Question

How do you perform transient analysis of an RL circuit with a step input using Laplace transforms?

Answer 1

To perform a transient analysis of an RL (Resistor-Inductor) circuit with a step input using Laplace transforms, you'll follow these steps:

Step 1: Obtain the circuit differential equation:
Start by writing down the governing differential equation for the RL circuit. For a series RL circuit, the differential equation is derived from Kirchhoff's voltage law:


(

)
=



(

)


+

⋅

(

)
v(t)=L
dt
di(t)
    ​

+R⋅i(t)

where:


(

)
v(t) is the input voltage (step input in this case, v(t) = V for t ≥ 0)


(

)
i(t) is the current flowing through the inductor


L is the inductance of the inductor


R is the resistance of the resistor

Step 2: Take the Laplace transform of the differential equation:
Apply the Laplace transform to both sides of the equation. The Laplace transform of

(

)
v(t) with respect to

t is denoted as

(

)
V(s), and the Laplace transform of

(

)
i(t) with respect to

t is denoted as

(

)
I(s). The Laplace transform of


(

)


dt
di(t)
    ​

 is


(

)
−

(
0
)
sI(s)−i(0), where

(
0
)
i(0) is the initial condition of the current (current at t = 0+).

After taking the Laplace transform and applying the initial condition

(
0
)
i(0), the equation becomes:


(

)
=

⋅
(


(

)
−

(
0
)
)
+

⋅

(

)
V(s)=L⋅(sI(s)−i(0))+R⋅I(s)

Step 3: Solve for

(

)
I(s):
Rearrange the equation to solve for

(

)
I(s):


(

)
=

(

)
+

(
0
)
⋅



+

I(s)=
sL+R
V(s)+i(0)⋅s
    ​


Step 4: Perform inverse Laplace transform:
Now, you need to take the inverse Laplace transform of

(

)
I(s) to get the time-domain current

(

)
i(t). The inverse Laplace transform of

(

)
I(s) can be found using tables or software tools like MATLAB or Wolfram Alpha.

The inverse Laplace transform of

(

)
+

(
0
)
⋅



+

sL+R
V(s)+i(0)⋅s
    ​

 is denoted as

(

)
i(t).

Step 5: Include initial conditions:
Finally, include the initial condition

(
0
)
i(0) to obtain the complete time-domain current

(

)
i(t).

In summary, follow these steps to perform transient analysis of an RL circuit with a step input using Laplace transforms:

Write the differential equation for the circuit.
Take the Laplace transform of the equation.
Solve for

(

)
I(s) in terms of

(

)
V(s) and

(
0
)
i(0).
Perform the inverse Laplace transform to get

(

)
i(t).
Include the initial condition

(
0
)
i(0) to get the complete time-domain current

(

)
i(t).