■ FIGURE 11.49
Instead of including a capacitor as indicated in Fig. 11.49, the circuit is erroneously constructed using two identical inductors, each having an impedance of j30 W at the operating frequency of 50 Hz. (a) Compute the complex power delivered to each passive component. (b) Verify your solution by calculating the complex power supplied by the source. (c) At what power factor is the source operating?
(a)
Find the complex power delivered to each passive element when
Answer to Problem 53E
The complex power delivered to
Explanation of Solution
Given data:
Refer to Figure 11.49 in the textbook for the given circuit.
Operating frequency is 50 Hz.
Formula used:
Write the expression for complex power delivered to the element as follows:
Here,
Write the expression for current in terms of voltage and impedance as follows:
Calculation:
Consider
Use the expression in Equation (2) and find the source current as follows:
Substitute
Use voltage division rule and find the voltage across
Substitute
Modify the expression in Equation (1) for the complex power delivered to
Substitute
Consider the node voltage across the shunt branches as
Substitute
Use current division rule and find the current through
Substitute
Modify the expression in Equation (1) for the complex power delivered to the
Substitute
Use voltage division rule and find the voltage across
Substitute
Use current division rule and find the current through
Substitute
Modify the expression in Equation (1) for the complex power delivered to the
Substitute
Use voltage division rule and find the voltage across
Substitute
Modify the expression in Equation (1) for the complex power delivered to the
Substitute
Conclusion:
Thus, the complex power delivered to
(b)
Verify the solution obtained in Part (a) by calculating the complex power supplied by the source.
Explanation of Solution
Calculation:
Modify the expression in Equation (1) for the complex power supplied by the source as follows:
Substitute
Write the expression for sum of complex delivered to (or absorbed by) each passive element as follows:
From Part (a), substitute
From the calculation, it is clear that, the complex power supplied by the source is equal to the complex power delivered to each passive element.
Conclusion:
Thus, the solution obtained in Part (a) is verified.
(c)
Find the power factor of the source.
Answer to Problem 53E
The power factor of the source is 0.253 lagging.
Explanation of Solution
Formula used:
Write the expression for complex power in the rectangular form as follows:
Here,
Write the expression for power factor as follows:
Calculation:
Rewrite the expression for complex power supplied by the source in rectangular form as follows:
Compare the complex power supplied by the source with the expression in Equation (3) and write the average and reactive power supplied by the source as follows:
Substitute 19.1482 W for
If the imaginary part of the complex power (reactive power) is positive value, then the load has lagging power factor. If the imaginary part is negative value, then the load has leading power factor.
As the imaginary part of the given complex power is positive value, the power factor is lagging power factor.
Conclusion:
Thus, the power factor of the source is 0.253 lagging.
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Chapter 11 Solutions
Loose Leaf for Engineering Circuit Analysis Format: Loose-leaf
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