On the Calculation of Electrodynamic Forces between Curvilinear Conductors in the End Parts of Single-Layer Windings of Electric Machines

G. A. Sipailov, V. I. Popov

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

In calculating the electrodynamic forces acting on the end part of a winding, it is convenient, in order to simplify the calculation procedure, to subdivide the latter into separate characteristic sections. In this case, the greatest difficulties are presented by determining the forces between the conductors of curvilinear sections. The difficulties of the calculation increase further when the influence of surrounding ferromagnetic surfaces is taken into account. In the present article, which is a continuation of [1], the first part of the problem is solved for the time being—the calculation of electrodynamic forces between curvilinear conductors of the end part without taking into account the influence of the surrounding surfaces.

To solve the stated problem, let us first consider the calculation of forces using the example of two conductors located in parallel planes at a distance \(h\) from one another (Fig. 1). In their shape, conductors 1 and 2 are arcs of radii \(R_1\) and \(R_2\), with central angles respectively equal to \(\varphi\) and \(\beta\), and arranged symmetrically with respect to the \(X\)-axis. Currents of equal magnitude flow through the conductors.

When determining the dynamic forces acting on a conductor, it is necessary to know the magnitude of the magnetic field in which this conductor is located. The nature of the calculation of the magnetic field of two conductors depends on the geometrical dimensions of the conductors and on their mutual arrangement. If the distance between the conductors substantially (by three to four times) exceeds the dimensions of their cross section, then the value of the mutual inductance between the conductors differs only slightly from the mutual inductance between their geometrical axes [2]. In this case the cross section of the conductors may be neglected, that is, it may be assumed that the currents flow along their geometrical axes. If, however, the distance between the conductors is small in comparison with the dimensions of their cross section, the calculation becomes considerably more complicated. In this case the field created by adjacent conductors may be represented as the sum of the fields of elementary filaments into which the conductors are divided.

Since in the case under consideration the principal attention is devoted to investigating the influence of the curvature of the conductor on the magnitude of the forces, the transverse dimensions of the conductors are assumed to be small in comparison with the distance between them.

The elementary force acting on the element \(dl_1\) of conductor 1, through which current \(i\) flows, can be expressed by the vector product

\[ dF=i(\overline{B}\,d\overline{l}_{1})=iBdl_{1}\sin(\widehat{Bdl_{1}}), \tag{1} \]

where

\(i\) is the current in conductor 1.
\(\overline{B}\) is the induction at point \(M\).

For the case in which the conductors are located in air, the most convenient way to determine the magnetic field is the Biot–Savart–Laplace law. This law makes it possible to express the component of the magnetic-field intensity at a given point \(M\) due to an element \(dl_{2}\) of the second conductor, through which a current \(i\) flows,

\[ dH=\frac{i}{r^{3}}(d\overline{l}_{2}\times \overline{r}). \tag{2} \]

Here \(r\) is the ray from the element \(dl_{2}\) to the element \(dl_{1}\).

After certain transformations, expression (2) can be written in a somewhat modified form,

\[ dH=i\,\frac{d\gamma}{r}\left|\frac{d\overline{l}_{2}}{dl_{2}}\times \frac{\overline{r}}{r}\right|, \tag{3} \]

where \(d\gamma\) is the elementary angle between the rays going from point \(M\) to the ends of the element \(dl_{2}\),

\[ \left|\frac{d\overline{l}_{2}}{dl_{2}}\times \frac{\overline{r}}{r}\right| \]

is the unit vector in the direction of \(dH\).

On the other hand, the elementary area in polar coordinates is

\[ dS=\frac{1}{2}\rho^{2}\,d\gamma. \tag{4} \]

If this area is represented as a vector having the direction \(dH\), then for \(\rho=\dfrac{1}{\sqrt{r}}\) we shall have

\[ dS=\frac{1}{2}\,\frac{d\gamma}{r}\left|\frac{d\overline{l}_{2}}{dl_{2}}\times \frac{\overline{r}}{r}\right|. \tag{5} \]

From comparison of expressions (3) and (5) it follows that

\[ dH=2i\,dS. \tag{6} \]

Thus, in order to determine the magnetic-field intensity it is necessary to find the surface \(S\) traced by the radius vector \(\rho=\dfrac{1}{\sqrt{r}}\) within the angle \(\gamma\) [3].

