Question

How do you calculate the transient response of an RL circuit to a step input using Laplace transforms?

Answer 1

To calculate the transient response of an RL (resistor-inductor) circuit to a step input using Laplace transforms, you can follow these steps:

Step 1: Formulate the Circuit Equations
Write the governing differential equation(s) for the RL circuit. In this case, it's the equation for the voltage across the inductor (voltage across the inductor is also the output of the circuit). For a series RL circuit, the differential equation is:




(

)


+


(

)
=

in
(

)
L
dt
di(t)
    ​

+Ri(t)=V
in
    ​

(t)

where:


L is the inductance in henries (H),

R is the resistance in ohms (Ω),

(

)
i(t) is the current through the inductor as a function of time

t,

in
(

)
V
in
    ​

(t) is the input voltage (step input in this case) as a function of time

t.

Step 2: Take the Laplace Transform
Take the Laplace transform of both sides of the differential equation using the properties of Laplace transforms. The Laplace transform of

(

)
i(t) is denoted as

(

)
I(s) and the Laplace transform of

in
(

)
V
in
    ​

(t) is denoted as

in
(

)
V
in
    ​

(s).


[


(

)
−

(
0
)
]
+


(

)
=

in
(

)
L[sI(s)−i(0)]+RI(s)=V
in
    ​

(s)

Where

(
0
)
i(0) is the initial current through the inductor at

=
0
t=0.

Step 3: Solve for

(

)
I(s)
Rearrange the equation to solve for

(

)
I(s):


(

)
=

in
(

)


+

+

(
0
)

I(s)=
sL+R
V
in
    ​

(s)
    ​

+
s
i(0)
    ​


Step 4: Apply Inverse Laplace Transform
Now, you need to find the inverse Laplace transform of

(

)
I(s) to obtain the transient response

(

)
i(t) in the time domain. To do this, use the Laplace transform table or an appropriate software/tool to find the inverse Laplace transform.

Step 5: Analyze the Transient Response
The inverse Laplace transform will give you the expression for the transient current

(

)
i(t) in the RL circuit as a function of time

t. You can analyze the behavior of the transient response based on this expression.

Keep in mind that the transient response will eventually decay and reach a steady-state value. The time it takes for the transient response to decay is determined by the time constant

=


τ=
R
L
    ​

 of the RL circuit.

Remember, Laplace transforms are a powerful tool for analyzing linear time-invariant systems like electrical circuits, and they are particularly useful in solving differential equations in the frequency domain.