EQUATIONS OF THE E.M.F. OF A UNIPOLAR ALTERNATING-CURRENT GENERATOR

A. D. CHESALIN

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

An analytical study of the operating modes of a unipolar alternating-current generator can be carried out on the basis of equations for the balance of the e.m.f.’s of the circuits. In setting up these equations we shall proceed from the following assumptions:

1) the magnetic permeability of the material of the magnetic circuit is equal to infinity;
2) eddy currents and losses in the steel are absent;
3) leakage fluxes are absent;
4) the windings on the generator rotor and the compensating winding on the stator are uniformly distributed;
5) the brushes on the commutator are uniformly distributed around its circumference.

In a unipolar alternating-current generator (UG) of cylindrical construction, two windings are located on the rotor: the armature winding and an additional winding [1]. These windings are made in the form of a squirrel cage of bars, short-circuited at the rotor ends by rings, and at the middle of the rotor brought out to a common commutator. Between the stator poles there is an excitation winding, made in the form of a cylindrical coil and supplied from an independent source with alternating current. On the stator poles there is a compensating winding, made in the form of a squirrel cage and connected in series with the armature winding through brushes located on the commutator.

In a unipolar alternating-current generator two main circuits can be distinguished: the excitation winding and the armature circuit, which includes the armature winding, the compensating winding, and the load. The axes of these circuits are mutually perpendicular; consequently, there is no transformer coupling between them. The coupling between these circuits is effected through the e.m.f. of rotation induced in the bars of the armature winding by the flux of the excitation winding. In addition to those considered, as a result of the shorting by a brush of two adjacent commutator plates, two identical short-circuited circuits arise on the rotor, one of which is formed by bars of the armature winding and the other by bars of the additional winding. Between these circuits, and between each of them and the excitation winding, there exists mutual-inductance coupling. The number $N_k$ of such pairs of circuits in the rotor is determined by the number of brushes on the commutator, and their dimensions by the tooth pitch of the rotor. Let us note that the circuits arising in the armature winding are not inductively coupled with one another. This also applies to the circuits arising in the additional winding. The flux produced by the short-circuited (s.c.) circuits will influence the magnitude of the e.m.f. of rotation in the bars of the armature winding. This influence can be taken into account through the e.m.f. of rotation due to the flux of the s.c. circuits.

For the UG circuits considered, one can write the emf balance equations:

for the field winding,

\[ u_{\mathrm{v}} = i_{\mathrm{v}} r_{\mathrm{v}} + L_{\mathrm{v}} \frac{d i_{\mathrm{v}}}{dt} + M_{\mathrm{vya}} \frac{d i_{\mathrm{ya}}}{dt} + M_{\mathrm{vd}} \frac{d i_{\mathrm{d}}}{dt}, \tag{1} \]

for the short-circuited circuit of the armature winding,

\[ 0 = i_{\mathrm{ya}} r_{\mathrm{ya}} + L_{\mathrm{ya}} \frac{d i_{\mathrm{ya}}}{dt} + M_{\mathrm{yav}} \frac{d i_{\mathrm{v}}}{dt} + M_{\mathrm{yad}} \frac{d i_{\mathrm{d}}}{dt}, \tag{2} \]

for the short-circuited circuit of the additional winding,

\[ 0 = i_{\mathrm{d}} r_{\mathrm{d}} + L_{\mathrm{d}} \frac{d i_{\mathrm{d}}}{dt} + M_{\mathrm{dv}} \frac{d i_{\mathrm{v}}}{dt} + M_{\mathrm{dya}} \frac{d i_{\mathrm{ya}}}{dt}, \tag{3} \]

for the armature circuit,

\[ u_{\mathrm{a}} = e_{\mathrm{a}} + e_{\mathrm{r}} - \left( i_{\mathrm{a}} r_{\mathrm{a}} + L_{\mathrm{a}} \frac{d i_{\mathrm{a}}}{dt} \right), \tag{4} \]

where \(u_{\mathrm{v}}, u_{\mathrm{a}}\) are, respectively, the voltage applied to the field winding and the generator voltage;

\(i_{\mathrm{v}}, i_{\mathrm{ya}}, i_{\mathrm{d}}, i_{\mathrm{a}}\) are the currents, respectively, in the field winding, in the short-circuited circuit of the armature winding, in the short-circuited circuit of the additional winding, and in the armature circuit;

\(e_{\mathrm{a}}, e_{\mathrm{r}}\) are the rotational emf, respectively, due to the flux of the field winding and due to the flux of the short-circuited circuits;

\(L_{\mathrm{v}}, L_{\mathrm{ya}}, L_{\mathrm{d}}, L_{\mathrm{a}}\) are the self-inductances of the circuits;

\(M_{\mathrm{vya}}, M_{\mathrm{vd}}\) are the mutual inductances of the field winding, respectively, with all the short-circuited circuits of the armature winding and with all the short-circuited circuits of the additional winding;

\(M_{\mathrm{yav}}, M_{\mathrm{dv}}, M_{\mathrm{yad}}, M_{\mathrm{dya}}\) are the mutual inductances of the short-circuited circuits with the field winding and with one another.

Since in a unipolar machine the magnetic induction is distributed uniformly in the air gap and does not depend on the position of the rotor \([2, 3]\), the self-inductances and mutual inductances in equations (1), (2), (3), and (4) will be constant.

The system of equations (1)--(4) can be simplified by determining the values of the rotational emf and of the mutual inductances.

The induction in the air gap of the UG due to the current of the field winding is equal to

\[ B_{\delta} = \frac{i_{\mathrm{v}} w_{\mathrm{v}}}{2}\lambda_{\delta}, \tag{5} \]

where \(w_{\mathrm{v}}\) is the number of turns of the field winding,

\(\lambda_{\delta}\) is the permeance of the air gap.