Since \(dS\) is a vector quantity, it can be resolved along the coordinate axes,

\[ dS=dS_{x}+dS_{y}+dS_{z}, \]

where

\[ \begin{aligned} dS_{x}&=dS_{yz},\\ dS_{y}&=dS_{xz},\\ dS_{z}&=dS_{xy}. \end{aligned} \]

Such a decomposition makes it possible to determine the components \(H_{x}\), \(H_{y}\), \(H_{z}\) of the magnetic-field intensity at any point of space produced by a conductor of arbitrary shape.

Using this method, let us determine the components of the magnetic-field intensity \(H_{x}\) and \(H_{z}\), produced by conductor 2 at a certain point \(M\) of conductor 1 (Fig. 1).

Fig. 1. For calculating the forces between two curvilinear conductors

The radius vector at a certain angle \(\alpha\) from the element \(dl_2\) to the point \(M\).

\[ r_\alpha = \sqrt{h^2 + R_1^2 + R_2^2 - 2R_1R_2\cos\alpha}. \]

Then

\[ \rho_\alpha = \frac{1}{\sqrt{r_\alpha}} = \frac{1}{\sqrt[4]{h^2 + R_1^2 + R_2^2 - 2R_1R_2\cos\alpha}}. \]

The projections \(\rho\alpha\) onto the planes \(YOZ\) and \(XOY\).

\[ \begin{aligned} \rho_{xyz}&=\rho_\alpha \cos \widehat{(\rho_\alpha \rho_{xyz})} =\frac{\left[h^2+\bigl(R_1\sin\omega+R_2\sin(\alpha-\omega)\bigr)^2\right]^{1/2}} {\left[h^2+R_1^2+R_2^2-2R_1R_2\cos\alpha\right]^{3/4}},\\ \rho_{axy}&=\rho_\alpha \cos \widehat{(\rho_\alpha \rho_{axy})} =\frac{\left[R_1^2+R_2^2-2R_1R_2\cos\alpha\right]^{1/2}} {\left[h^2+R_1^2+R_2^2-2R_1R_2\cos\alpha\right]^{3/4}} . \end{aligned} \tag{7} \]

Considering the points \(M_{xy}\) and \(M_{zy}\) as the poles of polar coordinates, about which the radius-vectors \(\rho_{axy}\) and \(\rho_{ayz}\), respectively, rotate, and the lines \(OM_{xy}\) and \(M_yM_{zy}\) as the lines of reference for the values of the angle \(\gamma\), the summary areas \(S_{xy}\) and \(S_{yz}\) are expressed as

\[ \begin{aligned} S_{xy}&=\frac{1}{2}\int_0^{\gamma_{1xy}}\rho_{axy}^2\,d\gamma_{xy} +\frac{1}{2}\int_0^{\gamma_{2xy}}\rho_{axy}^2\,d\gamma_{xy},\\ S_{yz}&=\frac{1}{2}\int_0^{\gamma_{1yz}}\rho_{ayz}^2\,d\gamma_{yz} +\frac{1}{2}\int_0^{\gamma_{2yz}}\rho_{ayz}^2\,d\gamma_{yz}. \end{aligned} \tag{8} \]

From expressions (7) it is evident that \(\rho_{axy}\) and \(\rho_{ayz}\) are functions of the angle \(\alpha\); therefore, when integrating expressions (8), it is necessary to express the angles \(\gamma_{xy}\) and \(\gamma_{yz}\) through \(\alpha\).

From Fig. 1 we obtain:

\[ d\gamma_{xy} = \frac{R_2^2-R_1R_2\cos\alpha} {R_1^2+R_2^2-2R_1R_2\cos\alpha}\,d\alpha, \]

\[ d\gamma_{yz} = \frac{R_2h\cos(\alpha-\omega)} {h^2+\bigl[R_1\sin\omega+R_2\sin(\alpha-\omega)\bigr]^2}\,d\alpha . \tag{9} \]

