To calculate the transient response of an RC circuit to an impulse input using Laplace transforms, you follow these steps:
Start with the circuit:
Consider an RC circuit, which consists of a resistor (R) and a capacitor (C) connected in series with an input voltage source (v(t)) as shown below:
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v(t) --///---| |---o
R | RC |
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The goal is to find the voltage across the capacitor (Vc(t)) as a response to an impulse input.
Write the differential equation:
The first step is to write the differential equation that describes the behavior of the circuit. The governing equation for the RC circuit is:
RC * d(Vc(t))/dt + Vc(t) = v(t)
Where:
Vc(t) is the voltage across the capacitor as a function of time.
v(t) is the input voltage (impulse input in this case).
R is the resistance in ohms.
C is the capacitance in farads.
Take the Laplace transform:
Take the Laplace transform of both sides of the differential equation. The Laplace transform of the left-hand side (LHS) will involve replacing the time-domain variable 't' with the Laplace variable 's', and the Laplace transform of the right-hand side (RHS) will be the Laplace transform of the impulse input function.
Applying the Laplace transform:
RC * [sVc(s) - Vc(0)] + Vc(s) = V(s)
Where:
Vc(s) is the Laplace transform of Vc(t).
Vc(0) is the initial condition of the capacitor voltage (at t=0+).
V(s) is the Laplace transform of the input voltage v(t) (impulse input).
Solve for Vc(s):
Rearrange the equation to solve for Vc(s):
Vc(s) [RC * s + 1] = V(s) + RC * Vc(0)
Vc(s) = (V(s) + RC * Vc(0)) / (RC * s + 1)
Find the inverse Laplace transform:
Use a table of Laplace transforms or knowledge of inverse Laplace transforms to find the time-domain expression for Vc(t). The inverse Laplace transform of Vc(s) will give you the transient response of the RC circuit to the impulse input.
The inverse Laplace transform of Vc(s) is the solution to the differential equation and will give you Vc(t), the voltage across the capacitor as a response to the impulse input.
Please note that the impulse input in the Laplace domain is represented as "V(s)," which is the Laplace transform of the impulse function. The Laplace transform of the impulse function is 1 (a constant). So, V(s) = 1. The initial condition of the capacitor voltage, Vc(0), represents the voltage across the capacitor just before the impulse is applied. If the capacitor is initially uncharged (at t=0+), Vc(0) will be 0.
Once you have the inverse Laplace transform of Vc(s), you will get the transient response of the RC circuit to an impulse input.