On the influence of humidity on the breakdown strength of air—with a case study on the PDIV of contacting enameled wire pairs

The partial discharge inception voltage (PDIV) in contacting enameled wire pairs exhibits a marked decrease with increased air humidity. While existing literature mentions several potential mechanisms for this reduction, a comprehensive quantitative assessment of the associated effects is lacking. This research paper addresses this knowledge gap by providing a quantitative estimation of the combined impact of water on the gas’s ionization yield (effective ionization coefficient) and the modification of the gap electric field caused by water absorption into the bulk of the insulating coating and the associated microscopic and macroscopic polarization processes (dielectric permittivity). However, a comparison of the theoretical predictions with experimental data reveals that these factors alone cannot fully account for the observed reduction in PDIV. Therefore, the study explores additional mechanisms mentioned in the literature, with particular focus on the development of a semi-conductive layer on the insulation coating in humid atmospheres. The numerical simulations of the surface charge dynamics within this layer suggests that the frequency-dependent decrease in PDIV under high-humidity atmospheres can indeed be attributed to the modification of the gap electric field due to the accumulation of surface charge in the semi-conductive layer.

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Introduction
Since the early work by Ritz in 1932 [1] it is known empirically that the breakdown strength of centimeter-sized, quasiuniform atmospheric pressure air gaps increases with the water vapor pressure p w (non-condensing, i.e. relative humidity RH = p w /p s (T) < 100%, see appendix A).Ritz observed a relative increase of about 4.5% for a d = 2 cm gap when p w was increased from 2 mmHg (RH = 8% at 25 • C) to 20 mmHg (RH = 84% at 25 • C).Moreover, he observed the relative increase of the electric strength to be less pronounced in smaller gaps.The basic trend of these early results-the increased breakdown strength of humid air in centimeter-sized uniform gaps and the gradual lessening of the effect with decreasing gap width-were confirmed and refined by a number of subsequent investigations [2][3][4][5].It was also recognized early on [4,6] that the increased electron attachment in humid air is likely a key factor for explaining the observed increase in the breakdown strength and its variation with gap width.The clarification of the effect of the polar H 2 O molecule on the electron energy distribution, the ensuing promotion of the dissociative attachment channels for O 2 and H 2 O, as well as the stabilization of the short-lived negative ions (O − , O − 2 , OH − ) by molecular water clusters emerged however only later (see e.g.[7][8][9]).In addition, the quantitative analysis of the complex interplay between electron impact ionization, (dissociative) electron attachment and detachment as well as ion conversion processes-with all the rate coefficients depending on the reduced electric field value-leads to the interesting consequence that the net electron avalanche growth rate can either be diminished ('low-field' regime) or enhanced ('high-field' regime) by the admixture of water molecules to air.A corresponding inversion point around a reduced electric field of ∼150 Td (∼4 kV mm −1 at 1 bar and 25 • C) has been identified both in measurement and numerical modeling [8,9].
In contrast to its effect in macro-gaps, the influence of (gaseous) water on the breakdown strength of quasi-uniform air micro-gaps (d ≪ 1 mm) is not well-studied empirically.No measurement data could be found on the effect of humidity on the breakdown voltage of such gaps at atmospheric pressure.Yet, micro-gap configurations are highly relevant in technical applications, such as in the inception of partial discharges (PDs) in the small air gaps in the turn-to-turn insulation of inverter-driven electric motor windings, as illustrated schematically in figure 12 (see e.g.[10,11]).The inception of PDs is the leading cause of premature insulation failures in these systems [12].The partial discharge inception voltage (PDIV) has been observed to drop significantly in air with increased humidity content [10,[13][14][15][16][17].While several hypotheses for this reduction in the PDIV have been put forward, a quantitative analysis of the expected relative contribution of the various hypothesized mechanisms is lacking.This is the research gap that is addressed in this paper.The contribution is of practical relevance because it provides strong evidence for identifying the major contributor to the observed decrease in the PDIV in humid air, and thus allows for targeted countermeasures.
The recent theoretical study by Li et al [9] uses a comprehensive ion kinetic scheme to calculate values of the effective ionization coefficient in humid air up to 300 Td.The availability of tabulated α eff (E, RH) data as a function of the electric field and the air relative humidity paves the way-under some further assumptions to be discussed-for quantitative predictions of the relative variation of the breakdown strength of ambient air of varying levels of humidity, and for gap widths ranging from the micro-to the macro-gap range.The derivation of analytic formulas using these input data and the comparison of the predicted variation of the breakdown strength with experimental data in the low-and high-field regime (macro-and micro-gap gap distances) is then one goal of this paper (sections 4. 1-4.3 and 4.4).Furthermore, the usefulness of the derived formulas is illustrated in a case study quantifying the expected effect of water in the gas phase on the inception of PDs in motor insulation.This case study highlights the most important results of this paper and is presented in section 5.

Relative humidity (RH), absolute humidity (ρ V ) and mole fraction of water (xw)
For a given (humid) air pressure p = 1 bar and temperature T, the electron swarm transport parameters depend on the mole fractions x i of the constituent gas molecules ( i x i = 1).In this paper, humid air is taken to be a N 2 -O 2 -H 2 O ternary mixture with x N2 /x O2 = 4.The mole fraction of water x w is varied within the limits typically encountered in ambient air, from antarctic ice deserts (x w ∼ 0%) to water-saturated hot air in the tropics (x w ∼ 4% to 6%).Often, the absolute humidity ρ V is used instead of the molar fraction.The two quantities are directly proportional, with a (slightly) temperaturedependent constant of proportionality (see equation (38) in appendix A), as illustrated in figure 1.The approximate value of 7 g m −3 water per 1% mole fraction may be used for temperatures between 20 • C to 40 • C. In contrast, the relative humidity is a strongly temperature-dependent quantity due to its dependence on the saturation vapor pressure, which is a strongly increasing function of temperature (see equations (39) and (40) in appendix A). Figure 1 shows, for instance, that x w = 2% in atmospheric pressure air at 20 • C corresponds to ρ V = 14.8 g m −3 and RH = 85.5%, while x w = 2% at 40 • C corresponds to ρ V = 13.8 g m −3 and RH = 27.1%.

Breakdown electric field in atmospheric pressure air
In this paper the air is assumed to be at atmospheric pressure (1 bar).Although the formulas derived in the following can be extended to consider pressure variations, this extension is not included here.Under these conditions, it will be illustrated in the following subsection that the effect of water molecules in air in a quasi-uniform field shows an opposing effect on the ionization dynamics: below an electric field strength of E x ≈ 4.15 kV mm −1 the ionization yield is reduced, while above it is increased.It is thus interesting to recall the breakdown electric field values E b of quasi-uniform air gaps (see e.g.[18]) in order to translate the crossover field strength E x into a corresponding crossover gap length d x , below/above which the ionization yield of an electron avalanche traversing the gap is decreased/increased. Figure 2 illustrates the three regimes.The indicated associated relative increase, approximate constancy or decrease of the breakdown voltage U b is derived in sections 4.1-4.3,respectively.It will be shown that the breakdown electric field in twisted enameled wire pairs with commonly encountered reduced coating thicknesses lies in the high-field range, and thus a certain reduction of the PDIV is expected in humid air.The magnitude of this reduction is quantified in section 5.

