CALCULATION OF HIGHER HARMONICS IN THE E.M.F. CURVE OF A SURGE GENERATOR

G. A. SIPAILOV, K. A. KHORKOV

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

One of the design features of a surge-power generator is the presence of slots for a damper winding on the large tooth of the rotor. Being less deep in comparison with the main slots occupied by the excitation winding, these slots exert the same influence on the permeance of the machine air gap as do the main rotor slots [1, 2]. Owing to this, the expression for the permeance of the rotor tooth layer with a smooth stator can be written as

\[ \lambda_{\mathrm{p}}=\frac{1}{k_{\delta \mathrm{p}}} -\sum_m (-1)^m \lambda_m \cos m z_2' \frac{\pi}{\tau} x . \tag{1} \]

The permeance of the stator tooth layer with a smooth rotor has an analogous expression:

\[ \lambda_{\mathrm{c}}=\frac{1}{k_{\delta \mathrm{c}}} -\sum_k (-1)^k \lambda_k \cos k z_1' \left(\frac{\pi}{\tau}x+\omega t\right), \tag{2} \]

where \(k_{\delta \mathrm{p}}, k_{\delta \mathrm{c}}\) are the air-gap coefficients of the rotor and stator,

\(z_2', z_1'\) are the numbers of tooth divisions of the rotor and stator,

\(\lambda_m, \lambda_k\) are the amplitudes of the permeance harmonics of the rotor and stator tooth layers,

\(m, k\) are the order indices of the permeance harmonics of the rotor and stator,

\(\tau\) is the pole pitch of the machine,

\(x\) is the running coordinate.

The complete relative permeance of the machine air gap with double-sided slotting can be calculated as the product of the permeances of the tooth layers of each side [3]; then:

\[ \lambda = \frac{1}{k_\delta} -\frac{1}{k_{\delta \mathrm{p}}} \sum_k (-1)^k \lambda_k \cos k z_1' \left(\frac{\pi}{\tau}x+\omega t\right) - \tag{3} \]

\[ -\frac{1}{k_{\delta \mathrm{c}}} \sum_m (-1)^m \lambda_m \cos m z_2' \frac{\pi}{\tau}x + \]

\[ +\sum_k \sum_m (-1)^{m+k}\lambda_k\lambda_m \cos k z_1'\left(\frac{\pi}{\tau}x+\omega t\right) \cos m z_2'\frac{\pi}{\tau}x . \]

The amplitudes of the permeances \(\lambda_k\) and \(\lambda_m\), as applied to asynchronous machines, which as a rule have semi-closed slots on the stator—

on the stator and on the rotor, are determined as the product of the coefficients \(\beta\) and \(\eta_i\) [3], where

\[ \beta=\frac{B_0}{B_m}=f\left(\frac{b_n}{\delta}\right), \qquad \eta_i=f\left(\frac{b_n}{t}\ \text{and}\ \frac{t}{\delta}\right). \]

The values of the coefficients \(\beta\) and \(\eta_i\) are given as curves in Fig. 1.

Fig. 1. For calculating the amplitudes of the harmonic permeances of toothed zones.

Since surge generators have predominantly open slots on the stator and on the rotor, for them the calculation of the permeances \(\lambda_k\) and \(\lambda_m\) by the method indicated above leads to overestimated values of the emf of the tooth harmonics. In this case the calculated values are obtained closer to the actual ones if, into the expressions by which \(\lambda_k\) and \(\lambda_m\) are determined, one introduces a coefficient \(\alpha\), taking into account the decrease in the amplitude of the tooth fluxes of one side due to the presence of open slots on the other side; then

\[ \lambda_{k(m)}=\alpha\beta\eta_i. \tag{4} \]

Here the value of the coefficient \(\alpha\) may, with a sufficient degree of accuracy, be taken equal to

\[ \frac{1}{k_\delta}. \]

The field in the air gap of the machine is determined from the expression

\[ B=\frac{\mu_0}{\delta}\lambda F, \tag{5} \]

where \(F\) is the magnetizing force of the excitation winding,
\(\delta\) is the magnitude of the one-sided air gap of the machine.

