LC ladder configuration. The general approach for synthesizing the networks N' and N" was described in section 4.4, and details were given there for the synthesis of RC ladders. In synthesizing LC ladders, the same general approach is used with respect to predicting a possible network configuration and synthesizing 22 (or y22) as a driving point impedance, section by section (1, 6]. Therefore, it is only necessary to intro- duce the basic LC sections which will make up the ladder, to indicate the procedure for synthesizing these sections, and to mention certain factors to be considered in prediction. Two basic LC sections are shown in Fig. 5.1. The sub- script or superscript, k, again indicates that this is the kth section in the predicted configuration, in which the Y* () (a) ~T (k)' r- b 01 ]c(k)j b Yk+l Z* Yk k+1 k k (b) Fig. 5.1 Basic LC sections (a) type 1 LC section (b) type 2 LC section sections are numbered from right to left consecutively. In each basic section of Fig. 5.1, a transmission zero is produced by the principal branch at s jkk. If, in Fig. 5.1 (a), L(k) is set equal to zero, or if C(k) is set equal a a to infinity, the kth section will contribute to a transmis- sion zero at s oo or s 0, respectively. Similarly, if, in Fig. 5.1 (b), L(k) is set equal to infinity, or C~k) is b b set equal to zero, this kth type 2 section will contribute to a transmission zero at s 0 or s -oo, respectively. Note that the auxiliary branches of each section may contrib- ute either to a transmission zero at a 0 or to a transmis- sion zero at s o . In order to synthesize a type 1 LC section, one must know the impedance looking into that section, Zk, indicated in Fig. 5.1 (a). One must also know the transmission zeros, s jkk, or their special cases, s = 0 and s = oo which are to be produced by this kth section. A knowledge of these two facts permits the synthesis of a type 1 LC section as follows: 1) Let jXk Zk(s) ), (5-23) s J^k then bk) X X k (if Xk > 0), (5-24) b k k k C(k) = /k b kk (if X < 0). (5-25) 2) The admittance Y* looking into the remaining portion k of the ladder, after ~k) or Cbk) is removed, is 1 Y* (5-26) k (k) Z sL k b or 1 Y* (5-27) k k Z /sCb respectively. 3) Y* has the form k Mks Y* -+ Y + 1 (5-28) k 2 2 k + + k where I = -- k Y*. (5-29) ks k s = Jk The elements of y(k) are then a (k) L 1/Mk (5-30) a and (k) 2 C a k/ k (5-31) C ak k 67 4) The impedance looking into the next basic section is then 1 Z = k+ 1 Mks Y* - k 2 2 S+ k k (5-32) The synthesis of a type 2 LC section requires a knowl- edge of Yk(s) and the transmission zeros which are to be produced by this section. The procedure for synthesizing this section is as follows: 1) Let Bk Yk(S) (5-33) then c(k) = BkAk a k k (k) La 1/(xklBkI) ak 2) The impedance Z* k of the ladder after C(k) a (if Bk 2 0) (if Bk < 0) (5-34) (5-35) looking into the remaining portion or L k) is removed, is ai 1 Z* k (k) k sCa (5-36) or Z* ,(5-37) Yk /sLa respectively. 3) Zk has the form NkS Z* + Z (5-38) k 2 2 k+l s + k where 2 2 s + \k) N Z* (5-39) k s k s = Jk (k) The elements of Zb are then b (k) Cb = 1/Nk (5-40) and (k) 2 Lb = Nkk (5-41) 4) The admittance looking into the next basic section is then Y (5-42) k+l Nks Z* k k 2 2 s +\ k Attention is now turned to those factors which are important in predicting possible LC network configurations. When the network parameters z21 and z22 (or y21 and y22) are chosen by the zero sharing methods above, it is certain that the parameters for the network N' (or N")will have the same denominator, and that z' and z' (or y' and y' ) will have 21 22 21 22 no common numerator factors. It is therefore assured that the zeros of z' (or y' ) are the transmission zeros of the 21 21 network N' (or N"). The procedure for predicting an LC network may then be stated as follows: 1) Examine z21 (or y21) to determine the transmission zeros of N' (or N"). 2) Arrange type 1 and/or type 2 LC sections (and pos- sibly some additional LC branches) in the ladder configura- tion of Fig. 4.2 for prescribed z' and z' or in the 21 22 configuration of Fig. 4.10 for prescribed y' and y'. 