Calculation of Currents in the Elements of a Circuit for Accelerated Energy Storage in an Inductive Load from a Surge Generator
Abstract Generated abstract
This paper presents calculation formulas for currents and voltages in a circuit intended to accelerate electromagnetic energy storage in an inductive load supplied by a shock generator. The circuit includes an auxiliary capacitor bank connected in parallel with the load and high-speed switching devices that alternate between energy accumulation and energy retention stages. Using operational equations and simplified time-domain expressions obtained by neglecting small active resistances, the authors derive relations for generator current, load current, and capacitor voltage over successive cycles. A numerical comparison indicates that, with a relatively small capacitor bank, the load current can reach a specified level in about half as many generator periods as in a circuit without capacitance, improving the rate and efficiency of energy transfer.
Full Text
CALCULATION OF CURRENTS IN THE ELEMENTS OF A CIRCUIT FOR ACCELERATED ENERGY STORAGE IN AN INDUCTIVE LOAD FROM A SHOCK GENERATOR
A. V. Loos, G. A. Sipaylov, K. A. Khorkov
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering).
A distinctive feature of the circuit under study for increasing the magnitude and rate of energy transfer from a shock generator into an inductive load is the presence of an auxiliary capacitor bank, connected in parallel with the load, and a special switching device that short-circuits the load at the moment when the load current passes through its maximum.
Since the physical nature of the increase in electromagnetic energy in the described circuit and the operation of the circuit in the process of stepwise accumulation of energy in the inductive load were considered earlier [1, 2, 3], the present article has as its purpose, without dwelling on the physical essence of the phenomena, to give calculation formulas for determining currents and voltages in the various elements of the circuit at all stages of its operation.
Fig. 1. Circuit for accelerated energy storage in an inductive load.
The circuit described is shown in Fig. 1, where $L_g$ is the equivalent subtransient inductance of the generator, $L_n$ is the load inductance, $r_g$, $r_n$ are the active resistances of the generator and the load, $C$ is the auxiliary capacitor bank, and $K$-1 and $K$-2 are high-speed arcless ionic-mechanical contact devices.
The process of energy storage in the load proceeds as follows. At the moment when the generator e.m.f. passes through zero, contact device $K$-1 is closed, with $K$-2 open. During the half-wave of the e.m.f., which we shall call positive, the generator transfers to the load a certain amount of electromagnetic energy. The load current
increases, and at the end of the positive half-period of the emf it reaches a maximum. At this moment, when the polarity of the emf changes, K-2 closes. The load is short-circuited, and the energy stored in it is preserved (except for losses) until the next storage cycle. At the same time the generator current drops sharply, since the generator is short-circuited under negative emf. At the moment when the generator current passes through zero, apparatus K-1 opens, and the circuit remains open for the entire remaining part of the negative half-period of the emf.
The next storage cycle begins with the closing of K-1 as the generator emf passes through zero into the positive half-wave, while contact apparatus K-2 remains closed. The generator operates in the sudden short-circuit mode and the generator current rises sharply. At the moment when the generator current reaches the value of the load current, apparatus K-2 opens. It should be noted that at this moment the current flowing through K-2, equal to the difference between the generator and load currents, is zero. From the moment K-2 opens, during the remaining positive half-period of the emf, the current in the load increases, i.e., energy storage takes place. At the moment when the generator emf becomes equal to zero, the load current reaches the next maximum and contact apparatus K-2 short-circuits the load. The stage of preserving the stored energy begins. The subsequent storage process proceeds similarly.
The character of the variation of the generator and load currents and of the voltage across the capacitance during the process of energy storage is clear from Fig. 2. For convenience
Fig. 2. Character of the variation of voltage and currents in the circuit elements.
of the mathematical description, it is expedient to divide the process of energy storage in the load into separate cycles, with durations equal to the emf period, each of which consists of two stages: the stage of energy storage and the stage of preserving the stored energy.
