Parallel RC circuit.

Cuthbert Nyack

The parallel RC circuit is shown schematically below.

The total admittance of the circuit can be written as:- Y = 1/R + jwC
The admittance can be represented as a magnitude and a phase with the magnitude of the admittance given by:-
|Y| = Ö {1/R² + w²C²}
and the phase of the impedance given by:-
Ðq = atan {wCR}

In this case the driving parameter is a current and the response is a voltage.

The applet below show the Frequency and transient response of the parallel resonant circuit.

Fn = 0 shows 2 frequency response characteristics for the 2 time constants T and T2.

Fn = 1 shows the transient response.
The frequency response looks like that of the series RC case. However here an alternating current is applied and the response shown is the voltage across both the capacitor and resistor. For frequencies above w = 1/T = 1/RC, the output decreases by 20dB/dec.
The following eg parameters
(0.5, 2.0, 0.0, 1.0, 0.0, 1.5, 1, na, 1.0, 1.0) show the output lagging the input by 45° but the magnitude depends on the value of the resistor R.
The parameters (10.0, 3.0, 0.0, 0.4, 1.0, 1.5, 1, na, 4.0, 1.0) show the input as a combination of a constant and alternating component. After an initial transient, the output voltage settles to a constant value given by R x the average current and the alternating component of the voltage is heavily attenuated.
This is what happens when the parallel RC combination is used between the emmitter and ground of a CE transistor amplifier.

Setting Fn = 2 shows how a rectified AM is smoothed by the RC circuit. In the time domain, a voltage is applied and the capacitor is assumed to charge through a series source resistor(series RC) with time constant T2 and discharges through a parallel larger load resistor(parallel RC) with time constant T.
eg parameters (2.0, 6.0, na, 0.3, 0.8, 0.02, 2, na, 2.0, 1.0) show the effect. In a real AM system the ratio of carrier to modulating frequency is much larger than it is here(~1E3), so the smoothed waveform looks a lot smoother than it does here.

Setting Fn = 3 shows how an alternating current is divided between R and C. At low frequencies most of the current goes through the resistor, while at high frequencies, most go through the capacitor, hence the name "bypass capacitor". As shown by the text, the currents do not add up arithmetically because of the phase difference between them.

When applet is enabled its appearance is illustrated by the gif image below.

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