Rotating magnetic fields

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1 - The discovery of the rotating magnetic field

In the fall of 1885 Galileo Ferraris, a physicist and engineer from Piedmont (1847-1897), one of the founders and the first president of the Italian Electrotechnical Association, showed publicly that by placing two coils perpendicular to each other and feeding them two strongly phase-shifted currents (almost in quadrature), a rotating magnetic field was produced in the space between them. The field could be detected, for example, with the device of Figure 1. When the coils carry currents with a large enough phase shift (in G. Ferraris's experiment the shift was about 1/5 of a period, 72°), the small copper cylinder hanging between O and O' tends to rotate, twisting the suspension wire. If the suspension is made with pivots and bearings, the cylinder rotates continuously. Ferraris experimented with alternating currents at about 40 Hz ("The reversals of the current were about 80 per second"), which corresponds to a theoretical field speed of 2400 revolutions per minute. In his experiments he reached speeds of up to about 900 rpm; beyond that, the primitive apparatus could not go. The phase shift between the currents was obtained by inserting a resistance of 15 to 18 Ω in coil 2BBB2' and making up for the lower current with a larger number of turns.  
In the same years Nikola Tesla (1856-1943), a Serbian-American physicist and engineer, was obtaining similar results. Galileo Ferraris's studies, which also described a small induction motor, were published in April 1888 in the journal "Il Nuovo Cimento" (download the paper).
On May 1 of the same year Tesla was granted the patent for the induction motor.   
With the introduction of three-phase power distribution, and thanks precisely to the discovery of the rotating magnetic field, the asynchronous (induction) motor quickly became widespread. It is still the most widely used rotating electrical machine in the world.
The rotating magnetic field is also the basis of the synchronous machine, which today is used mainly as a generator (alternator).

2 - The three-phase rotating magnetic field

The board on the left shows the stator ⁽¹⁾ of a hypothetical three-phase AC rotating machine. The gray part is the stator magnetic circuit which, together with the rotor magnetic circuit, amplifies and channels the magnetic fluxes the machine needs in order to work. The bodies projecting from the inner surface of the stator are the so-called poles, whose shaped ends are called pole shoes. They form the magnetic core of the windings (coils), each made of a number of turns of copper conductor. These windings, suitably connected, form the three phases of the system. A machine of this kind would be classified as a "salient-pole" or anisotropic machine. Although other solutions are used in practice (4), the salient-pole representation works well for teaching.

2.1 - Current direction convention

In the diagram, P₁ marks the start of phase 1 and F₁ its end. Let's follow the red conductor: we enter at P₁, wind N turns (3 in the example) around the pole, leave following the red conductor to the pole opposite the first one, wind another N turns around it in the same direction and then leave from F1. Each phase therefore uses two diametrically opposite poles. We follow the same procedure for the other two phases, in green and light blue. Suppose for now that we feed only phase 1 with the current I₁ and consider an instant when it reaches its positive peak value. We call the current "positive" if it is represented by a vector lying in the first or second quadrant, which therefore has a positive imaginary part⁽²⁾.
Note that the top pole becomes a magnetic south pole and the bottom one a north pole.
The arrows show the direction of the current in the windings, consistent with the so-called Maxwell's rule (or right-hand screw rule) shown in Figure 4.
By convention we call a current positive when it enters at the start of a winding and leaves at its end. The current I₁, at the instant shown, is therefore positive. When I₁ reaches its peak value in the opposite direction it will be considered negative: we will then see it enter at the end of the winding and leave at its start. The magnetic polarities obviously reverse over time, at the same frequency as the currents that produce the field. Note that as long as the vector I₁ stays in the 1st and 2nd quadrants, the current direction and the magnetic polarities of the phase do not change; what changes is the strength of the magnetic field B₁ produced by the current I₁ and represented by a vector along the polar axis of phase 1.
The same holds for the other two phases, carrying the currents I₂ and I₃.
The distance between one magnetic pole and the next pole of the same phase, measured along the inner circumference of the stator, is called the pole pitch τ (tau). If r is the inner radius of the stator, in the case shown τ = r π.  It is sometimes given as the central angle between two consecutive poles (here π radians), which amounts to setting r = 1. 

