On the Machine Constant of Parametric Generators of the Inductive Type
Abstract Generated abstract
This paper derives a design expression for the machine constant of inductive-type parametric generators, emphasizing its role in determining the principal machine dimensions, rotor diameter and active length. It argues that earlier formulations based on active power are incomplete for parametric generators, whose power factor can vary widely, and reformulates the constant in terms of apparent power. Through transformations using generated voltage, frequency, air-gap flux, and linear loading, the paper obtains a machine-constant formula including magnetic loading, electrical loading, leakage, and the rotor-stator pole ratio. The resulting expression is checked against an experimental 5 kVA generator and is proposed for use in sizing inductive parametric generators.
Full Text
On the Machine Constant of Inductive-Type Parametric Generators
V. M. VYSOTSKAYA
(Presented by the scientific seminar of the Physico-Technical Faculty)
The principal dimensions of an electric machine are considered to be the rotor diameter \(D\) and its length \(l\), which, in the main, determine all the other dimensions of the electric machine. The weight of the machine, its cost, and also its operating properties depend on the correct choice of \(D\) and \(l\) and of the ratio between them. Consequently, the determination of the principal dimensions \(D\) and \(l\) must be regarded as the basic stage in the design of an electric machine.
The dimensions \(D\) and \(l\) depend on the power of the electric machine, its rotational speed, and on the loadings chosen for its magnetic system and electrical part. It is known that the loading of the magnetic system is determined mainly by the value of the magnetic induction in the air gap \(B_\delta\), while the loading of the electrical part is determined by the value of the linear loading \(A\) \((a/\mathrm{cm})\).
The relation between the indicated quantities is established with the aid of the machine constant \(C\).
In view of the importance that the machine constant \(C\) has in the design of electric machines, it is necessary to obtain an expression for the machine constant that determines the principal dimensions of inductive-type parametric generators.
Academician N. D. Papaleksi, in one of his articles on parametric generation of electrical energy [1], gives the following value of the machine constant for an experimental inductive-type parametric generator of power \(1\ \mathrm{kW}\), manufactured in the workshops of LPI:
\[ \frac{D^2 \cdot l \cdot n}{P(\mathrm{kW})} = 31 \cdot 10^5. \]
It is obvious that the indicated expression for the machine constant is incomplete, since it does not show the relation of the principal dimensions to the loading of the magnetic system and of the electrical part. It is more correct to use for the machine constant a formula into which the apparent power \((P\ \mathrm{kVA})\) at the terminals of the parametric generator would enter, instead of the active power \((P\ \mathrm{kW})\).
The point is that for parametric machines the average value of \(\cos \varphi\), in the case when the parametric generator is intended to supply an ordinary power circuit, mainly with an active load, may be taken as…
\(\cos \varphi = 0.1 \div 0.3\). When an inductive parametric generator is used to supply an almost purely inductive load, \(\cos \varphi\) will be considerably smaller, which in the given case is a positive factor. As our investigations have shown [2], for the indicated operating mode the \(\cos \varphi\) of an inductive parametric generator has a value of \(0.011 \div 0.09\).
Thus, in order that the formula for the machine constant \(C\) may be used in all cases of application of an inductive parametric generator, it is necessary to make the following substitution:
\[ P_{\text{kW}} = P_{\text{kVA}} \cdot \cos \varphi \]
and
\[ \frac{D^{2}\cdot l \cdot n}{P_{\text{kVA}}} = C \cdot \cos \varphi \quad \text{or, for } \cos \varphi = 0.1 \div 0.3, \]
\[ \frac{D^{2}\cdot l \cdot n}{P_{\text{kVA}}} = 31 \cdot 10^{5}(0.1 \div 0.3) = 3.1 \cdot 10^{5} \div 9.3 \cdot 10^{5}. \]
The correctness of this relation was verified for an inductive parametric generator of 5 kVA made by us; in this case it was obtained that
\[ \frac{D^{2}\cdot l \cdot n}{P_{\text{kVA}}} = 4.18 \cdot 10^{5}. \]
To obtain a more complete expression for the machine constant \(C\), we shall carry out a series of transformations.
The apparent power of an inductive parametric generator is
\[ P = U \cdot I \cdot 10^{-3}. \tag{1} \]
The voltage of the parametric generator is
\[ U = 4.44 \cdot f \cdot W \cdot \sigma \cdot \Phi_{\delta} \cdot 10^{-8}, \tag{2} \]
where
\(\sigma\) is the leakage coefficient,
\(\Phi_{\delta}\) is the magnetic flux in the air gap,
\(W\) is the number of turns of the stator winding of the parametric generator.
The frequency of the current generated in the stator winding of the parametric generator is
\[ f = \frac{Z}{2}\cdot \frac{n}{60}. \]
Here \(Z\) is the number of teeth on the rotor of the parametric generator,
\(n\) is the speed of rotation of the primary motor. Denote the ratio
\[ \frac{Z}{2p_{1}} = \frac{t'}{t} = k, \]
where \(2p_{1}\) is the number of “conditional poles” on the stator of the parametric generator,
\(t'\) is the pole pitch of the stator,
\(t\) is the pole pitch of the rotor; then
\[ f = \frac{p_{1}\cdot k \cdot n}{60}. \tag{3} \]
The magnetic flux in the air gap is
\[ \Phi_{\delta} = b \cdot l \cdot B_{\delta} = \alpha \cdot t' \cdot l \cdot B_{\delta} = \alpha \frac{\pi D}{2p_{1}}\cdot l \cdot B_{\delta}. \tag{4} \]
Linear loading
\[ A=\frac{2\cdot AW}{\sqrt{2\cdot \pi D}}. \tag{5} \]
Substituting expressions 2, 3, 4, and 5 into equation 1 and replacing
\[ J\cdot W=\frac{AW}{\sqrt{2}}=\frac{A\cdot \pi D}{2}, \]
then the apparent power of the parametric generator is
\[ P=\frac{10^{-11}}{5.5}\cdot D^{2}\cdot l\cdot n\cdot A\cdot \kappa\cdot \alpha\cdot B_{\delta} \]
and the complete expression for the machine constant \(C\) of the parametric generator is obtained in the form:
\[ C=\frac{D^{2}\cdot l\cdot n}{P_{\mathrm{kVA}}} =\frac{5.5\cdot 10^{11}}{\alpha\cdot \kappa\cdot A\cdot B_{\delta}\cdot \sigma}, \tag{6} \]
where, in addition to the quantities ordinarily entering into the expression for the machine constant \(C\), new quantities appear that are specific to inductive parametric generators, namely:
\(\kappa\) — a coefficient depending on the ratio of the number of poles on the rotor and on the stator, and
\(\sigma\) — a coefficient taking into account the influence of the leakage field of the stator winding of the parametric generator.
Calculation of the right-hand side of expression (6) for the machine constant \(C\) for our experimental parametric generator showed that
\[ C=\frac{D^{2}\cdot l\cdot n}{P_{\mathrm{kVA}}} =\frac{5.5\cdot 10^{11}}{\alpha\cdot \kappa\cdot A\cdot B_{\delta}\cdot \sigma} =4.18\cdot 10^{5}. \]
The obtained expression (6) for the machine constant \(C\) is recommended for further use in determining the principal dimensions of inductive-type parametric generators.
LITERATURE
-
Papaleksi N. D. Parametric generation of alternating currents. Elektrichestvo, No. 11, 1938.
-
Vysotskaya V. M. Development and investigation of a parametric generator for the purpose of using it to supply the winding of a betatron electromagnet. Candidate dissertation, LPI, 1953.