Question

What is the step response of an RL series circuit?

Answer 1

The step response of an RL series circuit is how the circuit responds when a sudden change (step) in input voltage is applied, typically from 0 to a constant value (like applying a DC voltage source).

Components of an RL Series Circuit

1. R: Resistance (measured in Ohms, Ω)
   

1. L: Inductance (measured in Henrys, H)
   

When a step voltage (like a DC source) is applied to an RL circuit, the current does not instantly reach its final value. Instead, it gradually increases over time, due to the presence of the inductor. The inductor opposes sudden changes in current, which causes the gradual increase.

Differential Equation for RL Circuit

The behavior of the RL circuit is described by a first-order linear differential equation:

\[
V(t) = L \frac{dI(t)}{dt} + I(t)R
\]

Where:

1. \(V(t)\) is the applied voltage (step function, typically \(V_0\) for \(t \geq 0\)).
   

1. \(I(t)\) is the current through the circuit.
   

1. \(R\) is the resistance.
   

1. \(L\) is the inductance.
   

For a step input voltage \(V_0\), the equation becomes:

\[
V_0 = L \frac{dI(t)}{dt} + I(t)R
\]

Solution for Step Response

The solution to this differential equation gives the current \(I(t)\) over time:

\[
I(t) = \frac{V_0}{R} \left( 1 - e^{-\frac{R}{L}t} \right)
\]

Key Characteristics of the Step Response

1. Initial Condition: When the voltage is first applied (\(t = 0\)), the current \(I(0) = 0\). This is because the inductor initially resists any change in current.
   

   

1. Asymptotic Behavior: As time goes on, the current approaches its final steady-state value of \(I_{\text{final}} = \frac{V_0}{R}\). This is the final value the current will reach once the inductor's influence (resistance to changes in current) no longer affects the circuit.
   

1. Time Constant (\(\tau\)): The time constant \(\tau = \frac{L}{R}\) determines how fast the current rises. The larger the time constant, the slower the current increase.
   

1. Exponential Rise: The current increases exponentially from 0 towards the steady-state value, with the rate of increase determined by the time constant \(\tau\).
   

Time Behavior

1. At \(t = 0\), \(I(0) = 0\).
   

1. After \(t = \tau\), the current reaches approximately 63% of its final value.
   

1. After \(t = 5\tau\), the current is very close (over 99%) to its final value.
   

Summary

In an RL series circuit, the step response shows that the current gradually increases and asymptotically approaches a final steady value. The rate at which the current increases depends on the circuit's resistance and inductance, characterized by the time constant \(\tau = \frac{L}{R}\).