For the circuit in Fig. 15.55, (a) derive an algebraic expression for the transfer function H(jω) = iout/vin in terms of circuit components R1, R2, L1, and L2; (b) evaluate the the magnitude of H(jω) at frequencies of 10 kHz, 1 MHz, and 100 MHz for the case where R1 = 3 kΩ, R2 = 12 kΩ, L1 = 5 mH, and L2 = 8 mH; (c) qualitatively, explain the behavior of the transfer function magnitude frequency response.
■ FIGURE 15.55
(a)
Find an algebraic expression for the transfer function
Answer to Problem 5E
The transfer function
Explanation of Solution
Given data:
Refer to Figure 15.55 in the textbook.
Formula used:
Write the expression to calculate the impedance of the passive elements resistor and inductor.
Here,
Calculation:
The given circuit is redrawn as Figure 1.
The impedance circuit of the Figure 1 is drawn as Figure 2 using the equations (1) and (2).
Write the general expression to calculate the transfer function of the circuit in Figure 2.
Here,
Use nodal analysis on node
Rearrange the above equation to find
The output current
Substitute
Simplify the above equation to find
Substitute
Conclusion:
Thus, the transfer function
(b)
Find the value of magnitude of
Answer to Problem 5E
The value of magnitude of the transfer function
Explanation of Solution
Given data:
The value of the resistor 1
The value of the resistor 2
The value of the inductor 1
The value of the inductor 2
Formula used:
Write the expression to calculate the angular frequency.
Here,
Calculation:
From part (a), the transfer function is,
Substitute
Substitute
For frequency of 10 kHz:
Substitute
Take magnitude for above equation to find
For frequency of 1 MHz:
Substitute
Take magnitude for above equation to find
For frequency of 100 MHz:
Substitute
Take magnitude for above equation to find
Conclusion:
Thus, the value of magnitude of the transfer function
(c)
Explain the behavior of the transfer function magnitude frequency response.
Explanation of Solution
Discussion:
At lower frequencies
It is clear that the current
The magnitude approaches zero at low and high frequencies and at medium frequencies, the magnitude approaches maximum value. It implies the characteristics of the band-pass filter.
Conclusion:
Thus, the behavior of the transfer function magnitude frequency response is explained.
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