Digitized by the Internet Archive in 2013 http://archive.org/details/synthesismethodf02yaru A SYNTHESIS METHOD FOR BROAD-BAND ANTENNA IMPEDANCE MATCHING NETWORKS 310 f2 2 This volume Is bound without _ which is/are unavailable, ite Dat e: 1 February 1955 Approved by: R.H, DuHamel Research Assistant Professor E C Jordan Professor E.tAAc THE LIBRARY OF TKF ••' 1 6 1955 UNIVERSITY OF ILLINOIS ANTENNA SECTION ELECTRICAL ENGINEERING RESEARCH LABORATORY ENGINEERING EXPERIMENT STATION UNIVERSITY OF ILLINOIS URBANA. ILLINOIS ' ■? A SYNTHESIS METHOD FOR BROAD-BAND ANTENNA IMPEDANCE MATCHING NETWORKS Contract No. AF33(616)-310 RDO No, R- 112- 110 SR-6f2 TECHNICAL REPORT NO. 2 by Nicholas Yaru Research Associate Date: 1 February 1955 Approved by: R.H. DuHamel Research Assistant Professor ^KIa^^JU fur- E C Jordan Professor E.tA -JttU. THE LIBRARY OF TKF MAY 1 6 1955 UNIVERSITY OF ILLINOIS ANTENNA SECTION ELECTRICAL ENGINEERING RESEARCH LABORATORY ENGINEERING EXPERIMENT STATION UNIVERSITY OF ILLINOIS URBANA, ILLINOIS TABLE OF CONTENTS ^3 in Page 1. Introduction 1 2. Derivation of Transformation Equations 2 3. Application of Derived Open-Circuit Reactance Functions 12 3.1 Rational Function Approximation of Open Circuit Reactances 12 3.2 Positive-Real Conditions 17 4. Derivation of Useful Matching Procedure 19 5. Application of Matching Technique to Specific Antenna Impedances 37 6. Conclusions 81 References 83 Appendix I Bilinear Transformation and Properties of Two Terminal Pairs 84 Appendix II Positive-Real Functions and the One-Terminal Pair 87 II. 1 The Lossless One-Terminal Pair 93 Appendix III Properties of the Two-Pair Network 96 Appendix IV Cascade Matrices Representing a Line Transformer in Series with a Non-Dissapative Network 106 Appendix V Unsymmetrical Lattice Network 110 IV ABSTRACT The general problem of matching a given antenna impedance to a transmission line characteristic impedance to within a prescribed volt- age standing wave ratio over a band of frequencies is attacked. A system of equations representing the open-circuit driving-point and open-circuit transfer impedances of a lossless four terminal match- ing network is developed. These frequency-dependent relations are written in terms of the conditions existing at the terminals of the network. Because these prescribed conditions are not complete (that is, only the maximum voltage standing wave ratio is given along with the impedance to be matched), a major problem in the overall solution was that of determining a feasible functional form of the frequency- dependent reflection coefficient (H:) at the matched terminals. If this relation were known, sufficient independent conditions would be avail- able to permit solution for the three open-circuit reactance functions describing the impedance matching network. That a general mathematical description of the variation of the complex reflection coefficient at the matched terminals is unavoidably difficult is evident when one considers that the magnitude of the reflection coefficient can range in any manner as a function of frequency within the bounds of the prescribed VSWR circle. Furthermore, no restrictions whatsoever are imposed (for the receiving antenna case) on the form of the phase angle of the reflection coefficient. As a result, any comprehensive theory attempting to incorporate a general functional form for the r> soon becomes unwieldy. If, however, the variation of T^ is postulated (based on previous work and mathematical analysis), then a specific matching solution (one of an infinite number) dependent upon the above assumptions isavailable„ Simplicity in the resultant mathematics and in the synthesized network was another criterion upon which the above assumptions were based. V The aforementioned reactance functions which the theory yielded were approximated next by physically realizable functions (Foster's reactances in this case) so that a matching network closely fulfilling the prescribed conditions could be synthesized in the form of lumped inductances, capacitors, and line transformers. The synthesis procedure was applied to two practical antenna impedances, namely, a monopole through resonance and the electrically small partial sleeve antenna. For both cases it was required that the antenna impedances be matched to a standard cable impedance (50 ohms) to within a prescribed VSWR of 3:1 and 5:1 , respectively. Bandwidths of 30 and 50 percent, respectively, were attained with rather simple matching networks. 1 . INTRODUCTION The problem attacked in this work is both old and formidable. This is the problem of matching an antenna impedance (the resistive and reactive components of which are functions of frequency) to the characteristic impedance of the associated transmission line over a broad band of frequencies. Numerous matching techniques ' * varying from cut and try methods to more direct mathematical procedures ' 4 have been developed and are being employed currently. However, each direct method is usually restricted for use with impedances that lie in a specific region of the R X plane. Consequently the engineer may find the procedures applicable only to a limited portion of the impedance curve so that he is generally forced to rely on cut and try methods. The latter technique is time-consuming and often not too rewarding, especially when applied by a comparative neophyte. This same tool in the hands of an engineer experienced in impedance matching is a useful one which can yield results when adeptly applied It is apparent from the effort already expended on the problem by numerous contributors through the years that no simple solution exists. However, a more logical and flexible mathematical approach to the problem certainly would be useful and desirable. It is hoped that this work can be classed as a step toward such an ultimate solution. 2. DERIVATION OF TRANSFORMATION EQUATIONS The need for impedance matching networks arises from the de- sirability of efficiently transferring the available power from a signal source to a given load. The basic conditions for maximum power transfer from generator to load are common engineering knowledge, and the der- ivations o f these conditions are available in numerous text books. However, the more common techniques for fulfilling these conditions are limited to matching impedances at one frequency or at most over a very small frequency band. These narrow band matching techniques are limited further to impedances whose resistive components remain constant over a given frequency band. Applicable methods to the broad-band matching problem for impedances with resistive and reactive components both frequency dependent are not so readily available. A direct mathematical solution to the general problem must be capable of representing the variation of a number of frequency dependent parameters over a wide range. This is apparent when one considers the extreme degree of arbi- trariness for the allowable variation of the complex reflection coef- ficient at the matched terminals. That is, the magnitude of the re- flection coefficient can range in any manner as a function of frequency so long as this magnitude remains equal to or less than the prescribed maximum Furthermore, no restrictions whatsoever are imposed (for the receiving antenna case) on the form of the phase angle of the reflection coefficient. As a result, any comprehensive mathematical theory at- tempting to incorporate the general functional form of the complex reflection coefficient increases in complexity to such a degree that the expressions soon become unwieldy. Two mathematical attacks on this problem were attempted in which the expressions related the prescribed magnitude only of the reflection Other than those cited in references 1 and 3. -3- coefficient to the antenna impedance transformed through a loss les s four terminal network. The transformed impedance may be related to the characteristic impedance of a designated transmission line in terms of the com ple x voltage reflection coefficient. Analysis in terms of abso- lute magnitude alone led to quadratic forms involving undesirable cross product terms (functional expressions for lossy networks) so that the theory was further complicated by the necessity of transforming the principal axes of the representative quadric surface to eliminate the cross product terms. Because of the extreme complexity and unwieldiness of the resultant expressions, these approaches on the problem were discarded. It was then decided that a combination mathematical - cut- and-try method might yield the most practical solution. It is de- sirable of course to minimize the dependence of the overall match- ing technique on its cut and-try aspects. Modification of the expressions (described briefly above) to in- clude, in simplified form, the unprescribed phase angle (as well as the magnitude) of the reflection coefficient and to include a complex para- metric function resulted in comparatively manageable equations. As a consequence, expressions were derived which enabled the mathematical transformation (not necessarily physically realizable) of a general antenna impedance into a prescribed voltage standing wave ratio circle, Cut-and- try methods were applied to the system of equations to determine the unknown phase angle of the reflection coefficient for the trans- formed impedance referred to the prescribed standard cable impedance. However, a close first order approximation to the magnitudes of the phase angle is available froin the mathematical analysis so that the CUl and try process is not cumbersome In addition, the conditions for 1 uve reality were imposed on the phase angle determination so that the expressions for the open -circuit driving point and transfer im- pedance! of the two terminal pair were physically realizable.. This 4- approach resulted in the system of equations and the procedure which was finally adapted for the solution of the matching problem. In Fig. 1, the antenna impedance Z is connected across terminal pair 1 of the four terminal network. The transformed impedance TZ a at terminal pair 2 must be matched to within a prescribed voltage standing wave ratio (VSWR) with respect to the transmission line characteristic impedance, R Q . 