.-i V. FAR 1S. UNIVERSITY OF MICHIGAN ENGINEERING RESEARCH INSTITUTE ELECTRONIC DEFENSE GROUP TECHNICAL MEMORAJNDUM NO. 17 SUBJECT: A Minimum Transmission Loss Tschebycheff Two-Pole Matching Network BY: J. L. Dautremont, Jr., and P. H. Rogers DATE: October 13, 1954 Introduction When dealing with bandwidths in the order of 10 to 300 megacycles, the question of matching a partially reactive load to a transmission line presents a serious problem. The special case of a reactive load that consists of a resistance shunted by a capacitor is considered in the following sections. This problem is represented schematically in Figure 1. The minimum transmission ls rh__ w did Bd (e pn / loss for this problem was derived by Bode (See ppendix A). It may be expressed loss for this problem was derived by Bode (See Appendix A). It may be expressed 1 Bode, H. W., "Network Analysis and Feedback Amplifier Design," D. Van Nostrand, 1945. THE UNIVERSITY OF RM CHi-:GAN ENGINEERING LIBRARY

I f-. 2. L rj 2_

-2 - in db as Exp 1 (f2-f rRC o = 10 log0... (1) [Exp 1 ] - 1 (f 2-fl)RC where to is the minimum transmission loss over the passband in db f2 is the upper cutoff frequency fl is the lower cutoff frequency f2 - fl f. is the passband or bandwidth R and C are indicated in Figure 1. A plot of the minimum transmission loss versus cARC appears in Figure 2. The problem becomes one of finding the network whose response approaches this minimum transmission loss as closely as possible, with a given complexity. Networks can be synthesized to approach this theoretical optimum as closely as desired; however, in general, the number of circuit elements increases rapidly as the theoretical limit is approached. The two pole network shown in the following sections is rather simple as far as matching networks are concerned, and hence, a desirable type of network to use in electronic circuitry. The parameters for the two pole network have been determined so as to minimize the difference between its response and the theoretical minimum transmission loss curve. In general, it is very wasteful of gain bandwidth product to match perfectly at any frequency. It is possible to obtain considerable improvement by allowing a nominal mismatch over the band. This improvement for a two pole network can be seen in Figures 2 and 5.

56 - 52 48 44 o -MINIMUM TRANSMISSION LOSS 40 36:32 28___ 20 20 / fr22-2 POLE MATCHING NETWORK 16 7. 12 6 7 8 9 10 1 4 0 1 2 3 4 5 6 7 8 9 10 1II 12 13 TRANSMISSION LOSS (db FIG. 2 COARC VS. TRANSMISSION LOSS.

lo Derivation of Equations for the Two-Pole Minimum Transmission Loss Tschebycheff Matching Network This section outlines the derivation of a Tschebycheff matching network with a transmission loss equal to or less than a prescribed maximum within a bandwidth co. The network considered is shown in Figure 3. The frequency response of a Tschebycheff filter can be expressed as 2 2 P E A P2 (jo) | --- + 2 (2) a | E1 1+ eT n(co) where A2 is an arbitrary constant and Tn(co) is a Tschebycheff polynominal defined by: T (oo) cos(n cos co) (3) The solution to Equation 2 is given in Appendix B and results irz E A 2(p) -^ (4) n __ 1 fT (p-pk) where pk = p k + jco 2kl k k k Cr i -sin 2k- sinh k 2n k cos 2- T cosh ~K= 2n 1 -1 s= - sinh n For a two pole Tschebycheff response (n = 2), equation 4 becomes E2( ___ _ A (5) (P 2 '2 2 E1 P P V2 sinh * co' (2 sinh2 +1) -2 -

' VVVVV ~vT IsG. 3. O___ Ass ^/_7___e__, _ The response of the circuit shown in Figure 3 is given by E 22 2- = 1C (6) E" LC p2 P/ + 1 + 1...LC L~ RC LC Equating the coefficients of like powers of p in equations 5 and 6 yields a2R 1 o~^i sinh i -' -RC (7) and 1+a2 a_ (2 sinh2 + 1) -(8) 2g ~LC Solving these equations results in an expression for a2 a2.vcoRC sinh 7 - 1 a1~~......... * ___A/_______(9) ( 2 (2 sih2 +, 1) - iRC sinh i + 1 The response of the network shown in Figure 3 with the element values adjusted according to equations 7 and 8, and with the bandwidth normalized, is shown in Figure 4.

