8. Damping and the Natural Response in RLC Circuits

Series RLC circuit diagram

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 coefficient of the circuit

`omega_0 = sqrt(1/(LC)`is the resonant frequency of the circuit.

m1 and m2 are called the natural frequencies of the circuit.

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

There are three possibilities:

Case 1: R2 > 4L/C (Over-Damped)

Graph of over-damped case solution of RLC Circuit differential equation. t i
`A+B`

Graph of overdamped case.

Here both m1 and m2 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: R2 = 4L/C (Critically Damped)

Graph of damping and the natural response in RLC Circuit. t i A
`t=(2L)/R - A/B`

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: R2 < 4L/C (Under-Damped)

Graph of under-damped case in RLC Circuit differential equation. t i
`sqrt(A^2+B^2)\ e^(-Rt"/"2L)`
`-sqrt(A^2+B^2)\ e^(-Rt"/"2L)`

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.