On the Use of the Active Volume of Combined Machines
Abstract Generated abstract
This paper examines how effectively active materials are used in combined electrical machines whose windings share common slots in a magnetic core. It develops a general analytical formulation for machines combined in a common active volume, introducing distribution coefficients for electric and magnetic loadings and deriving conditions for extrema of the active volume utilization coefficient. The analysis indicates that utilization is maximized when electric and magnetic loadings are distributed equally for each component machine according to specified power and speed ratios, while the poorest utilization occurs when the component machines have approximately equal normalized power-speed parameters. Applied to frequency converters, the results suggest that single-machine converters have particularly poor active-volume utilization, and that increasing the number of machines combined in one active volume further worsens utilization.
Full Text
On the Use of the Active Volume of Combined Machines
V. G. Baum
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
Recently, increased interest has been shown in the application of electrical machines in which the windings of two or more machines are placed in the common slots of the magnetic core [1, 2, 3, 4, 5, 6].
Among other questions of interest in the creation of such machines, there arises the question of the utilization in them of active materials. At present, for the calculation of machines, the method [6] developed by V. S. Novokshenov is used mainly, taking into account the optimal use of active materials as applied to combined machines consisting of two machines.
However, this does not give a general idea of the influence of the design arrangements, the purpose, and the number of machines combined in a common active volume on the utilization of active materials.
Let us consider this question more broadly. For generality, let us assume that in the common slots of the magnetic core \(m\) machines are combined, and each of them has its own rated power \(P_i\), rotational speed of the synchronous field relative to the winding producing it \(n_i\), flux with maximum induction in air \(B_i\), and linear loading \(A_i\).
The fluxes of all machines act independently of one another in the active volume \(D^2l\). The degree of utilization of the active volume of each of the machines will be
\[ \frac{P_i}{n_i D^2 l} = \kappa_i A_i B_i, \tag{1} \]
where \(K_i\) is a coefficient depending on the coefficients of pole coverage \((\alpha_i)\), the shape of the field curve \((K_{\phi i})\), and the winding coefficient \(K_{0i}\), i.e.
\[ \kappa_i = \frac{\alpha_i}{n_i D^2 l}. \]
The magnitudes of the total electromagnetic loadings \(A\) and \(B\) of combined machines, as investigations [6] show, may be taken at the level of those for machines of normal design having the same rotational speeds and active dimensions. We introduce coefficients \(a_i\) and \(b_i\), characterizing the distribution of these loadings among the machines, i.e.
\[ a_i = \frac{A_i}{A}; \]
\[ b_i=\frac{B_i}{B}; \tag{2} \]
\[ \sum_{1}^{m} a_i=1; \]
\[ \sum_{1}^{m} b_i=1. \]
On this basis we transform formula (1):
\[ \frac{P_i}{n_i D^2 l}=\kappa_i a_i b_i AB. \]