Energy Storage Stage
During this stage the state of the circuit is as follows: contact apparatus K-1 is closed, and apparatus K-2 is open. At the moment K-1 is closed, the capacitor bank is not charged. The operation of the circuit in the energy storage stage of any cycle is described by the system of equations:
\[ e_{\mathrm{g}}(p)=i_{\mathrm{g}}(p)r_{\mathrm{g}}+pL_{\mathrm{g}}i_{\mathrm{g}}(p)+pL_{\mathrm{g}}I_{\mathrm{g}}(0)+\frac{1}{pc}\,[i_{\mathrm{g}}(p)-i_{\mathrm{l}}(p)], \]
\[ 0=i_{\mathrm{l}}(p)r_{\mathrm{l}}+pL_{\mathrm{l}}i_{\mathrm{l}}(p)+pL_{\mathrm{l}}I_{\mathrm{l}}(0)+\frac{1}{pc}\,[i_{\mathrm{l}}(p)-i_{\mathrm{g}}(p)]. \tag{1} \]
Solving system (1) with respect to the generator and load currents, we obtain, in operational form, the following expressions:
\[ i_{\mathrm{g}}(p)= \frac{ -e_{\mathrm{g}}(p)(p^{2}L_{\mathrm{l}}c+1)+pL_{\mathrm{g}}I_{\mathrm{g}}(0)+pL_{\mathrm{l}}I_{\mathrm{l}}(0) }{ p^{2}c(pL_{\mathrm{g}}L_{\mathrm{l}}-L_{\mathrm{l}}r_{\mathrm{g}}+L_{\mathrm{g}}r_{\mathrm{l}}) -p(L_{\mathrm{l}}+L_{\mathrm{g}})-(r_{\mathrm{l}}+r_{\mathrm{g}}) }, \tag{2} \]
\[ i_{\mathrm{l}}(p)= \frac{ -e_{\mathrm{g}}(p)+pL_{\mathrm{g}}I_{\mathrm{g}}(0)+pL_{\mathrm{l}}I_{\mathrm{l}}(0)(p^{2}L_{\mathrm{g}}c+pcr_{\mathrm{g}}+1) }{ p^{2}c(pL_{\mathrm{g}}L_{\mathrm{l}}-L_{\mathrm{l}}r_{\mathrm{g}}+L_{\mathrm{g}}r_{\mathrm{l}}) -p(L_{\mathrm{l}}+L_{\mathrm{g}})-(r_{\mathrm{l}}+r_{\mathrm{g}}) }. \tag{3} \]
After transition to the time form of notation, the expressions for the currents become so cumbersome that it is practically impossible to use them. Therefore, in order to simplify the expressions, we neglect the active-resistance quantities of the generator and the load. Such an assumption is fully justified, since the active resistances of the circuit are considerably smaller than the inductive ones. Taking the adopted assumptions into account, we obtain:
\[ i_{\mathrm{g}}(t)= \frac{E_m}{\omega L_{\mathrm{g}}} \left\{ \frac{\omega^2-\omega_{\mathrm{l}}^2}{\omega^2-\omega_{\mathrm{e}}^2} \left[ \sin\psi \sin\omega t+ \frac{\omega}{\omega_{\mathrm{e}}}\cos\psi(1-\cos\omega t) \right] + \right. \]
\[ \left. +\frac{\omega_{\mathrm{l}}^2-\omega_{\mathrm{e}}^2}{\omega^2-\omega_{\mathrm{e}}^2} \left[ \frac{\omega}{\omega_{\mathrm{e}}}\sin\psi\sin\omega_{\mathrm{e}}t+ \frac{\omega^2}{\omega_{\mathrm{e}}^2}\cos\psi(1-\cos\omega_{\mathrm{e}}t) \right] \right\} +I_{\mathrm{g}}(0). \tag{4} \]
