Vl E> RAHY OF THE UNIVERSITY Of ILLINOIS 510.84 IS6r no. 237-242 cop. 2 The person charging this material is re- sponsible for its return on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. University of Illinois Library Digitized by the Internet Archive in 2013 http://archive.org/details/optimizationofbr239bunt 57 U • o 7 **1 Jyuvu*) Report No. 239 r* *? .>* OPTIMIZATION OF THE BRIDGED-T NETWORK USING COMPUTER TECHNIQUES by Marcus L. Bunting August 15, 1967 THE LIBRARY OF THf OEC Z< 1967 UNIVERSITY OF ILLINOIS Report No. 239 OPTIMIZATION OF THE BRIDGED-T NETWORK USING COMPUTER TECHNIQUES* by Marcus L„ Bunting August 15, 1967 Department of Computer Science University of Illinois Urbana, Illinois 6l801 of'Mastel Tf iT^ ful £ illment ° f the requirements for the degree IllfnS; A°Z C t 1967? l6CtriCal **— ** ***»■«* of Ill ACKNOWLEDGMENT The author wishes to express his sincere thanks to Professor T. A. Murrell for his excellent counsel, support and encouragement in the preparation of this paper. ITianks are also extended to Mark Goebel for his assistance in providing the final drawings for this report and to Miss Bonnie Malcor for typing the manuscript. IV TABLE OF CONTENTS Page 1 . INTRODUCTION • • ....... 1 2. SOLVING FOR NETWORK PARAMETERS • •••-. 3 3. GENERALIZED NETWORK ANALYSIS • . •••«....,.. 7 h. OPTIMIZATION OF THE EQUIVALENT Q ' ' 17 5. SUMMARY '• 31 BIBLIOGRAPHY. •••••• • 4l 1 . INTRODUCTION R-C circuits are of great interest in that they offer frequency selectivity ana are easily fabricated using integrated circuit technics. A worthwhile goal is that of optimizing particular R-C circuits with respect to the number of components but more importantly with respect to certain circuit parameters. The optimization of certain parameters of a R-C circuit is necessary in order to be able to build tuned power amplifiers, very stable harmonic oscillators, and filters without the need of the usual L-C circuit. This is of great importance for integrated circuits in view of the difficulty of constructing small and efficient inductors. This paper will investigate one particular R-C circuit - the bridged-T- and will attempt to discover its usefullness and limitations as a filter and as an oscillator feedback network. Some general guide lines for the optimization of the network with respect to several performance characteristics will also be attempted. Although some studies on the bridged-T network have been done in the past, most authors have considered only the ideally loaded network configuration, that is, either an open or a short circuit load. Another constraint which is often placed upon bridged-T design is that of being a symmetrical network with both resistors equal and both capacitors equal. Consideration will be given here to a non-ideal loading, non-symmetrical network; however, due to time limitation, only resistive terminations will be investigated. 2 A high speed digital computer was used to generate solutions the network equations of the bridged-! network for various values of the variables. Due to the iterative procedures required in optimizing the network, the computer presented itself as the natural tool for this type of operation. Also, various plotting routines of the computer were used to provide easily understood graphs of the bridged-T's performance. 