CONTACTLESS SINGLE-PHASE SYNCHRONOUS GENERATOR

A. I. Skorospeleshkin, M. L. Kostyrev, A. P. Felzing

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

Single-phase synchronous generators are becoming increasingly widespread as autonomous sources of electrical energy in road construction, agriculture, transport, aviation, and also in various industrial and domestic devices.

As single-phase sources of increased frequency, inductor generators have become widely used; their substantial shortcomings are their considerable overall dimensions and weight, incomplete utilization of active materials, and also the dependence of the shape of the output-voltage curve on the magnitude and character of the load.

A contactless single-phase generator of increased frequency of the combined type appears highly promising. Combining two machines, an exciter and a generator, in one magnetic core makes it possible to construct the generator as a single-machine unit and to reduce the overall dimensions and weight of the machine.

It should be noted that the technical literature presents analytical studies mainly of single-phase contact generators, whereas for contactless generators only some experimental studies are given [1, 2, 3]. The theoretical analysis of single-phase machines is complicated by the fact that the differential equations describing even the steady-state operating regime of the generator are equations with periodic coefficients and are not amenable to linear transformations, as in ordinary machines.

The present article gives some theoretical and experimental studies of the operation of a contactless single-phase synchronous generator of increased frequency of the combined type. The generator studied (Fig. 1) is a cascade connection of an inverted-type synchronous generator and an asynchronous frequency converter combined in a common magnetic core. When voltage is applied to winding $W_f$ and the rotor is rotated, emf’s are induced in windings $W_i$ and $W_s$ with frequencies respectively

\[ f = p_1 \cdot n, \tag{1} \]

\[ f_s = \left(\frac{p_2}{p_1} \pm 1\right) f = (k \pm 1)f, \tag{2} \]

where $n$ is the rotational speed of the rotor,
$p_1, p_2$ are the numbers of pole pairs of the first and second cascades, respectively.

Fig. 1

Fig. 1. Diagram of a contactless single-phase synchronous generator.

\[ k=\frac{p_2}{p_1}. \]

The sign \((+)\) in (2) refers to the case when the rotor and the field rotate in the same direction; the sign \((-)\), in opposite directions. The first mode is the most expedient, since in this case there are no internal closed energy cycles [4]. In what follows it is taken as the principal mode.

One of the main features of the operation of a single-phase generator under load is the presence of a reverse synchronous field. This field induces in winding \(W_2\) an emf with frequency

\[ f^s=(2k+1)f, \tag{3} \]

under the action of which a current of the same frequency will flow through the rotor windings. In this case the field produced by winding \(W_1\) will rotate in the same direction as the rotor and induce in winding \(W_f\) an emf with frequency

\[ f_f=2(k+1)f \tag{4} \]

The current of this frequency, flowing through winding \(W_f\), creates a pulsating field, the backward-rotating component of which induces in winding \(W_1\) an emf with frequency

\[ f^f=(2k+3)f. \tag{5} \]

Winding \(W_2\) creates a field that induces in the output winding an emf with frequency

\[ f_{s3}=3(k+1)f. \tag{6} \]

Thus, higher harmonics appear in the output-voltage curve, caused by the presence of the reverse field.

Fig. 2. Coordinate axes of the generator

To formulate the differential equations it is convenient to choose the coordinate axes \((\alpha\) and \(\beta)\) on the rotor (Fig. 2). On the basis of generally accepted assumptions [6], we write the equilibrium equations for the machine circuits for instantaneous values:

\[ \begin{aligned} u_f&=(R_f+pL_f)i_f+pM_{1f}\cos\Theta_1 i_\alpha-pM_{1f}\sin\Theta_1 i_\beta,\\ 0&=pM_{f1}\cos\Theta_1 i_f+(R_R+pL_R)i_\alpha+pM_{s2}\cos\Theta_2 i_s,\\ 0&=-pM_{f1}\sin\Theta_1 i_f+(R_R+pL_R)i_\alpha+pM_{s2}\sin\Theta_2 i_s,\\ -u_s&=pM_{2s}\cos\Theta_2 i_\alpha+pM_{2s}\sin\Theta_2 i_\beta+(R_s+pL_s)i_s, \end{aligned} \tag{7} \]

where

\(R_f,\ R_R=R_1+R_2,\ R_s,\ L_f,\ L_R=L_1+L_2,\ L_s,\ M_{f1},\ M_{1f},\ M_{2s},\ M_{s2}\) are the active resistances and the coefficients of self- and mutual inductance of the machine windings;

\(\Theta_1,\ \Theta_2\) are the angles between the axes of the windings.

The system of equations (7) contains variable coefficients, which cannot be eliminated by means of known transformations [7]. For the steady-state condition, the solution of such a system of equations can be obtained if the character of the variation of the machine currents is known [6].

