Experimental comparison of partial discharge between fast-switching pulse waves and square waves

The properties of short rise-time voltage waveforms, representative of the short turn-turn voltage pulses that could be expected in fast switching converters, were examined under conditions representative of aerospace environments to attempt to learn more about the fundamental nature and behaviour of partial discharge. Two predominant models of partial discharge behaviour can describe much of what is observed here.

Firstly, expanding on the model of Patsch and Berton [21], Fabiani et al [22] outlined an idealised conceptual model for the temporal relationship between electric field and partial discharge activity for perfect square wave voltage waveforms (no higher frequency oscillations, ringing, and overshoots etc) in the air gaps between two wires in the presence of space charge. The model qualitatively describes the initiation of partial discharge requiring an electric field strength above the dielectric strength of air and subsequent inception of an electron avalanche following the Townsend breakdown process [18]. The increasing electric field during the rise of a voltage wavefront eventually meets this criterion and the partial discharge transfers charge between surfaces, partially reducing the net local electric field strength. The deposited heterocharge can recombine over time, gradually restoring the pre-partial discharge local electric field. Repeated partial discharge events during voltage waveform cycling are governed by the magnitude of the peak-to-peak voltage change being sufficient to restore the net electric field strength above the inception field threshold. Many other factors complicate space charge recombination and the temporal behaviour of the net local electric field strength, including charge trap depth, insulation surface conductivity, and charge mobility. Many of these relate to properties of the coating materials and surface condition.

While Paschen's law is often used to calculate breakdown voltage, it can only reasonably be applied in the presence of uniform fields and quasi-static conditions. This is often not representative of many systems of interest in which divergent fields are present and electrodes have insulation coatings with a greater work function than a bare electrode (9 eV compared with 4–5 eV for copper [23]), affecting the likelihood of secondary electron emission from positive ion or UV bombardment. As such, the more fundamental streamer inception criterion must be considered, and corresponding models incorporating this have been developed. The probability of partial discharge occurring can be quantitatively computed as a function of electrode geometry, pressure, and voltage waveform using Volume-Time theory. This physics first principles model derives from physical models of discharge breakdown processes [18, 24–28] and had initial application predicting breakdown in SF₆ gas from uncoated electrodes under lightning impulse and fast transient voltages [29–32]. While many subset models considering streamer inception allow for the spatial aspects of the breakdown phenomenon—typically seen with AC sine waves, Volume-Time theory allows the stochastic and temporal aspects of breakdown for short duration pulses and very short rise times to be captured. More recently, this has been developed further to incorporate the influence of surface emission [33]. The model considers collisional electron detachment processes from gas ions in the gap and is expressed in the form of a Weibull cumulative distribution function to estimate probability of the formation of a critical avalanche within a critical volume:

$$\begin{equation}P = 1 - {\text{exp}}\left[ { - \int\limits_0^t \left( {\int\limits_{{V_{{\text{cr}}}}} \frac{{{\text{d}}n}}{{{\text{d}}t}}\left( {1 - \frac{\eta }{\alpha }} \right){\text{d}}V + \int\limits_{{S_{{\text{cr}}}}} \frac{{{\text{d}}i}}{{{\text{d}}t}}\left( {1 - \frac{\eta }{\alpha }} \right){\text{d}}S} \right){\text{d}}t} \right]\end{equation} \tag{ 1 }$$

where:

• $P$ is the probability of generating a critical avalanche—one containing the critical electron number in the avalanche head, allowing survival and propagation to the electrode.
• ${V_{{\text{cr}}}}$ and ${S_{{\text{cr}}}}$ are the critical volume and critical surface, respectively.
• $\frac{{{\text{d}}n}}{{{\text{d}}t}}$ and $\frac{{{\text{d}}i}}{{{\text{d}}t}}$ are the rates of detachment within the critical volume and from the critical surface, respectively.
• $\alpha$ and $\eta$ are the ionisation and attachment coefficients, respectively.

