Question

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

Answer 1

To calculate the transient response of an RC circuit to a sinusoidal input using Laplace transforms, follow these steps:

Step 1: Write the differential equation for the circuit:
Consider an RC circuit with a resistor (R) and a capacitor (C) in series, driven by a sinusoidal voltage source (V_in) with angular frequency ω. The governing differential equation for the circuit can be written as:





out


+

out
=

in
(

)
RC
dt
dV
out
    ​

    ​

+V
out
    ​

=V
in
    ​

(t)

where V_out(t) is the output voltage across the capacitor.

Step 2: Take the Laplace transform of the differential equation:
The Laplace transform of a function f(t) is denoted as F(s) and is defined as:


(

)
=

{

(

)
}
=
∫
0
∞

−



(

)


F(s)=L{f(t)}=∫
0
∞
    ​

e
−st
f(t)dt

where s is the complex frequency variable. Taking the Laplace transform of the differential equation:





out
(

)
+

out
(

)
=

in
(

)
RCsV
out
    ​

(s)+V
out
    ​

(s)=V
in
    ​

(s)

Step 3: Solve for V_out(s):
Isolate V_out(s) on one side of the equation:


out
(

)
(



+
1
)
=

in
(

)
V
out
    ​

(s)(RCs+1)=V
in
    ​

(s)


out
(

)
=

in
(

)



+
1
V
out
    ​

(s)=
RCs+1
V
in
    ​

(s)
    ​


Step 4: Express V_in(s) in terms of Laplace transform:
For a sinusoidal input voltage, V_in(t) = A * sin(ωt). The Laplace transform of this function is:


in
(

)
=

{

sin
⁡
(


)
}
=



2
+

2
V
in
    ​

(s)=L{Asin(ωt)}=
s
2
+ω
2
Aω
    ​


Step 5: Substitute V_in(s) into V_out(s) equation:


out
(

)
=


(

2
+

2
)
(



+
1
)
V
out
    ​

(s)=
(s
2
+ω
2
)(RCs+1)
Aω
    ​


Step 6: Partial fraction decomposition (optional):
If needed, you can perform partial fraction decomposition on V_out(s) to express it in a form that can be inverted back to the time domain more easily.

Step 7: Inverse Laplace transform to find V_out(t):
The final step is to take the inverse Laplace transform of V_out(s) to obtain the time-domain expression for the output voltage V_out(t).

Keep in mind that the transient response focuses on the behavior of the circuit as it approaches its steady-state condition when the sinusoidal input is first applied. The steady-state response will be characterized by the circuit's frequency response, but the transient response tells us how the circuit behaves during the transient period before reaching the steady state.