Series RLC Circuits

frequency Hz

60

resistance Ω

5

inductance mH

50

capacitance μF

250

An RLC circuit consists of a resistor with resistance

R

, an inductor with inductance

L

, and a capacitor with capacitance

C

. The current

i(t)

in an RLC series circuit is determined by the differential equation

L

2

d

q

d

2

t

+R

dq

dt

+

q(t)

C

=e(t)

,

where

i(t)=

dq

dt

and

e(t)=

E

0

cos(ωt)

is the AC emf driving the circuit. The angular frequency ω is related to the frequency

f

in hertz (Hz) by

ω=2πf

. In this Demonstration, the amplitude

E

0

is set to 10 volts (V). You can vary the frequency

f

in Hz, the resistance

R

in ohms (

Ω

), the inductance

L

in millihenries (mH), and the capacitance

C

in microfarads (

μF

). The voltage V in volts and current

I

in milliamperes (mA) are shown in the plot over a 50-millisecond (msec) window.

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

ϕ

, where

tanϕ=

1

R

ωL-

1

ωC

.

When the effect of inductance is dominant, then

ϕ>0

, and the voltage leads the current. When the capacitance contribution is dominant (for small values of

C

), then

ϕ<0

, 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

ω=1

LC

is satisfied, then

ϕ=0

, meaning that the voltage and current are in phase.