When the computed probability from (equation (1)) reaches a value of 63.2% (the scale parameter of the Weibull distribution representing the modal peak of the corresponding derivative probability distribution function, and greatest likelihood), the corresponding voltage generating the fields is defined as inception and is associated with one initial detachment electron generated within the critical volume. The critical volume and critical surface represent regions—the boundaries of which can be determined using finite element analysis, as discussed below—where the electric field strength is high enough to allow multiple collisions and grow an avalanche. The area of the critical surface is assumed to be the region of the coated earth electrode in contact with the critical volume.

Electrical breakdown in air is generally assumed to occur, for all conditions under consideration, when the number of electrons in the tip of the avalanche reaches a critical value of 10⁸ [23]. The distance a growing avalanche travels along a line of electric force when it reaches the critical number of electrons, is called the critical electron drift path length, ${x_{{\text{cr}}}}$. In general, the number of electrons at the tip of the avalanche, $N\left( x \right)$, at distance $x$ from starting location, is [34]:

$$\begin{equation}N\left( x \right) = {\text{exp}}\left( {\int\limits_0^x \left( {\alpha - \eta } \right){\text{d}}x} \right)\end{equation} \tag{ 2 }$$

where $\left( {\alpha - \eta } \right)$ is the effective ionisation coefficient—the difference between ionisation and attachment coefficients, representing sources and sinks of electrons. Values above zero represent net electron contribution to the avalanche tip and thus propagation.

The critical volume, ${V_{{\text{cr}}}}$, must satisfy the Schumann criteria for streamer transition during the electron avalanche process [34]—namely that the integrated sum of the effective ionisation coefficients over the critical path length, is constant:

$$\begin{equation}\int\limits_{{x_{{\text{cr}}}}} \left( {\alpha - \eta } \right){\text{d}}\,x = K.\end{equation} \tag{ 3 }$$

${x_{{\text{cr}}}}$ is assumed to be equivalent to the gap length between electrodes and that electrons take a linear trajectory [33]. The effective ionisation coefficient is empirically determined for a given gas type by knowing the gas pressure, $p$ (MPa) and electric field strength, $E$ (kV mm⁻¹). Many studies have shown that $K$ does not have a value of 18 for air $\left(\approx {\text{ln}}\left( {{{10}^8}} \right)\right)$ [35–37], but varies depending on the specific system under test, indicating the importance of conducting corresponding experimental measurements of partial discharge inception voltage and computing this value [38] using finite element analysis [33, 39]. Many parameterisations necessary for this computation (or empirical data from which such expressions can be derived) exist for the effective ionisation coefficient, spanning the appropriate range of electric field strength and pressure [40–42], for example [41]:

$$\begin{equation}\frac{{\alpha - \eta }}{p} = 0.161{\left( {\frac{E}{p} - 21.65} \right)^2} - 2.87.\end{equation} \tag{ 4 }$$

The applied electric field strength must be above a critical electric field strength, ${E_{{\text{cr}}}}$, of approximately 21.65 kV mm⁻¹ MPa⁻¹ for dry air in this example [41].

In (equation (1)), the rate of detachment in the air volume is determined by the density of ${O_2}^ -$ ions, ${n^ - }$ (approximately 150 cm⁻³ at standard temperature and pressure [43]), scaled by the detachment coefficient, ${k_{\text{d}}}$ (µs⁻¹):

$$\begin{equation}\frac{{{\text{d}}n}}{{{\text{d}}t}} = {k_{\text{d}}} \times {n^ - }.\end{equation} \tag{ 5 }$$

${k_{\text{d}}}$ is determined by the overvoltage ratio, $(E/{E_{{\text{cr}}}})$, representing how many factors of the critical electric field strength are applied. This readily saturates as a function of electric field strength [33].

