RLC circuits

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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.

Table of contents

1 Serial RLC Circuit

1.1 The ZIR (Zero Input Response) solution
1.2 The ZSR (Zero State Response) solution

2 Parallel RLC Circuit
3 See also

Serial RLC Circuit

In the serial configuration, the power source is a voltage source and all three components are connected in serial: Image:RLC-serial-1.PNG

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": Image:RLC-serial-Over_Damping.PNG

In this case, the characteristic polynomial's solutions are identical negative real numbers. This is called "Critical Damping": Image:RLC-serial-Critical_Damping.PNG

In this case, the characteristic polynomial's solutions are complex conjugate and have negative real part. This is called "Under Damping": Image:RLC-serial-Under_Damping.PNG

The ZSR (Zero State Response) solution

This time we nullify the initials conditions and stay with the following equation:

Separate 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...)

Parallel RLC Circuit