## RLC Series Circuits with AC

11.3 RLC Series Circuits with AC

### Learning Objectives

By the end of this section, you will be able to:

- Describe how the current varies in a resistor, a capacitor, and an inductor while in series with an ac power source
- Use phasors to understand the phase angle of a resistor, capacitor, and inductor ac circuit and to understand what that phase angle means
- Calculate the impedance of a circuit

### RLC Circuits and Phasors

**Practice!**

Practice 11.3.1 |
---|

Consider the voltage phasors shown at three instants of time. Choose the figure that represents the instant of time at which the instantaneous value of the voltage has the largest magnitude. |

Practice 11.3.2 |
---|

Consider the voltage phasors shown at three instants of time. Choose the figure that represents the instant of time at which the instantaneous value of the voltage has the smallest magnitude. |

### Impedance

**Practice!**

Practice 11.3.3 |
---|

A resistor with resistance R and a capacitor with capacitance C are 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 happens to the rms current through the resistor as the angular frequency of the AC is increased? |

As the angular frequency of the AC source is increased, **the rms current in the resistor increases**. The

*rms*current will depend on the circuit’s impedance where

*X*is the inductive reactance and

_{L}*X*is the capacitive reactance. Using Ohm’s Law, the

_{C}*rms*current is , where

*V*is the

_{rms}*rms*voltage of the AC source.

In this circuit, *X _{L}* = 0 because the circuit does not contain an inductor, and . As is increased,

*X*decreases because the capacitive reactance is inversely proportional to . This decreasing

_{C}*X*results in a lower impedance, which in turn results in a greater

_{C}*rms*current.

Practice 11.3.4 |
---|

A resistor with resistance R and an inductor with inductance L are 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 happens to the rms current through the resistor as the angular frequency of the AC is increased? |

As the angular frequency of the AC source is increased, **the rms current in the resistor decreases**. The

*rms*current will depend on the circuit’s impedance where

*X*is the inductive reactance and

_{L}*X*is the capacitive reactance. Using Ohm’s Law, the

_{C}*rms*current is , where

*V*is the

_{rms}*rms*voltage of the AC source.

In this circuit, *X _{C}* = 0 because the circuit does not contain a capacitor, and . As is increased,

*X*increases because the inductive reactance is proportional to . This increasing

_{L}*X*results in a greater impedance, which in turn results in a greater

_{L}*rms*current.

**Discuss!**

A series RLC circuit has R = 400 Ω, L = 1.40 H, C = 3.9 μF. It is connected to an AC source with f = 60.0 Hz and ∆V_{max} = 150 V.

(a) Determine the inductive reactance, the capacitive reactance, and the impedance of the circuit.

(b) Find the maximum current in the circuit.

(c) Find the phase angle between the current and voltage.

(d) Find the maximum voltage across each element.