FEATURES OF SELECTING THE ROTOR MAGNETIC SYSTEM OF A LOW-POWER SYNCHRONOUS GENERATOR WITH TRAPEZOIDAL E.M.F.

G. A. SIPAILOV, A. B. TSUKUBLIN

(Recommended by the scientific seminar of the departments of electrical machines and general electrical engineering)

The use of a nonsinusoidal e.m.f. waveform for obtaining direct current by rectification makes it possible to obtain small values of the alternating component of the rectified voltage even with a relatively small number of generator phases [1].

An analysis of the operation of a three-phase bridge rectifier supplied by a generator with a trapezoidal e.m.f. showed that, for an e.m.f. waveform that is a trapezoid with \(\beta = 30^\circ\), the utilization factor of the power-supply unit decreases only slightly, while the rectified voltage obtained is of higher quality (the ripple factor did not exceed 5 percent even at relatively high generator loads). Therefore, the problem in calculating a generator operating into a rectifier unit is to obtain a trapezoidal e.m.f. with a selected slope angle. This problem reduces to determining such an induction curve in the air gap that, in the winding distributed on the stator, would give an e.m.f. of trapezoidal form.

The harmonic spectrum of the trapezoidal e.m.f. wave, whose form is shown in Fig. 1, is determined by expansion in a Fourier series:

\[ E(\omega t)=\frac{4E}{\pi \beta}\cdot \sum_{n=1}^{\infty} \frac{\sin(2n-1)\beta}{(2n-1)^2}\cdot \sin(2n-1)\omega t. \tag{1} \]

Fig. 1.

Using the method of superposition, we shall assume that in the air gap the induction consists of a series of harmonics and that each harmonic produces its own e.m.f. with the frequency of the harmonic of the magnetic induction, i.e.,

\[ E_{m\nu}=4.44\cdot \tau_\nu \cdot l \cdot W \cdot k_{\mathrm{ob}\nu}\cdot f_\nu \cdot B_{m\nu}, \tag{2} \]

\[ \nu = 2n-1. \]

To simplify obtaining a trapezoidal EMF, a full-pitch winding is usually used, i.e., the shortening coefficient of such a winding is \(k_{y1}=k_{y\nu}=1\), and, consequently, the winding factor is equal to the distribution factor

\[ K_{\mathrm{ob}1}=K_{\mathrm{r}1},\quad K_{\mathrm{ob}\nu}=K_{\mathrm{r}\nu}. \]

The distribution factor depends on the number of slots per pole and phase \(q\) and is determined by

\[ K_{\mathrm{r}\nu}= \frac{\sin \dfrac{\nu\pi}{2m}} {q\sin \dfrac{\nu\pi}{2mq}}. \]

This coefficient is different for different harmonics and, consequently, does not reduce the EMFs produced by the harmonics proportionally. This circumstance is the reason that the EMF waveform deviates from the induction curve.

A phenomenon that is positive from the standpoint of improving the sinusoidality of the EMF curve proves harmful in the case under consideration. Therefore, even before the final calculation of the magnetic circuit, it is necessary to determine the configuration of the inductor poles in such a way that in the air gap there will be an MMF curve which, with allowance for \(K_{\mathrm{ob}\nu}\), gives the required form of the EMF curve.

Let us express the amplitudes of the harmonics through the fundamental:

\[ \frac{E_{m\nu}}{E_{m1}}= \frac{K_{\mathrm{ob}\nu}}{K_{\mathrm{ob}1}}\cdot \frac{B_{m\nu}}{B_{m1}}. \tag{3} \]

Hence the amplitude of the induction of the \(\nu\)-th harmonic is

\[ B_{m\nu}= \frac{E_{m\nu}}{E_{m1}}\cdot \frac{K_{\mathrm{ob}1}}{K_{\mathrm{ob}\nu}}\cdot B_{m1} \tag{4} \]

or, if the value of the first harmonic is taken as the base, the amplitude of the higher harmonics of the induction in the air gap, in relative units, is expressed as:

\[ B^{*}_{m\nu}=E^{*}_{m\nu}\cdot \frac{1}{K^{*}_{\mathrm{ob}\nu}}, \tag{5} \]

where

\[ E^{*}_{m\nu}=\frac{E_{m\nu}}{E_{m1}};\quad K^{*}_{\mathrm{ob}\nu}=\frac{K_{\mathrm{ob}\nu}}{K_{\mathrm{ob}1}}. \]

Thus, knowing the harmonic spectrum of the required form of the EMF curve, it is possible, by specifying the type of winding, to determine the harmonic spectrum of the induction curve in the air gap

\[ B_{\delta}(x)=\sum_{\nu=1}^{s} B_{m\nu}\cdot \sin\left(\nu\frac{\pi}{\tau}x+\psi_{\nu}\right). \tag{6} \]

Then, considering the MMF distribution along the length of the pole pitch to be rectangular (a concentrated field winding on the rotor), and also neglecting the magnetic reluctance of the rotor and stator steel, it is possible, owing to the relatively small magnitudes of the gaps, to regard the induction at each point of the field as the quotient of the MMF at that point divided by the magnitude of the air gap. The magnitude of the gap, or the length of the line of force in air, can be determined as a segment of the radius of the armature circumference, measured at the given point between the pole and the armature.

Fig. 2.

