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In the circuit of Fig. 12-65(b), let R 1 ; C1 1:394 F; C2 0:202 F, and C3 3:551 F. Find H s V2 =V1 and show that it approximates the passive third-order Butterworth low-pass lter of Fig. 12-65(a). Ans: H s 1= 0:99992s3 1:99778s2 2s 1 Show that the half-power cuto frequency in the circuit of Fig. 8-42 is !0 1= RC and, therefore, frequency scaling may be done by changing the value of C or R. Ans: V2 2 2 1 ; !0  2   V1 R2 C2 s2 RCs 1 RC s s 1 !0 !0
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Find RLC values in the low-pass lter of Fig. 12-65(a) to move its half-power cuto frequency to 5 kHz. Ans: R 1 ; C 31:83 mF; L 63:66 mH
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13.1 TERMINALS AND PORTS In a two-terminal network, the terminal voltage is related to the terminal current by the impedance Z V=I. In a four-terminal network, if each terminal pair (or port) is connected separately to another circuit as in Fig. 13-1, the four variables i1 , i2 , v1 , and v2 are related by two equations called the terminal characteristics. These two equations, plus the terminal characteristics of the connected circuits, provide the necessary and su cient number of equations to solve for the four variables.
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Fig. 13-1
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The terminal characteristics of a two-port network, having linear elements and dependent sources, may be written in the s-domain as V1 Z11 I1 Z12 I2 V2 Z21 I1 Z22 I2 1
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The coe cients Zij have the dimension of impedance and are called the Z-parameters of the network. The Z-parameters are also called open-circuit impedance parameters since they may be measured at one terminal while the other terminal is open. They are V1 Z11 I1 I2 0 V Z12 1 I2 I1 0 2 V Z21 2 I1 I2 0 V2 Z22 I
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CHAP. 13]
TWO-PORT NETWORKS
EXAMPLE 13.1 Find the Z-parameters of the two-port circuit in Fig. 13-2. Apply KVL around the two loops in Fig. 13-2 with loop currents I1 and I2 to obtain V1 2I1 s I1 I2 2 s I1 sI2 V2 3I2 s I1 I2 sI1 3 s I2 3
Fig. 13-2 By comparing (1) and (3), the Z-parameters of the circuit are found to be Z11 s 2 Z12 Z21 s Z22 s 3 Note that in this example Z12 Z21 . 4
Reciprocal and Nonreciprocal Networks A two-port network is called reciprocal if the open-circuit transfer impedances are equal; Z12 Z21 . Consequently, in a reciprocal two-port network with current I feeding one port, the open-circuit voltage measured at the other port is the same, irrespective of the ports. The voltage is equal to V Z12 I Z21 I. Networks containing resistors, inductors, and capacitors are generally reciprocal. Networks that additionally have dependent sources are generally nonreciprocal (see Example 13.2).
EXAMPLE 13.2 The two-port circuit shown in Fig. 13-3 contains a current-dependent voltage source. Find its Z-parameters. As in Example 13.1, we apply KVL around the two loops: V1 2I1 I2 s I1 I2 2 s I1 s 1 I2 V2 3I2 s I1 I2 sI1 3 s I2
Fig. 13-3
TWO-PORT NETWORKS
[CHAP. 13
The Z-parameters are Z11 s 2 Z12 s 1 Z21 s Z22 s 3 With the dependent source in the circuit, Z12 6 Z21 and so the two-port circuit is nonreciprocal.
T-EQUIVALENT OF RECIPROCAL NETWORKS
A reciprocal network may be modeled by its T-equivalent as shown in the circuit of Fig. 13-4. Za , Zb , and Zc are obtained from the Z-parameters as follows. Za Z11 Z12 Zb Z22 Z21 Zc Z12 Z21 The T-equivalent network is not necessarily realizable. 6
Fig. 13-4
EXAMPLE 13.3 Find the Z-parameters of Fig. 13-4. Again we apply KVL to obtain V1 Za I1 Zc I1 I2 Za Zc I1 Zc I2 V2 Zb I2 Zc I1 I2 Zc I1 Zb Zc I2 By comparing (1) and (7), the Z-parameters are found to be Z11 Za Zc Z12 Z21 Zc Z22 Zb Zc 7
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