MATHEMATICAL MODEL OF A CIRCUIT FOR THE JOINT OPERATION OF A PULSED POWER GENERATOR AND A CAPACITOR BANK

A. V. Loos, G. A. Sipailov, K. A. Khorkov

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

An analytical study of the joint operation of a pulsed generator with a capacitor bank under an inductive load has shown that the use of an auxiliary capacitance leads to a considerable increase in the energy delivered by the generator to the load [1]. However, because of the unwieldiness of the formulas describing the processes during joint operation, the analysis of the circuit in [1] was carried out while neglecting the active resistances of the generator and the load and assuming equality of their inductances.

It is known that the magnitudes of the active and inductive resistances of the generator and the load, the value of the capacitance, and also the magnitude and sign of the initial voltage of the capacitance strongly influence the character of the transient process. Therefore, it is of great interest to investigate the influence of individual parameters on the magnitude of the load current in order to determine the optimal parameters of the circuit.

The use of the methods and means of mathematical modeling makes it possible to obtain a solution to this question while reducing to a minimum the required expenditure of time. The essence of modeling consists in replacing the real circuit by its mathematical model, which reproduces the operation of the real circuit. The processes occurring in the mathematical model can be observed, recorded, and checked for agreement with the results of theoretical analysis. Along with this, in mathematical modeling it is comparatively simple to change the parameters of individual elements of the circuit and to determine the influence of these changes on the quality of operation of the circuit as a whole.

The operation of the circuit was studied on an MNB-1 analog computer. Analog computers are characterized by the fact that each variable in the real system is brought into correspondence with a certain variable machine quantity. The original is the mathematical description of the processes in the circuit under study, while the analog computer realizes the prescribed mathematical relations.

In the present work, on the basis of the mathematical model created for the circuit of joint operation of a pulsed generator with a capacitor bank, studies were carried out of the influence of individual circuit parameters on the character of the transient process, in particular on the magnitude of the energy delivered to the load.

Mathematical Model of the Circuit

The circuit for increasing the energy obtained from a pulsed generator by means of a capacitor bank is shown in Fig. 1.

Fig. 1. Circuit for the joint operation of a pulsed generator (PG) and a capacitor bank. Here \(r_g, r_n, L_g, L_n\) are the active and inductive resistances of the generator and the load; K-1, K-2 are switching devices.

In the initial state the devices K-1 and K-2 are open, the capacitor bank is charged to the voltage \(U_{c0}\), and the generator is operating at no load. At the moment when the generator emf passes through zero \((e_g = 0)\), and with the direction of increase of \(e_g\) coinciding with \(U_{c0}\), K-1 is closed. The processes occurring in this case are described by the equation

\[ E_m \sin(\omega t+\psi_0)=L_g\frac{di_g}{dt}+r_g i_g+\frac{1}{C}\int i_c\,dt. \tag{1} \]

At the moment when the voltage across the capacitance passes through zero, K-2 is closed. The transient processes for this stage of circuit operation are described by equations (1) and (2)

\[ 0=L_n\frac{di_n}{dt}+r_n i_n+\frac{1}{C}\int i_c\,dt. \tag{2} \]

At the moment when the generator current first passes through zero, the generator is disconnected by opening K-1. The transient process in the circuit for this stage is described by equation (2). Then K-2 is opened at the moment when the load current passes through zero.

Using known mathematical methods, it is in principle possible to obtain expressions for the currents and voltages for the entire operating cycle of the circuit under study. But such a path to the solution is associated with considerable difficulties, which are explained by the changes in the circuit that occur during the cycle. Therefore, an analytical study of the circuit is practically accessible only by stages of circuit operation. However, investigation by stages does not give an overall picture of the variation of currents and voltages. An electronic model of the circuit under study makes it possible to overcome these difficulties, and also makes it possible to carry out investigations for a large number of variants.

Solving the equations of the transient processes by stages on an analog machine presents no difficulty. However, this increases the laboriousness of solving the problem and makes inevitable the errors introduced by determining and setting the initial conditions. Therefore, it is desirable—

was to find ways of solving the problem as a whole, without dividing it into stages. For this purpose we created an auxiliary circuit that automatically performs the switching in the mathematical model from stage to stage, which made it possible to investigate the entire operating process of the circuit at once, since in doing so a complete electronic model of the circuit under study was created. The structural diagram of the mathematical model is shown in Fig. 2. Blocks 1, 2, 3, 4, 5, 6, 10 model the differential

Fig. 2. Structural diagram of the mathematical model of the joint operation of the UG and the capacitor bank.

equations (1) and (2). An open contact \(1P_1\) and a closed \(1P_2\) correspond to the mathematical model of equation (1). Closed \(1P_1\) and \(1P_2\) correspond to the mathematical model of equations (1) and (2). Closed \(1P_1\) and open \(1P_2\) correspond to the mathematical model of equation (2).