21 22 In the arrangement described in 2, the principal branch of each LC section is assigned a complex conjugate pair of transmission zeros which it is to produce. In the event that all transmission zeros are at s 0, s oo, or a combination of these two, it is necessary that certain elements in the principal branches take on the limiting values discussed earlier. The principal branch will then produce a transmission zero at s 0 or s oo. The auxil- iary branch of each LC section will contribute to a trans- mission zero at either s 0 or s = oo As in the case of RC and RL networks, a number of 70 predicted configurations is possible for a given set of transmission zeros. Some of these predictions will be realizable, while others may contain negative circuit ele- ments and are therefore nonrealizable with passive networks. If a first prediction is nonrealizable, a rearrangement of the order of transmission zero assignments may cause the prediction to become realizable, or it may be necessary to predict a different configuration of basic sections. CHAPTER VI TRANSFER FUNCTIONS WITH COMPLEX CONJUGATE AND NEGATIVE REAL TRANSMISSION ZEROS A method will be developed in this chapter for synthe- sizing minimum phase transfer functions with networks which, in general, will consist of resistive, capacitive, and inductive elements. The realizability conditions, which a transfer function, T(s), must satisfy, are given in Chapter II. Although these are the only restrictions on the trans- fer functions, it will be found that the usefulness of the method under consideration is limited to transfer functions of low complexity. In fact, only those transfer functions of third degree or less in numerator and denominator will be considered in detail. The extension of the zero sharing method to higher degree transfer functions will be discussed but has not been placed on a firm basis. That is, in applying the technique to higher degree transfer functions, one cannot be sure that full advantage has been taken of the simplifying effects inherent in the zero sharing process. The trivial case, in which the denominator of T(s) is a first degree polynomial with the numerator either a constant or a first degree polynomial, is always positive real and therefore realizable with an RC or RL network in the L configuration of either Fig. 4.1 or Fig. 4.9. 6.1 Transfer Functions Whose Denominator Polynomial Is of Second Degree It will be shown in this section that every minimum phase transfer function, T(s), whose denominator polynomial is of second degree in s, can be synthesized using only two- terminal techniques. The resulting network will have the form indicated in Fig. 1.2 or Fig. 1.3, in which the networks N' or N" may be realized in the L configuration of Fig. 4.1 or Fig. 4.9 respectively. It is convenient to consider separately five different forms of T(s) which differ in their pole-zero configurations. 1) Let T(s) have the form H T(s) = (6-1) (s + c jd)(s + c + jd) where c > 0 and d > 0. z21 (or -y21) may be chosen as 1 21 (or -y21) = K (6-2) where 0 < e 2c. The product z21 Q(s)/P(s) (or -y21 Q(s)/P(s) ) then has the form (6-3) z2 Q(s) (or -1 Q(s) K 1 (s+c-Jd)(s+c+jd) L P(s) P(s) s+e 1 For ZT(s), A(), Y(s), or G(s), z22 (or y22) has the form (6-4) S (or y) = K/ + (2c-e) + c2+d2(2c-) L22 22)] s [ +e For A*(s) or G*(s), z22 (or y22) has the form (6-5) (or y ) K/H + (2 ) c2+d2-e(2c-e)] [22 (or Y22) K/H (-e) s+e Since z21 (or -y21) is a positive real driving point func- tion, a two-terminal realization may always be effected. A very simple sharing of transmission zeros is represented by the fact that one transmission zero at infinity is produced by z21 (or -y21), while the other transmission zero at infin- ity is produced as a private pole of z22 (or y22). 2) Let T(s) be of the form T(s) = H (6-6) (s + c jd)(s + c + jd) where a 0, c > 0, and d > 0. It is assumed that T(s) is not positive real, and therefore, that a > 2c. One possible choice of z21 (or -y21) is [-21 (or -y2)] K + (6-7) 121 21Zs + e where 0 < e < 2c. The product z21Q(s)/P(s) (or -y21 Q(s)/P(s) ) then has the form (6-8) (, ) (or Q(s))] K (s+c-jd)(s+c+jd) 21 P(s) 21 P(s) J s+e If ZT(s), A(s), YT(s), or G(s) is specified, then z22 (or y22) will have the form given in (6-4). For A*(s) or G*(s) specified, z22 (or y22) will have the form given in (6-5). Since z21 (or -y21) is a positive real driving point function, a two-terminal realization may always be effected. A sharing of transmission zeros is represented in that the finite transmission zero at s -a is produced by z21 (or -y21), while a transmission zero at infinity is produced as a private pole of z22 (or y22). 