Principal Dimensions of the Motor-Converter Section of an Asynchronous Brushless Frequency Converter and Optimal Distribution of Its Electromagnetic Loads

V. S. Novokshchenov

(Presented by the scientific and technical seminar of the EMF)

An asynchronous brushless frequency converter (ABFC) consists of an asynchronous drive motor, an asynchronous converter, and an inverted synchronous machine [1, 2]. The drive motor and the asynchronous converter are combined in a single magnetic circuit: in the slots of the stator core are located the primary winding (winding 1) of the asynchronous motor and the secondary winding (winding 3) of the asynchronous converter; in the slots of the rotor core are located the secondary winding (winding \(2'\)) of the asynchronous motor and the primary winding (winding 2) of the asynchronous converter; the magnetic flux of the asynchronous motor \(\Phi_1\) and the magnetic flux of the asynchronous converter \(\Phi_2\) are in the common magnetic circuit made up of the stator and rotor steel cores. Usually windings \(2'\) and 2 are combined into one winding performing the functions of both. The armature of the inverted synchronous machine, from which the primary winding of the asynchronous converter is supplied, is on the same shaft as the rotor core of the motor-converter section of the ABFC.

The weight of the active materials of the motor-converter section of the ABFC, like the weight of the active materials of an ordinary asynchronous machine, is determined by the product of the square of the internal diameter of the stator core and its length \((D^2 l)\).

According to the well-known basic design equation of an electric machine [3], the quantity \(D^2 l\) may be expressed, on the one hand, through the power and electromagnetic loads of the driving asynchronous motor,

\[ D^2 l = \frac{8.6 \cdot 10^{11} P'_1}{k_{w1} A_1 B_1 n_1} = \frac{8.6 \cdot 10^{11} P'_1}{k_{w1}(1-c_A)(1-c_B)A_{13}B_{12}n_1}, \tag{1} \]

and, on the other hand, through the power and electromagnetic loads of the asynchronous converter,

\[ D^2 l = \frac{8.6 \cdot 10^{11} P'_3}{k_{w3} A_3 B_2 n_3} = \frac{8.6 \cdot 10^{11} P'_3}{k_{w3} c_A c_B A_{13}B_{12}n_3}. \tag{2} \]

Here

\[ P'_1 = k_{E1} P_1, \tag{3} \]

\[ P'_3 = k_{E3} P_3 \tag{4} \]

are the calculated powers of windings 1 and 3 in kVA,

where \(P_1, P_3\) are the input and output powers of the ABFC in kVA;
\(k_{E1}=0.92 \div 0.96\); \(k_{E3}=1.04 \div 1.08\) are coefficients;
\(n_1, n_3\) are the rotational speeds of the magnetic fields of the asynchronous motor and the asynchronous converter relative to windings 1 and 3, in rpm;
\(k_{w1}, k_{w3}\) are the winding factors of windings 1 and 3;
\(A_1, A_3\) are the linear loadings of windings 1 and 3 in A/cm;
\(B_1, B_2\) are the magnetic inductions in the air gap of magnetic fields \(\Phi_1\) and \(\Phi_2\), in G;

\(A_{13}=A_1+A_3\) is the total linear loading;
\(B_{12}=B_1+B_2\) is the total magnetic loading;

\[ c_A=\frac{A_3}{A_{13}}, \qquad c_B=\frac{B_2}{B_{12}} \]

are quantities characterizing the distribution of the electric and magnetic loadings between the windings.

Equations (1) and (2) make it possible to eliminate one of the unknowns. Equating the right-hand sides of equations (1) and (2) and taking expressions (3), (4) into account, after certain transformations we obtain the dependence between the quantities \(c_A\) and \(c_B\):

\[ c_A=\frac{1-c_B}{1-c_B(1-k)}, \tag{5} \]

where the coefficient

\[ k=\frac{P_1 n_3 k_{w3} k_{E1}}{P_3 n_1 k_{w1} k_{E3}} . \tag{6} \]

Substituting expression (5) into equation (2), we obtain the final expression for the quantity \(D^2 l\):

\[ D^2l= \frac{8.6\cdot 10^{11} P_3 k_{E3}\,[1-c_B(1-k)]} {k_{w3} A_{13} B_{12} n_3 c_B(1-c_B)} . \tag{7} \]

