# The Design and Characterization of a Prototype Wideband Voltage Sensor Based on a Resistive Divider

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Voltage Sensor

#### 2.1. Physical Design

_{1}and R

_{2}, of 25 MΩ, each one connected in series (Figure 1). The low-voltage (LV) resistive branch of the divider, r, of 50 kΩ is composed of four 200 kΩ resistors arranged in parallel in a coaxial configuration. Two blocks of four capacitors, with a rated capacitance of 202 pF each form two capacitances, C

_{p}, of 808 pF, connected in series The HV resistor is placed on the axis of the set of capacitors (Figure 1). The first block of capacitors is connected in parallel with the first resistor of the HV branch through an upper electrode and a central electrode. The second block of capacitors is connected between the central electrode and the metallic enclose (see Figure 1). The central electrode serves as mechanical support for the two capacitor blocks and the two resistors of the HV branch. The configuration is designed to achieve a voltage distribution along each HV resistor for higher frequencies as close as possible to the voltage distribution obtained for 50 Hz. The voltage distribution along the HV resistors was determined by FEM simulation for the frequency range from 50 Hz to 5 kHz (see Figure 2a). The central electrode is connected to joint point of both HV resistances. The set is in a steel–aluminum casing to achieve a good shielding. The LV branch is also arranged in an aluminum compartment, different to the HV branch, although sharing the same gas insulation. The SF6 gas at 0.2 MPa is used as an internal insulation to pass dielectric tests corresponding to the insulation level of 24 kV. A plug-in connector is used to be connected to the cable entry of the enclosed metal box.

#### 2.2. Simplified Electrical Model

_{1}and R

_{2}, is modeled through an ideal resistance, R, in parallel with a capacitance: C

_{s}for the first resistor R

_{1}and C

_{s}′ for the second one R

_{2}. The parallel capacitance C

_{s}and C

_{s}′ includes not only the stray capacitance of the resistor but also the capacitance between the end electrodes of each HV resistor, R

_{1}and R

_{2}. Consequently, a different value of these capacitances C

_{s}and C

_{s}′ associated to each HV resistor is expected. The LV branch is represented by an ideal resistor r and a parallel capacitance C

_{c}′ (see Figure 3a) in which two additional capacitive effects are included: the coaxial cable C

_{c}and of the digital recorder C

_{r}(20 pF). In practice, the impedance of the recorder is considered a resistance of 1 MΩ, r*, that must be added to the value of the LV resistance. The capacitances, C

_{e}

_{1}, C

_{e}

_{2p}, and C

_{e}′, represent the earth capacitive effect between the metallic enveloping and the upper electrode, the central electrode, and the LV resistor respectively. The C

_{e}

_{2p}also includes any small difference between the first and the second capacitor blocks.

_{c}' + C

_{e}′)) with the second block (R//C

_{s}′) of the HV branch form the first RC divider, whose equivalent circuit is composed of two parallel impedances Z

_{eq}

_{1}and Z

_{eq}

_{2}shown in Figure 3b.

_{eq}

_{1}and Z

_{eq}

_{2}become R

_{eq}and C

_{eq}:

_{e}

_{2}and C

_{e}′ became a part of the total parallel capacitances of the circuit shown in Figure 3c. An appropriate selection of the capacities C

_{e}

_{2p}, C

_{e}′, C

_{s}, C

_{s}′, and C

_{c}is required to compensate the ratio and phase displacement errors of the divider. An improved model is shown in Figure 3d, in which the main difference from the simplified one is that the second resistor R of the HV branch is split into two parts and a stray capacitance is associated with each one. In addition, a parallel leakage resistance is introduced in each capacitor block to represent its insulation resistance. The behavior of this improved model is explained in detail by simulation in Section 4.

_{nd}(s = 0) = 1) by the following one:

_{s}and C

_{s}′, must be designed to comply with Equation (13), taking into account Equation (6):

_{c}is the capacitance per length unit of the coaxial cable of 66 pF/m, and C

_{r}is the capacitance of the digital recorder.

## 3. Frequency Response Analysis

#### 3.1. Frequency Response Analysis Using the Simplified Model

_{c}value. It can be slightly different from the theoretical value that satisfies Equations (10) and (11). The theoretical ratio error and the phase displacement error caused by the length of the coaxial cable can be determined using the transfer function, i.e., Equation (9), derived from the simplified circuit. In this simplified circuit, the theoretical maximum ratio error caused by the cable length of the coaxial cable can be determined through that equation for s → ∞:

_{r}and C

_{r}) are collected from the manufacturer’s data sheet. Other parameters were estimated by modeling in order to achieve the best fitting to the actual measured in the laboratory. All parameter values of the built HV sensor are shown in Table 1.

_{s}values referred to a percentage of C

_{p}. It justifies the change of the ratio error and angle error for lower frequencies due to changes of the C

_{s}value regarding the reference value given by Equation (11).

