MATHEMATICAL MODELING OF ELECTROMAGNETIC PROCESSES IN A SHOCK GENERATOR

A. V. Loos, G. A. Sipailov

Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering

The difficulty of physical modeling, associated with the need to create a new model for each change in the parameters of the object under study, brings the method of mathematical modeling to the forefront. Mathematical modeling is based on the identity of the mathematical description of the processes in the object under investigation and in its model.

The use of analog computing machines for mathematical modeling makes it possible to solve many problems of experimental research with ease.

In the mathematical modeling of a shock generator, the original is the mathematical description of the processes in the real machine. The main difficulty in the mathematical modeling of the electromagnetic processes of a pulse-power generator, as of all alternating-current machines, is determined by the time dependence of the mutual-inductance coefficients of the windings. The mutual-inductance coefficients of the stator and rotor windings are functions of the angular position of the rotor. This determines the appearance of variable coefficients in the differential equations describing the electromagnetic processes of the shock generator.

Modeling of a shock generator can be carried out in two ways:

1) Direct use of the differential equations describing the processes in the real windings of the generator; in this case, the presence of variable coefficients requires the introduction into the mathematical model of special multiplying-and-solving elements, which considerably complicates the solution circuit.

2) Transformation of the differential equations of the electromagnetic processes in such a way as to reduce the number of variable coefficients or to eliminate them altogether.

In this case the solution circuit is simplified; however, difficulties arise in analyzing the results obtained, connected with the transformation of the dependent variables.

Below the first method of modeling is considered, since it has greater clarity and convenience for investigation.

In the modeling, the following assumptions were adopted:
1. The active resistances of the windings are constant.
2. There are no losses in the steel.
3. The coefficients of mutual induction and self-induction of the windings do not depend on the magnitude of the current.
4. The rotational speed of the generator is constant.

The adopted assumptions can be justified by the fact that, in the case under consideration, the influence of individual winding parameters on the processes taking place is clarified under the condition that all other parameters remain constant.

In principle, any of the adopted assumptions can be abandoned. However, this leads to a complication of the solution scheme. Therefore, rejection of such inessential assumptions as the nonconstancy of the active resistance of the windings is generally inexpedient. In further studies of the operation of the shock generator, it is intended to take into account saturation of the magnetic circuit and variation of the rotational speed.

Under the assumptions adopted above, the coefficients of self-induction of the windings are constant; the coefficients of mutual induction of the stator and rotor windings vary according to a sinusoidal law; the mutual inductance between the excitation winding and the damper winding along the longitudinal axis is constant; owing to the symmetry of the rotor, there is no mutual inductance between the damper winding in the transverse axis and the windings in the longitudinal axis of the rotor. In all investigations, all quantities are expressed in relative units.

Equilibrium equations of the emf of the shock generator and their solution

The shock generator is a single-phase synchronous generator of the turbogenerator type. The generator has a powerful damper system along the longitudinal and transverse axes. Initially the generator operates at no load; at the moment when the generator emf passes through zero, the generator is connected to the load. When the generator current first passes through zero, the generator is disconnected. Fig. 1 shows the arrangement of the stator and rotor windings of the generator corresponding to the switching-on instant, and the notation for the parameters of the windings, the load, and the currents in the windings.

For each of the windings, the differential equations of emf equilibrium can be written in the following form:

\[ \begin{aligned} 0 &= (r_c + r_n)i_c + \frac{d\psi_c}{dt};\\ U_{\mathrm{B}} &= r_{\mathrm{B}} i_{\mathrm{B}} + \frac{d\psi_{\mathrm{B}}}{dt};\\ 0 &= r_{Dd} i_{Dd} + \frac{d\psi_{Dd}}{dt};\\ 0 &= r_{Dq} i_{Dq} + \frac{d\psi_{Dq}}{dt}. \end{aligned} \tag{1} \]