Using expressions (7) and (9), we obtain:

\[ S_{xy} = \frac{1}{2}\left[ \int_0^{\frac{\beta}{2}+\omega} \frac{(R_2^2-R_1R_2\cos\alpha)\,d\alpha} {\left(h^2+R_1^2+R_2^2-2R_1R_2\cos\alpha\right)^{3/2}} + \int_0^{\frac{\beta}{2}-\omega} \frac{(R_2^2-R_1R_2\cos\alpha)\,d\alpha} {\left(h^2+R_1^2+R_2^2-2R_1R_2\cos\alpha\right)^{3/2}} \right]. \tag{10} \]

\[ S_{yz} = \frac{1}{2}\left[ \int_0^{\frac{\beta}{2}+\omega} \frac{R_2h\cos(\alpha-\omega)\,d\alpha} {\left(h^2+R_1^2+R_2^2-2R_1R_2\cos\alpha\right)^{3/2}} + \int_0^{\frac{\beta}{2}-\omega} \frac{R_2h\cos(\alpha-\omega)\,d\alpha} {\left(h^2+R_1^2+R_2^2-2R_1R_2\cos\alpha\right)^{3/2}} \right]. \]

The integrals obtained do not reduce to elementary functions and are called elliptic integrals. Every elliptic in-

tegral can be represented as a linear combination of elementary functions and integrals of the 1st, 2nd, and 3rd kinds in Legendre normal form.

As a result of substituting \(\alpha=\pi-2\psi\), the integrals obtained can be reduced to normal form. After integration and transformations we obtain, in final form:

\[ \begin{aligned} S_{xy}={}& \frac{1}{2\sqrt{h^{2}+(R_{1}+R_{2})^{2}}} \Bigg[ 2K(\kappa)+ \left( \frac{R_{2}\kappa^{2}}{2R_{1}\kappa'^{2}} -\frac{\kappa^{2}}{2\kappa'^{2}}-1 \right) \times \\ &\times \Bigg( 2E(\kappa) - E\!\left(\frac{\pi-\left(\frac{\beta}{2}+\omega\right)}{2},\,\kappa\right) - E\!\left(\frac{\pi-\left(\frac{\beta}{2}-\omega\right)}{2},\,\kappa\right) + \\ &\qquad +\frac{\kappa^{2}\sin\left(\frac{\beta}{2}+\omega\right)} {2\sqrt{\,1-\kappa^{2}\cos^{2}\left(\frac{\beta}{4}+\frac{\omega}{2}\right)\,}} + \frac{\kappa^{2}\sin\left(\frac{\beta}{2}-\omega\right)} {2\sqrt{\,1-\kappa^{2}\cos^{2}\left(\frac{\beta}{4}-\frac{\omega}{2}\right)\,}} \Bigg) \\ &\qquad - F\!\left(\frac{\pi-\left(\frac{\beta}{2}+\omega\right)}{2},\,\kappa\right) - F\!\left(\frac{\pi-\left(\frac{\beta}{2}-\omega\right)}{2},\,\kappa\right) \Bigg]. \end{aligned} \tag{11} \]

\[ \begin{aligned} S_{yz}={}& \frac{h}{2R_{1}\sqrt{h^{2}+(R_{1}+R_{2})^{2}}} \Bigg\{ \Bigg[ F\!\left(\frac{\pi-\left(\frac{\beta}{2}+\omega\right)}{2},\,\kappa\right) + \\ &\quad + F\!\left(\frac{\pi-\left(\frac{\beta}{2}-\omega\right)}{2},\,\kappa\right) -2K(\kappa) + \frac{2-\kappa^{2}}{2(1-\kappa^{2})} \Bigg( 2E(\kappa) - \\ &\quad - E\!\left(\frac{\pi-\left(\frac{\beta}{2}+\omega\right)}{2},\,\kappa\right) - E\!\left(\frac{\pi-\left(\frac{\beta}{2}-\omega\right)}{2},\,\kappa\right) + \\ &\quad + \frac{\kappa^{2}\sin\left(\frac{\beta}{2}+\omega\right)} {2\sqrt{\,1-\kappa^{2}\cos^{2}\left(\frac{\beta}{4}+\frac{\omega}{2}\right)\,}} + \frac{\kappa^{2}\sin\left(\frac{\beta}{2}-\omega\right)} {2\sqrt{\,1-\kappa^{2}\cos^{2}\left(\frac{\beta}{4}-\frac{\omega}{2}\right)\,}} \Bigg) \Bigg]\cos\omega + \\ &\quad + \left[ \frac{1}{\sqrt{\,1-\kappa^{2}\cos^{2}\left(\frac{\beta}{4}-\frac{\omega}{2}\right)\,}} - \frac{1}{\sqrt{\,1-\kappa^{2}\cos^{2}\left(\frac{\beta}{4}+\frac{\omega}{2}\right)\,}} \right]\sin\omega \Bigg\}, \end{aligned} \]

