All the transformer theory I have reviewed uses sinusoidal waveforms, and concludes that a true sinusoidal input is optimal. But I'm wondering whether a sawtooth waveform would actually be optimal in practice?

Theory depends frequently on the differentiability of the waveform, so it's "convenient" to neglect sawtooths because they are not differentiable when voltage changes direction. However, I assume that:

  1. It would be relatively easy in practice to produce an input from a DC power source that is a clean sawtooth.
  2. A transformer would not suffer from the "harmonic noise" that might be suggested by an abstract attempt to approximate a sawtooth as a sum of higher-frequency sine curves.

On the other hand, one reason I imagine the sawtooth might not be optimal is the magnetic hysteresis of transformer cores: If that's large enough then the slow reversal of a sine wave could in fact optimize the transmission of power.

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    Optimal in what sense? An "ideal" transformer is already optimal, so obviously you're thinking of various secondary or parasitic effects in "real" transformers. Which ones do you have in mind, and why does a sawtooth address those specifically? – Dave Tweed Mar 5 '16 at 19:43
  • Why do you say that a transformer wouldn't have issues with the higher harmonics? A sawtooth (technically) implies an infinite sum of harmonics, and therefore requires a transformer with infinite frequency response. Any realistic transformer will not be able to handle higher harmonics, and there will be some loss there. – uint128_t Mar 5 '16 at 19:45
  • It's arguable that a square wave is optimal for a transformer, so 'always' working at maximum voltage and maximum current. The abrupt voltage step does not trouble the core magnetics, only the parasitic capacitances of the winding. – Neil_UK Mar 5 '16 at 20:23
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    @DaveTweed et. al. - Transformers don't "work" when voltage is held constant. Since the change in voltage drives the transformation it seems like the "first guess" would be that you want a constant change in voltage, and anything else would be sub-optimal. In fact, we know that in practice higher-frequency inputs allow for more efficient transformers (up to a point). Whatever rate of voltage change a transformer is optimized for, a sawtooth maintains that rate over its entire period, whereas a sine wave only hits it for an instant every half cycle. – feetwet Mar 5 '16 at 21:21
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    @Neil_UK - Now that is an excellent explanation! You should post that as an answer. – feetwet Mar 6 '16 at 17:42
up vote 6 down vote accepted

The optimal input waveform, as far as the core flux is concerned, is a square wave. It turns out that while sine wave is a sub-optimal, because it's the 'natural' waveform for generation and for 3 phase power distribution, it is used as is.

The key to understanding inductors and transformers is to recognise that voltage and flux are linked, not proportionally, but through an integral, so flux = \$\int\! V \, \mathrm{d}t\$ and voltage = d(flux)/dt

If we apply a finite voltage to the primary of a transformer, the flux will begin to slew, at a rate proportional to the voltage. The changing flux will induce a voltage in the secondary, which can be used to drive a current through the load.

If we had magical materials to hand, so a core of unobtanium with infinite permeability usable to infinite flux, we could leave this DC voltage connected indefinitely, and the transformer would continue to transmit power, while the flux ramped continually upward, the magnetising current held to reasonable levels by the unreasonable core permeability.

Unfortunately, we have the practical problem that we need to use real materials, iron or ferrite, for the core, and these will eventually saturate. When that happens, the flux stops increasing, and the output voltage collapses to zero.

In order to avoid saturation, we reverse the input voltage just before the core saturates. With the reversed voltage, the flux slews back down again, through zero, and approaches saturation in the opposite direction. Reverse and repeat.

If the core were the only consideration, we could reverse the input voltage as quickly as we liked. All it would do at the core is alter (very quickly) the direction that the flux was slewing, the flux itself would be continuous. In practice, nothing can generate a very sharp edge, the capacitance of the windings would not permit a very sharp edge, and as long as the edge is 'much steeper' (say 10x steeper) than a sine wave, we will have the bulk of the efficiency improvement that we are looking for over a sine waveform.

Running a transformer at best efficiency, we will slew the flux from near -max flux to near +max flux. As the change in flux is the integral of voltage, any waveform that does this will have the same mean voltage over the half cycle. If a sine wave with a peak of 1 swings the flux over this range, then a square wave would need to have a peak of around 0.7 to have the same average.

The power delivered by the transformer into a resistive load depends on both the output voltage and the output current. With sine excitation, the current and voltage vary between 0 and max during the cycle. With square wave excitation, the voltage and current stay at the maximum level for essentially the whole cycle, so have a slightly higher power throughput.

Iron transformers on power networks tend to use the approximately sine-wave voltage transmitted. High frequency ferrite transformers, powered by switching devices from a DC bus, use the naturally produced square waves. Much more significant than the input waveform is the operating frequency, and the size of the core. It is almost never worth the bother of changing the input waveform from "what's easy" for the small gain that would result.

Let's take your two 'advantages' and see if they hold:

It would be relatively easy in practice to produce an input from a DC power source that is a clean sawtooth.

Actually, that's very hard, especially if you need both a positive and a negative half. Nearly every active component has some non-linearity involved. Capacitors have an exponential voltage curve, inductors (of which transformers are a special case) have an exponentional current curve. Trying to make a linear increasing voltage or current is difficult. Even more difficult is to very quickly reverse the polarity at the falling edge of the sawtooth. Something your transformer won't like.

A transformer would not suffer from the "harmonic noise" that might be suggested by an abstract attempt to approximate a sawtooth as a sum of higher-frequency sine curves.

I've got a surprise for you: a perfect sine curve does not have harmonics. Any other waveform is made of a series of sines (by the Fourier theorem), so definetely suffers from harmonics.

  • Good clarification on the first point. I think you missed my second point: You can choose to analytically approximate (to an arbitrary epsilon) a sawtooth using a Fourier Transform, but is it certain that because an analytic approximation has harmonics (or any other characteristic) that the non-analytic function has those characteristics? Obviously one characteristic the sawtooth does not have that a Fourier Transform does is differentiability at all points. – feetwet Mar 6 '16 at 0:01

"Theory depends frequently on the differentiability of the waveform, so it's "convenient" to neglect sawtooths"

Not because it's convenient, but the only way of doing it is using a waveform that do not change between differentiations. You can take a look at Faraday's Law of induction. Differentiation is not optional.

One reason (the one I can think of) for using sinusoidal waves is that, although you can change phase and amplitude, you will always get the same waveform. You can take sin(x) as an example. It's derivative is cos(x). The latter's derivative is -sin(x). You can see that, as many times you derivate the waveform, it will always retain it's sinusoidal shape.

You can't do that with sawtooth. If you could derivate a sawtooth wave you would get a square wave, which can't drive another transformer. I guess the use of sinusoidal waves just makes everything easier. You can have many transformers in between the energy generation and the consumption.

Also,

abstract attempt to approximate a sawtooth as a sum of higher-frequency sine curves

That might seem abstract enough for real life but it is a thing. Every time you listen to music and tell between the violin and the guitar, that's because a sum of particular higher-frequency sine curves (overtones) produces that single sound you can later recognize as a distinct instrument.

As I haven't studied transformer theory yet, I won't assume anything else.

  • I think you're restating my premise: You can't analytically apply Faraday's Law to undifferentiable curves. I'm not negating that Law, but rather questioning whether an optimal solution to a system that involves it need be analytic. – feetwet Mar 5 '16 at 21:27

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