Concept explainers
Refer to the network in Fig. 14.96.
- (a) Find Zin(s).
- (b) Scale the elements by Km = 10 and Kf = 100. Find Zin(s) and ω0.
Figure 14.96
(a)
Find the value of the input impedance
Answer to Problem 79P
The value of the input impedance
Explanation of Solution
Given data:
Refer to Figure 14.96 in the textbook.
Formula used:
Write the expression to calculate the impedance of the passive elements resistor, inductor and capacitor in s-domain.
Here,
Calculation:
The given circuit is redrawn as Figure 1.
Use equation (1) to find
Use equation (1) to find
Use equation (2) to find
Use equation (3) to find
Insert a
Modify the Figure 2 with the representation of supernode and current direction as shown in Figure 3.
Apply Kirchhoff’s current law to the supernode in Figure 3.
The Figure 3 is reduced as following Figure 4 to show the voltage relation.
Apply Kirchhoff’s voltage law to the circuit in Figure 4 to obtain the relationship between voltages.
Apply voltage division rule on Figure 3 to find
Rearrange the above equation.
Rearrange the equation (6) to find
Substitute
Rearrange the above equation to find
Compare the equations (4) and (6) to obtain the following equation.
Substitute
Rearrange the above equation.
Simplify the above equation to find
Refer to Figure 3, the current
Substitute
The input impedance of the circuit in Figure 2 is calculated as follows.
Substitute
Substitute
Substitute
At resonance condition, the imaginary part of the impedance should be equal to zero. Therefore, equate the imaginary part of the above equation to zero.
Simplify the above equation to find
Take square root on both sides of the above equation to find
Substitute
Conclusion:
Thus, the value of the input impedance
(b)
Find the value of the input impedance
Answer to Problem 79P
The value of the input impedance
Explanation of Solution
Given data:
The value of the magnitude scaling factor
The value of the frequency scaling factor
Formula used:
Consider the equations used in magnitude and frequency scaling.
Write the expression to calculate the scaled resistor.
Here,
Write the expression to calculate the scaled inductor.
Here,
Write the expression to calculate the scaled capacitor.
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Conclusion:
Thus, the value of the input impedance
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Chapter 14 Solutions
EBK FUNDAMENTALS OF ELECTRIC CIRCUITS
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