where

\[ K(k) \quad \text{and} \quad E(k) \]

are complete elliptic integrals, respectively, of the 1st and 2nd kinds,

\[ F\!\left(\frac{\pi-\left(\frac{\beta}{2}\pm\omega\right)}{2},\,\kappa\right) \]

is an incomplete elliptic integral of the 1st kind,

\[ E\!\left(\frac{\pi-\left(\frac{\beta}{2}\pm\omega\right)}{2},\,\kappa\right) \]

is an incomplete elliptic integral of the 2nd kind,

\[ \kappa^2=\frac{4R_1R_2}{h^2+(R_1+R_2)^2} \]
—the principal modulus of the elliptic integrals,

\[ \kappa'^2=1-\kappa^2=\frac{h^2+(R_2-R_1)^2}{h^2+(R_1+R_2)^2} \]
—the complementary modulus of the elliptic integrals.

From the modulus of the elliptic integrals and the limits of integration, the values of the elliptic integrals can be found in the reference literature [5].

Thus, knowing the quantities \(S_{xy}\) and \(S_{yz}\), one can determine the components of the intensity \(H_z\) and \(H_x\) of the magnetic field at the point \(M\), produced by the current \(i\) flowing through conductor 2,

\[ \begin{aligned} H_x &= 2iS_{yz},\\ H_z &= 2iS_{xy}. \end{aligned} \tag{12} \]

Using expressions (1) and (12), one can find the axial and radial forces acting on conductor 1. In the SI system, the elementary forces acting on the element \(dl_1\), in its interaction with the whole of conductor 2, are:

axial

\[ dF_{a12}=10^{-7}i^2S_{yz}2R_1\cos\omega\,d\omega =10^{-7}i^2dc_{a12},\quad (\mathrm{N}) \tag{13} \]

radial

\[ dF_{p12}=10^{-7}i^2S_{xy}2R_1d\omega =10^{-7}i^2dc_{p12}\quad (\mathrm{N}). \]

It follows from (13) that the factor \(10^{-7}i^2\) in these expressions depends only on the magnitude of the currents, while the remaining factors \(dc_{a12}\) and \(dc_{p12}\) depend only on the mutual geometrical arrangement of the element \(dl_1\) and conductor 2. We shall call them the axial and radial increment of the geometrical coefficient of the contour; they have the following form:

\[ \begin{aligned} dc_{a12} &= \frac{h}{\sqrt{h^2+(R_1+R_2)^2}} \Bigg\{ \Bigg[ F\!\left(\frac{\pi-\left(\frac{\beta}{2}+\omega\right)}{2},\kappa\right) + F\!\left(\frac{\pi-\left(\frac{\beta}{2}-\omega\right)}{2},\kappa\right) -2K(\kappa) \\ &\quad +\frac{2-\kappa^2}{2(1-\kappa^2)} \Bigg( 2E(\kappa) - E\!\left(\frac{\pi-\left(\frac{\beta}{2}+\omega\right)}{2},\kappa\right) - E\!\left(\frac{\pi-\left(\frac{\beta}{2}-\omega\right)}{2},\kappa\right) \\ &\qquad + \frac{\kappa^2\sin\left(\frac{\beta}{2}+\omega\right)} {2\sqrt{1-\kappa^2\cos^2\left(\frac{\beta}{4}+\frac{\omega}{2}\right)}} + \frac{\kappa^2\sin\left(\frac{\beta}{2}-\omega\right)} {2\sqrt{1-\kappa^2\cos^2\left(\frac{\beta}{4}-\frac{\omega}{2}\right)}} \Bigg) \Bigg]\cos\omega \\ &\quad + \left[ \frac{1}{\sqrt{1-\kappa^2\cos^2\left(\frac{\beta}{4}-\frac{\omega}{2}\right)}} - \frac{1}{\sqrt{1-\kappa^2\cos^2\left(\frac{\beta}{4}+\frac{\omega}{2}\right)}} \right]\sin\omega \Bigg\}\cos\omega , \end{aligned} \tag{14} \]