2 T.L. a R + jX Matching Network TZ. FIGURE 1 Under the conditions that the two terminal pair is linear, passive, and bilateral, its steady state behavior is described by a linear trans * formation of the form AZ„ + B TZ, (2-1) CZ a + D A, B, C, D, are complex numbers independent of Z a and R Q and are constants at any given frequency. However, these network parameters are functions of frequency and of the circuit elements (resistance, capaci- tance, inductance, characteristic impedances, and lengths of lines) included in the network. A consideration of the possible insertion loss of the network indicates the desirability of making the two terminal pair lossless. Furthermore it is evident that a network with excessive loss could transform an antenna impedance into one matching the transmission line Appendix I -6 1 i * Pi e = gj + j g 2 (2-7) where gi = pj cos i , g 2 ■ p£ sin 6i so that the right side of Eq. 2-6 becomes TZ = R o [1 + gi + J g£J 3 1 ~ gl " j g2 (2-8) and A t ((j)[R(h )) + jX(h)) ] + jB 2 ((o) _ R n [l + gl (M) + jg 2 (u)j jC 2 (u)tB) + j g 2 (h))] [D a (w) - C(w)X(w)] + jC 2 (w)R(w) h((o)e j0t ' U) [1 - gi(u) - j g 2 (u)] (2 13) Since all terms in Eq. 2 13 except R Q are functions of frequency, that notation now will be discontinued as a simplification with the understanding that only R Q is a constant A,R + j[A t X * B 2 ] u he R Q [1 + gl + j g 2 ] ^ [D, C,X] ♦ jC 2 R heJ a [\ - gl - j g2 ] 8 The unknown function H(w) was introduced as an expedient to permit equating the numerators and equating the denominators of Eq. 2 14. Evidently four comparatively simple linear, independent equations may be derived from these equalities since the numerators and denominators have real and imaginary terms. That such a mathematical manipulation is proper may be deduced from the following analysis for real variables. Given that ■*■ = -§•,. then by the definition of positive rational numbers ad - be . However a ^ ~ - "j* implies that That is so that b = ka and d - kc, a b ka c d kc k . i . A K a c b ka Therefore with "a" and "c" known, the equality ~ = t — remains d kc completely general. Equating the two numerators and equating the two denominators imposes no restrictions on the unknowns (b and d) since the proportionality constant (k) is general. j a In expression 2 13, the complex function he represents the extension of the above arguments to the complex variable domain* The parametric function would have no utility however if its mathematical formulation remained unknown. Recall however that a fifth independent equation is available, namely, the non-linear equation, 2-11. As a con- sequence, the mathematical form of the magnitude of the function may be derived. -9- Summarizing then, an unknown complex function was introduced into expression 2-9 thereby greatly simplifying the mathematics for ob- taining four linear equations in terms of the network parameters A, B, C, D. A fifth independent equation (non-linear) makes possible a solu- tion for the magnitude of the complex parametric function, There remain therefore two more unknowns, the phase angle of the reflection coef- ficient and the phase angle of the complex function. * A relationship between the two phase angles can be postulated so that some criterion for specifying the variation of either phase angle remains to be found. Consider Eq . 2-14 which is equivalent to the expression below ■ hen he is further defined as he = hj + jh 2 A t R + j(A t X - B,) x P. Q [(1 + g 1 )h 1 - h 2 g 2 ] + jR Q [h 2 (l + gl ) + h lgg ] Dj - C 2 X + jCsB [(1 - gl ) ni > h 2g2 ] + jChaCl-gj- h lg2 ] (2-15) Equating the real parts and equating the imaginary parts of the numerator yields A t R ■ R [(l ! - gl )hi - h 2g2 ] A t X ► B 2 ■ B [(l - gl )h 2 + h ig2 ] By equating the real parts and by equating the imaginary parts of the denominator D x - C 2 X - (1 - gl )h t + h 2 g 2 C 2 B ■■ (1 - gi)h 2 - hig 2 So that by transposing A, .'■*! [(1 B, Ro [(1 + gi)h a ' h lg J - A t X \ ~f [(1 ■ gi )hi - h 2 g 2 ] (2-16) D C,X (1 - gl )h, * h 2g2 r - (1 ~ gi )"/ «.»i R • A* »u m(. > i ')»■» ribad in Chap 10- and from four terminal network theory AiDi + B 2 C 2 = 1 (2-11) From the determinant (A) of the four linear expressions for A 1( B 2 , C 2 , and Di it is evident that the equations are independent since the deter- minant is not equal to zero. A = 10 X 1 -X 1 10 - - l M From Eq. 2-16 expressions for the open-circuit driving point and open-circuit transfer reactances are derived k 22 (1 - gi)h 2 - hega (2-17) x - ~ Dl - (x i R[(1 " gl)hl * hag * ] ) C 2 (1 - gi)h 2 - higa (2-18) X12 = X 2 1 = — _ - R C2 (1 _ gi)h 2 - higs (2-19) and Subsituting into Eqs, 2-17, 2-18, and 2-19 the expressions he s hi + jh 2 a h cos a * jh sin a r i ■ Pi« J g? s Pi cos ©i + J Pi sin _ -R [cos a + pi cos (a + Q±)] A 22 ! ; , - i sin a - p£ sin (a + 9i) R[cos a - Pi cos (a + Oi)] An " — a - " — ■ — ; ; . , — r sin a - p^ sin (a + 6^ ) Xa -R h[sin a - ?£ sin(a + 0^] (2-20a) (2-20b) (2-20c) 11 An expression for determining "h " is obtained by substituting Eq. 2-16 into 2-11 so that R B (l - Pi) (2-21) 12- 3. APPLICATION OF DERIVED OPEN-CIRCUIT REACTANCE FUNCTIONS 3 1 Rational Function Approximation of Open-Circuit Reactances At this point, one method of attacking the synthesis problem might be to attempt polynomial approximations of the numerators and the de- nominators of each function Xn, X 22 , and Xi 2 (2-20). The polynomials must be of such form that Brune's 6,? * 8 conditions for positive reality (implying physical reali zabili ty ) are fulfilled. Consider that the terms in the reactance functions are transcendental functions so that their exact representation would require using infinite series. Choos- ing a finite number of terms from these series is of no practical assistance since the number of terms in the finite series must still be large (of the order of 15 terms). This condition is dictated by the fact that the phase angle of the reflection coefficient may have a magnitude of several radians. Hence, a large number of terms in an approximation of the sine and cosine series is required before the series converges to even approximate values for sine and cosine. However, if one sets up the expressions in high degree rational function form, cut-and-try techniques must be applied to the equations to determine the values of the phase angles 6^ and a to satisfy the conditions derived from the "positive-real" concept that are applicable. It is understood further that the terms in the open circuit reactance expressions must be approximated as functions of frequency. It becomes evident now that the cumbersome high degree approximation problem in conjunction with necessary cut-and-try technique definitely renders this attack futile. One important piece of information, though, which may be deduced from the rational function form of Xn, X 22 , and X 12 concerns the physical form of the impedance matching network. That is, the network -13- cannot be either a simple "T" or "n" two-terminal pair. In Chapter 5. matching networks are presented in the form of a "T" in cascade with an ideal transformer which differs, however, from a simple "T" consisting of inductors and capacitors alone. Figure 2 presents the "T" and "tc" networks and the relations for the series and shunt reactances in terms of the open-circuit reactances. o X a — | — X b o X c o 1 o o 1 X 2 1 o X l X 3 o 1 1 o Xii X. X 22 x c - x 1: 111 *22 2 X i2 X 22 X 12 X 11 X 2 2 - Xl2 Xis X X22 - Xis FIGURE 2 ii 12 The limitations are apparent for the "T" configuration by con- sidering that X - X12 so that Xi 2 ( a transfer reactance) must be physically realizable. Hence, the highest power in the numerator and in the denominator can differ at most by unity. Similarly, the lowest power in the numerator and in the denominator can differ at most by unity. Now X 12 ■■ -—-, =Ji — . _. (2- 20c) h[sin a - Pj sin (a + 0j)] :th so that a 15 degree approximation for the trancendental functions in the denominator re qui r t.hfit the approximation for the radiation re- 'Appendix II -14 sistance in the numerator be either 14 or 16 degree. That is, R must have 14 or 16 zeros on the real frequency axis. This is il- logical since the radiation resistance is regularly behaved and does not equal zero for any physical length. Hence, a high degree approxima- tion of R is practically impossible. The same argument applies for reactance arm X 2 in the network, K, because of the X i2 term in the denominator. Consequently the physical network must be a more complex, unsymmetrical type. The fact that de- rived expressions for X X1 and X 22 differ precludes any use of a sym- metrical network (i.e., in a symmetrical network Xii = X 22 ). At this point it seems advisable to check whether the derived expressions for Xn, X 2 2i and X i2 have any utility whatsoever. Is it possible, even mathematically, to transform a given antenna impedance to within a prescribed VSWR? Toward this end an impedance was selected and plotted in Fig. 3. It was prescribed that the curve be transformed into a VSWR circle of 5 to 1 with respect to a 50 ohm line. Refore the expression 2-20 can be used, however, some relationship between the phase angles indicated in the equations must be derived or postulated. One can assume the relation between the phase angle (9j) of the reflection coefficient and the phase angle (a) of the function H to be a + 0£ = qp(w) where the form of + 2 and X 12 . These values were used to compute (by slide rule) the values of X , Xl, and X , the reactance arms in a T matching network which is not physical . For a quick mathematical check, however, the T is desirable since its impedance transformation calculations are relatively simple and negative inductances and capacitors, though not physical, give the correct answers. The transformed curve is shown in Fig. 3 and Tables 1 and 2 pre- sent the initial and final data. It is evident that the transformed impedances lie at the proper magnitudes and phase angles for T^ that were assumed initially. Assumed Parameters Calculated Values = L/X Pi 9 i Xu 1 ^ X 12 X 2 2 0.1 0.667 200° 229.5 6.6 -8.8 0.125 0.470 100° 186.9 13.4 41.9 0.150 0.667 10° 156.3 65.1 283.5 0.175 0.470 60° 96.2 33.8 107.0 0.20 0.667 140° 66.5 15.1 18.2 TABLE 1 Calculating X , Xl, X for the T network at each frequency from the above X llf X 12 , and X 2 2 and placing the antenna impedance across terminal pair 1 yields the following TZ a : *a *h z a 1 • X c 2 TZ. FIGURE 4 -17 Calculated LA TZ_ Calculated Pi 6 i 0.1 10.8 -j 7.5 0.65 0.125 25.4 + j 34.4 0.46 0.15 186.1 + j 73.2 0.62 0.175 51.5 - j 60 0.50 0.20 11.3 - j 17,3 0.66 TABLE 2 This example shows therefore that it is mathematically possible to 198° 98° 11° 57° 139° transform a complex impedance Z a into a desired impedance TZ a lying within a prescribed VSWR circle using the derived equations for the open-circuit reactances Xu, X 22 and X 12 . 