Transformers

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Transformers are ubiquitous in our world. Our electricity distribution networks rely on them. Most mains powered devices will include one – either as a mains-frequency transformer or in a switched-mode power supply. Like most things in electronics, they are superficially fairly simple, but quickly get complex once you get beneath the surface.

Figure 1 shows a physical and schematic view of an ideal transformer. You just need two sets of windings on a common magnetically permeable core.  A changing current in the primary winding induces a magnetic flux in the core, per Faraday’s law. This flux couples to the secondary winding and induces a current in the secondary.

In an ideal transformer we assume the windings have no resistance, the core is infinitely permeable, and the flux produced by the primary winding perfectly links with the secondary winding.  Faraday’s law tells us that the instantaneous voltage in a winding is proportional to the number of turns and the rate of change of the linking flux. The equation below shows us how this law proves that the primary/secondary voltage ratio is equal to the primary/secondary turns ratio.

The ideal transformer is lossless, it has perfect regulation under load, and would draw zero primary current if the secondary was open circuited. Where can I buy one?

As we all know the world is never perfect so let’s take a look at the assumptions and build a more realistic model.

Firstly, the windings will have some resistance, however low, and this will cause I2R losses in the windings and will impact voltage regulation. We could represent this as a resistance in series with the windings of our ideal transformer.

There will be losses in the core due to the B-H hysteresis of the core material and due to eddy currents. This is the reason most mains transformer cores are made out of thin laminations of special steel with a small hysteresis characteristic, as illustrated in Figure 2. We can represent these losses by a resistor across the primary winding (although they are actually related to the core itself and are not completely linear).

Next, our core does not have infinite permeability which means it requires some energy to push the flux around the core. This would be represented in our model as an inductance in across the primary winding. This inductance means that there will be a (hopefully small) primary current even when the secondary is open circuit. This is known as the magnetising current.

In a transformer, the secondary winding produces a flux opposing that produced by the primary winding. In the ideal case these fluxes are perfectly linked and there will be zero net flux in the core, so the windings will exhibit no inductance. However, in reality, there will be a small amount of flux that does not perfectly link both windings as shown in Figure 3. There is always some flux leakage so each winding will have a so-called leakage inductance in series with the ideal transformer.

Putting it all together gives us a model for a transformer shown in Figure 4a. There is a resistance and inductance in series with the primary and secondary windings of the ideal transformer. Rp and Rs are the winding resistances, Ls and Lp are the leakage inductances due to imperfect flux coupling. There will also be a resistance and inductance across the primary; RC, which represents the core losses (hysteresis and eddy current) and LM which represents the magnetising current.

Figure 4b shows the same circuit but with Rs and Ls transposed to the primary side via the square of the turns ratio. You will often see transformer models drawn this way. Note that this model ignores capacitance completely. This is not too much of an issue for mains transformers but absolutely something you want to consider for high-frequency transformers.

How does all this help? If you can get hold of some of this data it can be very useful for understanding the transformer’s voltage regulation for example, but most manufacturers don’t publish it. You can however measure a lot of these parameters with a bit of care. Primary and secondary resistance is easy. You can measure leakage inductance by measuring the inductance of one winding with the other shorted. You can then calculate magnetising inductance by measuring the winding inductance with the other winding open, since this measurement will be the sum of the leakage and magnetising inductance.

As ever, there is much more to this topic than presented here, and many engineers have made a whole career out of designing transformers. However, everyone who designs a circuit with a transformer should be aware of their operating principles and basic non-idealities.

References
“Transformer.” In Wikipedia, May 7, 2022. https://en.wikipedia.org/w/index.php?title=Transformer&oldid=1086705931.

“Electrical Steel.” In Wikipedia, May 6, 2022. https://en.wikipedia.org/w/index.php?title=Electrical_steel&oldid=1086400062.

Voltech. “Measuring Leakage Inductance.” Accessed May 9, 2022. https://www.voltech.com/support/technical-articles/measuring-leakage-inductance/.

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