An article analyzing the harmonic induced potential of motor windings

In the previous issue, we discussed the fundamental induced potential of the motor winding. Now let’s talk about the harmonic induced potential of the motor winding. In actual AC motors, in addition to the fundamental wave, there are also a series of higher harmonics in the induced potential waveform of the winding. The higher harmonics in the no-load induced potential are mainly caused by two reasons. One is that the main pole magnetic field is non-sinusoidally distributed along the space; the other is caused by the slots on the stator surface. This issue will first discuss the potential higher harmonics caused by the first reason, that is, the harmonic potential caused by the non-sinusoidal distribution of the main pole magnetic field.

1 Spatial distribution of the main pole magnetic field

In a salient pole synchronous motor, the main pole magnetic field is usually distributed as a flat-top wave along the circumference of the armature surface, as shown in Figure 1. Assuming that the stator surface is smooth and not slotted, Fourier decomposition of this flat-top wave will yield a fundamental wave and a series of harmonics. Since each magnetic pole is symmetrical with respect to its center line, and the N pole and S pole are symmetrical in opposite directions with respect to their dividing lines, the main pole magnetic field contains only odd-order harmonics, i.e., only υ=1, 3, 5, 7... harmonics.

At this point, students often ask a question, why are there only odd harmonics? Why are there no even harmonics? Let's first teach these students some math knowledge. According to advanced math knowledge, any periodic function that meets certain conditions can be decomposed into a series of sine functions and cosine functions of different frequencies. This is the legendary Fourier decomposition. The Fourier decomposition of certain functions has some specific rules, among which the Fourier decomposition of odd harmonic functions and even harmonic functions has distinct characteristics. Let's talk about the concepts of odd harmonic functions and even harmonic functions first. Note that they are odd harmonic functions and even harmonic functions, not odd functions and even functions!

The so-called odd harmonic function means that if the image of a periodic function is translated along the horizontal axis (independent variable) by half a period, it is symmetrical with the original image relative to the horizontal axis image, that is, it satisfies:

f(x)=-f(x+T/2) (1)

Where T is the period of the function, the function is called an odd harmonic function or a half-wave symmetric function.

The so-called even harmonic function means that if the image of a periodic function is completely overlapped with the original function waveform after being translated along the horizontal axis for half a period, it satisfies:

f(x)=f(x+T/2) (2)

The function is called an even harmonic function or a half-period overlapping function.

The Fourier decomposition expansion of an odd harmonic function contains only the odd harmonic components of the sine and cosine terms, but not the even harmonic components; the Fourier decomposition expansion of an even harmonic function contains only the even harmonic components of the sine and cosine terms, but not the odd harmonic components. In addition to odd and even harmonic functions, there are also periodic functions that are neither odd nor even harmonic functions (periodic functions that do not meet the above definitions of odd and even harmonic functions), and their Fourier decomposition expansions contain both odd and even harmonic components of the sine and cosine terms.

The above conclusion can be proved by the calculation formula of the coefficients of each harmonic term in Fourier decomposition. Considering that we are not here to talk about advanced mathematics, we will not deduce and prove it in detail. Here we will only explain the principle in a popular way so that everyone can understand. It is not difficult to see from the waveforms of odd harmonics (1, 3, 5, 7... harmonics) in Figure 1 that the waveforms of all odd harmonics are symmetrical with the original waveform relative to the horizontal axis after half a period of translation along the horizontal axis, that is, odd harmonic functions are all odd harmonic functions; similarly, you can draw the image of even harmonic functions yourself, and you can find that even harmonics are all even harmonic functions. If only odd harmonic functions are added, the sum obtained must be an odd harmonic function; if only even harmonic functions are added, the sum obtained must be an even harmonic function. If the sum obtained by mixing odd harmonic functions and even harmonic functions is neither an odd harmonic function nor an even harmonic function. Therefore, the Fourier decomposition of odd harmonic functions must only include odd harmonics; the Fourier decomposition of even harmonic functions only includes even harmonics. For the main pole magnetic field distribution in the motor, since the N and S poles are symmetrically distributed and the magnetic field directions are opposite, it is obviously an odd harmonic function, so the main pole magnetic field only includes odd harmonics. If each pole is symmetrical relative to its own pole center line, then the initial phase angles of the contained odd harmonics are also the same, that is, the waveforms of each harmonic start from the initial phase angle position of the fundamental wave, as shown in Figure 1.