3.2 Positive-Real Conditions Since the previous reference has been made to the positive-real concept it seems desirable to include a brief discussion of the theory at this point. A more comprehensive treatment is given in the Appendi- ces but the salient points in the positive-real concept will be indi- cated in this section. Considering the complex frequency plane (p = 6 + jw), for p in the right half plane, general driving-point immittance (impedance or ad- mittance) functions of passive networks have positive-real parts, and their angles are bounded in magnitude by the angle of p. Every other property for driving-point immittance functions of passive networks is implied or may be derived from these conditions. Brune named a function having these properties as "positive-real" and his basis for setting up the positive-real property was derived from energy considerations for which positiveness has physical meaning. From a consideration of the P-R (positive- real ) property, notice that for driving-point immittances 1. When p is reaHw ■ 0) then Z(p) or Y(p) is also real. 2. When p is in the right half plane (6 > 0) then Z(p) or Y(p) is ■IsO in the right half plane. That is, when the real part of p is posi- -18- tive, the real part of the immittance is also positive. Further if p = jw (boundary line) then the real part of the immit- tance function is equal to or greater than zero for passive networks. It is necessary that a driving-point immittance function of a passive network be "positive-real". Any physical two-terminal network that is passive has driving-point impedances or admittances that are P-R. Suf- ficiency may also be shown. The notion of positive-reality contains implicitly almost all the important properties of driving-point immittance functions of a one- terminal pair. An idea due to Cauer made it possible to use one-terminal pair theory to determine an interrelation among Zii, Z 2 2 , and Z 12 of fundamental importance for two-terminal pairs. Hence, the necessary conditions for physical realizability which must be met by a set of open circuit driving-point and transfer impedances of a two-terminal pair were derived This material is presented in Appendix III. -19- 4. DERIVATION OF USEFUL MATCHING PROCEDURE The preceding mathematical spot check indicates the feasibility of the equations for determining the open-circuit driving-point and open- circuit transfer reactances of the two-terminal pair matching network. The ultimate utility of the system of equations is dependent upon whether or not boundary conditions may be applied which will insure that these relations are physically realizable. It is necessary further that such conditions lead to restrictions on the variation of the unknown reflection coefficient phase angle as a function of frequency at the matched terminals of the network That is, specific ranges of variation for 9^ quite possibly may result in order that the relations for X llt X 22 an d Xj. 2 be physically realizable. It may be shown that for a physical lossless, two-terminal pair the following relationships must be fullfilled: k!i > (4-1) KL K 22 - (K 12 ) 2 > . (4-2) s The symbol Ki i represents the residue of the pole at frequencies s corresponding to w = Wi,to 2 ,' , °' w s for the function X lx . The symbols K 2 2 s and K i2 are similarly defined for X 22 and X t 2 , respectively. Evidently the residues of the driving point reactance functions must be real and positive: Note further that it is necessary that all the poles in the function X 12 be present in both functions X xl and X 22 . X 11 and X 22 may have independent poles that are not present in X 12 but each pole in X 12 must be present in X xl and X 22 in order that KL K 22 - (K! 2 ) 2 > . (4 2) Appendix III -20- Recall from Eq . 2-20 that the denominators for X u , X 22 and X 12 are identical except for the multiplying parameter h in X ia - This term (h), however, can never be zero (see Eq. 2-21) unless the antenna radi- ation resistance is zero so that the h term introduces a pole only at zero frequency. Consequently all the poles in X 12 are present in both Xu and X 22 , excepting the pole at the origin. Because the conditions for physical realizability are stringent, it is hoped their application will lead to a relationship between 9^ and the other unknown phase angle, a. However, application of the conditions for physical realizability to the transcendental expressions (2-20) for X llt X 22 , and X 12 is ex- tremely difficult unless some starting point (as the variation of the a or 9j with frequency) is known, perhaps from the physical nature of the problem. In other words, upon examination of the Eq. 2-20, one can hardly expect to fix the boundary conditions on the three open-circuit reactances when the functional forms of the independent variables with respect to frequency are unknown. If one could successfully ascertain the functional form of either phase angle, or a relationship between the two, the above results would yield an elegant mathematical solution to the matching problem. However, one can readily conclude that such an ambitious undertaking would be prohibitively time-consuming. As a result of much analysis and calculation it appeared desirable to postulate general relationships, linear and non-linear, between a and 0- and to consider the oft times simplified forms for the open circuit reactances. This may be accomplished by assuming that the transformed impedance curve will be displaced about a terminal angle, $ • For example, if 9 i : $ ■■ 2q>(w) and a ■ -$ + 5 \ .22 75/ \ 250 .26; ^ ^22 X_<£(s) ^^^Pole - — _jS FIGURE 5. POLE AND ZERO CURVES FOR MONOPOLE IMPEDANCE WHEN e i = e - 2 . b. the zeros must be simple and conjugate. c. if the denominator is an odd function of frequency, the numerator must be an even function and vice-versa. d. the degree of the numerator can exceed that of the denominator by at most unity. e. the degree of the numerator can be any degree less than that of the denominator so long as (b) and (c) are fulfilled. Hence, the general form of X 12 ' need not be necessarily a Foster's form but will have the form Y , _ oj N 3 ((/ - h)g) 12 / 2 2 W 2 2 W 2 2 \ -30- where the zeros conform with properties (b), (c), (d), and (e). The problem is now one of approximating a non-physical reactance curve (transcendental form) with a physically realizable curve. Hence it is desirable to rewrite the transcendental relations (2-20) in their simplest forms in order to minimize calculations involved in the ap- proximation task. Toward this end, the following expressions can be derived: Xll - -x- BU -Pi) cot cp ( l + Pi) l 2 2 -Rn COt (4-5) if Xx ^F± h(l + p^) sin cp ) i = 2nn - 2cp(w) a = -2nrc + cp(w) so that where a + 9^ = -cp(u) $o = 2r\K . Since <$ wa s chosen equal to a constant, the resultant range of variation for cp(w) may be greater but this result is unimportant so long as pj (the reflection coefficient magnitude of the transformed impedance) is equal to or less than the prescribed value. It is evident from Eq 4 5 that the poles for the three functions occur at sin

c ) then X x u N 3 ••• ( •••••• ) (w 2 - Wi 2 )( ■■•■•'■• ) 35 where u < Ui < w L 22 u) N 2 (w 2 - w 2 2 ) / 2 2 \ / X.J i (it Ni (w 2 - w 4 2 )- • • ( • (w - Ui ) ( ) Now in any network configuration, be it a "T", "n", or lattice, the element values X , Xl, X , etc. , are found in terms of the factors Xi i — Xj. 2 and X 22 ~~ X 12 with the result that^ under the above conditions, poles occur in the series as well as the shunt arms of the network. In a "T", for example, (Eq. 3-1) X a ; Xi ! X x w Ni (w 2 - w 4 2 ) ■ • ■ ( • ) - w N 8 •■ ' ' ( / 2 2 \ ) \ - X* X x w N 2 (w 2 - w 2 ) ) - u No (w 2 - Ui 2 ) ■• • ( •••••• ) and the series arms have poles at U = Ui . Consequently the antenna impedance is disconnected from the match- ing network at this frequency and the impedance measured at the matched terminals of the network is in f ini te (theoretically) representing a complete mismatch (p^ = 1.0). FIGURE 7 o X q 1 * b ° VJ — Z o o in VV X c = °° at * = w . -36- Hence, in selecting the range of variation for = ±K. With 0j ■ n + 2cp(w) a = -k - cp(w) a + 9j_ = qp(w) RU + Pj) An - -T= T~ cot