2.2 - The rotating field

In any case, it is important to note that the currents produce three alternating magnetic fields, whose strength varies sinusoidally over time, following the currents that produce them.
The instant-by-instant sum ⁽³⁾ of the three vectors gives a resultant vector that rotates at constant angular velocity while also keeping a constant amplitude.
It is easy to see that the resultant field, at the instant when one of the phases carries its peak current, points along the polar axis of that phase. This is why the flux produced by the resultant field can in turn be represented by a rotating vector in phasor diagrams referred to a single phase of the electrical machine.
It is just as easy to find the amplitude of the resultant field. If we again take the situation in which  I₁ has its positive peak value, B₁ takes its maximum value, say B_M, while B₂ and B₃ are equal to B_M/2, since they are produced by currents whose instantaneous value is half the peak value. Note also that B₂ and B₃ form angles of 60° with B₁. Given the symmetry of the two fields with respect to B₁, the resultant field clearly points along B₁ itself. All we need to do, then, is add to B₁ the components of B₂ and B₃ along B₁: replacing B₁, B₂ and B₃ with the instantaneous values B_M, B_M/2 and B_M/2:
It can be shown that in any other situation the magnitude of the resultant field always keeps the same value, equal to 1.5 times the peak value of the field produced by one phase.
The rotational speed of the field depends on the frequency f of the three-phase current system.
Calling n₁ the rotational frequency of the field, in revolutions per minute, from n₁/60 = f we get the relation: n₁ =  60 f, loosely called the "speed" of the field.
If the phase currents are equal, that is, if the system is balanced, the field has a constant magnitude over time. In this case we speak of a circular field. An imbalance in the currents makes the magnitude of B vary continuously: the motion of the field then looks irregular and is called an elliptical field, because the tip of the resultant vector B traces an ellipse.

2.3 - Reversing the direction of rotation

If we swap two phases, which means, for example, sending current I₃ into phase 2 and current I₂ into phase 3, the direction of rotation of the field reverses. The example simulates exactly this swap of I₂ and I₃. Any pair of phases can be swapped to reverse the field.

3 - Pole pairs and speed of the rotating magnetic field

We have already seen that the speed of the rotating field is set by the frequency of the stator currents and is directly proportional to it: doubling the frequency, for example, doubles the rotational speed. But another factor affects the speed: the number of pole pairs of the rotating magnetic field.

3.1 - Pole pairs of the magnetic field

To understand the idea of the field's polarity, consider the following.
The resultant field produced by a system like the one on the board on the left can be thought of as produced by two rotating magnetic poles that form in the stator. The field lines leave the north pole and enter the south pole. A magnetic field like this is called two-pole. We can confirm this by looking at the instantaneous magnetic polarities of the poles: the stator always appears divided into two zones, each containing only south poles or only north poles.
The rotational frequency of a two-pole field equals the frequency of the currents that produce it: if the current frequency is 50 Hz, the "speed" of the field is 50 revolutions per second, or 3000 revolutions per minute, as the formulas above also give.

Now suppose that the system is built as shown on the left. Each phase has four poles. They must be wound so that the currents create magnetic poles alternating in the sequence north-south-north-south. For phase 1, from the top pole near the start P₁, we move to the pole 90° clockwise, then to those at 180° and 270°. By paying attention to the direction in which the turns are wound around the poles, the current I₁ of phase 1 creates four alternating magnetic poles N-S-N-S arranged as in the picture. The same goes for phase 2 and phase 3.
The overall effect of the three currents is then to divide the stator, magnetically, into four zones, each acting in turn as a north pole and a south pole. There are therefore two north-south pole pairs, and the field is called 4-pole or four-pole. Here too we can think of the rotating field as a set of 4 rotating magnetic poles that form in the stator. The lines of the resultant field leave the stator's north poles and enter its south poles.
Remarkably, this field rotates at half the speed of the previous case. If the current frequency is 50 Hz, the field makes 25 revolutions per second, 1500 revolutions per minute.