Effective ionization coefficient of humid air
The tabulated values of the effective ionization coefficient α eff (E, x w ) as a function of the electric field strength E for atmospheric pressure air at various levels of humidity x w were provided by Li et al (personal communication) as published in [9], though with an extended E/N range up to 300 Td instead of the upper limit of 200 Td used in the paper.They use the electron and ion kinetics of humid air within a spatial-temporal growth model to quantify the ionization dynamics in an electron avalanche.Figure 3 illustrates that the water molecules have opposite effects on the net ionization dynamics depending on the electric field strength.In the low-field region, that is, when the electric field strength is below the cross-over value E x ≈ 4.15 kV mm −1 , the effect of water molecules in the air is to decrease the effective ionization coefficient.This results The electric field in gas gap of twisted enameled wire pairs at PD inception lies in the indicated high-field region (also see figure 11).
in an increase of the critical electric field E crit with an increasing molar fraction x w of water in the air.The critical field is defined by α eff (E crit ) = 0, which means that below E crit attachment outweighs ionization (net reduction of the electron avalanche) whereas above E crit ionization outweighs attachment (net growth of the electron avalanche).The critical electric field scales with the air humidity as shown in figure 4. For the approximation derived in this paper, the following empirical parametrization is used: with E crit,0 = 2.43 kV mm −1 , b = 0.367 kV mm −1 and c = 35.86.The presence of water molecules in air increases both the ionization and the attachment coefficient.The increase of E crit (figure 4) as well as the decrease of α eff at low and moderate electric fields in humid air (figure 3, E < E x ) is due to the fact that in absolute terms, the increase in electron attachment and the formation of stable negative ions exceeds the increase in the ionization/detachment rate.Above the cross-over field, the dynamics tilts in favor of a larger net generation of electrons compared to dry air.The microphysical causes for these changes are beautifully intricate, and described in detail in [8,9].A brief outline is provided in appendix C.  Relative change of the effective ionization coefficient of atmospheric pressure air for different levels of air humidity for medium and large electric fields.In large fields, there is a relative increase which saturates to a value specified in figure 15.
From figure 5 it can be seen that the relative increase of α eff , saturates to a certain value r max (x w ) in large electric fields, which is displayed in figure 15 (appendix B).In the shown range 0 ⩽ x w ⩽ 5% the relation may be approximated by the second-order polynomial fit For electric field values exceeding the available electric field range of the tabulated data α eff (E, x w ) from Li et al (E > 7 kV mm −1 ), the extrapolation will be used, where α eff (E > 7 kV mm −1 ) for synthetic (dry) air is obtained from the Boltzmann solver BOLSIG+ [19] based on the cross section data of the SIGLO data base [20] (the detachment correction [21] can be neglected at these high fields).This amounts to assuming that the relative enhancement of the ionization coefficient in these large fields is equal to the saturated value observed in the mid-field region (see saturation in figure 5).The validity of this extrapolation required for the calculation of the breakdown voltage in micro-gaps smaller than about 300 µm (see figure 2) will be discussed in section 6.

Breakdown voltage of air with metallic electrodes in quasi-uniform electric field
While measured variations of the breakdown voltage of atmospheric pressure air in uniform electric fields at different humidity levels and macro-gap distances are available in the literature (e.g.[5], see figure 7), no such values could be found for the micro-gaps (10 µm < d < 1 mm).These values were thus measured within this work for comparison with the highfield model predictions.
A schematic of the measurement setup is shown in figure 6(a).The metallic electrodes are made from a coppertungsten alloy (50% Cu-50% W) for increased wear resistance.They are used in a quasi-uniform plane-sphere configuration (R = 10 mm).A 2 MΩ series resistor was used to limit the breakdown current and thus the electrode erosion due to multiple breakdowns.Comparison with results obtained with a 50 kΩ series resistor ensured that the measured breakdown voltage was not affected by the presence of the 2 MΩ resistor.Also, it was checked that the breakdown threshold reached a stable value after a few 'conditioning breakdowns' of slightly lower values, which probably removed surface imperfections or dust particles introduced during the gap setting procedure.The gap spacing was adjusted by inserting a precision gauge of the appropriate thickness in between the electrodes.
The relative humidity in the test chamber is continuously variable from <1% to about 80% at room temperature (25 • C) by a small purge flow of correspondingly conditioned ambient air.To this end, HEPA-filtered room air is divided into two air flows (RH 0%, RH 100%) and then mixed in the desired proportion to obtain the corresponding RH in the purge flow, as shown schematically in figure 6.The relative humidity and temperature were measured in the test cell with a Vaisala HMT310 sensor, which features a ±1% accuracy from 0% to 90% RH.
A low-pressure mercury vapor UV lamp with a peak irradiance at 254 nm was used to generate seed electrons, and hence to reduce the statistical time lag.The statistical time lag is the time between the time interval between the instant the applied voltage crosses the static breakdown threshold and the instant a seed electron becomes available.A temporary increase of the irradiance did not influence the value of the measured breakdown threshold, thus ensuring that the employed UV illumination indeed only influenced the statistical time lag.Although the statistical time lag is typically short enough in humid air (see section 6 and references therein), the UV illumination was used in all experiments to make sure that there is no unnecessary spread in the measured breakdown values due to a lack of seed electrons.
The DC breakdown threshold U b was determined for nine gap spacings between 20 µm and 1 mm by increasing the electrode voltage until a first breakdown was detected, then decreasing the voltage in steps of 2 V down to a voltage U b − 2 V, for which no breakdown could be detected for a time of at least one minute.
The largest contribution to the measurement uncertainty of the absolute value of the breakdown voltage U b originates from the uncertainty on the value of the actual gap spacing.As expected, the uncertainty is most pronounced in the smallest gaps.As an example, for d = 20 µm, when using twice the same precision gauge and gap setting procedure, the resulting absolute breakdown voltages were 462 V and 484 V (5% difference).In order to be able to measure relative changes in the breakdown voltage as a function of RH with a precision of ≲0.5% (see figures 9 and 10), it was thus instrumental to measure U b for a set gap spacing for the different RH values, such that the variation in the gap spacing procedure would not enter into the calculation of the relative variation of U b with RH.