(5-1) X 12 = _B h(l - P^ ) sin qp Recall that the expression for X t ^ lends itself to a determination of the range for 2R) and is zero near mid-band so that cp must be selected of such value at the ends of the band to make I cot cp I > 1 in order that Eq. 5-2 be fulfilled. Furthermore, cot cp has a positive slope with respect to cp = kw if cp varies clockwise or in the -cp direction. By substituting the values of R, X, and p- into Eq. 5-2 under the above considerations one can determine quite readily that $ can range linearly from -325° through -360° to -395° with the result that over the band dXii dw > . These data, along with the corresponding calculations, are presented in Table 4. The X 22 of necessity is a positive slope function because R cot a> CM ON CM CM •*& if) \£> ^O -rf \D r— I O i— I i— I CM -*tf" 5 CO r— a\ lo CM CM O CM CM CO o o o LO LO o o o CO •** LO LO CO CO lo LO 5* CO CM 8 CM 1— 1 I— i 00 *# ■>* CO 1—1 i— i o x t— o LO CM CM LO o c— 1— 1 r— I CO co I— I t— 1 Q. o lo LO LO CO CO O o CO \o O co CO 1 — 1 CM r- i— I LO LO CO 8 CO co CO CM o CM CO .-H i— 1 i— 1 CO co LO LO LO o t— CM + ** LO + CM CO On O co LO CM LO r~ t— i •*■ r- r— r— r— CO CO co CM •<* ■— I i— I -^ CO as as as LO CO CI LO CM CO LO r— co' co co CM CO LO CM LO CO LO t— LO o od r-^ vo lo r~ CO CO CO LO CM \o oo co LO as CO O •z- CO CI CN O LO vO lO LO oo LO CM o o CM LO LO LO LO 1— CM LO r~ ao i-H « f— 1 CO CO 3 CM e i i i CM CM CM LO LO t— o LO CM LO O LO CM LO LO lo CM CM CM -42- CM CO CM LO CM CM 0\ CM LO i-H CO CO CO o CM UJ _l CQ < -43- and cot cp was selected under this consideration. The form of X 12 is not required to be physical under the most general conditions. For the above choice of cp(w) which intersected only one pole the transfer re- actance is also of physical form. Having completed the first step in the synthesis method it is now necessary to find the functional forms for physical X 1± ' , X 22 ', an d X x 2 ' whose reactance values approximate those of Table 3. Because the net- work is lossless, the physical driving-point Xj. r ' and X 2 2 ' must be of the Foster's reactance type. Furthermore, X 12 ' is also a positive sloped function (for this case) so that its physical form too will be of the Foster's type. Only one pole was present in the reactances calcu- lated by the transcendental equations so that upon recalling that s = tw. N? s Y ' = * ~ Y A l 1 Aj j 2 2 s - Si n| s A 2 2 " A 2 2 \o~o) 2 2 S - S-l Y ' = ^3 S ~ v Ai 2 — ' " Ai2 2 ' 2 S - Si where s x = 0.23750. Proceed now to calculate the N? , Nf, and Nf by substituting values from Table 3 for Xj. 1 , X 22 , and X 12 at a number of discrete values of s into Eq. 5-3. These calculations are shown in Table 5. The variation of Ni , N 2 , and N 3 are plotted in Fig. 9, along with the selected N?, Nf , and Nf. The choice of Nf and Nf is easily made since the curves are fairly constant over the band; however, the best Ni is not so easy to determine. If, on the other hand, one recalls that * Appendix IV \ \ o c 2 / 7 I \ \ \ \ \ j 1 1 / i t 1 1 «yn 1 / o _ 1 am z Q z < COM z: OS o Ll. m CO z CM o H u LlI _l LU CO m Q h- z to < Q Z < u. o CO z o in __ CM CM < & < > in co Ld CM cc ID CD O o> oo r>- \ L where a t = 0.23750. (5-5) The above reactances can be calculated and are tabulated in Table 6- Recall now from Appendix IV that the, matching two -terminal pair can be derived from the three open circuit reactances in the form of a T net- X-1 1 '2 2 Xx .2125 102.5 73.67 86.89 .21875 138.7 99.73 117.6 .2250 211.2 151.8 179 .23125 428.3 307.8 363 1 .23750 00 00 00 .24375 - 439.7 - 316.0 - 372.8 .2500 - 222.6 - 160.0 - 1887 .25625 - 150.2 - 108.0 - 127.3 .2625 - 113.9 - 81.9 - 96.6 TABLE 6 PHYSICALLY REALIZABLE open -Circuit REACTANCES 46- work, including ideal transformers. Under these conditions the elements of the network are described in the following figure: I'.a + E. + E, X ll ~ Xi 2 Xu o X22 X12 'k "2 A22 D a 1 - X a 12 FIGURE 10 It is evident by finding X^'' = X 22 ~ Xi 2 f° r the "T" alone from Table 5 that Xjj' is not physical. The change in impedance level effected by the ideal transformer corrects this condition. Again, from Appendix IV, if K?! K| 2 - (K? 2 ) 2 > then |Kf g K s 2 2 - I/S KT (5-6) so that for this example 47 4^60_ 5.426 a = 3^ 4.60 since N? = 2Kfi, Nf . = 2Kf 2 , and Nf = 2K? 2 - As a result of Eq. 5-6 one calculates that 0.8478 = a. By substituting a = 0.8478 into the expressions for X a , X^, and X c in Fig. 10 one finds that X a = X^ = and these data are given in Table 7. X a ° 5 .426 4,6 0.847 s - n u 2 2 S - Si _— — i *b = i_9_ + 719 4,6 0.847 S - A 2 2 s - Si s Xi i X 12 X 2 2 a a \ X a 2125 102.5 102.5 102.5 21875 138.7 138.7 138,7 2250 211.2 211.1 211.2 23125 428.3 428.3 428.2 2375 00 00 00 24375 - 439.7 - 439.7 - 439.6 250 - 222.6 - 222.6 - 222.6 25625 - 150.2 - 150.1 - 150.2 2625 - 113.9 - 113.9 - 113.9 TABLE 7 Since the series reactances X a = X^ = 0, the pole at s = 0.2375 in the band is effective only in the shunt reactance X and is not detrimental in this matching network as is apparent from the following figure. The matching network takes the form FIGURE 11 : 0.8478 o - x c O — 48 where X„ = 1 X 12 = ~5 6 426 g s and Si = 0.23750. c a s - s- The next step is to synthesize the function X c , after X c is re written as a function of frequency. s = L/X where L is the antenna height in meters so that • •k* rhere v = 3 x 10 m/sec Hence, • - tW ■ (Lu/6rc) x 10 Therefore X = -5.42. 6 btf C .2,2 2 \ t (0J - Wi ) ._ / 2 2 \ t(w - Wi ) and it is apparent that X is a parallel resonant circuit. -5 . 4 2 6 m . TiV t G( U ) 2 2 v 2 2 t(w - Wi ) <•> - Wj. As u - », X c - 0, therefore, G(w) ■ 0. The negative sign in X_ can be dropped since it indicates only the sign of the reactance and it is known that X is physically realizable Ti _ 5.426 . - 1 t c, Lx = 5.426 2 -49 (5-7) and The elements C^ and L^ are written in terms of frequency so that some fi must be selected in order to calculate Ci and Li , which in turn are used to check the actual impedance transforming characteristics of the network. Evidently any frequency can be chosen and the network can be determined, but from a physical standpoint one should not select f 1 greater than 200 mc since lumped elements cease to act in their con- ventional manner for such frequencies. Furthermore it is desirable to replace the ideal transformer by a commercial type (whose character- istics approach those of an ideal transformer) and to investigate the effects of such a change on the impedance transformations. Since data on such a commercial transformer were available at 300 kc, this fre- quency was used to check the effect of the transformer. It is realized that for this frequency range the physical height of the antenna is very large (250 m), but this situation will be altered in later calculations where the frequency is increased and where the ideal transformer is replaced by transmission line elements With these thoughts in mind, an initial mathematical check on the matching network was carried out using an ideal transformer in the system. Next, the ideal transformer characteristics were replaced by the equivalent network for a commercially available transformer and this system was checked by calculating the transformed antenna impedance over the band. From Eq. 5-7 the values of L 1 and Ci can be calculated and for the 50 ideal transformer case the network takes the form: FIGURE 12 By substituting the values of Z a at various frequencies over the band and calculating X? and Xp ,. one may find TZ a and these data are tabulated below and plotted in Fig. 8 as the broken line curve. s & a Zj. Ro ! TCa .21250 218.4 - .1 120.9 3.14 - j 1.74 .21875 75.8 - J 87.9 1.09 - : 1.26 .225 44.0 - J 51.7 0.632 - : 0.740 .2375 31.6 - J 10.9 0.455 - : 0.157 .250 44.8 + J 18.6 0.644 + 0.267 .2625 106.7 + J 31.7 1.53 +j 0.456 .26875 167.7 - J 37.6 2.41 -j 0.54 .2750 111.2 - J 113.9 1.60 -j 1.64 TABLE 8 From Fig. 8 the location of the transformed curve with respect to Pi i Vi and to p^ £ 1/3 is apparent and is indicative of the capabilities of the procedure under the assumed conditions. If now one replaces the ideal transformer with the commercial one, then the network takes the form shown below. -51 P P a -vw — tikp — j-5i -f ^15W^- a a' 1:0 =FC — z, a TZ, FIGURE 13 /here the following data were measured at 300 kc: L P _ M s a z 4nh hi 2 a _ M = a : 4M-h J (5-8) Froa the criterion (Eq. 5-2) and the knowledge that cp(w) must be of the Lineal form uj Ld _l CO O _l o < < CL o Ll- Ld > a: cj o en Ld Nl Q Z < Ld _J O a. o u CD 8 o o 8 o o I o in CM I 60 Xii - . {2k) 2 (f 2 - f 2 ) N? = (f 2 - f 2 ) N9 (2n) 3 f (f 2 - f 2 ) 2iif(f 2 - f 2 ) PS X ' = 2Tcf Nf f Nf X 12 (2n) 2 (f 2 - f 2 ) 2n(f - f 2 ) 2nf Nf f Nf (2TO*(f* - fa) 2rc(f 8 - il) n fi = 81.88 mc -^ = 100 mc (5-10) By setting Xj. x ' = Xj. j. , X 22 ' = X 22 , and X x 2 ' = X 12 where Xn, X 22 , and X x 2 are available from Table 11, one may solve for Ni » N 2 and N 3 . These data are presented in Table 12. No 2k f f 2 - f 2 f(f 2 - il) Nii 5i X (mc) f f 2 - ■f! 2k 2n 70 -.7286 X 10 8 1.979 x 10 8 - 84.1 x 10 8 -26.4 x 75 -.5833 x 10 B 3.041 x 10 8 - 94.3 x 10 8 -29.2 x 80 -.450 x lO 8 9.472 x 10 8 - 82.4 x 10 8 -31.0 x 85 -.3265 x 10 8 -4.527 x 10 8 -211.0 x 10 8 -32.1 x 90 -.2111 x 10 8 -1.225 x 10 8 -172.0 x 10 8 -32.5 x 95 -.1026 x 10 8 -0.3990 x 10 8 -340 x 10 8 -32.3 x 100 o° t) co 110 +.1909 x 10 8 +0.427 x 10 8 TABLE 12 10' 10 £ 10 £ 10 s 10 £ 10 £ -29.9 -34.0 -36.7 -40.4 -41.0 -55.5 °° x 10 x 10' x 10 f x 10' x 10' x 10 f Curves indicating the variation of Ni , N 2 , and N 3 are plotted in Fig 20 where the choices for N?, N| , and Nf are also shown Again the values for Nf and Nf are not too difficult to choose while the selection for N? presents a problem. Note that N a remains fairly constant at the low end of the band, then rises rapidly to a higher level where again some degree of constan- cy is maintained. This behavior is due in part to the variation of the antenna's radiation resistance (a factor in the second term of X 1X ) which r;ui(/f:s from 5.5 ohms at 70 mc to 140 ohms at 110 mc (a ratio of 25 to 1). Wj id regard to the N a variation, the important point to be 300 200 100 g 80 CVJ *! 