Question

What is the step response of an RL circuit?

Answer 1

Are you asking about the theoretical step response of a basic RL circuit or a specific type of RL circuit?

Answer 2

The step response of an RL (resistor-inductor) circuit describes how the circuit's output evolves over time when subjected to a step input, typically a step voltage. Here's a detailed breakdown:

### RL Circuit Overview

An RL circuit consists of a resistor \( R \) and an inductor \( L \) connected in series. When subjected to a step voltage input, the circuit's behavior is characterized by its time constant and the response of the inductor and resistor to the sudden change in voltage.

### Step Response Analysis

1. **Initial Conditions:**
   - At \( t = 0 \), a step voltage \( V_{in} \) is applied to the circuit.
   - Initially, the current through the inductor is zero because an inductor resists sudden changes in current.

2. **Differential Equation:**
   - The voltage across the resistor \( V_R \) is \( i(t) \cdot R \).
   - The voltage across the inductor \( V_L \) is \( L \frac{di(t)}{dt} \).
   - Applying Kirchhoff’s Voltage Law (KVL) to the series circuit:
     \[
     V_{in} = V_R + V_L
     \]
     Substituting the expressions for \( V_R \) and \( V_L \):
     \[
     V_{in} = i(t) \cdot R + L \frac{di(t)}{dt}
     \]

3. **Solving the Differential Equation:**
   - Rearrange the equation to form a standard first-order linear differential equation:
     \[
     L \frac{di(t)}{dt} + i(t) \cdot R = V_{in}
     \]
   - The time constant \( \tau \) of the RL circuit is given by:
     \[
     \tau = \frac{L}{R}
     \]
   - The general solution to the differential equation is:
     \[
     i(t) = \frac{V_{in}}{R} \left(1 - e^{-\frac{t}{\tau}}\right)
     \]

4. **Step Response:**
   - When a step input \( V_{in} \) is applied, the current \( i(t) \) increases from 0 to \( \frac{V_{in}}{R} \) asymptotically.
   - The expression \( \frac{V_{in}}{R} \left(1 - e^{-\frac{t}{\tau}}\right) \) shows that initially (at \( t = 0 \)), the current is zero, and it approaches the steady-state value \( \frac{V_{in}}{R} \) as \( t \) approaches infinity.
   - The term \( e^{-\frac{t}{\tau}} \) describes the exponential approach to the steady-state value.

### Key Points:

- **Time Constant (\( \tau \)):** Determines how quickly the current reaches its final value. A larger \( \tau \) (larger \( L \) or smaller \( R \)) means a slower response.
- **Steady-State Value:** The final current \( I_{ss} \) after a long time is \( \frac{V_{in}}{R} \).
- **Transient Response:** The term \( e^{-\frac{t}{\tau}} \) governs the rate at which the current changes over time.

### Summary

In summary, the step response of an RL circuit shows how the current grows exponentially from zero to its final steady-state value in response to a sudden step in voltage. The time constant \( \tau = \frac{L}{R} \) dictates how quickly this transition occurs, with the current reaching its final value asymptotically.