\[ dc_{p12} = \frac{R_1}{\sqrt{h^2+(R_1+R_2)^2}} \left[ 2K(\kappa) + \left( \frac{R_2\kappa^2}{2R_1\kappa'^2} - \frac{\kappa^2}{2\kappa'^2} -1 \right) \left( 2E(\kappa)- \right. \]

\[ \begin{aligned} &-E\left(\frac{\pi-\left(\frac{\beta}{2}+\omega\right)}{2},\,\kappa\right) -E\left(\frac{\pi-\left(\frac{\beta}{2}-\omega\right)}{2},\,\kappa\right)+ \\ &\quad+\frac{\kappa^{2}\sin\left(\frac{\beta}{2}+\omega\right)} {2\sqrt{1-\kappa^{2}\cos^{2}\left(\frac{\beta}{4}+\frac{\omega}{2}\right)}} +\frac{\kappa^{2}\sin\left(\frac{\beta}{2}-\omega\right)} {2\sqrt{1-\kappa^{2}\cos^{2}\left(\frac{\beta}{4}-\frac{\omega}{2}\right)}}\Bigg) - \\ &\quad- F\left(\frac{\pi-\left(\frac{\beta}{2}+\omega\right)}{2},\,\kappa\right) - F\left(\frac{\pi-\left(\frac{\beta}{2}-\omega\right)}{2},\,\kappa\right)\Bigg]. \end{aligned} \]

To find the resultant axial and radial forces acting on conductor 1, expressions (13) must be integrated with respect to the angle \(\omega\), varying from \(-\frac{\varphi}{2}\) to \(+\frac{\varphi}{2}\). The integration of such expressions presents known difficulties. However, after constructing curves of the distribution of specific axial and radial loads along conductor 1 from expressions (13) as a function of the angle \(\omega\), it proves possible to replace expressions (13) in the interval \(-\frac{\pi}{2}<\omega<+\frac{\pi}{2}\) by the form

\[ \begin{aligned} dF_{a12}={}&10^{-7}i^{2}\frac{2h}{\sqrt{h^{2}+(R_{1}+R_{2})^{2}}} \Bigg[ \frac{h^{2}+R_{1}^{2}+R_{2}^{2}}{h^{2}+(R_{2}-R_{1})^{2}} \left( E(\kappa)-E\left(\frac{\pi}{2}-\frac{\beta}{4},\,\kappa\right)+ \right.\\ &\left.\qquad\qquad\qquad +\frac{\kappa^{2}\sin\frac{\beta}{2}} {2\sqrt{1-\kappa^{2}\cos^{2}\frac{\beta}{4}}} \right) +F\left(\frac{\pi}{2}-\frac{\beta}{4},\,\kappa\right)-K(\kappa) \Bigg]\cos^{2}\omega\,d\omega, \end{aligned} \tag{15} \]

\[ \begin{aligned} dF_{p12}={}&10^{-7}i^{2}\frac{2R_{1}}{\sqrt{h^{2}+(R_{1}+R_{2})^{2}}} \Bigg[ \frac{R_{2}^{2}-R_{1}^{2}-h^{2}}{h^{2}+(R_{2}-R_{1})^{2}} \left( E(\kappa)-E\left(\frac{\pi}{2}-\frac{\beta}{4},\,\kappa\right)+ \right.\\ &\left.\qquad\qquad\qquad +\frac{\kappa^{2}\sin\frac{\beta}{2}} {2\sqrt{1-\kappa^{2}\cos^{2}\frac{\beta}{4}}} \right) -F\left(\frac{\pi}{2}-\frac{\beta}{4},\,\kappa\right)+K(\kappa) \Bigg]\left(1-\frac{\omega^{2}}{\pi^{2}}\right)d\omega . \end{aligned} \]