In the general case the curve of the magnetizing force of the rotor of a surge generator is an isosceles trapezoid, symmetric with respect to the axis \(d\) of the poles. Expansion of the m.m.f. curve into a series gives:

\[ F=\frac{4}{\pi}F_m\sum_i \frac{\sin i\gamma \frac{\pi}{2}} {i^2\gamma\frac{\pi}{2}} \cos i\frac{\pi}{\tau}x, \tag{6} \]

where \(F_m = I_{\mathrm{в}} w_{\mathrm{в}}\) is the amplitude of the magnetizing force of the field winding,

\(i\) is the order of the spatial harmonic of the magnetizing force.

The resultant field of the machine may be regarded as a set of spatial harmonics of the magnetizing force, stationary relative to the rotor axis \(d\) and rotating together with the rotor. Integrating expression (5), taking (3) and (6) into account, within the limits from \(-\dfrac{\tau}{2}\) to \(+\dfrac{\tau}{2}\), gives the magnitude of the rotor flux per unit length of the machine.

\[ \Phi = \int\limits_{-\frac{\tau}{2}}^{+\frac{\tau}{2}} B\,dx = \frac{8\mu_0 F_m}{\pi^2 \delta \gamma} \sum_i \frac{1}{i^2}\, \sin i\gamma \frac{\pi}{2} \cdot \int\limits_{-\frac{\tau}{2}}^{+\frac{\tau}{2}} \lambda \cos i\frac{\pi}{\tau}x \cdot dx . \tag{7} \]

Let us consider the individual components of the total flux obtained after integrating expression (7).

1. The main flux of the machine is the result of the interaction of the m.m.f. of the field winding with the permeance of the reduced air gap, that is, the flux due to the spatial harmonics of the field-winding field with infinitely narrow stator and rotor slot openings:

\[ \Phi_0 = \frac{8\mu_0 F_m}{\pi^2 \delta k_\delta \gamma} \sum_i \frac{1}{i^2}\, \sin i\gamma \frac{\pi}{2} \int\limits_{-\frac{\tau}{2}}^{+\frac{\tau}{2}} \cos i\frac{\pi}{\tau}x \cdot dx . \]

Hence the flux of the \(i\)-th spatial harmonic of the machine is

\[ \Phi_i = \frac{16}{\pi^3}\, \frac{\tau}{\gamma}\, \frac{\mu_0}{\delta k_\delta}\, F_m\, \frac{1}{i^3}\, \sin i\gamma \frac{\pi}{2}\, \sin i\frac{\pi}{2} \tag{8} \]

and, in particular, the flux of the first harmonic is

\[ \Phi_1 = \frac{16}{\pi^3}\, \frac{\tau}{\gamma}\, \frac{\mu_0}{\delta k_\delta}\, F_m \sin \gamma \frac{\pi}{2}. \]

Expressing the fluxes of the \(i\)-th spatial harmonics through the flux of the first harmonic, we obtain

\[ \Phi_i = \Phi_1 \frac{1}{i^3} \frac{\sin i\gamma \dfrac{\pi}{2}} {\sin \gamma \dfrac{\pi}{2}} \sin i\frac{\pi}{2}. \]

If we denote by

\[ B_{mi} = \frac{\mu_0}{\delta k_\delta} F_{bi}, \qquad F_{bi} = \frac{8}{\pi^2 i^2} k_{pi} F_m, \qquad k_{pi} = \frac{\sin i\gamma \dfrac{\pi}{2}} {\gamma \sin i\dfrac{\pi}{2}}, \]

then expression (8) for the \(i\)-th harmonic is obtained in the form in which it is usually given in textbooks:

\[ \Phi_i = \frac{2}{\pi}\,\tau l\,\frac{1}{i}\,B_{mi}. \]

1. Tooth fluxes are the result of the interaction of the m.m.f. of the excitation winding with the permeance of the tooth layers of the real air gap, that is, fluxes associated with the permeance of the rotor tooth layer \(\Phi_{z'_2}\) and of the stator tooth layer \(\Phi_{z'_1}\):

\[ \Phi_{z'_2} = \frac{8}{\pi^2}\frac{\mu_0}{\delta k_\delta}\frac{F_m}{\gamma}k_{\mathrm{av}} \sum_i \frac{1}{i^2}\sin i\gamma\frac{\pi}{2} \times \sum_m \lambda_m(-1)^{m+1} \int_{-\frac{\tau}{2}}^{+\frac{\tau}{2}} \cos m z'_2\frac{\pi}{\tau}x\cdot \cos i\frac{\pi}{\tau}x\cdot dx, \tag{9} \]

