Determination of the Influence Coefficients of Input Parameters on Output Parameters for DC Machines Rated 0.4-30 kW
Abstract Generated abstract
This paper addresses the accuracy of small and medium DC machines by treating design and material quantities as variable input parameters rather than fixed calculated values. It develops error equations and influence coefficients linking variations in input parameters to armature current, rotational speed, and efficiency for 0.4 to 30 kW DC machines, deriving formulas from standard power balance, speed, resistance, and magnetic flux relations. Numerical coefficient ranges are reported for principal machine sizes, allowing negligible terms to be identified and showing how resistance and magnetic circuit variations propagate to operating characteristics. The resulting expressions are presented as a basis for tolerance calculation and for analyzing the effects of manufacturing and technological factors on machine performance.
Full Text
DETERMINATION OF THE INFLUENCE COEFFICIENTS OF INPUT PARAMETERS ON OUTPUT PARAMETERS FOR DC MACHINES WITH A POWER OF 0.4–30 kW
A. P. MURAVLEV, A. Ya. TSIRULIK
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
One of the most important tasks of modern technical progress is to improve the quality of products, including electrical machines. Only careful design and technological development during engineering makes it possible to improve an electrical machine effectively. The methods of engineering calculation of DC electrical machines that have been developed at present are based on the assumption that all machine parameters strictly correspond to the calculated ones, and that such quantities as specific resistance, turn length, wire diameter, and others have constant values determined from averaged characteristics. In reality, all quantities have a certain scatter, i.e., they are random quantities.
The quality of electrical machines is characterized by accuracy, i.e., by the degree to which the parameters of the manufactured machine correspond to the calculated parameters.
At present, increased accuracy requirements are imposed on DC machines, both for geometric dimensions and for technical characteristics.
The aim of the present work is to determine the influence of input parameters on output parameters for DC machines. Output parameters are parameters that characterize the operation of the machine. With constant power on the motor shaft, in this case the output parameters will be the armature current, efficiency, and rotational speed. The input parameters are the principal and local dimensions, as well as the characteristics of the materials used (geometric dimensions, wire diameter, specific resistance of copper, etc.).
For an electrical machine, the input and output parameters in the most general form may be related by the following expression:
\[ y_i = f_i(x_1, x_2, \ldots, x_j, \ldots, x_m), \tag{1} \]
where
\[ y_i \text{— output parameter,} \]
\[ x_i \text{— input parameter.} \]
From equation (1) one can obtain an expression for the relative error [1]
\[ \frac{\Delta y_i}{y_i} = \sum_{j=1}^{m} \frac{\partial f_i(x_1, x_2, \ldots, x_j, \ldots, x_m)}{\partial x_j} \, \frac{x_j}{f_i(x_1, x_2, \ldots, x_m)} \, \frac{\Delta x_j}{x_j}. \tag{2} \]
Expression (2), which establishes the relationship between the errors of the input and output parameters, is called the error equation in the theory of accuracy. The coefficients
\[ C_{ji}=\frac{\partial f_i(x_1,\ x_2,\ldots,\ x_j,\ldots,\ x_m)}{\partial x_j}\cdot \frac{x_j}{f_i(x_1,\ x_2,\ldots,\ x_m)} \tag{3} \]
are called the coefficients of influence of parameters \(j\) on output characteristic \(i\).
On the basis of equation (2), it may be concluded that a reduction in the scatter of the output parameter can be achieved by two principal methods:
- By reducing the scatter of the input parameters.
- By reducing the influence coefficients.
The greatest accuracy can be obtained when both methods are used simultaneously.
The deviations of the machine output parameters are described by a series of error equations, the number of which is determined by the number of output parameters. These equations are the initial ones for calculating tolerances and for analyzing manufacturing errors.
From the equations it is possible to calculate the deviations of the output parameter from the specified deviations of the input parameters (the direct problem of the theory of accuracy) and to calculate the permissible deviations of the input parameters from a specified tolerance for the output parameter (the inverse problem of the theory of accuracy).
