Concept explainers
If v(0) = 0, find v(t), i1(t), and i2(t) in the circuit of Fig. 6.63.
Figure 6.63
For Prob. 6.31.
Find the expression for voltage
Answer to Problem 31P
The expression of voltage
Explanation of Solution
Given data:
Refer to Figure 6.63 in the textbook.
The initial voltage at time
Formula used:
Write the expression to calculate the straight line equation for two points
Refer to Figure 6.63 in the textbook.
From the given graph, substitute
Write the expression to calculate the voltage across the capacitor.
Here,
Calculation:
The given circuit is redrawn as Figure 1.
Refer to Figure 1, the capacitors
Write the expression to calculate the equivalent capacitance for the parallel connected capacitors
Here,
Substitute
The given current waveform is redrawn as Figure 2.
Refer to Figure 2, split up the time period as three divisions
Case (i):
The two points
Substitute
Simplify the equation to find
Case (ii):
The two points
Substitute
Simplify the equation to find
Case (iii):
The two points
Substitute
Simplify the equation to find
Therefore, the current function of the signal in Figure 2 is,
For
Substitute
Simplify the equation to find
Substitute
For
Write the expression to calculate the capacitor voltage depends on the past history of the capacitor voltage.
Here,
Substitute
Simplify the equation to find
Substitute
For
Write the expression to calculate the capacitor voltage depends on the past history of the capacitor voltage.
Here,
Substitute
Simplify the equation to find
Simplify the equation to find
Therefore, the expression of the voltage
Write the expression to calculate the current through the capacitor
For
Substitute
Simplify the equation to find
For
Substitute
Simplify the equation to find
For
Substitute
Simplify the equation to find
Therefore, the expression of current through the capacitor
Write the expression to calculate the current through the capacitor
For
Substitute
Simplify the equation to find
For
Substitute
Simplify the equation to find
For
Substitute
Simplify the equation to find
Therefore, the expression of current through the capacitor
Conclusion:
Thus, the expression of voltage
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Chapter 6 Solutions
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