When a capacitive voltage divider (CVD) in a component is calibrated, the introduction of a standard high-voltage (HV) probe usually affects the original field distribution of this component. Therefore, the voltage ratio of the CVD is affected, especially when the distance between the HV probe and the CVD is close. A multiple-approximation method to calibrate the CVD of gas switches in Tesla-type generator without a standard HV probe is being put forward. The basic idea is to calculate the theoretical switch voltage uth based on the equivalent circuit of this type of generators and to use uth to substitute the real voltage ur acquired by the standard HV probe to calculate the voltage ratio β for the CVD of a gas switch. Since the gas-gap resistance Rs of the switch is not zero, uth is not equal to ur, and β therefore deviates from the real value β0. To solve this question, two steps are implemented: first, β(d) at different gas-gap distances d is obtained; second, the true voltage ratio β(0) when d = 0 is extrapolated. The theoretical basis of this method was analyzed first. Then, it is verified by experiments. In the end, the uncertainty of this method is evaluated.
I. INTRODUCTION
The measurement and diagnosis system in a pulse power generator is of great importance since it is the direct way to evaluate the performance of the generator.1,2 The measurement of a generator comprises voltage measurement and current measurement. The common voltage measurement devices include resistance dividers, capacitive dividers,3–6 and resistor–capacitor composite dividers,7 all of which are usually built inside the generators for convenience. The voltage ratio should be ascertained first after the establishment of the voltage measurement system, which is usually finished via a standard high-voltage (HV) probe, such as P6015A; that is, the real waveform ur is measured via P6015A, and the attenuated waveform up is measured via the built-in voltage probe. If the two waveforms agree with each other, the voltage ratio can be obtained by dividing ur by up.
The use of the standard HV probe has two requirements: (1) the bandwidth of the HV probe should be much wider than that of the signal sensor and (2) the introduction of the probe has no or a weak effect on the field distribution around the signal sensor. The first requirement can be easily met since the bandwidth of a standard probe can be as wide as 75 MHz (such as P6015A). However, the second requirement is sometimes hard to satisfy. For example, as to the capacitive voltage divider (CVD) used to monitor the switch voltage in Tesla-type generators, the standard HV probe needs to be connected to the switch to acquire the real waveform. However, it affects the original field distributions around the switch CVD.
Aiming to solve this dilemma, a method to calibrate the CVDs of the gas switches in Tesla-type generators based on an idea of “multiple approximations” is being put forward, which needs no standard HV probe during the calibration process. Section II introduces the background and the problem. Section III gives the theoretical basis for this method and the calibration results of two Tesla-type generators. Section IV verifies this method in experiments and evaluates the uncertainty. The last section is for the remarks and conclusions of this paper.
II. QUESTION FOR CONVENTIONAL CALIBRATION
A. Background
Tesla-type generators were put forward and constructed by Russian scientists2,8 in the last century, which were based on the Tesla transformer. The basic principle of a Tesla Transformer is to boost the low dc primary voltage to a μs kilo-volt (kV) pulse via a thousand-turn secondary winding.9 With a gas gap switch (main switch), the Tesla transformer can compress the μs pulse of the secondary winding to the nanosecond mega-volt (MV) scale. Figure 1(a) shows the schematic of this type of generator, which includes a Tesla transformer built in a coaxial pulse forming line (PFL, 1), a gas gap switch (2), a trigger built in a transmission line (TL, 3) and a load (4). The famous Tesla-type generators include the Sinus series10 and the Radan Series11 in Russia and the CHP series12 and the tesla-series pulsed generators (TPG) series13–15 in China.
Typical schematic of Tesla-type generators: (a) schematic and (b) the cross section of the gas switch from the cathode side—1: the Tesla transformer built in a PFL, 2: the main switch, 3: the trigger built in a TL, 4: the water load, A: the CVD of the PFL, B: the CVD of the switch, and C: the CVD of the load.
Typical schematic of Tesla-type generators: (a) schematic and (b) the cross section of the gas switch from the cathode side—1: the Tesla transformer built in a PFL, 2: the main switch, 3: the trigger built in a TL, 4: the water load, A: the CVD of the PFL, B: the CVD of the switch, and C: the CVD of the load.