After catching up on math knowledge, let's talk about the higher harmonics of the main pole magnetic field. As mentioned before, the main pole magnetic field contains a series of spatial odd harmonics. When the main pole rotates, the fundamental wave of the main pole magnetic field and this series of odd harmonics rotate with the main pole. Therefore, the speed of all harmonic magnetic fields is the same as the fundamental wave magnetic field speed, which is equal to the synchronous speed n1. As can be seen from Figure 1, the number of pole pairs of the υ subharmonic magnetic field is υ times that of the fundamental wave, and the pole pitch is 1/υ of the fundamental wave, that is, the higher harmonic magnetic field has the following characteristics:

nυ=n1

pυ=υ·p (3)

τυ=τ/υ

2 Higher harmonics of induced potential in windings

Like the fundamental wave, when the spatial harmonics of the main pole magnetic field rotate at the synchronous speed, they will also induce a harmonic potential with a frequency of fυ in the stator winding. Pay special attention! The harmonics of the induced potential are time harmonics! ! ! The calculation method of the harmonic potential is similar to that of the fundamental wave. The frequency of the potential harmonics is:

fυ=pυ•n1/60=υ•p•n1/60=υ•f1 (4)

The effective value of the harmonic potential is also calculated according to the famous 4.44 formula, namely:

Eφυ=4.44•fυ•Kdpυ•W•Φυ (5)

Where: Φυ is the magnetic flux of the υth harmonic.

Φυ=(2/π)•Bυ•τυ•l (6)

Kdpυ is the winding coefficient of the υth harmonic.

For the υ-th harmonic, the spatial electrical angle between the distributed coils is υ•α, and the potentials induced by them also differ in time by an electrical angle of υ•α; the distance between the two coil sides of the short-distance coil to the fundamental wave is Y1, and the distance to the υ-th harmonic is υ•Y1, so υ•α and υ•Y1 are used to replace α and Y1 in the formula of the fundamental wave short-distance coefficient and the distribution coefficient, respectively, to obtain the short-distance coefficient and distribution coefficient of the υ-th harmonic, namely:

Kdpυ=Kdυ•Kpυ

Kpυ=sin[υ•(Y1/τ)•90º] (7)

Kdυ=[sin(υ•q•α/2)]/[q•sin(υ•α/2)]

3 Effect of stator slotting on induced potential

The above analysis is based on the assumption that the stator surface is smooth and not slotted. In actual motors, the stator windings are usually embedded in the stator slots. Due to the influence of the stator slots, the air gap permeability per unit area becomes uneven. The air gap corresponding to the position of the teeth is smaller, and the permeability per unit area is larger; the air gap corresponding to the position of the slots is larger, and the permeability per unit area is smaller. It is the periodic change of the air gap permeability caused by the tooth slots that distorts the original magnetic flux density distribution, as shown in Figure 2. Figure 2a shows the rotating magnetic potential with a sinusoidal distribution, and Figure 2b shows the distortion of the air gap magnetic field due to the "modulation" effect of the tooth slots after slotting, and its magnetic flux density wave becomes a tooth permeability harmonic magnetic field superimposed on the sine fundamental wave. What effect will this distorted magnetic field have on the induced potential in the winding?

The analysis of the effect of stator slots on the induced potential of windings is very complicated, especially the quantitative calculation involves complicated mathematical derivation. I know that many students feel overwhelmed when math is mentioned. It doesn’t matter. Let us first state the conclusion of the effect of stator slots on the induced potential. If you are interested in the following mathematical derivation and proof, you can read it carefully. If you find it too confusing, you can ignore the derivation and proof part. The important thing is that you must remember the following conclusion!

Important conclusions! ! ! The effect of stator slotting on induced potential is like "you reap what you sow"! Although stator slotting distorts the waveform of the air gap magnetic field, its effect on the induced potential in the winding only affects the amplitude of the fundamental and harmonic induced potentials, and does not affect the frequency (order) of the harmonics! In other words, regardless of whether slotting is performed or not, the harmonic order of the potential in the winding corresponds to the harmonic order in the air gap magnetic potential. In the case of no slotting, the harmonic potential of the same order will be generated in the winding if there is a harmonic magnetic field of the same order in the air gap magnetic potential; the harmonic potential of the same order will not be generated in the winding if there is no harmonic in the air gap magnetic potential. After slotting, the harmonic potential of the same order will be generated after slotting if there is no harmonic magnetic potential before slotting; the harmonic potential of the same order will not be generated after slotting if there is no harmonic in the magnetic potential before slotting. Therefore, the harmonics of magnetic potential are like a kind of "genetic gene". The stator slots will not change this "genetic gene" to increase or decrease the number of harmonics induced in the winding potential. The stator slots only affect the size of the harmonics, not the number of harmonics. Therefore, we figuratively call this conclusion "you reap what you sow"!