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TWO-PORT NETWORKS

[CHAP. 13

Za ¼ Z11  Z12 ¼ 4s  3s ¼ s Zb ¼ Z22  Z21 ¼ 9s  3s ¼ 6s Zc ¼ Z12 ¼ Z21 ¼ 3s (b) We repeatedly combine the series and parallel elements of Fig. 13-22(b), with resistors being in k and s in krad/s, to find Zin in k as shown in the following. Zin ðsÞ ¼ Vs =I1 ¼ s þ

ð3s þ 6Þð6s þ 12Þ ¼ 3s þ 4 9s þ 18

and i1 ¼ 0:2 cos ð1000t  36:98Þ

or

Zin ð jÞ ¼ 3j þ 4 ¼ 5 36:98 k

(mA).

13.10 Two two-port networks a and b, with open-circuit impedances Za and Zb , are connected in series (see Fig. 13-12). Derive the Z-parameters equations (31a). From network a we have V1a ¼ Z11;a I1a þ Z12;a I2a V2a ¼ Z21;a I1a þ Z22;a I2a From network b we have V1b ¼ Z11;b I1b þ Z12;b I2b V2b ¼ Z21;b I1b þ Z22;b I2b From connection between a and b we have I1 ¼ I1a ¼ I1b

V1 ¼ V1a þ V1b

I2 ¼ I2a ¼ I2b

V2 ¼ V2a þ V2b

Therefore, V1 ¼ ðZ11;a þ Z11;b ÞI1 þ ðZ12;a þ Z12;b ÞI2 V2 ¼ ðZ21;a þ Z21;b ÞI1 þ ðZ22;a þ Z22;b ÞI2 from which the Z-parameters (31a) are derived.

13.11 Two two-port networks a and b, with short-circuit admittances Ya and Yb , are connected in parallel (see Fig. 13-13). Derive the Y-parameters equations (32a). From network a we have I1a ¼ Y11;a V1a þ Y12;a V2a I2a ¼ Y21;a V1a þ Y22;a V2a and from network b we have I1b ¼ Y11;b V1b þ Y12;b V2b I2b ¼ Y21;b V1b þ Y22;b V2b From connection between a and b we have V1 ¼ V1a ¼ V1b V2 ¼ V2a ¼ V2b

I1 ¼ I1a þ I1b I2 ¼ I2a þ I2b

Therefore, I1 ¼ ðY11;a þ Y11;b ÞV1 þ ðY12;a þ Y12;b ÞV2 I2 ¼ ðY21;a þ Y21;b ÞV1 þ ðY22;a þ Y22;b ÞV2 from which the Y-parameters of (32a) result.

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