Analysis of the synthetic magnetic potential of three-phase windings

In the previous issue, we discussed the magnetic potential generated when a single-phase winding is passed through a single-phase alternating current. This issue continues to discuss the magnetic potential generated when a multi-phase symmetrical winding is passed through a multi-phase symmetrical alternating current. It should be stated that the content described in this issue is only applicable to integer-slot multi-phase symmetrical windings, not fractional-slot windings. We take three-phase symmetrical windings as an example for analysis, and then generalize it to any m (m≥2)-phase symmetrical winding. There are many ways to analyze the synthetic magnetic potential of three-phase windings. Here we mainly introduce the three commonly used methods of analytical method, graphical method and double rotation theory method to analyze the synthetic magnetic potential of three-phase windings.

1 Analytical method

The three-phase symmetrical windings are 120 degrees apart in space. If a three-phase symmetrical alternating current is passed through the three-phase symmetrical windings, the three-phase currents are also 120 degrees apart in time. If the origin of the spatial coordinates is taken at the phase axis of phase A (the symmetrical center line of the winding); in terms of time, the time when the current A reaches its maximum value is taken as the starting point of time (i.e., iA is at its maximum value at t=0), then the expression of the fundamental wave of the pulsating magnetic potential generated by each of the three-phase windings is:

Using the product-to-difference formula of trigonometric functions cosα•cosβ=(1/2)[cos(α+β)+cos(α-β)], the expressions in equation (1) are decomposed into:

The synthetic fundamental magnetic potential of the three-phase winding should be the sum of the three formulas in formula (2). It can be seen from formula (2) that the first term in each formula is the same, and the second term is three cosine functions with a difference of 120 degrees. The sum of the three equals 0, so the synthetic fundamental magnetic potential of the three-phase winding is only the sum of the three first terms.

f1(θ,t)=fA1+fB1+fC1=(3/2)•FΦ1•cos(ωt-θ)=F1•cos(ωt-θ) ⑶

in:

F1=(3/2)•FΦ1=(3/2)•0.9•(W•Kdp1/p)•IΦ=1.35•(W•Kdp1/p)•IΦ ⑷

The above formula (3) is the expression of the synthetic fundamental magnetic potential of the three-phase winding.

Next, we will introduce how to understand the meaning of this expression. From formula (3), we can see that: when time t=0, f1(θ,0)=F1•cos(-θ), and the wave crest is at the phase axis position of phase A; when time t=t1, f1(θ,t1)=F1•cos(ωt1-θ), and the wave crest is at the position of θ=ωt1. Plotting the magnetic potential waves at these two moments for comparison, we can see that the amplitude of the magnetic potential has not changed, but f1(θ,t1) has advanced an angle β compared to f1(θ,0), β=ωt1, as shown in Figure 1. As time goes by, the β angle continues to increase, that is, the magnetic potential wave continues to move in the +θ direction, so f1(θ,t) is a sinusoidal traveling wave with a constant amplitude. Since the stator inner cavity is cylindrical, f1(θ,t) is essentially a rotating magnetic potential wave with a constant amplitude that continuously moves along the circumference of the air gap, as shown in Figure 2.

As mentioned above, after t1 time, the rotating magnetic potential wave rotates through an electrical angle of β=ωt1, then the electrical angular velocity of the rotating magnetic potential wave is ω=β/t1. Since each circle of the air gap is p•2π electrical angle, the mechanical speed of the rotating magnetic potential wave is:

n=ω/(p•2π)=f/p (revolutions/second)

=60f/p (revolutions per minute)

=ns ⑸

Where: f is the frequency of the stator current; p is the number of pole pairs; ns is the synchronous speed. It can be seen from formula (5) that the speed of the stator fundamental wave rotating magnetic potential is the synchronous speed. In addition, it can be seen from formula (3) that the above derivation process is the rotation direction obtained when the middle symbol in (ωt-θ) is "-". If the middle symbol is "+", the same method can be used to deduce that the rotation direction is the opposite direction (the derivation process is omitted). Therefore, the middle symbol in (ωt-θ) represents the direction of rotation of the rotating magnetic potential wave. The "-" sign represents the direction from the A phase axis to the B phase axis and then to the C phase axis..., which is called forward rotation. If the symbol is "+", it means that the direction is A-C-B, which is called reverse rotation. When the current of a certain phase reaches the maximum value, the rotating magnetic potential just turns to the position of the phase axis.

