A series RLC circuit in which R = l.00 Ω, L = 1.00 mH, and C = 1.00 nF is connected to an AC source delivering 1.0 V (rms). (a) Make a precise graph of the power delivered to the circuit as a function of the frequency and (b) verify that the full width of the resonance peak at half-maximum is R/2πL.
(a)
To draw: A precise graph of the power delivered to the circuit as a function of the frequency.
Answer to Problem 33.76AP
Explanation of Solution
Given info: The value of resistance is
Formula to calculate the inductive reactance of the circuit is,
Here,
Formula to calculate the inductive reactance of the circuit is,
Here,
Formula to calculate the impedance of the circuit is,
Here,
Substitute
Formula to calculate the rms current in the circuit is,
Here,
Write the expression for the power deliver to the circuit.
Substitute
Substitute
Substitute
Draw the table for the power for different values of frequency.
|
|
|
|
|
0.9991 | 999.1 | 1000.9 | 2.06 | 0.23569 |
0.9993 | 999.3 | 1000.7 | 1.72 | 0.33768 |
0.9995 | 999.5 | 1000.5 | 1.41 | 0.49987 |
0.9997 | 999.7 | 1000.3 | 1.17 | 0.73524 |
0.9999 | 999.9 | 1000.1 | 1.02 | 0.96153 |
1.0000 | 1000 | 1000.0 | 1.00 | 1.00000 |
1.0001 | 1000.1 | 999.9 | 1.02 | 0.96154 |
1.0003 | 1000.3 | 999.7 | 1.17 | 0.73535 |
1.0005 | 1000.5 | 999.5 | 1.41 | 0.50012 |
1.0007 | 1000.7 | 999.3 | 1.72 | 0.33799 |
1.0009 | 1000.9 | 999.1 | 2.06 | 0.23601 |
Draw precise graph of the power delivered to the circuit as a function of the frequency.
Figure (1)
(b)
To verify: The full width of the resonance peak at half maximum is
Answer to Problem 33.76AP
Explanation of Solution
Given info: The value of resistance is
Write the expression for the term
Substitute
From the graph the half maximum power occurs at two values of the angular frequencies that are
Formula to calculate the angular bandwidth is,
Substitute
Write the expression for the frequency bandwidth.
Rearrange the equation for
Substitute
From equation (1) and equation (2), the RHS values are same that verify the full width of the resonance peak at the half power maximum is
Conclusion:
Therefore, the full width of the resonance peak at half maximum is
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Chapter 33 Solutions
Physics For Scientists And Engineers, Technology Update, Loose-leaf Version
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