To calculate the natural response of an RL circuit, you need to analyze the behavior of the circuit when it is disconnected from any external sources (i.e., batteries, generators, etc.) and the initial energy stored in the inductor is allowed to dissipate over time.
The natural response of an RL circuit can be described by a first-order linear differential equation, which relates the current through the inductor to the voltage across it. The differential equation governing the behavior of an RL circuit is:
(
)
+
(
)
=
0
L
dt
di(t)
β
+Ri(t)=0
where:
L is the inductance of the inductor in henries (H).
R is the resistance of the resistor in ohms (Ξ©).
i(t) is the current through the inductor as a function of time t.
To solve this differential equation, follow these steps:
Step 1: Write the differential equation for the circuit.
Step 2: Assume a solution of the form:
(
)
=
i(t)=Ae
st
Here, A is the initial current through the inductor (at t=0) and s is a complex constant.
Step 3: Substitute the assumed solution into the differential equation and solve for s.
Step 4: Once you have the value of s, find the natural frequency (Ο_n) and damping ratio (ΞΆ) from s.
The natural frequency (Ο_n) is the absolute value of the imaginary part of s:
=
β£
(
)
β£
Ο
n
β
=β£Im(s)β£
The damping ratio (ΞΆ) is the negative real part of s divided by the natural frequency:
=
β
(
)
/
ΞΆ=βRe(s)/Ο
n
β
Step 5: Based on the values of Ο_n and ΞΆ, determine the type of response: overdamped, critically damped, or underdamped.
The general natural response of an RL circuit can be expressed as a decaying exponential:
(
)
=
β
(
+
)
i(t)=Ae
βΞΆΟ
n
β
t
cos(Ο
d
β
t+Ο)
where:
A is the initial current through the inductor.
ΞΆ is the damping ratio.
Ο_n is the natural frequency.
Ο_d is the damped angular frequency (Ο_d = Ο_n * sqrt(1 - ΞΆ^2)).
Ο is the phase angle.
The specific values of A, ΞΆ, Ο_n, Ο_d, and Ο depend on the initial conditions and the values of L and R in the circuit.
Note that this solution assumes that the inductor's initial energy has been fully dissipated, and the response is governed only by the inductor's inductance (L) and resistance (R).