PDIV: partial breakdown voltage of wedge-shaped, dielectric-bounded air gaps
The measurement results on the variation of the PDIV with relative humidity were recorded with the setup shown in figure 6(b).The wedge-shaped air gap is formed by contacting a plane and a semi-spherical (D = 2 • R = 10 mm) electrode coated with a (55 ± 3) µm layer of silicone polyester.Its dielectric permittivity equals 8.35 − j • 0.042 under dry conditions at 1 kHz and is only slightly dependent on RH and frequency such that the reduced coating thickness varies in the range 5.9 µm < s/ε r < 6.4 µm (at 50 Hz) and 6.1 µm < s/ε r < 6.4 µm (at 1 kHz) for 0% < RH < 80%.Although ε ′ ′ r increases with relative humidity, ε ′ ′ r /ε ′ r stays below 2.5% and hence ε ′ ′ r can be neglected for the calculation of the gap electric field.
The PDIV was determined under UV illumination in order to provide seed electrons (as explained in section 3.1).The PD current was measured with a high-frequency current transformer (HFCT, Tektronix CT-1) as well as a series capacitor (C series ≈ 1.7 nF).The detection threshold was around 5 pC.It is interesting to note that under 'dry' conditions (< 30% at 50 Hz and < 50% at 1 kHz), the onset of PDs is abrupt (PD charges well above the detection threshold), while under 'humid' conditions, the onset is gradual (PD pulses dis/appear into/from the noise floor).This observation is illustrated in figure 13 by the vertical lines under the corresponding data points, which indicate that the PDs do not physically disappear, but rather seem to slide below the detection threshold of ∼5 pC.

Results
An overview on the effect of humidity on the breakdown strength of atmospheric pressure air is provided in table 1.Three characteristic regimes are identified and described in the respective subsections.The results of the high-field regime Table 1.Overview of the effect of humidity on the electric breakdown strength of atmospheric pressure air for low, medium, and high electric field values. Approx.
will be illustrated in the case study quantifying the expected effect of water in the gas phase on the PDIV of twisted enameled wire pairs (section 5).

Low-field approximation
In this article, 'low' electric field strengths refer to values that do not significantly exceed the critical field strength.More precisely, it will be shown that the derived approximation is in good agreement with experimental results for gap widths d ≳ 50 mm, for which the breakdown electric field does not exceed the critical electric field by more than approximately 10%.According to equation ( 1), the effect of humidity on the effective ionization coefficient can-within a first order approximation-be attributed to its influence on the critical electric field strength, and the effective ionization coefficient may thus be written as within the low-field quadratic approximation.The limits of this simple parametrization will be discussed below.
In a quasi-uniform electrode configuration the breakdown electric field E b is defined by a Townsend-Schumann type criterion [18]: where K is the ionization threshold.From equations ( 5) and ( 6) it follows that The breakdown voltage is thus given by Let ∆U b be the absolute difference in the breakdown voltage in a humid atmosphere (water mole fraction x w > 0) vs. a dry atmosphere (x w = 0): Using equations ( 1) and ( 8), the relative change in the breakdown voltage (or breakdown electric field) of air in large quasiuniform gaps as a function of air humidity can be written as with b = 0.367 kV mm −1 and c = 35.86.Note that the ionization threshold K and the parameter a are assumed to be independent of the air humidity content.The latter assumption restricts the approximation to electric fields slightly above the critical field, and hence one may as well use the simplified asymptotic expression with E crit,0 = 2.43 kV mm −1 .Using the limit d → ∞ instead of d = 60 mm only leads to a relatively marginal (calculated) further increase of the humidity correction, as can be seen on figure 7.This justifies the use of the simpler equation ( 11) instead of (10) as a general expression for the impact of humidity on the breakdown strength of air for low electric fields/large gaps.Figure 7 also compares the predictions of equations ( 10) and (11) to experimental data from literature.The linear humidity correction factor given in IEC-60052 is also shown for comparison.

Mid-field approximation
As seen in figure 3, the calculated effective ionization coefficient of air at atmospheric pressure is relatively independent of the amount of water in the air at electric field values around E x ≈ 4.15 kV mm −1 (the only constraint being x w ≲ 10% ≪ 1).This implies that the average net number of new electrons created by a drifting electron is not significantly altered by the amount of water present in the air, and the breakdown voltage should thus not be affected by humidity in the gas phase:

High-field approximation
Above the cross-over field strength E x , the effective ionization coefficient of dry, atmospheric pressure air can be parameterized by Only the value of the parameter E th will be required in this section, and it can be obtained by the condition In the following derivation, it will be assumed that E > 6.5 kV mm −1 (d < 0.25 mm), which means that α eff (x w ) ≈ r max (x w ) • α eff (x w = 0) can be used (see figure 5).Since E th = 26.2kV mm −1 > 6.5 kV mm −1 it follows that E th is not expected to be affected by the humidity level, because the factor r max (x w ) cancels out in the defining equation (14).
Let E b and E b (x w ) = E b + ∆E b be the breakdown field in dry and humid air, respectively.By formulating the Townsend-Schumann criterion (equation ( 6)) for dry and humid air, and subtracting the two equations so obtained, the following relation between the corresponding breakdown strengths can be derived: Assuming the correction to be small, ∆E b ≪ E b , the first term on the right can be developed as a Taylor expansion, which leads to The breakdown voltage is predicted to decrease proportionally to ln(1 + r max (x w )) ≈ r max (x w ) ≪ 1 and the breakdown electric field E b .Based on the latter dependency (see figure 2),  16).The hatched area for d < 10 µm is meant to indicate that the breakdown in these gaps at atmospheric pressure starts to be dominated by field emission of electrons (see e.g.[18]), and hence is outside the scope of this paper.
one would expect to see the most pronounced effects in the smallest gaps, which can indeed be observed in figure 8. Also, since the relationship between r max and x w is close to a direct proportionality (figure 15), one expects the breakdown voltage to roughly scale as U b ∝ −x w .

Measurement of RH and gap width dependence of breakdown voltage in mid-and high-field regime
Measurements of the breakdown voltage in atmospheric pressure air at various levels of humidity were performed to validate the above model in the mid-and high-field regime.To this end, the breakdown voltage in a quasi-uniform electrode configuration (R = 10 mm sphere-plane, W-Cu) was recorded for gap widths ranging from d = 20 µm to 1 mm.The measurement results are shown in figure 9.The model curve is based on equation ( 16) and only predicts satisfactorily the relative change in breakdown voltages if the gap distance in the small range from about 0.2 mm to 0.5 mm.For gaps smaller than 0.2 mm, the model increasingly underestimates the measured relative decrease in the breakdown voltage.
The inversion point for the breakdown voltage was observed to lie between 0.5 mm and 1 mm, which is only slightly lower than expected from the sole consideration of the ionization coefficient on the breakdown threshold (see figure 2).
The empirical evidence of a physical phenomenon occurring at low humidity levels, which is not yet accounted for in the presented model, is clearly visible in figure 10.The model curve here corresponds to a vertical cut at d = 30 µm in figure 8.While the model predicts well the quasi-linear decrease of the breakdown voltage above  16)) of the relative reduction of the PDIV as a function of gap width, when the relative humidity is increased from about 2% to 78%.It can be seen that in smaller gaps, there is an increasing discrepancy with measured values, which can be related to a significant drop of the breakdown voltage in the low-humidity range 1% → 20% (see figure 9), whose underlying physics is not captured by the presented model.RH = 20%, it does not predict the strong increase of the breakdown voltage observed when the RH is lowered from 20% to <1%.Under these conditions, the presence of water molecules exerts an influence on the breakdown voltage that scales strongly non-linearly with the H 2 O molar fraction.

