Relative permittivity is a fundamental material parameter which affects the propagation of electrical fields.
The relative permittivity is always greater than or equal to 1. Permittivity is a measure of how much the molecules oppose an external electric field. Thus, the relative permittivity is a measure of reduction in electric field compared to if the electric field would propagate in vacuum [18]. Higher relative permittivity means more reduction in electric field. In dielectrics such as insulating transformer oil, the permittivity is often considered complex [19]. With the real part being related to the stored energy within the material.
The imaginary part relates to the loss of energy in the material. However, in this study the real part is mainly investigated as the loss in the material is indirectly investigated using the tangent delta.
6. Experimental setup & hardware
6.1 IDA 200 Insulation Diagnostic System
IDA 200 is a system used in measurement and analysis of insulating material [20]. By applying a relatively low voltage, it is possible to acquire parameters like resistivity and permittivity of the insulating material.
The only parameters IDA directly measures are the load voltage and current. From this the various param-eters are calculated. IDA is useful for many applications and uses different models to do the calculations depending on the circumstance. The different impedance models used in this thesis and corresponding pa-rameters are shown in table6.1.
Impedance model Parameter Resistive , ρ, σ Tangent Delta C, tanδ, P F
Table 6.1: IDA 200: Impedance models and corresponding parameters
In the field of transformer oils, the dielectric and resistive properties are of interest. These include, resis-tivity, loss factor and permittivity. While IDA 200 has more impedance models available. To measure said parameters the resistive and tangent delta models are enough. Since insulation diagnostics is based on ma-terial characterization, the geometry of the measuring cell is relevant when calculating mama-terial parameters from the measured current and voltage [20]. In other words, before any measurement is done on the oil the measuring cell has to get its geometric capacitance defined. This is done by doing measurement when only air (or vacuum) is between the electrodes. Since no “material” is between the electrodes, the capacitance of the sample is the geometric capacitance of the electrodes. The material is then inserted between the electrodes and this will influence the current that passes through. This influence is then used in the various models to do the calculations.
6.1.1 Equivalent circuit
Figure 6.1: IDA 200: Simplified circuit
When a voltage is applied to the electrodes the ions in the oil start to drift to their respective counterpart, negatively charged ions drift to the positive electrode and vice versa. This brings up a few issues. Firstly, to do a measurement an applied voltage is required. But applied voltage will cause stress to the system thus not keeping it in ideal thermodynamic equilibrium. A state which the oil is assumed to be in when extrapolating the measured data to use in simulations via the ion drift model. Further, due to the fact that the ions in the oil will start drifting apart under DC stress, the measurement could be affected. Because of this, it is recommended to apply a low AC voltage with very low frequency as the voltage source. The low peak voltage will not affect the equilibrium of the oil in any significant way and applying an AC voltage will keep the ions in the oil from drifting apart (since the positive and negative electrode switch polarity regularly). For most of the experiments related to this study, measurements are done at 1-2 V (rms) and a frequency sweep from 1 kHz to 1 mHz. The measured parameter to be used in future simulations would be the data point at the lowest frequency, 1 mHz, as it best resembles DC.
6.1.2 Sine correlation technique
According to Peter Werelius, the inventor of the diagnostic tool [21], IDA uses the sine correlation technique to achieve a complex representation of the current and voltage. Both input voltages (one representing the voltage, another the current) are multiplied by a sine and cosine respectively and then averaged over an integer multiple (N) of the interval of time (T). The sine, cosine and the applied voltage have the same exact frequency.
Consider channel zero to be the voltage measurement. The real and imaginary part of the voltage are calculated according to equations6.1and6.2.
Re(Ch0P eak) = 2
Since impedance requires both current and voltage to be calculated. A second channel is used. The layout of the sine correlation implementation is shown below.
Figure 6.2: IDA 200: Sine correlation layout
The result is a complex voltage (Ch 0) and a complex current (Ch 1) both with a phase referring to the internal sine wave generator. Calculating the impedance by ohm’s law Z=U/I means that the phase of the impedance will be [φ(U ) + φinternal] - [φ(I) + φinternal]= φ(U )-φ(I).
Thus the impedance is represented as a complex number (or equivalent amplitude and phase) with the real part being the resistance and the imaginary part being the reactance.
Z = R + iXc (6.3)
|Z| =p
R2+ (iXc)2 (6.4)
φ(Z) = arctanXc
R (6.5)
Since the measured current can be quite small (nA) as a consequence of the low applied voltage (to measure the equilibrium resistivity) and the high resistivity of the oils. Noise is of course an issue. This issue is greatly reduced by the integrating and averaging part of the sine correlation technique.
6.1.3 Calculations of parameters
Capacitance
IDA considers the insulating material to have its capacitance modelled in parallel to the resistance. The capacitance is calculated from the impedance as follows
C = Re( 1
jωZ) (6.6)
The geometric capacitance, as mentioned above in section 6.1, is calculated using this equation when no material (or air) is between the electrodes. Capacitance is measured in Farad.
Tangent delta
IDA calculates the loss factor using the following equation:
tanδ = −Re(Z)
Im(Z) (6.7)
The parameter is unitless. Higher value means worse insulator.
Resistivity
IDA measures the resistivity in ohm meter [Ωm], the value is calculated from the geometric capacitance together with the measured impedance. IDA calculates the resistivity using the following equation:
ρ =C0
0
1
Re(Z1) (6.8)
with 0 being the permittivity in vacuum at 8.854E-12[F/m]. The equation is derived from the more com-monly seen formulas for resistance and capacitance
R = ρd
Since the geometry of the cell is taken into account when there is no material, or air, in between the elec-trodes. The relative permittivity equates to one and can be neglected as it is shown in equation6.8
The resistance is given by the real part of the impedance. The reason as to why IDA uses Re(11
Z) instead of Re(Z) could be to clarify that the impedance model considers the capacitance to be in parallel with the resistance. Thus the resistance is given by Re(Z1) and the total resistance is given by Re(11
Z) since in parallel circuits the total impedance is generally given by
Ztot= ( 1
Because the equation is derived from other equations it is good to confirm the validity of the equation by checking the SI-units.
The relative permittivity is calculated using the impedance and the geometric capacitance.
r= Re( 1
jωC0Z) (6.14)
Worth noting is that IDA considers the relative permittivity to be a complex entity. As mentioned previously, this is not uncommon when talking about dielectrics. In this thesis only the real part of the relative permittivity is interesting as the imaginary part relates to losses due to high frequency. Something not relevant to this study as the transformer oil is to be used in HVDC applications. Instead the dielectric losses are investigated using the tangent delta parameter.