Question

What is the zero-state response of an RL circuit?

Answer 1

The **zero-state response** of an RL circuit refers to the response of the circuit when the initial conditions (such as the current in the inductor) are zero, and the system is subjected to an external input like a voltage or current source. This type of response depends solely on the applied input and not on any stored energy from previous states.

For an RL circuit consisting of a resistor \( R \) and an inductor \( L \), with an input voltage \( v(t) \), the differential equation governing the circuit is:

\[
v(t) = L \frac{di(t)}{dt} + Ri(t)
\]

### Solving for Zero-State Response:
To find the zero-state response \( i_{zs}(t) \), assume that the circuit starts with no initial current, i.e., \( i(0) = 0 \).

1. **Input is a Step Voltage \( V_u(t) \)** (e.g., \( v(t) = V \cdot u(t) \), where \( u(t) \) is the unit step function):
   
   The differential equation becomes:
   
   \[
   V = L \frac{di(t)}{dt} + Ri(t)
   \]
   
   Solving this for \( i(t) \) yields:
   
   \[
   i_{zs}(t) = \frac{V}{R} \left(1 - e^{-\frac{R}{L} t}\right) \quad \text{for} \ t \geq 0
   \]

   Here, \( \frac{R}{L} \) is the time constant \( \tau = \frac{L}{R} \), which determines how quickly the current reaches its steady-state value.

### Key Points:
- The zero-state response describes the current's behavior due to the applied input.
- The current increases exponentially with time, approaching a steady-state value of \( \frac{V}{R} \).
- The rate at which the current changes is governed by the time constant \( \tau = \frac{L}{R} \).

Answer 2

To better understand the zero-state response of an RL circuit, could you clarify whether you’re referring to a series or parallel RL circuit, and if there's a specific type of input or initial condition you're considering?