sensors Article Principles of Charge Estimation Methods Using High-Frequency Current Transformer Sensors in Partial Discharge Measurements Armando Rodrigo-Mor *, Fabio A. Muñoz and Luis Carlos Castro-Heredia Electrical Sustainable Energy Department, Delft University of Technology, 2628 CD Delft, The Netherlands; f.a.munozmunoz-1@tudelft.nl (F.A.M.); L.C.CastroHeredia@tudelft.nl (L.C.C.-H.) * Correspondence: A.RodrigoMor@tudelft.nl Received: 10 March 2020; Accepted: 27 April 2020; Published: 29 April 2020 Abstract: This paper describes a simplified model and a generic model of high-frequency current transformer (HFCT) sensors. By analyzing the models, a universal charge estimation method based on the double time integral of the measured voltage is inferred. The method is demonstrated to be valid irrespective of HFCT sensor, assuming that its transfer function can be modelled as a combination of real zeros and poles. This paper describes the mathematical foundation of the method and its particularities when applied to measure nanosecond current pulses. In practice, the applicability of the method is subjected to the characteristics and frequency response of the sensor and the current pulse duration. Therefore, a proposal to use the double time integral or the simple time integral of the measured voltage is described depending upon the sensor response. The procedures used to obtain the respective calibration constants based on the frequency response of the HFCT sensors are explained. Two examples, one using a HFCT sensor with a broadband flat frequency response and another using a HFCT sensor with a non-flat frequency response, are presented. Keywords: high-frequency current transformer; HFCT; Rogowski coils; magnetic loop antenna; partial discharge; PD; magnetically coupled sensors; charge estimation 1. Introduction Partial discharge (PD) measurements are a fundamental tool for insulation diagnostics. The conventional method, defined in the standard IEC 60270 [1], presents the concept of apparent charge. Moreover, it sets the procedure for charge calibration and the bandwidths for charge estimation. As presented in [2], the charge estimation method defined in the IEC 60270 is based on the impulse response of bandpass filters—the bandwidth of which is stated in the standard. On the other hand, there is a variety of unconventional partial discharge measurements that rely on high-frequency current transformer (HFCT) sensors, Rogowski coils and magnetic sensors [3–9]. Recently, it has been shown that HFCT can also be used in gas-insulated systems (GISs) [10]. Alternative charge estimation methods are required if the bandwidth of the HFCT sensor does not fall within the IEC60270 frequency ranges or if it is needed to resolve the PD pulses in the µs range. As described in [2], several techniques are available for charge estimation methods in unconventional partial discharge measurements. One of the proposed charge estimation techniques is based on the frequency domain analysis, while others rely on the evaluation of the time integral of the measured voltage. The method based on the evaluation of the time integral of the measured voltage is suitable when the HFCT sensor has a flat and broadband frequency response. However, if the HFCT sensor has a non-flat frequency response, then the application of the time integral of the measured voltage leads to significant charge estimation errors. Therefore, a charge estimation method is needed for HFCT sensors with a non-flat frequency response. Sensors 2020, 20, 2520; doi:10.3390/s20092520 www.mdpi.com/journal/sensors Sensors 2020, 20, x FOR PEER REVIEW Sensors 2020, 20, 2520 2 of 16 2 of 16 leads to significant charge estimation errors. Therefore, a charge estimation method is needed for HFCT sensors with a non-flat frequency response. This paper shows aa simplified simplified and and aa generic genericmodel modelthat thatdescribes describesthe thefrequency frequencyresponse responseofofa aHFCT. HFCT.Using Usingthe thegeneric genericmodel, model,aabasic basicprinciple principleof ofcharge charge estimation estimation based based on the evaluation of the measured voltage voltage is is inferred. inferred. This method is proven to be theoretically theoretically double time integral of the measured response. However, its practical application depends on suitable irrespective of the HFCT frequency response. the required requiredintegration integration time, which is related the current pulse duration and the frequency the time, which is related to thetocurrent pulse duration and the frequency response response of the sensor. of the sensor. Finally, two two examples examples are are presented. presented. The first one uses a HFCT sensor with a non-flat frequency Finally, response where the charge of a current pulse estimated using double time integral method. In response where the charge of a current pulse is is estimated using thethe double time integral method. In the the second example, a HFCT sensor with a broadband flatfrequency frequencyresponse responseisisused usedalong along with with the second example, a HFCT sensor with a broadband flat time integral integral of the measured measured voltage. voltage. The calculation of the calibration constants for each sensor and time explained. method is also explained. 2. Equivalent Model of HFCT Sensors 2.1. Simplified HFCT 2.1. Simplified HFCT Model Model HFCT modelled as magnetically coupled coils.coils. FigureFigure 1 shows a simplified electric HFCT sensors sensorscan canbebe modelled as magnetically coupled 1 shows a simplified circuit where Lwhere of the of primary coil, Lcoil, the self-inductance of the 1 is the 2 is L electricmodel, circuit model, L1 self-inductance is the self-inductance the primary 2 is the self-inductance of secondary coil, M is the mutual inductance, C represents the parasitic capacitance the secondary the secondary coil, M is the mutual inductance, C represents the parasiticofcapacitance of coil the and R is a loading resistor. secondary coil and R is a loading resistor. Figure 1. Electric circuit model of a magnetically coupled sensor. sensor. For relationship For the the above above electric electric circuit circuit model, model, the the differential differential equation equation that that describes describes the the √ relationship