3) Let T(s) have the form T(s) = (s + a jb)(s + a + jb) (6-9) (s + c jd)(s + c + jd) where a > 0, b > 0, c > 0, and d 0. The single necessary and sufficient condition for a function of this form to be a positive real driving point function is given by S2 + b2- c2 +d2 4 ac. (6-10) It will be assumed that T(s) is not positive real and there- fore that the condition given in (6-10) is not satisfied. If z21 (or -Y21) is chosen as (s + a jb)(s + a + jb) [z (or -y )] K (6-11) 21 21 s + e where 0 s, e 2a and 0 e < 2c, a realization may be easily effected using only two-terminal techniques. The product z21Q(s)/P(s) (or -y21Q(s)/P(s) ) then has the form given in (6-8). If ZT(s), A(s), YT(s), or G(s) is specified, z22 (or y22) will have the form given in (6-4). If A*(s) or G*(s) is specified, z22 (or y22) will have the form given in (6-5). Here, z21 (or -y21) is positive real, and, therefore, a realization may always be effected using only two-terminal techniques. No sharing of transmission zeros is possible since both members of the pair of complex conjugate trans- mission zeros must be assigned to z21 (or -y21). 4) Let T(s) have the form T(s) (s + a)(a + b) (6-12) (s + c jd)(s + c + jd) where a 0, b > 0, c > 0, and d > 0, with a and b distinct. It will be assumed that T(s) is not a positive real driving point function and therefore that the condition Vab V2 + d22 2c (a +b) (6-13) is not satisfied. A realization may always be effected by choosing z21 (or -Y21) as (s + a)(s + b) [z21 (or -y21) ] K + b) (6-14) in which 0 e e 2c and 0 o e (a + b). The product z21 Q(s)/P(s) (or -y21Q(s)/P(s) ) then has the form given in (6-8). In the synthesis of ZT(s), A(s), YT(s), or G(s), z22 (or y22) will have the form given in (6-4). In the synthe- sis of A*(s) or G*(s), z22 (or y22) will have the form given in (6-5). Since z21 (or -y21) is positive real, the complete realization may always be carried out using two-terminal techniques. A realization may frequently be effected in which the transmission zeros are shared. In order to produce this sharing, the numerator polynomial, N(s), of z21 (or -y21) should be chosen as either s + a or a + b. Without loss of generality s -a is chosen as the zero of N(s). z21 (or -y21) is assigned a simple pole at s -e, which gives s + a [z21 (or -y21)] K a (6-15) s+e The selection of e is made in such a way that the expression ( reb- Vc2 + d2)2 2c (e + b) (6-16) is satisfied. It will be found convenient to assure that e satisfies the restriction, [e (e b)]2 > (c e)2 for e > b, or (6-17) [b (b e)]2 > (c b)2 + d2 for b > e. The product z 21(s)/P(s) (or -y Q(s)/P(s) ) then has the form [z Q(s) (or y (s)) = 21 P(s) 21 P(s) J (s+c-jd)(s+c+Jd) (s+e)(s+b) In the synthesis of ZT(s), A(s), YT(s) or G(s), z22 (or y22) will have the form [z22 (or y22)] K/H (s+c-d)(s+c+d) . 22 22 (s+e)(s+b) J (6-19) In the synthesis of A*(s) or G*(s), z22 (or y22) will have the form 22 (or = K/ (s-c-jd)(/s+c+d) 22 22 1 (s+e)(s+b) I 5) Let T(s) have the form S (s+a-Jb)(s+a+jb) +c)T(s)(s+d) (s+c)(s+d) (6-20) (6-21) (6-18) where a > 0, b > 0, c > 0, and d > 0 with c and d distinct. It will be assumed that T(s) is not a positive real driving point function, and therefore, that the condition ( -d -a2 + b2) < 2a(c + d) (6-22) is not satisfied. A realization may always be effected by choosing z21 (or -y21) as (s+a-jb)(s+a+jb) [z21 (or -Y21)] = K (6-23) s+e in which 0 < e < 2a and 0 < e < (c+d). The product z21 Q(s)/P(s) (or -y21Q(s)/P(s) ) then has the form QF Q(s) Q (s)) K (s+c)(s+d) Lz2 (or -ys K (6-24) 21 P(s) -Y21 P(s) J s+e In the synthesis of ZT(s), A(s), YT(s), or G(s), z22 (or y22) will have the form (6-25) [ cd-e(c+d-e)1e [z (or y )] = K/I s + (c+d-e) + cd-e(c1 22 22 s+e while in the synthesis of A*(s) or G*(s), z22 (or y22) may be written as (6-26) [z22 (or y22)] = K/H s + (c+d-e) + cd-e(cd-e) 2s+e Since z21 (or -y21) is positive real, a realization using only two-terminal techniques is possible. No sharing of transmission zeros is possible in this case since both members of the complex conjugate pair of transmission zeros must be assigned to z21 (or -Y21). The five cases considered above include all physically- realizable transfer functions, whose denominator is of sec- ond degree, which were not included in the synthesis methods given in Chapter IV and Chapter V. The transfer functions considered in Chapters IV and V, which have denominators of second degree are found to belong to the special classes, which were defined in section 4.2 and section 5.1, respec- tively. It may, therefore, be concluded that all transfer functions which have denominators of second degree may be synthesized using only two-terminal techniques. Methods have been given for the synthesis of every physically-realizable minimum phase transfer function whose denominator is of second degree. However, only those choices of z21 (or -y 21) have been considered in which the zeros of