70 60 50 40 30 20 N / >^ NvX a N 3 •- _ tf, 70 75 80 85 1 in mc 90 I00 Figure 20. variation of n x . n 2 . and n 3 with frequency -62- made here concerns the information the matching procedure yields when one considers the problem of determining how much bandwidth is theoreti- cally available from a given matching network in cascade with Z a . That is, if one assumes a finite number of elements (in this method by pre- scribing the form of 2 2 substituting for 2K 2 2 a nd 2K 1; (2^^(32 x io 8 x 2ti) - (40 x io 8 x 2n) 2 1 -63 or 2Ki t > 50 x 2k x io 8 . (5-10a) Now Ki x can be found in terms of N? from X x x ' ra/, 2 ,2 , N f(h) -Mi) K 2Kix u Ax J o— (0(a) 2 ~ w 2 ) w 50 x 2rt x io' or N a > 151.7 x 2k x io u . (5-11) As a result N? must be selected to be equal to or greater than (151.7) 2k x 10 in order that the network be physically realizable. It appears, therefore, that N a must be chosen to approximate the variation of Ni at the upper end of the band where Ni is greater then the lower limit of Eq. 5-11. Hence, one cannot possibly expect to match the given impedances for any frequency below that at which Ni - 150 x 2k x 10 under the speci fie conditions for match ( p^ is equal to 0.60 and with the given number of elements in the network). It must be realized, furthermore, that even this lower frequency limit is contingent upon how well N? approximates the Ni variation in 64- this range. If N? differs considerably from Nj. , obviously the matching network will not transform the impedances in the desired manner. Hence, the value N? = 190 * 2k x 10 must be selected rather than N? = 90 x 2n x 10 8 . There is, however, another choice for N a which satisfies the residue relation, namely, N? such that Ki i K22 _ vKi 2) s . Recall that under this equality, no poles occur in the series arm of the "T" matching section so that pole limitation on bandwidth is re- moved Consequently there exist two possible matching configurations depending upon which N a is used. One has definite limitations because of the pole in desired frequency band while the other network may not transform the impedances properly over the entire band depending upon whether or not N a is off in the correct direction from N lt That is, N a , quite different from Ni, does not constitute, necessarily, a poor match- ing condition; on the contrary, this situation only indicates that p^ will not be constant over the band at the matched terminals and the possibility exists that p^ can be much less than pj = c over a portion of the band. As a consequence, two networks were synthesized in which the same values for Nf and Nf were used and in which the N a values corresponded to the two aforementioned figures. For the first network where the effect of the pole at 100 mc is marked, the calculations for the physical open-circuit reactances with N a - 190. x 2k x 10 8 Nf ' 32. x Ik x 10 8 N a 40 x 2n x 10 8 •> r >• ci vh 1 n Tab] e 13. 65- X 41 = -(190)(2n x 10 8 )(f 2 - f 2 ) -190 x I0 8 (f 2 - f 2 ) 2ti f[f 2 - f 2 ] f [f 2 - f 2 ] X 22 = ^132112* x If) f . ~32 2 x iQ f 2n[f 2 - f 2 ] [f 2 - f 2 ] fx = 81.88 mc f 2 = 100 mc Xlfl = d- 4Q)( 2rt x i n e ) f = -40 x i ; f 2k[? - fj] [f y - fj] Xu' l 2 2 75 80 85 90 100 110 62.5 54.8 20.1 71.1 41.5 98.0 153.5 151.6 00 00 439.2 - 167.6 TABLE 13 Ai 2 68.6 88.9 122.5 189.5 00 209.5 From these data one can set up the same type matching network described in Appendix IV and used in the previous example, that is, a " T" in cas- cade with an ideal transformer with the understanding that the trans- former will be replaced by other elements. The network form is: i:a o X a 1 X b Xc o figure 20 a 66- where "a = An a * 2 X b = ~V A 2 2 ~ a Xi a a X c = ix ia . The transformer turns ratio is given by I K» 9 | K2: < a < Ki 1 |Ki 2 I and &LD. < a < 22^0. 62.0 ~ ~ 40.0 0=645 £ a < 0.80 , Let a = 0.80 so that or . -190 x 10°(f° - f«) 4fl x , n . f 3 f(f 2 - fg) 0.80(f - ff) X - -140 x 10 s [f 2 - (95.5 x 10 6 ) 2 1 f(f 2 - 10 16 ) X 32 x,10 e f 2 + - 40 x 2 10 6 f = b 0.64(f f 2 ) 0.8(f - fz) X c - -40 * 1Q 6 | . -5Q a * 10; f c o.8(f f 2 ) r - f 2 Hence from conventional synthesis identities, taking into account a A" factor since the reactances were written in terms of frequency rather than w, one finds for X. -67- - = ^0_£_10l_f Lc f 2 - f 2 -1 50 x 10 50 x 10° f jwC c + t-L- or wC„ = c 50 x io f so that 10. — 8 c 2k x 50 31.83 nuf and wL c 50 x io 8 f so that L c ■ V h&f - ^^ - 00796 ^ h c f 2 (2n;f) 2k x 10 Similarly for X. x = 140 x iQ 8 rf 2 2 - (Q.S 6 5 x 1Q 6 ) 2 1 = To + 21, f^ f(f - 10 ) f r - io Tn = 140(95-5 x 10,11 and Co = r-^r ■ 12.63 nnf 2rc T 2T X 140 x lQ 8 (f 2 - Q 913 x IQ 16 ) - (140 x 10 )(0.087) 8 f ■ 10 : = l ' a 2Tt(2Ti) 132.6 nnf 68 2T L fl = -^ = 0.0191 nh a w 2 The matching network is therefore L FIGURE 21 A mathematical check on the matching network was carried out using the ideal transformer in the system with the understanding that it can be replaced by line transformers. These data are presented (the dashed curve) in Fig. 18 and the effect of the pole at 100 mc is now apparent. The remainder of the curve over the frequency band fulfills the pre- scribed conditions. The overall impedance curve is narrow band so that its utility is limited but the calculations serve the purpose of indi- cating the effect of the pole in the desired matching band. Return now to the alternative case where so that N? ■ 151.7 x 2rc x io' Kfx K s 22 (Kf 2 ) 2 - and Kj 2 K?7 2 2 Kf K?, 0.80 Then in Fig. 20 a 69- X-i i -151.7 x 10 8 (f 2 - f 2 ) f(f 2 - f 2 ) y ' - -3 2 x lQ 8 f f 2 - f ? y ' - -40 x lQ 8 f r 2 _ r 2 I — I 2 r> £i = 81.88 mc £ 2 = 100 mc So that Y = Y ' - 1 Y ' = A A 1X a A 12 -151. 7(f 2 - fi) * 50 f 2 f(f 2 - fa) or 16v 1A 8 X = (-101.7 r + 101.7 x 1Q-)1Q° = -101.7 x 1Q- f(f 2 - 10 16 ) f -32 x 1Q 8 + 4Q + 1Q 8 0.64 0.80 -2- 1 — 2 = f 2 - fo - 40 * 1 0!£- = ^5J0 x 1Q? f 0.8(f - fa) f - 10 Synthesizing X, 101.7 x 1Q° f 1 2rc f C. c = 10"! a 271(101.7) 15.65 niif -70- The constants for X c remain the same as those of Fig. 21, namely, C c = 31.83 Wif L c = 0.0796 [ih The complete matching network is Co FIGURE 22 |:Q -*-T? f With Z a connected across the input terminals, the calculated TZ with the above matching network is very useful and these data are pre- sented as the solid curve in Fig. 23 and Table 14. The VSWR circles, a = 4.0 and o = 5.0 are also shown. f(mc) z, 2 7 a Z x Ro 75 80 85 90 95 100 110 524.6 + J 192 71.0 - J 98 3 35.3 - J 57.2 25.7 - J 35.6 39.0 - J 5.94 62.5 + J 7.0 111 2 - J 59.5 TABLE 14 6.71 + J 2.46 0.91 - J 1.26 0.45 - J 0.73 0.33 - .1 0.45 0.50 - J 08 80 + J 09 1.42 - J 0.76 Since it is desirable to replace the ideal transformer, for practi- cal purposes, by a cascaded quarter- wavelength line configuration, the -71 Figure 23. partial sleeve antenna ideal transformer IN NETWORK AND N a (151 7) (2rc) 1 O e -72- input impedance Z x presented to the ideal transformer (Fig. 22) was calculated and plotted in Fig. 23 (dashed curve). It is this imped- ance data which will be transformed by the line elements. From the previous data a = 0.80 = Z 02 /Z i so that RG-62/u and RG- 13/u, whose characteristic impedances are Z i = 93 ohms and Z 02 = 74 ohms, respectively, can be used. Let the lines be quarter-wavelength resonant at 97 mc (slightly higher than geometric mid-band = 91 mc ) , then the impedances at the lower end of the band will be rotated more and the bandwidth may be increased over that obtained with the ideal trans- former circuit. These data are presented in Table 15 and the resultant matching network takes the form of Fig. 24. u Zi I TZ 01 TZa TZ a (mc) X 50 75 5.64 +j 2.06 0.193 17 2 -j 39.7 21.8 +j 60.7 0.44 +j 1.21 80 0.76 -j 1.06 0.206 32.5 +j 30.7 168.7 -j 70.3 3.37 -j 1.41 85 38 -j 0.62 0.219 47 9 +j 79.0 42.5 -j 72.2 0.85 -j 1.44 90 0.28 -j 38 232 80.0 +j 136.7 19.8 -j 39.8 0.40 -j 0.80 95 0.42 -j 0.064 0.245 208.3 +j 46.5 25.8 -j 0.80 0.52 -j 0.016 100 67 +j 0.075 0.258 134.8 -j 20.9 40.7 +j 8.88 0.81 +j 0.18 110 1 20 -j 0.64 0.282 71.6 +j 44.6 46.6 -j 23.9 0.93 -j 0.48 TABLE 15 FIGURE 24 73- FIGURE 25 PARTIAL SLEEVE ANTENNA IMPEDANCE Nf = (151 .7) l2Tt)10 8 AND IDEAL TRANSFORMER REPLACED BY LINE ELEMENTS -74- The transformed impedance is plotted in Fig. 25 and a bandwidth in excess of 45 percent is obtained which represents a marked improve- ment over the original antenna impedance. The percentage bandwidth is defined as (f 2 - fj 100/fi .* An additional set of calculations was carried through for the same partial sleeve antenna impedance in which qp(k>) was chosen to move the pole in the reactance functions as far toward the lower end of the fre- quency band (80 mc) as possible. This is accomplished i f cp = 62.5 x 1(T 10 f - 130. For this cp(u)) variation Xn, X 22 , an d X 12 now contain only one critical frequency, a pole. This choice of cp is shown in Fig. 19 where the location of the pole is apparent and where it is evident further that no zero curves are intersected in the 80-110 mc band. It should be added that the above phase angle relation was obtained by more cut-and-try attempts because K X 1Q» u 5— u - U1 X 4 ' - -1 QQ x 2^ * 1Q > 71 2" 1 (1) - OJ, S fi = 80 mc These data were used to calculate the elements in a matching net- (Wt- work of the form shown belov where FIGURE 26 •75 K.O \V S I'M 2 100 x 10 120 x 10 8 ~ " ~ 100 x 10 < a < -90 x 1Q 0.833 < a < 0.90 Let a = 0.90 so that X a = Xll ' - 1 Xl2 ' = (-120 ♦ J^L) 2* I K " 0.90 w -Uj or -8.9 x 2ti x 1Q 8 gj ■ st * -tf — i — ~ — 2 w - u x \ - 90 x 100 2n x 1Q° u) 0.81 0.90 u 2 -u? -100 x 2ti x in" (j = -1 11.1 x 2k x iq° op 0.90(u 2 - Wi) U - Wi Therefore in X s -nsw^- ■76- FlGURE 26A C a = » IP n = 178.8 nnf a 8.9 x 2n i = 8.9 x 2k x 1Q 2 = 0.02214 |ih ' 80 mc Similarly in X f 1Q _e C (111.1X2TC) 14. 3 niif 1 . 11 1 . 