between (L11·L between the the input, input,i(t), i(t),and andthe theoutput, output,u(t), u(t),can canbe bewritten writtenasasininEquation Equation(1), (1),being beingMM==k·k·√(L ·L22)) and and kk the coupling coefficient. the coupling coefficient. di(t) L C d2 u(t) L du(t) u(t) = 2 + 2 + (1) 2 dtππ(π‘) MπΏ πΆ dt dt M πΏ ππ’(π‘) π’(π‘) π π’(π‘) RM (1) = + + ππ‘ π ππ‘ of theπ π ππ‘ is described π In the Laplace domain, the transfer function sensor by the following equation. In the Laplace domain, the transfer function U (s) of the sensor s is described by the following equation. H (s) = = LC (2) L2 1 2 I (s) s2 + RM s+ M M π π(π ) = π»(π ) = √ (2) πΏ πΆ πΏ 1 πΌ(π ) If R < 0.5· (L2 /C), the transfer function of the sensor, π as+modelled π + by Equation (1), can be rewritten π π π π as a generic transfer function with real zeros and poles as follows. If R < 0.5·√(L2/C), the transfer function of the sensor, as modelled by Equation (1), can be rewritten U (s) α·s as a generic transfer function withHreal and (s) zeros = =poles as follows. (3) (s + p1 )(s + p2 ) I (s) πΌ·π π(π ) where p1 and p2 are poles representing the=lower and is = upper cutoff frequencies of the sensor, and α(3) π»(π ) πΌ(π ) (π + π )(π + π ) a constant. The frequency response H(ω) of the canupper be written in Equation (4), sensor, where ω is α the p2 are poles representing the sensor lower and cutoffas frequencies of the and is where p1 and angular frequency. a constant. jωM U (ω) V H (ω) = = (4) L2as in The frequency response H(π) of Ithe can be written 2 (ω)sensor (1 − L2 Cω ) + jω R A Equation (4), where π is the angular frequency. π»(π) = π(π) π πππ = πΌ(π) (1 − πΏ πΆπ ) + ππ πΏ π΄ π (4) Sensors 2020, 20, 2520 3 of 16 According to Equation (4), the maximum sensor gain occurs at frequency fo and accounts for Hmax, as described in the following equations. According to Equation (4), the maximum sensor gain occurs at frequency fo and accounts for Hmax , as described in the following equations. 1 π = (5) 2π1 πΏ β πΆ (5) fo = √ 2π L2 ·C π βπ π = π» (6) R·MπΏ V π΄ Hmax = (6) L2 A This simplified simplified model model of of HFCT HFCT sensors, sensors, and, and, in in general, general, of of any any sensor sensor based based on onmagnetically magnetically This coupled inductors, describes the sensor response when the lower and upper cutoff frequencies are coupled inductors, describes the sensor response when the lower and upper cutoff frequencies are separated enough, which means that the poles do not interfere with each other. separated enough, which means that the poles do not interfere with each other. To illustrate illustrate this this behavior, behavior, aa HFCT HFCT sensor sensor based based on on aa N30 N30 magnetic magnetic material material core core has has been been To characterized; see Figure 2. The sensor has a 5-turn secondary winding, made with a flat copper characterized; see Figure 2. The sensor has a 5-turn secondary winding, made with a flat copper ribbon. ribbon. Theissensor loaded with a R, resistor, R, inwith parallel with acapacitor, ceramic capacitor, C. The of The sensor loadediswith a resistor, in parallel a ceramic C. The values of values R and C R and C are shown in Table 1. The terminals of the secondary coil are connected to an isolated BNC are shown in Table 1. The terminals of the secondary coil are connected to an isolated BNC connector. connector. ofBNC the isolated BNC connectortoisthe connected to the central conductor of a BNC The shield ofThe theshield isolated connector is connected central conductor of a BNC connector—the connector—the shield of which is connected to the HFCT enclosure. By BNC usingcap, a shorted BNC cap, the shield of which is connected to the HFCT enclosure. By using a shorted the terminal of the terminal of the coil is connected to the enclosure. A teflon ring is used to keep the magnetic core in coil is connected to the enclosure. A teflon ring is used to keep the magnetic core in place. The whole place. The whole arrangement is in a metallic enclosure for electric shielding purposes. Table 1 shows arrangement is in a metallic enclosure for electric shielding purposes. Table 1 shows the parameters of thesensor parameters ofHFCT_LG. the sensor named HFCT_LG. the named Figure Figure 2. 2. High-frequency High-frequency current current transformer transformer (HFCT) (HFCT) enclosure enclosureand anddetailed detailedinner innerview. view. Table 1. HFCT_LG sensor parameters. Table 1. HFCT_LG sensor parameters. HFCT Sensor Parameters HFCT Sensor Parameters Sensor L (µH) L C (pF)R (Ω) 1 L1 (μH) L22 (µH) (μH) C (pF) HFCT_LGHFCT_LG 5.25 5.25 130.64 130.64 28.2628.2655.61 Sensor R (β¦) 55.61 The sensor sensor has with a maximum gaingain of 10.3 mV/mA and aand bandwidth from 67from kHz The hasaaflat flatresponse, response, with a maximum of 10.3 mV/mA a bandwidth to kHz 67 MHz; Figure Using3.theUsing measured frequencyfrequency response,response, the two poles p1 and p2, 2πβ67 10,3 67 to 67see MHz; see3.Figure the measured the two poles p1 and× p 2 rad/s × and 106 2π·67 rad/s,×respectively, are estimated. The poles correspond to the lower upper 2π·67 1032πβ67 rad/s×and 106 rad/s, respectively, are estimated. The poles correspond toand the lower 9. The cutoff frequencies, 67 kHz and 67 MHz, The value α isofset 4.34 × 10× and upper cutoff frequencies, 67 kHz and 67correspondingly. MHz, correspondingly. The of value α istoset to 4.34 109 . simulated frequency response obtained using Equation (4) is depicted in Figure 3 along with the The simulated frequency response obtained using Equation (4) is depicted in Figure 3 along with the measured frequency response. The figure shows the good agreement between the measurement and measured frequency response. The figure shows the good agreement between the measurement and thesimulation, simulation,which whichvalidates validatesthe thesimplified simplifiedmodel modelfor forthis thisparticular particularsensor. sensor. the Sensors 2020, 20, 2520 Sensors 2020, 20, x FOR PEER REVIEW 4 of 16 4 of 16 Figure Figure3.3.HFCT_LG HFCT_LGsensor sensorfrequency frequencyresponse, response,measured measuredand andsimulated. simulated. 