-6 - db /scerOsA/ wohSS -/ a o/ vSFZ 0 /_ dCb, ^ e 7/ fir,' S ZC/, F-SLS otz ',vo/_.L u /.,VL ~J..o /S ae. S ' TR w^M/SSL/O USS ^V D2&^?dS Fd6/Q ' /'..v. o )-,;e,/OSAka J C 'L V W APSs 7,'-Wb o,,, AY.7C.$ / /vW a VrWf PA/e$ From equation 2, the points of maximum transmission loss occurs when T 2 n (co) = 1, or a= +1, 0 for n 2 2. However, from the circuit of Figure 3 at zero frequency P (1 + a2)2 4a2 (10) (1 +a2)2 1 or 2 - 10 logo[ (1 a2 a 2 where P is the available power from the generator and P2 is the power delivered to the load. 2 has its minimum value for a2 " 1 a2 is

-7 - restricted to values less than or equal to one; hence for a small 2' a2 should be as large as possible. a2 is a function of oARC and sinh as is seen from equation 9. The maximum value of a2 for a given o&RC subject to the restraints imposed on a2 by equation 9 may be found by evaluating da2.. = O (n1) sinh This results in the following two equations 2 1 + a.- (12) A/1 + (c RC) and sinh2 = _ _ 2 (13) 2 1 - a1 (See Appendix C for this solution to equation 11) Solving equation 8 for L one has L m — 1 + a2 (lh) 2 Substituting the value of sinh2. from equation 13 into equation 14 results in 1 - a4 L (15) A2 C We have now determined the parameters of the circuit so that its response has a minimum transmission loss for a given value of coRC. A measure of the efficiency of this network is obtained from Figure 5 where the ratio of the theoretical maximum load power to the load power for this network has been plotted against co RC

100% 90 0 - 80 x0 X 4 0 Is w 60 30 0 a. - w 0 40 w 30 4d 0 _J - 20 a: L 10. I" I I I I I I I I I -*1_ ___ I I I I I 2 POLE MATCHING NETWORK. l l l FIGURE 5 NETWORK EFFICIENCY VS <WARC I 0 0 _ I 2 POLE FILTERI I I I I I I I I I I I~~ I I I I I m I 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 wARC 34 36 38 40

-9 - The efficiency of the normal two pole filter network is also plotted for comparison. 2. Design Procedure for Two Pole Bandpass Matching Networks with a Tschebyscheff Response Figures 10 and 11 have been included in this section for the purpose of evaluating the design before it is made. For a required coRC, the transmission loss for the network may be determined from Figure 10 and the ripple in its response from Figure 11. In the following design procedure it is assumed that the bandwidth, resistance levels, and capacitance are given. Design Procedure Givens: A, R, C, and <p2 which are defined by Figure 6. Gve R,/ Ca. 6a STEP 1: Determine the value of a2 for the given value of o RC from Figure 12. STEP' 2: Calculate L from Equation 15 1 - a4 Lj C- o C

-10 - This determines the values in the network shown in Figure 7. P/:;-. 7 STEP 3: The low pass network is changed to a bandpass network by resonating the reactive elements at the geometric mean of the upper and lower cutoff frequencies (See Appendix D). --- t -- t - L -- 1( --- | R AF- 8 STEP 4= The output impedance (a2R) is changed to the desired output impedance level (See Appendix E). f \G _ 4} 3 i < R^ JIt fa.>< A/l-. 9 where L a Lc(b-l) 1 -b LC L2 b L L + (1-b)LC 3 b2 b2 2 a2 b ' C2 b2CL 2 L Example of Design Procedure R = 5000 ohm = 20 me 02R m 50 ohm C - 7 Mt4f: 200 mc