1 x j n x 1Q 8 •- 0.276 nh Summarizing then .022i/xh fO— l78B/z/zf E, 0276/i.h -o- 1:090 14.3/X/lf « 2, FIGURE 27 -o + -o- -77- Again the network was checked mathematically with the antenna im- pedance placed across terminal 1. The network transformed Z a to the position shown in Fig. 28, where it is evident that less bandwidth is available from the above network configuration in comparison to the previous network because of the pole at f = 80 mc. Again it is desirable to replace the ideal transformer, for practi- cal purposes, by a line configuration similar to that of Fig. 14 so that the input impedance Z x presented the ideal transformer (Fig. 27) was calculated and plotted in Fig. 29 (solid curve). It is these data which will be transformed by the cascaded A/4 lines. Note however that the impedance curve (Z t ) of Fig 29 s where the matching network did not include the transformer is as useful as the impedance plot of Fig. 28, for which data the transformer was used Recall however that the turns ratio of the transformer was a = 9 so that the transformer effect is simply to decrease each impedance of Fig 29 by a =0 81. This effect is not too appreciable, especially since Zj lies well within the prescribed VSWR circle for that portion of the curve which is matched; so that a Zi shifts this same portion slightly in the R = direction in accordance with the 0.81 factor. The fact that there exists no choice between the transformation in Fig. 28 and that in Fig. 29 is due to the approximations made in the technique. That is, if the procedure were a precise mathematical method, the impedance transformed by the overall network would lie exactly at the prescribed values of p- and Q- so that the partial network (no transformer) would transform only a portion of the impedance curve (low frequency end for "a" less than 1.0) into the desired VSWR circle. The approximation step allows the overall network to shift the curve to the prescribed location only for that frequency range in which the approximations are good For the other frequencies in the band one takes what one gets, with the result that one range (where the approxi- •78- FIGURE 28 PARTIAL SLEEVE ANTENNA cp — 62 5 x 10" 8 f - 130 •79- FIGURE 29, Z x - INPUT IMPEDANCE TO IDEAL TRANSFORMER AND TZ USING CASCADED LINE TRANSFORMERS a -80- mations are poor) may lie closer to R than to the prescribed VSWR circle while another such range will lie outside this circle. Hence, the effect of adding to the network an ideal transformer whose turns ratio is nearly unity is simply a minor impedance level shift over the given band with no marked overall improvement in bandwidth. Reconsidering the network in Fig. 27, the ideal transformer can be replaced by cascaded quarter-wavelength lines and the overall match- ing configuration will yield results quite similar to those of Fig. 28. The general effect of the line elements is to spiral the extremities of the impedance curve about the center of the Smith chart so that an in- crease in bandwidth usually is effected. In this example, however, the pole at 80 mc is reactive and when it is transformed by the line sec- tions it remains reactive so that not too much improvement in bandwidth can be gained in this frequency range. In any case, the lines were chosen quarter-wavelength resonant at 100 mc (midband = 94 mc) causing the lower frequency impedances to shift more than the high frequency impedance. Further a ■ ^ = o.90 so that RG-58/u and RG-25/u, whose characteristic impedances are Z 0i * 53.3 and Z 02 ■ 48.0 ohms, respectively, can be used. The transformed antenna impedance curve is presented in Fig. 29 as the dashed curve and somewhat larger bandwidth is available from this curve in comparison to that of Fig. 28. -81- 6. CONCLUSIONS One may conclude from this work, that the procedure presented does offer a direct attack on the synthesis problem for antenna impedance matching networks That is to say, for bandwidths of 25 to 50 percent, a straightforward analytical -graphical technique has been developed which certainly is an improvement over the conventional " cut-and- try" method. Extensive calculations are required in the new method to de- termine the variation of numerous parameters over a given frequency band, but these orderly calculations involve considerably less time than that required in the " cut-and- try" procedure Recall further that the general problem of dealing with complex impedances over a band of frequencies is unavoidably laborious, The ideal transformer condition has proven to be no limitation when it exists in the design of matching networks for either the ultra-high frequency range or the lower band of frequencies In the UHF band the cascaded line transformers (short physical length and high "Q") are adequate replacements for the ideal transformer because the line con- figuration transforms the impedances in very much the same manner as does the ideal transformer over the limited bandwidths. Similarly, for the lower range of frequencies, present-day manufacturers can design transformers (most probably with ferrite cores) whose characteristics will approach the ideal over very large bandwidths (100 percent or more ) „ One main limitation of the method is the upper frequency limit for which the procedure is directly applicable and this limit stems from fulfillment of the original hypothesis that the matching network consist of lumped constants (inductances and capacitors). Directly, the utility of the procedure is limited to matching impedances at lower frequencies (say below 30 mc ) where the synthesized lumped constants have reasonable values and where they retain their design characteristics because their -82- physical size is still small in terms of the wavelength. Indirectly, this limitation can be circumvented for the ultra-high frequencies by carrying out the network synthesis in terms of lumped constants and then replacing the calculated element reactances by their transmission line equivalents. In some cases (the intermediate frequencies, 30-100 mc or so, where neither lumped constants nor line elements alone are feasible from practical aspects) it is more desirable to utilize a combination lumped constant-line element network. The desirability of using such a combination network was made evident in one of the examples presented in the text where it was indicated that a synthesized shunt inductance of low value could be replaced by a short-circuited line element which is physically more practical in the 30 mc range. The bandwidth limitations discussed in the previous chapter are severe. However, it has been shown that at least two alternatives exist, namely: 1. Choosing By equating (1) and (2) A ' "2 2 Zi 2 ' j i l R - "22 *n : ? "2 1 D Zi 2 C ' 1 Zl 2 "15 (1-3) 86- and Zn C ^55 = — z 12 = c Zii Z22 " Zi 2 Z 2 1 ~~7T (1-4) If all the elements of the network are lossless, then Z i± - j Xn, Z22 :: j X 2 2» and Zi 2 ''' j X12 general, complex so that '12 jx 1 2 d + jC. Hie network parameters A, B, C, D, are, in Z 2 2 := c JX, Ai + j A; d + j d 7 --£- Zn - c jX 1 1 Pi + j Pi " d + j c 5 and ■vd ■ Xa 1 c 2 ••A, d ■ Ax ••D< Xii = -C Ci = A 2 Pi ■ In summary: (1-5) -87- APPENDIX II POSITIVE - REAL FUNCTIONS FOR A ONE - TERMINAL PAIR Since Positive - Real (P-R) function theory is of fundamental importance in the study of electrical network synthesis, a brief resume of the concepts will be presented here. More comprehensive proofs and theory are available from the works of several authors 8 and from the original work by Brune. A rational function w = f(z) is called "P-R" if the real part of f(z) is positive for all z with a positive real part and, if the func- tion is real, for real values of z. That is, Re[f(z)] > for Re [z] ± and f(z) real for z real. In the network synthesis problem, the rational function is an im~ mittance (impedance or admittance) function where the independent variable is the complex frequency p = a + jw. The immittance functions are quotients of two polynomials (for a network of finite, lumped elements) and the zeros of the polynomials cannot lie in the right half plane. They may lie on the boundary under certain conditions. Polynomials whose zeros have negative (non-zero) real parts are Hurwitz polynomials which are useful also in the study of stability of active feedback systems. Immittance functions of passive networks must have both numerators and denominators which are Hurwitz polynomials multiplied by factors which vanish at purely imaginary values of "p" and are simple, such as p 2 + 1 = (p + j)(p - j). It can be shown from energy considerations that the "P-R" prop- erty is a necessary and sufficient condition if an immittance func- -88- tion is to represent the driving-point immittance of a physical one- terminal pair. A brief proof is given below. Fundamentally the energy functions may be written as W = R i t = L i . 2 »& (II-I) where energy is positive from physical nature. Consider a coil, n L", which is the only coil in a network of two loops and where this coil is not coupled to any other coil. The in- stantaneous energy stored is o T = Y 2 L i 2 = l A L (i! - i 2 ) 2 ' = K L (i t 2 - 2i!i 2 *■ i 2 2 ) FIGURE 31 T - % [Lii i i i 2 _ L 12 iii2 -L 2i i 2 i i + L 22 i2*2] so that for all coils in the network, including mutuals, i r i T = % 2 / > m = 1 A 1 * ; x m 1 s Similary for condensers, the instaneous stored energy is v = y 2 c v 2 = l A i C ms v m v s For resistors, half the power dissipated is [ F = % R i 2 = % H I E« s=l ms 1 m 1 s (II-2) (II-3) (II-4) The F, T, and V functions are positive because they represent energy stored (T and V) and energy dissipated (F) . The mathematical expressions are of quadratic form, i.e., £ a XX, and are positive ids m s m s de f mite. 6 -89 If one analyzes a bilateral network on the loop basis from the initial rest conditions, the transform equations for loop number "m" is: L [l p + r + om I - R £~f ms V n ms p s ^ where E m is the net driving voltage in loop "m". When multiplied by I m (conjugate loop current) ^ms ^m Is P + Rms \ h + 3at ^ = ** \ The equation applies to each of the "Z" values which "m" can assume. Writing the "Z" equations and adding yields m= 1 + (H S mc I m I c )-— v m s ms m s' p - r P + f' +-y (ii -5) The functions T' , F' , V, are defined by the coefficients in the previous equations and are of the same form as the energy functions. Each is real and positive and of quadratic form with the "I" complex ^• e - - J m = a m + J b m and ^m = a m " J b m } Consider the T ' ' m 2 s ^ s Im Is V = &K>s < a m J b m>< a s + J b s> = m 2 s L ms < a m a s + b m b s> + J m 2 s L ms < a m b s - a s b m> I ( L ms a m a s + L ms b m b s> + ° ms therefore T' is real, similarly for F', and V. For a one terminal pair, E 2 = E 3 = Ej - -90- Passive Linear Bilateral FIGURE 32 Divide by Ejlj - T' p + F' +-y- I1I1 = \U Ei T' p + F' + 2fl- V -i = Z( P ) E— - e_= To P + F + -f 11 llit 2 P < where the coefficients are redefined to absorb the positive denominator, As a result, with p = a + ju V Z(p) ■ F + T (a + jw) +- Z(p) T n a Vo a 2 2 a + w a + ju + J [To u (II-6) V w 2 2- a + u) Note that (1) When p is real (go - 0) then Z(p) is real (2) When p is in the right half plane (o > 0) then Z(p) is also in the right half plane. (i.e., when the real part of p is positive, the real part of Z(p) is also positive). Therefore, based on the energy concept, the "P-R" property was derived for a network made up of a finite number of resistances, in- ductances and capacitances. From the positive reality notion, almost all the important prop- erties of driving-point immittances of a one-terminal pair can be derived. These properties will be listed without proof, since this information is readily available in the reference cited previously. Necessary properties of a driving-point immittance function that represents a passive physical network are: -91- 1. For a network of finite, lumped elements, the immittance function is restricted to the rational class of "P-R" functions. 