2.2. Generic HFCT Model 2.2. Generic HFCT Model In some situations, the frequency response of a sensor based on magnetically coupled inductors In some situations, the frequency response of a sensor based on magnetically coupled inductors cannot be represented by the simplified model, particularly when the sensor does not exhibit a flat and cannot be represented by the simplified model, particularly when the sensor does not exhibit a flat wide frequency response. The deviations from the simplified model are due to different aspects not and wide frequency response. The deviations from the simplified model are due to different aspects taken into account in the simplified model. Some of them are the permeability frequency dependence not taken into account in the simplified model. Some of them are the permeability frequency behavior, the non-linear losses in the magnetic core, the multiple coil resonances or, for instance, dependence behavior, the non-linear losses in the magnetic core, the multiple coil resonances or, for the effect of parasitic capacitances. instance, the effect of parasitic capacitances. To illustrate this fact, another sensor has been built using different resistor values while using the To illustrate this fact, another sensor has been built using different resistor values while using same magnetic core material and number of turns in the secondary coil as the previous sensor. Table 2 the same magnetic core material and number of turns in the secondary coil as the previous sensor. shows the parameters of the sensor named HFCT_HG. Table 2 shows the parameters of the sensor named HFCT_HG. Table 2. HFCT_HG sensor parameters. Table 2. HFCT_HG sensor parameters. HFCT Sensor Parameters HFCT Sensor Parameters Sensor L1 (µH) C (pF) L1 (μH) LL2 2(µH) (μH) C (pF) R (Ω) HFCT_HGHFCT_HG4.90 4.90 127.14 26.65 127.14 26.65 996.4 Sensor R (β¦) 996.4 Themeasured measuredfrequency frequencyresponse responseofofthe thesensor sensorisisshown shownininFigure Figure4.4.Due Duetotothe theparameters parametersofof The thesensor, sensor,its itsfrequency frequencyresponse responseexhibits exhibitsaapeak peakresponse responseatatapproximately approximately11MHz, MHz,with with100 100mV/mA mV/mA the peak gain. peak gain. In this case, the sensor cannot be modelled by the simplified model, since the shape of the measured frequency response does not match with the transfer function shown in Equation (3). This is illustrated in Figure 5, where a simulated frequency response according to Equation (4) indicates a proper fit in the low-frequency range but fails for frequencies above 1 MHz in both the magnitude and the phase frequency response. Figure 4. HFCT_HG sensor frequency response, measured and simulated. Sensor HFCT_HG L1 (μH) 4.90 L2 (μH) 127.14 C (pF) 26.65 R (Ω) 996.4 The measured frequency response of the sensor is shown in Figure 4. Due to the parameters of the sensor, its2520 frequency response exhibits a peak response at approximately 1 MHz, with 100 mV/mA Sensors 2020, 20, 5 of 16 peak gain. Sensors 2020, 20, x FOR PEER REVIEW 5 of 16 In this case, the sensor cannot be modelled by the simplified model, since the shape of the measured frequency response does not match with the transfer function shown in Equation (3). This is illustrated in Figure 5, where a simulated frequency response according to Equation (4) indicates a proper fit in the low-frequency range but fails for frequencies above 1 MHz in both the magnitude measured and and simulated. simulated. Figure 4. HFCT_HG sensor frequency response, measured and the phase frequency response. . Figure 5. HFCT_HG sensor frequency response, measured and simulated, using the simplified model Figure 5. and HFCT_HG sensor frequency response, measured and simulated, using the simplified model (one zero two poles). (one zero and two poles). In order to obtain a good fitting model, the number of zeros and poles must be increased. In this In order obtain goodbeen fitting model, adjusted the number of zeros and be increased. In this case, the zerosto and polesahave manually by inspection. A poles more must elaborated and automatic case, the zeros and poles have been manually adjusted by inspection. A more elaborated procedure can be found in [11]. For the given sensor, the transfer function has been determined toand be automatic can found in [11]. For the given sensor, the transfer function been as describedprocedure in Equation (7)be with the following values: α = 3.78, z1 = 2π·3.9 × 106 rad/s, z2 = has 2π·130 × 6 rad/s, p to 6 rad/s 6 rad/s. determined be as described in Equation (7) with the following values: α = 3.78, z 1 = 2πβ3.9 × 106 rad/s, 10 = 2π·1.1 × 10 and p = 2π·9 × 10 1 2 z2 = 2πβ130 × 106 rad/s, p1 = 2πβ1.1 × 106 rad/s and p2 = 2πβ9 × 106 rad/s. U (s) α·s(s + z1 )(s + z2 ) H (s) = π(π )= πΌ · π (π +2 π§ )(π + π§ ) (7) π»(π ) =I (s) = (s + p1 ) (s + p2 ) (7) πΌ(π ) (π + π ) (π + π ) The simulated frequency response according to Equation (7) is shown in Figure 4 along with the The simulated according to Equation is similar shown in 4 along the measured frequencyfrequency response.response Since both frequency responses(7) are in Figure magnitude andwith phase, measured frequency response. Since both frequency responses are similar in magnitude and phase, the model shown in Equation (7) is therefore validated. the model shown in Equation (7) is therefore validated. 3. Charge Estimation Theory 3. Charge Estimation Theory 3.1. Charge Estimation Theory Using the Simplified HFCT Model 3.1. Charge Estimation Theory Using the Simplified HFCTthe Model Considering Equation (1), it is possible to derive calculation of the charge, q(t), knowing that the definition of charge is the integral of the current i(t). Considering Equation (1), it is possible to derive the calculation of the charge, q(t), knowing that the definition of charge is the integral of the current i(t). To calculate the evolution of charge over time, a double integration in the time domain is applied to Equation (1). Proceeding as described, Equation (1) can be written as follows. πΏ πΆ ππ(π‘) ππ‘ = π ππ‘ π π’(π‘) πΏ ππ‘ + π π ππ‘ ππ’(π‘) 1 ππ‘ + π ππ‘ π’(π‘) ππ‘ (8) Sensors 2020, 20, 2520 6 of 16 To calculate the evolution of charge over time, a double integration in the time domain is applied to Equation (1). Proceeding as described, Equation (1) can be written as follows. x di(t) L2 C x d2 u(t) 2 L2 x du(t) 2 1 x dt2 = + dt u(t)dt2 dt + dt M RM dt M dt2 | {z } | {z } {z } | q(t) L2 RM L2 C M u(t) R (8) u(t)dt Assuming no initial conditions and simplifying Equation (8), the final equation for q(t) is: Zt t L2 L2 C 1 x q(t) = u(t) + u(t)dt2 u(t)dt + M RM M 0 | {z } 0 {z } | {z } | f irst term second term (9) third term Considering that i(t) is a current pulse of limited duration, u(t) will tend to zero after the current pulse i(t) has been extinguished. Therefore, the first term of q(t) tends to zero when t tends to infinite. After a certain time, tint , the time integral of u(t) tends to zero, since an inductive measuring system does not measure the DC component, which in turn nulls the second term. Therefore, simplifying Equation (9), the total charge, Q, of the input current pulse i(t) can be calculated as follows. tint 1 x Q ≈ q(tint ) ≈ u(t)dt2 M (10) 0 This mathematical analysis reveals that, theoretically, any partial discharge sensor based on magnetically coupled inductors that can be modelled as a second-order differential equation can measure the input charge independently of sensor design and frequency response. According to the previous equation, the charge Q can be calculated via the double time integral of the measured voltage u(t), knowing the mutual inductance M. Similar results for Rogowski coils have been presented in [12]. 3.2. Charge Estimation Theory Using the Generic HFCT Model It has been shown that the simplified HFCT model has limited applicability depending upon the sensor frequency response. A generic model for HFCT transformers can be created by a combination of zeros and poles, as shown in Equation (11), if no complex poles are needed to model the frequency response of the sensor. The number and location of the zeros and the poles depend on the frequency response to be modelled. P =m ( s + zi ) α·s· ii= U (s) H (s) = = P j=n 1 (11) I (s) s + p j j=1 Using Equation (11), it is possible to express the double time integral of the measured voltage in the Laplace domain as follows. P m α· ii= U (s) 1 (s + zi ) = I ( s ) P j= (12) =n s2 s· s + pj j=1 Applying a partial fraction expansion, it is possible to break Equation (12) down into simple terms expressed as a function of the poles and the coefficients an . U (s) a a1 an = I (s) 0 + +...+ s (s + p1 ) (s + pn ) s2 ! (13) Sensors 2020, 20, 2520 7 of 16 Using the inverse Laplace transform, Equation (13) can be written in the time domain as follows. xt Zt 2 u(t)dt = ao 0 i(t)dt + i=n X Zt ai · i=1 0 | {z } −pi· t epi· t ·i(t)dt e · |{z} (14) →0 as t→∞ 0 | q(t) {z } addends Equation (14) shows that for any arbitrary model based on the generic model of the HFCT, the value of the double time integral contains information about the charge of the current pulse. Since a partial discharge current pulse has a limited duration, the integrals in the addends reach a constant value after the time duration of the current pulse. Once an integral has reached its constant value, it is damped by the exponential decay with the time constant determined by its pole. Hence, over time, all the addends in the summation tend to zero, and therefore the summation. The speed of the convergence of the summation to zero is dominated by the smaller pole, named p1 , which imposes the slowest dynamic. Assuming that the duration of the current pulse is shorter than the time constant determined by p1 , the total charge Q could be approximated by Q ≈ s 4 p1 0 u(t)dt2 (15) a0 An integration time of 4/p1 has been selected, since the first exponential decay has already decreased by 98% of its initial value at this moment in time. Therefore, 4/p1 is a good estimation of the necessary integration time for current pulses with a pulse duration shorter than it. This result proves that the double time integral of the measured voltage is proportional to the total charge of the current pulse after a certain integration time. This statement is valid irrespective of the sensor frequency response (determined by the number of zeros and poles), provided that the sensor exhibits a derivative behavior in the low-frequency range. This required characteristic is typical of sensors based on magnetically coupled inductors, which applies to HFCT sensors, Rogowski coils and to the recently developed magnetic loop antennas used to measure partial discharges in gas-insulated systems (GISs) [13,14]. According to Equation (15), to estimate the charge value, the first constant value a0 of the partial fraction expansion shown in Equation (13) must be determined. Using Equation (13), it is possible to rewrite the transfer function of the sensor as follows. U (s) a s2 an ·s2 = a0 ·s + 1· +...+ (s + p1 ) (s + pn ) I (s) (16) In the frequency domain, Equation (16) is equivalent to the following equation. U (ω) a1· ω2 an ·ω2 = jωa0 − −...− (p1 + jω) (pn + jω) I (ω) (17) When ω <<< p1 , then (pi + j ω) ≈ pi . Hence, Equation (17) can be simplified as follows H (ω) = U (ω) a ω2 an ·ω2 ≈ jωa0 − 1· −...− p1 pn I (ω) (18) By applying the derivative with respect to ω in Equation (18), the following result applies dH (ω) 2a ω 2an ω ≈ ja0 − 1· − . . . − dω p1 pn ! (19) Sensors 2020, 20, 2520 8 of 16 Therefore, when ω tends to zero, the following result holds for the derivative of the transfer function. dH (ω) dω ≈ ja0 (20) ω→0 This result means that the slope of the Bode magnitude plot is a good approximation of the value of ao . Since the derivative of H(ω) when ω tends to zero is a complex number, the value of the slope must be determined in the low-frequency ranges, where the Bode phase plot is close to 90β¦ . A circuital analysis of the HFCT basic circuit helps to give this slope a physical meaning. Figure 6 shows the 20, equivalent circuit of the secondary of a HFCT in the frequency domain. Sensors 2020, x FOR PEER REVIEW 8 of 16 Figure 6. circuit in the domain of theof secondary coil of the coupled 6. Equivalent Equivalent circuit in Laplace the Laplace domain the secondary coilmagnetically of the magnetically sensor. coupled sensor. In the to to zero, thethe impedance of the inductor tends to zero and In the low-frequency low-frequencyrange, range,when whenωπtends tends zero, impedance of the inductor tends to zero the impedance of the capacitor to infinite. Therefore, the voltage that appears across the resistor equals and the impedance of the capacitor to infinite. Therefore, the voltage that appears across the resistor the induced