Estimation of gas gap voltages and electric fields
Considering the twisted wire pair in the quasi-uniform field approximation has been shown to be a good approximation for PDIV calculations, unless the wire radius R is smaller than about 0.15 mm [18,23].The voltage across the gas gap at a position x (= ordinate of the field line start/end point) is then given by [18] which shows that only a fraction of the applied electrode voltage U el is available to drive the discharge in the gas gap.In low-loss dielectrics, such as the one employed in the present study, the complex part of the relative permittivity is negligible, such that the real part ε r ≈ ε ′ r ≡ ε r may be used.The average electric field along the field line in the gas then reads The exact field line arc length d may be approximated by the two line segments d ′ (see figure 12).The calculation of the segment length leads to the following two-segment approximation of the field line length: For x ≲ R + s, equation ( 19) can be considerably simplified to which geometrically corresponds to approximating the field line arc d by the vertical line d ′′ .For practical enameled wires geometries, s ≪ R, and the discharge location x PDIV is observed to satisfy the condition x PDIV ≲ R + s.Equation (20) will thus be used in the following.The electric field in the gas gap at location x is then obtained from equations ( 18) and ( 20):

Breakdown criterion: estimated PDIV and discharge location
For the gas gap at the location x to break down (partially), the Townsend-Schumann criterion (6) has to be fulfilled.As will be shown below, the discharging gap lengths for practical enameled wire geometries fall in the range 10 µm < d PDIV < 100 µm, which, according to figure 2, corresponds to electric fields in the gas gap of 10 kV mm −1 < E g < 40 kV mm −1 .
In this range, the high-field parametrization ( 13) can be used.The parameter α ∞ = 882 mm −1 is obtained by fitting equation ( 13) with E th = 26.2kV mm −1 to α eff data obtained in this high-field range from the Boltzmann solver BOLSIG+ [19] based on the cross section data of the SIGLO data base [20] (the detachment correction [21] can be neglected at these high fields).The ionization integral I thus reads Since I(0) = 0, lim x→∞ I(x) = 0 and I(x) > 0 ∀x ∈ (0, ∞), it follows that I(x) has a maximum value I max > 0 in the interval (0, ∞).By setting it follows (because K(x) ̸ = 0) that the ionization integral is maximal at a field line length of which is located at The PDIV is reached when I max reaches the ionization threshold K.For the range 10 µm < d < 1 mm, K = ln(γ −1 ) = − ln(2.5 • 10 −3 ) ≈ 6 may be used as a satisfactory approximate value of the ionization threshold, although there is a decrease towards a value of ∼4 when d changes from ∼25 µm to ∼10 µm [18].This trend can be taken into account by adding an iterative correction if d = PDIV/E th is smaller than ∼25 µm, where PDIV is the inception voltage calculated using K = 6.The electric field in the gas gap where the ionization yield is maximal (I max ) for a given electrode voltage U el is given by where E max = εrU el 2s is the maximum electric field in the gas (occurring at the triple point x → 0).The criterion for U el = PDIV now reads The inverse of the function e −x /x is given by W(1/x), where W is the (upper branch) of the product log function (also known as the Lambert function).In the programming language Python, the Lambert function lambertw is included in the SciPy library (from scipy.specialimport lambertw).It follows that and finally The electric field in the gas gap at the location of maximal electron amplification at an electrode voltage equal to the PDIV is obtained by inserting (29) into equation ( 26): By using U el = PDIV in equations ( 24) and ( 25), it follows that the partial breakdown at PDIV takes place at the location where the gas gap discharging at PDIV has a length of From equations (30), (32) and (33) the interesting relation follows, from which the discharge location and the discharging gap length may easily be estimated for a given PDIV without knowing any further details of the system (other than the wire radius R for x PDIV ).

Estimated effect of the water in the gas phase on the PDIV
In the large electric fields encountered in the gas gap of twisted enameled wire pairs (see figure 11), the effect of water is described by the saturated relative increase of the effective ionization coefficient (see figure 15 and equation ( 4)).With the high-field parametrization (13) used in the previous section, this amounts to a corresponding increase of the parameter The PDIV including the effect of water in the gas phase can thus be written as and is shown in figure 14 for two levels of humidity.The result shows that the PDIV indeed decreases due to presence of water molecules in the gas phase.In figure 13 the calculated relative decrease of the PDIV as a function of the air water content is displayed for different values of the reduced coating thickness of the enameled wire.As discussed in the following section 5, the slightly increased ionization yield in humid air cannot explain the measured decrease of the PDIV.

