Series RLC Circuits

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 (((d^2)q)/(d (t^2)))+R ((d q)/(d t))+((q(t))/C)=e(t),

where i(t)=((d q)/(d t))and e(t)=(Subscript[E, 0])cos(ω t) is the AC emf driving the circuit. The angular frequency ω is related to the frequency f in hertz (Hz) by ω=2π f. In this Demonstration, the amplitude (Subscript[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

tan ϕ=(1/R)(ω L-(1/(ω C))).

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/(Sqrt[L C])is satisfied, then ϕ=0, meaning that the voltage and current are in phase.

THINGS TO TRY

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DETAILS

Snapshot 1: effect of induction dominates; ϕ>0 and voltage leads current (ELI)

Snapshot 2: effect of capacitance dominates; ϕ<0 and current leads voltage (ICE)

Snapshot 3: circuit fulfills resonance condition; ϕ=0 and current and voltage are in phase

RELATED LINKS

S. M. Blinder

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