-11 - o)RC = 2n From Fig. 10 From Fig. 10 o x 20 x 106 x 5 x 103 x 7 x 10'12 = 4.4 = 2.3 db From Fig. 11 db = 0.655 STEP 1e From Fig. 12 for o RC AI - 4.14, a2 = 0.225 STEP 2: STEP 3: 1 - a4 "A C LC o 3A20 1 C- o L (a 2L = 8.6Ah. * 0.0907/A h - 0.0738 /sf?5 = 22.5 f STEP 4U b2 2 0- 2 0.22 S5C Tooc b = 4.75 L (b-l) 1 b 0.0907(3.75) 4.75 = 0.0716/ h = LC b.0907 '.-7 - 0.019/4 h L + (1-b)Lc 3 b2 8.6-3.75 x 0.0907 22.5 = 0.367/ h C = bC 1.66 /'f 2 = L '11f

100 80::: "L: -~ — — i.... ill.-. r. LI 1~~~1~~'~~::1 - — ~-1- *~ r- f -— f i-I I: r*- tl:-~ — _... 60 40 20 O 8 10 a 8 6 4 2 I I r - -- -;-1 ---- i I ~. -.. - _t 1-i -: 1-. ti ~-J1lf;- t-g - I U: t1 L-.,.-..-~ -..1 -..t....... I 4-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r.jt *:..: -:.- - --- =- - - - - - -- --! I _ _ __ - — 4-: — 7 — - ---- -1,-IIII - l- -r 1 -.f, -, -. - _.. -f. _. -::-; —, j- q --- -X- — W 1 —1 { I -, i A 1 -r-+~:.:-~...-.. — -— '-: ' -— r- ~-1~.- 1 H- ___ ---1 — --!-1 1 #I -i I I& I Iit'-l- _ -_ - r -:-4 —' —::-: - 1:- il: ' X | |. | | | 1: - |::: J:?- 1 l:.;::::::.:... __ _ t X _.:+.4 --- —t - 4................ v. - - z o *i r —1- a _ _ -~t-1 —t v = -:.-1 '-r -. -. - I - L k lf:l - I-: — -1::...1..,.... H-H1- 1 fi-1 I -L4 I. i - -~. t iii v I......- *- ---- -> -. - v. - l| I L - I 0.2 0.4 0.6 8 1.0 2 4 6 8 10 TRANSMISSION LOSS (db) FIG. 10 20 40 wARC VS. TRANSMISSION LOSS

1.4 wJ 1.2 1.0 0.8.w 5; 0.6 a. 0.4 02 0 a 10 WA RC FIG. II cWRC VS. RIPPLE.

I00 410 - ---; * g ~ ** ~..4 4 - 8 0 60 40 S T 20 iflii lt t| -l i I 7l:!.!N i 4 l i 4:0n~~~~, Ig iljt~~~off~itiiMlf 1W141lii I- i l, 2..._.T 0.01 0.02 04 006 0.08 0.10 t FIG. 1 2 IRC VS. I a WARCV.c? 0.2 0.4 0.6 0.8 1.0

-15 -APPENDIX A DERIVATION OF MINIMUM THEORETICAL TRANSMISSION LOSS The equation due to Bode, representing the minimum realizable reflection coefficient for the network of Figure 1, is = [- ((2 c (A) C) 3,E2. r 1 -- f - E~p [ (f 2fl ) Rc The absolute value of the reflection coefficient is defined as: 12. Pa - P2 (A.2) Pa Where P is the power available from the source and P2 is the a 2 power which gets to the load. Hence, from equation 2 Pa_ __ _ IP _ (A.3) P2 ^ ~- _4 -_ - 12.. 2 So 0O = 10 logl d.| 1 - 1 L and A,4 4 g (A.4) But, from equations A.] = 10 log10 Eu, [f^)1RC I Eil [(f2-f1)RC 1 EXP4w 1 Op, Cit.; p. 1*

