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Assume that the two circuits in Fig. 19.135 are equivalent. The parameters of the two circuits must be equal. Using this factor and the z parameters, derive Eqs. (9.67) and (9.68).
Figure 19.135
Derive the expressions in Equations (9.67) and (9.68) in the textbook.
Explanation of Solution
Given Data:
Refer to Figure 19.135 in the textbook given circuits.
Consider the parameters of two circuits are equal.
Calculation:
Refer to Figure 19.135 (a) in the textbook and write the expression for
Refer to Figure 19.135 (b) in the textbook and write the expression for
From Equation (1), substitute
Refer to Figure 19.135 (a) in the textbook and write the expression for
Refer to Figure 19.135 (b) in the textbook and write the expression for
From Equation (3), substitute
Refer to Figure 19.135 (a) in the textbook and write the expression for
Refer to Figure 19.135 (b) in the textbook and write the expression for
From Equation (5), substitute
Subtract Equation (4) from Equation (2) and obtain the expression as follows:
Add Equations (6) and (7) and obtain the expression as follows:
Simplify the expression as follows:
Subtract Equation (8) from Equation (6) and obtain the expression as follows:
Subtract Equation (8) from Equation (2) and obtain the expression as follows:
From Equations (8), (9), and (10), the expressions in Equation (9.68) are derived.
Note that, the obtained expressions are not same as the expressions in the textbook, since the position of the impedances are changed in the given circuits.
Use expressions in Equations (8), (9), and (10) and obtain the expression as follows:
Divide Equation (11) by Equation (8) and obtain the expression as follows:
Divide Equation (11) by Equation (9) and obtain the expression as follows:
Divide Equation (11) by Equation (10) and obtain the expression as follows:
From Equations (12), (13), and (14), the expressions in Equation (9.67) are derived.
Note that, the obtained expressions are not same as the expressions in the textbook, since the position of the impedances are changed in the given circuits.
Conclusion:
Thus, the expressions in Equations (9.67) and (9.68) in the textbook are derived.
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Chapter 19 Solutions
Fundamentals of Electric Circuits
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