Determine the center frequency and bandwidth of the band-pass filters in Fig. 14.88.
(a)
Find the center frequency and bandwidth of the band-pass filter shown in Figure 14.88(a).
Answer to Problem 57P
The value of the center frequency
Explanation of Solution
Given data:
Refer to Figure 14.88(a) 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 drawn as Figure 1.
Use equation (1) to find
Use equation (1) to find
Use equation (3) to find
Use equation (3) to find
The s-domain circuit of the Figure 1 is drawn as Figure 2.
Write the general expression to calculate the transfer function of the circuit in Figure 2.
Here,
Refer to Figure 2, the series connected impedances
Therefore, the equivalent impedance is calculated as follows.
Substitute
Refer to the given data. The value of the resistors
Substitute
Simplify the above equation to find
The reduced circuit of Figure 2 is drawn as Figure 3.
Refer to the Figure 3, the current through the circuit is expressed as,
Apply current division rule on Figure 2 to find
Substitute
Substitute
Refer to Figure 2, the output voltage
Substitute
Rearrange the above equation to find
Substitute
Compare the denominator factor of above equation with the standard quadratic equation
Substitute
Take square root on both sides of the above equation to find
Write the expression to calculate the bandwidth of the band-pass filter.
Substitute
Conclusion:
Thus, the value of the center frequency
(b)
Find the center frequency and bandwidth of the band-pass filter shown in Figure 14.88(b).
Answer to Problem 57P
The value of the center frequency
Explanation of Solution
Given data:
Refer to Figure 14.88(b) in the textbook.
Calculation:
The given circuit is drawn as Figure 4.
Use equation (1) to find
Use equation (1) to find
Use equation (2) to find
Use equation (2) to find
The s-domain circuit of the Figure 4 is drawn as Figure 5.
Write the general expression to calculate the transfer function of the circuit in Figure 5.
Refer to Figure 5, the series connected impedances
Therefore, the equivalent impedance is calculated as follows.
Substitute
Refer to the given data. The value of the resistors
Substitute
Simplify the above equation to find
The reduced circuit of Figure 5 is drawn as Figure 6.
Refer to the Figure 6, the current through the circuit is expressed as,
Apply current division rule on Figure 5 to find
Substitute
Substitute
Refer to Figure 5, the output voltage
Substitute
Rearrange the above equation to find
Substitute
Compare the denominator factor of above equation with the standard quadratic equation
Substitute
Take square root on both sides of the above equation to find
Substitute
Conclusion:
Thus, the value of the center frequency
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Chapter 14 Solutions
Fundamentals of Electric Circuits
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