-16 - APPEODTIX. B LOCATIOCi OF POLES FOR A TSCHEBYCFEFF P]iSPONSE E 2 - (jco) E1 T (o) = < n A2 1 + Tn (co) cos (n cos' co) Set:. Coslco = 9a Tn(co) = cos n 1 Let p be complex: sp = 4 + i7 T (o) n - cos n(4 + j/) T (o) n = cos n ~ cosh ny + j sin n4 sinh n? But T 2() 1 - n e. = ~I I T (co) n Hence: = sin n sinh n cos n 4 Cosh np - 0 Since": Then: Hence cosh n > 1 Cos n = w 0 n 3 r 5n 2 ' 2' 29,, (2k-l)n for k 2n 7n 2_...' = 1, 2, 3,... 2n

-17 -Hence: sin n Cm -1 sinh nu -/ and -(2k- 1) r + " sinh and 2n..... Remember that: co cos pD = p/j - * 4! + ji co cos(4 + jY) co a cos 8 cosh t + j sin ( sinh. C o cos 2n cosh * J sin2k- sinh (Bl) Define p 6 + jco The poles of 2(jc) are found by substituting p/j for w in E1 equation B.1 and identifying the real and imaginary parts. Hence: (2k - ) ak " sin 2 sinh k 2n (2k - l) k = cos - 2n cosh (B*2) for k 1, 2,..., 2n The poles specified by equation B.2 lie on an ellipse with foci at ~J. This follows from the fact that: 2 2 _,k + k sinh2 / cosh t The poles in the left half plane are given by Pk where Pk = k + jck and - 2 k -siln (2k- l)n s.nh k 2n

-18 -2k - 1 co w cos - cosh v k 2n for k = 1, 2,..* n Thus, (p) may be expressed by: E1 m(E n (....) E- ( ) (Bk3) E1 k-l k

-19 - APPENDIX C DETERMINATION OF PARAMETERS FOR MINIMUM TRANSMISSION LOSS FOR A GIVEN coARC Equation (9) of the Introduction was 2 L,IRC sinh l - a (ARC)2(2 sinh2 + ) -V ARC sinh v + 1 2 It was shown in the Introduction that it was desirable to have a2 as large as possible in order to minimize the transmission loss. We can do this mathematically by 2 a - o sinh 2 (RC (2 sinh2 + 1' -,2 oARC sinh * 1 (C.i) 2 L (,,RC)2 2 -(VRC sinh p - 1) [, sinh ) - 2< RC] 2 (A2 (2 sinh 2 + l) - oRC sinh, + 1 2 $ Since this equation is zervo we cs-i equate the numerator to zero and solve for sinh s. ini _[j (c2) sinsb b -^- | 1 + J1 + (oRC)2 (C.2) Substituting this value for sinh in the original equation for a2 yields a2 l 1 (0.3)

-20 -Solving equation C.3 for c RC and substituting in equation C.2 gives 1 (c.a2 sinh i =.-.. (c.*)

-21 -APPENDIX D LOW PASS TO BAND PASS TRANSFORMATION A scheme for transforming from low pass to bandpass is given in Fig. D.l. The resulting two pole bandpass circuit and typical response curve appears in Figure D.2.

LOW PASS 0 C *( -o BANDPASS b 0 (L~0 c C L CL L C 0-nBooboop ---o WHERE: ( = -1 ' - 0C LcC LCL| 2 w2 —1 = WA FIG. D.I LOW PASS TO BANDPASS TRANSFORMATION. 2 E1 4A2 4A2 1 I +e W R FIG. D.2 POLE MATCHING NETWORK. BANDPASS TWO 22

-23 -APPENDIX E IMPEDANCE LEVEL. TRANSFORMATION To change the ipedance leel of the two pole bandpass Tchebycheff network, follow the steps in Figure E.1.

L a R E2 L I b2CL -- I I"Lc t I at^ | j L/b b'CL C LCL L _ R E I5 L+(l-b)Lc | b(b-1) b_ b k,. ~~I ZOCL TII | Lc(b-l) L+(I-b)LC I I: 1 b T t FIG. E.I FINAL CIRCUIT FOR TWO POLE BANDPASS MATCHING NETWORK.

UNIVERSITY OF MICHIGAN 3 9015 02844 0744 THE UNIVERSITY OF MICHIGAN DATE DUE T C)Lh qiGAN 10 li'm - I*,. 0 -