Method for Calculating Forces in the End Windings of a Single-Layer Winding Using an Electronic Digital Computer

A. I. Verkhoturov, G. A. Sipailov, K. A. Khorkov

(Presented by the scientific seminar of the Electromechanical Faculty)

In certain emergency operating conditions of electrical machines, for example, during short circuits or closures through a small external resistance, large electrodynamic forces arise in the region of the end windings of the stator winding. These forces are especially dangerous in a surge generator, whose normal operating mode is the mode of a sudden short circuit or a sudden connection to a load whose parameters are commensurate with those of the generator. For example, in a surge-power generator the amplitude of the first half-wave of current may exceed by 30–40 times the amplitude value of the steady-state current and, consequently, the forces to which the conductors of the generator stator will be subjected increase by 900–1600 times.

The best type of stator winding for a single-phase surge-power generator is a single-layer, subdivided, concentric winding. For calculating the forces in the end portions of the winding, it is convenient to divide the entire overhang of the end windings into three zones (Fig. 1, a).

Fig. 1. For determining the geometric dimensions of the winding and the forces on a conductor.

The forces on the end portion are considered as the sum of the forces between segments both within each zone and between zones. In

In this case, the forces arising between the end winding and the end face of the stator iron are taken into account as forces between segments of the end winding and their mirror images. The force between two segments parallel to one another and separated from one another by a distance \(h\) is determined by the equation

\[ F_{ab}=\frac{\psi_0}{4\pi} i_1 i_2 \int_0^a \int_0^b \frac{db_1\, da_1 \sin \gamma}{\rho^2}, \tag{1} \]

where \(\rho\) is the distance between the segments \(db_1\), \(da_1\) under consideration; \(\gamma\) is the acute angle enclosed between the segments \(db_1\), \(\rho\).

The general solution of equation (1) is complicated; however, for particular positions of the segments, comparatively simple calculation formulas can be obtained. Using the method of [2], for the cases of interest to us we obtain the following relations:

a) the segments emanate from one boundary plane (Fig. 2,a)

\[ F_{ab}=1.02\cdot 10^{-8} I_{\text{ud}}^2 \left[ \frac{q+r\pm h-p}{h} \right], \tag{2} \]

Fig. 2. For the calculation of mechanical forces between segments.

b) segments of unequal length are arranged symmetrically (Fig. 2,b)

\[ F_{ab}=2.04\cdot 10^{-8} I_{\text{ud}}^2 \left[ \frac{r-p}{h} \right], \tag{3} \]

where \(I_{\text{ud}}\) is the maximum value of the transient short-circuit current;

\[ r=\sqrt{(a-\delta)^2+h^2}, \qquad q=\sqrt{a^2+h^2}, \qquad p=\sqrt{\delta^2+h^2}. \tag{4} \]

In known works devoted to the calculation of mechanical forces in the end windings of a stator winding, in order to simplify the calculations a number of simplifications are allowed which introduce substantial errors into the calculation. For example, in [3] the force of attraction to the stator iron is not taken into account, and the conductors are assumed to be infinitely long. The most complete method is [4], in which finite segments are considered and the influence of the stator iron is taken into account. However, in this method the end windings are assumed to be planar, which is valid only for multipole machines, when the length of the arc of the end winding is considerably smaller than the diameter of the stator bore, and the rectilinear segments of the end winding are arranged perpendicular to the end face of the stator. Naturally, for two-pole machines with a concentric winding, method [4] gives errors. In the present article a method is given for calculating the mechanical forces in the end windings of a single-layer concentric stator winding, taking into account the actual arrangement of the winding conductors.

It is convenient to carry out the calculation of the forces acting on an individual conductor of the end winding by means of digital computing machines,

Fig. 3. Flowchart for calculating mechanical forces in the end-winding parts of the stator winding on an electronic digital computer.

specifying the order of alternation of the segments and the law of variation of their lengths and the distances between them. In order to develop the method, calculations of the mechanical forces in the end-winding parts were carried out on the Minsk-1 electronic digital computer for a number of machines with different stator-bore diameters, different projections of the end winding, and different numbers of stator-winding coils. The algorithm for solving the problem is clear from the block diagram of the program shown in Fig. 3. Briefly, the program contains the following solution sections:

a) the geometrical dimensions of the coils are determined;

b) the radial and tangential forces acting on the rectilinear sections of the coil overhang are determined;

c) the forces of attraction of the current-carrying conductors to the stator iron are determined;

d) the forces acting on the bent parts of the coil overhang are determined;

e) the resultant forces acting on the conductor and the fastening details are determined.

Principal regularities in the variation of the dimensions of the end-winding part of the coils in space

Assign to the coil located closest to the end face of the stator the number 1. The length of the rectilinear section of the overhang of this coil (Fig. 1, a) is

\[ l_{1\min}=c_2D+0.5b_n, \tag{5} \]

where \(b_n\) is the width of the stator slot.