The values of the flux linkages depend on the mutual arrangement of the windings,

\[ \begin{aligned} \psi_c &= i_c (L_c + L_n) + i_{\mathrm{B}} M\cos\omega t + i_{Dd}M\cos\omega t + i_{Dq}M\sin\omega t,\\ \psi_{\mathrm{B}} &= i_c M\cos\omega t + i_{\mathrm{B}} L_{\mathrm{B}} + i_{Dd}M,\\ \psi_{Dd} &= i_c M\cos\omega t + i_{\mathrm{B}}M + i_{Dd}L_{Dd},\\ \psi_{Dq} &= i_c M\sin\omega t + i_{Dq}L_{Dq}. \end{aligned} \tag{2} \]

Fig. 1. Arrangement of the stator and rotor windings corresponding to the instant of switching on.

Substituting the values (2) into equations (1) and reducing them to a form convenient for constructing a block diagram, we obtain the equations of the electromagnetic processes of the shock generator:

\[ \frac{di_c}{dt} = -\frac{r_c+r_{\mathrm{н}}}{L_c+L_{\mathrm{н}}}\,i_c - \frac{M}{L_c+L_{\mathrm{н}}}\cos\omega t\cdot \frac{di_{\mathrm{в}}}{dt} + \frac{\omega M}{L_c+L_{\mathrm{н}}}\sin\omega t\cdot i_{\mathrm{в}} - \]

\[ - \frac{M}{L_c+L_{\mathrm{н}}}\cos\omega t\cdot \frac{di_{Dd}}{dt} + \frac{\omega M}{L_c+L_{\mathrm{н}}}\sin\omega t\cdot i_{Dd} - \]

\[ - \frac{M}{L_c+L_{\mathrm{н}}}\sin\omega t\cdot \frac{di_{Dq}}{dt} - \frac{\omega M}{L_c+L_{\mathrm{н}}}\cos\omega t\cdot i_{Dq}; \]

\[ \begin{gathered} \frac{di_{\mathrm{B}}}{dt} = -\frac{r_{\mathrm{B}}}{L_{\mathrm{B}}} i_{\mathrm{B}} +\frac{1}{L_{\mathrm{B}}}U_{\mathrm{B}} -\frac{M}{L_{\mathrm{B}}}\cos\omega t \cdot \frac{di_{\mathrm{C}}}{dt} +\frac{\omega M}{L_{\mathrm{B}}}\sin\omega t \cdot i_{\mathrm{C}} -\\[6pt] -\frac{M}{L_{\mathrm{B}}}\cdot \frac{di_{Dd}}{dt}; \\[10pt] \frac{di_{Dd}}{dt} = -\frac{r_{Dd}}{L_{Dd}} i_{Dd} -\frac{M}{L_{Dd}}\cdot \frac{di_{\mathrm{B}}}{dt} -\frac{M}{L_{Dd}}\cos\omega t \cdot \frac{di_{\mathrm{C}}}{dt} +\frac{\omega M}{L_{Dd}}\sin\omega t \cdot i_{\mathrm{C}}; \\[10pt] \frac{di_{Dq}}{dt} = -\frac{r_{Dq}}{L_{Dq}} i_{Dq} -\frac{M}{L_{Dq}}\sin\omega t \cdot \frac{di_{\mathrm{C}}}{dt} -\frac{\omega M}{L_{Dq}}\cos\omega t \cdot i_{\mathrm{C}}. \end{gathered} \tag{3} \]

The obtained system of differential equations has a large number of variable coefficients; therefore an analytical investigation of system (3) by ordinary methods without additional simplifications is practically impossible. The use of an analog computer makes it possible to solve this problem and permits a large research program to be carried out with small expenditures of time.

The solution of the system of equations (3) was carried out on the IPT-5 analog computer. The block diagram of the solution is presented in Fig. 2. The modeling circuit for each of the differential equations of system (3) is identical.

To integrate differential equations with variable coefficients on the IPT-5, electromechanical variable-coefficient units are used, which multiply the dependent variable by the variable coefficient. In the solution circuit, 10 variable-coefficient units are used, on which the dependence of the mutual inductance of the windings on the angular position of the rotor is set.

