REVIEW OF SCIENTIFIC INSTRUMENTS 86, 035114 (2015)
A novel design of Rogowski coil for measurement of nanosecond-risetime high-level pulsed current Ruo-Yu Han, Wei-Dong Ding,a) Jia-Wei Wu, Hai-Bin Zhou, Yan Jing, Qiao-Jue Liu, You-Chuang Chao, and Ai-Ci Qiu State Key Laboratory of Electrical Insulation and Power Equipment, School of Electrical Engineering, Xi’an Jiaotong University, Xi’an, 710049, China
(Received 21 November 2014; accepted 11 March 2015; published online 26 March 2015) In pulsed power systems, pulsed currents with risetimes from nanosecond to microsecond can be effectively measured by self-integrating Rogowski coils. Appropriate design of the structure and the integrating resistor is crucial to the high-frequency response of a coil. In this paper, several novel designs of Rogowski coil’s integrating resistors were proposed and tested. Experimental results showed that the optimized coil could response square waves with fronts of ∼1.5 ns and had a sensitivity of ∼0.75 V/kA. The maximal peak current was designed as 100 kA. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4916094]
I. INTRODUCTION
II. A BRIEF THEORETICAL ANALYSIS
Rogowski coil is a toroidal winding widely used in current measurement. It has been researched for more than 100 years since it was proposed in 1912 by Rogowski.1 This kind of sensor possesses many advantages.2 For example, it has no direct electrical connection with the main circuit. Up to now, numerous Rogowski coils have been designed for different applications, such as pulsed current measurement, relay protection, and partial discharge monitoring.3–7 These coils can be divided into self-integrating coils and differentiating coils by their operation mode. Also, the cores have two types, “air core” and “magnetic core.”8 Because of stray capacitance and parasitic inductance, it is more difficult to measure a nanosecond-risetime pulsed current than a microsecond-risetime one. In some articles, the coil was considered a delay line and adopted distributed parameter model to calculate its response under different input currents.9–12 While in other articles, some Rogowski coils which could measure the pulsed currents with nanosecond risetime were made and tested by experiments.13–16 According to these papers, it could be considered that, for measuring fast (risetime ∼1 ns), high-level (magnitude more than 100 kA) pulsed currents, air core self-integrating Rogowski coils with few turns are often used. Unlike magnetic core Rogowski coils, air core selfintegrating Rogowski coils are much harder to be achieved when measuring fast and high-level pulsed currents. Although some researchers made coils with good high-frequency response, few investigated the effects of the integrating resistor and structure of coils.8,13 In the present article, novel integrating resistors and an optimized structure of the coil are proposed and tested by experiments.
a)Electronic mail: [email protected].
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In this study, the cross section of the core was rectangular for manufacture convenience, which is shown in Fig. 1. Besides, the coil had a return counter-turn, also called a return loop, aiming to diminish the effect of stray magnetic field. A detailed explanation of this feature is presented in the Sec. V. In the distributed model, the coil is regarded as a distributed-element transmission line and the integrating resistor as a RLC circuit, which can be seen in Fig. 2.9,11 According to Fig. 2, the voltage and current in this model can be described by −
∂u(x,t) ∂i(x,t) ∂i 1(t) = R0i(x,t) + L 0 ± M0 , ∂x ∂t ∂t
(1)
−
∂i(x,t) ∂u(x,t) = G0u(x,t) + C0 . ∂x ∂t
(2)
The “plus” or “minus” of M0 is determined by the direction of the main current flowing through the coil. With zero initial condition, equations mentioned above can be solved by the method of Laplace transformation,9,17 I(l, s) = Iout(s) = ±
1 Z0
l 0
sM0(x)I1 · cosh γ(l − x) · dx sinh γl + ( ZZ ) · cosh γl
,
(3)
0
where Z, Z0, γ, and I1 represent terminal impedance, wave impedance, propagation constant, and Laplace transformation of primary current, respectively.9 The accuracy of this model depends much on the calculation accuracy of these parameters. They are obtained by equations in previous studies.8,10,18 Nevertheless, accurate estimation of the parameters in distributed model is difficult. Even the meaning of some parameters is unclear. For example, M0 is used by Dubickas11 to represent induced electromotive force caused by primary current while Cooper9 used u(t). Some essential simplifications were also made in these simulations.
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FIG. 1. Structure of the Rogowski core in this study.
III. DESIGN OF ROGOWSKI COILS
A typical self-integrating Rogowski coil has three parts, toroidal coil (including main spiral winding and return counter-turn), integrating resistor, and electrostatic shield. Design of the coil and the integrating resistor will be discussed in detail, respectively. A. Design of the coil
Many factors should be considered during the design of the coil. The size of the coil is usually determined by its application. In this study, inner radius a was 40 mm and outer radius b was 50 mm. Height h and distance H were both 10 mm. The material of the coil should have fine thermo-stability, small coefficient of thermal expansion, high mechanical strength, and sufficient dielectric strength. Moreover, the permittivity should be as small as possible to reduce stray capacitance (between turns and the return-loop) and increase propagation velocity. Then, Teflon was selected for its conformance to requirements. The number of turns N decides the length of the coil wire. Transit time T depends much on the length of the wire. Neglecting inter-turn capacitance, it has the following form:8 √ T = l ε µ, (4) where l, ε, and µ are total wire length around the coil, permittivity, and magnetic permeability of the coil’s material. Less transit time leads to better response to fast wave, while worse response to slow wave.4,10 Since the size of the coil is determined by the environmental factors, there are only two parameters that need to be settled, N and d (wire diameter). Supposing the terminal impedance Z is pure resistance of 0.1 Ω. Simulations are helpful to select proper parameters in the distributed model. First, the wire diameter d is assumed as 0.6 mm. Then, change the number of turns N with values of 10, 50, and 100. The results are shown in Fig. 3. It can be easily found that the upper cutoff frequency f H is about 109 rad/s, when N is 10. As N increases, f H decreases to 4 × 107 rad/s quickly. Besides, the
FIG. 3. Bode diagram of different N when d is 0.6 mm.
