An R-L-C series circuit has R = 4 Ω, L = 2 H and C = 2 F. What type of transient current response is offered by the circuit for step function voltage input? 

Concept:

The characteristic equation defining the behavior of a source-free series RLC circuit is given by:

\({s^2} + 2α s + ω _c^2 = 0\)

Where:

α = damping factor given by

\(α = \frac{R}{{2L}}\)

ω0 = Undamped natural frequency or resonant frequency is given by:

\({ω _0} = \frac{1}{{\sqrt {LC} }}\)  

Analysis:

The roots of the characteristic equation will be:

\(s = \frac{{ - 2α \pm \sqrt {{{\left( {2α } \right)}^2} - 4\left( 1 \right)\left( {ω _c^2} \right)} }}{2}\)

\(s = \frac{{ - 2α \pm \sqrt {4{α ^2} - 4ω _c^2} }}{2}\)

\(s = \frac{{ - 2α \pm \sqrt {{α ^2} - ω _c^2} }}{2}\)

\(s = - α \pm \sqrt {{α ^2} - ω _0^2} \)

\({s_1} = - α + \sqrt {{α ^2} - ω _0^2} \)

\({s_2} = - α - \sqrt {{α ^2} - ω _0^2} \)

From the above two equations, we can have three types of solutions:

1) If α > ω0, we have the overdamped case

2) If α = ω0, we have the critically damped case

3) If α < ω0, we have the underdamped case.

Calculation:

With R = 4 Ω, L = 2 H, and C = 2 F, we get:

\(α = \frac{R}{{2L}} = \frac{4}{2 \times 2}\)

α = 1

Also:

\({ω _0} = \frac{1}{{\sqrt {LC} }} = \frac{1}{\sqrt{2\times2}}\)

ω0 = 0.5 rad/sec

Since α > ω0, we conclude that the resulting current response will be over-damped.