voltagevoltage in the secondary coil of the coupling. Under this condition, the transfer equals the induced in the secondary coilmagnetic of the magnetic coupling. Under this condition, the function can be approximated by the following equation. transfer function can be approximated by the following equation. U (ω) (ω) == π(π)≈≈jωM Hπ»(π) πππ when π€βππ ωπ→ →00rad/s πππ/π I (πΌ(π) ω) (21) (21) Equation (21) reveals that the value of a matches the mutual inductance M. The mutual inductance Equation (21) reveals that the value0 of a0 matches the mutual inductance M. The mutual can then be experimentally determined by the slope of the Bode magnitude plot in the low-frequency inductance can then be experimentally determined by the slope of the Bode magnitude plot in the range, where the Bode phase plot is 90β¦ . This result, obtained for a generic HFCT model, is in low-frequency range, where the Bode phase plot is 90°. This result, obtained for a generic HFCT accordance with Equation (10), which was obtained for the simplified model using electrical parameters, model, is in accordance with Equation (10), which was obtained for the simplified model using thus demonstrating that the simplified model is a simple particular case of the generic model. electrical parameters, thus demonstrating that the simplified model is a simple particular case of the generic model. 4. Charge Estimation Methods The charge evolution over time and the integration time strongly depends on the inductive sensor 4. Charge Estimation Methods characteristics as defined in Equation (14). As explained in Section 3.2 and shown in Equation (15), The charge evolution over time and the integration time strongly depends on the inductive the double time integral of the measured voltage converges to the charge value after a certain integration sensor characteristics as defined in Equation (14). As explained in Section 3.2 and shown in Equation (15), time. When the duration of the current pulse is smaller than the time constant determined by 4/p1 , the double time integral of the measured voltage converges to the charge value after a certain the integration time can then be approximated by 4/p1 , p1 being the first pole of the transfer function. integration time. When the duration of the current pulse is smaller than the time constant determined An estimation of p1 can be calculated using the sensor parameters as R/L2 . by 4/p1, the integration time can then be approximated by 4/p1, p1 being the first pole of the transfer Table 3 shows the estimated integration time using both methods for the HFCT_LG and function. An estimation of p1 can be calculated using the sensor parameters as R/L2. HFCT_HG sensor. Table 3 shows the estimated integration time using both methods for the HFCT_LG and HFCT_HG Table 3.sensor. Estimation of integration times for charge calculation using the double time integral of the measured voltage. Table 3. Estimation of integration times for charge calculation using the double time integral of the HFCT_LG HFCT_HG measured voltage. Estimated integration times for current pulse 4/p1 4·L2 /R 4/p1 4·L2 /R durations <<< 4/p1 HFCT_LG HFCT_HG Estimated integration times 9.5 µs 9.4 µs 578 ns 510 ns 4/p1 4·L2/R 4/p1 4·L2/R 9.5 μs 9.4 μs 578 ns 510 ns To illustrate the convergence of the double time integral, a 5 pC triangular pulse of 10 ns duration has been used to simulate the HFCTof responses to the current pulse.a 5 pC triangular pulse of 10 ns To illustrate the convergence the double time integral, duration has been used to simulate the HFCT responses to the current pulse. The responses of the sensors have been simulated using the transfer functions defined in Section 2.1 and Section 2.2, using the zeros and poles. The mutual inductance M has been estimated by the slopes of the Bode magnitude plots between 1 kHz and 4 kHz, since both Bode phase plots show phase values close to 90° between these two frequencies; see Figure 3 and Figure 4. The estimated value of for current pulse durations <<< 4/p1 Sensors 2020, 20, 2520 9 of 16 The responses of the sensors have been simulated using the transfer functions defined in Sections 2.1 and 2.2, using the zeros and poles. The mutual inductance M has been estimated by the slopes of the Bode magnitude plots between 1 kHz and 4 kHz, since both Bode phase plots show phase values close to 90β¦ between these two frequencies; see Figures 3 and 4. The estimated value of M for the HFCT_LG sensor is 24.4 µH, and 28 µH for the HFCT_HG sensor. Figures 7 and 8 show the current pulse, the simulated voltage outputs and the charge evolution. In this case, since the current pulse duration is smaller than the estimated integration times shown in Table 3, the convergence of the charge estimation approaches the charge value within the estimated Sensors 2020, 20, x FOR PEER REVIEW 9 of 16 integration times. AFigure longer pulse duration response has been and simulated illustrate the onand the convergence Sensors 2020, 20,9xcurrent FOR PEER REVIEW 9 of of 16 shows the HFCT_LG chargetoestimation to aeffects 500 pC 1000 ns current the double time integral for the charge estimation. pulse. Since the current pulse duration is still smaller than the estimated integration time of 9.5 μs, Figure 99 shows shows thedouble HFCT_LG response charge estimation pC and andintegration 1000 ns ns current current Figure the HFCT_LG and estimation to the a 500 pC 1000 the convergence of the timeresponse integral of thecharge voltage falls within expected time. pulse. Since the current pulse duration is still smaller than the estimated integration time of 9.5 μs, pulse. µs, the convergence convergence of of the the double double time time integral integral of of the thevoltage voltagefalls fallswithin withinthe theexpected expectedintegration integrationtime. time. the Figure 7. HFCT_LG sensor response and charge estimation of a 5 pC and 10 ns triangular current pulse. Figure and charge estimation of aof 5 pC ns 10 triangular currentcurrent pulse. Figure 7.7. HFCT_LG HFCT_LGsensor sensorresponse response and charge estimation a 5and pC 10 and ns triangular pulse. Figure response andand charge estimation of a 5ofpC ns triangular currentcurrent pulse. Figure8.8.HFCT_HG HFCT_HGsensor sensor response charge estimation a 5and pC10and 10 ns triangular pulse. Figure 8. HFCT_HG sensor response and charge estimation of a 5 pC and 10 ns triangular current pulse. Figure 8. HFCT_HG sensor response and charge estimation of a 5 pC and 10 ns triangular current 10 of 16 pulse. Sensors 2020, 20, 2520 Figure HFCT_LG sensor sensorresponse responseand and charge estimation a pC 500and pC 1000 and ns 1000 ns triangular Figure9.9. HFCT_LG charge estimation of aof500 triangular current Sensors 2020, pulse. 