3.2 - Speed of the field

What we have just seen holds in general: the speed of the rotating field is inversely proportional to the number of pole pairs. So we define the quantity p = number of pole pairs. We will then speak of a field, a winding or a machine with p pole pairs. A two-pole machine has p = 1, a four-pole machine p = 2, and so on. Clearly 2p is the number of poles: if p = 2 the field has 2p = 4 poles, if p = 3 it has 2p = 6 poles, and so on.
We can then write a more general law for the speed of the rotating field:
from which we get the speed in revolutions per minute:
Remember once again that the speed is, strictly speaking, a rotational frequency, which should be measured in Hz (hertz), that is, s^-1. The unit min^-1 (or revolutions per minute), widely used in practice, is not part of the International System of Units. 
For a four-pole machine, the pole pitch τ  is given by \( \tau= \frac{r \pi}{2}=\frac{r \pi}{p} \).

4 - Air-gap magnetic field in isotropic machines

A construction often used for electrical machines based on the three-phase rotating field (induction motors, synchronous motors and generators) is the so-called "non-salient-pole" (cylindrical-rotor) configuration. A non-salient-pole machine, also called isotropic, has a uniform air gap⁽⁴⁾.
By contrast, salient-pole or anisotropic machines have a non-uniform air gap, which makes the reluctances of the magnetic circuit non-uniform as well.
In an isotropic machine the conductors that make up the windings are placed in channels, or slots, cut into the stator and rotor magnetic circuits. The slots are evenly distributed along the cylindrical surfaces of the stator and rotor (Figure 5, which however does not show the rotor slots).
Even though the air gap is uniform, the reluctance of the circuits is not strictly constant, because the field lines tend to concentrate at the stator and rotor teeth; by shaping slots and teeth suitably, however, we can get close to the ideal condition.
Figure 5 shows the stator of a two-pole isotropic machine.
The winding of each phase occupies 10 slots, so we have q = 5 slots per pole per phase. We have introduced here the parameter q, the number of slots per pole per phase. The colors show which phase each conductor belongs to.

4.1 - Air-gap field of an isotropic machine

To find the magnetic field in the air gap of an isotropic machine we will use Ampère's circuital law and a few simplifying assumptions that make the calculation easier without taking away the validity and generality of the results.

4.2 - Simplifying assumptions

To study the magnetic field in the air gap it helps to rely on a few assumptions that simplify the analysis:

1. The magnetic reluctance of the paths in the material the stator and rotor are made of is zero: ℜ_m = 0 or, equivalently, their magnetic permeability is infinite, μ = ∞
2. The field has the same configuration in every plane perpendicular to the machine axis
3. The field lines leave and enter the infinitely permeable material perpendicularly and can be considered radial in the air gap