Case study: expected effect of water in the gas phase on the PDIV of contacting enameled wire pairs
After describing in detail the results on the derived approximations quantifying the variation of the electric strength of air as a function of its water vapor pressure for both macroand micro-gaps, an example using the new results on microgaps is presented here in order to illustrate its application to answer the open research question mentioned in the introduction and described in more detail in the following two paragraphs.
It is reported in the literature [10,[13][14][15][16][17] that the air humidity can reduce the PDIV of wedge-shaped gas gaps, as they occur for example in the turn-to-turn insulation of motor windings.A prototypical insulation system that is often used to model these systems in laboratory tests is a twisted pair of insulated copper wires as shown in figure 12 (also commonly known as 'magnet wires' due to their use in electromagnets).Although the measured dependencies of PDIV vs. humidity vary between investigators, reductions of the PDIV in these systems from a few % up to 40% are observed when the relative humidity is increased from dry to humid conditions (from ∼20% to ∼90% RH at 25 • C) [10,[13][14][15][16][17].The effect is thus of great practical relevance to avoid the inception of PDs during service, and only a thorough understanding of the underlying physical mechanisms will allow for targeted countermeasures when the humidity level cannot be controlled under operating conditions.
Various physical mechanisms have been suggested to explain the observed decrease of the PDIV in humid air, including the effect of the increased ionization yield due to the lower ionization potential of water molecules as compared to its average in dry air, the increase of the dielectric permittivity of the insulating coating, as well as the increase of its surface conductivity (see e.g.[13,14,16]).To our knowledge, however, a quantitative analysis of the relative importance of the suggested mechanisms is lacking.The goal of this section is thus to apply the model presented in the main part of the paper to the twisted enameled wire pair as shown in figure 12 in order to evaluate the expected variation of the PDIV as a function of the water vapor content of the air, and compare the predictions to measurement data.The effect of the change of the coating's permittivity in humid air is also taken into account, such that the comparison between model predictions and experimental results allows to quantify how much of the observed decrease of the PDIV in humid air may be attributed to the combined effect of the water in the gas phase as well the change in the coating's permittivity.The remaining 'unexplained' difference must then be attributed to other mechanisms, such as the above-mentioned increase in the surface conductivity.The evidence in support of the latter mechanism is discussed in section 6 and the underlying physical mechanism is studied numerically in appendix D.
In figure 13, the measured relative variation of the PDIV of the depicted electrode configuration (with R = 5 mm, s = (55 ± 3) µm) is shown for sinusoidal voltage excitations of 50 Hz and 1 kHz, respectively.The PDIVs are normalized to the IEC 60 058 standard humidity of 8.5 g m −3 .The influence of the insulation coating is determined solely by its reduced coating thickness s/ε r .The polymeric insulating coating employed in this study (silicone polyester) features a reduced coating thickness within the range indicated in figure 13 for the two excitation frequencies and changes of relative humidity from < 1% to 80% at 25 • C.
The comparison of the model and measurement data shows that the water in the gas phase leads to only a small decrease of the PDIV of approximatively 2% when the RH is increased from 20% to 80% at 25 • C, and hence cannot explain the significantly larger decrease which is observed experimentally.Also, for the employed insulation material (silicone polyester), the increase in the dielectric permittivity due to absorbed water leads to another decrease of the same magnitude (∼2%), see figure 13.Note that this contribution depends on the insulation material and thus can be more or less pronounced for different materials (and excitation frequencies).Comparing the combined effect of the increased ionization yield (∼2%) and the increased permittivity (∼2%) with the observed relative change of the PDIV (∼13% at 1 kHz and ∼30% at 50 Hz) leads to the conclusion that another physical mechanism is responsible for the bulk of the observed reduction in the PDIV  in humid air.From the data shown in figure 13 it is also visible that this mechanism is frequency-dependent, with a smaller decrease of the PDIV at the larger frequency.
As shown in section 4.3, the PDIV of contacting magnet wires can be parametrized explicitly as a function of the reduced coating thickness s/ε r by equation ( 30), in the practically relevant range of reduced coating thicknesses of 4 µm < s/ε r < 30 µm.The function W is the product-log (Lambert) function, e ≈ 2.72 is Euler's number, the parameter d PDIV is the length of the discharging gas gap at PDIV and γ ≈ 2.5 • 10 −3 is the secondary electron emission coefficient (number of secondary electrons per ionization event in the primary avalanche).The used constant value of γ is taken from [18] (figure 3), and corresponds to the plateau value of photo-feedback in the gap distance range from about 20 µm to 1 mm.The parameters α ∞ = 882 mm −1 and E th = 26.2kV mm −1 characterize the ionization coefficient in atmospheric pressure air according to the Townsend approximation It is interesting to note the parallels to Paschen's law, which for atmospheric pressure reads The equivalent of Paschen's law for dielectric-bounded, wedge-shaped gas gaps is seen to derive from Paschen's law by the substitutions d → 2s/ε r and ln(x) → W(e −1 • x).
According to equation (33), the discharge location on the horizontal axis is predicted to occur at where R is the radius of the curved electrode.For the employed silicone polyester coating, s/ε r is equal to 6.4 µm (at 50 Hz, RH = 20% and 25 • C), which leads to d PDIV = 29 µm and x PDIV = 0.38 mm (for R = 5 mm).The predicted PDIV is E th • d PDIV = 754 V, the measured PDIV is (735 ± 10) V, which amounts to an overestimation by 2.6%.
Note that the effect of water is included in the above equations only via its possible effect on the dielectric permittivity (this includes space charge polarization).Its effect on the electron kinetics could as well be included (see equation (35)), but the change in the predicted PDIV is within its measurement uncertainty, as illustrated in figure 14. Figure 14 compares the predictions of the derived parametrization (30) with a large number of measurements.The added measurement points were obtained in dry air with aluminum electrodes covered with the polymers silicone polyester (s/ε r = 6.7 µm, 1 kHz) and perfluoroalkoxy alkane (s/ε r = 32 µm, 1 kHz), respectively.The satisfying agreement between the model prediction and the measurements Predicted PDIV of twisted enameled wire pairs as a function of the reduced coating thickness compared to own measurements and measurements collected from literature.The calculated variation of the PDIV with air humidity is seen to be within scatter of measured values.Note that only the effect of water molecules on the ionization coefficient is included in the calculation for the shown model curves.Water can have a significant effect on the PDIV as shown in figure 13.The measurements shown here were presumably obtained in relatively dry atmospheres (≲ 8.5 g m −3 ), except possibly for the points that are systematically below the model prediction.
suggests that the literature measurements were performed under ambient conditions and/or materials for which the significant reduction of the PDIV under more humid conditions due to the above-identified main influencing factor is not yet activated.A possible explanation of the significant reduction of the PDIV in the more humid ambient air is given in section 6 and appendix D. Thus, to conclude this section, it is important to reiterate that the formulas presented in this section are expected to show gradually larger deviations to measured values in more humid atmospheres (RH≳ 40%).
A more detailed discussion of the results presented in the above case study is provided in the following section.

