Parallel Connected Coils

Parallel Connected Inductors are said to be connected in parallel when both terminals are connected to each terminal of another inductor or inductor, respectively.

All parallel connected coils will have the same voltage value.

VL1 = VL2 = VL3 = VAB

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

parallel connected inductors
Parallel Connected Inductors

In previous series-bound inductors training, we found that the total inductee of the circuit, LT,was equal to the sum of all individual inductors added together. For parallel connected inductors, equivalent circuit inductive LT is calculated differently.

The sum of individual currents flowing from each inductor can be found using Kirchoff's Current Act (KCL), where we know from previous tutorials on IT = I1 + I2 + I3 and inductace that self-induced emk is given throughout an inductor: V = L di/dt

Then, by taking the values of the individual currents flowing from each inductor in our circuit above and replacing the i current with i1 + i2 + i3, the voltage in the parallel combination is given as follows:

parallel connected inductors

If we replace di/dt with v/L in the equation above, it returns:

parallel connected inductors

When connecting inductors in parallel, we can reduce this to give a final statement to calculate the total inductee of a circuit:

parallel connected inductors

Here, the mutual (1/Ln) value of individual inducings, such as calculations for parallel resistors, is collected in place of the induced ones themselves. But again, as with serially connected inducings, the above equation applies only when there is no mutual induced or magnetic connection between two or more inductors (they are magnetically isolated from each other). When there is coupling between the coils, the total inducing is also affected by the amount of coupling.

This calculation method can be used to calculate any number of individual inductees connected to each other within a single parallel network. However, if there are only two separate inductors in parallel, a much simpler and faster formula can be used to find the total inductace value:

parallel connected inductors

An important point to remember about inductors in parallel circuits is that the total inductee (LT)of any two or more inductors connected in parallel will always be LESS than the value of the smallest inductee in the parallel chain.

Parallel Connected Inductors Question Sample 1

Three inductors of 60mH, 120mH and 75mH, respectively, are connected in a parallel combination without mutual inducing between them. Calculate the total inductee of the parallel combination in millihenry.

parallel connected inductors

Interconnected Inductors

When inductors are connected in parallel with each other so that one's magnetic field is connected to the other, the effect of mutual inducing increases or decreases the total inductee depending on the amount of magnetic coupling that exists between the coils. The effect of this mutual inducing depends on the distance between the coils and their orientation to each other.

Inductors that are mutually connected in parallel can be classified as "auxiliary" or "opposite" rather than total inductive, with parallel auxiliary connected coils increasing total equivalent inductives compared to coils with zero mutual inducings and parallel opposing coils that reduce total equivalent inductiveness.

Cross-paired parallel coils can be shown as auxiliary or connected in a contrasting configuration using polarity points or polarity marks, as shown below.

Parallel Auxiliary Inductors

parallel connected inductors

The voltage on the two parallel auxiliary inductors above must be equal because they are parallel, so the two currents, i1 and i2, must change so that the voltage between them remains the same. Then the total inducing for two parallel auxiliary inductors, LT is given as follows:

parallel connected inductors

Where: 2M represents the effect of the L1 coil on L2 and similarly the effect of the L2 coil on L1.

If the two induced inducts are equal and the magnetic coupling is excellent, as in a toroidal circuit, the equivalent inductive of the two parallel connected inductors is L in LT = L1 = L2 = M. However, if the mutual inducing between them is zero , the equivalent induced will be L ÷ 2, as for two self-induced inductors in parallel.

If one of the two coils was inverted relative to the other, we would have two parallel opposite inductors, and the mutual inductance, the M located between the two coils, would have a canceling effect on each coil instead of an auxiliary effect, as shown below.

Parallel Counter-Inductors

parallel connected inductors

For two parallel opposing inductors, the total inducing, LT is given as follows:

parallel connected inductors

This time, if the two induced values are equal and the magnetic coupling between them is excellent, the equivalent induced, as well as the self-induced emk along the inductors, will be zero, since the two inductors cancel each other out.

This is because the total mutual flow produced between them is zero, as the two currents flow from each inductor in rows i1 and i2, since both of the two flows produced by each inductor are equal in size but in opposite directions.

Then, the two coils are effectively short-circuited to the flow of the current in the circuit, so that the equivalent inducing equals LT ( L ± M ) ÷ 2.

Parallel Connected Inductors Question Sample 2

The two inductors, whose inducings are 75mH and 55mH respectively, are connected by a parallel auxiliary connection. Subsequently, their mutual inducing was calculated as 22.5mH. Calculate the total inductee of the parallel combination:

parallel connected inductors

Parallel Connected Inductors Question Sample 3

Calculate the equivalent inductive inductive of the inductive circuit below.

parallel connected inductors
  • Calculate the first inductor branch (LA),(in parallel with L5 inductor L6 and L7 inductors)
parallel connected inductors
  • Calculate the second inductor branch (LB),(in parallel with L3 inductor l4 and LA inductors)
parallel connected inductors
  • Calculate LEQ equivalent circuit induct,(L1 inductor parallels L2 and LB inductors)
parallel connected inductors


As with resistance, inductors that connect together in parallel, between them V have the same voltage. In addition, connecting inductors in parallel reduces the effective induction of the circuit, the equivalent induct of "N" inductors connected in parallel is the opposite of the sum of the reciprocals of individual inductees.

As with serially connected inductors, inductors that are connected in parallel are classified as "auxiliary" or "opposite" rather than this total inductive, depending on whether the coils are cumulatively connected (in the same direction) or differentiatedly connected (in the opposite direction).

So far we have examined the inductor as a pure or ideal passive component. In the next lesson about inductors, with resistance, we will serially connect the inductor and examine the time constant of such a circuit.