D(s) are assigned to points on the negative o axis or on the ju axis. The possibility of choosing a complex conjugate pair of poles for z21 (or -y21) has not been con- sidered. As previously described, one should assign as few poles and zeros to z21 (or -y21) as possible and still assure that the product z21Q(s)/P(s) (or -y21Q(s)/P(s) ) is positive real. In general, when more poles and zeros than the minimum required number are assigned to z21 (or -y 21), an excessive number of circuit elements is required in the realization, Since the synthesis of all transfer functions, T(s), whose denominator is of second degree can be carried out with z21(or -y21) having a single pole on the negative o axis, it would be expected that the assignment of a pair of complex poles to z2 (or -y 21) will cause an excessive number 21 21 of circuit elements to be necessary in the realization. Although this is true, consideration of the complex conjugate pole assignment is important from a theoretical standpoint, and will be found useful in the synthesis of transfer func- tions whose denominator polynomial is of third degree or higher in s. A procedure will now be given for synthesizing any of the five forms of T(s) given above, in which one has the freedom to choose the poles of z21 (or -y21) and z22 (or y22) as a complex conjugate pair. Attention is first turned to a preliminary subject, the problem of constructing a positive real function, F(s), whose poles are to appear in complex conjugate pairs. With the form of F(s) given as F(s) = (s + c jd)(s + c + jd) (6-27) (s + e jf)(s + e + jf) the problem may be stated as follows. If the zero locations, s = -c jd, are known, what values may be chosen for e and f so that F(s) will be posi- tive real? Permissible values for e and f are determined by a simple geometric construction which is explained in the following four steps: (A proof of the validity of this procedure is given in section 6.2.) 1) On a piece of graph paper construct a rectangular coordinate system to represent the s plane. With a compass, mark the point where a circle, whose center is at the origin and which passes through the point -c +jd, intersects the positive jw axis. (This intersection is, of course, at s j 4c2 + d2.) Call this point jucl 2) From the specified value of c, and u1 as just deter- mined, calculate r, where r is given by r 2c + 1. (6-28) (It is convenient to perform the addition in (6-28) by sum- ming lengths on the jw axis.) With a compass mark the points at which a circle, whose radius is r and whose center is at s = jul, intersects the real axis. Call the point at the intersection of this circle with the negative o axis -x0, and the point at the intersection with the positive a axis xg. 3) Construct a circle, C1, whose center is at s -x0, and whose radius is r, as given in (6-28). Construct a second circle, C2, whose center is at s x0, and whose radius is also r. This construction is illustrated in Fig. 6.1 for c 1 and d 2. Fig. 6.1 The construction of a region in which poles of F(s) may be selected. 4) A pair of complex conjugate poles may be placed anywhere within or on the closed region whose interior is to the left and outside C2, and inside C1. The shaded area in Fig. 6.1 indicates such a region. For this assignment of poles, F(s), as given in (6-27), is a positive real driving point function. The case will now be considered in which the zeros of F(s) lie on the negative portion of the a axis. F(s) is to have the form F(s) (s + c)(s + d) F(s) (6-29) (s + e jf)(s + e + jf) The zeros at s = -c and s -d are considered to be previ- ously specified, and e and f are to be chosen so that F(s) will be positive real. A region in which poles of F(s) may be selected can be determined by following steps 1 through 5 with two modifications. First, the point, jul, is now given by ju j e (6-30) and, second, the radius r is now given by r = u1 + c + d. (6-31) An illustration of this construction is shown in Fig. 6.2, for F(s) as given in (6-29), with c 1 and d 2. If a pair of complex conjugate poles is chosen for F(s) within one of the regions just defined, it is certain that F(s) will be positive real. Although such a choice is sufficient to assure that F(s) is positive real, it is not necessary that the poles be chosen within these regions. Fig. 6.2 A region in which poles of F(s) may be selected, fpr F(s) as given in (6-29) Larger regions have been shown to exist. However, these regions are much more difficult to construct [7]. The above procedures for choosing the poles of a posi- tive real