Equivalent Inductance of Series and Parallel Inductors (With Mutual Inductance)

When inductors are connected in series, and mutual inductance can be ignored, the equivalent inductance is the sum of the individual inductances. Series resistance of series-connected resistors uses the same addition rule.

For inductors that share flux, add or subtract the mutual inductance between those inductors.

The effective inductance of each coil then includes both self-inductance and mutual inductance.

Mutual inductance is added to or subtracted from self-inductance based on the polarity of the inductors.

The later sections show how that mutual term is added or subtracted.

If mutual inductance is neglected, the equivalent inductance of series-connected inductors is

When inductors are connected in parallel, and mutual inductance can be ignored, the reciprocal of the equivalent inductance is the sum of the reciprocals of the individual inductances.

Parallel resistance of parallel-connected resistors uses the same reciprocal rule. Mutual inductance is added or subtracted in the same polarity sense when the coils share flux.

The parallel mutual-inductance case is treated later. If mutual inductance is neglected,

An inductor is a passive circuit element. The next sections derive the equivalent inductance of series-connected and parallel-connected inductors.

Adding Inductors in Series

Take n inductors connected in series as shown below.

Label the coils as follows.
the inductance of inductor 1 and voltage drop across it is L₁ and v_1, respectively,
the inductance of inductor 2 and voltage drop across it is L₂ and v_2, respectively,
the inductance of inductor 3 and voltage drop across it is L₃ and v_3, respectively,
the inductance of inductor 4 and voltage drop across it is L₄ and v_4, respectively,
the inductance of inductor n and voltage drop across it are L_n and v_n, respectively.

Apply Kirchhoff’s Voltage Law. The total voltage drop (v) across the series combination of the inductors is

The voltage drop across an inductor of inductance L can be expressed as,

Where i is the instantaneous current through the inductor.

Because the inductors are in series, the same current i flows in each coil. The KVL equation then becomes

Rewrite that as

L_eq is the equivalent inductance of the series combined inductors. Hence,

So the equivalent inductance of series-connected inductors, with no mutual inductance, is the arithmetic sum of the individual inductances.

Adding Inductors in Parallel

Take n inductors connected in parallel, as shown below.

Label the coils as follows.
the inductance of inductor 1 and current through it is L₁ and i_1, respectively,
the inductance of inductor 2 and current through it is L₂ and i_2, respectively,
the inductance of inductor 3 and current through it is L₃ and i_3, respectively,
the inductance of inductor 4 and current through it is L₄ and i_4, respectively,
the inductance of inductor n and current through it is L_n and i_n, respectively.

Apply Kirchhoff’s Current Law. The total current (i) entering the parallel combination of the inductors is

The current through an inductor of inductance L can be written as

Where v is the instantaneous voltage across the inductor.

Because the inductors are in parallel, the voltage drop across each coil is the same voltage v. The KCL equation then becomes

Rewrite that as

L_eq is the equivalent inductance of the parallel combined inductors. Hence,

So the reciprocal of the equivalent inductance of parallel-connected inductors, with no mutual inductance, is the arithmetic sum of the reciprocals of the individual inductances.

Effect of Mutual Induction in Series Connected Inductors

When coils sit close enough for flux from one to link another, mutual induction is present. For series inductors that share flux, include mutual inductance in the equivalent inductance.

Use the dot convention. Each inductor is marked with a dot at one end.

Current that enters the dotted terminal of one inductor induces a voltage in the other inductor with positive polarity at that second inductor’s dotted terminal. The next figure shows one series case.

Because the inductors are in series, the same current flows in both coils.

When current enters the dotted terminal of inductor 1, it also enters the dotted terminal of inductor 2.

Inductor 2 will induce the voltage across inductor 1 with positive polarity at the dotted end of inductor 1.

Current that enters the dotted terminal of inductor 1 induces a voltage across inductor 2 with positive polarity at the dotted end of inductor 2.

Both mutually induced EMFs then sit in the same sense as the self-induced EMFs, so mutual inductance is added to the self-inductances when you form the equivalent inductance.

In the second series example below, the dot marks are opposite. Current enters the dotted terminal of one inductor and leaves the dotted terminal of the other.

The mutually induced EMF then opposes the self-induced EMF. The equivalent inductance of that combination is

Effect of Mutual Induction in Parallel Connected Inductors

The same rule applies to parallel inductors. In the first parallel example, both dots sit on the same side of the two coils.

When current enters the dotted terminal of inductor 1, the EMF induced in inductor 2 is positive at the dotted end of inductor 2.

When current enters the dotted terminal of inductor 2, the EMF induced in inductor 1 is positive at the dotted end of inductor 1. The equivalent inductance is

When two parallel inductors are dotted as in the next figure, the equivalent inductance is