Concept explainers
A square-wave generator produces the voltage waveform shown in Fig. 6.94(a). What kind of a circuit component is needed to convert the voltage waveform to the triangular current waveform shown in Fig. 6.94(b)? Calculate the value of the component, assuming that it is initially uncharged.
Figure 6.94
For Prob. 6.85.
Find the circuit component that is needed to convert the voltage waveform into the triangular current waveform and to calculate the value of the circuit component.
Answer to Problem 85CP
The circuit component inductor is needed to convert the given voltage waveform into the triangular current waveform and the value for the inductor
Explanation of Solution
Given data:
Refer to Figure 6.94 in the textbook.
Formula used:
Write the expression to calculate the straight line equation for two points
Refer to Figure 6.94(b) in the textbook.
From the given graph, substitute
Calculation:
The given voltage waveform is redrawn as Figure 1.
The given triangular current waveform is redrawn as Figure 2.
Refer to Figure 1 and Figure 2. Generally, integration of the square waveform gives the triangular waveform. That is, the integration of the voltage waveform gives the current waveform. Such relation can be obtained in following relation.
Here,
Refer to equation (3), the circuit component inductor is needed to convert the square voltage waveform to the triangular current waveform.
Differentiate the equation (3) with respect to
Rearrange the above equation to find
Refer to Figure 1. The voltage function is expressed as,
Refer to Figure 2, split up the time period as four 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
Case (iv):
The two points
Substitute
Simplify the equation to find
Therefore, the current function of the signal in Figure 2 is,
For
Substitute
For
Substitute
For
Substitute
For
Substitute
Therefore, the voltage function of the Figure 2 is expressed as,
The voltage function of the signal in Figure 1 is equal to the voltage function that is obtained in equation (6).
Compare the equations (5) and (6) for any of the time limits. Assume the comparison is made for
Rearrange the above equation to find
Therefore, the circuit component inductor is needed to convert the given voltage waveform into the triangular current waveform and the value for the inductor
Conclusion:
Thus, the circuit component inductor is needed to convert the given voltage waveform into the triangular current waveform and the value for the inductor
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Chapter 6 Solutions
Fundamentals of Electric Circuits
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