In the context of electrical circuits, the natural response of an RL circuit refers to the behavior of the circuit after a sudden change in its initial conditions or input, such as switching on or off a voltage source. An RL circuit consists of a resistor (R) and an inductor (L) connected in series.
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.