# How do you calculate the natural response of an RL circuit?

How do you calculate the natural response of an RL circuit?

To calculate the natural response of an RL (resistor-inductor) circuit, you need to consider the behavior of the circuit after a sudden change in the input (for example, if a voltage source is disconnected or a current source is connected). The natural response occurs due to the energy stored in the inductor and the resistance in the circuit. The natural response of an RL circuit is characterized by the current in the inductor as it decays over time.

The natural response of an RL circuit is governed by the following differential equation:

(

)

+

β

(

)
=
0
dt
di(t)
β

+
L
R
β

βi(t)=0

where:

(

)
i(t) = current in the inductor at time

t,

R = resistance in the circuit (in ohms),

L = inductance of the inductor (in henrys).

The general solution to this differential equation is of the form:

(

)
=

β

β

β

i(t)=Aβe
β
L
R
β

βt

where

A is the initial current in the inductor at

=
0
t=0 (right after the sudden change in input).

To find the constant

A, you need to know the initial condition of the inductor current (

(
0
)
i(0)), which is usually given in the problem or can be obtained from the initial conditions of the circuit.

So, the steps to calculate the natural response of an RL circuit are as follows:

Determine the resistance (

R) and inductance (

L) values of the circuit.
Determine the initial current in the inductor (

(
0
)
i(0)).
Write the general equation for the natural response:

(

)
=

β

β

β

i(t)=Aβe
β
L
R
β

βt
.
Use the initial condition (

(
0
)
i(0)) to solve for the constant

A.
Once you have the value of

A, you can find the current in the inductor (

(

)
i(t)) at any given time

t during the natural response.

Remember that the natural response only represents the behavior of the inductor's current due to its inherent energy storage. In practical scenarios, you might also have a forced response (due to external sources) or a combined response of both forced and natural components.
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=βRe(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
βΞΆΟ
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).

## Related questions

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