The voltages in different components of Tesla-type generators need to be monitored when the generators are constructed, which are realized via different CVDs assembled in the generators. These probes comprise a PFL CVD (Probe A in Fig. 1), a switch CVD (Probe B in Fig. 1), and a load CVD (Probe C in Fig. 1).
A CVD is a kind of polyimide film (48 µm), both sides of which are covered with copper foils (12 µm). The front-side foil, as seen in Fig. 2(a), which faces the HV conductor of the generator, usually connects to the HV conductor of a cable connector and is used to induce the HV voltage. The back-side copper foil, as seen in Fig. 2(b), is usually stuck on the inner surface of the GND conductor of the Tesla-type generator. Figure 3(c) shows the schematic of a CVD connected to a cable connector and stuck on the inner surface of an HV device. The equivalent circuit of this CVD is shown in Fig. 2(d), where Ui represents the voltage between the HV and the LV conductors, Ui represents the voltage on the cable connector, and R represents the sampling resistor connecting to the cable connector. With Fig. 2(d) and the Laplace transformation, one can obtain the transmission function H(s) for this circuit as follows:
where s is the Laplace argument. From H(s), the impulse response function can be obtained,
In addition, the zero-state circuit response function can also be obtained in the time domain, which is
where Δt = R(C1 + C2), which is the time constant of the circuit.
CVD applied in Tesla-type generators: (a) front copper of the CVD, (b) back copper of the CVD, (c) the schematic of the CVD stuck on an HV device and connected to a cable connector, and (d) the equivalent circuit of a CVD.
CVD applied in Tesla-type generators: (a) front copper of the CVD, (b) back copper of the CVD, (c) the schematic of the CVD stuck on an HV device and connected to a cable connector, and (d) the equivalent circuit of a CVD.
Calibration for the PFL CVD and the load CVD: (a) the position of the PFL CVD and the HV probe and (b) the position of the TL CVD and the HV probe.
Calibration for the PFL CVD and the load CVD: (a) the position of the PFL CVD and the HV probe and (b) the position of the TL CVD and the HV probe.
Usually, there are two cases for practical application of Eq. (3). Case 1: the pulse width τ is much longer than the circuit constant Δt. In this case, the first part Ui(t) can be neglected, and Eq. (3) can be simplified to be
Equation (3) degrades to the integration form, which is just the D-dot sensor. More details on a D-dot sensor can be seen in Ref. 19. Case 2: the pulse width τ is much shorter than the circuit constant Δt. In this case, the second part of the integration of Ui(t) can be neglected, and Eq. (3) can be simplified to be
Equation (3) degrades to the linear form, which means that the pulse to test is linear to the output pulse. Some points need to be stressed for the two cases of the equivalent circuit: (1) The transition case, when the pulse width τ is close to the circuit constant Δt, is usually avoided by designers since the mathematical relation between Uo(t) and Uo(t) is complicated, (2) the linear case is more simple than the integration case because the latter needs an extra circuit for re-constructing the waveform of Ui(t), whereas the former needs no circuit, and (3) due to the simple form of the linear case, CVDs of such a case are widely used for measuring the voltage in TPG-series Tesla-type generators.3,7,15,17,20,21 The measurement method based on linear-case CVDs has the advantages of easy fabrication, a high voltage ratio, and a large time constant.
After fabrication and assembly, the CVDs need to be calibrated. Conventional calibration for a CVD usually involves acquiring the real waveform ur using a standard HV probe P6015A. As mentioned above, the introduction of P6015A should not affect the original field distribution around the CVD. When the PFL CVD and the load CVD are calibrated with P6015A, as shown in Fig. 3, it is found that the waveforms from the CVD and that from the P6015A can overlap each other well. Hence, the calibration results are acceptable.
B. Question for calibrating a switch CVD with a standard probe
However, when the switch CVD is calibrated with the standard HV probe P6015A, as shown in Fig. 4(b), it is found that the waveform acquired by the switch CVD and that from P6015A cannot overlap each other, as shown in Fig. 4(b). Hence, the voltage ratio of the switch CVD cannot be ascertained. The photo of the switch CVD in a Tesla-type generator named 20 GW is shown in Fig. 1(b). We monitored the waveforms of the switch CVD in two cases: (1) P6015A connected to the cathode of the switch and (2) P6015A disconnected to the cathode of the switch. Through comparison, we found that the two waveforms were not the same. The waveform to be monitored by the switch CVD for the 20 GW generator is negative and trapezoid with a top of 40 ns and a rise time and a fall time of 5 ns. Another generator is TPG700L, which has a width of 200 ns, a rise time of 30 ns, and a fall time of 50 ns.