After integration of expressions (15), the resultant forces acting on conductor 1 in the axial and radial directions are written as the expressions

\[ \begin{aligned} F_{a12}={}&10^{-7}\cdot i^{2}\frac{h}{\sqrt{h^{2}+(R_{2}+R_{1})^{2}}} \Bigg[ \frac{h^{2}+R_{1}^{2}+R_{2}^{2}}{h^{2}+(R_{2}-R_{1})^{2}} \left( E(\kappa)-E\left(\frac{\pi}{2}-\frac{\beta}{4},\,\kappa\right)+ \right.\\ &\left.\qquad\qquad\qquad +\frac{\kappa^{2}\sin\frac{\beta}{2}} {2\sqrt{1-\kappa^{2}\cos^{2}\frac{\beta}{4}}} \right) +F\left(\frac{\pi}{2}-\frac{\beta}{4},\,\kappa\right)-K(\kappa) \Bigg](\varphi+\sin\varphi). \end{aligned} \tag{16} \]

\[ \begin{aligned} F_{p12}={}&10^{-7}\cdot i^{2}\frac{R_{1}}{\sqrt{h^{2}+(R_{2}+R_{1})^{2}}} \Bigg[ \frac{R_{2}^{2}-R_{1}^{2}-h^{2}}{h^{2}+(R_{2}-R_{1})^{2}} \left( E(\kappa)-E\left(\frac{\pi}{2}-\frac{\beta}{4},\,\kappa\right)+ \right. \end{aligned} \]

\[ + \frac{\kappa^2 \sin \dfrac{\beta}{2}} {2\sqrt{1-\kappa^2\cos^2\dfrac{\beta}{4}}}\right) - F\left(\frac{\pi}{2}-\frac{\beta}{4},\kappa\right)+K(\kappa)\right] \left(\varphi-\frac{\varphi^3}{12\pi^2}\right). \]

When the radii of the conductors are equal, \(R_1=R_2=R\), we obtain:

\[ \begin{aligned} F_{a12} &= 10^{-7}\, i^2 \frac{h}{\sqrt{h^2+4R^2}} \left[ \frac{h^2+2R^2}{h^2} \left( E(\kappa)-E\left(\frac{\pi}{2}-\frac{\beta}{4},\kappa\right) + \frac{\kappa^2 \sin \dfrac{\beta}{2}} {2\sqrt{1-\kappa^2\cos^2\dfrac{\beta}{4}}} \right) \beta +F\left(\frac{\pi}{2}-\frac{\beta}{4},\kappa\right)-K(\kappa) \right] (\varphi+\sin\varphi), \\[1.2em] F_{p12} &= 10^{-7}\, i^2 \frac{R}{\sqrt{h^2+4R^2}} \left[ K(\kappa)- \left( E(\kappa)-E\left(\frac{\pi}{2}-\frac{\beta}{4},\kappa\right) + \frac{\kappa^2 \sin \dfrac{\beta}{2}} {2\sqrt{1-\kappa^2\cos^2\dfrac{\beta}{4}}} \right) -F\left(\frac{\pi}{2}-\frac{\beta}{4},\kappa\right) \right] \left(\varphi-\frac{\varphi^3}{12\pi^2}\right). \end{aligned} \tag{17} \]

In the case of a curvilinear conductor carrying current, its individual sections interact with one another, creating forces coinciding with \(F_{p12}\). This force can be found using the preceding expression. If two curvilinear conductors of identical form and cross section are superposed until they coincide completely, then the distance \(h\) between the conductors becomes the quantity \(g_{11}\), called the mean geometric distance of the area of the conductor cross section from itself [4]. The modulus of the elliptic integrals in this case is determined as

\[ \kappa_{11}^{2}=\frac{4R_1^2}{4R_1^2+g_{11}^{2}}. \]

The total radial force acting on curvilinear conductor 1 from itself is

\[ F_{p11}=10^{-7} i^2 c_{p11}, \]

where

\[ c_{p11}= \frac{R_1}{\sqrt{4R_1^2+g_{11}^{2}}} \left[ K(\kappa_{11})-E(\kappa_{11}) -F\left(\frac{\pi}{2}-\frac{\varphi}{4},\kappa_{11}\right) + E\left(\frac{\pi}{2}-\frac{\varphi}{4},\kappa_{11}\right) - \frac{\kappa_{11}^{2}\sin\dfrac{\varphi}{2}} {2\sqrt{1-\kappa_{11}^{2}\cos^2\dfrac{\varphi}{4}}} \right] \left(\varphi-\frac{\varphi^3}{12\pi^2}\right). \tag{18} \]

The total radial force acting on conductor 1 in the radial direction is

\[ F_{p1}=F_{p12}+F_{p11}. \]

The geometric coefficients \(c_{a12}\), \(c_{p12}\), \(c_{p11}\) are dimensionless quantities.