#### 3.2. Measuring the Response Frequency of the Built Sensor

_{s}= 2.2 pF (measured) when it should be 14.3 pF to satisfy Equation (11). An improved design of the upper electrode and the central electrode would permit a reduction in this discrepancy in order to obtain a class of 0.2 from 1 Hz to 5 kHz with the same divider ratio value, as is shown in Figure 5a.

_{c}values (different cable lengths) in the LCOE calibration laboratory in order to check Equation (9) of the simplified model. It was also observed that, for lengths of the coaxial cable larger than the reference value (97 cm), the tendency of the ratio error and the angle error is negative for higher frequencies (>200 kHz) according to the simplified model, but a preliminary oscillation is observed in the interest frequency range (2–100 kHz). This effect is not detectable by the simplified model. For this reason, the improved model shown in Figure 3d is introduced in Section 4.

_{c}values analyzed (85.0–101.1% C

_{c}).

## 4. Improved Electrical Model

_{s}′′ and C

_{s}′′′ in parallel depending on the geometrical location between the resistance and the central electrode (see Figure 2a), which it is considered in the PSPICE circuit shown in Figure 9 by C

_{sp}

_{1}(C

_{s}′′) and C

_{sp}

_{2}(C

_{s}′′′), respectively. For the built sensor, an equivalent capacitance of C

_{s}′′ is in parallel with the first part of the resistance part k·R and another capacitance of C

_{s}′′′ is in parallel with the other resistance part (1 − k) · R. The values of the coefficient k (0.96) and of the capacitances C

_{s}′′ (0.20 pF) and C

_{s}′′′ (0.87 pF) have been determined by an iterative process by means of circuit analysis and synthesis using PSPICE modeling and MATLAB in order to fit the theoretical curve of the frequency response to the real one measured in the laboratory. Furthermore, two additional resistors (R

_{sp}and R

_{e}

_{2p}) that mainly represent leakage resistances of both capacitor blocks are also added to simulate in a better way the real behavior of the built divider. The R

_{sp}magnitude also includes any difference between the 1st and the 2nd HV resistances. The improved model achieves a good fitting to the measured frequency response (see Figure 10) for the set values of the parameters of Table 1, while the simplified model does not fit the high frequency range (Figure 8). The frequency response measured in the built divider with a small change in the emplacement of the 2nd resistance is also included in Figure 10. The emplacement change moved vertically 2 mm from the relative position of the 2nd resistance regarding the central electrode. A significant influence in the frequency response curve can be observed, which justifies the inclusion of stray capacitances C

_{s}′′ and C

_{s}′′′ in the improved circuit model shown in Figure 3d and in its simulate PSPICE circuit shown in Figure 9.

## 5. High-Voltage Calibration and Insulation Testing

#### 5.1. Ratio and Angle Errors

#### 5.2. Insulation Tests

_{peak}/√2) for a minute was applied without any breakdown (see Figure 12).

## 6. Conclusions

_{s}, of the first HV resistor, R, is increased. The insulation tests demonstrate that the built sensor can be used in power grids up to 24 kV.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Voltage divider for MV switchgear under metal enclosure: (

**a**) The cut-away view of the voltage divider enclosed in a metallic enveloping; (

**b**) schematic electrical circuit; (

**c**) general view of the HV divider sensor to be connected in a MV cabin.

**Figure 2.**Voltage distribution: (

**a**) Cut-away view of equipotential lines in the voltage divider; (

**b**) voltage distribution along the two resistors of the HV branch for frequency range 50 Hz–5 kHz.

**Figure 3.**Equivalent electric schemes: (

**a**) simplified divider model; (

**b**) equivalent circuit replacing the 1st RC divider by two impedances in parallel Z

_{eq}

_{1}and Z

_{eq}

_{2}; (

**c**) equivalent model 2nd RC divider; (

**d**) improved divider model analyzed in Section 4.

**Figure 4.**Errors due to difference of length of coaxial cable C

_{c}regarding to the theoretical value regarding the simplified model of Figure 3: (

**a**) ratio error; (

**b**) phase displacement error.

**Figure 5.**Errors due to different C

_{s}values using the simplified model of Figure 3: (

**a**) ratio error; (

**b**) phase displacement error.

**Figure 6.**Measured errors in the real design of Figure 1 due to the difference in the length of coaxial cable C

_{c}with respect to the theoretical value: (

**a**) ratio error; (

**b**) phase displacement error.

**Figure 7.**Measured errors of the built HV sensor in the low frequency range for difference lengths of the coaxial cable C

_{c}: (

**a**) ratio error; (

**b**) phase displacement error.

**Figure 8.**Frequency response curves: the red curve is obtained via Equation (9) and the blue curve was measured in the LCOE laboratory: (

