Question

A.C. Fundamentals - Step Response of R-L-C Circuit

Answer 1

The step response of an R-L-C circuit refers to how the circuit's output voltage or current responds when a step input voltage is applied to it. An R-L-C circuit is a combination of resistors (R), inductors (L), and capacitors (C) connected together. The step response analysis helps us understand how the circuit behaves over time after a sudden change in input.

Let's consider a simple series R-L-C circuit driven by a step voltage source. The circuit consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series with a voltage source (V_in) that suddenly changes from 0V to a constant value at time t = 0.

The differential equation that governs the behavior of this circuit can be derived using Kirchhoff's voltage law (KVL):




=


+


+


V
in
    ​

=V
R
    ​

+V
L
    ​

+V
C
    ​


Where:




V
in
    ​

 is the input step voltage.


V
R
    ​

 is the voltage across the resistor (R) which is

⋅

I⋅R, where

I is the current flowing through the circuit.


V
L
    ​

 is the voltage across the inductor (L) which is

⋅




L⋅
dt
dI
    ​

, where




dt
dI
    ​

 is the rate of change of current with respect to time.


V
C
    ​

 is the voltage across the capacitor (C) which is
1

∫

 


C
1
    ​

∫Idt, where
∫

 


∫Idt is the integral of current with respect to time.

Substituting these expressions into the KVL equation:




=

⋅

+

⋅




+
1

∫

 


V
in
    ​

=I⋅R+L⋅
dt
dI
    ​

+
C
1
    ​

∫Idt

This equation can be rearranged and simplified into a second-order linear differential equation:


⋅

2



2
+

⋅




+
1

⋅

=






L⋅
dt
2
d
2
I
    ​

+R⋅
dt
dI
    ​

+
C
1
    ​

⋅I=
dt
dV
in
    ​

    ​


The solution to this differential equation gives us the current

(

)
I(t) flowing through the circuit as a function of time. The form of the solution depends on the values of the circuit components (R, L, and C) and the initial conditions (initial current and voltage). The response can be categorized into three types based on the damping factor (

ζ) of the circuit:

Underdamped (

<
1
ζ<1): The current oscillates around its final steady-state value before settling down.

Critically Damped (

=
1
ζ=1): The current approaches its final steady-state value without oscillation as quickly as possible.

Overdamped (

>
1
ζ>1): The current takes a longer time to reach its steady-state value without oscillation.

The step response of an R-L-C circuit can be further analyzed by its natural frequency (


ω
n
    ​

) and time constants (

τ). The expressions for these parameters are:



=
1


ω
n
    ​

=
LC
    ​

1
    ​



=
1

⋅


τ=
ζ⋅ω
n
    ​

1
    ​


Understanding these parameters and the behavior of the circuit helps in designing and analyzing circuits for various applications.