Investigation of the Characteristics of an Electromechanical Equivalent of Autonomous Energy Generators
Abstract Generated abstract
The paper studies an electromechanical equivalent of autonomous energy generators, focusing on how variations in correction-circuit parameters, feedback coefficients, and the electromechanical amplifier time constant affect its behavior. Using a signal-graph representation, Mason’s formula, and sensitivity theory, the authors derive the equivalent transfer function and relative sensitivity functions for the main variable parameters, then construct frequency characteristics and modeling circuits. The analysis indicates that most considered parameters, except the combined coefficient related to current and feedback paths, substantially influence output stability in the operating frequency range, with correction-device time constants especially important at low frequencies. Simulation of step responses with 10 percent parameter changes suggests that correction-device parameters and the amplifier time constant affect transient quality, while the voltage-feedback coefficient most strongly affects static accuracy.
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STUDY OF THE CHARACTERISTICS OF AN ELECTROMECHANICAL EQUIVALENT OF AUTONOMOUS ENERGY GENERATORS
G. G. KONSTANTINOV, M. V. LUKYANENKO, V. A. SARYCHEV
Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering
In the development of electric-power systems for autonomous facilities, equivalents of autonomous energy generators are widely used. Questions concerning the application and circuit design of equivalents are considered in [1, 2]. The present work is devoted to the study of the characteristics of an electromechanical equivalent of autonomous energy generators. The block diagram of the equivalent is given in Fig. 1, where $W_{\text{k.u}}(p)$ is the transfer function of the correcting device, $K_{\text{u.n}}$ is the gain of the voltage amplifier in the direct circuit, $W_{\text{emu}}(p)$ is the transfer function of the electromechanical amplifier with a smooth armature, $Z_{\text{n}}$ is the load resistance, $K_{\text{i.u}}$ is the gain of the measuring device, $K_{\text{n}}$ is the voltage-feedback coefficient, $K_{\text{u.o.s}}$ is the gain of the feedback amplifier, $K_{\text{f.p}}$ is the coefficient of the functional converter, and $U_{\text{sm}}$ is the bias voltage.
In designing the equivalent, the task arises of refining the parameters of the correcting circuits and of estimating the influence on the operation of the equivalent of those parameters that may vary in the course of design, adjustment, and operation.
This task may be solved by an analytical method or by the method of mathematical modeling. Mathematical modeling using sensitivity theory and the signal-graph method is preferable in this case, since modern analog machines make it possible, with sufficient accuracy, to take into account a large number of factors affecting the course of the processes under investigation when the parameters of the equivalent are varied over wide ranges.
Let us investigate the sensitivity of the equivalent to variations in the transfer coefficients $K_{\text{n}}$, $K_{\text{T}} = K_{\text{f.p}} \cdot K_{\text{u.o.s}} \cdot K_{\text{i.u}} \cdot \dfrac{1}{Z_{\text{n}}}$, the time constants of the correcting device $T_1, T_2, T_3, T_4$, and the time constant of the electromechanical amplifier $T_{\text{emu}}$.
According to the block diagram (Fig. 1), a graph of the equivalent was constructed (Fig. 2). The parameters of the equivalent, expressed in terms of the transfer coefficients of the graph, are written as
\[ k_1 = \frac{1}{T_1}\left(\frac{T_1}{T_2} - 1\right);\quad k_2 = \frac{1}{T_2};\quad k_3 = \frac{T_1}{T_2};\quad k_4 = \frac{1}{T_4}\left(1 - \frac{T_3}{T_4}\right); \]
\[ k_5=\frac{1}{T_4};\quad k_6=\frac{T_3}{T_4};\quad k_7=\frac{K_{\mathrm{EMU}}}{T_{\mathrm{EMU}}^2};\quad k_8=\frac{1}{T_{\mathrm{EMU}}^2};\quad k_9=\frac{2\xi}{T_{\mathrm{EMU}}};\quad k_{10}=k_{\mathrm{un}}; \]
\[ k_{11}=k_T;\quad k_{12}=k_n. \]
Fig. 1. Structural diagram of the equivalent
According to Mason’s formula [3], the transfer function of the equivalent has been determined; for convenience in modeling it is written in the form of a fractional-rational function:
\[ W(p)= \frac{\displaystyle\sum_{i=1}^{n} P_i(p)\cdot \Delta_i(p)} {\Delta(p)} = \frac{ 1.6\,\frac{1}{p^4}+8.58\,\frac{1}{p^3}+11.24\,\frac{1}{p^2} }{ 1.6\,\frac{1}{p^4}+9.3\,\frac{1}{p^3}+15.8\,\frac{1}{p^2}+7.3\,\frac{1}{p}+1 }, \tag{1} \]
where \(P_i(p)\) is the \(i\)-th path of the graph; \(\Delta(p)\) is the determinant of the graph; \(\Delta_i(p)\) is the minor of the determinant of the \(i\)-th path of the graph.
Fig. 2. Graph of the equivalent
To determine the functions of relative sensitivity with respect to the transfer function \(k(p)\) of some element of the graph, we use the expression [4]
\[ S_{k(p)}^{W(p)}(p)= \frac{\Delta_k(p)}{\Delta(p)} - \frac{ \left[ \displaystyle\sum_{i=1}^{n} P_i(p)\cdot \Delta_i(p) \right]_k }{ P_i(p)\cdot \Delta_i(p) }, \tag{2} \]
where the index “\(k\)” denotes exclusion of branch \(k\) from the determinant.