20, x FOR PEER REVIEW current pulse. 10 of 16 However,the the1000 1000nsnslong longtriangular triangularcurrent currentpulse pulsehas hasa apulse pulseduration durationthat thatisisalmost almosttwice twicethe the However, estimatedintegration integrationtime timeofofthe theHFCT_HG HFCT_HGsensor, sensor,asasindicated indicatedininTable Table3.3.As Asshown shownininFigure Figure10, 10, estimated therequired requiredintegration integrationtime timeisisapproximately approximately2000 2000ns, ns,which whichmatches matchesthe thevoltage voltagepulse pulseduration. duration. the Forthat thatreason, reason, partial discharge measurements, where current pulse duration is expected For inin partial discharge measurements, where thethe current pulse duration is expected to beto be bigger than the integration as calculated bypole the of first of the transfer bigger than the integration time, astime, calculated by the first thepole transfer function, it isfunction, advisableittois advisable estimate the time via themeasured analysis of the measured voltage. The estimated estimate theto integration timeintegration via the analysis of the voltage. The estimated integration time integration in this case, by be the approximated by the time between thesecond starting andcrossing the second could, in thistime case,could, be approximated time between starting and the zero of zero crossingvoltage of the measured the measured signal. voltage signal. Figure10. 10.HFCT_HG HFCT_HGsensor sensorresponse responseand andcharge chargeestimation estimationofofaa500 500pC pCand and1000 1000nsnstriangular triangular Figure current currentpulse. pulse. ItItisisworth worthnoticing noticingthat thatfor forthe theHFCT_LG HFCT_LGsensor, sensor,the theintegration integrationtime timeisisremarkably remarkablylonger longerthan than the thecurrent currentpulse pulseduration, duration,since sincethe thelower lowercutoff cutofffrequency frequencyisisininthe thekHz kHzrange. range.Moreover, Moreover,for forthis this particular sensor, thethe undershoot produced in the voltage is veryissmall. In practice, this could particular sensor, undershoot produced inmeasured the measured voltage very small. In practice, this lead to integration errors due to offsets and the vertical scale digitalization resolution. Moreover, could lead to integration errors due to offsets and the vertical scale digitalization resolution. the longer thethe required time, the higher the the probability to be affected by or by external Moreover, longer integration the required integration time, higher the probability tonoise be affected by noise disturbances. Therefore, if the sensor requires a long integration time using the double time integral or by external disturbances. Therefore, if the sensor requires a long integration time using the double method, but the sensor has broadband frequency response that allows for athat reliable current pulse time integral method, but athe sensor hasflat a broadband flat frequency response allows for a reliable shape measurement, an alternative for charge estimation is toestimation estimate the current pulse current pulse shapethen measurement, thenmethod an alternative method for charge is to estimate the current pulse using the sensor gain and the measured voltage. Once the current is estimated, the charge can be evaluated by the time integration between the voltage signal zero crossings, as explained in [15]. 5. Test Measurements Two cases of study are presented to illustrate the principles of the charge estimation methods Sensors 2020, 20, 2520 11 of 16 using the sensor gain and the measured voltage. Once the current is estimated, the charge can be evaluated by the time integration between the voltage signal zero crossings, as explained in [15]. 5. Test Measurements Two cases of study are presented to illustrate the principles of the charge estimation methods and the calibration methods presented before. For that purpose, the HFCT_LG sensor and the HFCT_HG sensor are used. A partial discharge calibrator is used to inject a nanosecond current pulse and to test the performance of each method. 5.1. Sensor Characterization To characterize the HFCT sensors, they have been provided with two panel-mounted BNC connectors, one in each side of the enclosure, internally connected using a copper wire and fixed to the enclosure using a clamp; see Figure 11. The setup allows a continuous coaxial arrangement in which a voltage source or a partial discharge calibrator is connected to one BNC connector, and the other BNC connector to an oscilloscope to measure the injected signals. In this case, an external 50 β¦ resistor is connected to a 1 Mβ¦ and 500 MHz Sensors 2020, 20, x FOR PEER REVIEW 11 of 16 bandwidth input channel of an oscilloscope. The injected currents are evaluated as the measured voltage over the 50over β¦ external load resistance. HFCT sensor output is directly measured voltage the 50 RF Ω external RF load The resistance. The HFCT sensor outputconnected is directlyto aconnected 1 Mβ¦ andto500 bandwidth channelinput of the oscilloscope. a 1 MHz MΩ and 500 MHzinput bandwidth channel of the oscilloscope. External 50 Ω RF load Voltage measurement Coax cable current injection Panel-mounted Cooper BNC connector wire Clamp Figure Figure 11. 11. HFCT HFCTtesting testingsetup setupand andsensor sensorconnection connectiontotothe theoscilloscope. oscilloscope. determine its its frequency response using a The same arrangement arrangementisisused usedwith withboth bothsensors sensorstoto determine frequency response using asignal signalgenerator, generator,and andlater latertotoinject injectcalibrator calibratorPD PDsignals signalstotocheck checkeach eachofofthe thecharge chargeestimation estimation methods. This arrangement methods. arrangement is is similar similarto tothe theone onedescribed describedinin[16]. [16]. 