The output power \(P_3\) of the ABFC and the rotational speed \(n_3\) are prescribed quantities. The total linear loading \(A_{13}\) is determined mainly by the permissible overheating of the windings and by the specified efficiency of the ABFC. Since the magnetic fluxes \(\Phi_1\) and \(\Phi_2\) are in a common magnetic core, the magnetizing currents of the asynchronous motor and the asynchronous converter depend on the magnitude of the magnetic inductions of both fields [2]. Therefore the magnetic loading \(B_{12}\) is determined mainly by the specified values of the power factor and efficiency. For a given output power of the ABFC, the electromagnetic loadings \(A_{13}\) and \(B_{12}\), like the electromagnetic loadings of ordinary asynchronous motors, may vary within rather narrow limits.

The coefficient $k$ is determined by the quantities entering expression (6).

The quantity $c_B$, characterizing the distribution of electromagnetic loads between the asynchronous motor and the asynchronous converter, could be chosen arbitrarily by the designer. However, the choice of one or another value of $c_B$ has a very great influence on the volume and weight of the active materials. Indeed, when the quantity $c_B$ varies from zero to unity, the quantity $D^2 l$ changes from infinity to a certain minimum value, and then again, as $c_B \to 1$, tends to infinity. The dependence $D^2 l = f(c_B)$, obtained for constant values of the other quantities entering equation (7), has the form of the U-shaped curve $a$ shown in Fig. 1. It is obvious that the quantity $c_B$ must be chosen on the basis of the condition of minimum weight of the active materials.

The optimal value of the quantity $c_B$, corresponding to the minimum weight of the active materials for given values of the other quantities entering equation (7), can be obtained from the equation

$$ \frac{\partial}{\partial c_B}(D^2 l)=0. \tag{8} $$

Substituting into equation (8) the value $D^2 l$ expressed by equation (7), after certain transformations we obtain:

$$ c_{BO}=\frac{1\pm \sqrt{k}}{1-k}, \tag{9} $$

where $c_{BO}$ is the optimal value of the quantity $c_B$. Since $0<c_B<1$, expression (9) with the “plus” sign before the radical loses physical meaning. Consequently,

$$ c_{BO}=\frac{1-\sqrt{k}}{1-k}. \tag{10} $$

Fig. 1.

Since the losses of electrical machines are a function of the electromagnetic loads and of the weight of the active materials, it is obvious that, for given electromagnetic loads $A_{13}$, $B_{12}$, the minimum weight of the ABFC corresponds to the minimum of its losses. When the quantity $c_B$ deviates from the optimum, the losses increase, which leads to an increase in the consumed power $P_1$ and, consequently, to an increase in the coefficient $k$. Therefore the dependence $D^2 l = f(c_B)$, constructed with allowance for the change in the coefficient $k$, will have the form of curve $b$ shown in Fig. 1, having the same minimum as curve $a$, but passing above it for $c_B \ne c_{BO}$. Curves $a$ and $b$ convincingly show how profound the influence of the choice of the quantity $c_B$ is on the weight of the active materials and how important it is to determine the optimal value of the quantity $c_B$.

Conclusions

1. Equation (7), relating the principal dimensions of the motor-converter section of the ABFC to its power, electromagnetic loadings, and the rotational speeds of the magnetic fields, may be regarded as one of the basic design equations of the ABFC.

2. The quantity $c_B$, characterizing the distribution of electromagnetic loadings in the motor-converter section of the ABFC, must be chosen from the condition of minimum weight of active materials and minimum losses, in accordance with equation (10).

3. The expressions (7) and (10) given above may be used in designing any electrical machines based on the application of the principle of combining two rotating magnetic fields in one magnetic circuit.

References

1. Author’s certificate No. 106209.

2. Novokshenov V. S. Some questions in the theory of an asynchronous brushless frequency converter. Proceedings of the Tomsk Polytechnic Institute, vol. 94, GEI, 1958.

3. R. Richter. Electrical Machines, vol. 1, GONTI, 1939.