To derive the coefficients of influence on the armature current, the formula for the supplied power [2] is used:
\[ P_1=P_2+P_{\mathrm{v}}+P_{\mathrm{m}}+P_{\mathrm{sh}}+P_{\mathrm{st}}+P_{\mathrm{add}}, \tag{4} \]
from which the quantities depending on \(I_a\) are separated:
\[ P'=P_1-P_{\mathrm{v}}=I_aU=I_a^2R_a+2\Delta U_{\mathrm{sh}}I_a+p''. \tag{5} \]
From equation (5) the expression for the armature current is obtained:
\[ I_a=\frac{U-2\Delta U_{\mathrm{sh}}}{2R_a} -\sqrt{\left(\frac{U-2\Delta U_{\mathrm{sh}}}{2R_a}\right)^2-\frac{p''}{R_a}}. \tag{6} \]
On the basis of equations (6) and (3), for \(C_{ji}\) we find:
\[ C_{R_a I_a}=-1+ \frac{2R_a p''}{\sqrt{a}\,(1-\sqrt{a})(U-2\Delta U_{\mathrm{sh}})^2}; \tag{7} \]
\[ C_{p'' I_a}=- \frac{2R_a p''}{\sqrt{a}\,(1-\sqrt{a})(U-2\Delta U_{\mathrm{sh}})^2}; \tag{8} \]
\[ C_{U I_a}= \frac{U}{U-2\Delta U_{\mathrm{sh}}} -\frac{4R_a p''U}{\sqrt{a}\,(1-\sqrt{a})(U-2\Delta U_{\mathrm{sh}})^3}; \tag{9} \]
\[ C_{\Delta U_{\mathrm{sh}} I_a}= -\frac{2\Delta U_{\mathrm{sh}}}{U-2\Delta U_{\mathrm{sh}}} +\frac{8R_a p''\Delta U_{\mathrm{sh}}}{\sqrt{a}\,(1-\sqrt{a})(U-2\Delta U_{\mathrm{sh}})^3}, \tag{10} \]
where
\[ a=1-\frac{4p''R_a}{(U-2\Delta U_{\mathrm{sh}})^2}. \]
The coefficients of influence on the rotational speed are obtained on the basis of the expression:
\[ n=\frac{U-I_aR_a-2\Delta U_{\mathrm{sh}}}{C_e\Phi}. \tag{11} \]
\[ C_{I_a n}=-\frac{R_a I_a}{U-I_aR_a-2\Delta U_{\mathrm{sh}}}; \tag{12} \]
\[ C_{U n}=\frac{U}{U-I_aR_a-2\Delta U_{\mathrm{sh}}}; \tag{13} \]
\[ C_{\Delta U_{\mathrm{sh}} n}=-\frac{2\Delta U_{\mathrm{sh}}}{U-I_aR_a-2\Delta U_{\mathrm{sh}}}; \tag{14} \]
\[ C_{R_a n}=-\frac{I_aR_a}{U-I_aR_a-2\Delta U_{\mathrm{sh}}}; \tag{15} \]
\[ C_{\Phi n}=-1. \]
The expression for the efficiency is derived from equation (4)
\[ \eta=\frac{P_2}{I_a^2R_a+2\Delta U_{\mathrm{sh}}I_a+\dfrac{U^2}{r_{\mathrm{sh}}}+p''}. \tag{16} \]
The coefficients of influence on the efficiency have the form:
\[ C_{I_a\eta}=-\frac{(2I_aR_a+2\Delta U_{\mathrm{sh}})I_a} {I_a^2R_a+2\Delta U_{\mathrm{sh}}I_a+\dfrac{U^2}{r_{\mathrm{sh}}}+p''}; \tag{17} \]
