# RC circuit, Transient Response.

Cuthbert Nyack
The basic series RC circuit is shown schematically below. For a unit step input the capacitor voltage is Vc = (1 - e-t/RC).
For a sinusoid the amplitude of the voltage across the capacitor is reduced by 1/Ö(1 + w2C2R2) and the phase is delayed by f = atan(-wCR)
The applet below shows the RC response for different inputs. When setting parameters Fn should be changed first.

Fn = 0 shows the unit step and impulse response.
The plot shows 2 responses with time constants set by (1) and (14).

Fn = 1 shows the transient response to a ramp input with time constant set by (1).

Fn = 2 shows the transient response to a parabolic input.

Fn = 3 shows the response to a sinusoid input whose frequency and phase is set by (4) and (9).

Fn = 4 shows the response to a signal which has an average part vo (set by (11)) and an oscillating part(frequency and phase set by (4) and (9)). In this case the RC circuit acts as a separator with the average part appearing across the capacitor and the oscillating part across the resistor.

Fn = 5 shows the response to a signal which can be adjusted by "a"(20) to be a triangle, a square or to a triangle via a different path or anything in between.

Fn = 6 shows the response to a signal which can be changed by "a"(19) from a square to a +ve ramp to a -ve ramp and to a square.

Fn = 7 shows the response to a nonlinear sinusoid whose nonlinearity can be changed by "a"(21).

Fn = 8 shows the response to a positive pulse sequence whose duty cycle can be changed by "a"(7).

Fn = 9 shows the response to a positive pulse amplitude modulated signal whose modulation index can be changed by "m"(18).

Fn = 10 shows the response to a positive pulse width modulated signal whose modulation index can be changed by "m"(18).

Fn = 11 shows the response to a rectangular whose positive part can be changed by "a"(7).

Fn = 12 shows the response to a chopped sine.

Fn = 13 shows the response to a quantised sine.

Fn = 14 shows the response to a rectified sine.

Fn = 15 shows the response to 2 sinusoids, the lower frequency is set (4) and (10). The higher frequency is ten times the lower frequency and its amplitude is set by (17).

Fn = 16 shows the response to a "noisy" sinusoid.

Fn = 17,18 shows the response to PWL signals whose shape is set by (22) to (27) with different interpolated values.

Fn = 19 shows the response of an RC circuit with time constant T2 to an impulse response from a circuit with time constant T.
Fn = 20, 21 and 22 show special cases of 19.

Fn = 23 shows the power relations in the series RC circuit. The average power consumed by the capacitor is zero. When the voltage is increasing, power is used to increase the energy in the electric field of the capacitor. As the voltage reduces, the energy in the electric field is returned to the source. At any instant of time, the power supplied by the source is equal to the power dissipated in the resistor and the power going into/out of the capacitor.

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