function may be used in synthesizing any one of the five forms of T(s) given in equations (6-1), (6-6), (6-9), (6-12), or (6-21). After choosing an acceptable pair of poles for z21 (or -Y21), the synthesis is carried out by first, selecting a value of K/H, and second, synthesizing the network N' or N". If z21(or -y21) is chosen as a positive real driving point function, the synthesis may be carried out on a two-terminal basis, with the resulting net- work having an L configuration. For such a realization, K/H should be chosen in such a way that z' z' (or y' y' ) 22 21 22 21 is a positive real driving point function. If, however, a general four-terminal realization of N' or N" is to be effected, then in the selection of K/H, it is only neces- sary that property 4 for the network parameters, as given in Chapter II, be satisfied. Illustrations will be given in section 6.5 for those transfer functions, T(s), whose denominator polynomial is of second degree in s. 6.2 The Derivation of Regions In Which A Positive Real Function, F(s), May Have Poles The validity of the construction procedure, given in steps 1, 2, 3, and 4 of section 6.1, will now be established. F(s) is to be a biquadratic driving point function of the form given in (6-27) or (6-29). For specified zero loca- tions, if the poles of F(s) are chosen to lie within a certain region of the s plane, it will be assured that F(s) is positive real. A derivation of these regions will be given in the following eight steps: 1) Definition of a C path [7]: Consider a circle which lies partly in the left-hand half of the a plane and partly in the right-hand half of the s plane, and whose center is on the real axis. A C path is, by definition, that portion of the circle which lies in the closed left-hand half of the s plane. Also by definition, the negative direction on a C path always points to the left. Fig. 6.3 shows a typical C path with its negative direction indicated. Fig. 6.3 A typical C path with arrows indicating its negative direction 2) The phase function of a quadratic: Let Q(s) be a real quadratic given by 2 2 Q(s) 2 a + 2as + a + 2 with by (6-32) a > 0 and B 0. The phase function of Q(jw) is given (6-33) -1 2au (w) tan- 2 Q 2 2 2 a +8 - 3) A property of the phase function as the zeros of Q(s) are moved along a C path [7]: Let the zeros of Q(s) move in the negative direction on a C path that intersects the ju axis at s ju (WI > 0). Then the phase function SQ(w) of Q(jw) decreases monotonically for every fixed w in the interval wl < w < oo and increases monotonically for every fixed w in the interval 0 < w < u This has been shown by Steiglitz and Zemanian [7]. 4) A property of a biquadratic driving point function, F(s), whose poles are restricted to lie on the ju axis: Let F(s) have the form Q1(s) 92 + 2as + a2 + 2 (6-34) F(s) (6-34) Q2(s) 2 + 2 s + W1 If values of a and B (a > 0, B > 0) are specified, only one permissible value of wl exists for which F(s) is positive real. A proof of this fact using the general procedure given in Chapter V is as follows: Q1 () has the form Q1(s) = s2 + 2as + a2 + B2 (6-35) From (5-14) the permissible values of w1 are the roots of the equation 0 -R2 + a2 + 2 (6-36) which gives S- R a2 + B2 (6-37) Similarly, if F(s) has two distinct zeros on the negative a axis at s -c and s -d, then F(s) has the form s2 + (c + d) s + cd 2 2 s +WJ 1 Again, only one permissible value of w' 1 of w' may be found by applying equation 1 as given in (6-38), this gives (6-38) exists. This value (5-14). For Q (s) 1 0 -R2 + cd 0 -R + cd (6-39) (6-40) S- R -cd . 1 5) A property of a biquadratic driving point function, F(s), which has a double pole on the negative portion of the a axis: First, let F(s) have the form Q1(*) 52 2 2 F(s) 8s2 + 2as +a2 B2 F(s) Q2(s) (g + T)2 where values of a and B (a > 0, B >0) are specified. From F(s) Q2(s) (6-41) the single condition of physical realizability for a biquad- ratic driving point function, it is necessary that the condition [ a2 +2 2 < 4a (6-42) be satisfied [8]. This inequality determines a segment of the negative a axis on which a double pole of F(s) may be placed, with the result that F(s) is a positive real driving point function. The end points of this segment are deter- mined by assuming equality in (6-42), and then solving for the roots, Tl and r2, where T1 > T2. The roots, T1 and r2' are given by (6-43) I ( a+ B2 + 2a) + /[ a 2 + 2a2 (a2 + B2) and (6-44) 2 ( a + 2 + 2a) + a2 B2 + 2a]2 (a2 +B2), respectively. From (6-37), yl and r2 become T1 (1 + 2a) + 1(w1 + 2a)2 2 (6-45) 1 1 1 |