Basic Electronics On The Go - Inductors in Parallel

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Inductors are said to be connected in parallel when two terminals of an inductor respectively connected to each terminal of other inductors or inductor. Similar to the parallel connection of resistors, the total inductance in parallel connection of inductors is somewhat lesser than smallest inductance of an inductor in that connection.

When the inductors are connected in parallel, the current flow through each inductor is not exactly equal to the total current, but the sum of each individual current through the  parallel inductors gives the total current (as it divides among parallel inductors).

If the current flow through the each inductor is less than the total current, the magnetic field generated by each inductor is also less than that of field generated by total current through it.

In case of resistors in parallel, most of the current flows through the smallest resistor as it offer the least opposition to the current flow than larger resistor.

Likewise, if the inductors are connected in parallel, the current chooses the least opposition path of the inductor when current in that circuit is decreased or increased while each inductor individually opposes that change (increase or decrease of current).

Inductors Connected in Parallel (Without Magnetic Coupling)

As we discussed above, one end of inductors is connected to a node and other ends of inductors are collectively connected to another node in parallel connection. The parallel connection of n inductors is shown in below figure.

Consider that there is no magnetic coupling between inductors and hence the total inductance equal to the sum of reciprocals of the individual inductances. Let us discuss how this statement can be obtained.

We know that, in a parallel network the voltage remains constant and the current divides at each parallel inductor. If IL1, IL2, IL3 and so on ILn are the individual currents flowing in the parallel connected inductors L1, L2 and so on Ln, respectively, then the total current in the parallel inductors is given by

ITotal = IL1 + IL2 + IL3 . . . . + In

If the individual voltage drops in the parallel connection are VL1, VL2, VL3 and so on VLn, then the total voltage drop between the two terminals VT is

VTotal = VL1 = VL2 = VL3 . . . . = Vn

The voltage drop in terms of self inductance can be expressed as V = L di/ dt. This implies total voltage drop,

VT = LT di/dt

⇒ LT  d/dt (IL1 + IL2 + IL3 . . . . + In)

⇒ LT ( (di1)/dt + (di2)/dt + (di3)/dt . . . .)

Substituting V / L in place of di/dt, the above equation becomes

VT = LT (V/L1+ V/L2 + V/L3 . . . .)

As the voltage drop is constant across the circuit, then v = VT. So we can write

1/LT = 1/L1 + 1/L2 + 1/L3 . . . . .

This means that the reciprocal of total inductance of the parallel connection is the sum of reciprocals of individual inductances of all inductors. The above equation is true when there is no mutual inductance between the parallel connected coils.

For avoiding complexity in dealing with fractions, we can use product over sum method to calculate the total inductance of two inductors. If two inductors are connected in parallel, and if there is no mutual inductance between them, then the total inductance is given as

LT = (L1× L2)/(L1+ L2)

Mutually Coupled Inductors in Parallel

When there exist magnetic coupling between the inductors, the above derived formula for total inductance must be modified because total inductance can be more or less depending on the magnetic field directions from each inductor. The magnetic flux produced by the parallel connected inductors will link with 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.

Consider that two inductors are connected in parallel with self inductances L1 and L2, and which are mutually coupled with mutual inductance M as shown in the  figure.below

Parallel Aiding Inductors

Consider the figure (a), in which inductors L1 and L2 are connected in parallel with their magnetic fields aiding. The total current through the circuit is given as

i = i1 + i2

di/dt = (di1)/dt + (di2)/dt …………. (1)

The voltage across the inductor or parallel branch is given as

V = L1 (di1)/dt + M (di2)/dt or L2 (di2)/dt + M (di1)/dt

L1 (di1)/dt + M (di2)/dt = L2 (di2)/dt + M (di1)/dt

(di1)/dt (L1– M) = (di2)/dt (L2– M)

(di1)/dt = (di2)/dt ((L2 – M))/((L1 – M)) …………. (2)

Substituting equation 2 in equation 1, we get

di/dt = (di2)/dt ((L2– M))/((L1– M)) + (di2)/dt

di/dt = (di2)/dt { (L2– M))/((L1– M)) + 1} …………. (3)

If LT is the total inductance of the parallel inductor circuit , then voltage is given by

V = LT di/dt

LT di/dt = L1 (di1)/dt + M (di2)/dt

di/dt = 1/ LT { L1 (di1)/dt + M (di2)/dt }

Substituting equation 2 in above equation, we get

di/dt = 1/ LT { L1 (di2)/dt (L2– M))/((L1– M)) + M (di2)/dt }

di/dt = 1/ LT { L1 (L2– M))/((L1– M)) + M }(di2)/dt …………. (4)

Equating equations 3 and 4, and simplifying the results, we get

LT = (L1 L2– M2)/(L1+ L1 )-2M)

Here 2M represents the magnetic flux of L1 on L2 or L2 on L1.

Parallel Opposing Inductors

Similarly, if we consider the figure (b), in which inductors L1 & L2 are connected in parallel with their magnetic fields opposing, the total inductance is given as

LT = (L1 L2– M2)/(L1+ L2)+ 2M)

Summary

• When two terminals of the  inductor are connected respectively to other inductor or inductors terminals, then that connection is referred to as “a parallel connection of inductors”.
• When the fluxes produced by individual inductors are in the same direction, the mutual inductance will be increased; then these coils are called “Aiding” coils. Total inductance for aiding coils is LT = (L1 L2– M2)/(L1+ L2 )-2M).When the fluxes produced by individual inductors are in the opposite direction of magnetic flux, the mutual inductance will be decreased; then these coils are called “opposing” coils. Total inductance for aiding coils is LT = (L1 L2– M2)/(L1+ L2 )+2M)