# Principles of Charge Estimation Methods Using High-Frequency Current Transformer Sensors in Partial Discharge Measurements

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Equivalent Model of HFCT Sensors

#### 2.1. Simplified HFCT Model

_{1}is the self-inductance of the primary coil, L

_{2}is the self-inductance of the secondary coil, M is the mutual inductance, C represents the parasitic capacitance of the secondary coil and R is a loading resistor.

_{1}·L

_{2}) and k the coupling coefficient.

_{2}/C), the transfer function of the sensor, as modelled by Equation (1), can be rewritten as a generic transfer function with real zeros and poles as follows.

_{1}and p

_{2}are poles representing the lower and upper cutoff frequencies of the sensor, and α is a constant.

_{o}and accounts for H

_{max}, as described in the following equations.

_{1}and p

_{2}, 2π∙67 × 10

^{3}rad/s and 2π∙67 × 10

^{6}rad/s, respectively, are estimated. The poles correspond to the lower and upper cutoff frequencies, 67 kHz and 67 MHz, correspondingly. The value of α is set to 4.34 × 10

^{9}. 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 the simulation, which validates the simplified model for this particular sensor.

#### 2.2. Generic HFCT Model

_{1}= 2π∙3.9 × 10

^{6}rad/s, z

_{2}= 2π∙130 × 10

^{6}rad/s, p

_{1}= 2π∙1.1 × 10

^{6}rad/s and p

_{2}= 2π∙9 × 10

^{6}rad/s.

## 3. Charge Estimation Theory

#### 3.1. Charge Estimation Theory Using the Simplified HFCT Model

_{int}, 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.

#### 3.2. Charge Estimation Theory Using the Generic HFCT Model

_{n}.

_{1}, which imposes the slowest dynamic. Assuming that the duration of the current pulse is shorter than the time constant determined by p

_{1}, the total charge Q could be approximated by

_{1}has been selected, since the first exponential decay has already decreased by 98% of its initial value at this moment in time. Therefore, 4/p

_{1}is a good estimation of the necessary integration time for current pulses with a pulse duration shorter than it.

_{0}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.

_{1}, then (p

_{i}+ j$\omega $) ≈ p

_{i}. Hence, Equation (17) can be simplified as follows

_{o}. Since the derivative of H($\omega $) when $\omega $ 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 90°.

_{0}matches the mutual inductance M. The mutual inductance can then be experimentally determined by the slope of the Bode magnitude plot in the low-frequency range, where the Bode phase plot is 90°. This result, obtained for a generic HFCT model, is in accordance with Equation (10), which was obtained for the simplified model using electrical parameters, thus demonstrating that the simplified model is a simple particular case of the generic model.

## 4. Charge Estimation Methods

_{1}, the integration time can then be approximated by 4/p

_{1}, p

_{1}being the first pole of the transfer function. An estimation of p

_{1}can be calculated using the sensor parameters as R/L

_{2}.

## 5. Test Measurements

#### 5.1. Sensor Characterization

#### 5.2. Case Study: HFCT_HG

^{3}) ≈ 2.38 × 10

^{−5}H.

#### 5.3. Case Study: HFCT_LG

_{max}is the sensor gain, equal to 10.3 mV/mA, and t

_{zc}

_{1}and t

_{zc}

_{2}are the u(t) zero-crossing times.

_{zc2}—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.

## 6. Discussion on HFCT Sensor Design Considerations

_{−3db_low}, and the higher cutoff frequency, f

_{−3db_high}, are separated enough, the maximum gain, H

_{max}, and the cutoff frequencies can be estimated by:

_{−3db_low}, f

_{−3db_high}, and H

_{max}, 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 H

_{max}gain than a HFCT sensor with a non-flat frequency response.

_{max}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.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**HFCT_HG sensor frequency response, measured and simulated, using the simplified model (one zero and two poles).

**Figure 6.**Equivalent circuit in the Laplace domain of the secondary coil of the magnetically coupled sensor.

**Figure 7.**HFCT_LG 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 pulse.

**Figure 9.**HFCT_LG sensor response and charge estimation of a 500 pC and 1000 ns triangular current pulse.

**Figure 10.**HFCT_HG sensor response and charge estimation of a 500 pC and 1000 ns triangular current pulse.

**Figure 13.**HFCT_HG: (

**a**) injected calibrator current pulse; (

**b**) measured pulse; (

**c**) charge estimation using double integral.

**Figure 14.**HFCT_LG: (

**a**) injected calibrator current pulse; (

**b**) measured pulse; (

**c**) charge estimation using peak integration.

Sensor | HFCT Sensor Parameters | |||
---|---|---|---|---|

L_{1} (μH) | L_{2} (μH) | C (pF) | R (Ω) | |

HFCT_LG | 5.25 | 130.64 | 28.26 | 55.61 |

Sensor | HFCT Sensor Parameters | |||
---|---|---|---|---|

L_{1} (μH) | L_{2} (μH) | C (pF) | R (Ω) | |

HFCT_HG | 4.90 | 127.14 | 26.65 | 996.4 |

**Table 3.**Estimation of integration times for charge calculation using the double time integral of the measured voltage.

Estimated integration times for current pulse durations <<< 4/p_{1} | HFCT_LG | HFCT_HG | ||

4/p_{1} | 4∙L_{2}/R | 4/p_{1} | 4∙L_{2}/R | |

9.5 µs | 9.4 µs | 578 ns | 510 ns |

Charge (pC) | Pulse Duration (ns) | Current Pulse Shape | $\frac{{{\mathbf{V}}_{\mathbf{p}\mathbf{e}\mathbf{a}\mathbf{k}}|}_{\mathbf{H}\mathbf{F}\mathbf{C}\mathbf{T}\_\mathbf{H}\mathbf{G}}}{{{\mathbf{V}}_{\mathbf{p}\mathbf{e}\mathbf{a}\mathbf{k}}|}_{\mathbf{H}\mathbf{F}\mathbf{C}\mathbf{T}\_\mathbf{L}\mathbf{G}}}$ |
---|---|---|---|

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 |

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**MDPI and ACS Style**

Rodrigo-Mor, A.; Muñoz, F.A.; Castro-Heredia, L.C.
Principles of Charge Estimation Methods Using High-Frequency Current Transformer Sensors in Partial Discharge Measurements. *Sensors* **2020**, *20*, 2520.
https://doi.org/10.3390/s20092520

**AMA Style**

Rodrigo-Mor A, Muñoz FA, Castro-Heredia LC.
Principles of Charge Estimation Methods Using High-Frequency Current Transformer Sensors in Partial Discharge Measurements. *Sensors*. 2020; 20(9):2520.
https://doi.org/10.3390/s20092520

**Chicago/Turabian Style**

Rodrigo-Mor, Armando, Fabio A. Muñoz, and Luis Carlos Castro-Heredia.
2020. "Principles of Charge Estimation Methods Using High-Frequency Current Transformer Sensors in Partial Discharge Measurements" *Sensors* 20, no. 9: 2520.
https://doi.org/10.3390/s20092520