4.3 - The air-gap field: circulation and solenoidality

In the lower right part of the board we see a portion of stator and rotor, unrolled into a straight line. The current in conductor 1 flows out of the page (toward the observer), the current in conductor 2 flows into it. We call δ (delta) the thickness of the air gap.
To compute the circulation we choose a closed path in the magnetic field, for example an oriented closed line around conductor 1.
The circulation can be computed by dividing the path into small segments \(\Delta l \), computing the scalar product \(H \cdot \Delta l\) for each of them and adding up until the whole closed line is covered. In our case, since we set \(H= 0\) in the stator and rotor, the only contributions to the circulation come from the segments crossing the air gap, of length \(\delta \). This length is usually small enough for the field to be considered constant along the whole segment \(\delta \). The field at points A and B due to conductor 1 has, given the current direction, the pattern shown: \(H_A \) pointing down, \(H_B \) pointing up. For a correct calculation of the circulation, however, it is convenient to fix a positive direction for the field⁽⁵⁾ in the air gap. Let us take as positive, for example, a field directed from the stator to the rotor. A contribution to the circulation is then positive if the line's orientation agrees with the positive direction of H, negative otherwise ⁽⁶⁾. The orientation of the line is arbitrary too; here we chose counterclockwise (the same as the field lines of conductor 1).
Then, with these assumptions: The minus sign in front of \(H_B \) therefore comes from the choice of positive direction for H made above.
Now consider a second circulation path lying between the two conductors. The circulation is: It is zero because the closed line encircles no current.
This last relation tells us that \(H_D = H_E \); since the only constraint we set is that the line lies between conductors 1 and 2, we can say that \(H_D = H_E = H_B\) and conclude that between conductors 1 and 2 the field must be constant and equal, for example, to \(H_B \).
Finally, consider a third closed line enclosing both conductors. Since, from our point of view, the two conductors are the active sides of one turn (or of one winding), the two currents are equal and opposite. Then: This relation tells us that \(H_A = H_C \), so the field outside the two conductors is constant as well.
We now already know a good deal about the field in the air gap: under our assumptions it has a rectangular shape, like the one shown in the upper part of the board.
The second condition comes from the solenoidality of the magnetic field: the flux entering the stator cylinder equals the flux leaving it. Together, the two conditions give the value of the field in the two segments, which depends on the angle α subtended by the coil at the center. When the conductors divide the stator into two equal parts (α = π) the two fields are equal except for their direction: With 4 conductors (four-pole machine) the two coils give a rectangular, symmetric resultant field.

4.6 - Field of a distributed winding

Now let's look at a winding spread over several slots. In the example, imagine that the conductors of one phase are distributed over 3 slots, each at an angular distance θ from the next. Assuming α = π we get rectangular fields, as we have already seen. Here too we compute the field for each pair of corresponding conductors (each coil): pair 1-1' produces the highlighted rectangular field H₁, similar to the field H₂ of pair 2-2' and to H₃ of pair 3-3', which however are shifted by a distance r⋅θ from one another (in the graph the radius is set to r = 1). Superposing the three fields gives a stepped resultant field.

These rectangular fields can be obtained as a superposition of sinusoids, that is, they can be expanded in a Fourier series. The field H₁, for example, can be approximated as:
which contains the first three terms of the Fourier series. The phase β accounts for the shift with respect to the origin of the axes.
As a first approximation, each rectangular field can be represented by the fundamental sinusoid, or first harmonic; for H₁:
which greatly simplifies the calculations. The price to pay, however, is that we neglect the harmonic content of these waveforms, which matters most for synchronous generators, where the generated voltages must be sinusoidal to a high degree of accuracy.
In motors, harmonics in the rotating field mainly mean higher losses and parasitic torques, which in some situations can make operation irregular.
The resultant field is the superposition of the three rectangular waves H₁, H₂, H₃, and it too can be expanded in a Fourier series. The first harmonic of H is then obtained as the superposition of the three fundamentals of H₁, H₂ and H₃, or it can be found directly from the expression:
where q is the number of slots per pole per phase (3 in the example) and k_a, known as the winding factor, is a correction coefficient (never greater than 1) that accounts for the phase difference between the sinusoids (and the phasors that represent them) caused by the angular shift θ from slot to slot. k_a is essentially the ratio between the vector sum of the fields and the algebraic sum of their magnitudes \(k_a=\frac{\sum \vec H_i}{\sum H_i}\) and can be computed quickly with the formula shown on the board.

5 - Rotating magnetic field in an isotropic machine

Having seen how to find the magnetic field of a distributed winding, we still have to study the rotating field produced by a stator with a distributed three-phase winding. The board shows a cross-section of the stator of a 4-pole (p = 2) isotropic machine. The winding uses 3 slots per pole per phase. As usual, the phases are shown in different colors. With the same current-direction conventions adopted earlier, we can show the sequence of the instantaneous directions of the currents in the phase conductors. They produce 4 stator magnetic poles rotating at a speed:
which in turn produce a rotating magnetic field, as shown on the board.