The emf of sinusoidal form is obtained from a special generator (blocks 7, 8, 9). The necessary switchings in the mathematical

models occur as a function of the voltage on the capacitor bank and of the generator current. The operation of the circuit is controlled automatically and is assembled on blocks 11, 12 and lamps \(L_1\) and \(L_2\). The operation of the relays is controlled by means of blocks 11 and 12, which have diode limiters in the feedback circuit. When the input signal is zero, a voltage jump is produced at the output of these blocks (relay characteristic). This control voltage is applied to the grids of \(L_1\) and \(L_2\). Under the action of this signal the lamps open (each at the corresponding moment), current flows through the relay windings, and the relays operate and perform the necessary switching in the circuit of the mathematical model. The processes in the mathematical model are described by the equations:

\[ pU_1=-k_{11}U_1-\frac{1}{p}k_{12}k_{61}k_{51}k_{41}k_{21}U_1+ \frac{1}{p}k_{12}k_{61}k_{51}k_{42}U_3-k_{13}U_0, \tag{3} \]

\[ pU_3=-k_{31}U_3+\frac{1}{p}k_{21}k_{32}k_{51}k_{41}U_1- \frac{1}{p}k_{32}k_{51}k_{42}U_3. \tag{4} \]

Here

\[ U_1,\ U_3,\ U_0 \]

are the voltages at the outputs of the blocks;

\[ k_{11},\ k_{12},\ldots \]

are the transfer coefficients of the blocks, where the first digit denotes the block number and the second the input number.

The initial charge of the capacitor bank in the mathematical model corresponds to the initial condition at block 5. Comparing equations (1), (2) and (3), (4), we express the transfer coefficients of the blocks through the parameters of the actual circuit. Changes in the parameters of the actual circuit correspond in the model to changes in only four coefficients \((k_{11}, k_{12}, k_{31}, k_{32})\), which depend on the parameters of the actual circuit as follows:

\[ k_{11}=0.016\,\frac{r_{\mathrm{g}}}{L_{\mathrm{g}}}, \]

\[ k_{12}=0.0000256\,\frac{1}{L_{\mathrm{g}}C}, \]

\[ k_{31}=0.016\,\frac{r_{\mathrm{n}}}{L_{\mathrm{n}}}, \tag{5} \]

\[ k_{32}=0.0000256\,\frac{1}{L_{\mathrm{n}}C}. \]

Studies of the operation of the circuit were carried out in order to determine the influence of individual circuit parameters on the amount of energy delivered to the load by the impulse generator during joint operation with the capacitor bank.

Results of investigations of the circuit operation

1. Investigation of the influence of the active resistance of the stator windings and of the load on the magnitude of the current flowing through the load.

The investigations were carried out for a circuit with natural angular frequency of oscillation

\[ \omega_k=\sqrt{3.5}\,\omega \]

with an initial charge of the capacitance \(U_{c0}=0.5E_m\), for ratios of active resistance to reactance

\[ \frac{r}{x}=0;\ 0.01; \]

\[ 0.02;\ 0.05;\ 0.1;\ 0.15;\ 0.2. \]

The results of the investigations are presented in Table 1.

Table 1

As follows from Table 1, the magnitude of the current in the load decreases as the active resistance increases. For pulsed generators, the ratio $\dfrac{r}{x}$ has, on average, a value of the order of 0,05; in this case the attenuation coefficient

$$ k_z = \frac{i(r)}{i(r = 0)} $$

will be equal to $\sim 0,92$.

When the circuit operates into an inductive load that does not contain a steel core, the attenuation coefficient due to an increase in the ratio $\dfrac{r}{x}$ will be still smaller.

On average, it may be considered that neglecting the active resistance introduces into the results of calculating the current magnitude an error of the order of 10%. Since the energy is proportional to the square of the current, neglecting the active resistance introduces into the results of calculating the energy magnitude an error of the order of 20%. The dependence characterizing the influence of the active resistance on the magnitude of the energy transferred to the load is presented in Fig. 3, a. From the data obtained it follows that, in the joint operation of a pulsed generator and a capacitor bank into an inductive load, the active resistance should not be neglected.