\[ \Phi_{z'_1} = \frac{8}{\pi^2}\frac{\mu_0}{\delta k_\delta}\frac{F_m}{\gamma}k_{\delta c} \sum_i \frac{1}{i^2}\sin i\gamma\frac{\pi}{2} \times \sum_k \lambda_k(-1)^{k+1} \int_{-\frac{\tau}{2}}^{+\frac{\tau}{2}} \cos i\frac{\pi}{\tau}x\cdot \cos k z'_1\left(\frac{\pi}{\tau}x+\omega t\right)\cdot dx. \tag{10} \]

Hence the flux of some harmonic due to the \(m\)-th order of the permeance of the rotor tooth layer and the \(i\)-th harmonic of the m.m.f. will be

\[ \Phi_{\nu m} = \frac{1}{2}\Phi_i k_{\mathrm{av}}\lambda_m(-1)^{m+1} \left[ \frac{i}{i-mz'_2} \frac{\sin (i-mz'_2)\frac{\pi}{2}} {\sin i\frac{\pi}{2}} + \frac{i}{i+mz'_2} \frac{\sin (i+mz'_2)\frac{\pi}{2}} {\sin i\frac{\pi}{2}} \right], \tag{11} \]

and the flux of the \(\nu\)-th harmonic due to the \(k\)-th order of the permeance of the stator tooth layer and the \(i\)-th harmonic of the m.m.f. is equal to

\[ \Phi_{\nu k} = \frac{1}{2}\Phi_i k_{\delta c}\lambda_k(-1)^{k+1} \left[ \frac{i}{i-kz'_1} \frac{\sin (i-kz'_1)\frac{\pi}{2}} {\sin i\frac{\pi}{2}} + \frac{i}{i+kz'_1} \frac{\sin (i+kz'_1)\frac{\pi}{2}} {\sin i\frac{\pi}{2}} \right]\cos kz'_1\omega t. \tag{12} \]

It should be noted that, due to the permeance harmonics of the rotor tooth layer, we obtain flux harmonics that are stationary with respect to the rotor axis \(d\), that is, rotating synchronously with the rotor. Of all the harmonics associated with the rotor slotting, the most significant are the harmonics obtained from the first harmonic of the m.m.f. The number of these harmonics, which are customarily called tooth harmonics, is

\[ \nu_m = 1 \mp mz'_2. \]

Of the tooth harmonics, the harmonic components of the first order, \(m=1\), have the greatest amplitude. In this case

\[ \nu_m = 1 \mp z'_2 . \]

When strongly pronounced harmonics of the third, fifth, seventh, etc. orders are present in the mmf field of the excitation winding, harmonic fluxes appear:

\[ 3 \mp mz'_2,\quad 5 \mp mz'_2 \quad \text{etc.} \]

Owing to the harmonics of the conductance of the stator tooth layer, rotating relative to the rotor axis \(d\) with angular frequency \(\omega k z'_1\), flux harmonics \(\Phi_{z'_1}\) are produced. Of all the harmonics of this flux, the most dangerous are the harmonics from the first mmf harmonic, \(i=1\), which are usually called tooth harmonics. Their number is

\[ \nu_k = 1 \mp kz'_1 . \]

The harmonics due to conductance of the first order, \(k=1\), have the greatest amplitude. In this case

\[ \nu_k = 1 \mp z'_1 . \]

When significant harmonics of the third, fifth, etc. orders are present in the mmf curve, it is also necessary to consider the harmonic fluxes:

\[ 3 \mp kz'_1,\quad 5 \mp kz'_1 \quad \text{etc.} \]

If a damper system is present on the rotor, its damping action must be taken into account. In [4] the damping coefficient of the higher harmonics of the flux \(\Phi z'_1\) by the damper winding is given as:

\[ D_\nu = 1 - \frac{f_{2\nu}^{\,2} f_{\mathrm{sk}\nu}^{\,2}}{1 + \sigma_{2\nu}}, \tag{13} \]

where \(f_{2\nu}\) is the winding factor of the \(\nu\)-th harmonic,
\(f_{\mathrm{sk}\nu}\) is the slot-skew factor,
\(\sigma_{2\nu}\) is the rotor leakage coefficient.