The degree of utilization of the active volume by all machines is determined from the expression
\[ \frac{\displaystyle \sum_{1}^{m}\frac{P_i}{n_i}}{D^2 l} = \sum_{1}^{m}\kappa_i a_i b_i AB \]
or
\[ \frac{\displaystyle \sum_{1}^{m}\frac{P_i}{n_i}}{D^2 l} = NAB, \]
where we shall call the coefficient of utilization
\[ N=\sum_{1}^{m}\kappa_i a_i b_i. \tag{3} \]
From the condition that there is a common volume for all machines it follows that
\[ D^2 l AB = \frac{P_1}{n_1\kappa_1 a_1 b_1} = \frac{P_2}{n_2\kappa_2 a_2 b_2} = \cdots = \frac{P_m}{n_m\kappa_m a_m b_m}, \]
whence, denoting
\[ \frac{P_i}{n_i}=\gamma_i, \]
we obtain
\[ \gamma_1:\gamma_2:\ldots:\gamma_m = \kappa_1 a_1 b_1:\kappa_2 a_2 b_2:\ldots:\kappa_m a_m b_m. \tag{4} \]
If the dependence
\[ \gamma_1:\gamma_2:\ldots:\gamma_m \]
is known and remains unchanged, then the values \(a_i, b_i\) for which the function
\[ N(a_1,b_1,a_2,b_2,\ldots,a_m,b_m) \]
has a conditional extremum can be found from the solution of the system of equations:
\[ \frac{\partial F}{\partial a_1}=0; \]
\[ \frac{\partial F}{\partial b_1}=0; \]
\[ \frac{\partial F}{\partial a_2}=0; \]
\[ \frac{\partial F}{\partial b_2}=0; \]
\[ \cdots \]
\[ \frac{\partial F}{\partial a_m}=0; \]
\[ \frac{\partial F}{\partial b_m}=0; \]
\[ a_1+a_2+\cdots+a_m=1; \]
\[ b_1+b_2+\cdots+b_m=1; \]
\[ \kappa_1 a_1 b_1-\frac{\gamma_1}{\gamma_2}\kappa_2 a_2 b_2=0; \]
\[ \kappa_1 a_1 b_1-\frac{\gamma_1}{\gamma_3}\kappa_3 a_3 b_3=0; \]
\[ \cdots \]
\[ \kappa_1 a_1 b_1-\frac{\gamma_1}{\gamma_m}\kappa_m a_m b_m=0, \]
where
\[ \begin{aligned} F={}&\kappa_1 a_1 b_1+\kappa_2 a_2 b_2+\cdots+\kappa_m a_m b_m+\lambda_1(a_1+{}\\ &+a_2+\cdots+a_m-1)+\lambda_2(b_1+b_2+\cdots+b_m-1)+{}\\ &+\lambda_3\left(\kappa_1 a_1 b_1-\frac{\gamma_1}{\gamma_2}\kappa_2 a_2 b_2\right)+\cdots+\lambda_{m+1}\left(\kappa_1 a_1 b_1-\frac{\gamma_1}{\gamma_m}\kappa_m a_m b_m\right). \end{aligned} \]
Here \(\lambda_1,\lambda_2,\lambda_3,\ldots,\lambda_{m+1}\) are Lagrange multipliers.
After analyzing the results of the solution, one can verify that the coefficient of utilization \(N\) will have a maximum value when
\[ a_1=b_1= \frac{1}{ 1+\sqrt{\frac{\kappa_1\gamma_2}{\kappa_2\gamma_1}} +\sqrt{\frac{\kappa_1\gamma_3}{\kappa_3\gamma_1}} +\cdots+ \sqrt{\frac{\kappa_1\gamma_m}{\kappa_m\gamma_1}} }; \]
\[ a_2=b_2= \frac{1}{ \sqrt{\frac{\kappa_2\gamma_1}{\kappa_1\gamma_2}} +1+ \sqrt{\frac{\kappa_2\gamma_3}{\kappa_3\gamma_2}} +\cdots+ \sqrt{\frac{\kappa_2\gamma_m}{\kappa_m\gamma_2}} }; \tag{5} \]
\[ \cdots \]
\[ a_m=b_m= \frac{1}{ \sqrt{\frac{\kappa_m\gamma_1}{\kappa_1\gamma_m}} +\sqrt{\frac{\kappa_m\gamma_2}{\kappa_2\gamma_m}} +\cdots+ \sqrt{\frac{\kappa_m\gamma_{m-1}}{\kappa_{m-1}\gamma_m}} +1 }. \]
Taking into account the dependences (5), formula (3) can be given another form:
\[ N=\kappa_1 c_1^2+\kappa_2 c_2^2+\cdots+\kappa_m c_m^2, \tag{6} \]
where
\[ a_i=b_i=c_i. \]
If the condition of constancy of the ratios (4) is removed, then the values \(c_i\) at which the function \(N(c_1,c_2,\ldots,c_m)\) has a conditional extremum will be found after solving the system of equations:
\[ \frac{\partial F}{\partial c_1}=0; \]
\[ \frac{\partial F}{\partial c_2}=0; \]
\[ \cdots \]
\[ \frac{\partial F}{\partial c_m}=0; \]
\[ c_1+c_2+\cdots+c_m=1, \]
where
\[ F=\kappa_1 c_1^2+\kappa_2 c_2^2+\cdots+\kappa_m c_m^2+\lambda(c_1+c_2+\cdots+c_m-1), \]
Here \(\lambda\) is the Lagrange multiplier.
