Experiment 12: AC Circuits - RLC Circuit
Introduction
An inductor (L) is an important component of circuits, on the same level as resistors (R) and capacitors
(C). The inductor is based on the principle of inductance - that moving charges create a magnetic field (the
reverse is also true - a moving magnetic field creates an electric field). Inductors can be used to produce
a desired magnetic field and store energy in its magnetic field, similar to capacitors being used to produce
electric fields and storing energy in their electric field. At its simplest level, an inductor consists of a coil
of wire in a circuit. The circuit symbol for an inductor is shown in Figure 1a.
So far we observed that in an RC circuit the charge, current, and potential difference grew and decayed
exponentially described by a time constant τ. If an inductor and a capacitor are connected in series in a
circuit, the charge, current and potential difference do not grow/decay exponentially, but instead oscillate
sinusoidally. In an ideal setting (no internal resistance) these oscillations will continue indefinitely with a
period (T) and an angular frequency ωgiven by
ω=1
√LC (1)
This is referred to as the circuit’s natural angular frequency.
A circuit containing a resistor, a capacitor, and an inductor is called an RLC circuit (or LCR), as
shown in Figure 1b. With a resistor present, the total electromagnetic energy is no longer constant since
energy is lost via Joule heating in the resistor. The oscillations of charge, current and potential are now
continuously decreasing with amplitude. This is referred to as damped oscillations. The oscillations in the
RLC circuit will not damp out if an external emf source supplies enough energy to account for the energy
lost from the resistor. This energy is supplied from an oscillating emf source with an alternating current
(AC).
If the applied voltage in an RLC is of the form
E=E0sin(ωt +φ) (2)
where E0is the maximum amplitude of the emf, then the potential difference across each component can
be written as
VR=V0Rsin(ωt) (3)
VC=V0Csin(ωt −π
2) (4)
VL=V0Lsin(ωt +π
2) (5)
where V0R,V0C, and V0Lare the maximum (amplitude) voltages across the resistor, capacitor, and inductor
components respectively. The ωhere is the driving angular frequency. Notice that the voltage across the
capacitor VClags VRby 90◦, and the voltage across the inductor VLleads VRby 90◦. All three voltages
are plotted in Figure 2.
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