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CIRCUIT ANALYSIS: PORT POINT OF VIEW
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[CHAP. 1
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The Norton admittance is found from the result of Example 1.3 as YN 1 1 0:2 S ZTh 5
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We shall sometimes double-subscript voltages and currents to show the terminals that are of interest. Thus, V13 is the voltage across terminals 1 and 3, where terminal 1 is at a higher potential than terminal 3. Similarly, I13 is the current that ows from terminal 1 to terminal 3. As an example, VL in Fig. 1-6(a) could be labeled V12 (but not V21 ). Note also that an active element (either independent or controlled) is restricted to its assigned, or stated, current or voltage, no matter what is involved in the rest of the circuit. Thus the controlled source in Fig. 1-6(a) will provide VL A no matter what voltage is required to do so and no matter what changes take place in other parts of the circuit.
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The network of Fig. 1-8 is a two-port network if I1 I10 and I2 I20 . It can be characterized by the four variables V1 ; V2 ; I1 , and I2 , only two of which can be independent. If V1 and V2 are taken as independent variables and the linear network contains no independent sources, the independent and dependent variables are related by the open-circuit impedance parameters (or, simply, the z parameters) z11 ; z12 ; z21 ; and z22 through the equation set V1 z11 I1 z12 I2 V2 z21 I1 z22 I2
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I1 1 + V1 _ I1 Linear network I2 I2 + V2 _ 2
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1:8 1:9
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Each of the z parameters can be evaluated by setting the proper current to zero (or, equivalently, by open-circuiting an appropriate port of the network). They are V z11 1 1:10 I1 I2 0 V z12 1 1:11 I2 I1 0 V z21 2 1:12 I1 I2 0 V z22 2 1:13 I
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2 I1 0
In a similar manner, if V1 and I2 are taken as the independent variables, a characterization of the two-port network via the hybrid parameters (or, simply, the h-parameters) results: V1 h11 I1 h12 V2 I2 h21 I1 h22 V2 1:14 1:15
CHAP. 1]
CIRCUIT ANALYSIS: PORT POINT OF VIEW
Two of the h parameters are determined by short-circuiting port 2, while the remaining two parameters are found by open-circuiting port 1: V1 h11 1:16 I1 V2 0 V h12 1 1:17 V
2 I1 0
h21 h22
I2 I1 V2 0 I 2 V
1:18 1:19
2 I1 0
Example 1.7. Find the z parameters for the two-port network of Fig. 1-9. With port 2 (on the right) open-circuited, I2 0 and the use of (1.10) gives V R R R3 R1 k R2 R3 1 2 z11 1 R1 R2 R3 I1 I2 0
I1 + V1 _ R1 R2 I2 + V2 _
Fig. 1-9 Also, the current IR2 owing downward through R2 is, by current division, IR2 But, by Ohm s law, V2 IR2 R2 Hence, by (1.12), z21 V2 R1 R2 I1 I2 0 R1 R2 R3 R1 R2 I R1 R2 R3 1 R1 I R1 R2 R3 1
Similarly, with port 1 open-circuited, I1 0 and (1.13) leads to V R R R3 R2 k R1 R3 2 1 z22 2 R1 R2 R3 I2 I1 0 The use of current division to nd the current downward through R1 yields IR1 and Ohm s law gives V1 R1 IR1 R1 R2 I R1 R2 R3 2 R2 I R1 R2 R3 2
CIRCUIT ANALYSIS: PORT POINT OF VIEW
[CHAP. 1
Thus, by (1.11), z12 V1 R1 R2 I2 I1 0 R1 R2 R3
Example 1.8. Find the h parameters for the two-port network of Fig. 1-9. With port 2 short-circuited, V2 0 and, by (1.16), V R1 R3 h11 1 R1 kR3 I1 V2 0 R1 R3 By current division, I2 so that, by (1.18), h21 I2 R1 I1 V2 0 R1 R3 R1 I R1 R3 1
If port 1 is open-circuited, voltage division and (1.17) lead to V1 and h12 R1 V R1 R3 2 V R1 1 V2 I1 0 R1 R3
Finally, h22 is the admittance looking into port 2, as given by (1.19): h22 I2 1 R R2 R3 1 V2 I1 0 R2 k R1 R3 R2 R1 R3
The z parameters and the h parameters can be numerically evaluated by SPICE methods. In electronics applications, the z and h parameters nd application in analysis when small ac signals are impressed on circuits that exhibit limited-range linearity. Thus, in general, the test sources in the SPICE analysis should be of magnitudes comparable to the impressed signals of the anticipated application. Typically, the devices used in an electronic circuit will have one or more dc sources connected to bias or that place the device at a favorable point of operation. The input and output ports may be coupled by large capacitors that act to block the appearance of any dc voltages at the input and output ports while presenting negligible impedance to ac signals. Further, electronic circuits are usually frequency-sensitive so that any set of z or h parameters is valid for a particular frequency. Any SPICE-based evaluation of the z and h parameters should be capable of addressing the above outlined characteristics of electronic circuits.
Example 1.9. For the frequency-sensitive two-port network of Fig. 1-10(a), use SPICE methods to determine the z parameters suitable for use with sinusoidal excitation over a frequency range from 1 kHz to 10 kHz. The z parameters as given by (1.10) to (1.13), when evaluated for sinusoidal steady-state conditions, are formed as the ratios of phasor voltages and currents. Consequently, the values of the z parameters are complex numbers that can be represented in polar form as zij zij ij . For determination of the z parameters, matching terminals of the two sinusoidal current sources of Fig. 1-10(b) are connected to the network under test of Fig. 1-10(a). The netlist code below models the resulting network with parameter-assigned values for I"1 and I"5 . Two separate executions of <Ex1_9.CIR> are required to determine all four z parameters. The .AC statement speci es a sinusoidal steady-state solution of the circuit for 11 values of frequency over the range from 10 kHz to 100 kHz.
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