As a result of analyzing the solution of this system of equations, we obtain that the smallest value of the utilization coefficient will occur when
\[ c_1=c_2=\cdots=c_m=\frac{1}{m}. \tag{7} \]
From expressions (4, 5, and 7) we obtain that, in this case,
\[ \kappa_1\gamma_2=\kappa_1\gamma_3=\cdots=\kappa_m\gamma_s, \]
where \(s\) is any number from the normal sequence of \(m\) numbers not equal to \(m\).
The coefficients \(\kappa_i\) may differ only slightly from one another. For such machines one may take
\[ \kappa_1=\kappa_2=\cdots=\kappa_m=\kappa. \]
Then the worst utilization of the active materials will be for machines for which
\[ \gamma_1=\gamma_2=\cdots=\gamma_m. \]
In this case formula (6) takes the form
\[ N=\frac{\kappa}{m}. \]
A considerable number of known combined machines have windings of two machines in common slots [1, 2, 3, 4, 6]. For these cases, Fig. 1 shows the graphical dependence
Fig. 1
\[ N\left(\frac{\gamma_2}{\gamma_1}\right), \]
where
\[ \gamma_1 > \gamma_2 . \]
The coefficient of utilization of the active volume of a normal machine having the same dimensions and rotational speed as the combined machine at \(\gamma_2 = 0\) is taken as unity \((N = 1)\). From the graph it is evident that, when a second machine is combined with one machine in common slots, the utilization coefficient decreases sharply, reaching a minimum
\[ N_{\min}=0.5 \]
when
\[ \gamma_1=\gamma_2 . \]
A sharp decrease occurs at relatively small values of \(\gamma_2\).
The utilization of the active volume remains low over rather broad limits of variation of
\[ \frac{\gamma_2}{\gamma_1}. \]
Using the above, one can assess the rationality of two types of combined machines: single-machine (Fig. 2) and contactless (Fig. 3) frequency converters.
Fig. 2
In a single-machine frequency converter, the windings of an asynchronous motor \((w_1, w_2)\) and of a synchronous generator \((w_3, w_4)\) are combined in a common magnetic circuit. When the converter operates from a network of normal frequency \(f_1\), the increased frequency \(f_2\) is taken from the stator winding of the generator.
The contactless converter is a motor-generator unit in which the windings of the synchronous generator \((w_2, w_3)\) are combined in common slots with the windings of the exciter \((w_4, w_5)\). The rotor winding of the exciter \((w_5)\), through rotating rectifiers, is connected to the excitation winding of the generator \((w_3)\). During operation of the conver-
Fig. 3
... of the converter from a network of normal frequency \(f_1\), an increased frequency \(f_2\) is taken from the stator winding of the generator; (\(w_1\) is the stator winding of the squirrel-cage motor).
For a single-machine converter [1, 2], \(\gamma_1\) and \(\gamma_2\), respectively, for the motor and the generator differ only slightly from one another. In a contactless converter, \(\gamma_1\) and \(\gamma_2\), respectively, for the exciter and the generator may differ significantly from one another, since the rated power of the exciter may be negligible in comparison with the power of the generator.
On the basis of the foregoing, the single-machine frequency converter may be classified as a type of machine with the poorest utilization of active volume.
An increase in the number of machines [5] combined in a common active volume worsens its utilization.
REFERENCES
-
A. S. Bykov. Asynchronous frequency converter in single-machine design. Dissertation, Kiev Polytechnic Institute, 1953.
-
V. N. Pavlinin, N. S. Siyunov. Properties and effectiveness of a single-machine frequency converter. Proceedings of Higher Educational Institutions, “Electromechanics”, No. 1, 1960.
-
V. G. Baum. Circuits of contactless frequency converters. Industrial Power Engineering, No. 9, 1961.
-
G. E. Pukhov, V. A. Borkovskii, M. A. Sutormin. Contactless induction machines with an internal cascade. Collection of Scientific Papers of the Kiev Institute of Civil Air Fleet, issue 1, 1963.
-
V. S. Nersesyan. Elements of the calculation of single-machine contactless frequency converters. Proceedings of the Gorky Polytechnic Institute, vol. XIX, issue 3, 1963.
-
V. S. Novokshenov. Investigation of an asynchronous brushless frequency converter. Dissertation, Tomsk Polytechnic Institute, 1961.