Discussion
The agreement between the derived approximate formulas and experimental results regarding the relative change of the breakdown strength in the low-, mid-, and high-field regime will be discussed first.Starting with low electric fields (E ≲ 4 kV mm −1 ) and large metal-bounded gaps (d ≳ 1 mm), figure 7 illustrates that the asymptotic (d → ∞) low-field approximation (11) agrees well with experimental data for a 60 mm gap.On the other hand, the non-asymptotic approximation (10) is not accurate in describing the transition towards smaller gaps.This can be understood by the restricted validity of the employed approximation (5) of the effective ionization coefficient: assuming the proportionality factor a to be independent of x w must restrict the validity of (5) to electric field above but close to E crit (x w ) (see figure 3).
The model prediction for the mid-field regime (around the inversion point at E ∼ 4 kV mm −1 , d ∼ 1 mm) is also in agreement with experimental results.The literature values shown in figure 7 confirm the expected decrease of ∆U b /U b towards the inversion point, while our own measurements additionally confirm the presence of an actual inversion of ∆U b /U b in the gap distance interval from 0.5 mm to 1 mm (see figure 10).
Literature data on the variation of the electric breakdown strength of micro-gaps in air between metallic electrodes at varying levels of humidity is scarce.In [24] a 100 µm air gap is investigated in ambient (i.e.humid) vs. dry air gap at 0.2 Torr • cm ≲ p • d ≲ 4 Torr • cm values (pressure: 30 mbar ≲ p ≲ 500 mbar).Their measured breakdown strength of ambient humid air (24 • C, 40% RH) is lower than for dry air, although the reduced electric field is clearly above the inversion point.No explanation can be inferred here about the reason for the contradicting results.
Before moving on to the PDIV of dielectric-coated electrodes, the observed large variation of the breakdown strength in the low-humidity range (0%-20%) warrants discussion.The effect is not predicted by the variation in the effective ionization coefficient with air humidity.The discrepancy increases for smaller gaps (d ≲ 0.1 mm) and larger electric field values (E ≳ 10 kV mm −1 ).With the present data, one can only speculate about the possible causes of the discrepancy.One possibility is that the assumed saturation of the relative increase of the effective ionization coefficient with the electric field strength (figure 5) does not hold.Alternatively, it is also possible that the assumed constancy of the ionization threshold K for the derivation of the approximation (8) does not apply.It is known that the field-emission of electrons from the cathode plays an increasingly important role as the gap width decreases (ion-enhanced field emission below a few tens of micrometers in atmospheric air [25,26]).The onset of field-emission depends on the work function of the cathode surface.The work function of metal oxides is lowered by the water layer (first layer chemisorbed by -OH termination, second layer hydrogen-bonded water, further layers physisorbed water mixed with small organic molecules) that forms on these surfaces in air [27].A qualitative hypothesis for explaining the relatively large drop in the breakdown strength of small metal-bounded air gaps by the addition of less than 0.5% of water (mole fraction) could thus be a lowering of the cathode work function by the presence of water, which lowers the electric field strength needed to extract electrons from the surface and hence increases the contribution of ion-enhanced field emission on the secondary electron emission coefficient.The process being a surface (and not a volume) effect would also explain the observed saturation of the drop after the addition of a small volume (mole) fraction of water.More research is required to test these hypotheses.
In the remainder of the discussion, the focus is on the partial breakdown in the dielectric-bounded air gaps (PDIV).First, it is astonishing to observe that the relative change of the PDIV in the low-humidity range (0%-20%) is opposite to the trend observed with metallic electrodes: the PDIV increases with increasing humidity in this regime.No conclusive explanation for this phenomenon can be offered here, except for drawing the qualitative but striking parallel to measurements of the secondary electron yield (SEY) of metal and insulator surfaces exposed to ambient (humid) air [28].'The SEY of air-exposed metal surfaces is generally higher than the SEY of the corresponding atomically clean metals.The SEY increase during air exposure is mainly caused by the adsorption of an organic surface contamination with embedded water molecules.'( [28], p 1091) and 'While the SEY of metals is increased, the SEY of insulating materials is reduced during air exposure.'( [28], p 1090).These statements are in line with the above picture of the chemi-/physisorbed water modifying the work function of metal oxides, and could explain why the relative change of the (partial) breakdown voltage goes in different directions for a metal (oxide) cathode and a polymeric cathode.Further tests are needed here as well to corroborate these explanatory approaches.
One conclusion from the theoretical part of this paper is that the impact of water in the gas phase is not expected to lead to a practically significant reduction of the PDIV.With regard to the effect of water absorbed into the polymer bulk, the impact on the PDIV is mainly determined by the ensuing change of the polymer's dielectric permittivity.Hereby, it is emphasized that space charge polarization due to enhanced bulk conductivity (long-range charge migration with accumulation at the electrode or gas interface) can significantly increase the gas electric field [29].For the polymer used in the present study (silicone polyester) the effect of the air humidity on the dielectric permittivity is however much too small to explain the observed reduction in the PDIV above 20% RH (25 • C).
In the literature, several explanations are put forward to explain the drop of the PDIV in twisted pairs with increasing air humidity in the >20% RH range (at ∼25 • C): (i) Increased ionization coefficient due to H 2 O's lower ionization potential as compared to the average in dry air (e.g.[14]) (ii) Increased dielectric permittivity of the coating (e.g.[15]) (iii) Field enhancement caused by condensed water on insulator surface [30] (iv) Larger abundance of starting electrons (e.g.[17]) (v) Increased surface conductivity of the coating (e.g.[14,16]) The present study suggests that there is indeed an increase of the effective ionization coefficient due to the presence of water molecules (by the physical mechanism described in appendix C), but that the corresponding reduction of the PDIV is typically smaller than the achievable precision of PDIV measurements (see figure 14).The effect of the dielectric permittivity may or may not contribute significantly to the reduction of the PDIV, depending on the material characteristicsin the present investigation, the increase of the permittivity in humid air could only explain a small fraction of the observed drop in the PDIV (see figure 13).The hypothesized occurrence of local field enhancements in [30] due to condensed water (droplet formation) on the insulator surface is not quite clear, because if the insulator surface is not cooler than the ambient air, no condensation of macroscopically relevant amounts of water ('droplets') should occur for the indicated conditions of RH < 95%.The reduced statistical time lag in humid air relating to the point (iv) above had been known for a long time for metallic electrodes (e.g.[31,32]), and is also observed with dielectric-bounded gaps (e.g.[33]).The detachment of electrons from the more abundant negative ions in humid air (also see appendix C) is a possible cause for the observed reduction of the statistical time lag [32].If seed electrons are not provided at a sufficient rate (thus increasing the statistical time lag t s ), the PDIV will indeed be overestimated by t s • ∆U/∆t, where ∆U/∆t is the rate of rise of the electrode voltage.The effect would be especially significant with fast transient voltages, such as occurring during inverter surges.
During the experiments of this study, the lack of seed electrons was noticeable in the low-humidity air, even at 50 Hz sinusoidal voltages (expected due to the small stressed air volume in the employed electrode geometry).This is why UV illumination was used to exclude the possibility that the larger measured PDIVs in low-humidity air could be due to a low provision of seed electrons.
Item (v) of the above list remains as a possible candidate of literature-compiled explanations for the reduction of the PDIV at larger RH.There is indeed strong experimental evidence from [16] that water adsorbed to an enamel surface 'contaminated' with ions of alkaline earth metals (N + , K + , Ca 2+ , . ..) leads to the formation of an electrolyte film, which increases the enamel's surface conductivity.When the enamel surface is initially cleaned with pure water (presumably removing the contaminants), no reduction of the PDIV with increased air humidity was observed in [16].In the present study, rinsing the polymer surface with distilled water also led to a significantly less pronounced decrease of the PDIV at larger RH.The effect of RH did not vanish completely, which may be due to an incomplete removal of the ionic contaminants (silicone polyester is hydrophobic and not easily wetted).
It is however not quite clear from the cited literature how the increased surface conductivity due to the electrolyte layer formed by water and ionic contaminants would lower the PDIV.If there is a semi-conductive layer on the insulating coating with a 'good' electrical contact between the wires, the expected effect would be a partial expulsion of the electric field from the gas gap (the field would be completely expelled with a metallized coating).Applying a semi-conductive layer (resistive paint with a surface resistance of R □ ∼ 3 kΩ) onto the coating has in fact been suggested as a means to increase the PDIV in the turn-to-turn insulation of enameled wire coils [34].Although the apparent surface resistance of enameled wire under humid conditions (80% RH at 25 • C) is several orders of magnitude larger, it is not initially clear how a semiconductive layer due to increased RH would reduce the gas electric field.An order of magnitude estimation of the surface resistance of an enameled wire (polyamide-imide overcoat) can be obtained from I-V-measurements provided in [14]: at 80% RH (25 • C), the apparent surface resistance is R □ ∼ 20 MΩ, while at 20% RH (25 • C) it is R □ > 200 GΩ, i.e. an increase of RH by 60% decreases the apparent surface resistance by a factor of >10 4 .
The authors in [35] consider a 2 µm thick 'moisture film' of electrical conductivity 1 mS m −1 (resulting in R □ = 500 MΩ) and show by numerical simulation that-as expected-the electric field in the air gap is reduced (hence larger PDIV) if the water films on the two enameled conductors touch at the mechanical contact of the wires.Charge can then flow from one conductor surface to the other, and hence reduce the gap electric field.In order to explain the decrease of the PDIV with the increased surface conductivity due to the moisture layer, the authors in [35] assume the layers to form initially only around the location where PDs occur in the absence of moisture, such that the two moisture layers have no electrical connection.In their theory, the water layers are a consequence of the PD activity, and presumably form after the 'pre-discharge' PDs have made the polymer surface more hydrophilic in a limited area which does not include the area around the wire contact points.Since here we are only interested in explaining the reduction of PDIV with increasing RH and decreasing frequency, a simpler model is presented and validated semiquantitatively in appendix D. It is based solely on the surface charge dynamics in the semi-conductive layer, which is assumed to form independently of the action of any prolonged pre-discharge (which was absent in the present investigation).It is argued that charge transfer between wire surfaces is not efficient on time scales of the excitation voltage period, and hence the gap electric field is increased by the presence of the semi-conductive layer.It is thus shown that the charge motion in the semi-conductive layer can produce reductions in the PDIV in humid atmospheres which are compatible with the significant reductions observed experimentally (see figure 13).

