316

TWO-PORT NETWORKS

[CHAP. 13

Z22 DZZ Z12 ¼ DZZ Z21 ¼ DZZ Z11 ¼ DZZ

Y11 ¼ Y12 Y21 Y22

ð20Þ

Given the Z-parameters, for the Y-parameters to exist, the determinant DZZ must be nonzero. versely, given the Y-parameters, the Z-parameters are Y22 DYY Y12 ¼ DYY Y21 ¼ DYY Y11 ¼ DYY

Con-

Z11 ¼ Z12 Z21 Z22

ð21Þ

where DYY ¼ Y11 Y22  Y12 Y21 is the determinant of the coeﬃcients in (9). For the Z-parameters of a two-port circuit to be derived from its Y-parameters, DYY should be nonzero. EXAMPLE 13.6 Referring to Example 13.4, ﬁnd the Z-parameters of the circuit of Fig. 13-5 from its Y-parameters. The Y-parameters of the circuit were found to be [see (14)] Y11 ¼

sþ3 5s þ 6

Y12 ¼ Y21 ¼

s 5s þ 6

Y22 ¼

sþ2 5s þ 6

Substituting into (21), where DYY ¼ 1=ð5s þ 6Þ, we obtain Z11 ¼ s þ 2 Z12 ¼ Z21 ¼ s

ð22Þ

Z22 ¼ s þ 3 The Z-parameters in (22) are identical to the Z-parameters of the circuit of Fig. 13-2. The two circuits are equivalent as far as the terminals are concerned. This was by design. Figure 13-2 is the T-equivalent of Fig. 13-5. The equivalence between Fig. 13-2 and Fig. 13-5 may be veriﬁed directly by applying (6) to the Z-parameters given in (22) to obtain its T-equivalent network.

13.8

h-PARAMETERS

Some two-port circuits or electronic devices are best characterized by the following terminal equations: V1 ¼ h11 I1 þ h12 V2 I2 ¼ h21 I1 þ h22 V2

ð23Þ

where the hij coeﬃcients are called the hybrid parameters, or h-parameters. EXAMPLE 13.7 Find the h-parameters of Fig. 13-9. This is the simple model of a bipolar junction transistor in its linear region of operation. By inspection, the terminal characteristics of Fig. 13-9 are V1 ¼ 50I1

and

I2 ¼ 300I1

ð24Þ

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An