Everything You Ever Wanted to Know about Coupled Inductors but Were Too Afraid to Ask!


About the writer: Harvey Morehouse is a contractor/consultant with many years of experience using circuit analysis programs. His primary activities are in Reliability, Safety, Testability and Circuit Analysis. He may be reached at harvey.annie@verizon.net. Simple questions for which I know the answer are free. Complex questions, especially where I am ignorant of the answers, are costly!!!

Summary: An often overlooked and often neglected element is the inductor coupling device. There is a device model for two mutually coupled inductors, but only two, and the device symbol shows two inductors.

Often it is convenient to couple two inductors that are independently locatable with respect to each other. And often it is desirable to more than two inductors coupled together. While the use of ideal transformers with leakage inductances and a magnetizing inductance will allow multiple windings to be coupled together, any topology change can only be made with new symbology.

This article discusses the use of the 'K' inductive coupling element which alleviates this difficulty.

'K' inductive coupling element:

The 'K' inductive coupling element is shown in Figure 1 following:


Figure 1
K element symbol

Interestingly enough, there are NO pins on this device model. The coupled inductors are passed to the device as parameters, as is the coupling coefficient between the two inductors.

This device links two inductors. But ALL coupled inductors must be 'coupled' to all other coupled inductors via a 'K' element. Thus, there must be 2n-1 of these blocks if there are 'n' coupled inductors. For 2 coupled elements, 2 'K' elements are required, for 3 coupled elements, 4 'K' elements are required.

This device is a carry-over from SPICE2. Some SPICE variants have made modifications to this device to add some capabilities. However, this is what is present in B2SPICETM .

The parameters window for this device is shown in Figure 2 following:


Figure 2
'K' device model parameter entry window

It is virtually impossible to make a perfectly universal parameter entry window, as the possible parameters that could be entered into a device could be mechanical units, length, weight, temperature, or constants (within a simulation), unit-less quantities - the possibilities are almost endless.

But it does make sense. One enters the name of the inductors to be coupled, and the units of the inductors entered are in Henries.

Now, the symbology shown in Figure 1 shows two inductors with polarity marking. Devices such as resistors, inductors and capacitors do have inherent polarities within SPICE. By this it is meant that IF one were to examine the current through a device, or the voltage across a device, such as i(L1), or v(R2), the sense of the reported currents through, and the voltages across many devices is dependent on their orientation within a circuit.

Consider the test circuit of Figure 3 following:


Figure 3
'K' device model

In Figure 3 the resistors are all 1K, and the inductors are all 1mHy. V1 is a 1kHz, 1V sine wave source. The coupling of the inductors, one to each other, is unity.
With the connection shown, one would expect that the same voltage would be induced into L2, L3 and L4. A transient graph of the circuit is shown in Figure 4 following:


Figure 4
'K' device model test

The bold curve is vin. The remainder of the waveforms save for v5 are slightly delayed, The variant curve is v5, reversed in sense as the polarity of L4 is reversed compared to the other inductors. It was deliberately flipped horizontally, end for end.

If the sense of the inductors is all the same, but one output is negative, then each coefficient of coupling from a 'positive output' inductor to a 'negative output' inductor should be negative. The coupling factor between two inductors of same sense circuits should be positive.

Now all of this begs the question of how to determine the coupling factor between windings, which is always a problem. You should read the Beigebag Resource section articles on transformers, particularly the discussion of perfect and ideal transformers and their differences. Every pair of coupled inductors with a non-unity, non-zero coupling coefficient does have an equivalent leakage inductance. They can be converted one to the other form.

Now this can lead to a discussion of the use of coupled inductors at the input side of a full wave rectifier to reject common mode noise, the effects of different inductor values connected among different levels of a DC-DC converter output and more. The purpose was to introduce the device.

Conclusions:

The 'K' factor device is a convenient way to mutually couple inductors without the schematic display limitations of a transformer device model.

References:

associated circuit files (v5)