Conclusion
The research gap motivating this paper is the question of whether the increase in the effective ionization coefficient α eff with air humidity levels (molar fraction x w ) in large electric fields could explain the magnitude of the observed drop in the PDIV of contacting enameled wire pairs.Starting from recently published data of the parametrization α eff (x w ), approximation formulas describing the relative change of the breakdown voltage U b with air humidity x w of quasi-uniform air gaps are derived (at atmospheric pressure).Approximation formulas are derived to cover not only the regime where the breakdown voltage is predicted to decrease by the enhanced net ionization yield (E ≳ E x ≈ 4.2 kV mm −1 , d ≲ d x ≈ 1 mm, see equation ( 16)), but also the regime where the breakdown voltage is increased with increasing air humidity (E ≲ E x ≈ 4.2 kV mm −1 , d ≳ d x ≈ 1 mm, see equation (11)).The latter case has been extensively studied in previous literature, and the newly derived approximation formula aligns well with available literature data.
For the high-field regime (E > E x ), own measurements in metal-bounded air gaps confirm the decrease of the breakdown voltage with increasing humidity.The measured relative decrease is in line with the predictions of the approximation formula for relative humidity values above ∼20% (at 25 • C).The measurements indicate the presence of a high-field process which is strongly influenced by small amounts of air humidity, and not captured by the present model.
The application of the high-field approximation to the determination of the PDIV in wedge-shaped, dielectricbounded air gaps reveals that the impact of water in the gas phase cannot explain the magnitude of the measured reduction in the PDIV with increasing air humidity.The reduction caused by the enhanced effective ionization coefficient is on the same order of magnitude as the measurement uncertainty of the PDIV itself.Moreover, the coating material used in this study shows a relatively small increase of its dielectric permittivity with air humidity, such that it cannot explain the measured magnitude of the PDIV reduction either.The increased air humidity correlates with an increased surface conductivity of the coating, and it is shown by numerical modeling of the surface charge dynamics in the semi-conductive surface layer that a significant reduction of the PDIV can indeed be explained by this process.Further empirical studies are required to corroborate the surface charge theory.
The dominant processes for the increased attachment coefficient in the presence of water vapor are the following.Firstly, the efficient moderation of the electron energy by H 2 O's lowlying rotational and vibrational modes, as well as the more efficient elastic energy transfer due to H 2 O's smaller mass compared to O 2 and N 2 , leads to a reduced average electron energy up to a few 10 Td, which in turn increases the three-body attachment rate to molecular oxygen.This effect leads to a much more pronounced minimum of the net ionization coefficient in humid as compared to dry air (not shown in figure 3).Secondly-and more importantly for the effect of water on the critical field and the value of the net ionization coefficient above the critical field value-at moderate and large electric fields, the average electron energy in humid air is larger than in dry air.At moderate field values, between about 80 Td to 110 Td, the increase is large enough to open up the dissociative attachment channels for the O 2 and H 2 O molecules (which possess energy thresholds of a few eV [37,38]), yet still low enough to keep ionization at bay.Moreover, at these electric field values, the generated ions (O − , O − 2 and OH − ) are efficiently stabilized through hydration, i.e. the formation of hydrated negative ion clusters.These effects entail the overall decrease of the net ionization coefficient at low and moderate electric field values, as shown in figure 3.
As the electric field is further increased, the rate coefficients for dissociative attachment start to saturate, while those for impact ionization are strongly increasing.The increase of the ionization coefficient in humid air is on the one hand due to the lower ionization potential of H 2 O (12.6 eV) as compared to the average dry air molecule (∼0.8 • 15.6 eV + 0.2 • 12.1 eV = 14.9 eV), and on the other hand due to the abovementioned increase of the number of electrons with energies above the ionization thresholds.This increase is explained by the fact that H 2 O-as opposed to O 2 and N 2 -possesses a pronounced Ramsauer minimum around 2 eV, which increases the electrons' mean free path, and thus allows more electrons to reach the ionization threshold than in dry air (the water molecules become largely 'transparent' for electrons of a kinetic energy around 2 eV).Moreover, in larger electric fields, the ions acquire larger kinetic energies, which promotes collisional electron detachment and impedes ion hydration.In net terms, the electron avalanche growth rate starts to exceed the value in dry air when the electric field is above the cross-over value E x , as shown in figure 3.

Appendix D. Modeled influence of the surface conductivity on the PDIV
In the following, a finite element method (FEM) simulation is performed to quantify the effect of the charge dynamics in the surface layer of the dielectric coating on the electric field in the gas gap.Since the dielectric permittivity of the insulating coating varies by less than 2.5% from 10 Hz to 10 kHz, a constant average value of 8.3 will be assumed here.The thickness of the conductive surface layer is assumed to be negligible compared to the insulation coating thickness, i.e. in the range from some nanometers to some tens of nanometers.Then the surface resistance R □ fully characterizes the surface layer, and together with the frequency f of the sinusoidal excitation voltage they constitute the variable input parameters in the simulation (the exact functional variation R □ (RH) is not currently known for the employed insulation material).
For electric charges to move in the conductive surface layer, a tangential electric field must be present.Figure 16(a) shows the tangential field as a function of the excitation frequency for R □ = 300 MΩ and an electrode voltage of 500 V. Beyond f = 10 kHz, the curves saturate rapidly to the tangential electric field of a non-conductive surface layer, because the charges move a negligible distance during a half-period of the excitation voltage.This tangential field is present before the charges start drifting in the semi-conductive surface layer, i.e. the layer developing on the surface of the polymer coating due to the presence of air humidity.If the charges are given enough time to reach equilibrium conditions, that is, if the frequency of the external driving field is low enough, the tangential field approaches zero (quasi-static situation).
As expected from symmetry reasons, the tangential electric field vanishes at the contact point under all conditions.Depending on the contact pressure and the coating material's elastic modulus, the actual contact is a small circular area (see e.g.[39] appendix A).Nonetheless, the strong decrease of the tangential electric field towards the contact point is expected to make the contact area a bottleneck for charge transport.Hence it is 'difficult' for charges from within the conductive surface layer to reach the contact area, where they could potentially transition to the other coating to shield the electric field in the gap (as is observed with relatively conductive layers as described in section 6).This process of charge transfer between wire coatings is expected to be inefficient compared to charge migration along the surfaces.For these reasons, the shown model results do not consider charge transport between the coatings/semi-conductive layers, such that the gap electric field is always enhanced by the surface charge building up in the semi-conductive layer.Note that the surface charge referred to in this appendix originates by charge separation in the semi-conductive layer, and is not related to the charge deposited by the PD after inception.Taking into account the latter would be important when considering the evolution of the PD activity over prolonged times (e.g. up to 3 min in [35]).
Figure 16(c) illustrates how the calculated PDIV varies as a function of the excitation frequency.At the lowest frequencies, the electrode potential is fully transferred to the gap close to the contact point, such that the PDIV approaches the Paschen minimum.
From the simulation results of the air gap electric field (figure 16(b)), the presence of the relaxation phenomenon is also clearly visible.It is characterized by the relaxation frequency f rel , which is related to the surface resistance by It can be seen that the observed significant reduction of the PDIV (see figure 13) can be in principle explained by the mechanism of charge transport in the semi-conductive surface layer.For example, at R □ = 300 MΩ, the predicted PDIV thresholds are 370 V and 610 V for 50 Hz and 1 kHz, respectively.The measured thresholds were limited by the noise floor of the equipment (∼5 pC) and amount to <510 V and <650 V (at 70% RH).If the proposed theory is correct, the PDs indeed started significantly below the measurement threshold in the humid atmosphere.This was observed experimentally as a gradual disappearance of the PD signal in the noise floor.The calculation also predicts that the discharge inception location ('corona ring radius' [39]) is decreased from 0.45 mm to 0.1 mm under humid conditions (for f ≲ 100 Hz at R □ = 300 MΩ).It is illustrated by the shift of the indicated discharge locations (small circles) in the direction of the arrow in figures 16(c) and (d).Such a reduction was observed experimentally by [35].This could also explain why the discharge type changes from dry to humid conditions.Under humid conditions, the discharging gap is exposed to a larger electric field (>50 kV cm −1 ) as shown in figure 16(d), such that starting electrons for a breakdown near the Paschen minimum are made available very efficiently by field emission (FE).This presumably leads to a large number of smaller (subthreshold) PDs instead of the 'classical' distinguishable single PD pulses.More detailed measurements are required to fully corroborate the model predictions and hypotheses presented in this appendix.This is left to future investigations.