5.1 - How the rotating field behaves in the air gap

Let's now try to visualize the rotating field in the air gap. To do so, we have drawn the stator in a schematic, linearized form with its conductors, also marking the start of each phase (P₁, P₂, P₃).
To begin with, we feed the system only with the current I₁. We immediately see the stepped field we already studied in 4.6. It is an alternating field, because the current producing it varies sinusoidally over time, but it is fixed in space. The same applies to the fields produced by the current I₂ and by the current I₃.
The magnetic field created by one phase therefore has a time period \(T = \frac 1 f\) and a spatial period \(T_S=\frac{2\;\pi\;r}{p}\), where r is the radius of the inner circumference of the stator, f is the frequency of the stator currents and p is the number of pole pairs.
In the example (r = 1, p = 2) we see two complete oscillations, so the spatial period is T_S = π.
The instant-by-instant superposition of these three fields along the whole air gap produces a resultant field that travels (along the air-gap circumference) at constant speed toward the right. We see at once that the field does not keep its shape exactly as it moves. A basic periodic shape is clearly recognizable, but its details (the peak value, the relative position of the steps) change continuously. This observation matters for some of the remarks that follow.
This field too is a periodic function of time, with \(T = \frac 1 f\), and of space, with period \(T_S=\frac{2\;\pi}{p}\) (r = 1). The period is twice the pole pitch τ.
The number of spatial oscillations contained in the angular extent (2π) of the stator depends on the number of pole pairs. A 4-pole field must show two complete spatial oscillations N-S-N-S. This is why p, the number of pole pairs, is exactly the spatial "oscillation" frequency of the field.
Likewise, if the trigonometric function cos(x) shows a single oscillation in its spatial period 2π, in the same period the function cos(p x) shows p oscillations.
As a first approximation we can describe the field with a sinusoid, which we will call the fundamental sinusoid (or simply the "fundamental").
The fundamental sinusoid must match the stepped function as closely as possible. So, instant by instant, it must "follow" the magnetic field.
What, then, is the analytical expression of this "traveling" sinusoid?

Activity 1: find the traveling sinusoid

The result of the activity above should be a sinusoid that travels with the stepped magnetic field, at the same speed, with as many spatial oscillations as the field. Things to consider:

• all four variables x, t, ω, p come into play
• it can help to look separately at the effect of the spatial and time variables: for example, how does a function that depends only on t behave? And one that depends only on x?

6 - Counter-rotating fields and Leblanc's theorem

Here on the left we see a vector B drawn at a given position in space (an x-y plane). Once its direction is fixed, this vector can be constant in direction and magnitude, in which case a (t, B) graph describing the vector as time goes by would simply be a straight line parallel to the time axis. Alternatively, the vector can vary over time according to some law f(t). Of particular interest for applications, and for this study, is the case where f(t) is sinusoidal. The magnitude and direction of the vector then change over time, giving what is called an alternating vector. As a function of time, a vector of this kind is expressed by \( B(t)=B_M cos(\omega t +\varphi) \).
Leblanc's theorem⁽⁷⁾ states that an alternating vector can be obtained, instant by instant, as the sum of two vectors of constant magnitude ¹/₂B_M that rotate in opposite directions at the same angular velocity: B = B_CW + B_CCW, where B_CW is the clockwise field and B_CCW the counterclockwise field. At the initial instant they form the angles −φ and φ with B. The result holds for any direction of B, as you can see by varying the angle θ.

6.1 - The three-phase rotating field as a sum of counter-rotating vectors

Each of the alternating magnetic fields produced by the stator phases can be split into two counter-rotating fields, called D_k and S_k (k = 1, 2, 3) for simplicity. So, for example, D₁ is the clockwise (CW) field produced by phase 1, while S₁ is the corresponding counterclockwise (CCW) field. Splitting the field B₁ gives the two vectors D₁ (solid line) and S₁ (dashed). Look also at how the field B₂ and the field B₃ split. Now let's show only the six counter-rotating fields.
Here we see that the D fields are in phase (they are drawn side by side rather than on top of one another, so that they are all visible), and their sum gives the resultant field B, which clearly has magnitude 1.5 B_M (see 2.2) and rotates at the same angular velocity as the D fields. The sum of the S fields, which form a symmetric star of vectors, is zero, as is well known⁽⁸⁾.

6.2 - Books by the author