2. The zeros and poles of the rational function must occur in con- jugate pairs, if they are not real. 3. No zero or pole of a "P-R" function can have a positive real part. 4. Poles and zeros on the imaginary axis must be simple and the res- idue must be real and positive. 5. When large, real values of "p" are considered, Z(p) or Y(p) behaves as Z(p) -Kp n - m where "n" and "m" are the largest exponents in the rational form of Z(p). The multiplier "K" must be real and positive for Z(p) to be real with "p" real 6. The highest power of p in the numerator and denominator of a "P-R" immittance function can differ at most by unity. 7. The lowest power of "p" in numerator and denominator can differ at most by unity. 8. For p = jd) (i.e., real frequencies), the real part of the immittance function is an even function of "to", and the imaginary part of Z(p) is an odd function of "w". Because of the fundamental importance of the positive-real property, it is desirable to examine a given function Z(p) to determine whether or not it is positive real. One must 1. test whether or not Z(p) is real when p is real. Just inspect the coefficients of Z(p) and they must be real. 2. test whether the Re [Z(p)] > when o > 0. This is a more difficult procedure; however, by using the Maximum Modulus theorem one can show that the minimum value attained by Z(p) as p wanders over a region occurs for p on the boundary of the region. -92- Consider the right half region of the "p" plane as outlined with semicircles about the poles on the imaginary axis. FIGURE 33 Let R "- ® and r ~* Then the region inside C is effectively the right half plane where for "p" values the Re [Z(p)] should be greater than zero. (2a) Determine whether Z(p) has any poles in the right half plane or on the "jw" axis. If any exist for a > 0, then Z(p) is not "P-R". If none exist in the right half plane but some lie on the imaginary axis, these must be simple and have a real, positive residue. If not, Z(p) is not "P-R". A pole at infinity (a point of the imaginary axis) can be checked by Z(p) - K p where K must be positive and real. All of this theory follows from the Maximum Modulus theorem which states that the Re [Z(p)j attains its minimum value of all values at- tained in the right half plane for "p" somewhere on "C" From the proofs of properties 5, 6, and 7 it can be shown for poles on the "jw" -93 axis that for large |p|, Z(p) either vanishes, is constant, or is pro- portional to p. Hence values for Re [Z(p)] for "p " on the large semi- circle approaching infinity never drop below a value attained on the real-frequency (jw) axis, so that if it is shown that the Re [Z(p)] - for all values of V\ then the Re [Z(p)] - for a - 0. 3. test the Re [Z(jw)] for all values of w to determine its sign; if it never goes negative, then Z(p) is "P-R". Therefore the basic principles of the "P-R" tests are: (a) see whether Z(p) is analytic in the right half plane and, if so, make sure that on the boundary its singularities are restricted to the kind allowed there. (b) see whether the real part of Z(p) is ever negative at real frequencies (jw axis). Il-I Lossless One-Terminal Pair The case in which the one-terminal pair has no resistance in it (i.e , non-dissipative ) is expressed mathematically by Eq.~II-6 where the "F" function is zero. Z(p) = T p + — P Y(p) - V P + — P FIGURE 34 For no resistances in the network, the driving-point impedance at real frequencies will be purely reactive and will have a zero real part, hence, is called reactive or non-dissipative (set p - ju in Eq. II-7.) Consider the properties of a reactive network: 1. Driving-point immittance functions Z(p) or Y(p) must be "P-R" and rational for a finite network. 2. The coefficients must be real numbers and the zeros and poles must occur in conjugate pairs. L,C,M only •94- When 7/ \ - r\ - t x ^2. 2 - ^° Z(p) - - T p + D or p = o so that + fa When Z(p) ■ co or Y(p) ■ = V p + — Since V and T are real and positive, the zeros and poles of Z(p) are purely imaginary and occur in conjugate pairs. 3. The zeros and poles must be simple with the derivative at each zero real and positive and with the residue at each pole real and positive. 4. The highest and lowest powers of "p " in the numerator and de- nominator can differ at most by unity. 5. At real frequencies (p = jw), Z(p) is an odd function. This fact is evident since the constituent elements are of the form "Lp" and — — only odd functions of real frequency, hence, the whole immittance cp must be an odd function of p - jw. 6. From energy concepts it can be shown that the slope of the reactance curve at real frequencies is always positive for the L-C case. In a simpler fashion this may be shown by using the fact that Z(p) is "P-R". Let Z(p) - u + j y where u and y are functions of p = o + j U, When p = j u) then u = 0, but when a > then u > 0, since Z(p) is "P-R". Hence, at points on the imagninary axis, — — is increasing, that is ao -SSL> 0. Z(p) is analytic at all points except those imaginary values of p lich it has pole urn equations but t which it has poles Excluding these points^"- = — -^ by the Cauchy- oa ooj 95 |2 = |* (for p = jtt, u -R « y = X reactance therefore 3u = 3y_ = 9X > 3a 3w 3go or the slope of the reactance curves is always positive with respect to real frequency. 7. From the slope property follows the fact that the zeros and poles separate each other. Note that the origin must be either a pole or a zero: this follows from the separation property combined with the requirement that X(w) be odd (i.e., X(w) ~ -X(-w)). The physical nature of the DC impedance of a coil and condenser also makes this property evident. Similarly, at p = °°, the reactance function must have either a pole or a zero. These properties will be applicable to the driving-point impedance functions of the two- terminal pair. -96- APPENDIX III PROPERTIES OF THE TWO- PAIR NETWORK Certain necessary conditions which must be met by the set of open - circuit network impedances (Z llf Z 22 , a nd Z 12 ) for physical realiza- bility of a two-terminal pair will be discussed. An idea due to Caure makes it possible to apply to a two- terminal pair the conditions for positive- reali ty encountered for a one-terminal pair. It can be expected that some condition interrelating all three open-circuit impedances exists. To find this relation connect two ideal transformers to the network in the manner of Fig. 35. ♦ o - a o ♦ o - FIGURE 35 An ideal transformer insures that the input and output voltages have exactly the ratio of transformation and so also do the input and output currents. The ideal transformer is extremely useful in Network theory and is defined as a purely-inductive transformer which has no magnetizing current (loss current), has infinite mutual coupling, has no leakage reactance. The ratio of transformation may be positive or negative (a matter of poling the terminals), may have any magnitude, but must be a real number. Hence, Ei' = aE x T ' = 1 T E 2 ' - bE 2 V = a" 1 (III-l) -97- By substituting into the impedance mesh equations E 1 ' a Zi 1 I x ' + ab Z 12 I2' E 2 ' ~ ab Z 12 Ii' + b Z 22 I 2 ' (III-2) With the transformers in place, connect the two ends of the new T-T-Pair in series (Brune's connection) thereby reducing the setup to a one-terminal pair driving -point impedance. Figure 36 shows the con- - E + so that FIGURE 36 E = E x ' + E 2 ' and I = I 1 ' = I 2 ' Z ■ E * a 2 Z lt + 2ab Z 12 + b 2 Z : (III-3) This interrelation of all three open circuit impedances is of funda- mental importance in the theory of T T pairs. It is also a P-R function (a driving point impedance of a one terminal -pai r) about which conclu- '"'' car be formed which specify the form of a set of open-circuit im- ■■ < ' if they are to represent a physical network. Equation III - 3 -98- must hold for any pair of values a, and b (real numbers). Certainly Z = a 2 Z u + 2ab Z 12 + b 2 Z 22 (III-3) must be P-R. As a result Z(p) of Eq. III-3 can have no poles in the right half plane, also the Re[Z(jw)] > for all w, and any poles which Z(p) may have on the imaginary axis must be simple with positive, real residues. These properties are already true for the first and third terms of Eq. Ill - 3 since a and b are real and positive coefficients of Z x j. and Z 22 which are P-R driving-point impedances. If Z(p) is a physical structure (PR) then Z 12 can have no poles in the right half plane so that Z 12 must be analytic in the right half of the P-plane. Further Re[Z(jw)] = a 2 -Re[Z 11 (jw)] + 2ab Re[Z 12 (jw)] + b 2 'Re [Z 22 (j w)] >0 where a 2 Re[Zn(ju)] 1 b 2 Re[Z 22 (jw)] > but 2ab-Re[Z 12 (jw)] = ? Thus one can conclude that the Re[Z 12 (jw)] need not obey the non- negative rule. It may be negative, so long as the sum of all three terms remains positive. Z 12 may have poles on the imaginary axis, but these poles must be simple. For finite networks Z 12 is a rational function of "p" with real coefficients, hence, its zeros and poles