magnitude increases slightly as N increases. The results also reveal the law that the smaller number of turns leads to better high-frequency response and worse low-frequency response. Similarly, N is assumed as 50 and d is assigned to 0.6 mm, 1.2 mm, and 1.8 mm. The results are shown in Fig. 4. It can be seen that different values of d have little influence on the coil’s frequency characteristics. Both the upper and lower cutoff frequencies decrease as d increases. In other words, smaller d means better high-frequency response and worse low-frequency response. These phenomena could be explained by the variation of stray capacitance. Smaller N and d give longer distance between turns, which results in reducing the inter-turn capacitance. Considering the results above and the previous simulations, the number of turns N is adopted as 30 and the wire diameter d is adopted as 1 mm in this study. The material object is shown in Fig. 5. B. Design of integrating resistors
The inductance of the coil is quite low because of the air coil and the small number of turns. It is advisable to select an integrating resistor with low impedance, which means that terminal impedance is low. Low impedance leads to better high-frequency response,9 so a integrating resistor is similar to a current shunt, or saying current viewing resistor (CVR). Coaxial structure is most commonly used in Rogowski coils.8,19 The structure of the coaxial integrating resistor is illustrated in Fig. 6.
FIG. 2. Distributed model of Rogowski coil.
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FIG. 6. Structure of the coaxial integrating resistor.
the high frequency will distort the output voltage because of the inductance, and the output voltage is supposed as a form of di out + Rti out. dt Actually, the front of square wave has a form of uout = L t
FIG. 4. Bode diagram of different d when N is 50.
It seems like any ordinary coaxial structure can meet the requirements in this study. However, if the coil is used to measure a large amplitude current with nanosecond front, its structure must be specifically designed. The large amplitude and the steep front require low resistance and low inductance of the integrating resistor. The inductance of insulation layer is the main part and has a form of Lt ≈
µ 0l t r t2 ln( ), 2π r t1
(5)
where l t, r t1, and r t2 represent the insulation layer’s length, inner radius, and outer radius separately. The resistance of coaxial structure has a form of Rt ≈
ρl t , 2 2 π(r t4 − r t3 )
(6)
where l t, r 3, and r 4 represent the outer conductor layer’s length, inner radius, and outer radius separately. The coil could be considered as a current source and the output current of this source is supposed as a square wave. So
FIG. 5. Real object of the coil.
(7)
At, 0 < t < t r I1 = , (8) At r, t ≥ t r where t r is the risetime of the square wave and A is a constant representing the steepness of the wave. It is also supposed that the coil has a flat frequency response, and the output voltage will have a form of K AL t + K ARtt, 0 < t < t r uout = , (9) K ARtt r, t ≥ t r where K is a coefficient. Then at t r, the wave is discontinuous. In order to diminish the distortion and measuring error, L t and Rt should satisfy the following equation: K AL t ≤ 10%. (10) K ARtt r This equation also represents the voltage overshoot caused by inductance. If the risetime of the square wave is 1 ns, the ratio Rt/L t should be at least 1 × 1010. Thus, the design of insulation layer and outer conductor layer is extremely important. The insulation layer should be as thin as possible to reduce the inductance. The outer conductor layer should be thick enough for low resistance and little increase of L t. Considering the low resistivity of commonly used metal, such as aluminum, copper, and stainless steel, the outer conductor layer should be as thin as possible. For example, Pellinen used 0.0125 mm thick stainless steel to make a 0.1 Ω resistor.13 In this study, metal spraying method and surface mounted device (SMD) resistors were also used as outer conductor layer. Besides, the inductance could be limited by reducing the thickness of insulation layer. And the method of anodic oxidation was helpful to make an extremely thin insulation layer. The material objects (without shield) are shown in Fig. 7. The inner conductor layer is an aluminum cylinder. The insulation layer is the light gray part in Fig. 7(b) with an average thickness of 0.01 mm. The outer conductor layer is error =
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FIG. 8. Structure of the coaxial integrating resistor.
FIG. 7. Structure of the coaxial integrating resistors. (a) Gold. (b) Stainless steel. (c) SMD resistors.
made of gold, stainless steel, and SMD resistors, which is shown in Fig. 7, respectively. The parameters of them are presented in Table I. The results in Table I are gained by calculation.