\[ C_{R_a\eta}=-\frac{I_a^2R_a} {I_a^2R_a+2\Delta U_{\mathrm{sh}}I_a+\dfrac{U^2}{r_{\mathrm{sh}}}+p''}; \tag{18} \]
\[ C_{\Delta U_{\mathrm{sh}}\eta}=-\frac{2I_a\Delta U_{\mathrm{sh}}} {I_a^2R_a+2\Delta U_{\mathrm{sh}}I_a+\dfrac{U^2}{r_{\mathrm{sh}}}+p''}; \tag{19} \]
\[ C_{U\eta}=-\frac{2\dfrac{U^2}{r_{\mathrm{sh}}}} {I_a^2R_a+2\Delta U_{\mathrm{sh}}I_a+\dfrac{U^2}{r_{\mathrm{sh}}}+p''}; \tag{20} \]
\[ C_{r_{\mathrm{sh}}\eta}=-\frac{\dfrac{U^2}{r_{\mathrm{sh}}}} {I_a^2R_a+2\Delta U_{\mathrm{sh}}I_a+\dfrac{U^2}{r_{\mathrm{sh}}}+p''}; \tag{21} \]
\[ C_{p''\eta}=-\frac{p''} {I_a^2R_a+2\Delta U_{\mathrm{sh}}I_a+\dfrac{U^2}{r_{\mathrm{sh}}}+p''}. \tag{22} \]
For the principal standard sizes of DC machines with a power of 0.4–30 kW, the coefficients of influence were calculated. The numerical values of the coefficients of influence are given in Table 1.
Table 1
| \(C_{R_a I_a}\) | \(C_{U I_a}\) | \(C_{\Delta U_{\mathrm{sh}} I_a}\) | \(C_{p'' I_a}\) | \(C_{U n}\) | \(C_{I_a n}=C_{R_a n}\) | \(C_{\Delta U_{\mathrm{sh}} n}\) |
|---|---|---|---|---|---|---|
| 0.022 ÷ 0.300 |
−2.295 −1.043 |
0.0038 −0.06768 |
1.022 ÷ 1.3 |
1.034 ÷ 1.245 |
−0.0261 ÷ −0.260 |
−0.0094 ÷ −0.0226 |
| \(C_{I_a\eta}\) | \(C_{R_a\eta}\) | \(C_{U\eta}\) | \(C_{r_{\mathrm{sh}}\eta}\) | \(C_{p''\eta}\) | \(C_{\Delta U_{\mathrm{sh}}\eta}\) | \(C_{\Phi n}\) |
|---|---|---|---|---|---|---|
| −0.056 ÷ −0.324 |
−0.0236 ÷ −0.1585 |
−0.0344 ÷ −0.374 |
0.0232 ÷ 0.187 |
−0.680 ÷ −0.948 |
−0.007377 ÷ −0.0241 |
−1 |
On the basis of an analysis of the results obtained, it was decided to neglect the influence coefficients \(C_{\Delta U_{\mathrm{sh}} I_a}\) and \(C_{\Delta U_{\mathrm{sh}} n}\) because of their smallness, and also to assume that the variance of the supply voltage is equal to zero, since, when taking nominal characteristics, the voltage at the machine must be nominal.
In the influence coefficients obtained above, quantities such as \(R_a\), \(r_{\mathrm{sh}}\), \(p''\), and \(\Phi\) are intermediate quantities depending on the input parameters (resistivity, wire diameter, mean turn length, geometrical dimensions of the magnetic circuit, etc.).
