Find the bandwidth and center frequency of the band-stop filter of Fig. 14.89.
Figure 14.89
Find the bandwidth and center frequency of the band-stop filter shown in Figure 14.89.
Answer to Problem 59P
The value of the bandwidth
Explanation of Solution
Given data:
Refer to Figure 14.89 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 (2) 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.
Simplify the above equation to find
The reduced circuit of Figure 2 is drawn as Figure 3.
Apply voltage division rule on Figure 3 to find
Rearrange the above equation to find
Substitute
Substitute
Refer to Figure 3, the input impedance is expressed as,
Substitute
Substitute
Simplify the above equation to find
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
Simplify the above equation to find
Substitute
Substitute
Substitute
Simplify the above equation to find
At corner frequency
Substitute
Simplify the above equation.
Substitute
Simplify the above equation.
Square on both sides of the above equation to simplify it.
Simplify the above equation.
Simplify the above equation.
Assume
Write a general expression to calculate the roots of quadratic equation
Compare the equation (8) with the quadratic equation
Substitute
Substitute the roots of characteristic equation in equation (8).
Substitute
Simplify the equation (10) to find
Take square root on both sides of the above equation to find
Simplify the equation (10) to find
Take square root on both sides of the above equation to find
Write the expression to calculate the bandwidth of the band-stop filter.
Here,
Substitute
Conclusion:
Thus, the value of the bandwidth
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