2 SOLVING FOR NETWORK PARAMETERS The bridged-T can be represented by the following topology, i G>H -o~ < Z L The loop equations are v i - V z i + S ] + W + VV v 2 - 1,(23) + i 2 (z 3 + z 4 ) - i 3 ( Z ^) = 1^) - i 2 ( Zji ) +i 3 (z 1 + z 2 + z 4 ) Solving Equation (2-3) for I yields (2-1) (2-2) (2-3) J 3 = (2-k) Substituting Equation (2-k) into Equations (2-1) and (2-2) Z? Z l\ i = WV VVV ^ J 2 (z 3 + v^' V 2 Z.Z, z ) The standard two-port nomenclature is V l = Vl + Z 12 X 2 V 2 = Wl + Z 22 I 2 A direct comparison yields hh < (2-< v 2 ■ h'-h ♦ r£ft * W* " vW (2 " 6) (2-7) (2-8) (2-9) Since current gain is a more useful parameter for transistor circuits, A rather than A will be investigated. The standard forms for input i v impedance and current gain are Z 12 Z 21_, Z in - Z ll " Z 22 + Z L A. = Z -- 1 ' Z 22 + Z L From these two equati ons Z. => Z - Z A ln n 12 i (2-lM s ubstltutlng the open clrcuit lmpeaance p8rameters intQ ^ ^^ Ration ana reduclng , an expresslQn ^ ^ ^ ^ ^ ^ Z n Z - ; Z 7 z + z + -A-l + _£_^ Z k \ can be simplified by looking at a specific -idged-T newer* with restrictions upon the element values, if the impedance values are chosen such that 1 Z = Z, = -i- 1 4 jwc Z 2 = R 2 = aR Z„ = R = 5 3 3 Z L = (idealized loading) (2-16) (2-1?) (2-18) (2-19) and RC (2-20) then by substition into f?-Kl a „ to ^ ^/j A i c an be written as 2 W 00 A . 1 . , - ^ 2-21) - + a + ,H77 a From this equation it can be seen that at the radian frequency o> = <* Q , the current gain goes through a minimum, referred to as the null point, and the phase shift (difference in angle between the input current to the bridged-T network and the current out of the network) is degrees. Figure 1 is the plot of the magnitude and the phase angle of A ± versus UL for Equation (2-2l). It can be seen that by choosing various values of the variable a, the equivalent Q of the network can be made more or less sharp. By increasing the value of a, the Q is also increased; however, it is seen that the off -resonance phase shift also increases in magnitude „ Referring to Figure 1, as a goes from 1 to 10, the null sharp- ness also increases . At the normalized frequency value — - 2, the phase angle increases from a value of approximately 11 degrees for a equal to 1 to 7^ degrees for a equal to 10. Because of this effect of rapid phase shift with respect to frequency change near the circuit null point, the bridged-T is limited in its applicability as a negative feedback network. 7 3, GENERALIZED NETWORK ANALYSIS To proceed to the more general case, the restrict- c ; ^ne restrictions on the capacitor values and on the loading re,ic + oaamg resistor are now relaxed The impedances of Equations ( 2 -l6) through ( 2 20) irougn (2-20) can be redefined as Z 2 = aR (3-1) 3 8 (3-2) Z . =^- 1 jwbC , . (3-3) Z = - b . k ' JMC (3-4) RC . (3-5) z T = R r = as L (3-6) allows for more c ° m ™ e — manlPUlation and for ; asy determination of thp fnn^ n ui xne imal element valuer -p^ -p values . The freedom of the designer to ae Vel0 p an efflcient net „ ork is ^ greater using ^^ ^ definitions than that of -fv, an that of the symmetrical network ideal lv i ,, „ uia j laeaily-ioaded case, A PPlying the above definitions ^ ■, aeimitions of element values to A and reducing the resulting equation gives 2 w oj 1 + b + jab (-j- - -0) (3-7) The increased complexity of this equation Equation (2-2l) is quite evident especially when the phase shift is considered. No longer is the phase shift equal to zero at w nor will the null necessarily occur exactly at the defined » - w Q because of the added terras. Two important questions arise. How sharp a null can be achieved without exceeding a certain phase shift? At what point will the overall current gain be so low tnat the network can no longer be practical value? The consideration of the maximum phase shift and the minimum current gain creating a restriction on the upper value of Q will be defined as the "limiting conditions" in the optimization procedure. The frequency at which these limiting conditions are evaluated defined as * = F X u) For convenience , F is set equal to 2, To define the "sharpness of the null" or the equivalent Q of the circuit, a ratio can be formed !i i ':^L E i^o (3-8) 1 o This suggests the use of a normalized frequency variable u (3-9) E = — The equivalent Q, is the value of the magnitude of the current gain j evaluated at u> = E X o> Q divided by the value of the magnitude of current gain at w = w Finding the magnitude of A. in Equation (3-7) ^d substituting ',3-9) into this result gives 2 "~ 2 (!