2. Investigation of the influence of the circuit’s natural frequency on the magnitude of the energy transferred to the load. The investigations were carried out for the following values of the ratio of the squares of the natural frequency of the circuit to the frequency of the generator emf:

$$ \nu^2 = \frac{\omega_k^2}{\omega^2} = 1,25;\ 1,5;\ 2;\ 2,5;\ 3;\ 3,5;\ 4; $$

4,5; 5 for $\dfrac{r}{x} = 0,05$, $U_{c0} = 0,5E_m$, $L_n = L_g$. The results of the investigations, expressed in relative units, are collected in Table 2.

Table 2

In this case the following were adopted as base units:

\(I_m=\dfrac{E_m}{L_{\mathrm{н}}+L_{\mathrm{г}}}\) — the value of the periodic component of the current when the generator operates directly into the load, i.e., without capacitance, for \(L_{\mathrm{п}}=L_{\mathrm{г}}\);

\(E_m\) — base emf, equal to the amplitude of the generator emf;

\(Q_1=0.5L_{\mathrm{н}}(2I_m)^2\) — base energy, which would be delivered to the load when the pulsed generator operates into the load according to the usual circuit, i.e., without capacitance.

In determining the base energy, the magnitude of the shock current was taken equal to the sum of the periodic and aperiodic components of the current \((2I_m)\), without taking damping into account.

The maximum energy in the load and the maximum energy of the capacitor bank were calculated from the formulas:

\[ Q_{\mathrm{н}}=0.5L_{\mathrm{н}}I_{\mathrm{н}}^2, \]

\[ Q_c=0.5cU_{c\max}^2, \]

\[ c=\frac{L_{\mathrm{н}}+L_{\mathrm{г}}}{\omega_{\mathrm{к}}^2L_{\mathrm{н}}L_{\mathrm{г}}}. \]

As follows from Table 2, with the aid of an auxiliary capacitance it is possible to increase severalfold the energy delivered to the load. The limits of acceptable values of \(\nu^2\) can be determined on the basis of two conditions.

1. In order to obtain maximum energy in the load, one should strive for smaller values of \(\nu^2\); in this case the lower limit may be taken to be \(\nu^2=2\), for which \(Q_c=0.5Q_{\mathrm{н}}\). For \(\nu^2<2\) the generator changes over to motor operation, and operation of the circuit becomes impractical.

2. From economic considerations, i.e., from considerations of the cost of 1 joule of energy delivered to the load, one should strive for the largest values of \(\nu^2\); in this case the upper limit may be taken to be \(\nu^2=6\), for which the energy taken from the generator into the load is

\[ Q_{\mathrm{н}}=1.5Q_1. \]

Thus, the possible limits of the ratio \(\nu^2\) will be

\[ 2\leqslant \nu^2\leqslant 6. \]

As a result of the investigations carried out, narrower limits for the optimal values of \(\nu^2\) may be recommended, namely

\[ 3\leqslant \nu^2\leqslant 3.5. \]

1. Investigation of the influence of the initial charge of the capacitor bank. The investigations were carried out for values of \(U_{c0}\) equal to 0; 0.25; 0.5; 0.75; 1 of the magnitude \(E_m\), with \(\dfrac{\kappa}{x}=0.05\) and \(\nu^2=3.5\). The results of the investigations are presented in Table 3. As follows from Table 3, precharging the capacitance leads to an increase in the energy delivered by the generator to the load; however, at the same time the size of the capacitor bank increases, i.e., the cost of the energy produced by the circuit increases.

The question of choosing the optimal initial voltage on the capacitor bank should also be considered from the conditions of obtaining maximum energy and the minimum cost per unit of energy.

Table 3

In our opinion, an acceptable value of $U_{c0}$ lies within the limits $0,25 \div 0,75$ of the value $E_m$.

1. Investigation of the effect of the load magnitude. The investigations were carried out for values of the load inductance equal to 0,5; 1; 2; 3; 5; 10 of the value $L_{\mathrm{g}}$ at $\dfrac{r}{x}=0,05$, $U_{c0}=0,5E_m$, and values of $\nu^2$ equal to 3; 3,5; 4. The results of the investigations are summarized in Table 4. As follows from Table 4, with deviation of the inductive resistance of the load from a value equal to the resistance of the generator, the amount of energy transferred to the load decreases, while at the same time the energy for which the capacitor batte-

Table 4

battery. Increasing the inductive resistance of the load with an unchanged capacitor bank may lead to the fact that, at the moment of the maximum load current, a negative surge of the generator current appears. This indicates an overestimated capacitance of the bank. For example, at \(\dfrac{L_{\text{н}}}{L_{\text{г}}}=3\), \(\nu^2=3;\ 3.5;\ 4\), we have, respectively, the generator current equal to \(-1.20;\ -0.30;\ -0.25\). With a decrease in the capacitance of the bank, the generator does not pass into the motor mode, and a greater part of the energy is transferred to the load. If at \(\nu^2=3\), \(Q_{\text{н}}=1.31\), then at \(\nu^2=4\), \(Q_{\text{н}}=1.49\), with the ratio \(\dfrac{Q_{\text{с}}}{Q_{\text{н}}}\) equal to \(0.66\) and \(0.39\), respectively.