5.2. HFCT_HG 5.2. Case Study: HFCT_HG This case study uses uses aa HFCT HFCT with withthe theparameters parametersdescribed describedininTable Table2.2.The Thefrequency frequencyresponse response of AccordingtotoTable Table3,3,the theintegration integrationtime timefor forthis thissensor sensorisis of the the sensor sensor is shown in Figure 4. According approximately 578 578ns. ns.Since Sincethe thesensor sensorhas hasa ashort shortintegration integration time, it decided is decided to use double approximately time, it is to use thethe double time time integral to estimate the charge. integral to estimate the charge. A detailed detailed Bode magnitude magnitude plot plot at at low low frequencies frequencies isis depicted depictedin in Figure Figure12. 12.The Thefrequency frequency response of the sensor 100 kHz. response sensor clearly clearly shows shows aa linear linearbehavior behaviorininthe thelow-frequency low-frequencyrange rangebelow below 100 kHz. The mutual mutual inductance inductance is The is therefore therefore calculated calculatedas asthe theslope slopeofofthe themagnitude magnitudefrequency frequencyresponse, response, −5 −5 accounting for 15/(2π·100 15/(2πβ100 × × 10 H.H. accounting 1033))≈≈2.38 2.38××1010 approximately 578 ns. Since the sensor has a short integration time, it is decided to use the double time integral to estimate the charge. A detailed Bode magnitude plot at low frequencies is depicted in Figure 12. The frequency response of the sensor clearly shows a linear behavior in the low-frequency range below 100 kHz. The mutual inductance is therefore calculated as the slope of the magnitude frequency response, Sensors 2020, 20, 2520 12 of 16 accounting for 15/(2πβ100 × 103) ≈ 2.38 × 10−5 H. Figure12. 12. HFCT_HG HFCT_HG sensor sensor low-frequency low-frequency magnitude magnituderesponse. response. Figure A been injected to validate the charge estimation method. The injected pulse Acalibration calibrationpulse pulsehas has been injected to validate the charge estimation method. The injected is shown in Figure 13a. Figure 13b shows the measured voltage. Figure 13c clearly shows that that the pulse is shown in Figure 13a. Figure 13b shows the measured voltage. Figure 13c clearly shows charge estimation method using thethe double time integral and the the charge estimation method using double time integral and themutual mutualinductance inductanceisisaasuitable suitable charge method for charge estimation method forthis thisHFCT HFCTsensor. sensor. Sensorsestimation 2020, 20, x FOR PEER REVIEW 12 of 16 Figure 13.13. HFCT_HG: (a)(a) injected calibrator current pulse; (b)(b) measured pulse; (c)(c) charge estimation Figure HFCT_HG: injected calibrator current pulse; measured pulse; charge estimation using double integral. using double integral. 5.3. Case Study: HFCT_LG 5.3. Case Study: HFCT_LG This case study uses a HFCT with the parameters described in Table 1. The frequency response This case study uses a HFCT with the parameters described in Table 1. The frequency response of the sensor is shown in Figure 3. According to Table 3, the integration time for this sensor is of the sensor is shown in Figure 3. According to Table 3, the integration time for this sensor is approximately 9.4 µs. Since the sensor has a long integration time but a flat and broadband frequency approximately 9.4 μs. Since the sensor has a long integration time but a flat and broadband frequency response, it is decided to estimate the charge using the sensor gain and the first integral of the measured response, it is decided to estimate the charge using the sensor gain and the first integral of the voltage. In this case, the charge is evaluated as measured voltage. In this case, the charge is evaluated as π= π(π‘)ππ‘ ≈ 1 π» π’(π‘)ππ‘ (22) where Hmax is the sensor gain, equal to 10.3 mV/mA, and tzc1 and tzc2 are the u(t) zero-crossing times. Sensors 2020, 20, 2520 13 of 16 Z∞ i(t)dt ≈ Q= 0 1 Hmax Ztzc2 u(t)dt (22) tzc1 where Hmax is the sensor gain, equal to 10.3 mV/mA, and tzc1 and tzc2 are the u(t) zero-crossing times. The sensor has been used to measure a calibrator pulse. The injected charge is obtained by direct integration of the current through the oscilloscope input impedance. The injected charge is of 2022 pC. Figure 14b depicts the measured voltage pulse. As shown, the measured pulse shape is very accurate, with almost no pulse distortion. The measured pulse has an almost unnoticeable low-level and long-lasting pulse undershoot. This undershoot is responsible for the integral value decrease after tzc2 —see Figure 14c—which will finally reach zero. This behavior is due to the broadband and flat frequency response of the sensor. Moreover, the peak value of the integral occurs at nearly the same time as the current pulse has been extinguished. Figure 14c shows how the peak of the first time integral of the measured voltage approximated the injected charge with a small error, therefore validating the charge evaluation method for this sensor. In this case, the peak value of the time integral is equal to the integral value between the zero crossings ofSensors the measured voltage signal. 2020, 20, x FOR PEER REVIEW 13 of 16 Figure HFCT_LG: injected calibrator current pulse; measured pulse; charge estimation Figure 14.14. HFCT_LG: (a)(a) injected calibrator current pulse; (b)(b) measured pulse; (c)(c) charge estimation using peak integration. using peak integration. 