Inductors in Parallel

Inductors are said to be connected together in "Parallel" when both of their terminals are respectively connected to each terminal of the other inductor or inductors. The voltage drop across all of the inductors in parallel will be the same. Then, Inductors in Parallel have a Common Voltage across them and in our example below the voltage across the inductors is given as:

V_L1 = V_L2 = V_L3 = V_AB ...etc

In the following circuit the inductors L₁, L₂ and L₃ are all connected together in parallel between the two points A and B.

Inductors in Parallel Circuit

In the previous series inductors tutorial, we saw that the total inductance, L_T of the circuit was equal to the sum of all the individual inductors added together. For inductors in parallel the equivalent circuit inductance L_T is calculated differently.

The sum of the individual currents flowing through each inductor can be found using Kirchoff's Current Law (KCL) where, I_T = I₁ + I₂ + I₃ and we know from the previous tutorials on inductance that the self-induced emf across an inductor is given as: V = L di/dt

Then by taking the values of the individual currents flowing through each inductor in our circuit above, and substituting the current i for i₁ + i₂ + i₃ the voltage across the parallel combination is given as:

By substituting di/dt in the above equation with v/L gives:

We can reduce it to give a final expression for calculating the total inductance of a circuit when connecting inductors in parallel and this is given as:

Parallel Inductor Equation

Here, like the calculations for parallel resistors, the reciprocal ( 1/Ln ) value of the individual inductances are all added together instead of the inductances themselves. But again as with series connected inductances, the above equation only holds true when there is "NO" mutual inductance or magnetic coupling between two or more of the inductors, (they are magnetically isolated from each other). Where there is coupling between coils, the total inductance is also affected by the amount of coupling.

This method of calculation can be used for calculating any number of individual inductances connected together within a single parallel network. If however, there are only two individual inductors in parallel then a much simpler and quicker formula can be used to find the total inductance value, and this is:

One important point to remember about inductors in parallel circuits, the total inductance (L_T) of any two or more inductors connected together in series will always be LESS than the value of the smallest inductance in the parallel chain.

Example No1

Three inductors of 60mH, 120mH and 75mH are connected together in a parallel combination with no mutual inductance between them. Calculate the total inductance of the parallel combination.

Mutually Coupled Inductors in Parallel

When inductors are connected together in parallel so that the magnetic field of one links with the other, the effect of mutual inductance either increases or decreases the total inductance depending upon the amount of magnetic coupling that exists between the coils. The effect of this mutual inductance depends upon the distance apart of the coils and their orientation to each other. Mutually connected inductors in parallel can be classed as either "aiding" or "opposing" the total inductance with parallel aiding connected coils increasing the total equivalent inductance and parallel opposing coils decreasing the total equivalent inductance compared to coils that have zero mutual inductance.

Mutual coupled parallel coils can be shown as either connected in an aiding or opposing configuration by the use of polarity dots or polarity markers as shown below.

Parallel Aiding Inductors

The voltage across the two parallel aiding inductors above must be equal since they are in parallel so the two currents, i₁ and i₂ must vary so that the voltage across them stays the same. Then the total inductance, L_T for two parallel aiding inductors is given as:

Where:

2M represents the influence of coil L ₁ on L ₂ and likewise coil L ₂ on L ₁.

If the two inductances are equal and the magnetic coupling is perfect such as in a toroidal circuit, then the equivalent inductance of the two inductors in parallel is L as L_T = L₁ = L₂ = M. However, if the mutual inductance between them is zero, the equivalent inductance would be L ÷ 2 the same as for two self-induced inductors in parallel.

If one of the two coils was reversed with respect to the other, we would then have two parallel opposing inductors and the mutual inductance, M that exists between the two coils will have a cancelling effect on each coil instead of an aiding effect as shown below.

Parallel Opposing Inductors

Then the total inductance, L_T for two parallel opposing inductors is given as:

This time, if the two inductances are equal in value and the magnetic coupling is perfect between them, the equivalent inductance and also the self-induced emf across the inductors will be zero as the two inductors cancel each other out. This is because as the two currents, i₁ and i₂ flow through each inductor inturn the total mutual flux generated between them is zero because these two flux's produced by each inductor are both equal in magnitude but in opposite directions. Then the two coils effectively become a short circuit to the flow of current in the circuit so the equivalent inductance, L_T becomes equal to ( L ± M ) ÷ 2.

Example No2

Two inductors whose self-inductances are of 75mH and 55mH respectively are connected together in parallel aiding. Their mutual inductance is given as 22.5mH. Calculate the total inductance of the parallel combination.

Example No3

Calculate the equivalent inductance of the following inductive circuit.

Calculate the first inductor branch, L_A

Calculate the second inductor branch, L_B

Calculate the equivalent circuit inductance, L_EQ

Then the equivalent inductance is 15mH.

Inductors in Parallel Summary

As with the resistor, inductors connected together in parallel have the same voltage, V across them. Also connecting together inductors in parallel decreases the effective inductance of the circuit with the equivalent inductance of "N" inductors connected in parallel being the reciprocal of the sum of the reciprocals of the individual inductances. As with series connected inductors, mutually connected inductors in parallel are classed as either "aiding" or "opposing" this total inductance depending whether the coils are cumulatively coupled (in the same direction) or differentially coupled (in opposite direction).

Thus far we have examined the inductor as a pure or ideal passive component. In the next tutorial about Inductors, we will look at non-ideal inductors that have real world resistive coils producing the equivalent circuit of an inductor in series with a resistance and examine the time constant of such a circuit.