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.
Serial RLC Circuit
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 three possible cases:
In this case, the characteristic polynomial's solutions are both negative real numbers. This is called "Over Damping":
In this case, the characteristic polynomial's solutions are identical negative real numbers. This is called "Critical Damping":
In this case, the characteristic polynomial's solutions are complex conjugate and have 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.
The equation will be therefore, for t>0:
Assuming  are the roots of  , then as in the ZIR solution, we have 3 cases here:
- Over Damping - two negative real roots, the solution is:
- Critical Damping - the two roots are identical (
 ), the solution is:
- Under Damping - two conjugate roots (
 ), the solution is:
(to be continued...)
See also
Referenced By
Circuit theory | Electric circuit | Electrical circuit | Electrical network | Electronic circuit | Electronic oscillator | RLC circuit | RLC series circuit
|
