Define the natural response of an RL circuit.

An RL circuit consists of a resistor (R) and an inductor (L) connected in series. When a voltage is applied to the circuit, the current starts to build up through the inductor. However, due to the inherent property of inductors to resist changes in current, the current does not reach its steady-state value instantaneously.

The natural response of the RL circuit can be described by the transient behavior of the current (i(t)) through the inductor and the voltage (v(t)) across the inductor over time. The natural response is governed by the following differential equation:

L di(t)/dt + R i(t) = 0

Where:

L is the inductance of the inductor in henries (H).

di(t)/dt represents the rate of change of current with respect to time, which is the current's derivative.

R is the resistance of the resistor in ohms (Ω).

i(t) is the current through the inductor as a function of time.

The solution to this differential equation is an exponential decay or growth, depending on the initial conditions of the circuit. The natural response can be expressed as:

i(t) = Ie^(-Rt/L)

Where:

i(t) is the current at time t.

I is the initial current through the inductor at t = 0 (initial condition).

The natural response of an RL circuit is transient, meaning it diminishes over time and eventually approaches zero, allowing the circuit to reach its steady-state condition. The time it takes for the transient response to decay depends on the values of the inductance (L) and resistance (R) in the circuit. Larger inductances or resistances result in slower decay, and vice versa.

When an RL circuit is energized or de-energized (due to a switch being turned on or off, for example), a transient response, also known as the natural response, occurs. This transient response arises from the inductor's property to oppose changes in current. When the circuit is energized, the current in the inductor begins to rise, but the inductor's self-induced back-emf opposes this change, causing the current to increase gradually over time.

The natural response in an RL circuit can be described mathematically using the following equation:

(

)

=

initial

⋅

−

/

i(t)=I

initial

⋅e

−t/τ

where:

(

)

i(t) is the current in the circuit at time

t.

initial

I

initial

is the initial current flowing in the circuit just before the change occurred (initial condition).

e is the base of the natural logarithm, approximately equal to 2.71828.

t is the time after the change occurred.

τ is the time constant of the RL circuit, given by

=

τ=

R

L

, where

L is the inductance of the inductor and

R is the resistance of the resistor.

The time constant

τ represents the time it takes for the current to reach approximately 63.2% of its final steady-state value. As time progresses, the current in the RL circuit approaches a constant value equal to the steady-state current, which is determined by the DC voltage source and the resistance in the circuit. The natural response is transient and decays over time until the circuit reaches a steady-state.