From the compiled block diagram, one can write, for each operational amplifier, the expression for the output voltage:

\[ U_1=-(k_{11}U_5+k_{12}U_8+k_{13}U_4+k_{14}U_9+k_{15}U_{10}+k_{16}U_{11}+k_{17}U_{15}). \]

\[ U_2=-\frac{1}{p}\,k_{21}U_1. \]

\[ U_3=-(k_{31}U_7+k_{32}U_{\mathrm{B}}+k_{33}U_1+k_{34}U_2+k_{35}U_9). \]

\[ \begin{gathered} U_4=-\frac{1}{p}k_{41}U_3,\\[4pt] U_5=-k_{51}U_2,\\[4pt] U_7=-k_{71}U_4,\\[4pt] U_9=-(k_{91}U_{13}+k_{92}U_3+k_{93}U_1+k_{94}U_2),\\[4pt] U_{10}=-\frac{1}{p}k_{101}U_9,\\[4pt] U_{11}=-(k_{111}U_{15}+k_{112}U_1+k_{113}U_2),\\[4pt] U_{12}=-\frac{1}{p}k_{121}U_{11},\\[4pt] U_{13}=-k_{131}U_{10},\\[4pt] U_{15}=-k_{151}U_{12}, \end{gathered} \tag{4} \]

Fig. 2. Structural solution scheme.

where

\(U_1, U_2, \ldots, U_{15}\) are the voltages at the amplifier output;

\(k_{11}, k_{12}, k_{13}, \ldots\) are the transfer coefficients of the amplifiers, in which the first digit of the subscript corresponds to the block number, and the second to the input number.

Solving the system of equations (4) with respect to \(U_2, U_4, U_{10}, U_{12}\), substituting into the resulting equations, in place of \(U_4, U_2, U_{10}, U_{12}\), their expressions through the dependent variables and scale factors, and transforming the resulting equations to the form of system (3), we obtain:

\[ \begin{aligned} \frac{di_c}{dt}={}&-\frac{k_{21}k_{11}k_{51}}{M_t}\,i_c -\frac{k_{21}k_{12}}{k_{41}}\cdot\frac{di_{\mathrm{B}}}{dt} +\frac{k_{21}k_{13}}{M_t}\,i_{\mathrm{B}} -\frac{k_{21}k_{14}}{k_{101}}\cdot\frac{di_{Dd}}{dt}+{}\\ &+\frac{k_{21}k_{15}}{M_t}\,i_{Dd} -\frac{k_{21}k_{16}}{k_{121}}\cdot\frac{di_{Dq}}{dt} -\frac{k_{21}k_{17}k_{151}}{M_t}\,i_{Dq};\\[6pt] \frac{di_{\mathrm{B}}}{dt}={}&-\frac{k_{41}k_{31}k_{71}}{M_t}\,i_{\mathrm{B}} +\frac{M_i k_{41}k_{32}}{M_t M_u}\,U_{\mathrm{B}} -\frac{k_{41}k_{33}}{k_{21}}\cdot\frac{di_c}{dt}+{}\\ &+\frac{k_{41}k_{34}}{M_t}\,i_c -\frac{k_{41}k_{35}}{k_{101}}\cdot\frac{di_{Dd}}{dt};\\[6pt] \frac{di_{Dd}}{dt}={}&-\frac{k_{101}k_{91}k_{131}}{M_t}\,i_{Dd} -\frac{k_{101}k_{92}}{k_{41}}\cdot\frac{di_{\mathrm{B}}}{dt} -\frac{k_{101}k_{93}}{k_{21}}\cdot\frac{di_c}{dt} +\frac{k_{101}\cdot k_{94}}{M_t}\,i_c;\\[6pt] \frac{di_{Dq}}{dt}={}&-\frac{k_{121}k_{111}k_{151}}{M_t}\,i_{Dq} -\frac{k_{121}k_{112}}{k_{21}}\cdot\frac{di_c}{dt} -\frac{k_{121}k_{131}}{M_t}\,i_c, \end{aligned} \tag{5} \]