IV. EXPERIMENTAL RESULTS
The Rogowski coils in this study were calibrated and tested by a square wave generator.20,21 The equipment is shown in Fig. 8. The self-made generator could produce a square wave with a front of 1-5 ns and full width at a half maximum (FWHM) of ∼100 ns. The voltage of the square wave was 13 kV. The voltage and front could be adjusted by changing the parameters of the generator. The output square wave was transmitted along a 50 Ω cable with a length of ∼2 m. At the end of the cable, a 50 Ω matched resistor and a coaxial structure (made by electrostatic shield) were used. The main current flowed through the center of the coil and then back through the outer conductor of the coaxial structure. The coaxial structure ensured a relatively uniform main magnetic field coupled with the coil. This structure with a 50 Ω matched resistor also made a low inductance terminal, meaning the voltage and current had almost the same waveform at the end of the cable. Since the air core coil has no magnetic saturation, it could be calibrated by a low current. The output of the generator was a voltage square wave with amplitude of 1 kV by a 50 Ω coaxial cable. So the primary current was a square wave with amplitude of 20 A. The voltage waveform was presented in Fig. 9 as a reference. In the figure, the risetime of the square wave front is ∼1.5 ns. In fact, the bandwidth of the square wave in this experiment was more than 200 MHz, while the voltage probe in our laboratory only had a bandwidth of 75 MHz. In order to measure a square wave voltage with ∼1.5 ns front, the coaxial structure was removed and the output cable was directly
connected to oscilloscope (Tektronix MDO3054) through a 40 dB coaxial attenuator with 4 GHz bandwidth. The square wave generator used a gas switch, which could cause the uncertainty of the output signal. A series of voltage waveforms (10 times) was acquired. The results showed that the mean value was 1.003 kV and the standard deviation was 0.008 72 kV. The 50 Ω matched resistor had a value of 50.87 Ω by a LCR meter (TH2817B) at the frequency of 100 kHz. Supposing the coaxial structure was a pure resistor (50.87 Ω), the “real” output signal was estimated as 19.72 ± 0.17 A. The output waveforms of these three coils were shown in Fig. 10. In the figure above, the resistor using stainless steel possessed the best performance, while the SMD resistors performed the worst. Since the main signal was in several tens of millivolts, noise was inevitable. Neither the stainless steel nor the gold layer distorted the waveform. However, SMD resistors distorted the front of the wave because of its larger inductance. Although gold layer provided a larger Rt/L t than stainless steel foil, it seemed that the latter had a better performance, which disaccorded with the results in Sec. III. The reason might be the uneven thickness of the gold layer, which resulted in an inhomogeneous distribution of current. So the value of Rt/L t was not exactly equal to 4.26 × 1010. Square wave with a faster front would be used in following studies to determine a better one between stainless steel foil and gold layer. The sensitivities of these three coils were 1.6 mV/A, 0.75 mV/A, and 15 mV/A. Calculated by modified distributed model with the data in Table I, the sensitivities turn out to be 5.60 mV/A, 3.16 mV/A, and 7.08 mV/A. These results did not match the experimental results because of the inaccurate estimation of the coil’s parameters, which would be discussed in Sec. V.
TABLE I. Parameters of the objects (considering skin depth).
Thickness Resistance R t Inductance L t R t/L t
Gold layer
Stainless steel foil
SMD resistors
∼40 nm 0.429 Ω 10.08 pH 4.26 × 1010
0.01 mm 0.1322 Ω 11.25 pH 1.175 × 1010
>0.3 Ω >11.25 pH FIG. 9. Voltage waveform front.
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FIG. 12. A typical waveform when used aluminum foil.
A. The insulation and conductor layers
At first, a 0.01 mm thick aluminum foil was used as the outer conductor layer while a 0.05 mm Teflon film as the insulation layer. Using the aluminum foil could decrease the sensitivity of the coil. The resistance could be even less than 0.01 Ω, so the sensitivity might reach 0.1 mV/A. However, the influence of inductance appeared as discussed in the Sec. III. A typical waveform with aluminum foil was shown in Fig. 12. The output waveform was the voltage of the resistor’s inductance. The resistive voltage is much lower than the inductive voltage. Even though the insulation layer was thin enough and L t reached ∼10 pH, Rt/L t might still be far less than 1010. There are two ways to solve this problem. One is to use a thinner foil. The other one is to use a foil with larger resistivity. All things considered, metal spraying and 0.01 mm stainless steel foil were used. However, the lowest sensitivity in this study was 0.75 mV/A, which made it hard to measure the current larger than 500 kA. In our following studies, modified resistors would be designed and investigated, aiming to reach the sensitivity of 0.1 mV/A. FIG. 10. Voltage waveform front. (a) Resistor using gold film. (b) Resistor using stainless steel foil. (c) Resistor using SMD resistors.
B. Influence of the connection part
A typical current waveform measured by the coil with stainless steel foil was shown in Fig. 11. In the figure, the risetime of the square wave front was also ∼1.5 ns, which fitted the front of voltage front in Fig. 9.
V. DISCUSSION
It is necessary to discuss the interesting phenomena that appeared in these experiments.
FIG. 11. Current waveform front.
Another interesting phenomenon was the influence of the connection part between the coil and the integrating resistor. Previously, few researchers concerned the connection part. Usually, the connection part consists of two leads. Fig. 13 provided a model with an integrating resistor. This model was just the terminal part in Fig. 2. If the connection part was short or the integrating resistor possessed a relatively large resistance, the influence of L t1, L t2,
FIG. 13. The modified model of terminal of the coil.
