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CHAP. 13]

EXAMPLE 13.5

315

TWO-PORT NETWORKS

The Z-parameters of a two-port network are given by Z11 ¼ 2s þ 1=s

Z12 ¼ Z21 ¼ 2s

Z22 ¼ 2s þ 4

The network is connected to a source and a load as shown in Fig. 13-8.

Find I1 , I2 , V1 , and V2 .

Fig. 13-8 The terminal characteristics are given by V1 ¼ ð2s þ 1=sÞI1 þ 2sI2 V2 ¼ 2sI1 þ ð2s þ 4ÞI2

ð17Þ

The phasor representation of voltage vs ðtÞ is Vs ¼ 12 V with s ¼ j. From KVL around the input and output loops we obtain the two additional equations (18) Vs ¼ 3I1 þ V1 0 ¼ ð1 þ sÞI2 þ V2

ð18Þ

Substituting s ¼ j and Vs ¼ 12 in (17) and in (18) we get V1 ¼ jI1 þ 2jI2 V2 ¼ 2jI1 þ ð4 þ 2jÞI2 12 ¼ 3I1 þ V1 0 ¼ ð1 þ jÞI2 þ V2 from which I1 ¼ 3:29 10:28 V1 ¼ 2:88 37:58

13.7

I2 ¼ 1:13 131:28 V2 ¼ 1:6 93:88

CONVERSION BETWEEN Z- AND Y-PARAMETERS

The Y-parameters may be obtained from the Z-parameters by solving (1) for I1 and I2 . Applying Cramer’s rule to (1), we get Z22 Z V1  12 V2 DZZ DZZ Z21 Z I2 ¼ V þ 11 V DZZ 1 DZZ 2 I1 ¼

ð19Þ

where DZZ ¼ Z11 Z22  Z12 Z21 is the determinant of the coefficients in (1). By comparing (19) with (9) we have

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