+b ) + (ab(E--)) Cl+b +a b +ab d+ ad) 2 + (ab(E-i) +a 2 bdE) 2 Applying Equation (3-10) to the (3-10) (3-8) result Q definition of equivalent Q gi s m given by (3-11) - previous state*. u . Uq will not be ^ ^ ^^ ^ A, *>r the non- ld ea lly termiMted case; ^ ^ the considered here it m-n u -, wUl be SU mcie„tl y near to the nl „ lnuB for deslgn considerations . Thus, as the value of the rat-in n - the ratio Q increases, the null becomes sharper; and as Q decreases the null h« b, the null becomes broader. Figure 2 and Figure 3 illustrate this point . Throughout the rest n-p +w^ rest of this paper, the frequency variable E is chosen to be equal to 2, The current B ■ me current gam ratio Q will then be determined by the radian frequencies « Q and 2 x ^ " C ° mPUter Pr ° gram W3S - 1 "- to look 1 certain circuit —tens such as current gain at « Q and at 2 x „„ the input impedance ^ various loading i^dances, and the phase shift of the network for these values of the frequency variable. This program, shown on pages 10 anrl n t v y 8^b iu and Intakes input data of «* ™ lues of ., bi a , and E and substitutes these ^^ ^ on ^ to solve for the current galn ana into Equatlon fc^) fQr input CM UJ cm o c\j CO CD cm ^O _J < UJ > a < r- a. UJ CM ii ii UJ x z ^}- »— I « is) cm • X I s - GO « UJ • < II lT\ CD » CD en CM £> X UJ LD CM < CM < CD < CD II II < CD • • i-l CM tl II rH CM UJ UJ UJ UJ CM KD <-. < CM 5! o < < II 5; cm < o w < u. s: < o uj IT* |SJ ^" I— r-4 *»» * 3 * ■4- a o ^ (- x mD h • c — UJ a • < r- H UJ li t- < D Q. (_ x * ID 4" CM O- • CO tsl X co ii UJ is) X 00 •I en UJ tl < z O •— ' UJ M • X O CO < » s: CM o CM cmq: cmo •> u_ %o L0 UJ UJ CLlD <_l r- < > I— "D ax h- r- D rH i— ■ (Ti fsj rH ». UJ UJ rH a CM II UJ < z 2: < < - X co r- CM »• z < UJ < •> ITITN \D • • e^en r— 1 1— I LULU II II o c < Z> >- ZZ CO ZZ _l oo < CM CO o -^ CM CM CM ITiCM en CM vor- < Z NJ 11 NJ LU or z •— t N » < < * < ft LU 0T < LU * Q •• 03 «> < < LU z D • O O or in 2: 2: t-H 1— 1 > > < < + + LU LU or dc > > < < co D CD CD cm (\i 11 LU •s. • e— I I t\j LU GO — — X LU * Q * - + Q CO O * "N 4. < • < w rH + + — 10 I- (I II II <\J CO * Q 9 1—1 + *vLU •* + * (NCD CO— << + I tOl— II II > > + * D NJ NJ * * Nl NJ + + >• >- V V V v ^<^HD^^ > + CO 11 II LU 5: < < - nj nj • * * 1- 10 » + I >- >- 10 I— II tl lu 5: OT «-h > > < < > > << * * 2:lu ft-, [x << + I LU LU DC QT > > < < LU ar < < -I _J DC DC II II LU 5; or m * r\j * in * < + NJ DC * 2 % »— . LU NJ or — < ii 81 < N f eg * LU NN < DC Z h- Q w|u z Nl OT LU LU o 00 LU DC < LU LO < X C\J LU Q\ Z iTi LU DC < in < o oo o_ 000 00 00 00 CO CO » LU CM _J 15 I- o- H D cm co O • LU DC CO _| CD II O D ~ CO Q_ < LU £ < < CO LU _J ID ZU-lLZ < IT < M II LU O LU —If— _J o z < o< * en LU _l 12 Z m < 11 OLu l-_l O oz LU D Z2 *-. a: I- Z) ZhQ O LU 2 u or lu CM a very stable oscillator seems achievable. 2 ' Princeton. \£o) .~^S^^^ ^ Van N °^and =0., 1, I960), pp 174-176 16 The drawback comes, however, from the large amount of attenuation inherent in the bridged-! network, particularly in this high Q -- large phase shift region. Unlike the typical L-C network for which loading conditions are not too severe, a bridged-T network loads the input circuit even when operating at its so called resonant or null point 3 This was pointed out for the parallel-T case by Leonard Stanton. 