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

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.

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).