Question

How do you calculate the transient response of an RC circuit to a step input using time-domain analysis?

Answer 1

To calculate the transient response of an RC circuit to a step input using time-domain analysis, follow these steps:

Understand the RC Circuit:
An RC circuit consists of a resistor (R) and a capacitor (C) connected in series or parallel. The step input is a sudden change in the input voltage from one constant value to another constant value.

Derive the Differential Equation:
Apply Kirchhoff's voltage law to the RC circuit to derive the differential equation governing its behavior. For a series RC circuit, the equation can be derived as follows:

V_in(t) - V_R(t) - V_C(t) = 0

where:

V_in(t) is the step input voltage.
V_R(t) is the voltage across the resistor R, which is equal to i(t) * R, where i(t) is the current through the circuit.
V_C(t) is the voltage across the capacitor C, which is equal to q(t) / C, where q(t) is the charge on the capacitor.
Find the Capacitor Current:
The next step is to find the current through the circuit, which is equal to the derivative of the capacitor voltage with respect to time:

i(t) = C * dV_C(t)/dt

Substitute Current and Capacitor Voltage into the Differential Equation:
Replace i(t) and V_C(t) in the differential equation using their respective expressions:

V_in(t) - R * C * dV_C(t)/dt - V_C(t) = 0

Solve the Differential Equation:
Now, solve the differential equation for V_C(t). This will involve integrating and manipulating the equation to isolate V_C(t) on one side.

Apply Initial Condition:
The transient response requires an initial condition. For a step input, the initial condition is the voltage across the capacitor just before the step occurs (i.e., t = 0^-). If the capacitor is uncharged initially (V_C(0^-) = 0), the initial condition becomes V_C(0^+) = V_in(0^+), where V_in(0^+) is the value of the step input just after the step occurs.

Calculate Time Constant (τ):
The time constant (τ) of the RC circuit is given by τ = R * C. It represents the time it takes for the capacitor voltage to reach approximately 63.2% of the final value.

Evaluate the Transient Response:
With the solution for V_C(t) and the value of τ, you can evaluate the transient response of the RC circuit at any given time t.

Please note that the transient response decays over time until it reaches its steady-state value, which occurs when t >> τ. At that point, the capacitor behaves like an open circuit, and the voltage across the capacitor becomes constant.