# RLC CIRCUIT

Cuthbert Nyack

The RLC circuit is shown schematically above.

The Impedance of the circuit is

Z = R + jwL + 1/jwC
= R + j(wL - 1/wC)

The magnitude of the impedance of the circuit is given by:-

|Z| = Ö{R^{2} + (wL - 1/wC)
^{2} }

The phase of the impedance of the circuit is given by:-

Ðq = atan
{(wL - 1/wC)
/R}

The phase of the current and V_{r} is
Ð-q.

V_{c} is delayed
relative to V_{r} by 90° and V_{l} is
advanced relative to V_{r} by 90°.

The magnitudes and phases of V_{r} (orange curve),
V_{c}, (red curve) V_{s} (green curve) and
V_{l} (blue curve) are shown by the sinusoidal
waves in the applet above. The
variation of the magnitudes and phases as a function of frequency
and Q can be demonstrated by the applet. A frequency of 1
corresponds to the resonant frequency.

The above applet shows the magnitudes and phases of V_{r},
V_{c} and V_{l} represented in phasor form. The
The sum of V_{c} and V_{l} and the parallelogram
showing the resultant of V_{l} - V_{c} and
V_{r} are shown by the purple lines. Note that at low
frequencies most of the voltage appears accross the resistor
while at higher frequencies the voltage is mostly accross the inductor.

The variation of the magnitudes of V_{r}, V_{c}
and V_{l} is shown above as a function of frequency.
The horizontal frequency scale goes from 0.5w
_{o} to 1.5w_{o}.
A plot of the phase of V_{r} is also shown. The vertical
scale for phase varies from -p at
the bottom to +p at the top. The
magnitude of V_{l}-V_{c} is shown by the
purple line.

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COPYRIGHT © 1996 Cuthbert A. Nyack.