By J. McDonald, N. Weiss
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Additional resources for A Course in Real Analysis
If we choose to reverse t h e as sumed current flow of 7 2 , then we m a y rewrite t h e equilibrium equations with a change in sign of I2 only; V1 = h11I1 + h12V29 -I2 = h21I1^rh22V2. 32) I t is now further required t h a t eqns. 32) be transformed. The variables will be then be related as dictated by t h e structure of t h e transmission matrix. From eqn. 34) "21 Substituting now from eqn. 34) for I1 into eqn. -fc'.. Fj = >A-*iAi Fj _*ii /l 2 1 p") 7j (3 38) ft21 Multiplying t h e numerator and denominator of t h e first term on t h e right-hand side b y — 1, Fi = fhnh22-h12h21\ \ ^21 y2_hi / h (3 39) ^21 I t will now be recognized t h a t t h e numerator of t h e first term represents t h e determinant of t h e h matrix, t h a t is (hnh22-h12h21) = Ah.
I t will be shown t h a t indeed, by observing a few simple rules in effecting the interconnections, the basic law of algebraic summation applies. The key to valid interconnection of twoports is shown to require the adherence to consistent matrix notations. (i) The Z matrix interconnection Consider a pair of arbitrary twoports which are mathema tically defined by the Z parameter matrix. We can represent the twoports as "black boxes" and interconnected as shown in Fig. 10. The matrices[Z] A and[Z]B are completely gene ral and may stand for, passive or active twoports.
Essentially, the use of tables for obtaining matrix interrelations is also quite simple and warmly recommended once the student has acquired the proper comprehension of algebraic manipulations leading to them. By referring to Table II, each of the intersections of a vertical column and a horizontal row yield the desired trans form pair. 4 From B D 1 1 Zll 221 2 2l 2 22 4 1 -2/22 2ll *11 ^21 2/22 4 «21 221 2/22 -2/21 «11 Zll 2/21 -A 2/21 ^r -z12 1 2/ii Z22 22 2 2/2i 1 1 2 2 2/21 -2/n 2/21 = 0U022-012021- ^21 ^ z — 2 u 2 2 2 — 212^21» h2i — h22 h2i A -1 -hu h2l -1 4 -4 A hn 4 -h2 4 1 021 4 021 ~" 2/ll2/22 ~ 2/12021» 021 011 021 022 4 1 011 4 4 -021 -012 4 022 022 022 -021 4 hn 022 012 4 4 011 -012 011 g 011 021 022 -h2l 2/22 1 0n hn -&12 ^22 r h12 h22 1 ri22 h21 hn 1 hu h22 -h2l 4 h22 & 2/22 2/12 2/ii 4 2/11 -2/12 4 4 - 2 21 4 2l2 4 4 2/ii -2/21 2/ii Zll — Z2\ 4 22 4 A.
A Course in Real Analysis by J. McDonald, N. Weiss