Sensitivity Analysis of an Electromechanical Amplifier with a Smooth Armature to Parameter Variations
Abstract Generated abstract
This paper analyzes the sensitivity of a transverse-field electric machine amplifier with a smooth armature to variations in its internal parameters during operation with an active-inductive load. Using a dynamic equation system, signal-graph representation, Mason’s formula, and sensitivity theory, the authors derive transfer and relative-sensitivity functions and construct sensitivity models for key resistances and winding turns. Amplitude-frequency characteristics and additional-motion curves show that control resistance, shunt resistance, transverse-circuit resistance, and winding turns are critical in the operating frequency range, while longitudinal and transverse circuit resistances most affect transients and armature and compensating winding turns most affect steady state. The results are intended to support design-stage tolerance selection and parameter accuracy requirements for amplifiers with prescribed operating quality.
Full Text
STUDY OF THE SENSITIVITY OF AN ELECTRIC MACHINE AMPLIFIER WITH A SMOOTH ARMATURE TO PARAMETER VARIATIONS
G. A. Sipailov, G. G. Konstantinov, V. M. Lukyanenko,
V. V. Slyusar
Presented by the scientific seminar of the departments of electrical machines and general electrical engineering
In automatic-control and tracking electric-drive systems, transverse-field electric machine amplifiers (EMAs) are widely used. The stringent requirements imposed on the static and dynamic characteristics of elements of automatic systems require the creation of EMAs characterized by high quality indicators. At present, work is being carried out to improve the characteristics of series-produced EMAs and to create new, more efficient EMA designs. Such amplifiers include, in particular, a transverse-field EMA with a smooth armature, which is characterized by higher reliability, faster response, and a low level of output-voltage pulsations.
The complexity of the amplifier structure and the interrelation of parameters, while ensuring high efficiency and the quality of certain characteristics, may lead to deterioration of the stability and sensitivity characteristics, which are the most important indicators and must be evaluated at the design stage of an EMA in order to provide the desired dynamic properties.
At the operating stage of an EMA, as a result of aging of elements, overloads, temperature, etc., changes in EMA parameters occur, which disrupts its normal functioning.
The study of the quality indicators of EMA operation makes it possible, already at the design stage, to identify the parameters that must be determined most accurately in calculation, and also to establish permissible limits for deviations of EMA parameters within the specified quality of operation. This makes it possible to design an amplifier that is weakly sensitive to changes in internal parameters and to establish justified limits for technological tolerances on EMA parameters.
In the present work, on the basis of sensitivity theory using the topological method, the influence of variations of the most important parameters of an EMA with a smooth armature on quality indicators in the operating frequency range is determined.
As a basis, we take the system of equations (1), describing the dynamics of an EMA with a smooth armature when operating into an active-inductive load [1].
\[ p \Phi_1 = \frac{1}{\sigma_y w_y}\, U_y - \frac{r_y}{\sigma_y w_y}\, i_y ; \]
\[ p \cdot \Phi_2=\frac{c}{\sigma_2 \cdot w_2}\cdot \Phi_1-\frac{r_2}{\sigma_2 \cdot w_2}\cdot i_2; \]
\[ p \cdot i_3=\frac{c}{L_{\mathrm{n}}}\cdot \Phi_2-\frac{r_3}{L_{\mathrm{n}}}\cdot i_3-\frac{r_{\mathrm{k}}}{L_{\mathrm{n}}}\cdot i_{\mathrm{k}}+ \frac{\sigma_{\mathrm{a}}\cdot w_3-\sigma_{\mathrm{k}}\cdot w_{\mathrm{k}}}{L_{\mathrm{n}}}\cdot p\cdot \Phi_1; \tag{1} \]
\[ i_{\mathrm{u}}=\frac{1}{\lambda_1\cdot w_{\mathrm{u}}}\cdot \Phi_1-\frac{w_{\mathrm{k}}}{w_{\mathrm{u}}}\cdot i_{\mathrm{k}}+\frac{w_3}{w_{\mathrm{u}}}\cdot i_3; \]
\[ i_2=\frac{1}{\lambda_2\cdot w_2}\cdot \Phi_2; \]
\[ i_{\mathrm{k}}=\frac{r_{\mathrm{sh}}}{R_{\mathrm{k}}}\cdot i_3-\frac{\sigma_{\mathrm{k}}\cdot w_{\mathrm{k}}}{R_{\mathrm{k}}}\cdot p\cdot \Phi_1, \]
where \(\Phi_1,\lambda_1\) are the resultant flux and magnetic permeance along the longitudinal axis;
\(\Phi_2,\lambda_2\) are the resultant flux and magnetic permeance along the transverse axis;
\(\sigma_{\mathrm{a}}, \sigma_{\mathrm{k}}\) are the leakage coefficients of the armature and compensating windings;
\(c\) is the proportionality coefficient between flux and emf;
\(r_{\mathrm{u}}, r_{\mathrm{k}}\) are the resistances of the control and compensating windings;
\(r_{\mathrm{sh}}\) is the resistance of the shunt of the compensating winding;
\(R_{\mathrm{k}}\) is the total resistance of the circuit: compensating winding, shunting resistance;
\(r_{\mathrm{n}}, L_{\mathrm{n}}\) are the load resistance and inductance;
\(w_{\mathrm{u}}, w_{\mathrm{k}}, w_2, w_3\) are the numbers of turns of the control, compensating, transverse-circuit, and longitudinal-circuit windings;
\(i_{\mathrm{u}}, U_{\mathrm{u}}\) are the control current and voltage;
\(i_{\mathrm{k}}\) is the current of the compensating winding;
\(i_2, i_3\) are the currents of the transverse and longitudinal circuits.