must occur in conjugate pairs, -99- if complex Z 12 cannot have any real frequency poles of higher order for if it did, such higher order poles must appear in Z(p), which is outlawed Consider the restrictions on the residues of the simple poles in Z 12 . Suppose that p = jw g is a pole of Z 12 an d that its residue there is K? 2 - F° r generality let this pole appear also in Z llt Z 22 , and Z with residues Kfi, Kf 2 , and K s , respectively. The residues Kfi, K 22 , and K s must be real and positive or zero In the immediate vicinity of the pole the predominant term in the Lauiont expansion for each of the impedance functions will be of the form K mn P " JW. and by going close enough to the pole, all other terms are negligible Hence, one concludes 7 = Ki 7 £ Ki i etc P " JW S P - jw s so that in Eq. Ill -3 K s - a 2 Kfi + 2ab Kf 2 + b 2 Kf 2 (III 4) where K s > This will be true if the 3 residues (Ki lf Kf 2 , K? 2 ) are coef- ficients of a positive definite quadratic form. Hence, it is found that certain simple conditions must be met by the coefficients (the K^ n ) for the form of K s to be always positive, namely, K?» > Kfi Kf 2 - (K? 2 ) 2 > (III-5) -100- Since Kf ! and Kf 2 must be positive or zero, one may conclude that K? 2 need not be positive, but it is restricted in magnitude by the values of the other residues. Using the positive definite theory on Eq . III-3 it can be shown that since Re[Z(j(*))] > that this property can be expressed as: rn > Tn r 22 - (r 12 ) > where Zn = r x ! + j Xn Z 2 2 r 2 2 + J A 2 2 ^12 r 12 + J ^12 In summary the conditions for a two terminal pair are: (a) the open-circuit network impedances (Z 11; Z 22 , Z 12 ) must be regular in the right half "p" plane. (b) for p = jw, the real parts of these impedances must satisfy Re[Zn(ju)] 1 Re[Zii(ju)]-Re[Z 22 (ju)] - (Re [Z a 2 (j w)] ) 2 > (c) for p = jw, the poles of Z llt Z 22 , and Z 12 must be simple and the residues must satisfy Kfi > K?! Kf 2 - (K? 2 ) 2 > (III-5) It is evident that all poles in Z 12 at real frequencies must be in both Z x 1 and Z 22 in order that the above residue relation be satisfied. -101- An interesting property of Z 12 pertains to its zeros. Now Z 12 is the quotient of two polynomials in "p" (assuming a finite number of elements in the T-T-pair). Suppose Z 12 = P(p)/Q(p) where the zeros of Q(p) occur in conjugate pairs and are all simple Let the degree of Q(p) be even, so that P(p) is odd (for a reactance network) and cannot exceed Q(p) in degree by more than unity, otherwise the pole at infinity would not be simple. On the other hand, there is no lower bound on the degree of P(p). That is, the degree of P(p) can be less than the degree of Q(p) by any number permitted by the requirement that P(p) must still be an odd polynomial. If P(p) = (b + b 2 p + ---)p the " b • " may be positive, negative, or zero (but real) so long as at least one does not vanish This amounts to the fact that Z 12 or Z 2 i is definitely not the same in behavior as its reciprocal 1/Z 12 The zeros and poles obey quite different laws. For driving point impedances there is no difference between the impedance or admittance functions: for transfer impedances there is a great difference. For actual L-C synthesis of the two-terminal pair whose open- circuit driving point and transfer reactances obey the necessary con- ditions for physical realizability, consider the simple T network. This network is the simplest of the two terminal pairs but is restrictive as far as physical realization theory is concerned. FIGURE 37 -102- That is, Z 12 = Z must be "P-R" in this case while generally Z 12 is not "P-R". Hence, one must conclude that this scheme will not work in general. Further, if Z 12 is not "P-R", it is not obvious that Z a and Z^ are "P-R" because of the subtraction involved. Therefore, the "T" as it sits is unsatisfactory. If one employs ideal transformers, the sign of the residue at a pole of Z 12 can be adjusted (reverse polarity) in an effort to make the T physical. This setup may change the signs of the residues at other poles from positive to negative for 2. 11 and Z 22 but the situation can be remedied by using an ideal transformer for each of the various poles in- volved, and then removing the unnecessary ones. Mathematically, consider the three functions of frequency which are given and make partial fraction expansions of each. In the reactive case, all poles will be imaginary and in conjugate pairs; combine the two members of a pair into a term. Further, all poles of X 12 will appear in both \ X1 and X 22 so that Ki i 2Ki i w oo An + -, 2- + — + Kn u K 22 2K 22 w .woo /ttt ^\ X 22 - + -^ 2- + --- + K 22 w (III-6) v _ Ki 2 2K 12 w co A 12 - + —2 2- + +K 12 w w w - Wi Each set of K' s in a vertical column, must by hypothesis, satisfy Eq. III-5 (namely) K?i Kf 2 - (K? 2 ) 2 > The procedure now is to take each column in turn and synthesize a T network which realizes three elements of the set. Later, all the com- ponent "T"s" are connected in a series and it can be shown that the re- -103- sult is correct Denote three elements in any one vertical column by X? i , Xf 2 > Xf 2 and seek to realize this component set by a network of Fig. 38. o 1 X Q 1 X b .. E x c E o 1 v. FIGURE 38 I: a For an ideal transformer so that E 2 /E, L p and L s - °° aE = Ii Z 12 or Z 1 2 aE Ii therefore from which one derives Z. 12 :: a Z c Zli = Z a + Z c Z 22 - a 2 (Z b + Z c ) I Z s a **U Zb s a "i2 1_ 7S _. 1 vs ? ^22 a ^i a 7S _ 1 7S ^ii a ^i rs a "12 (III-7) which represent the expressions for the elements in the "T". The con- stant "a" can be chosen so that Eq. III-7 gives three realizable L-C im- pedances if Eq III 5 holds -104- Consider the element X in the T network: for X to be "P-R", it is necessary that "a" have the sign of Kf 2 If it does not, then Z c = 1/a Z? 2 would not be "P-R", assuming that Eqs . III-6 are "P-R". Furthermore, limits on the magnitude of "a" can be established by substituting in Eq, III-7, the predominant term in the Laurent ex- pansion for Zf 2 , Zf 2 , an d Zf x (in the immediate vicinity of the pole K s Z 1X = ; — , etc.) So that from P - J w s 7, = 1 7S _ 1 7S "a one derives or and IV 2 2 "112 . —2- - 2p z * _a a U 2 2 — P + W « 1*2 2 IVl 2 „ 2 _ _ U a a K s I "-1 2 From Z„ - Z n - r Z 1; i£-"> lal (III-8) can be derived K?! - i Kf 2 > a>^|ii (III-9) Kfi combining Eqs. III-8 and III -9 105- Th e r e f o r e the component set of "Z's" can be realized if "a" is chosen to have the sign of Kf 2 and if "a" has any magnitude that satis- fies Eq. Ill- 10 . This is always possible if Kfi Kf 2 - (K? 2 ) 2 > for Ki i K 2 2 — (Ki 2 ) or lKf 2 K 2 ! - \u s Hence, each column by itself can be realized as a "X" network plus an ideal transformer. Because each component U T" ends in a transformer, there can be no interaction between components if connected as in Fig. 39 below, and the overall "Z's" will be simply the sum of the com- ponent "Z' s". r-TTTP-l r-nnnp— i 1 J i_ innr*— — ,— -nnnr^ FIGURE 39 -106- APPENDIX IV CASCADE MATRICES REPRESENTING A LINE TRANSFORMER IN SERIES WITH A NON-DISSIPATIVE NETWORK Since the derived expression for X u , X 22 , and X 12 (Eq. 2-20) are transcendental functions, it would be desirable to express the prop- erties of the matching network in the same form. The input reactances of transmission line elements are transcendental functions but the general combination of such elements in a complex circuit is very dif= ficult to handle because of the numerous unknowns (i.e., each line may have a different Z Q and length). About the only combination that is not too cumbersome to handle is the cascade connection of transmission ele- ments, and the cascaded units of necessity must be small in number since each unit presents two unknowns requiring two independent conditions if a solution is to be had. With this in mind, consider a lossless line transformer whose characteristic impedance is Z' Q and whose length is I. The input impedance of the line is Z' Q |f cos 0Z + j sin 0Z in J Z p sin 3Z + cos (3 1 a; z a * j b 2 j c 2 z a + d; (IV-l) and the expression is quite similar to Eq 3-7, Hence it appears feasible to assume that the overall matching net- work could consist of a line transformer in cascade with L-C network as is indicated in Fig. 40. L-C Net. Ai vc: A 2 Line •*-TZ i 'fa Transformer 107- Note now, however, that two more unknowns, Z q and I, have been in- troduced into the overall problem, with the result that two more inde- pendent conditions must be known in order to solve the problem. That is to say, some method of prescribing the proper Z' and I is necessary before the solution for the L-C cascade network can be effected. Of course one could arbitrarily select Zg and I and take whatever form results for the L-C network. This situation is not feasible, since one has made no direct gain in the solution of the problem by intro- ducing the line transformer. On the contrary, the overall network has been complicated with more elements. A more reasonable approach would be to determine graphically the approximate values for Z Q and I and then to continue the solution for the overall network. For the L-C lossless, two-terminal pair the impedance transfor- mation is TZ, . A 1 Z i + jB 2 jC 2 Zi + Dt (IV- 2) Consider the transformation for the overall network (shown dotted) in the preceding figure TZ, TL r whe 1 e / A, A / J C 2 a: jB" 2 • f> II jC 2 D\ a;' AiA( - B 2 C 2 A ,. jB 2 jC 2 Di C 2 ■ CoAi + Di C 2 Di' ■ dDi - C a B 2 108- Nov M 2 2 Xfi XI. ii. C 2 C 2 Zq sin 3^ — A t cos 3^ ^ « , a i n I C 2 cos 31 + Aj 71 ^o sin BZ B 2 z f - Di cos pi C 2 cos 3* + A i sin gj Z o J_ ^1 C2 C 2 cos 3^ + T^ s i n $ l ~\ > (IV 3) The above equations express the overall open-circuit reactances in terms of the A it B 2 , C 2 , D x parameters of the lossless L-C network and the Z' Q and length of the line transformer. By equating these relations to the transcendental functions derived in Eq. 4-6, one can solve for the A t , B 2 , C 2 , Di parameters which des- cribe the L-C network. Assume that 3^ = k i