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FIG. 15. Shapes of the leads. (a) Shape of leads at first. (b) Shape of leads finally.
FIG. 14. Two output waveforms in the experiment. (a) An output waveform in the experiment. (b) Another output waveform in the experiment.
Ct, Rc, and e(t) could be neglected. However, in this study, resistance and inductance of the integrating resistor were quite low, and the connection part was 10 mm long. The calculation results showed that the resistor’s inductance was much lower than L t1 or L t2. The interesting phenomenon was shown in Fig. 14(a). At first, two sharp pulses at 0 ns and 100 ns were regarded as the inductive voltage. With the direction of the coil reversed and yet the integrating resistor remaining at the same position, another interesting phenomenon appeared, as shown in Fig. 14(b). The two sharp pulses (indicated in Fig. 14) were in the opposite direction of the main signal, which proved that the two sharp pulses were not caused by the inductance of the resistor. Then, the shape of the connecting leads was changed. At first, the leads were presented in Fig. 15(a). The distance d was 5 mm. Finally, it turned into the shape in Fig. 15(b). After this change, the results in Fig. 10(b) could be obtained. Using transfer function of the modified model in Fig. 2, bode diagrams were plotted in Fig. 16 with terminal impedance Z regarded as a pure resistance or impedances (defined by two shapes in Fig. 15). In Fig. 16, the connection part mainly affected the lower cutoff frequency of the coil. This might have explained the distortion and oscillation after the front. It is also obvious that the connection part had little influence on the upper cutoff frequency, explaining that sharp pulse was generated by the electromagnetic interference rather than the inductive voltage.
For example, when stainless steel was used, the calculated sensitivity was 3.16 mV/A, while the experimental result was 0.75 mV/A. The upper cutoff frequency was also lower than that in the experiments. One reason might be the estimation of the parameters in the model did not fit the real value. Especially in a highfrequency case, various factors should be considered. The Rogowski coil in this study used a return counter-turn. This structure has two major advantages, avoiding the influence of magnetic flux normal to the face of return counter-turn and putting both terminals at the same end of the spiral.22,23 The main spiral winding, the return counter-turn, and the electrostatic shield constructed a three-conductor transmission line. However, in this study, the electrostatic shield was neglected and the coil was considered as a two-conductor transmission line, which consisted of the main spiral winding and the return counter-turn. The per-unit-length parameters were calculated
C. Disagreement of simulations and experiments
The third interesting phenomenon was that the calculation results did not agree with the experimental results well.
FIG. 16. Bode diagrams with different connections.
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according to the geometry of this two-conductor transmission line. Nevertheless, the estimations of these parameters were based on the previous papers, which might not be the real values. The effect of the return counter-turn was not fully considered. Another reason was the transfer function did not consider the skin depth effect, which would also lead some errors between the calculation and experiment. Besides, the uncertainty was inevitable in the experimental process. It was obviously that the model adopted in this study could not well describe the characteristics of the Rogowski coil.
VI. CONCLUSION
Three Rogowski coils with different integrating resistors and were designed and tested in this article. The experimental results showed that the Rogowski coils could measure the current with a ∼1.5 ns rise-time. The lowest sensitivity was 0.75 mV/A, which can help to measure high-level current. Novel designs of integrating resistors used in this study had been proved to be suitable for fast pulsed currents measurement. Improvements, such as new structure of the integrating resistors with lower resistance and larger ratio of Rt/L t as well as better estimations of coil parameters, will be adopted in the future work. ACKNOWLEDGMENTS
This study is supported by The National High Technology Research and Development Program of China (2013AA064502), Key Projects in the National Science & Technology Pillar Program during the Twelfth Five-year Plan
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Period (2012BAK04B02) and The National Basic Research Program of China (2013CB228004). The authors would like to thank Professor Yongmin Zhang and Mr. Yunfei Liu for the many useful suggestions and discussions. 1W.