From equations (1), a graph has been constructed (Fig. 1) and, corresponding to it, a simulation circuit for an EMA with a smooth armature when operating into an active-inductive load (Fig. 2).
Fig. 1. Graph of an EMA with a smooth armature
Fig. 2. Modeling diagram of an EMA with a smooth armature.
Equations (1), expressed in the coefficients of the model, are written in the following form:
\[ \begin{gathered} p \cdot \Phi_1 = k_1 \cdot U_y - k_2 \cdot i_y; \\ p \cdot \Phi_2 = k_4 \cdot \Phi_1 - k_8 \cdot i_2; \\ p \cdot i_3 = k_{11} \cdot \Phi_2 - k_{13} \cdot i_3 - k_{12} \cdot i_k + k_7 \cdot p \cdot \Phi_1; \\ \\ i_y = k_3 \cdot \Phi_1 - k_6 \cdot i_k + k_9 \cdot i_3; \\ i_2 = k_{10} \cdot \Phi_2; \\ i_k = k_{14} \cdot i_3 - k_5 \cdot p \cdot \Phi_1 . \end{gathered} \tag{2} \]
Using Mason’s formula [2], we determine the transfer functions of the paths and contours of the EMA graph:
\[ P_1(p) = \frac{1}{p^3} \cdot k_1 \cdot k_4 \cdot k_{15}; \tag{3} \]
\[ L_1(p) = -\frac{1}{p^3} \cdot k_2 \cdot k_4 \cdot k_{11} \cdot k_9; \tag{4} \]
\[ L_2(p) = \frac{1}{p^3} \cdot k_2 \cdot k_4 \cdot k_{11} \cdot k_6 \cdot k_{14}; \tag{5} \]
\[ L_3(p) = \frac{1}{p} \cdot k_2 \cdot k_3; \tag{6} \]
\[ L_4(p) = -\frac{1}{p} \cdot k_8 \cdot k_{10}; \tag{7} \]
\[ L_5(p) = -\frac{1}{p} \cdot k_{13}; \tag{8} \]
\[ L_6(p) = -\frac{1}{p} \cdot k_{12} \cdot k_{14}; \tag{9} \]
\[ L_7(p) = -k_2 \cdot k_5 \cdot k_6 . \tag{10} \]
Accordingly, the transfer function of the EMA is equal to
\[ W_{\text{EMA}}(p) = \frac{E_3(p)}{U_y(p)} = \frac{P_1(p) \cdot \Delta_1(p)}{\Delta(p)}, \tag{11} \]
where \(\Delta(p)\) is the determinant of the graph;
\(\Delta_1(p)\) is the minor of the determinant of the graph.
For the analysis of the sensitivity of the EMA to changes in internal parameters, we use the method of signal graphs, since the direct determination of the relative-sensitivity functions is hindered by cumbersome calculations associated with the differentiation of complex expressions.