The quantity \(c_2\) is taken according to [1], depending on the voltage of the stator winding. When the number of slots filled by the winding is equal to \(\frac{2}{3}z'_1\), the number of coils in the zone is determined as

\[ n_{\max}=\frac{z'_1}{6}, \tag{6} \]

where the quantity \(n_{\max}\) is at the same time the number of the largest coil. The length of the rectilinear section of this coil is

\[ l_{1\max}=l_{1\min}+h_1\left(\frac{z'_1}{6}-1\right). \tag{7} \]

The logic of the program is based on the order of successive variation of the coil number \(n\) from 1 to \(n_{\max}\); in this case the lengths of the rectilinear and bent segments are determined from the formulas:

\[ b_{1n}=l_{1\min}+h_1(n-1), \tag{8} \]

\[ b_{2n}=\frac{\pi}{z'_1}(D+k_1k_4b_n)\left(\frac{z_1}{6'_1}+2n-1\right), \tag{9} \]

where \(h_1\) is the distance between the mean lines of adjacent bent segments;

\(k_1\) is the ratio of the height of the stator slot to its width;

\(k_4\) is a coefficient taking into account the angle of inclination of the end-winding part to the axis of the machine \(\beta\):

\[ k_4=1+\frac{2b_{1n}\tg\beta}{k_1b_n}. \tag{10} \]

Calculation of Forces Acting on Straight-Line Sections

Let us consider successively the straight-line sections \(1, 2, 3\ldots\) of the coils of the first and second zones (Fig. 1, a). The forces on a straight-line section of the first coil are composed of the forces between this section and the adjacent sections, whose numbers we denote by \(k\). The quantity \(k\) characterizes the number of tooth divisions between sections within the zone, from \(k=1\) to \(k=n_{\max}-n\) in the direction of increasing sections \(a_{1k}\), and from \(k=-1\) to \(k=-(n-1)\) in the direction of decreasing sections \(a_{1k}\), relative to the length of the \(n\)-th section under consideration \(b_{1n}\). The length of the section of the \(k\)-th coil is equal to

\[ a_{1k}=b_{1n}+kh_1 . \tag{11} \]

The distance between the \(n\)-th and \(k\)-th sections along a straight line (Fig. 1, b) is equal to

\[ h=(D+k_1 k_t b_n)\sin k\frac{\alpha}{2}, \tag{12} \]

where \(\alpha\) is the angle between the axes of adjacent slots,

\[ \alpha=\frac{2\pi p}{z'_1}. \tag{13} \]

For each \(n\)-th section of the first zone, the force acting on it from the sections of zone 1a is determined simultaneously; in this case the quantity \(k\) correspondingly takes larger values, up to \(k=2n_{\max}-n\).

It is meaningful to resolve the forces acting on the straight-line sections into tangential \(F_{\mathrm{tn}}\) and radial \(F_{\mathrm{rn}}\) components; for this purpose the data obtained from formula (2) must be multiplied respectively by \(\cos k\frac{\alpha}{2}\) and by \(\sin k\frac{\alpha}{2}\). The total radial force from the straight-line sections of the first zone per unit length of the machine is determined as the sum

\[ f_R=\sum_{n=1}^{n_{\max}}\frac{2F_{\mathrm{rn}}}{b_{1n}} \cos\left(\frac{z'_1}{3}-2n+1\right)\frac{\pi}{z'_1}. \tag{14} \]

Calculation of Forces Acting on Curvilinear Sections

It is advisable to divide the calculation of the forces acting on the sections of the third zone into two parts: calculation of the forces due to the interaction of the magnetic fields of the sections within the zone, and calculation of the forces due to the interaction of the magnetic fields of the sections with the ferromagnetic surface of the stator end face. In both cases the calculation procedure is analogous to the procedure for calculating the forces in the first zone. This makes it possible to link all the calculations into a single algorithm for varying \(n\) and \(k\).

The calculation of the forces between sections \(n\) and \(k\) of the third zone, \(F_n\), is carried out according to formula (3). In this case the length of the \(n\)-th section is determined by (9), and the length of the \(k\)-th section by (15)

\[ a_{2k}=\frac{\pi}{z'_1}(D+k_1 k_t b_n) \left(\frac{z'_1}{6}+2n+2k-1\right). \tag{15} \]

The distance between the axial lines of the \(n\)-th and \(k\)-th sections is equal to

\[ h_1=k(b_n+c_1). \tag{16} \]

The forces between the sections of the third zone and the ferromagnetic surface are

\[ F_{cn}=2{,}04\cdot 10^{-8} I_{\mathrm{yx}}^2 \left[\frac{q_2-h_2}{h_2}\right], \tag{17} \]

where \(h_2\) is the distance between the \(n\)-th segment and its mirror image (Fig. 2b).

The total axial force acting on the end winding as a result of the interaction of the segments within the zone and of the forces attracting the current-carrying segments to the iron, per unit length of the end winding, is determined as the sum

\[ f_A = \sum_{n=1}^{n_{\max}}\left[\frac{F_n}{b_{2n}}+\frac{F_c}{b_{2n}}\right]. \tag{18} \]

As an example, one may cite the results of calculating the forces on the end winding of a machine with \(D = 90\) cm, \(c_2 = 0.14\), and \(z'_1 = 48\), performed on the Minsk-1 electronic digital computer (Table 1).

Table 1

Note: To obtain the forces in kg per meter of end-winding length, the values given in Table 1 must be multiplied by \(I_{\mathrm{ud}}^2 \cdot 10^{-8}\).

From the calculations presented, the following conclusions may be drawn:

1. The greatest forces per unit length of conductor act on the first conductor. Decomposition of the force into tangential and radial components makes it possible to calculate the conductor for shear and to determine the resultant radial force on the inner cap of the stator bracing assembly.

2. The axial force found makes it possible to calculate the curved sections of the end winding for bending and strength. The force of attraction of the end-winding segments to the stator iron that has been found makes it possible to calculate the outer ring of the stator cap.

REFERENCES

1. A. E. Alekseev. Construction of Electrical Machines. GEI, 1958.
2. G. B. Kholyavskii. Calculation of Electrodynamic Forces in Electrical Apparatus. GEI, 1962.
3. I. F. Calvert. Forces in Turbine Generator Stator Windings, Trans. of the A. I. E. E., 1931, 50, p. 173.
4. D. Harrington. Forces in Machine End Winding, Trans. of the A. I. E. E., 1952, 71, pt. III.