Resonance in an AC Circuit
11.5 Resonance in an AC Circuit
Learning Objectives
By the end of this section, you will be able to:
- Determine the peak ac resonant angular frequency for a RLC circuit
- Explain the width of the average power versus angular frequency curve and its significance using terms like bandwidth and quality factor
Resonance
Practice!
| Practice 11.5.1 |
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In an L-R-C series circuit as shown, there is a phase angle between the instantaneous current through the circuit and the instantaneous voltage vad across the entire circuit. For what value of the phase angle is the greatest power delivered to the resistor? |
| Practice 11.5.2 |
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A resistor with resistance R, a capacitor with capacitance C, and an inductor with inductance L are all connected in series with an AC source that provides a sinusoidal voltage of  , where V is the maximum voltage,  is the angular frequency, and t is the time. If the capacitive reactance, XC, is equal to the inductive reactance, XL, what is true about the impedance of this R-L-C series circuit? |
If the capacitive reactance, XC, is equal to the inductive reactance, XL, then the impedance is equal to R. Impedance is equal to 
where R is the resistance of the resistor, XL is the inductive reactance, and XC is the capacitive reactance. If XL = XC, then (XL – XC)2 = 0 and 
.
When XL = XC and the impedance is equal to R, we say that the R-L-C circuit is in resonance. When a circuit is in resonance, the impedance is a minimum value and the rms current is a maximum value. Resonance occurs at one special AC source angular frequency called the resonant frequency.
| Practice 11.5.3 |
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| A resistor with resistance R, a capacitor with capacitance C, and an inductor with inductance L are all connected in series with an AC source that provides a sinusoidal voltage of , where V is the maximum voltage, is the angular frequency, and t is the time. For what angular frequency will this R-L-C series circuit be in resonance? |
A.  |
B.  |
C.  |
D.  |
E.  |
This R-L-C series circuit will be in resonance when the angular frequency is . When XL = XC and the impedance is equal to R, we say that the R-L-C circuit is in resonance. Resonance occurs at one special AC source angular frequency called the resonant frequency. To determine this frequency, you can equate XL to XC : 
and solve for .
| Practice 11.5.4 |
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| A resistor with resistance R, a capacitor with capacitance C, and an inductor with inductance L are all connected in series with an AC source that provides a sinusoidal voltage of , where V is the maximum voltage, is the angular frequency, and t is the time. What is true about the average power loss in the resistor when this R-L-C series circuit is in resonance? |
When this series R-L-C series circuit is in resonance, the average power loss in the circuit is a maximum value. When a circuit is in resonance, the impedance is a minimum value and the rms current is a maximum value. Since the power loss in the resistor is calculated with P = I2R, a maximum current results in maximum power loss in the resistor.
| Practice 11.5.5 |
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| In an L-R-C series circuit as shown, the current has a very small amplitude if the ac source oscillates at a very high frequency. Which circuit element causes this behavior? |
| Practice 11.5.6 |
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| In an L-R-C series circuit as shown, suppose that the angular frequency of the ac source equals the resonance angular frequency. In this case, the circuit impedance… |
Discuss!
An RLC circuit is used in a radio to tune into an FM station broadcasting at f = 99.7 MHz. The resistance in the circuit is R = 17.0 Ω, and the inductance is L = 1.10 μH. What capacitance should be used?
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