The relative-sensitivity functions with respect to the transfer function of any element of the graph are determined by the expression [3]:
\[ S_k^{W(p)}(p) = \frac{\Delta_k(p)}{\Delta(p)} - \frac{ \left[ \displaystyle\sum_{i=1}^{n} P_i(p) \cdot \Delta_i(p) \right]_k }{ \displaystyle\sum_{i=1}^{n} P_i(p) \cdot \Delta_i(p) }, \tag{12} \]
where the index \(k\) denotes the exclusion of branch \(k\); \(P_i(p)\) is the \(i\)-th path of the graph; \(\Delta_i(p)\) is the minor of the determinant of the \(i\)-th path of the graph.
In view of the fact that the parameters being investigated enter into several branches of the EMA graph, to determine the relative sensitivity functions with respect to the parameters we shall use the expression [3]:
\[ S_q^{w(p)}(p)=\sum_{i=1}^{r} S_{K_i(p)}^{w(p)}(p)\cdot S_q^{K_i(p)}(p), \tag{13} \]
where \(q\) is the parameter under investigation;
\(r\) is the number of branches of the graph containing the parameter;
\(K_i(p)\) is the transfer function of the \(i\)-th branch.
As the variable parameters of the EMA we take the resistances \(r_y, r_{\text{sh}}, r_{\text{sh}}, r_2, r_3\) and the turns \(w_y, w_z, w_k\).
Using the expressions obtained from formula (13), graphs of the relative sensitivity of the EMA and the corresponding sensitivity models were constructed. When a harmonic signal was applied to the inputs of the EMA sensitivity models, the amplitude-frequency characteristics (AFC) of the relative sensitivity functions in the operating frequency range were obtained. Figure 3 shows the AFCs of the relative sensitivity functions with respect to variations of the EMA parameters:
\[ 1-\left|S_{w_3}^{w(p)}(j\omega)\right|,\ \left|S_{w_k}^{w(p)}(j\omega)\right|;\quad 2-\left|S_{r_2}^{w(p)}(j\omega)\right|;\quad 3-\left|S_{r_{\text{sh}}}^{w(p)}(j\omega)\right|; \]
\[ 4-\left|S_{r_3}^{w(p)}(j\omega)\right|;\quad 5-\left|S_{w_y}^{w(p)}(j\omega)\right|;\quad 6-\left|S_{r_y}^{w(p)}(j\omega)\right|;\quad 8-\left|S_{r_{\text{sh}}}^{w(p)}(j\omega)\right|. \]
From analysis of the sensitivity curves it is evident that, in the operating frequency range, the critical parameters are: \(r_y, r_{\text{sh}}, r_2, w_y, w_z, w_k\).
Fig. 3. Amplitude-frequency characteristics of the sensitivity of an EMA with a smooth armature
More complete information on the influence of parameter variations on the static and dynamic characteristics of the EMA can be obtained by analyzing the additional motions of the output parameter. To obtain the additional motion with respect to some parameter, the EMA model is connected with the sensitivity model for the corresponding parameter. By applying a step action to the input of the EMA model, at the output of the sensitivity model one obtains the additional motion from the vari-
of variation of an EMA parameter. Figure 4 shows the curves of additional motions \(\delta_i(t)\) when the EMA parameters are increased by 10% and the transient characteristic of the unvaried EMA \(y(t)\):
\(1\) — \(\delta_{w_k}(t)\), — \(\delta_{w_3}(t)\); \(2\) — \(\delta_{r_2}(t)\); \(3\) — \(\delta_{r_{\sh}}(t)\); \(4\) — \(\delta_{r_3}(t)\); \(5\) — \(\delta_{w_y}(t)\);
\(6\) — \(\delta_{r_y}(t)\); \(7\) — \(y(t)\).
From the curves of additional motions of the output parameter of the EMA, the following conclusions can be drawn:
Fig. 4. Transient characteristic and additional motions when the parameters of an EMA with a smooth armature are changed
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The character of the transient process is affected most by variations in the resistances of the longitudinal and transverse circuits.
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The steady state of the transient process is affected most by variations in the numbers of turns of the armature winding and the compensating winding.
Thus, proceeding from the specified quality of operation of the EMA, one can, using the obtained curves of additional motions, select the permissible limits of variation of the principal parameters of the EMA.
References
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G. G. Konstantinov. Development and study of an electric machine amplifier of a transverse field with a smooth armature. Dissertation for the degree of Candidate of Technical Sciences. Tomsk, 1968.
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L. Robisho, M. Buaver, J. Kober. Directed graphs and their application to electrical circuits and machines. Moscow–Leningrad, “Energiya,” 1964.
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R. Tomovich, M. Vukobratovich. General theory of sensitivity. Moscow, “Sovetskoe Radio,” 1972.