Calibration for the switch CVD in a TL with a water load: (a) schematic and (b) comparison of the waveforms with and without the standard HV probe.
Calibration for the switch CVD in a TL with a water load: (a) schematic and (b) comparison of the waveforms with and without the standard HV probe.
The disagreement for the measured switch waveforms of the 20 GW is explained as follows: During calibration of the switch CVD, P6015A was placed very close to the switch CVD that the field distribution around the switch CVD was strongly influenced by P6015 A. Hence, the waveform monitored by the switch CVD was distorted. However, during calibration of the PFL CVD and the load CVD, P6015A and the CVD in each calibration were placed at the two ends of a component, i.e., they were far from each other. Hence, the existence of P6015A has little effect on electric field distributions near the PFL CVD or the load CVD.
Hence, there is a dilemma that if the switch CVD is to be calibrated, the real waveform in the switch should be obtained, which involves P6015A; However, the use of P6015A will in turn affect the monitored waveform by the switch CVD and therefore affects the voltage ratio β0 of the switch CVD.
III. MULTIPLE-APPROXIMATION CALIBRATION METHOD
A. Theoretical basis
In order to solve this question, a multiple-approximation calibration method without P6015A is being put forward. The theoretical basis for this method is the equivalent circuit of the Tesla-type generator, as shown in Fig. 5. In Fig. 5(a), the components of the PFL and TL are represented by two segments of cables. The load is considered as a part of the TL. The gas gap switch is represented by a variable resistor. The characteristic impedances of the PFL and TL are zPFL and zTL, respectively, which are designed equally for the Tesla-type generators. The resistance of the switch is Rs, which is time dependent. Figure 5 (b) shows the series discharging model of the Tesla-type generator. According to this figure, the voltage on TL uTL when the switch gets closed is as follows:
where u0 is the charge voltage. Rs is a function of time t and gap distance d, which can be expressed as follows:16
Equivalent circuit of Tesla-type generators when the switch is closed: (a) the transmission line model and (b) the circuit model.
Equivalent circuit of Tesla-type generators when the switch is closed: (a) the transmission line model and (b) the circuit model.
Figure 6 shows the typical impedance curve of the switch in Tesla-type generators. As to the top of the waveform, the corresponding impedance is basically a constant. Hence, Eq. (7) can be simplified into
where C is constant.
When d = 0 and zPFL = zTL, uTL would be equal to u0/2. Assuming the real voltage ratio of the switch CVD is β0, then,
where utest is the voltage tested by the CVD of the switch.
If d approaches 0, the voltage on the TL approaches u0/2. Here, u0/2 is used to represent the real voltage of the TL when d is close to 0. The voltage ratio β(d) at different d would approximately be
That is,
From Eq. (13), it can be seen that β(d) is linearly dependent on d. In addition, for the Tesla-type generator, ZPFL = ZTL. Hence, β(0) = β0. Based on this result, β(0) can be fitted linearly and extrapolated in order to obtain the real ratio β0 of the switch CVD.
B. Calibration steps
Equation (13) is the theoretical basis to calibrate the switch CVD without the standard HV probe. As to this method, one first needs to know β(d) at different d; secondly he or she needs to fit out β(0) by extrapolation. Generally, the calibration steps are as follows:
Set a switch distance d. Record the voltage of the PFL u0 and the test voltage utest via the CVD for the switch. Calculate β(d) by dividing u0/2 by utest.
Repeat step 1 at least six times in order to obtain an averaged voltage ratio βav(d) at d.
Change the switch distance d for different values. Repeat steps 1 and 2 in order to obtain βav(d) at other switch distances.
Fit different data of βav(d) v.s. d in a coordinate system linearly; then, the intercept β(0) is the real voltage ratio β0.