Rogowski and W. Steinhaus, Arch. Elektrotech. 1, 141 (1912). Bluhm, Pulsed Power Systems Principles and Applications (Springer, Berlin, 2006). 3I. Biganzoli, R. Barni, and C. Riccardi, Rev. Sci. Instrum. 84(1), 016101 (2013). 4I. A. Metwally, IEEE Trans. Instrum. Meas. 59(2), 353–360 (2010). 5L. Kojovic, IEEE Comput. Appl. Power 10(2), 47–52 (1997). 6Z. Zhang, D. Xiao, and Y. Li, IET Sci., Meas. Technol. 3(3), 187-196 (2009). 7Y. Liu, F. Lin, and Q. Zhang, IEEE Sens. J. 11(1), 123-130 (2011). 8D. G. Pellinen, M. S. Di Capua, S. E. Sampayan, H. Gerbracht, and M. Wang, Rev. Sci. Instrum. 51(11), 1535-1540 (1980). 9J. Cooper, J. Nucl. Energy, Part C 5, 285-289 (1963). 10W. Stygar and G. Gerdin, IEEE Trans. Plasma Sci. 10(1), 40–44 (2009). 11V. Dubickas and H. Edin, IEEE Trans. Instrum. Meas. 56(6), 2284–2288 (2010). 12X. Zhai, Y. Geng, Z. Song, and J. Wang, High Voltage Appar. 43(2), 102–105 (1997). 13D. G. Pellinen and P. W. Spence, Rev. Sci. Instrum. 42(11), 1699-1701 (1971). 14J. Zhu, Q. Zhang, J. Jia, F. Tao, L. Yang, and L. Yang, Plasma Sci. Technol. 8(4), 457-460 (2006). 15V. Nassisi and A. Luches, Rev. Sci. Instrum. 50(7), 900-902 (1979). 16Y. Zhang, J. Liu, G. Bai, and J. Feng, Measurement 45, 1277-1285 (2012). 17A. Smith, Coupling of External Electromagnetic Fields to Transmission Lines (Wiley, New York, 1977). 18M. Shafiq, G. A. Hussain, L. Kütt, and M. Lehtonen, Measurement 49, 126137 (2014). 19D. Pellinen, Rev. Sci. Instrum. 42(5), 667-670 (1971). 20R. Liu, X. Wang, X. Zou, J. Yuan, N. Zou, and L. He, Rev. Sci. Instrum. 78(8), 084702 (2007). 21L. Lu, J. Huang, and Y. Li, Rev. Sci. Instrum. 79(8), 085106 (2008). 22J. D. Ramboz, Proceedings of 1995 IEEE Instrumentation and Measurement Technology Conference (IEEE, 1995), p. 329. 23P. N. Murgatroyd and D. N. Woodland, IEE Colloquium on Low Frequency Power Measurement and Analysis (IET, 1994). 2H.
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A novel design of Rogowski coil for measurement of nanosecond-risetime high-level pulsed current Ruo-Yu Han, Wei-Dong Ding,a) Jia-Wei Wu, Hai-Bin Zhou, Yan Jing, Qiao-Jue Liu, You-Chuang Chao, and Ai-Ci Qiu State Key Laboratory of Electrical Insulation and Power Equipment, School of Electrical Engineering, Xi’an Jiaotong University, Xi’an, 710049, China
(Received 21 November 2014; accepted 11 March 2015; published online 26 March 2015) In pulsed power systems, pulsed currents with risetimes from nanosecond to microsecond can be effectively measured by self-integrating Rogowski coils. Appropriate design of the structure and the integrating resistor is crucial to the high-frequency response of a coil. In this paper, several novel designs of Rogowski coil’s integrating resistors were proposed and tested. Experimental results showed that the optimized coil could response square waves with fronts of ∼1.5 ns and had a sensitivity of ∼0.75 V/kA. The maximal peak current was designed as 100 kA. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4916094]
I. INTRODUCTION
II. A BRIEF THEORETICAL ANALYSIS
Rogowski coil is a toroidal winding widely used in current measurement. It has been researched for more than 100 years since it was proposed in 1912 by Rogowski.1 This kind of sensor possesses many advantages.2 For example, it has no direct electrical connection with the main circuit. Up to now, numerous Rogowski coils have been designed for different applications, such as pulsed current measurement, relay protection, and partial discharge monitoring.3–7 These coils can be divided into self-integrating coils and differentiating coils by their operation mode. Also, the cores have two types, “air core” and “magnetic core.”8 Because of stray capacitance and parasitic inductance, it is more difficult to measure a nanosecond-risetime pulsed current than a microsecond-risetime one. In some articles, the coil was considered a delay line and adopted distributed parameter model to calculate its response under different input currents.9–12 While in other articles, some Rogowski coils which could measure the pulsed currents with nanosecond risetime were made and tested by experiments.13–16 According to these papers, it could be considered that, for measuring fast (risetime ∼1 ns), high-level (magnitude more than 100 kA) pulsed currents, air core self-integrating Rogowski coils with few turns are often used. Unlike magnetic core Rogowski coils, air core selfintegrating Rogowski coils are much harder to be achieved when measuring fast and high-level pulsed currents. Although some researchers made coils with good high-frequency response, few investigated the effects of the integrating resistor and structure of coils.8,13 In the present article, novel integrating resistors and an optimized structure of the coil are proposed and tested by experiments.
a)Electronic mail: [email protected].
0034-6748/2015/86(3)/035114/7/$30.00
In this study, the cross section of the core was rectangular for manufacture convenience, which is shown in Fig. 1. Besides, the coil had a return counter-turn, also called a return loop, aiming to diminish the effect of stray magnetic field. A detailed explanation of this feature is presented in the Sec. V. In the distributed model, the coil is regarded as a distributed-element transmission line and the integrating resistor as a RLC circuit, which can be seen in Fig. 2.9,11 According to Fig. 2, the voltage and current in this model can be described by −
∂u(x,t) ∂i(x,t) ∂i 1(t) = R0i(x,t) + L 0 ± M0 , ∂x ∂t ∂t
(1)
−
∂i(x,t) ∂u(x,t) = G0u(x,t) + C0 . ∂x ∂t
(2)
The “plus” or “minus” of M0 is determined by the direction of the main current flowing through the coil. With zero initial condition, equations mentioned above can be solved by the method of Laplace transformation,9,17 I(l, s) = Iout(s) = ±
1 Z0
l 0
sM0(x)I1 · cosh γ(l − x) · dx sinh γl + ( ZZ ) · cosh γl
,
(3)
0
where Z, Z0, γ, and I1 represent terminal impedance, wave impedance, propagation constant, and Laplace transformation of primary current, respectively.9 The accuracy of this model depends much on the calculation accuracy of these parameters. They are obtained by equations in previous studies.8,10,18 Nevertheless, accurate estimation of the parameters in distributed model is difficult. Even the meaning of some parameters is unclear. For example, M0 is used by Dubickas11 to represent induced electromotive force caused by primary current while Cooper9 used u(t). Some essential simplifications were also made in these simulations.