From expressions (14) and (18) it follows that, when the scale of the geometrical dimensions of the model is changed (Fig. 1), the geometrical coefficients remain constant. This means that, in accordance with the law of geometrical similarity, conclusions drawn from the results of experiments on a model of end parts can be extended to real, geometrically similar end parts of the windings of electrical machines.

It should be noted that, in the case of symmetrical curvilinear conductors, the axial geometrical coefficient does not change when the acting conductor is replaced by the conductor being acted upon, and vice versa, that is,

\[ c_{a12}=c_{a21}. \]

This can be justified by the fact that the forces in the axial direction applied to the first conductor due to the action of the second, as well as the forces applied to the second conductor due to the action of the first, are parallel to one another. The change in the mutual inductance of the conductors when either of them is displaced in the direction of the force vector should not depend on which of the conductors is displaced. Consequently, the electrodynamic forces, proportional to the change in mutual inductance, do not depend on the action of which conductor on the other is being considered.

For a system of conductors corresponding to the arrangement of the end part of a concentric single-layer distributed winding (Fig. 2), the total geometrical coefficient for any conductor of the system, in its interaction with all the conductors, is equal to the sum of the partial geometrical coefficients. The sign of a partial coefficient is determined by the direction of the currents in the conductors. For the conductors (Fig. 2), when the current direction in them is the same, the total

Fig. 2. Diagram of the arrangement of conductors of the end part of a single-layer concentric winding

the axial geometric coefficients for the first, second, etc. conductors are determined as

\[ c_{a1}=c_{a12}+c_{a13}+c_{a14}+c_{a15}+c_{a16}. \]

\[ c_{a2}=-c_{a21}+c_{a23}+c_{a24}+c_{a25}+c_{a26}. \]

\[ c_{a3}=-c_{a31}-c_{a32}+c_{a34}+c_{a35}+c_{a36}. \]

\[ \cdots \]

\[ c_{a6}=-c_{a61}-c_{a62}-c_{a63}-c_{a64}-c_{a65}. \]

The total axial forces acting on the first, second, etc. conductors when a current \(i\) flows through them are

\[ F_{a1}=10^{-7} i^2 c_{a1}, \]

\[ F_{a2}=10^{-7} i^2 c_{a2}, \]

\[ \cdots \]

\[ F_{a6}=10^{-7} i^2 c_{a6}. \]

To assess the acceptability of the described method for calculating electrodynamic forces between the curvilinear parts of the conductors of the end portion of a single-layer concentric winding, without taking the surrounding surfaces into account, calculations were made of axial forces, which were checked experimentally on a model by means of strain-gauge sensors [1]. The forces were determined for a section \(\varphi=57^\circ\) of each conductor. The radii of the curvilinear parts for the model conductors were \(R=24\) cm. The distance between the axes of adjacent conductors was \(h=1.7\) cm. Figure 3 gives the distribution of axial forces along the end portion of the model at a current of 4 kA. For comparison, the results of the experimental investigation are also given there.

Fig. 3. Distribution of axial forces along the end portion of a single-layer winding.
— calculated curve.
— — — experimental curve.

studies on a model whose parameters were used in the calculation. The satisfactory agreement between the calculated and experimental results confirms the possibility of applying the method considered for calculating electrodynamic forces in the end portions of the windings of electrical machines.

REFERENCES

1. V. I. Popov, G. A. Sipaylov. An installation for the experimental investigation of electrodynamic forces in the end portions of the windings of an impulse generator. Izvestiya TPI, vol. 190, 1968.

2. G. B. Kholyavskii. Calculation of electrodynamic forces in electrical apparatus. GEI, 1962.

3. A. A. Halacsi. Practical Calculation of Magnetizing Force. Proc. IEE, Vol. 97, Part I, No. 104.

4. L. A. Tseitlin. Inductances of wires and circuits. GEI, 1950.

5. E. Jahnke, F. Emde, F. Lösch. Special Functions. “Nauka,” 1964.