The paper presents an electrotechnical analysis of circuits based on an inductive energy store and an opening switch for operation with no load, with an inductive load, and with a resistive load, and also with two-stage pulse sharpening and upstream-of-switch load connection. The function of a switch is to cut off the load of a pulse generator during energy storage and to provide fast energy delivery to the load on approaching a certain critical current. The analysis suggests simple and useful formulae to estimate the load pulse parameters at a linearly rising switch resistance. The approximation of a linear resistance rise at the phase of current cutoff is a useful tool to assess the energetics of pulse generators both with plasma opening switches and with exploding wire switches. The estimates are compared with experimental data. The use of a more complex resistance approximation can improve the agreement between calculations and experiments, but it inevitably deprives the relations of their simplicity, clarity, and promptitude.
an electrotechnical analysis, an energy store, energetics
I. Introduction
Inductive energy stores, being ten times superior to capacitive ones in energy density, can greatly decrease the weight, dimensions, and cost of pulsed power systems [1]. However, their efficient use needs a switch capable of providing high-current interruption, many-fold pulse compression, and power amplification at a load. Among the most widespread types of switch are exploding wire switches [2], [3] and plasma opening switches [4], [5]. The function of a switch is to cut off the load of a pulse generator during energy storage and to provide fast energy delivery to the load on approaching a certain critical current.
This paper analyzes the operation of circuits based on an inductive energy store and an opening switch (shortly, inductive store–switch circuits) with no load, with an inductive load, and with a resistive load, and also with two-stage pulse sharpening and upstream-of-switch load connection. The analysis suggests simple and useful formulae to promptly estimate the load pulse parameters at a linearly rising switch resistance. The estimates and respective current and voltage waveforms are compared with experimental data.
II. Inductive Store–Switch Circuits
Figure 1 shows three main circuits with an inductive energy store and a switch. When the switch is conducting, the inductance 
is charged from the primary capacitive energy store to a current 
. The voltage at the instant of current cutoff depends on the circuit parameters and on the rate of rise of the switch resistance.
When the switch also serves as a load (Fig. 1a), the discharge current is determined by the equation 
, where (like in all further expressions) the dot over the current symbol stands for a time derivative. The solution of this equations has the form 
with for the inductance current at the instant of switch operation. Because the current decays exponentially, the switch voltage 
reaches its peak subject to 
. For the resistance rising linearly 
with a constant rate 
, the peak voltage
, (1)
where 
is the base of natural logarithms, is attained at the point in time
. (2)
The voltage full width at half maximum 
, where the numerical factor is equal to the difference of roots of the equation 
, 
. Because the maximum energy store current is 
, where 
is the output voltage of the primary capacitive store with a discharge capacitance 
and 
is the wave impedance, the voltage multiplication factor according to (1) is 
.
The dissipated switch power reaches its extremum 
at the time point 
determined from the equation 
. From this we have 
and 
. For =100¸400 
, 
1 MA, and 
0.1 W/ns, the voltage rises to 
2¸4 MV in 
30¸60 ns at 
1.4¸2.8 TW.
When the switch operates into an inductance 
(Fig. 1b), the store current 
, the switch current 
, and the load current 
are determined from the system of equations
where the initial conditions are 
. Thus, we have the load current 
and the switch current 
, where 
. The parameters 
, 
are given by (1) and (2) with 
. At 
, these parameters decrease 
times compared to their values for the circuit in Fig. 1a.
For full opening of the switch, the total energy in the circuit elements is 
, where 
is the stored energy. The ratio 
decreases from unity at 
to zero at 
. On the contrary, the dissipated energy 
increases from zero at to unity at . The load energy 
is maximal at for which 
and 
.
The load pulse power 
, where 
, is maximal at the time point 
determined from the equation 
, where 
. From this we have 
, 
, where 
. At 
, the power reaches its absolute extremum 
. At , the power is 
.
With a resistive load (Fig. 1c), the currents are determined from the equations
where the initial conditions are 
. For 
, the switch current is
.
This current can be expressed as 
, where 
. For , we have the function 
and the switch current is
, (3)
where 
, 
.
The peak voltage across the load 
is reached at the time point determined from (2), which is independent of 
. However, the value of the peak depends both on the load resistance 
(Fig. 2a, 
200 nH, 1 MA) and on the switch resistance rise rate 
(Fig. 2b). At 
, the voltage tends to because, in view of (3), 
, where 
, and at 
, the limit is known and is 
.
