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

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