RC SERIES CIRCUIT

Impulse Response of a LTIS

Consider a LTIS modelled by the input-output relation in block diagram:

In the diagram and in the expression that follow, the arguments of the variables are represented by “.”. This representaion can mean:

– for continuos time systems
– for discrete time systems

The Impulse Response of a LTIS is obtained when a unit impulse is applied to the input of the system with zero initial conditions.

The symbol of the impulse response is h(.).

Kirchhoff Voltage Law

Kirchhoff Voltage Law is based on the Principle of the Conservation of Energy and it states that the algebraic sum of all voltages (potential differences) along any closed loop of a circuit is zero.

SIMULAÇÕES EM ENGENHARIA ELÉTRICA

RC SERIES CIRCUIT

THE MATHEMATICAL MODEL OF THE RC SERIES CIRCUIT

Consider the RC series circuit shown in the figure:

Since it is a series circuit, the source current is the same of the resistor and the capacitor.

Using Kirchhoff Voltage Law, the time domain mathematical model can be written. It is the following first order ordinary linear differential equation with constant coefficients:

v(t) = R.i(t) + \frac{1}{C} \int_{0}^{t}i(\tau) d\tau; ~i(0) = 0

In case you want to remember Kirchhoff Voltage Law, click on the magnifying glass.

The equation can be solved for three variables – the current, the voltage in the resistor and the voltage in the capacitor. The current and the voltage in the resistor have the same shape with different amplitude, since their relation is:

v_R(t) = R.i(t)

Thus, it is possible to solve for one of the variables and compute the other either by dividing or by multiplying by R. When filters are studied, the variables are all voltages, for this reason, the equation will be solved for both voltages.

Initially, the object addresses the voltages. The starting point is the impulse response. If you want to remember the concept of the impulse response of a LTIS – Linear Time Invariant System, click on the magnifying glass.