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FIG. 1. Structure of the Rogowski core in this study.
III. DESIGN OF ROGOWSKI COILS
A typical self-integrating Rogowski coil has three parts, toroidal coil (including main spiral winding and return counter-turn), integrating resistor, and electrostatic shield. Design of the coil and the integrating resistor will be discussed in detail, respectively. A. Design of the coil
Many factors should be considered during the design of the coil. The size of the coil is usually determined by its application. In this study, inner radius a was 40 mm and outer radius b was 50 mm. Height h and distance H were both 10 mm. The material of the coil should have fine thermo-stability, small coefficient of thermal expansion, high mechanical strength, and sufficient dielectric strength. Moreover, the permittivity should be as small as possible to reduce stray capacitance (between turns and the return-loop) and increase propagation velocity. Then, Teflon was selected for its conformance to requirements. The number of turns N decides the length of the coil wire. Transit time T depends much on the length of the wire. Neglecting inter-turn capacitance, it has the following form:8 √ T = l ε µ, (4) where l, ε, and µ are total wire length around the coil, permittivity, and magnetic permeability of the coil’s material. Less transit time leads to better response to fast wave, while worse response to slow wave.4,10 Since the size of the coil is determined by the environmental factors, there are only two parameters that need to be settled, N and d (wire diameter). Supposing the terminal impedance Z is pure resistance of 0.1 Ω. Simulations are helpful to select proper parameters in the distributed model. First, the wire diameter d is assumed as 0.6 mm. Then, change the number of turns N with values of 10, 50, and 100. The results are shown in Fig. 3. It can be easily found that the upper cutoff frequency f H is about 109 rad/s, when N is 10. As N increases, f H decreases to 4 × 107 rad/s quickly. Besides, the
FIG. 3. Bode diagram of different N when d is 0.6 mm.
magnitude increases slightly as N increases. The results also reveal the law that the smaller number of turns leads to better high-frequency response and worse low-frequency response. Similarly, N is assumed as 50 and d is assigned to 0.6 mm, 1.2 mm, and 1.8 mm. The results are shown in Fig. 4. It can be seen that different values of d have little influence on the coil’s frequency characteristics. Both the upper and lower cutoff frequencies decrease as d increases. In other words, smaller d means better high-frequency response and worse low-frequency response. These phenomena could be explained by the variation of stray capacitance. Smaller N and d give longer distance between turns, which results in reducing the inter-turn capacitance. Considering the results above and the previous simulations, the number of turns N is adopted as 30 and the wire diameter d is adopted as 1 mm in this study. The material object is shown in Fig. 5. B. Design of integrating resistors
The inductance of the coil is quite low because of the air coil and the small number of turns. It is advisable to select an integrating resistor with low impedance, which means that terminal impedance is low. Low impedance leads to better high-frequency response,9 so a integrating resistor is similar to a current shunt, or saying current viewing resistor (CVR). Coaxial structure is most commonly used in Rogowski coils.8,19 The structure of the coaxial integrating resistor is illustrated in Fig. 6.
FIG. 2. Distributed model of Rogowski coil.
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FIG. 6. Structure of the coaxial integrating resistor.
the high frequency will distort the output voltage because of the inductance, and the output voltage is supposed as a form of di out + Rti out. dt Actually, the front of square wave has a form of uout = L t
FIG. 4. Bode diagram of different d when N is 50.
It seems like any ordinary coaxial structure can meet the requirements in this study. However, if the coil is used to measure a large amplitude current with nanosecond front, its structure must be specifically designed. The large amplitude and the steep front require low resistance and low inductance of the integrating resistor. The inductance of insulation layer is the main part and has a form of Lt ≈
µ 0l t r t2 ln( ), 2π r t1
(5)
where l t, r t1, and r t2 represent the insulation layer’s length, inner radius, and outer radius separately. The resistance of coaxial structure has a form of Rt ≈
ρl t , 2 2 π(r t4 − r t3 )
(6)
where l t, r 3, and r 4 represent the outer conductor layer’s length, inner radius, and outer radius separately. The coil could be considered as a current source and the output current of this source is supposed as a square wave. So
FIG. 5. Real object of the coil.
