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a.
Find the roots of the characteristic equation that describes the voltage response of the circuit.
a.
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Answer to Problem 1P
The roots of the characteristic equation are
Explanation of Solution
Given data:
The values of resistance, inductance, and capacitance of a parallel RLC circuit are given as follows:
Formula used:
Write the expression for roots of characteristic equation for a parallel RLC circuit as follows:
Here,
Write the expression for Neper frequency as follows:
Here,
Write the expression for resonant radian frequency as follows:
Here,
Calculation:
Substitute
Substitute 12.5 H for
Substitute
The roots are determined as follows:
Conclusion:
Thus, the roots of the characteristic equation are
b.
Find whether the response is over-, under-, or critically damped response.
b.
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Answer to Problem 1P
The response for the given system is an over-damped response.
Explanation of Solution
Given data:
The value of capacitance is adjusted to give a neper frequency of
Formula used:
Write the expression for over-damped response for a parallel RLC circuit as follows:
Write the expression for under-damped response for a parallel RLC circuit as follows:
Write the expression for critically damped response for a parallel RLC circuit as follows:
Calculation:
From Part (a), the values of
Substitute
The expression is satisfied. Therefore, the response is the over-damped response. As the Equation (4) is satisfied, it is not required to check the Equations (5) and (6).
Conclusion:
Thus, the response for the given system is an over-damped response.
c.
Calculate the value of resistance that yields a damped frequency of
c.
![Check Mark](/static/check-mark.png)
Answer to Problem 1P
The value of resistance that yields a damped frequency of
Explanation of Solution
Given data:
Formula used:
Write the expression for damped frequency for a parallel RLC circuit as follows:
Calculation:
Rearrange the expression in Equation (7) as follows:
Substitute
Rearrange the expression in Equation (2) as follows:
Substitute
Conclusion:
Thus, the value of resistance that yields a damped frequency of
d.
Find the roots of the characteristic equation for the value of resistance obtained in Part (c).
d.
![Check Mark](/static/check-mark.png)
Answer to Problem 1P
The roots of the characteristic equation are
Explanation of Solution
Calculation:
From Part (c), the value of resistance is obtained as
Substitute
Substitute
The roots are determined as follows:
Conclusion:
Thus, the roots of the characteristic equation are
e.
Calculate the value of resistance
e.
![Check Mark](/static/check-mark.png)
Answer to Problem 1P
The value of resistance that makes the voltage response critically damped is
Explanation of Solution
Calculation:
From Equations (2) and (3), substitute
Rearrange the expression for resistance of the capacitor.
Substitute 12.5 H for
Conclusion:
Thus, the value of resistance that makes the voltage response critically damped is
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EBK ELECTRIC CIRCUITS
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