According to (1–6), the currents in the circuits of the machine for the steady-state condition should be sought in the following form:

\[ \begin{aligned} i_f&=I_f=I_{f2}\cos[(k+1)2\omega t+\varphi_{f2}]+\cdots,\\ i_s&=I_{s1}\cos[(k+1)\omega t+\varphi_{s1}] +I_{s3}\cos[3(k+1)\omega t+\varphi_{s3}]+\cdots,\\ i_\alpha&=I_{\alpha1}\cos(\omega t+\varphi_{\alpha1}) +I_\alpha^{2k+1}\cos[(2k+1)\omega t+\varphi_\alpha^{2k+1}]+\\ &\quad+I_\alpha^{2k+3}\cos[(2k+3)\omega t+\varphi_\alpha^{2k+3}]+\cdots,\\ i_\beta&=-I_{\beta1}\sin(\omega t+\varphi_{\beta1}) +I_\beta^{2k+1}\sin[(2k+1)\omega t+\varphi_\beta^{2k+1}]-\\ &\quad-I_\beta^{2k+3}\sin[(2k+3)\omega t+\varphi_\beta^{2k+3}]+\cdots . \end{aligned} \tag{8} \]

Taking into account only the constant component of the current in winding \(W_f\), the expressions for the currents will have the form

\[ \begin{aligned} i_f&=I_f,\\ i_s&=I_{s1}\cos[(k+1)\omega t+\varphi_{s1}],\\ i_\alpha&=I_{\alpha1}\cos(\omega t+\varphi_{\alpha1}) +I_\alpha^{2k+1}\cos[(2k+1)\omega t+\varphi_\alpha^{2k+1}],\\ i_\beta&=-I_{\beta1}\sin(\omega t+\varphi_{\beta1}) +I_\beta^{2k+1}\sin[(2k+1)\omega t+\varphi_\beta^{2k+1}]. \end{aligned} \tag{9} \]

Substituting (9) into (7), after certain transformations we obtain the following expressions for the equations of voltage balance of the machine circuits:

\[ u_f=R_fI_f-M_{1f}2(k+1)\omega \sin[2(k+1)\omega t+\varphi^{2k+1}+\delta_{01}]I^{2k+1}, \]

\[ \begin{aligned} 0={}&M_{f1}\omega\sin(\omega t+\delta_{01})I_f +\{R_R\cos(\omega t+\varphi_{\alpha1}) -L_R\omega\sin(\omega t+\varphi_{\alpha1})\}I_{\alpha1}+\\ &+\{R_R\cos[(2k+1)\omega t+\varphi_\alpha^{2k+1}] -L_R(2k+1)\omega\sin[(2k+1)\omega t+\varphi_\alpha^{2k+1}]\}I_\alpha^{2k+1}-\\ &-M_{s2}\left(k+\frac12\right)\omega \sin[(2k+1)\omega t+\varphi_{s1}+\delta_{02}]I_{s1}-\\ &-M_{s2}\frac{\omega}{2}\sin(\omega t+\varphi_{s1}-\delta_{02})I_{s1}, \end{aligned} \]

\[ \begin{aligned} 0={}&-M_{f1}\omega\cos(\omega t+\delta_{01})I_f -\{R_R\sin(\omega t+\varphi_{\beta1}) +L_R\omega\cos(\omega t+\varphi_{\beta1})\}I_{\beta1}+\\ &+\{R_R\sin[(2k+1)\omega t+\varphi_\beta^{2k+1}] +L_R(2k+1)\omega\cos[(2k+1)\omega t+\varphi_\beta^{2k+1}]\}I_\beta^{2k+1}+\\ &+M_{s2}\left(k+\frac12\right)\omega \cos[(2k+1)\omega t+\varphi_{s1}+\delta_{02}]I_{s1}- \end{aligned} \tag{10} \]

\[ - M_{S2}\frac{\omega}{2}\cos(\omega t+\varphi_{S1}-\delta_{02}) I_{S1}, \]

\[ \begin{aligned} 0={}&-M_{S2}(k+1)\omega\sin[(k+1)\omega t+\varphi_1+\delta_{02}]I_1\\ &-M_{eS}(k+1)\omega\sin[(k+1)\omega t+\varphi^{2k+1}-\delta_{02}]I^{2k+1}\\ &+R\cos[(k+1)\omega t+\varphi_{S1}]I_S -L(k+1)\omega\sin[(k+1)\omega t+\varphi_{S1}]I_{S1}, \end{aligned} \]

where

\[ R=R_s+R_n, \]

\[ L=L_s+L_n, \]

\(R_n, L_n\) are the active and inductive resistance of the load.

