Inductance Formula Circuits – Series and Parallel Inductors

Inductance Formula Circuits

If you have read the previous post about what is an inductor, let us proceed to the next level, what is the inductance formula circuits.

Now that the inductor has been added to our list of passive elements, it is necessary to extend the powerful tool of series-parallel combination. We need to know how to find the equivalent inductance of a series-connected or parallel-connected set of inductors found in practical circuits.

Series Inductors

Consider a series connection of N inductors, as shown in Figure.(1a), with the equivalent circuit shown in Figure.(1b).

Inductance Formula Circuits
Figure 1. (a) A series connection of N inductors, (b) equivalent circuit for the series inductors.

The inductors have the same current through them. Applying KVL to the loop,

Inductance Formula Circuits
(1)

Substituting vk​=​ Lk di/dt results in

Inductance Formula Circuits
(2)

where

Inductance Formula Circuits
(3)

Thus,

The equivalent inductance of series-connected inductors is the sum of the individual inductances.

Inductors in series are combined in exactly the same way as resistors in series.

Parallel Inductors

We now consider a parallel connection of N inductors, as shown in Figure.(2a), with the equivalent circuit in Figure.(2b). The inductors have the same voltage across them. Using KCL,

Inductance Formula Circuits
(4)

But

Inductance Formula Circuits

hence,

Inductance Formula Circuits
(5)

where

Inductance Formula Circuits
(6)

The initial current i(t0) through Leq at t​=​ t0 is expected by KCL to be the sum of the inductor currents at t0. Thus, according to Equation.(5),

Inductance Formula Circuits

According to Equation.(6),

The equivalent inductance of parallel inductors is the reciprocal of the sum of the reciprocals of the individual inductances.

Note that the inductors in parallel are combined in the same way as resistors in parallel.

For two inductors in parallel (N = 2), Equation.(6) becomes

Inductance Formula Circuits
(7)

Inductance Formula Circuits Examples

1. Find the equivalent inductance of the circuit shown in Figure.(3).

Inductance Formula Circuits
Figure 3

Solution:
The 10-H, 12-H, and 20-H inductors are in series; thus, combining them gives a 42-H inductance. This 42-H inductor is in parallel with the 7-H inductor so that they are combined, to give

Inductance Formula Circuits

This 6-H inductor is in series with the 4-H and 8-H inductors. Hence,

Inductance Formula Circuits

 

2. For the circuit in Figure.(4),
i(t) = 4(2 − e−10t) mA. If i2(0) = −1 mA, find:
(a) i1(0);
(b) v(t), v1(t), and v2(t);
(c) i1(t) and i2(t).

Inductance Formula Circuits
Figure 4

Solution:
(a) From i(t) = 4(2 − e−10t) mA, i(0) = 4(2 − 1) = 4 mA.
Since i = i1 + i2,

Inductance Formula Circuits

(b) The equivalent inductance is

Inductance Formula Circuits

Thus,

Inductance Formula Circuits

and

Inductance Formula Circuits

Since v = v1 + v2,

Inductance Formula Circuits

(c) The current i1 is obtained as

Inductance Formula Circuits

Similarly,

Inductance Formula Circuits

Note that i1(t) + i2(t) = i(t).