6.6.Discussion DiscussionononHFCT HFCTSensor SensorDesign DesignConsiderations Considerations The Thedesign designchoices, choices,constructive constructiveaspects aspectsand andmagnetic magneticmaterial materialproperties propertiesofofa aHFCT HFCTsensor sensor determine and shape its frequency response. determine and shape its frequency response. According toto the simplified model of of a HFCT, when thethe lower cutoff frequency, f−3db_low , and the According the simplified model a HFCT, when lower cutoff frequency, f−3db_low , and the higher cutoff frequency, f , are separated enough, the maximum gain, H , and the cutoff higher cutoff frequency,−3db_high f−3db_high, are separated enough, the maximum gain, max Hmax, and the cutoff frequencies can bebe estimated by: frequencies can estimated by: R 1 RM π f−3db = 1 Hmax = π π f−3dblow = high π = π = π» 2πL2 2πRC L= 2 2ππΏ 2ππ πΆ πΏ (23) (23) As shown, for a given secondary inductance and capacitance (accounting for the added capacitor and parasitic capacitances), an increase in the load resistance increases the maximum gain, increases the lower cutoff frequency, and decreases the higher cutoff frequency. In practice, this is translated into a peaky frequency response that heavily distorts the pulse shape, since the frequency components of the current pulse are amplified and phase shifted at different levels. This phenomenon According to the simplified model of a HFCT, when the lower cutoff frequency, f−3db_low, and the higher cutoff frequency, f−3db_high, are separated enough, the maximum gain, Hmax, and the cutoff frequencies can be estimated by: Sensors 2020, 20, 2520 π = π 2ππΏ π = 1 2ππ πΆ π» = π π πΏ 14 of 16 (23) AsAsshown, shown,for fora agiven givensecondary secondaryinductance inductanceand andcapacitance capacitance(accounting (accountingfor forthe theadded addedcapacitor capacitor and parasitic capacitances), an increase in the load resistance increases the maximum gain, and parasitic capacitances), an increase in the load resistance increases the maximum gain,increases increases the lower cutoff frequency, and decreases the higher cutoff frequency. In practice, this is translated into the lower cutoff frequency, and decreases the higher cutoff frequency. In practice, this is translated a into peakya frequency response that heavily distorts the pulse shape, since the frequency components peaky frequency response that heavily distorts the pulse shape, since the frequency ofcomponents the current of pulse are amplified and phase shifted at different This phenomenon can be the current pulse are amplified and phase shifted atlevels. different levels. This phenomenon appreciated in the comparison of the frequency responsesresponses of the HFCT_LG and the HFCT_HG sensors can be appreciated in the comparison of the frequency of the HFCT_LG and the HFCT_HG shown in Figure 15. sensors shown in Figure 15. Figure 15. Comparison of Bode plots of the HFCT_LG and the HFCT_HG sensors. Due to the physical relationships between f−3db_low , f−3db_high , and Hmax , it is impossible to optimize one characteristic of the sensor without affecting the other ones. In practice, this means that a HFCT sensor with a flat and broadband frequency response will have a smaller Hmax gain than a HFCT sensor with a non-flat frequency response. Regarding partial discharge measurements, a bigger Hmax gain is an interesting property, since it increases the measured voltage peak values. Table 4 shows the ratio of the measured voltage peaks to the same current pulses when using the HFCT_LG and the HFCT_HG sensors. However, the higher peak voltages measured with the HFCT_HG sensor are at the expense of pulse shape accuracy, which jeopardizes the extraction of any pulse shape-related feature, such as rise time and tail time, or charge estimation using the first time integral of the measured voltage. This paper has demonstrated that one important current pulse feature—that is, the charge of the current pulse—can be properly estimated using the double time integral of the measured voltage, and, moreover, that the calibration constant can be obtained from the frequency response of the sensor. This statement is valid irrespective of the frequency response of the HFCT sensor, assuming that its transfer function can be modelled as a combination of real zeros and poles. Table 4. Ratio of peak values for the same current pulses using the HFCT_LG and HFCT_HG sensors. Charge (pC) Pulse Duration (ns) Current Pulse Shape Vpeak |HFCT_HG Vpeak |HFCT_LG 5 10 Triangular (Figure 7, Figure 8) ≈1.8 500 1000 Triangular (Figure 9, Figure 10) ≈5.5 2022 100 Calibrator (Figure 13, Figure 14) ≈4.1 From a practical point of view, this result leads to the conclusion that in partial discharge measurements where the pulse shape accuracy is not a relevant factor, the sensitivity of the measurement can be increased by designing a high-gain HFCT sensor. Along with the gain, the expected integration Sensors 2020, 20, 2520 15 of 16 times must be then considered in the design depending upon the partial discharge expected time duration. Current pulse durations can be in the tens of nanosecond range in compact partial discharge-measuring test setups or in GISs, and in the microsecond range in electrical machines or long cables. Therefore, design choices are needed depending upon the application. 7. Conclusions This paper shows a simplified and a generic model of HFCT sensors. Using the generic model, the principles of charge evaluation when using inductive sensors based on magnetically coupled inductors in partial discharge measurements have been explained. The mathematical analysis of the generic transfer function model reveals that, theoretically, the double time integral of the measured voltage is an exact method for charge estimation irrespective of the sensor frequency response and characteristics. Moreover, the calibration of the sensor for charge estimation purposes can be carried out using the Bode plots of the sensor. The practical limitations of this charge estimation method rely on the necessary integration times. Depending upon the application, the expected current pulse duration and the sensor, a suitable charge estimation method must be selected. The charge estimation method based on the first integral of the measured voltage proves to be a suitable method when the current pulse waveform is accurately measured. This is normally the case when a HFCT sensor with a flat and broadband frequency response is used. In this case, any other pulse shape-related features, such as rise time and tail time, can also be properly estimated. The charge estimation method based on the evaluation of the double time integral of the measured voltage is a suitable method when the sensor has an expected short integration time. Since both methods are based on integration, it is important to determine the integration limits to minimize the contribution of signal offsets and noise in the charge evaluation. Therefore, it is recommended to use the voltage zero-crossing points as integration limits. Author Contributions: Conceptualization, methodology, validation and formal analysis, A.R.-M.; draft preparation, review and editing, F.A.M. and L.C.C.-H. 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