Find v(t) in the RLC circuit of Fig. 9.48.
Figure 9.48
Find the value the voltage
Answer to Problem 41P
The value of the voltage
Explanation of Solution
Given date:
Refer to Figure 9.48 in the textbook.
Formula used:
Write the expression to convert the time domain expression into phasor domain.
Here,
A is the magnitude,
t is the time, and
Write the expression to calculate the total phasor current.
Here,
Write the expression to calculate the impedance of the passive elements resistor, inductor and capacitor.
Here,
Calculation:
The given RLC circuit is redrawn as Figure 1.
Refer to Figure 1, the source voltage is,
Here, angular frequency
Use the equation (1) to express the above equation in phasor form.
Substitute
Substitute
Substitute
Substitute
The Figure 1 is redrawn as impedance circuit in the following Figure 2.
Refer to Figure 2, the series combination of the impedances
Write the expression to calculate the equivalent impedance 1 for the parallel combination as follows.
Here,
Substitute
The reduced circuit of the Figure 2 is drawn as Figure 3.
Refer to Figure 2, the impedances
Write the expression to calculate the equivalent capacitance for the series connected impedances
Here,
Substitute
The reduced circuit of the Figure 3 is drawn as Figure 4.
Substitute
Refer to Figure 2, the current through the capacitor is expressed as,
Refer to Figure 2, the voltage across the capacitor is expressed as,
Substitute
Substitute
Use the equation (1) to express the above equation in time domain form.
Substitute
Conclusion:
Thus, the value of the voltage
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