Since in steady state only EMFs of one and the same frequency can balance one another, each equation for the rotor circuits can be represented by two equations. Taking this into account, and also the fact that in steady state

\[ I_{\alpha1}=I_{\beta1}=I_1,\qquad I_{\alpha}^{2k+1}=I_{\beta}^{2k+1}=I^{2k+1},\qquad \delta_{01}=\delta_{02}=0, \]

we obtain the following system of equations in complex form:

\[ \dot U_f=R_f\dot I_f+jX_{f1}2(k+1)\dot I^{2k+1}, \]

\[ 0=jX_{f1}\dot I_f+(R_R+jX_R)\dot I_1+j\frac{1}{2}X_{S2}\dot I_{S1}, \]

\[ 0=[R_R+jX_R(2k+1)]\dot I^{2k+1}+jX_{S2}\left(k+\frac{1}{2}\right)\dot I_{S1}, \tag{11} \]

\[ 0=jX_{S2}(k+1)\dot I_1+jX_{S2}(k+1)\dot I^{2k+1}+[R+jX(k+1)]\dot I_{S1}. \]

The resulting system of equations (11) makes it possible to calculate all the characteristics of the contactless single-phase synchronous generator in steady state.

Experimental investigations were carried out on a prototype of a 3-phase contactless electric machine manufactured on the basis of an AO2-42-4 asynchronous motor.

Data of the prototype:

\[ P_n=2400\ \text{W}, \qquad U_s=127\ \text{V}, \]

\[ I_s=6.7\ \text{A}, \]

\[ 2p_1=2 \qquad 2p_2=6 \]

\[ n=3000\ \text{rpm}. \]

Figure 3 shows an oscillogram of the current in the rotor during operation of the test prototype in the single-phase load mode, from which it is evident,

Fig. 3. Oscillogram of the current in the rotor under single-phase load

that the frequency defined by expression (3) is superimposed on the current curve of the fundamental frequency. Figures 4, 5, and 6 show oscillograms of the voltage at the generator output, recorded at no load and in the single-phase load mode without a choke and with a choke in the circuit of the field winding \(W_f\). As noted above, in the single-phase load mode an emf is induced in winding \(W_f\) with a frequency determined by (4). In the absence of a choke in the field circuit, currents of this same frequency, closing through the power source and winding \(W_f\), create a pulsating field which, according to (5) and (6), leads to distortion of the waveform of the output voltage curve (Fig. 5).

Fig. 4. Oscillogram of the output voltage in no-load operation

Fig. 5. Oscillogram of the output voltage under single-phase load without a choke in the field-winding circuit.

The inclusion of a choke in the field circuit significantly improves the waveform of the voltage curve (Fig. 6). Table 1 gives data from harmonic analysis of the rotor current curves and of the voltage at the generator output, obtained using a type S4-7 harmonic analyzer.

Fig. 6. Oscillogram of the output voltage under a single-phase load. A choke is connected in the field-winding circuit

Table 1

Harmonics as % of the fundamental harmonic

Figure 7 gives the regulation characteristics calculated from (11) and obtained experimentally. The calculated curve lies somewhat below the experimentally obtained one, since losses in the steel were not taken into account in the calculation.

From the results of the investigations carried out, the following conclusions may be drawn:

1. The reverse-synchronous field arising in the single-phase mode leads to a whole series of undesirable phenomena: overvoltage in the rotor windings, pulsations in the field winding, and distortion of the waveform of the output-voltage curve.

2. The sufficiently good agreement between the calculated and experimental regulation characteristics shows that the theoretical investigations carried out...

Fig. 7. Regulation characteristics:
a) calculated, b) experimental

studies correctly reflect the principal processes taking place in a single-phase contactless synchronous generator when it operates in the steady-state mode.

1. Analysis of the oscillograms presented and of Table 1 shows that the inclusion of a choke in the excitation-winding circuit sharply reduces the percentage content of higher harmonics in the output-voltage curve.

REFERENCES

1. M. V. Antonov. Contactless single-phase synchronous generator using a counter-synchronous field. Elektrotekhnika, 1966, No. 8.

2. V. M. Petrov and N. S. Siyunov. Experimental study of the voltage and current curves of three-phase synchronous generators under single-phase loads. Elektromekhanika, 1964, No. 1.

3. L. M. Palastin. Calculation of steady-state load conditions of single-phase synchronous generators. VEI, No. 9, 1958.

4. V. S. Novokshonov. Study of an asynchronous brushless frequency converter. Dissertation, Tomsk, 1960.

5. M. L. Kostyrev. Development and study of contactless electrical-machine amplifiers of increased frequency. Dissertation, Tomsk, 1968.

6. L. N. Gruzov. Methods of mathematical investigation of electrical machines. GEI, 1953.

7. N. Hancock. Matrix analysis of electrical machines. Energiya, Moscow, 1957.