Based on these steps, the CVDs of the main switches in two Tesla-type generators, namely, 20 GW and TPG700L, in the Northwest Institute of Nuclear Technology are calibrated. The 20 GW generator can output intense electron beams with a power of 20 GW and a pulse width of 40 ns at a repetition rate of 100 Hz, the schematic of which can be seen in Fig. 1. Details of the 20 GW generator can be seen in Ref. 13. TPG700L is a long pulse generator by cascading a pulse forming network (PFN) at the left end of the PFL of the 20 GW generator. The key components of these two generators are the main switches (also gas gap switches). There are CVDs assembled near the main switch to monitor the switch voltage, as shown in Fig. 1(a). The switch voltage is also the incident TL pulse in the perspective of wave theory, as shown in Fig. 4(b). Figure 7 shows the sectional enlarged view of the main switches. In this figure, there is a screw connecting the HV conductor of the TL. Hence, the distance between the two electrodes of the main switches can be adjusted conveniently.
Calibration for the CVD of the switch with the multiple-approximation method.
Figures 8(a) and 8(b) show the calibration results for the 20 GW and TPG700L generators, respectively. From Fig. 8(a), it is seen that β(0), which is the intercept of the fitting line, is 11.64 kV/V for the 20 GW generator. Based on Eq. (13), 11.64 kV/V is just the real voltage ratio β0 of the switch CVD. From Fig. 8(b), it is seen that β0 of TPG700L is 10.75 kV/V.
Calibration results for the switch CVD of the main switch for (a) 20 GW and (b) TPG700L.
Calibration results for the switch CVD of the main switch for (a) 20 GW and (b) TPG700L.
IV. EXPERIMENTAL VERIFICATION
A. Verification results
The calibration result for the CVD of the 20 GW generator via the multiple-approximation method is verified by a Rogowski coil assembled in the outer conductor of the TL near the load. The Rogowski coil can monitor the nanosecond current through the load with a long response time and high accuracy.18 When the load is resistant, the current waveform monitored by the Rogowski coil can totally agree with the voltage waveform. Furthermore, when the impedance of the resistant load matches the characteristics of the TL, the load voltage waveform is just the same as that of the TL, only with 2–3 ns time delay.
Based on this principle, the experimental setup to verify the multiple-approximation method to calibrate the switch CVD was designed, as shown in Fig. 9(a). The schematic is shown in Fig. 9(b). In this figure, a 50-Ω resistor is connected to the inner conductor of the TL. Since the characteristic impedance of the TL is also 50 Ω, this resistor matches the TL. In addition, the amplitude and the waveform of the current through this 50-Ω resistor can be monitored by the Rogowski Coil. Hence, the amplitude of the load voltage, which is also the TL voltage, can be obtained by multiplying the current with 50 Ω.
Verification of the multiple-approximation method to calibrate the switch CVD: (a) experimental setup, (b) schematic, and (c) experimental results.
Verification of the multiple-approximation method to calibrate the switch CVD: (a) experimental setup, (b) schematic, and (c) experimental results.
Figure 9(c) shows the waveforms monitored by the Rogowski Coil and the switch CVD. From this figure, it is seen that the two waveforms basically agree with each other. This figure demonstrates the feasibility of the verification method. As shown in Fig. 9(c), the voltage ratio of the switch CVD can be obtained indirectly. For example, for the 20 GW generator, the current monitored by the Rogowski Coil is 1.8 kA/V. The amplitude of the monitored switch voltage is 7.7 V. Hence, the voltage ratio via the Rogowski coil is 11.68 kV/V (=1.8 kA/V × 50 Ω ÷ 7.7 V). This value agrees with 11.64 kV/V obtained by the multiple-approximation method, only with a deviation smaller than 1%. Hence, the correctness of the multiple-approximation method is verified.
It is worth mentioning that in these verification experiments, the role of the Rogowski coil as well as the matched resistor is the same as that of the HV probe in the conventional calibration method. Even the voltage ratio of the switch CVD can be obtained by this verification method; the calibration method is considered to be complicated since it involves a matched resistor and a conical outer conductor for this resistor.
B. Uncertainty analysis
Many factors may affect the accuracy of the voltage ratio of the switch CVD by the multiple-approximation method.
