RC Circuit - Step Response

The RC circuit can act as a simple integrator or a first order low-pass filter. It is analysed here.

We start by considering the following circuit:

Figure 1

If a voltage (V_in) is applied at t = 0, what is the form of the output?

For a capacitor:

			dVc
i	=	C
			dt

Since there is a single path for the current, we can write:

VR					dVc
	=	i	=	C
R					dt

Substituting and rearranging:

			dVc
Vin - Vc	=	RC		(1)
			dt

Now separate the variables:

			1
1.dt	=	RC		dVc
			Vin - V c

Integrating:

					1
	1.dt	=	RC			dVc
					Vin - V c

Thus:

t

=

- CR . ln(Vin - Vc )

+

const

When: t = 0,  V_c = 0 :

0

=

- CR . ln(Vin )

+

const

Therefore:

const

=

CR . ln(Vin )

Thus:

t

=

CR . ln(Vin )

- CR . ln(Vin - Vc )

t
	=	ln(Vin )	-  ln(Vin - Vc )
CR

			Vin
=	ln
			Vin - Vc

And:

		t			Vin
exp				=
		CR			Vin - Vc

Rearranging:

			t
(Vin - Vc )	exp				=	Vin
			CR

Rearranging:

		Vin
Vin - Vc	=
		exp{ t / CR }

Rearranging:

					- t
Vc	=	Vin	- Vin . exp
					CR

Giving the result:

						- t
Vc	=	Vin		1 - exp					(2)
						CR

for t = 0, V_c = 0
and, as t ∞, V_c V_in

A plot of this response (for: C = R = V_in = 1) is given in figure 2 below.

Figure 2

CR - the product of resistance (Ohms) and capacitance (Farads) - has the unit of seconds, and is referred to as the time constant. The Greek letter t (tau) is usually used to denote this variable.

The output voltage (V_c) reaches 63.2% of its final value in 1 time constant (1 second in this case). In general, the time taken to reach a particular value is related to the number of time constants given in the table below.

Number of time constants required to reach a proportion of the final value
t	2t	3t	4t	5t	6t	7t
63.2%	86.5%	95.0%	98.2%	99.3%	99.7%	99.9%

Reducing the value of t (i.e. reducing R or C) means that the output will change faster and that any given voltage will be reached sooner.

• Go to rise time.