where

\[ M_t \quad \text{is the time scale;} \]

\[ M_i=\frac{i_c}{U_2}=\frac{i_{\mathrm{B}}}{U_4} =\frac{i_{Dd}}{U_{10}}=\frac{i_{Dq}}{U_{12}} \quad \text{is the current scale;} \]

\[ M_u=\frac{U_{\mathrm{B}}}{U_{\mathrm{B}}} \quad \text{is the voltage scale.} \]

Equating the coefficients of the unknowns in systems (3) and (5), and taking
\(k_{21}=k_{51}=k_{61}=k_{41}=k_{71}=k_{81}=k_{101}=k_{121}=k_{131}=k_{141}=k_{151}=k_{161}=1\), we obtain expressions for the transmission coefficients of the summing blocks:

\[ k_{11}=M_t\frac{r_c+r_{\mathrm{H}}}{L_c+L_{\mathrm{H}}},\qquad k_{12}=\frac{M}{L_c+L_{\mathrm{H}}},\qquad k_{13}=\frac{M_t\omega M}{L_c+L_{\mathrm{H}}}, \]

\[ k_{14}=\frac{M}{L_c+L_{\mathrm{H}}},\qquad k_{15}=\frac{M_t\omega M}{L_c+L_{\mathrm{H}}},\qquad k_{16}=\frac{M}{L_c+L_{\mathrm{H}}}, \]

\[ k_{17}=\frac{M_t\omega M}{L_c+L_{\mathrm{H}}},\qquad k_{31}=M_t\frac{r_{\mathrm{B}}}{L_{\mathrm{B}}},\qquad k_{32}=\frac{M_t M_u}{M_i L_{\mathrm{B}}}, \]

\[ k_{33}=\frac{M}{L_{\mathrm{H}}},\qquad k_{34}=\frac{M_t\omega M}{L_{\mathrm{B}}},\qquad k_{35}=\frac{M}{L_{\mathrm{B}}}, \]

\[ k_{91}=M_t\frac{r_{Dd}}{L_{Dd}},\qquad k_{92}=\frac{M}{L_{Dd}},\qquad k_{93}=\frac{M}{L_{Dd}}, \]

\[ k_{94}=\frac{M_t\omega M}{L_{Dd}},\qquad k_{111}=M_t\frac{r_{Dq}}{L_{Dq}},\qquad k_{112}=\frac{M}{L_{Dq}}, \]

\[ k_{113}=\frac{M_t\omega M}{L_{Dq}}. \]

By changing these coefficients by successively varying the corresponding parameters that determine these coefficients, one can study the operation of the shock generator for any parameter values.

The circuit shown in Fig. 2 also makes it possible to study the operation of a single-phase synchronous generator without damper windings or with a damper winding along only one axis. To do this, it is necessary to disconnect from the solution circuit the blocks that model the differential equation of the processes in the corresponding winding.

Fig. 3. Oscillogram of the solution.

Figure 3 shows an oscillogram of the solution of the differential equations (3) for the following generator parameters in relative units:

\[ M = 1.00; \qquad L_c = 1.05; \qquad L_{\mathrm{B}} = 1.1; \]

\[ L_{Dq} = L_{Dd} = 1.05; \qquad r_c = 0.005; \qquad r_{\mathrm{B}} = 0.005; \]

\[ r_{Dd} = r_{Dq} = 0.005. \]

CONCLUSION

Mathematical modeling of the electromagnetic processes of a shock generator on the basis of the developed mathematical model is of special importance, since computational investigation by ordinary methods is practically impossible because of the complexity of the original system of equations.

The developed mathematical model of the electromagnetic processes of a shock generator makes it possible to carry out a large program of studies; moreover, in order to pass to new parameter values it is necessary only to change the values of the input resistances at blocks 1, 3, 9, 11 (Fig. 2).