CALCULATION OF THE MAGNETIC CIRCUIT OF DC ELECTRICAL MACHINES WITH COMBINED HETEROPOLAR INDUCTORS

Yu. N. KRONEBERG, A. S. ZHIBINOV, S. N. MARUKHIN

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

Electrical machines with combined excitation are known in which part of the pole pairs is made of permanent magnets (PM); in this case the armature winding must be series-connected [1, 2]. They possess a number of important advantages: in comparison with electromagnetic machines they have reliable self-excitation and increased overload capacity (owing to the small influence of the armature reaction on the PM), and require a smaller excitation-winding (EW) power; unlike machines with PM, they have good controllability.

As is known, the advantages of PM are most fully manifested in low-power machines, which are usually made four-pole. However, four-pole inductors of the combined type are magnetically asymmetrical, which leads to an imbalance of the magnetic attraction forces with all the resulting unfavorable consequences. The minimum number of poles at which magnetic symmetry can be ensured is six. Such machines are of the greatest interest.

In the present article, using as an example a six-pole machine with one pair of PM (Fig. 1), the features of calculating the magnetic

Fig. 1. Magnetic circuit of a machine with a combined inductor

circuit. It is based on the usual assumptions, which reduce distributed parameters to lumped ones. Since the magnetic circuit has only four identical parts (a pole with an excitation winding and half of a permanent magnet), the equivalent circuit (Fig. 2) is relatively complex.

Fig. 2. Equivalent circuit of the magnetic circuit

The purpose of the calculation is, for a given magnetomotive force of the excitation winding \(F_f\), to find the magnetic fluxes \(\Phi_{\delta m}\) (of half the permanent magnet) and \(\Phi_{\delta f}\) (of the pole with the excitation winding), and from them to compute the emf of the machine:

\[ E=\frac{pn}{60a}\,N\cdot 10^{-8}\cdot \frac{2}{3}\,(\Phi_{\delta f}+\Phi_{\delta m}), \tag{1} \]

where \(p, N, a\) are the number of pole pairs, conductors, and parallel branches of the armature winding;

\(n\) is the rotational speed of the armature, rpm.

The equivalent circuit contains a number of nonlinear elements; the PM branch \(5\!-\!1\) is nonlinear only when operating on the demagnetization curve (switch position “C”), and after stabilization (position “m”) it is replaced by the magnetomotive force \(F_m\) (corresponding to the point of intersection of the extension of the recoil line with the mmf axis) and by the linear resistance \(R_m\). By graphical addition of the ampere-turn characteristics of branches \(5\!-\!3\) and \(5\!-\!11\), the number of nonlinear elements is reduced to six. But since more than three nonlinear elements remain, an exact graphical solution is impossible [3]. Additional simplifications are necessary.

Taking into account that the magnetic flux of the PM varies within small limits, the nonlinear resistances of branches \(5\!-\!3\) and \(5\!-\!11\) may be replaced by linear resistances \(R_{5-3}\), \(R_{5-11}\) and mmfs \(F_{5-3}\), \(F_{5-11}\); however, for this it is necessary at least approximately to determine the position of the operating regions on the ampere-turn characteristics. For this purpose it is most convenient to assume temporarily that the magnetic fluxes of the excitation winding and the PM are independent, i.e., in the equivalent circuit, to connect points 2, 6, and 10. Then the left-hand part will be an ordinary equivalent circuit of a machine with a PM, which

is solved by known methods. This determines one point on each Weber–ampere characteristic and makes it possible to linearize them.

A further simplification is based on the small value of the reluctance of the armature yoke \(R_{\mathrm{aff}}\), which makes it possible, without great error, to modify the circuit by connecting the reluctance \(R_{\sigma ff}\) in parallel with the reluctance \(R_{\infty f}\). This means that the magnetic flux through the reluctance \(R_{\mathrm{aff}}\) will become larger than the actual one by the amount of the leakage flux, while the magnetic flux through the reluctance \(R_{\sigma ff}\) will decrease, since the difference of magnetic potentials between points 7 and 8 will be reduced by the amount of the drop of magnetic voltage in the reluctance \(R_{\mathrm{aff}}\). The resulting errors (especially the second) are usually small, but, if necessary, they can be reduced further.

The first refinement consists in using, when calculating the reduced Weber–ampere characteristic of branch 11—12, not the actual height of the armature yoke, but one increased in proportion to the leakage coefficient. The second consists in reducing the reluctance \(R_{\sigma ff}\) in proportion to the ratio of the drops of magnetic voltages between points 7—11 and 7—12 for the operating mode adopted as the calculated one.

The modified right-hand part of the circuit is easily simplified (graphical addition of the Weber–ampere characteristics of the parallel-connected reluctances \(R_{\delta f}\) and \(R_{\sigma ff}\) and the series-connected \(R_{\mathrm{aff}}\) and \(R_{\mathrm{iff}}\)), being transformed into an active triangle. The linear part of the circuit is also reduced to the same form by known methods. Two parallel-connected active triangles are obtained: a nonlinear one and a linear one. The graphical solution of such a circuit no longer presents difficulties, and as a result the dependences of the magnetic fluxes \(\Phi_{\delta f}\) and \(\Phi_{\delta m}\) on the excitation m.m.f. \(F_f\) are found, and from equation (1) the required value of the e.m.f. \(E\).

By the method described, a six-pole direct-current generator with two PM poles was calculated (armature diameter 5.5 cm, armature length 8.5 cm, speed 8000 rpm). Figure 3 gives the calculated and experimental no-load characteristics. Their agreement is quite satisfactory.

Fig. 3. No-load characteristic of a direct-current generator: — experimental; — — — calculated.

Certain procedures of the method presented may be used for the calculation of machines with combined excitation and with a different pole-system structure. These include, first of all: linearization of the Weber–ampere characteristics of branches whose principal effect is exerted by PM; temporary assumption of independence

magnetic fluxes of the OB and PM to determine the parameters of the equivalent linear branches; a modification of the circuit based on an approximate allowance for small drops in magnetic potentials.

LITERATURE

1. Yu. N. Kroneberg. Basic principles of constructing systems of combined excitation. Izvestiya TPI, vol. 242, Tomsk, TTU Publishing House, 1972.

2. Yu. N. Kroneberg, A. S. Zhibinov, S. N. Marukhin. Electrical machines with combined unlike-pole inductors. Izvestiya TPI, vol. 200, Tomsk, TTU Publishing House, 1974.

3. L. R. Neiman, P. L. Kalantarov. Theoretical Foundations of Electrical Engineering. Part 1, Moscow–Leningrad, Gosenergoizdat, 1959.