It follows from Table 4 that a deviation of the inductive resistance of the load from the inductive resistance of the generator by more than a factor of two is disadvantageous.

Fig. 3. Curves of the dependence of the load energy and the energy of the capacitor bank on: a) active resistance; b) natural frequency of oscillations of the circuit; c) initial charge of the capacitor bank; d) magnitude of the load resistance.

Based on the results of the investigations (Tables 1, 2, 3, 4), curves have been constructed (Fig. 3) characterizing the dependence of the energy in the load and of the energy for which the capacitor bank must be designed on various factors. The curves make it possible to trace visually the influence of the active resistance, the natural frequency of oscillations of the load-resistance circuit, the initial charge, and the capacitance on the magnitude of the energy.

Figure 4 gives an oscillogram of the solution of the process of switching a pulsed generator and a capacitor bank onto an inductive lo-

load at \(\nu^2=3,\ U_{c0}=0.5E_m,\ L_{\mathrm{n}}=L_{\mathrm{g}},\ \dfrac{r}{x}=0.05\). The oscillogram shows the curves of variation of the voltage \(U_c\) and current \(I_c\) of the capacitor bank, the currents of the generator \(I_{\mathrm{g}}\) and the load \(I_{\mathrm{n}}\), and the generator e.m.f. \(e_{\mathrm{g}}\) as functions of time. Qualitatively, the oscillogram curves coincide with the curves obtained by calculation when active resistance is neglected, while quantitatively they differ from them by an amount determined by the damping of the current [1].

Fig. 4. Oscillograms of currents and voltages in various elements of the circuit.

1. Switching off the generator and the load. Switching off the generator by opening \(K\)-1 occurs at the moment when the current in the generator is equal to zero. The remaining part of the circuit is a series connection of \(r,\ L,\ c\). In this circuit, after the generator is switched off, energy remains, determined by the current through the inductance and by the voltage on the capacitance. During prolonged operation of such a circuit, the energy would be redistributed between the capacitance and the inductance, gradually dissipating in the active resistance. The load is switched off at the moment when the current in the load is equal to zero. This moment corresponds to the transfer of all the energy into the capacitor bank. In this case the charge on the capacitor bank may be greater or less than the initial value.

For repeated use of the circuit it is important to select the parameters of the circuit and the charge of the capacitor bank so that, after one cycle, the initial charge remains on the capacitor bank.

On an analog machine it is not difficult to find such ratios by changing \(U_{c0}\) or \(\nu^2\). For example, equality of the charge of the capacitor bank at the beginning and at the end of the cycle takes place for \(U_{c0}=0.25E_m,\ L_{\mathrm{n}}=L_{\mathrm{g}},\)

\[ \frac{r}{x}=0.05;\quad \nu^2=3.5. \]

In this case there will be no need to have a powerful charging device for the capacitor bank, and the time required to prepare the circuit for the next cycle will also be reduced.

Conclusions

The electronic model developed made it possible to investigate the joint operation of a pulse generator and a capacitor bank for various ratios of the circuit parameters and initial conditions.

From the results of the investigations it may be concluded that the use of an auxiliary capacitance leads to a significant (2–4-fold) increase in the energy delivered by the generator to the load.

Acceptable limits for the variation of the parameters, within which operation of the circuit will be especially effective and expedient, are as follows:

\[ \nu^2 = 3 \div 3.5, \]

\[ L_{\text{n}} = (0.5 \div 2)L_{\text{g}}, \]

\[ U_{c0} = (0.25 \div 0.75)E_m. \]

This mathematical model can be used in designing a real circuit for the joint operation of a pulse generator and a capacitor bank. In this case it will not be difficult to find the parameters of the circuit elements and the initial conditions under which the speed of response of the circuit and the greatest efficiency of its application will be ensured.

REFERENCES

1. G. A. Sipaylov, K. A. Khor’kov. On the joint operation of a pulse generator and a capacitor bank. Izvestiya TPI, vol. 132, 1964.