RLC circuits
An
RLC circuit is a kind of
electrical circuit composed of a
resistor (R), a
capacitor (C) and an
inductor (L). See
RC circuit for the simpler case. It is called a second order circuit, for mathematical reasons to do with the underlying
differential equations.
There are two types of common configuration of RLC circuits: parallel and serial.
In the serial configuration, the power source is a voltage source and all three components are connected in serial:
Where the notations in the figure above are:
 V  the voltage of the power source (measured in Volt)
 I  the current in the circuit (measured in Ampere)
 R  the resistance of the Resistor (measured in Ohm)
 L  the inductance of the Inductor (measured in Henry)
 C  the capacitance of the Capacitor (measured in Farad)
Given the parameters V,R,L,C, the solution for the current (I) using
Kirchoff's Voltage Law (or KVL) is:

Rearranging the equation will result in the following second order differential equation:
The ZIR (Zero Input Response) solution:
Nullifying the Input(i.e voltage sources) we get the equation:

with the initial conditions for the inductor current () and the capacitor voltage (). However, in order to solve the equation properly, the initial conditions needed are .
The first one we already have since the current in the main branch is also the current in the inductor, therefore .
The second one is obtained employing KVL again:


We have now a
homogeneous second order differential equation with two initial conditions. Usually second order differential equations are written as:

In case of an electrical circuit and therefore, there are 3 possible cases:
In this case, the charectaristic polynom solutions are both negative real numbers. This is called "Over Damping":
In this case, the charectaristic polynom solutions are identical negative real numbers. This is called "Critical Damping":
In this case, the charectaristic polynom are conjugated and have a negative real part. This is called "Under Damping":
The ZSR (Zero State Response) solution:
This time we nullify the initials conditions and stay with the following equation:

Seperate solution for every possible function for V(t) is impossible, however, there is a way to find a formula for I(t) using
convolution. In order to do that, we need a solution for a basic input  the
Dirac delta function.
In order to find the solution more easily we will start solving for the Heaviside
step function and then using the fact our circuit is a linear system , its derivative will be the solution for the delta function.
(to be continued...)