(7)
At, 0 < t < t r I1 = , (8) At r, t ≥ t r where t r is the risetime of the square wave and A is a constant representing the steepness of the wave. It is also supposed that the coil has a flat frequency response, and the output voltage will have a form of K AL t + K ARtt, 0 < t < t r uout = , (9) K ARtt r, t ≥ t r where K is a coefficient. Then at t r, the wave is discontinuous. In order to diminish the distortion and measuring error, L t and Rt should satisfy the following equation: K AL t ≤ 10%. (10) K ARtt r This equation also represents the voltage overshoot caused by inductance. If the risetime of the square wave is 1 ns, the ratio Rt/L t should be at least 1 × 1010. Thus, the design of insulation layer and outer conductor layer is extremely important. The insulation layer should be as thin as possible to reduce the inductance. The outer conductor layer should be thick enough for low resistance and little increase of L t. Considering the low resistivity of commonly used metal, such as aluminum, copper, and stainless steel, the outer conductor layer should be as thin as possible. For example, Pellinen used 0.0125 mm thick stainless steel to make a 0.1 Ω resistor.13 In this study, metal spraying method and surface mounted device (SMD) resistors were also used as outer conductor layer. Besides, the inductance could be limited by reducing the thickness of insulation layer. And the method of anodic oxidation was helpful to make an extremely thin insulation layer. The material objects (without shield) are shown in Fig. 7. The inner conductor layer is an aluminum cylinder. The insulation layer is the light gray part in Fig. 7(b) with an average thickness of 0.01 mm. The outer conductor layer is error =
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FIG. 8. Structure of the coaxial integrating resistor.
FIG. 7. Structure of the coaxial integrating resistors. (a) Gold. (b) Stainless steel. (c) SMD resistors.
made of gold, stainless steel, and SMD resistors, which is shown in Fig. 7, respectively. The parameters of them are presented in Table I. The results in Table I are gained by calculation.
IV. EXPERIMENTAL RESULTS
The Rogowski coils in this study were calibrated and tested by a square wave generator.20,21 The equipment is shown in Fig. 8. The self-made generator could produce a square wave with a front of 1-5 ns and full width at a half maximum (FWHM) of ∼100 ns. The voltage of the square wave was 13 kV. The voltage and front could be adjusted by changing the parameters of the generator. The output square wave was transmitted along a 50 Ω cable with a length of ∼2 m. At the end of the cable, a 50 Ω matched resistor and a coaxial structure (made by electrostatic shield) were used. The main current flowed through the center of the coil and then back through the outer conductor of the coaxial structure. The coaxial structure ensured a relatively uniform main magnetic field coupled with the coil. This structure with a 50 Ω matched resistor also made a low inductance terminal, meaning the voltage and current had almost the same waveform at the end of the cable. Since the air core coil has no magnetic saturation, it could be calibrated by a low current. The output of the generator was a voltage square wave with amplitude of 1 kV by a 50 Ω coaxial cable. So the primary current was a square wave with amplitude of 20 A. The voltage waveform was presented in Fig. 9 as a reference. In the figure, the risetime of the square wave front is ∼1.5 ns. In fact, the bandwidth of the square wave in this experiment was more than 200 MHz, while the voltage probe in our laboratory only had a bandwidth of 75 MHz. In order to measure a square wave voltage with ∼1.5 ns front, the coaxial structure was removed and the output cable was directly
connected to oscilloscope (Tektronix MDO3054) through a 40 dB coaxial attenuator with 4 GHz bandwidth. The square wave generator used a gas switch, which could cause the uncertainty of the output signal. A series of voltage waveforms (10 times) was acquired. The results showed that the mean value was 1.003 kV and the standard deviation was 0.008 72 kV. The 50 Ω matched resistor had a value of 50.87 Ω by a LCR meter (TH2817B) at the frequency of 100 kHz. Supposing the coaxial structure was a pure resistor (50.87 Ω), the “real” output signal was estimated as 19.72 ± 0.17 A. The output waveforms of these three coils were shown in Fig. 10. In the figure above, the resistor using stainless steel possessed the best performance, while the SMD resistors performed the worst. Since the main signal was in several tens of millivolts, noise was inevitable. Neither the stainless steel nor the gold layer distorted the waveform. However, SMD resistors distorted the front of the wave because of its larger inductance. Although gold layer provided a larger Rt/L t than stainless steel foil, it seemed that the latter had a better performance, which disaccorded with the results in Sec. III. The reason might be the uneven thickness of the gold layer, which resulted in an inhomogeneous distribution of current. So the value of Rt/L t was not exactly equal to 4.26 × 1010. Square wave with a faster front would be used in following studies to determine a better one between stainless steel foil and gold layer. The sensitivities of these three coils were 1.6 mV/A, 0.75 mV/A, and 15 mV/A. Calculated by modified distributed model with the data in Table I, the sensitivities turn out to be 5.60 mV/A, 3.16 mV/A, and 7.08 mV/A. These results did not match the experimental results because of the inaccurate estimation of the coil’s parameters, which would be discussed in Sec. V.
TABLE I. Parameters of the objects (considering skin depth).
Thickness Resistance R t Inductance L t R t/L t
Gold layer
Stainless steel foil
SMD resistors
∼40 nm 0.429 Ω 10.08 pH 4.26 × 1010
0.01 mm 0.1322 Ω 11.25 pH 1.175 × 1010
>0.3 Ω >11.25 pH FIG. 9. Voltage waveform front.
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FIG. 12. A typical waveform when used aluminum foil.