**a**) ratio error; (

**b**) phase displacement error.

**Figure 9.**Improved model of the sensor: (

**a**) equivalent electric circuit; (

**b**) PSPICE model to fit the frequency response to the measured one.

**Figure 10.**Frequency response curves measured with the best emplacement of the 2nd HV resistance (blue curve), with a vertical displacement of 2 mm (red curve), and obtained by PSPICE simulating the frequency response of best emplacement of the 2nd HV resistance (green curve) using the coefficient k = 0.96 to the frequency response measured: (

**a**) ratio error; (

**b**) phase displacement error.

**Figure 11.**(

**a**) Ratio errors for HV calibration from 1 to 14 kV, 50 Hz at 20 °C and 40 °C, (

**b**) ratio errors for HV calibration from 11 to 14 kV, 60 Hz at 20 °C.

**Figure 12.**(

**a**) Testing layout of the voltage divider prototype connected to a medium voltage cable, (

**b**) negative lightning impulse (125 kV) applied during the withstand test.

Parameter | Primary Value | Determined By | Simplified Model Figure 3b | Simplified Model Figure 3b | Improved Model Figure 3d |
---|---|---|---|---|---|

R | 25 MΩ | Measurements | 25 MΩ | 25 MΩ | 25 MΩ |

r | 50 kΩ | Measurements | - | - | - |

Z_{r} | 1 MΩ | Data Sheet | - | - | - |

r* = r·Z_{r}/(r + Z_{r}) | - | Derived | 47.6 kΩ | 47.6 kΩ | 47.6 kΩ |

C_{p} | 809.2 pF | Measurements | 809.2 pF | 809.2 pF | 809.2 pF |

C_{c} ^{(1)} | 63.8 pF | Measurements | - | - | - |

C_{r} | 20.0 pF | Data sheet | - | - | - |

C_{c}′ = C_{c} + C_{r} | 83.8 pF | Derived | - | - | - |

C_{e}′ | 12.1 pF | Measurements | 12.1 pF | 12.1 pF | 12.1 pF |

C_{c}′′ = C_{c}′+ C_{e}′ | - | Derived | 95.9 pF | 95.9 pF | 95.9 pF |

C_{e}_{1} ^{(3)} | 29.5 pF | Measurements | - | - | 29.5 pF |

C_{e}_{2p} | 12.5 pF | Measurements | 12.5 pF | 12.5 pF | 12.5 pF |

C_{s} | 2.2 pF | Measurements | 2.2 pF ^{(2)} | 2.2 pF ^{(2)} | 2.2 pF ^{(2)} |

C_{s}′ | 0.18 pF | Modeling | 0.18 pF | 0.18 pF | - |

C_{s}′′/C_{s}′′′ | - | Modeling | - | - | 0.20/0.87 pF |

R_{equ} | - | Derived (5) | 25 048 kΩ | 25 048 kΩ | - |

C_{equ} | - | Derived (6) | 0.15 pF | - | - |

C_{t} | - | Derived (12) | 811.4 pF | - | |

C_{t’} | - | Derived (12) | - | 821.9 pF | - |

R_{s} | - | Modeling | - | - | 865 MΩ |

R_{e}_{2} | - | Modeling | - | - | 658 MΩ |

k | - | Modeling | - | - | 0.96 |

^{(1)}This capacitance corresponds to a length of the coaxial cable 97 cm (66 pF/m).

^{(2)}The measured value is different to the reference value 14.3 pF given by Equations (11) and (12).

^{(3)}The capacitance difference between the magnitudes of the 2nd capacitor block C

_{p}and the 1st one is included in the C

_{e}

_{2p}value.

Restriction of 1st RC Divisor | Equation (1) R·C_{s}′ = r*·(C_{c}′ + C_{e}′) | Restriction of 2nd RC Divisor | R·C_{t} = R_{eq}·C_{t}′ |
---|---|---|---|

R·C_{s}′ (kΩ·pF) | 4545 | R·C_{t} (kΩ·pF) | 20,284 |

r*·(C_{c}' + C_{e}′) (kΩ·pF) | 4568 | R_{eq}·C_{t}' (kΩ·pF) | 20,587 |

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**MDPI and ACS Style**

Garnacho, F.; Khamlichi, A.; Rovira, J.
The Design and Characterization of a Prototype Wideband Voltage Sensor Based on a Resistive Divider. *Sensors* **2017**, *17*, 2657.
https://doi.org/10.3390/s17112657

**AMA Style**

Garnacho F, Khamlichi A, Rovira J.
The Design and Characterization of a Prototype Wideband Voltage Sensor Based on a Resistive Divider. *Sensors*. 2017; 17(11):2657.
https://doi.org/10.3390/s17112657

**Chicago/Turabian Style**

Garnacho, Fernando, Abderrahim Khamlichi, and Jorge Rovira.
2017. "The Design and Characterization of a Prototype Wideband Voltage Sensor Based on a Resistive Divider" *Sensors* 17, no. 11: 2657.
https://doi.org/10.3390/s17112657