Let us consider the variance of the resistance of the armature circuit
\[ R_a = r_{\mathrm{ya}} + r_{\mathrm{d}} + r_{\mathrm{c}} . \tag{23} \]
From equation (23) we obtain
\[ \sigma^2_{R_a0} = C^2_{r_{\mathrm{ya}}R_a}\sigma^2_{(r_{\mathrm{ya}})0} + C^2_{r_{\mathrm{d}}R_a}\sigma^2_{(r_{\mathrm{d}})0} + C^2_{r_{\mathrm{c}}R_a}\sigma^2_{(r_{\mathrm{c}})0}, \tag{24} \]
where
\[ C_{r_{\mathrm{ya}}R_a}=\frac{r_{\mathrm{ya}}}{R_a}; \qquad C_{r_{\mathrm{d}}R_a}=\frac{r_{\mathrm{d}}}{R_a}; \qquad C_{r_{\mathrm{c}}R_a}=\frac{r_{\mathrm{c}}}{R_a}. \]
Calculations showed that the values of the influence coefficients on resistance for all frame sizes lie within the limits given in Table 2.
Table 2
| Coefficient | Range |
|---|---|
| \(C_{r_{\mathrm{ya}}R_a}\) | 0.594–0.805 |
| \(C_{r_{\mathrm{d}}R_a}\) | 0.136–0.377 |
| \(C_{r_{\mathrm{c}}R_a}\) | 0.0117–0.139 |
It is seen from Table 2 that the influence coefficients \(C_{r_{\mathrm{c}}R_a}\) may be neglected because of their small magnitude.
In turn, the variances of the resistances of the armature windings and the auxiliary poles depend on certain quantities. In general form, the expression for the resistance has the form
\[ r=\rho\frac{4l_{\mathrm{v.sr}}w}{\pi d^2(2a)^2}. \tag{25} \]
On the basis of expression (25), the influence coefficients on the resistance are obtained as constants and are numerically equal to
\[ C_{\rho r}=1;\qquad C_{l_{\mathrm{v.sr}}r}=1;\qquad C_{dr}=-2;\qquad C_{wr}=1. \]
Then the influence coefficients of the input parameters \((\rho,\; l_{\mathrm{v.sr}},\; d,\; w)\) on the current \(I_a\) will be equal to
\[ C_{\rho I_a}=C_{r I_a}\cdot C_{\rho r}; \qquad C_{l_{\mathrm{v.sr}} I_a}=C_{r I_a}\cdot C_{l_{\mathrm{v.sr}}r}; \]
\[ C_{d I_a}=C_{r I_a}\cdot C_{dr}; \qquad C_{w I_a}=C_{r I_a}\cdot C_{wr}. \]
The expression for the magnetic flux has the form
\[ \Phi= \frac{F_{\mathrm{rez}}} {\displaystyle \sum_{i=1}^{n}\frac{l_i}{\mu_i\cdot S_i}}. \tag{26} \]
The variance of the flux from this expression is
\[ \sigma_{\Phi_0}^{2} = \sum_{i=1}^{n} \alpha_i^{2} \left( \sigma_{l_{i0}}^{2} + \sigma_{a_{i0}}^{2} + \sigma_{b_{i0}}^{2} + \sigma_{\mu_{i0}}^{2} \right) + \sigma_{(F_{\mathrm{res}})_0}^{2}, \tag{27} \]
where
\[ \alpha_i = \frac{l_i}{\mu_i a_i b_i} \cdot \frac{1}{\displaystyle \sum_{i=1}^{n} \frac{l_i}{\mu_i S_i}} \]
and represents the relative share of the magnetic resistance of the given section of the magnetic system.
In the magnetic system, six sections of the magnetic circuit are distinguished: the frame, the armature yoke, the armature teeth, the air gap, the pole core, and the gap between the pole core and the frame.
In expressions (27), \(a\), \(b\), \(l\) are the geometric dimensions of each section, and \(\mu\) is the magnetic permeability.
The formulas presented for calculating the influence coefficients of input parameters on output parameters for DC machines with a power of 0.4–30 kW make it possible to analyze the influence of various technological factors on the characteristics of the machine.
REFERENCES
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I. P. Isaev. Tolerances for the Characteristics of Electric Locomotives. Moscow, Transzheldorizdat, 1958.
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N. N. Rodinov and I. G. Shubov. Design of DC Electrical Machines. Leningrad, “Energiya,” 1967.