By Gibbs R. C.

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93) will yield t h e desired expansion for voltage transfer ratio. Thus from eqn. 93) b y transposition, 4, V _ F * 1 A BY . 95) L 1 By definition Y L == l/ZL' therefore eqn. 95) can be rewritten ZL AZL+B Ay T A B L E I. T H E T E R M I N A L PROPERTIES I N TERMS OF THE MATRIX A< Av Z2lZ>L A. 96) «»l ^22 + %L PARAMETERS Zo Zi Az + z22Zg *n + Z, A, + zuZL *22 + ZL -y2iYL yu+Yg Ay+ynYL Ay+y22Yg Ay+ynYL -h2lYL h22 + YL hn + Z, Ah + h22Zg dh + hnYL h22 + YL 921 021 Ag + 922Yg 922+Z 1+022*^ At + 9n%L 9n + Yg ^g+9nZL DZ0+B CZG+A AZL + B CZL+D -1/21 y 22 +YL -hi hnYL + Ah ZL AZL+B 1 CZL + D Definitions: z0 = z,= IV D e t e r m i n a n t s of m a t r i c e s : ^z ~ dy = Ah = Ag = ZZ \\ 22~ Z \2Z2\i yuy22-yi2y2i, hnh22-h12h2l, 9ii922-9i293i' y22+YL L 3.

69) Av=^ V2 ^1 (iii) Input impedance Consider again t h a t the load impedance ZL is connected across t h e output terminals 3-4. We again solve eqns. 72) (iv) Output impedance If in Fig. 73) 28 BASIC MATRIX ANALYSIS AND SYNTHESIS Substitute now for I±, into eqn. 60): -V1YG = g11V1 + g12I2. 74) From eqns. 76) ^ 2 = ^ 2 1 ^ 1 + ^22^2- Solve eqns. 77) g12\ ffn 0221 By transposition and inversion of eqn. 78) gn+Yo (e) THE TERMINATED TWOPOBT AND THE TRANSMISSION MATRIX When manipulating our linear, terminated twoports in the preceeding chapters, we have, in fact practised impli­ citly some of the basic techniques of matrix synthesis.

This time we revert to Fig. 14). By adding the admittance element Y2 to the Y22 position, we have in fact synthesized the mathematical mo­ del of our n network: The elements in the array are now identified as follows: Y11=(Y1+Y3), raa = (F a +r 8 ). (d) T H E ABCD OR T R A N S M I S S I O N M A T R I X A P P L I E D TO T H E S Y N T H E S I S OF S I M P L E N E T W O R K MODELS A closer look at the linear models of L, T or n networks will suggest the definition t h a t these are synthesized by cascading of series impedances and shunt admittances.

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