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Fig. 2. Peak voltage |
Fig. 3. Load and switch powers (a) and energies (b) versus |
As can be seen in Figure 2a, the voltage first rises quasi-linearly with the load resistance. Such a behavior can be interpreted as load-limited operation [6] in which the load current depends weakly on the load resistance. Further increasing the load resistance slows down the voltage rise, and this can be interpreted as switch-limited operation in which the load current is approximately inversely proportion to the load resistance. For the curves in Figure 2а, the conditional boundary between the two operation modes lies at ~1.5 W.
The dissipated switch power reaches its peak at 
. The load power shows its absolute extremum 
at an optimum resistance 
. The coefficient 
is the solution of the equation 
. For such a load, 
, 
, and the peak voltage 
is two times lower than (1). The voltage FWHM is 
. Here, the numerical factor is the difference of roots of the equation 
, where 
, 
.
At other values of , the energy characteristics behave as expected (Fig. 3). Increasing (e.g., at 
200 nH, 
1 MA) increases the dissipation in the switch and hence decreases the energy extraction. Increasing increases the load energy. Because 
, its value tends to increase as is increased (Fig. 3a).
For a load with 
, the overvoltage coefficient is 
. For 
as a parameter, we have 
, and
, (4)
where 
, 
, 
.
With no switch, the load current is 
, where 
, and the damping decrement 
is less than critical [7]. The current amplitude 
, where 
, is attained at the time point 
. The load power is maximal at 
, which is the solution of the equation 
. The peak power
(5)
is dissipated at 
. From comparison of (4) and (5) it follows that the use of a switch which provides an overvoltage 
can increase the load power about 15 times.
III. Two–Stage Pulse Sharpening
Figure 4 shows two series-connected inductive store–switch circuits. On opening of the first switch 
with the second switch 
closed, the energy from 
is extracted to 
. On opening of the energy is switched from to the load. Increasing the rate of rise of the current in the switch increases the rate of rise of its resistance and hence the output generator voltage and the load power. For example, in experiments [8], the voltage reached ~1 MV on operation of the first plasma opening switch (conduction time 
1.2 ms, conduction current 
1.7 MA), and on operation of the second one (130 ns, 0.55 MA), the voltage across the diode load reached ~4 MV. The resistance rose to about 1¸1.5 
in 50 ns, and 
to about 10¸15 
in 10 ns.
For an inductive load with fully open switches, the intermediate store current is 
and the load current is 
. The maximum load current 
is reached at 
. The load energy 
with 
is maximal at 
, measuring 
=1/16 or 6.25% of the inductive store energy. This casts doubts on the efficiency of multistage pulse sharpening for liner loads.
The switch voltage ratio for linearly rising switch resistances 
at 
is equal to 
. Hence, for increasing the load voltage, the rate of rise of the resistance 
should be four times higher than that of 
.
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Fig. 4. Two-stage pulse sharpening circuit. |
Fig. 5. Dependences |
The peak load power 
is attained at the time point 
with 
and with 
and 
being the same as those for the one-stage circuit in Fig. 1b. The absolute extremum 
falls on 
, 
but the energy extraction efficiency, in this case, decreases to 1/24. At 
, the peak power is equal to 
. The power ratio for the two- and one-stage circuits is 
. Thus, for the most efficient energy extraction, the second stage is reasonable only if 
.
For other values of 
, 
, 
, the power ratio is given by 
, where 
, and its extremum is reached at 
(Fig. 5).
IV. Upsream–of–Switch Load Connection
One of the ways of eliminating the adverse effect of plasma from switch to load is to connect the load upstream of the switch [9]. Such a circuit (Fig. 6) includes a separating closing switch 
which cuts off the load from current till the opening switch gets open. Let the switch turns on at the onset of current interruption, and after fast switching, its resistance drops to zero. Before opening, the currents through the inductances 
and are equal to 
. After opening, the currents are determined from the system of equations
Solving the system gives the same time dependences as those for the circuit in Fig. 1b but with 
. The presence of increases 
and decreases the peak voltage across the load 
. At 
, the peak voltage 
is 
of the switch voltage. The maximum load power is 
, where 
.
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Fig. 6. Upstream-of-switch load connection. |
Fig. 7. Dependences |
The dependences 
at different values of are shown in Figure 7. As can be seen, varying by an order of magnitude influences 
only slightly. At the same time, the presence of greatly decreases the load power at a load inductance of less than 100 nH. The power 
reaches its absolute extremum at 
, where 
. At 
, we have 
and 
.
The ratio of the load energy 
, where 
, to its value in the conventional circuit is 
. At 
~10, the decrease in the energy extraction efficiency is less than ~10%.