The above analysis shows that when a three-phase symmetrical winding is connected with a three-phase symmetrical current, the fundamental wave synthetic magnetic potential generated is a rotating magnetic potential wave with constant amplitude, sinusoidal distribution, and rotating at a synchronous speed. The amplitude F1 of the magnetic potential wave is 3/2 times the amplitude of the single-phase magnetic potential. The direction of rotation depends on the phase sequence of the three-phase current. If the phase sequence of the three-phase current is A-B-C (positive sequence), the direction of rotation is from the A-phase axis to the B-phase axis and then to the C-phase axis... (forward rotation); if the phase sequence of the three-phase current is A-C-B (negative sequence), the direction of rotation is from the A-phase axis to the C-phase axis and then to the B-phase axis... (reverse rotation). Since the amplitude of the rotating magnetic potential wave is constant, the trajectory of the magnetic potential wave amplitude F1 is a circle, as shown in Figure 2. We call this magnetic potential wave and the corresponding magnetic field circular rotating magnetic potential and circular rotating magnetic field.

2 Graphical method

The above conclusions can also be obtained from the graphical method. Figure 3 shows the graphical synthesis process of the three-phase magnetic potential.

The five figures on the left in Figure 3 represent the three-phase current phasors at five different moments, the five figures in the middle represent the fundamental magnetic potential and the synthetic magnetic potential generated by each phase winding at the corresponding moment on the left, and the five figures on the right represent the corresponding magnetic potential space vector diagram. In the figure, the three-phase windings A, B, and C are represented by three concentrated coils, and the fundamental pulse magnetic potential generated by each phase winding is represented by a pulsed sine wave in the middle figure and by the corresponding space pulse vector in the right figure.

Figure 3(a) shows the moment of ωt=0, when the current of phase A reaches its maximum value, and the amplitude of the magnetic potential of phase A also reaches its maximum value FΦ1. At this time, the instantaneous value of the current of phases B and C is just half of the amplitude of the negative phase current, that is, iB=iC=-(1/2)Im. Correspondingly, the magnetic potential of phases B and C is -(1/2)FΦ1. The three-phase composite magnetic potential wave can be obtained by adding the three magnetic potential waves point by point, as shown by the thick solid line in the middle of Figure 3(a). At this time, the amplitude of the three-phase composite magnetic potential coincides with the axis of the phase A winding, which is 3/2 times of FΦ1.

As time goes by, at the moment ωt=60º, the current of phase A gradually decreases from the maximum value to iA=(1/2)Im, the current of phase B changes from negative to positive to iB=(1/2)Im, and the current of phase C gradually changes to the negative maximum value iC=-Im. The magnetic potentials of each phase also change accordingly. In this process, the amplitude position of the synthetic magnetic potential will approach the axis of phase A from the axis of phase B, and at this time it just reaches the opposite position of the axis of phase C, as shown in Figure 3(b).

When ωt=120º, the current of phase B reaches its maximum value, and the amplitude of the synthetic magnetic potential turns to the axis position of phase B, as shown in Figure 3(c), and so on.

In this way, when a symmetrical positive sequence current is passed through the three-phase winding, the amplitude of the synthetic magnetic potential rotates in the positive direction. After one cycle of the three-phase current alternation, the synthetic magnetic potential rotates through 360 degrees of electrical angle, that is, 1/p turns, and its speed is exactly equal to the synchronous speed. Similarly, when a symmetrical negative sequence current is passed through the three-phase winding, the rotation direction of the synthetic magnetic potential will be reversed, which will not be repeated.

The above uses the graphical method to analyze that the fundamental wave of the magnetic potential generated by the three-phase winding passing through the three-phase symmetrical alternating current is a circular rotating magnetic potential with a constant amplitude, which is consistent with the conclusion drawn by the analytical method. In fact, the principle of the graphical method is very simple. In layman's terms, it is to pass current through the coils at different positions in sequence, and then the magnetic potential obtained must be where the current passes, and the magnetic potential will go to where the current passes. If this explanation is not easy for you to understand, then we will use a more vivid metaphor to describe this phenomenon. I believe that many students have played human waves in the football field. Even if you have not played it, you should have seen it on TV.

For each audience, they are distributed in different positions of the stands, just like multi-phase coils are distributed in different positions of the air gap circumference (three-phase windings are 120 degrees apart in electrical angle in spatial distribution). Each audience only needs to repeat the action of "standing up - sitting down" at his own position, that is, each audience only "pulses" at his own position. As long as the "pulsation" actions of adjacent audiences have a certain time difference, which is equivalent to the current flowing in each phase winding having a phase difference in time, then the total effect of all the audiences superimposed is that waves of people rotate on the stands of the stadium, and the speed of rotation obviously depends on the "pulsation" frequency of each person. Do you understand this explanation?