3, Stanton, Leonard, "Theory and Application of Parallel-T Resistance- Capacitance Frequency-Selective Network," Proc. I.R.E ., Vol. 6*, (July, 19^6), pp. kkl-ktf. "». OPTIMIZATIOM OP THE EQUIVALENT Q * starting point for optimization can be establishes bv an investigation o f the Q versus normalised network ^^ ^ previously found. Figures h q a„x c u g k, 5, and 6 show specifically how the variables can be manipulated to increase the value of Q. Usin , thlq . 1 «%. using this knowledge and the limiting conditions as defined on page ft n P + u « Page ft, a computer program was written to compute the circuit element values. *» a practical standpoint, somewhat limited ranges of element values are useful. We WO uld l ike resistor values to be from several tens of ohms to several tens of megohms. Capacitors shouid be in the micro- to pico-farad range. From the prions chapter; the general ^ ^ ^^ ^ ^ was seen to be between .01 and loon is. is and 1000. From the definition Equations (3-1) and (3-so) and the knowledge of the obtainable resistor values, a starting value of B was chosen as 5 X lO^ ohms. Because this network will probably be used ^ ^.^^ ^ a transistor circuit of low input impedance, R L „ as chosen to be approximately 50 ohmq im,-,- «, ~ y yv onms. This corresponds to a d of R- 50 t = -4 S IT " .ooi R 5xio 4 " C^-i) The value of d was kept small throughout this optimization procedure because of element value restrictions. From Figure 5 , the optimum value of b was seen to be approximately 1. p* om a mathematlca] _ ^^ ^ ^^ fc ^ found as follows. Simplifying Equation (3-ll) with the restriction that d be small with respect to a, b, and E, it follows that 2 N 2 [l+b ) + (ab(E-i)) 2 22 1+b +a b (U-2) (l+b 2 +a 2 b 2 ) + (ab(E-^)) 1+b' If a is taken to be much larger than 1.0 in order to get large values, of Q, and E is near 1.0, which agrees with the definition of E, then (1 + b 2 + a 2 b 2 ) 2 » (ab(E - h) 2 (^-3) and Q = l + b 2 ) + (ab) 2 ab l+b l+b (k-k) Forming the derivative of Q with respect to b gives db 1 + fab \ l+b 2 ab l+b X a (l+b 2 ) - 2ab (l+b 2 ) 2 - (+-5) To maximize equivalent Q, this derivative is set equal to zero and the resulting equation is then solved for b„ ^ = = 1 + b 2 - 2b 2 db 2 ~ . b = 1 Thus, a good starting value for b in the optimization program is (U-6) (+-T) (U-8) b = 1. 19 The value of = was found by lncreaslng a frQm ^ ^^ ^ value (.0! was chosen) to a point at which either the phase shift ana/or current magnituae limits are vlolated . ^ ^ ^ ^ ^ ^^ a can then he reaucea ana . new value of h computed. I„ tMs case> „ „, reduced by one-half of the difference between its present value ana its preceaing value. This nmi-prinr. „-p • procedure of increasing a until the limiting conaitiona are violate* then facing it h y a specific factor and finaing an optimum „ „ m lead to a final value for a, b , and ,. For „ reference angle of , 5 degrees, convergence to the final values took place after 13 iterations. The term "optimum b - aescribea that set of network parameters resulting in the largest Q for a b at the peak of the - * r\J Z o u C\J z o u m in 10 z UJ O rH o O o • • OO II <-H IT ^ II II I- js ct < o o o o O .-I o o o o • «— I • r-H II II II X z ~ <£ •— I t— I r ^ w < CD X v0 UJ CL < *H O rH O C • • • H II H (\J m - > fMi-l OO uu • II H _J «~^ rv — z < < s: u. u_ UJ UJ ct: a: <* i- • D r-t Q. 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•H d,E = constant Figure 8. Curves of Equivalent Q for Two Values of a la BIBLIOGRAPHY Sulzer, Peter G,, "A Note, on a Bridged t iw , r, „ Vol. 39, No. 7 (July, 1951)! J3ridged ~ T Network/' Proc^JC^E. Terman, Frederick E. and Joseph M Pettit fipm-,. • M McGraw-Hill Book Company, Inc!, New y^ lllf^^ Tuttle, W. N., "Bridged-T and Parallel-T Null n- KStar^'^*^»a, P rent lce -„ all Inc ., r * J Uf, ?0 %