V. Comparison with Experiments
According to empirical data [10], the voltage multiplication factor provided by an exploding wire switch at a load with 
is 
. For this value of 
, the optimum resistance is 
. The fact that 
is close to 
justifies the assumption of a linear resistance rise on current cutoff for calculations of the load pulse parameters.
Let us refer to experimental data. Figure 8 shows waveforms recorded on operation of one of the generators built around an inductive energy store and a exploding wire switch for high-power microwave sources [11]. The store, having an inductance of ~4.6 mH, and the switch represent a series connected circuit into which a capacitor of ~4 mF (
70 kV) is switched. The voltage 
that arises on interruption of the current 
30 kA is applied via a sharpening switch to a resistive load of ~40 
. The rate of rise of the load current is ~0.5 kA/ns, the voltage is 
600 kV, and the pulse power is 
9 GW.
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Fig. 8. Shot with wire switch. Interpretation of traces is given in respective paragraphs. |
Fig. 9. Load voltage (a) and switch and load currents (b). |
Figure 9 compares the experimental and calculated waveforms of the load voltage, switch current, and load current. The calculation is by formula (3) for 
1.7 W/ns, which approximates the steep rise of resistance on current cutoff. As can be seen, the experimental and calculated current waveforms, being coincident in amplitude, show some difference during the pulse rise time, and this is due to the transient response of the sharpening switch whose resistance decreases to ~1.5 W only within ~60 ns after the onset of current flow through the load. The finite resistance of this switch delays the extraction of energy. The increase in 
within ~100 ns after the instant of current cutoff suggests that the exploding wire switch recovers its conductivity, which shows up as a decay of 
in Figure 8.
The assumption of a linear resistance rise can also be justified by the example of experiments on GIT-4 and GIT-12 mega-joule setups with mega-ampere plasma opening switches [10]. In the GIT-4 setup, the stored energy is ~550 kJ at a charge voltage 40 kV; the discharge current rises to ~3 MA in ~1.1 ms. In the GIT-12 setup, the stored energy is ~5 MJ at 70 kV; the current rises to ~6 MA in ~1.7 ms.
Figure 10a shows waveforms for one of the shots on the GIT-12 setup with a radial plasma opening switch [12]. In its energy store with ~160 nH, the current 
reaches ~1.2 MA in ~550 ns. The switch operates into a load of ~50 nH. On current cutoff, the switch resistance increases with a rate of ~17 mW/ns. As can be seen in Figure 10b, the curves 
and 
calculated for this value of 
and for its two times lower value lie above and below the amplitude of the experimental curve 
.
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Fig. 10. Shot on GIT-12 setup at |
Fig. 11. Shot on GIT-4 setup. |
Figure 11a shows waveforms for one of the shots on the GIT-4 setup in open-cathode mode [12]. In its energy store with ~270 nH, the current reaches 
1 MA in ~800 ns. The voltage that arises on current cutoff is 
2 MV. The switch resistance 
increases with a rate of ~85 mW/ns (Fig. 11b). At this value of 
, the switch fully open into a load with an unlimited impedance would provide a peak voltage of 
~3 MV. However, in the shot considered, the increment of the discharge circuit inductance after opening of the switch is limited to 
230 nH. For this load, the calculated voltage 
has its amplitude close to the peak of 
. The current 
calculated for experimental values of and finite value of 
decays more appreciably than the switch current 
, suggesting that is prevented from decay by certain processes which are discussed, e.g., elsewhere [13-15].
The foregoing demonstrates that the approximation of a linear resistance rise at the phase of current cutoff is a useful tool to assess the energetics of pulse generators both with plasma opening switches and with exploding wire switches. Certainly, the use of a more complex resistance approximation can improve the agreement between calculations and experiments, but it inevitably deprives the relations of their simplicity, clarity, and promptitude.
VI. Conclusion
Thus, the electrotechnical analysis of inductive store–switch circuits provides simple analytical formulae which are useful both for experimental data interpretation and for predication of load pulse parameters in generators based on this type of circuit. The load pulse parameters depend on the rate of rise of the switch resistance. For its estimation, which is left untouched in the paper, one should consider physical processes responsible for current interruption in one or another switch. For example, as applied to plasma opening switches, such a consideration gives as a function of the switch parameters and rate of current rise [16]. As applied to exploding wire switches, the rate of rise of the resistance can be estimated from empirical similarity criteria [10].
The work was performed under State Assignment of the Ministry of Science and Higher Education of the Russian Federation (project No. FWRM-2021-0001).
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