# 8. Damping and the Natural Response in RLC Circuits

Consider a series RLC circuit (one that has a resistor, an inductor and a capacitor) with a **constant **driving electro-motive force (emf) *E*. The current equation for the circuit is

`L(di)/(dt)+Ri+1/Cinti\ dt=E`

This is equivalent: `L(di)/(dt)+Ri+1/Cq=E`

Differentiating, we have

`L(d^2i)/(dt^2)+R(di)/(dt)+1/Ci=0`

This is a second order linear homogeneous equation.

Its corresponding auxiliary equation is

`Lm^2+Rm+1/C=0`

with roots:

`m_1=(-R)/(2L)+(sqrt((R^2-4L"/"C)))/(2L)`

`=-alpha+sqrt(alpha^2-omega_0^2`

`m_2=(-R)/(2L)-(sqrt((R^2-4L"/"C)))/(2L)`

`=-alpha-sqrt(alpha^2-omega_0^2`

Now

`alpha=R/(2L)` is called the

damping coefficientof the circuit`omega_0 = sqrt(1/(LC)`is the

resonant frequencyof the circuit.

m_{1}andm_{2}are called thenatural frequenciesof the circuit.

The nature of the current will depend on the relationship between *R*,* L *and *C*.

There are three possibilities:

## Case 1: *R*^{2} > 4*L*/*C* (Over-Damped)

Graph of overdamped case.

Here both *m*_{1} and *m*_{2} are real, distinct and negative. The general solution is given by

`i(t)=Ae^(m_1t)+Be^(m_2t)`

The motion (current) is not oscillatory, and the vibration returns to equilibrium.

## Case 2: *R*^{2} = 4*L*/*C* (Critically Damped)

Graph of critcally damped case.

Here the roots are negative, real and equal,

i.e. `m_1= m_2= -R/(2L)`.

The general solution is given by

`i(t)=(A+Bt)e^(-Rt"/"2L)`

The vibration (current) returns to equilibrium in the minimum time and there is just enough damping to prevent oscillation.

## Case 3: *R*^{2} < 4*L*/*C* (Under-Damped)

Graph of RLC under-damped case.

Here the roots are complex where

`m_1=alpha+jomega`, and `m_2=alpha-jomega`

The general solution is given by

`i(t)=e^(-alpha t)(A\ cos\ omegat+B\ sin\ omegat)`

where

`\alpha = R/(2L)` is called the **damping coefficient**, and `omega` is given by:

`omega=sqrt(1/(LC)-R^2/(4L^2)`

In this case, the motion (current) is oscillatory and the amplitude decreases exponentially, bounded by

`i=+-sqrt(A^2+B^2)\ e^(-Rt"/"2L)`

as we can see in the diagram above.

When *R* = 0, the circuit displays its natural or resonant frequency, `omega_0=sqrt(1/(LC))`.

### Example

In a series RCL circuit driven by a constant emf, the natural response of the circuit is given by

`(d^2i)/(dt^2)+4(di)/(dt)+4i=0`

for which the initial conditions are *i*(0) = 2 A and `(di)/(dt)` at *t* = 0 is 4.

State the nature of response of the current and hence solve for *i*.

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