## 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 v_{ad} 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, X, is equal to the inductive reactance, _{C}X, what is true about the impedance of this _{L}R-L-C series circuit? |

If the capacitive reactance, *X _{C}*, is equal to the inductive reactance,

*X*, then

_{L}**the impedance is equal to**. Impedance is equal to where

*R**R*is the resistance of the resistor,

*X*is the inductive reactance, and

_{L}*X*is the capacitive reactance. If

_{C}*X*=

_{L}*X*, then (

_{C}*X*–

_{L}*X*)

_{C}^{2}= 0 and .

When *X _{L}* =

*X*and the impedance is equal to

_{C}*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 *X _{L}* =

*X*and the impedance is equal to

_{C}*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

*X*to

_{L}*X*: and solve for .

_{C }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* = *I*^{2}*R*, 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?