Concept explainers
For the RL circuit in Fig. 15.52, (a) determine the transer function defined as H(jω) = vout/vin; (b) for the case of R = 200 Ω and L = 5 mH, construct a plot of the magnitude and phase as a function of frequency; and (c) evaluate the magnitude and phase at a frequency of 10 kHz.
FIGURE 15.52
(a)
Find the transfer function
Answer to Problem 1E
The transfer function
Explanation of Solution
Given data:
Refer to Figure 15.52 in the textbook.
Formula used:
Write the expression to calculate the impedance of the passive elements resistor and inductor.
Here,
Calculation:
The given RL circuit is drawn as Figure 1.
The Figure 1 is redrawn as impedance circuit in Figure 2 using the equations (1) and (2).
Write the general expression to calculate the transfer function of the circuit in Figure 2.
Here,
Apply Kirchhoff’s voltage law on Figure 2 to find
Rearrange the above equation to find
Substitute
Conclusion:
Thus, the transfer function
(b)
Plot the magnitude and phase as a function of frequency.
Explanation of Solution
Given data:
The value of the resistor
The value of the inductor
Calculation:
From part (a), the transfer function is,
Substitute
Simplify the above equation to find
Re-write the transfer function
From equation (4), the magnitude function of
Write the above equation in decibel (dB).
From equation (4), the phase angle is expressed as follows:
Substitute
Substitute
Similarly, by substituting various values for
Table 1
0.1 | 1 | 2 | 10 | 20 | 50 | 200 | |
–112 | –92 | –86 | –72 | –66 | –58 | –46 |
Table 2
0.1 | 10 | 104 | 105 | 106 | |
90 | 90 | 75.96 | 21.8 | 2.3 |
The Figure 1 is the magnitude plot of the given transfer function obtained using Table 1.
The Figure 2 is the phase plot of the given transfer function obtained using Table 2.
Conclusion:
Thus, the magnitude and phase as a function of frequency is plotted.
(c)
Find the value of the magnitude and phase at a frequency of
Answer to Problem 1E
The value of the magnitude and phase at a frequency of
Explanation of Solution
Given data:
The value of the frequency
Formula used:
Write the expression to calculate the angular frequency.
Here,
Calculation:
From part (a), the transfer function is expressed as,
From equation (7), the magnitude function is expressed as,
Substitute
Substitute
From equation (7), the phase function is expressed as,
Substitute
Substitute
Conclusion:
Thus, the value of the magnitude and phase at a frequency of
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