# RC circuit - state space representation

A state space representation and a transfer function designating for a low-pass filter. The Low-pass filter is an electric circuit which contains a resistor and a capacitor. The Resistor and the capacitor are connected in series. The input voltage (the input signal) is between resistor and capacitor. The output voltage (the output signal) is the capacitor's voltage. A one of capacitor's terminals is connected to the ground.

## The subject RC circuit electric diagram i.e. a low-pass RC filter

For the RC circuit shown above a dynamic model will be created. To be more precise, two dynamic models will be created: - the first dynamic model as circuit’s transfer function H(s)
- the second dynamic model as circuit’s state space representation
The input signal in the considered circuit is voltage u_1(t).
The output signal in the considered circuit is voltage u_2(t).

## A dynamic model of circuit in form transfer function H(s)

Note that input and output signals are in time domain. Now we write Kirchhoff’s voltage law (KVL) for circuit.

Take into account following relationships and formulas:

Passing to Laplace transformation representation for start conditions equal to zero (ST=0).

Note that after Laplace transformation domain changed from time t to variable s.

## A Dynamic model in form state space representation equations

General form of state space representation equations is following:

Where:
[A]- state matrix, [B]- input matrix, [C]- output matrix, [D]- feedthrough matrix

## Using received dynamic models in MATLAB software

The received dynamic models can be used in MATBLAB which has got built in functions for simulating circuits answers. A circuit's answer is understood as a trace/shape/form of circuit's output signal. The MATLAB provides a bunch of functions which allow to simulate variatey of extortions (input signals): an impulse signal, a step signal. It is also possible to plot a Bode diagram.
lsim(l,m,uin,t)
impulse(A,B,C,D)
step(A,B,C,D)
bode(A,B,C,D)