Using formula (2), the functions of relative sensitivity have been determined for the elements of the equivalent graph that include the variable parameters:
\[ S_{K_1}^{w(p)}(p);\ S_{K_2}^{w(p)}(p);\ S_{K_3}^{w(p)}(p);\ S_{K_4}^{w(p)}(p);\ S_{K_5}^{w(p)}(p);\ S_{K_6}^{w(p)}(p);\ S_{K_7}^{w(p)}(p); \]
\[ S_{K_8}^{w(p)}(p);\ S_{K_9}^{w(p)}(p);\ S_{K_{10}}^{w(p)}(p);\ S_{K_{11}}^{w(p)}(p);\ S_{K_{12}}^{w(p)}(p). \]
From the expressions obtained, graphs and simulation circuits of the sensitivities with respect to the elements of the equivalent graph can be constructed. In most cases the variable parameters enter into several branches of the graph. The functions of relative sensitivity with respect to the parameters are determined by the expression [4]:
\[ S_q^{w(p)}(p)=\sum_{i=1}^{r} S_{K_i(p)}^{w(p)}(p)\cdot S_q^{K_i(p)}(p), \tag{3} \]
where \(q\) is the parameter under investigation,
\(r\) is the number of branches of the graph containing the parameter \(q\),
\(k_i(p)\) is the transfer function of the \(i\)-th branch.
Using formula (3), the functions of relative sensitivity directly with respect to the parameters of the equivalent have been determined:
\[ S_{T_1}^{w(p)}(p)=1.07\,S_{K_1}^{w(p)}(p)+S_{K_3}^{w(p)}(p); \tag{4} \]
\[ S_{T_2}^{w(p)}(p)=-\left[2.07\,S_{K_1}^{w(p)}(p)+S_{K_2}^{w(p)}(p)+S_{K_3}^{w(p)}(p)\right]; \tag{5} \]
\[ S_{T_3}^{w(p)}(p)=S_{K_6}^{w(p)}(p)-0.0656\,S_{K_4}^{w(p)}(p); \tag{6} \]
\[ S_{T_4}^{w(p)}(p)=-\left[0.935\,S_{K_4}^{w(p)}(p)+S_{K_5}^{w(p)}(p)+S_{K_6}^{w(p)}(p)\right]; \tag{7} \]
\[ S_{T_{\mathrm{emu}}}^{w(p)}(p)=-\left[2S_{K_7}^{w(p)}(p)+2S_{K_8}^{w(p)}(p)+S_{K_9}^{w(p)}(p)\right]; \tag{8} \]
\[ S_{K_{\mathrm{n}}}^{w(p)}(p)=S_{K_{12}}^{w(p)}(p); \tag{9} \]
\[ S_{K_T}^{w(p)}(p)=S_{K_{11}}^{w(p)}(p). \tag{10} \]
Fig. 3. Amplitude-frequency characteristics of the sensitivities of the equivalent:
\[ 1 — \left|S_{T_1}^{w(p)}(p)\right|;\quad 2 — \left|S_{T_2}^{w(p)}(p)\right|;\quad 3 — \left|S_{T_3}^{w(p)}(p)\right|; \]
\[ 4 — \left|S_{T_4}^{w(p)}(p)\right|;\quad 5 — \left|S_{T_{\mathrm{emu}}}^{w(p)}(p)\right|;\quad 6 — \left|S_{K_{\mathrm{n}}}^{w(p)}(p)\right|. \]
On the basis of the obtained expressions for the relative-sensitivity functions, amplitude-frequency characteristics of the equivalent’s sensitivity to parameter variations have been constructed (Fig. 3). From an analysis of the curves it is evident that, in the operating frequency range, the parameters considered, except for the coefficient \(k_T\), have a significant influence on the stability of the equivalent’s output characteristics. In the low-
Fig. 4. Equivalent modeling circuit
frequency region, [[unclear: continuation is on the next page or obscured by figure boundary]]
Fig. 5. Transient characteristic and additional motions when the equivalent parameters are varied:
\(1\) — \(\delta_{T_1}(t)\); \(2\) — \(\delta_{T_2}(t)\); \(3\) — \(\delta_{T_3}(t)\); \(4\) — \(\delta_{T_4}(t)\); \(5\) — \(\delta_{T_{\mathrm{emu}}}(t)\); \(6\) — \(\delta_{k_n}(t)\); \(7\) — \(y(t)\).
...frequencies, the greatest influence is exerted by the time constants of the correcting device \(T_1\), \(T_2\), and \(T_4\). However, the influence of the time constant \(T_4\) decreases sharply as the frequency increases. The sensitivity with respect to the coefficient \(k_T\) is close to zero.
In order to be able to quantitatively assess the influence of changes in the parameters on the static and dynamic characteristics of the equivalent, it is necessary to obtain an additional motion of the output voltage when one or another parameter is varied.
The additional motions are obtained by applying a step action to the input of the equivalent model (Fig. 4), which is connected to the sensitivity model for the corresponding parameter. At the output of the sensitivity model, the additional motion is obtained.
Figure 5 shows the curves of the additional motions \(\delta_i(t)\) when the parameters of the equivalent are increased by 10%, and the unvaried transient process \(Y(t)\).
From the curves of the additional motions of the output voltage of the equivalent, the following conclusions may be drawn:
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Variations in the parameters of the correcting device and the time constant of the electromachine amplifier have a significant effect on the qualitative indicators of the transient process of the equivalent.
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The static accuracy of the equivalent is affected most strongly by variation of the coefficient \(k_{\mathrm{n}}\).
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