The magnetic flux produced by the field winding is equal to

\[ \Phi_{\delta} = \pi D l B_{\delta} = \frac{i_{\mathrm{v}} w_{\mathrm{v}}}{2}\lambda_{\delta}\pi D l, \tag{6} \]

where \(D\) is the rotor diameter,

\(l\) is the calculated length of the rotor.

Knowing the flux of the field winding and taking into account the assumptions adopted earlier, one can determine the inductance of the field winding:

\[ L_{\mathrm{v}} = \frac{\Phi_{\delta} w_{\mathrm{v}}}{i_{\mathrm{v}}} = \frac{1}{2}\pi D l \lambda_{\delta} w_{\mathrm{v}}^{2}. \tag{7} \]

It is known that the rotational emf of a UG is determined by the equation \([3]\)

\[ e = \Phi_{\delta} n. \tag{8} \]

Therefore, for the rotational emf due to the flux of the field winding, taking expression (7) into account, one can write the equation

\[ e_{\mathrm{a}} = \frac{1}{2}\pi D l \lambda_{\delta} w_{\mathrm{v}} n i_{\mathrm{v}} = \frac{n}{w_{\mathrm{v}}} L_{\mathrm{v}} i_{\mathrm{v}}, \tag{9} \]

where \(n\) is the rotational speed of the generator rotor. We shall determine the value of the emf of rotation due to the flux of the s.c. circuits of the rotor through the average value of the induction of the flux of these circuits in the air gap.

The magnetic flux of one s.c. circuit is equal to

\[ \Phi_{\mathrm{t}}=\frac{1}{Z}\pi D l \lambda_{\delta} i_{\mathrm{t}}, \tag{10} \]

where \(Z\) is the number of rotor teeth.

The inductance of an s.c. circuit will then be determined by the expression

\[ L_{\mathrm{t}}=\frac{1}{Z}\pi D l \lambda_{\delta}. \tag{11} \]

The average value of the induction in the air gap from all s.c. circuits is

\[ B_{\delta \mathrm{t}}=\frac{2N_{\mathrm{k}}\Phi_{\mathrm{t}}}{\pi D l} =\frac{2N_{\mathrm{k}}}{Z}\lambda_{\delta} i_{\mathrm{t}}. \tag{12} \]

Then the emf of rotation due to the magnetic flux of the s.c. circuits will be equal to

\[ e_{\mathrm{t}}=2N_{\mathrm{k}}nL_{\mathrm{t}}i_{\mathrm{t}}. \tag{13} \]

Since all s.c. circuits on the rotor are identical in their parameters and are under identical conditions, then, denoting the mutual inductance of one s.c. circuit with the excitation winding by \(M\) and the mutual inductance of an s.c. circuit of the armature winding with an s.c. circuit of the additional winding by \(M_{\mathrm{t}}\), we may write the equalities

\[ \begin{aligned} M_{\mathrm{av}}&=M_{\mathrm{dv}}=M,\\ M_{\mathrm{ad}}&=M_{\mathrm{da}}=M_{\mathrm{t}}. \end{aligned} \tag{14} \]

Then the mutual inductances of the excitation winding with all s.c. circuits of the armature winding and the additional winding will be equal to

\[ M_{\mathrm{va}}=M_{\mathrm{vd}}=N_{\mathrm{k}}M. \tag{15} \]

Since the parameters of all s.c. circuits are identical, we can write for them the equalities

\[ \begin{aligned} i_{\mathrm{a}}&=i_{\mathrm{d}}=i_{\mathrm{t}}, \qquad L_{\mathrm{a}}=L_{\mathrm{d}}=L_{\mathrm{t}},\\ r_{\mathrm{a}}&=r_{\mathrm{d}}=r_{\mathrm{t}}, \qquad M_{\mathrm{t}}=L_{\mathrm{t}}. \end{aligned} \tag{16} \]

Transforming equations (1), (2), (3), and (4), taking into account expressions (9), (13), (14--16), we obtain the system of equations

\[ u_{\mathrm{v}}=i_{\mathrm{v}}r_{\mathrm{v}}+L_{\mathrm{v}}\frac{di_{\mathrm{v}}}{dt} +2N_{\mathrm{k}}M\frac{di_{\mathrm{t}}}{dt}, \]

\[ 0=i_{\mathrm{t}}r_{\mathrm{t}}+2L_{\mathrm{t}}\frac{di_{\mathrm{t}}}{dt} +M\frac{di_{\mathrm{v}}}{dt}, \tag{17} \]

\[ u_{\mathrm{a}}=\frac{n}{\omega_{\mathrm{v}}}L_{\mathrm{v}}i_{\mathrm{v}} +2N_{\mathrm{k}}nL_{\mathrm{t}}i_{\mathrm{t}} -\left(i_{\mathrm{a}}r_{\mathrm{a}}+L_{\mathrm{a}}\frac{di_{\mathrm{a}}}{dt}\right). \]

The resulting system of equations can be used for an analytical investigation of the operating regimes of an alternating-current unipolar generator.

Literature

1. R. Klaudy. Unipolarmaschine fur Wechselstrom. Austrian patent, class 21d, 85, No. 210514, 1960.

2. B. S. Khlusevich. A Unipolar Alternating-Current Machine and Certain Questions of Its Design, “Proceedings of LODKVAIU,” 1958, vol. 3.

3. A. I. Bertinov, B. L. Alievsky, S. R. Troitsky. Unipolar Electrical Machines with a Liquid-Metal Current Collector, 1966.