The value \(\sigma_{2\nu}\) may be taken equal to zero; then, for \(f_{\mathrm{sk}\nu}=1\),

\[ D_\nu = 1 - f_{2\nu}^{\,2}, \]

where

\[ f_{2\nu} = \frac{ \sin \nu \dfrac{\pi p}{z'_2} }{ \nu \dfrac{\pi p}{z'_2} }. \]

The dependence \(D_\nu = f\!\left(\dfrac{\nu p}{z'_2}\right)\) is presented in Fig. 2, from which it follows that the lower harmonics are almost completely damped by the damper winding. Harmonics whose number is close to the number of rotor teeth are practically not damped. The experimental data given in Table 2 confirm that the emf curve of the stator winding contains clearly pronounced tooth harmonics with numbers \(1 \mp z'_1\).

Fig. 2. Dependence of the damping coefficient of higher harmonics on the ratio between the harmonic number and the number of rotor damper bars.

1. The flux associated with the permeance of the toothed zones of the stator, rotor, and the d.c. field winding

\[ \Phi_{z'_1 z'_2} = \frac{8}{\pi^2}\frac{\mu_0}{\delta k_\delta}\frac{F_m}{\gamma} k_\delta \sum_i \frac{1}{i^2}\sin i\gamma \frac{\pi}{2} \sum_k \sum_m (-1)^{k+m}\lambda_k \lambda_m \times \]
\[ \times \cos kz'_1 \omega t \int_{-\tau/2}^{\tau/2} \cos i\frac{\pi}{\tau}x \cdot \cos mz'_2 \frac{\pi}{\tau}x \cdot \cos kz'_1 \frac{\pi}{\tau}x \cdot dx . \tag{14} \]

Whence the flux of any one harmonic due to the \(k\)-th and \(m\)-th orders of the permeances of the stator and rotor tooth layers and the \(i\)-th harmonic of the d.c. field winding will be equal to

\[ \Phi_{\nu km} = \frac{1}{4}\Phi_i k_\delta \lambda_k \lambda_m (-1)^{k+m} \left[ \frac{i}{i \pm kz'_1 \mp mz'_2} \frac{\sin (i \pm kz'_1 \mp mz'_2)\frac{\pi}{2}} {\sin i\frac{\pi}{2}} + \right. \]
\[ \left. + \frac{i}{i \pm kz'_1 \pm mz'_2} \frac{\sin (i \pm kz'_1 \pm mz'_2)\frac{\pi}{2}} {\sin i\frac{\pi}{2}} \right] \cos kz'_1 \omega t . \tag{15} \]

Of the entire frequency spectrum of the flux \(\Phi_{\nu km}\), only the difference harmonics play any role, their frequency being equal to:

\[ \nu_{km}= i \pm (kz'_1 - mz'_2). \]

If a damping winding is present on the rotor, these fluxes will be strongly damped; therefore the fluxes \(\Phi_{\nu km}\) may practically be neglected.

Table 1

According to the method set forth, the fluxes and the e.m.f.’s induced by them in a coil with diametral pitch, located on the stator, were calculated.

Table 2

The calculations were carried out for a machine with a stator having 24 semi-closed slots with a 3 mm opening and a tooth pitch of 13.6 mm, and with a set of rotors having numbers of tooth divisions of 20, 26, and 28. The geometric parameters of the rotor tooth zones and the calculated coefficients are presented in Table 1. The results of the calculation of the tooth harmonics and the experimental data are given in Table 2. As follows from Table 2, the agreement of the calculated data with the experiment is satisfactory.

REFERENCES

1. G. A. Sipaylov, K. A. Khorkov. On higher harmonics in the e.m.f. curve of a surge generator. Interuniversity Collection of Works, issue 3. Questions of the Theory and Design of Electrical Machines, Novosibirsk, 1963.

2. G. A. Sipaylov, K. A. Khorkov. On the choice of the rotor winding of a surge generator. Izvestiya TPI, vol. 138, 1965.

3. A. I. Voldek. The influence of nonuniformity of the air gap on the magnetic field of an asynchronous machine. Elektrichestvo, No. 12, 1951.

4. T. G. Soroker. Differential leakage of polyphase synchronous motors. VEI, No. 6, 1956.