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...)
See also