\[ i_{\mathrm{l}}(t)= \frac{E_m}{\omega L_{\mathrm{l}}} \left\{ \frac{\omega_{\mathrm{g}}^2}{\omega_{\mathrm{e}}^2-\omega^2} \left[ \sin\psi\sin\omega t+ \cos\psi(1-\cos\omega t) \right] + \right. \]
\[ \left. +\frac{\omega_{\mathrm{g}}^2}{\omega^2-\omega_{\mathrm{e}}^2} \left[ \frac{\omega}{\omega_{\mathrm{e}}}\sin\psi\sin\omega_{\mathrm{e}}t+ \frac{\omega^2}{\omega_{\mathrm{e}}^2}\cos\psi(1-\cos\omega_{\mathrm{e}}t) \right] \right\} +I_{\mathrm{l}}(0). \tag{5} \]
\[ U_c(t)= E_m\frac{\omega_{\mathrm{g}}^2}{\omega_{\mathrm{e}}^2-\omega^2} \left[ \sin\psi(\cos\omega t-\cos\omega_{\mathrm{e}}t)+ \right. \]
\[ \left. +\cos\psi\left(\sin\omega t-\frac{\omega}{\omega_{\mathrm{e}}}\sin\omega_{\mathrm{e}}t\right) \right] + \left[ I_{\mathrm{l}}(0)L_{\mathrm{l}} \left(\omega-\frac{\omega_{\mathrm{g}}^2}{\omega_{\mathrm{e}}}\right) - \right. \]
\[ \left. -I_{\mathrm{g}}(0)L_{\mathrm{g}}\frac{\omega_{\mathrm{g}}^2}{\omega_{\mathrm{e}}} \right]\sin\omega_{\mathrm{e}}t - U_c(0)\cos\omega_{\mathrm{e}}t, \tag{6} \]
where \(\omega\) is the angular frequency of the generator emf,
\[
\omega_{\mathrm{e}}=\sqrt{\frac{L_{\mathrm{l}}+L_{\mathrm{g}}}{cL_{\mathrm{l}}L_{\mathrm{g}}}}
\]
is the equivalent frequency of the \(L_{\mathrm{l}}L_{\mathrm{g}}c\) circuit,
\[
\omega_{\mathrm{g}}=\sqrt{\frac{1}{cL_{\mathrm{g}}}}
\]
is the natural frequency of the \(L_{\mathrm{g}}c\) circuit,
\[
\omega_{\mathrm{n}}=\sqrt{\frac{1}{cL_{\mathrm{n}}}}
\]
is the natural frequency of the \(L_{\mathrm{n}}c\) circuit,
\(E_m\) is the amplitude of the generator emf.
For the first cycle the initial conditions are as follows:
\[ U_c(0)=0,\qquad I_{\mathrm{g}}(0)=0,\qquad I_{\mathrm{n}}(0)=0,\qquad \psi=\psi_0=0, \]
where \(\psi_0\) is the initial switching angle.
For all subsequent cycles:
\[ U_c(n)=0,\qquad I_{\mathrm{g}(n)}=I_{\mathrm{n}(n)}=I_{\max(n-1)}e^{-\frac{1}{T_{\mathrm{n}}}\left(\frac{\pi+\psi_n}{\omega}\right)},\qquad \psi=\psi_n, \]
where \(\psi_n\) is the angle from the beginning of the positive half-period of the emf to the moment when the short circuit is removed (opening of K-2).
Energy-retention stage
The stage begins with the closing of apparatus K-2 and ends at the moment this contact apparatus is opened. During this stage the load current varies according to an exponential law
\[ i_{\mathrm{n}}(t)=I_{\max(n)}e^{-\frac{t}{T_{\mathrm{n}}}}, \]
where \(T_{\mathrm{n}}=\dfrac{L_{\mathrm{n}}}{r_{\mathrm{n}}}\) is the decay time constant of the load current.