1. Linear fitting
The first factor comes from the multiple-approximation method itself. Since the final real voltage ratio is obtained by fitting and extrapolation, the uniformity of the fitting may directly determine the intercept, i.e., the real voltage ratio. This point can be demonstrated through Figs. 8(a) and 8(b). By comparing the two figures, once can see the uniformity of the linear fitting in Fig. 8(b) is better than that in Fig. 8(a). Therefore, the extrapolated β0 shown in Fig. 8(b) is considered more reliable than that shown in Fig. 8(a). Table I summarizes two key indices to evaluate the linear fitting shown in Figs. 8(a) and 8(b). From this table, it is seen that no matter what the perspective of Adjusted R-Square or Standard Error be, the fitting shown in Fig. 8(b) is better than that shown in Fig. 8(a), which agrees with the general recognition. Hence, a better linear fitting is desired in the multiple-approximation method.
Evaluation of the multiple-approximation method in perspective of linear fitting.
. | Intercept . | Adjusted . | Standard . | Standard error/ . |
---|---|---|---|---|
. | or β0 . | R-squarea . | errorb . | intercept (%) . |
20 GW | 11.64 | 0.865 | 0.1055 | 0.1 |
TPG700L | 10.75 | 0.953 | 0.0845 | 0.78 |
. | Intercept . | Adjusted . | Standard . | Standard error/ . |
---|---|---|---|---|
. | or β0 . | R-squarea . | errorb . | intercept (%) . |
20 GW | 11.64 | 0.865 | 0.1055 | 0.1 |
TPG700L | 10.75 | 0.953 | 0.0845 | 0.78 |
Adjusted R-square usually demonstrates goodness of the fitting, which usually varies from 0 to 1. The larger the adjusted R-square is, the better is the fitting.
Standard error is usually used to describe the reliability of the fitting. The smaller the ratio of standard error divided by intercept is, the more reliable is the fitting.
2. Dispersion effect of the CVD
Another factor comes from the CVD itself since the extrapolation is involved in different data of [d, βav(d)] and each datum of [d, βav(d)] is related to the voltage ratio of other CVDs. The main component of the CVD is the double-layer-copper-covered film. The capacitance per unit area C0 of this kind of film is not constant but decreases as the frequency increases. For example, C0(10 kHz) = 65 pF/cm2 and C0(100 MHz) = 60 pF/cm2, as shown in Fig. 10. The waveform to test is not a sine waveform but a trapezoid one which is superimposed by many sine wavelets. Once there is a slight change in the trapezoid waveform, the superimposed wavelets many be different, and the contribution from different wavelets to the voltage ratio of the switch CVD may be different, which may lead to the variation in the fitted switch voltage ratio.
Frequency response of the double-layer-copper-covered film used as the CVD.
In addition to these two factors, other factors such as the environment temperature and the measurement of the gap distance may also affect the accuracy of the voltage ratio of the switch CVD since temperature may affect C0 and the measurement of the gap distance is related to the point of [d, βav(d)]. It considered that the values of C0 and the fitting of the data on [d, βav(d)] are the two main factors leading to errors in the voltage ratio in the switch CVD, which may be ±4%. This error range is usually acceptable in the pulsed power field.
V. REMARKS AND CONCLUSIONS
The multiple-approximation calibration method to calibrate the switch CVD in Tesla-type generators can reflect the real voltage ratio. Compared with the direct calibration method via P6015A, this method has the following advantage: it does not need a standard HV probe, so the measurement result of the CVD cannot be affected, and the voltage ratio of the CVD is therefore accurate. However, this method also has the following disadvantages: it is only suitable for the case of the gas switch, and the voltage relation between the front and the back components of the switch are known in advance. In addition, it involves a great deal of work, i.e., to obtain one voltage ratio via this method, tens of calibrations on βav(d) should be performed. Even though the multiple-approximation calibration method is indirect and involves plenty of pre-calibrations, it is a good complement for the direct method via P6015A.
As a conclusion, a multiple-approximation method to calibrate the CVDs of the gas switches in Tesla-type generators is being put forward, which does not involve a standard HV probe. The core idea of this method is to approach β(0) by different points of [d, β(d)]. The calibration result is accurate and reliable. It can be used in the case where the CVD is close to the gas gap switch.
DATA AVAILABILITY
The data that support the findings of this study are available within the article.