Thus, under the assumptions made,

\[ F=\frac{B_{\delta(x)}}{\mu_0}\cdot \delta_{(x)}=\text{const.} \tag{7} \]

Let us take as the unit of the air gap its value under the center of the pole \(\left(x=\frac{\tau}{2}\right)\); then

\[ F=\frac{B_{\delta\left(\frac{\tau}{2}\right)}}{\mu_0}\cdot \delta_{\left(\frac{\tau}{2}\right)} = \frac{B_{\delta(x)}}{\mu_0}\cdot \delta_{(x)} . \tag{8} \]

From this equation, the air gap within the pole division is determined as:

\[ \delta_{(x)}= \frac{B_{\delta\left(\frac{\tau}{2}\right)}}{B_{\delta(x)}}\cdot \delta_{\left(\frac{\tau}{2}\right)} \]

or

\[ \delta^*_{(x)}= \frac{B_{\delta\left(\frac{\tau}{2}\right)}}{B_{\delta(x)}}, \quad \text{where} \quad \delta^*_{(x)}= \frac{\delta_{(x)}}{\delta_{\left(\frac{\tau}{2}\right)}} . \tag{9} \]

Assuming, in a first approximation, that the stator is ideally smooth, the magnitude of the air gap under the pole determines the configuration of the pole shoe.

As for the calculation of the field outside the pole shoe, it can be carried out by the usual method for calculating synchronous salient-pole machines, taking into account the prescribed spectrum of harmonics of the magnetic induction [2].

The form of the field in the interpolar space is determined only by the distribution of the m.m.f. and by the ratio of the maximum gap (under the edge of the pole shoe) to the distance between adjacent pole shoes,

\[ \frac{\delta_{\max}}{\tau(1-\alpha)} . \]

Constructions of these fields [2] have shown that the field in the space between the poles depends very little on the shape of the edges of the pole shoes. When the gap under the middle of the pole, i.e., in our case the basic gap, is decreased, the relative magnitude of the fluxes in the interpolar space decreases. Therefore, taking into account that the gaps in low-power synchronous generators are small, one may—

but with sufficient accuracy for practice, to construct the field in the interpolar gap.

The method for determining the pole configuration and its width should be as follows:

1. From the conditions for rational placement of the winding, the number \(q\) and the type of winding are selected. Usually, for low-power three-phase synchronous generators a double-layer winding is selected. From the condition of obtaining a trapezoidal emf, a diametral pitch is selected. The choice of the value \(q\) is determined by the pole pitch of the machine and by the number of phases. Since the pole pitch, for the selected number of poles, depends on the generator power, \(q\) also depends on the power.

For magnetoelectric generators of ordinary application [3], it is recommended, at \(P = 100 \div 250\) W, \(m = 3\), and \(2p = 6\), to select \(q = 1\frac{1}{2}\); and at \(2p = 4\), \(q = 2\frac{1}{2}\); at a power on the order of \(P = 750\) W and \(2p = 6\), \(q = 2\frac{1}{2}\). But in ordinary application these recommendations proceed from the condition of obtaining the best shape of the emf curve, since a winding with fractional \(q\), from this point of view, is equivalent to a winding with an integral \(q\) equal to the numerator of the fraction. In addition, increasing \(q\) reduces the leakage of the stator winding and improves the conditions for its cooling. However, in our application, the fractional character of \(q\) will adversely affect generator operation, for the emf curve will approach a sinusoidal one. Therefore, in low-power synchronous generators with a trapezoidal emf waveform it is better to select integral values of \(q\), the value of \(q\) being determined, as in ordinary applications, by the power and the number of poles. The most commonly used values of \(q\) for low-power synchronous generators operating with a rectifier are \(q = 1 \div 2\). Larger values of \(q\) lead to a considerable complication of the shape of the inductor pole (sometimes even practically impossible) in order to obtain a trapezoidal emf. From this point of view, a concentrated winding (\(q = 1\)) is preferable where, according to other conditions, the choice \(q = 1\) or \(q = 2\) is equivalent.

After selecting the winding, the winding factors for the fundamental and higher harmonics are determined.

1. The harmonic spectrum of the emf curve is specified and, from expression (5), the spectrum of harmonic inductions in the air gap is determined (Fig. 2).

2. Taking into account the spectrum of harmonic magnetic induction in the gap, the configuration of the air gap over the length of the pole pitch is determined according to expression (9).

3. In accordance with the configuration of the air gap, the pole width and its configuration are selected.

4. The field pattern in the air gap is constructed, taking into account the selected pole configuration, and the induction at the stator surface is checked. The necessary corrections are introduced, taking into account the actual distribution of induction and the influence of armature reaction.

The further calculation of the stator winding and of the magnetic circuit is carried out in the usual manner, depending on the type of magnetic circuit and the type of generator. It must be kept in mind, however, that saturation of the steel of the magnetic core may distort the field pattern in the gap; therefore the induction values should not be greatly increased, especially in the poles or pole shoes.

REFERENCES

1. J. T. Duane. A brushless direct-current generator for aviation applications. Appl. and Industry, No. 39, 1958.

2. T. G. Soroker, I. N. Gorazhevskii. Calculation of magnetic fields in the gap of an explicitly salient-pole machine. Elektrichestvo, No. 6, 1952.

3. I. N. Pleshchunov. Magnetoelectric synchronous generators with a “star” type rotor, dissertation, VNIIEM, Moscow, 1964.