The change in the generator current during this stage is described by several equations. Initially, after K-2 is closed, the generator current decreases according to the law
\[ i_{\mathrm{g}}(t)= \frac{E_m}{\omega L_{\mathrm{g}}} \left[ \cos(\omega t+\psi)-\cos\psi\,e^{-\frac{t}{T_{\mathrm{g}}}} \right] + I_{\max(n)}e^{-\frac{t}{T_{\mathrm{g}}}}, \tag{8} \]
where \(T_{\mathrm{g}}=\dfrac{L_{\mathrm{g}}}{r_{\mathrm{g}}}\) is the generator time constant.
At the moment when the generator current passes through zero, apparatus K-1 is opened and remains open as long as the generator emf has negative polarity. When the generator emf changes to the positive half-wave, contact apparatus K-1 is closed and the generator is short-circuited through K-2.
The change in the generator current is described by equation (9) up to the moment when apparatus K-2 is opened, that is, until the circuit transfers to the accumulation mode:
\[ i_{\mathrm{g}}(t)= \frac{E_m}{\omega L_{\mathrm{g}}} \left( e^{-\frac{t}{T_{\mathrm{g}}}}-\cos\omega t \right). \tag{9} \]
The maximum current that can be obtained in the load is equal to the amplitude of the first maximum half-wave of the current of a sudden short circuit of the shock generator.
Using the method presented, the currents in the elements of the circuit during the process of energy accumulation in an inductive load were calculated for the following data:
\[ \begin{aligned} E_m&=22\ \text{kV}, & \omega&=314,\\ L_{\mathrm{g}}&=0.001\ \text{H}, & L_{\mathrm{n}}&=2.5\,L_{\mathrm{g}},\\ c&=0.00405\ \text{F}, & r_{\mathrm{n}}&=0.05\ \Omega, \end{aligned} \]
In this case, \(\omega_g = 483,\quad \omega_n = 315,\quad \omega_e = 587\).
The results of the calculation are presented graphically in Fig. 2.
It is of interest to compare the rate of energy storage in the load in the described circuit with the rate of accumulation in a circuit without capacitance.
As the quantity to be compared, the number of periods of the generator EMF was taken, after which a current of \(0.9 I_n\) is established in the load. In the calculations, the value of the capacitance is determined from the condition of obtaining the equivalent frequency of the circuit \(\omega_e=\sqrt{3.5}\,\omega\). In this case, as the calculations showed, the energy for which the capacitor bank must be designed (for \(L_n \geqslant 2.5 L_g\)) does not exceed \(5.5\%\) of the energy transferred to the load. The calculation results are presented in Table 1. It is evident that the process of energy accumulation using
Table 1
| \(\dfrac{L_n}{L_g}\) | 1 | 2.5 | 5 | 7.5 | 10 | Note |
|---|---|---|---|---|---|---|
| Number of periods of EMF | 4 | 8 | 14 | 20 | 25 | according to the circuit without capacitance |
| Number of periods of EMF | 2 | 4 | 7 | 10 | 12 | according to the circuit with capacitance |
| \(\dfrac{Q_c}{Q_n}\,\%\) | 18 | 5.5 | 2.2 | 1.4 | 1.1 | — |
a comparatively small capacitor bank proceeds twice as fast, which significantly increases the efficiency of the installation.
REFERENCES
-
G. A. Sipaylov, K. A. Khorkov. On the joint operation of a shock generator and a capacitor bank on an inductive load. Izvestiya TPI, vol. 132, 1965.
-
A. V. Loos, G. A. Sipaylov, K. A. Khorkov. Mathematical model of the circuit for joint operation of a shock-power generator and a capacitor bank. Izvestiya TPI, vol. 138, 1965.
-
V. V. Ivashin, A. V. Loos, G. A. Sipaylov, K. A. Khorkov. A circuit for increasing the energy and the rate of its transfer from a shock generator to an inductive load. Izvestiya TPI, vol. 139, 1965.