#### 2.1. Physical Design

The voltage divider consists of a 50 MΩ HV resistive branch, composed of two HV film resistors,

R_{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

The simplified electric circuit of the double RC voltage divider is shown in

Figure 3a. No inductance effect is considered because the cylindrical configuration of the divider and its size lead to an inductance less than 1 μH, which does not affect the frequency operation range of the divider. Each resistor of the HV branch,

R_{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.

The circuit of

Figure 3a is a simplified circuit for a specific frequency range to be determined. In this circuit, the LV branch (

r*//

C_{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.

where

if the following condition is met:

Thus, both impedances

Z_{eq}_{1} and

Z_{eq}_{2} become

R_{eq} and

C_{eq}:

Consequently, the circuit of

Figure 3b leads to the circuit of

Figure 3c in the second RC divider, in which the first RC divider is therein. It justifies the name “double RC divider” that the authors have given to this voltage sensor. Using this design, the earth capacitances

C_{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.

The transfer function of the divider is given by the following formula:

For direct voltage (

s = 0) the transfer function is transformed in the following:

and the normalized Laplace transfer function (

G*_{nd} (

s = 0) = 1) by the following one:

Each RC divider should be designed to meet the following requirements:

Requirement of the 1st RC divider:

Requirement of the 2nd RC divider:

where

If both RC dividers meet these requirements, the normalized transfer function will be 1 p.u. The capacitances,

C_{s} and

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

and the length of the coaxial cable must be chosen to comply the following condition:

where

c_{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.