# Series RLC Circuits

Series RLC Circuits

An RLC circuit consists of a resistor with resistance , an inductor with inductance , and a capacitor with capacitance . The current in an RLC series circuit is determined by the differential equation

R

L

C

i(t)

Lq+R+=e(t)

2

d

d

2

t

dq

dt

q(t)

C

where and is the AC emf driving the circuit. The angular frequency ω is related to the frequency in hertz (Hz) by . In this Demonstration, the amplitude is set to 10 volts (V). You can vary the frequency in Hz, the resistance in ohms (), the inductance in millihenries (mH), and the capacitance in microfarads (). The voltage V in volts and current in milliamperes (mA) are shown in the plot over a 50-millisecond (msec) window.

i(t)=

dq

dt

e(t)=cos(ωt)

E

0

f

ω=2πf

E

0

f

R

Ω

L

C

μF

I

The sinusoidal curves for voltage and current are out of phase by an angle , where

ϕ

tanϕ=ωL-

1

R

1

ωC

When the effect of inductance is dominant, then , and the voltage leads the current. When the capacitance contribution is dominant (for small values of ), then , and the current leads the voltage. The mnemonic "ELI the ICEman" summarizes these relationships. When the circuit has a pure resistance or when the resonance condition is satisfied, then , meaning that the voltage and current are in phase.

ϕ>0

C

ϕ<0

ω=1

LC

ϕ=0