A. The insulation and conductor layers
At first, a 0.01 mm thick aluminum foil was used as the outer conductor layer while a 0.05 mm Teflon film as the insulation layer. Using the aluminum foil could decrease the sensitivity of the coil. The resistance could be even less than 0.01 Ω, so the sensitivity might reach 0.1 mV/A. However, the influence of inductance appeared as discussed in the Sec. III. A typical waveform with aluminum foil was shown in Fig. 12. The output waveform was the voltage of the resistor’s inductance. The resistive voltage is much lower than the inductive voltage. Even though the insulation layer was thin enough and L t reached ∼10 pH, Rt/L t might still be far less than 1010. There are two ways to solve this problem. One is to use a thinner foil. The other one is to use a foil with larger resistivity. All things considered, metal spraying and 0.01 mm stainless steel foil were used. However, the lowest sensitivity in this study was 0.75 mV/A, which made it hard to measure the current larger than 500 kA. In our following studies, modified resistors would be designed and investigated, aiming to reach the sensitivity of 0.1 mV/A. FIG. 10. Voltage waveform front. (a) Resistor using gold film. (b) Resistor using stainless steel foil. (c) Resistor using SMD resistors.
B. Influence of the connection part
A typical current waveform measured by the coil with stainless steel foil was shown in Fig. 11. In the figure, the risetime of the square wave front was also ∼1.5 ns, which fitted the front of voltage front in Fig. 9.
V. DISCUSSION
It is necessary to discuss the interesting phenomena that appeared in these experiments.
FIG. 11. Current waveform front.
Another interesting phenomenon was the influence of the connection part between the coil and the integrating resistor. Previously, few researchers concerned the connection part. Usually, the connection part consists of two leads. Fig. 13 provided a model with an integrating resistor. This model was just the terminal part in Fig. 2. If the connection part was short or the integrating resistor possessed a relatively large resistance, the influence of L t1, L t2,
FIG. 13. The modified model of terminal of the coil.
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FIG. 15. Shapes of the leads. (a) Shape of leads at first. (b) Shape of leads finally.
FIG. 14. Two output waveforms in the experiment. (a) An output waveform in the experiment. (b) Another output waveform in the experiment.
Ct, Rc, and e(t) could be neglected. However, in this study, resistance and inductance of the integrating resistor were quite low, and the connection part was 10 mm long. The calculation results showed that the resistor’s inductance was much lower than L t1 or L t2. The interesting phenomenon was shown in Fig. 14(a). At first, two sharp pulses at 0 ns and 100 ns were regarded as the inductive voltage. With the direction of the coil reversed and yet the integrating resistor remaining at the same position, another interesting phenomenon appeared, as shown in Fig. 14(b). The two sharp pulses (indicated in Fig. 14) were in the opposite direction of the main signal, which proved that the two sharp pulses were not caused by the inductance of the resistor. Then, the shape of the connecting leads was changed. At first, the leads were presented in Fig. 15(a). The distance d was 5 mm. Finally, it turned into the shape in Fig. 15(b). After this change, the results in Fig. 10(b) could be obtained. Using transfer function of the modified model in Fig. 2, bode diagrams were plotted in Fig. 16 with terminal impedance Z regarded as a pure resistance or impedances (defined by two shapes in Fig. 15). In Fig. 16, the connection part mainly affected the lower cutoff frequency of the coil. This might have explained the distortion and oscillation after the front. It is also obvious that the connection part had little influence on the upper cutoff frequency, explaining that sharp pulse was generated by the electromagnetic interference rather than the inductive voltage.
For example, when stainless steel was used, the calculated sensitivity was 3.16 mV/A, while the experimental result was 0.75 mV/A. The upper cutoff frequency was also lower than that in the experiments. One reason might be the estimation of the parameters in the model did not fit the real value. Especially in a highfrequency case, various factors should be considered. The Rogowski coil in this study used a return counter-turn. This structure has two major advantages, avoiding the influence of magnetic flux normal to the face of return counter-turn and putting both terminals at the same end of the spiral.22,23 The main spiral winding, the return counter-turn, and the electrostatic shield constructed a three-conductor transmission line. However, in this study, the electrostatic shield was neglected and the coil was considered as a two-conductor transmission line, which consisted of the main spiral winding and the return counter-turn. The per-unit-length parameters were calculated
C. Disagreement of simulations and experiments
The third interesting phenomenon was that the calculation results did not agree with the experimental results well.
FIG. 16. Bode diagrams with different connections.
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according to the geometry of this two-conductor transmission line. Nevertheless, the estimations of these parameters were based on the previous papers, which might not be the real values. The effect of the return counter-turn was not fully considered. Another reason was the transfer function did not consider the skin depth effect, which would also lead some errors between the calculation and experiment. Besides, the uncertainty was inevitable in the experimental process. It was obviously that the model adopted in this study could not well describe the characteristics of the Rogowski coil.
VI. CONCLUSION
Three Rogowski coils with different integrating resistors and were designed and tested in this article. The experimental results showed that the Rogowski coils could measure the current with a ∼1.5 ns rise-time. The lowest sensitivity was 0.75 mV/A, which can help to measure high-level current. Novel designs of integrating resistors used in this study had been proved to be suitable for fast pulsed currents measurement. Improvements, such as new structure of the integrating resistors with lower resistance and larger ratio of Rt/L t as well as better estimations of coil parameters, will be adopted in the future work. ACKNOWLEDGMENTS
This study is supported by The National High Technology Research and Development Program of China (2013AA064502), Key Projects in the National Science & Technology Pillar Program during the Twelfth Five-year Plan
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Period (2012BAK04B02) and The National Basic Research Program of China (2013CB228004). The authors would like to thank Professor Yongmin Zhang and Mr. Yunfei Liu for the many useful suggestions and discussions. 1W.
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