Figure 1 .
Figure 1.Relation between the molar fraction xw of water in air at the indicated temperature and pressure, and the absolute and relative air humidity.

Figure 2 .
Figure 2. Gap electric field E b = U b /d at the static breakdown voltage U b in uniform atmospheric pressure air gaps (continuous line).The expected effect of humidity on the breakdown voltage is indicated for the three regimes 'low-field'(E ≪ Ex, d ≫ dx), 'mid-field' (E ∼ Ex, d ∼ dx) and 'high-field' (E ≫ Ex, d ≪ dx).The electric field in gas gap of twisted enameled wire pairs at PD inception lies in the indicated high-field region (also see figure11).

Figure 3 .
Figure 3. Effective ionization coefficient of humid air at atmospheric pressure as a function of the electric field.The indicated levels of humidity can be translated into absolute or relative humidity values by using figure 1.

Figure 4 .
Figure 4. Increase of the critical electric field of atmospheric pressure air with humidity.This effect controls the increase of its breakdown strength in low electric fields (large gaps).

Figure 5 .
Figure 5.Relative change of the effective ionization coefficient of atmospheric pressure air for different levels of air humidity for medium and large electric fields.In large fields, there is a relative increase which saturates to a value specified in figure15.

Figure 6 .
Figure 6.Schematics of the experimental setup used to measure (a) the breakdown voltage of air gaps with metallic electrodes (b) the PDIV of contacting dielectric-coated electrodes.

Figure 7 .
Figure 7.Comparison between the calculated relative change of the breakdown voltage as a function of the absolute air humidity with measurement data from literature [5] as well as the humidity correction factor given in the IEC standard 60052 [22].The data is referenced to the humidity value ρ V = 8.5 g cm −3 specified in the IEC standard 60052.

Figure 8 .
Figure 8.Predicted relative change of the breakdown strength of small atmospheric pressure air gaps based on the high-field approximation(16).The hatched area for d < 10 µm is meant to indicate that the breakdown in these gaps at atmospheric pressure starts to be dominated by field emission of electrons (see e.g.[18]), and hence is outside the scope of this paper.

Figure 9 .
Figure 9.The continuous line shows the model prediction (equation (16)) of the relative reduction of the PDIV as a function of gap width, when the relative humidity is increased from about 2% to 78%.It can be seen that in smaller gaps, there is an increasing discrepancy with measured values, which can be related to a significant drop of the breakdown voltage in the low-humidity range 1% → 20% (see figure9), whose underlying physics is not captured by the presented model.

Figure 10 .
Figure 10.The measured relative variation of the breakdown voltage of atmospheric pressure air with relative humidity agrees well with the model in the range from 20% to 80%, whereas the strong increase below 20% is not captured by the model or the underlying parameter values.The values are referenced to the standard humidity of ρ V = 8.5 g m −3 (IEC 60058), which is equal to RH ≈ 37% at 25 • C.

Figure 11 .
Figure 11.Calculated discharge location (x PDIV ), discharge gap length (d PDIV ) and average electric field (E g,PDIV ) of the gas gap breaking down at PDIV.The discharge location is shown for the sphere-plane configuration depicted in figure 6(b) (R = 5 mm).

Figure 12 .
Figure 12.The wedge-shaped air gap in a twisted enameled wire pair is bounded by insulating dielectric layers of thickness s and relative permittivity εr.

Figure 13 .
Figure13.Relative variation of the PDIV as a function of the water vapor content in air at atmospheric pressure.The values are referenced to the standard humidity of ρ V = 8.5 g m −3 (IEC 60 058), which is equal to RH ≈ 37% at 25 • C. The continuous lines are model predictions including the effect of water on the ionization coefficient in the gas as well as the variation of the coating's dielectric permittivity.The dashed lines through the measured data points provide guides to the eye.

Figure 14 .
Figure 14.Predicted PDIV of twisted enameled wire pairs as a function of the reduced coating thickness compared to own measurements and measurements collected from literature.The calculated variation of the PDIV with air humidity is seen to be within scatter of measured values.Note that only the effect of water molecules on the ionization coefficient is included in the calculation for the shown model curves.Water can have a significant effect on the PDIV as shown in figure13.The measurements shown here were presumably obtained in relatively dry atmospheres (≲ 8.5 g m −3 ), except possibly for the points that are systematically below the model prediction.

Figure 16 .
Figure 16.(a) Tangential electric field (peak value) at the surface of the coating for an electrode voltage of 500 V (peak) for R □ = 300 MΩ and a number of sinusoidal excitation frequencies.(b) PDIV as a function of frequency for three values of the surface resistance.The relaxation frequency approximately follows the relationship given in equation (42) (c) Gas gap voltage for electrode voltage = PDIV and R □ = 300 MΩ.(d) Gas gap electric field at for electrode voltage = PDIV and R □ = 300 MΩ.
Threshold electric field value below which ionic space charge enhances the ionization yield of secondary avalanches Ex 4.15 • 10 6 V m −1 Crossover field strength above which α eff is larger in humid air compared to dry air B • T barGas pressure (note: T = 300 K is used for the conversion between p and N)