The rate of electron detachment from the earth electrode insulation surface is calculated from field emission theory, relying on emissions from both protrusions in the cathode surface (described by Fowler–Nordheim theory in quantum mechanics), as well as from insulation impurities, deposits, or trap levels in the presence of a high electric field [43]:

$$\begin{equation}\frac{{{\text{d}}i}}{{{\text{d}}t}} = \frac{{{e^3}{{\left( {\beta {E_{\text{i}}}} \right)}^2}}}{{8\pi h\left( {e{\varphi _D}} \right)}}\exp \left( { - \frac{{8\pi \sqrt {2m} }}{{3he\beta {E_{\text{i}}}}}{{\left( {e{\varphi _D}} \right)}^{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ \right.} \!\lower0.7ex\hbox{$2$}}}}} \right)\end{equation} \tag{ 6 }$$

where:

• $e$ is the electron charge.
• $\beta$ is the field enhancement factor.
• ${E_{\text{i}}}$ is the electric field strength applied to the insulation.
• $h$ is Planck's constant.
• ${\varphi _D}$ is potential barrier height.
• $m$ is the effective electron mass.

All terms of (equation (1)) are ultimately functions of the electric field strength and air pressure, and it is this understanding which informs interpretation of the experimental observations presented.

Partial discharge behaviour is observed to be stochastic for both pulse and square waves, manifesting in three cases:

• Case 1.  
  Partial discharge only sometimes occurs after voltage change.
• Case 2.  
  Multiple partial discharge events occur on some cycles (not explicitly investigated here).
• Case 3.  
  Partial discharge occurs at variable times after voltage change has occurred.

Case 1 is principally accounted for by considering both magnitudes of resulting local electric fields and the influence of space charge, as well as sources of initial electrons. A partial discharge event transfers charge between surfaces, setting up a counter electric field and reducing net local electric field strength. When voltage polarity reverses (or returns to zero in the case of unipolar waveforms), the electric field component from the charge transferred supplements the external field contributed by the voltage change [22]. If the net electric field strength remains below the inception field, a discharge event will not occur during that cycle, accounting for some variability in discharge occurrence. The magnitude of charge transferred during the partial discharge event, as well as the factors affecting the deposited space charge decay, are the controlling factors. Furthermore, for unipolar waveforms, as used here, it is already established that the deposited heterocharge can reduce net fields, raising inception voltage [22, 44, 45], and reducing subsequent discharge likelihood. Long term exposure of a sample to unipolar waveforms, giving a substantial DC field component, can inject homocharge from the electrode into deeper traps where depolarisation takes longer, acting to partially reverse the influence of heterocharge and reduce inception voltage [22], although testing was not sufficiently lengthy for that effect to be noted here.

Fabiani et al [22] also occasionally observed two partial discharge events in a given half cycle of a square wave (case 2) at low frequency (approximately 10 Hz) when square wave peak voltage is sustained for a relatively longer time. After a discharge event, space charge decay from shallow traps—which is more rapid for materials with higher surface conductivity—restores the local field within the half cycle, taking the net electric field above the inception field and allowing a second discharge.

The presence of space charge (electrons and positive and negative ions) has strong influence on the repetition of partial discharge events. Often partial discharge activity is observed to be intermittent when fields are first applied [44], particularly for bipolar waveforms, and as space charge is subsequently deposited from earlier discharge activity, the frequency of partial discharges may increase. This is frequently observed in the laboratory during initial experimental testing when inception voltage is higher than for subsequent tests within a short space of time, and the reason why partial discharge extinction voltage magnitudes are lower than inception. The increased presence of negative ions in the gaps and on the surface can serve as sources of initial electrons [46], which if present within the critical volume, will incept an avalanche. Case 3 is discussed in sections 5.2 and 5.3.

Several studies have examined partial discharge activity using square wave voltage waveforms (e.g. [7, 22, 46]) with a range of frequencies and duty cycles. With sustained peak voltages of the order of at least 100 µs, it was observed that frequency and total duration of exposure to voltage had little impact on partial discharge inception voltage, leading to the conclusion they did not have first order influence. Fabiani et al [22] noted no inception voltage dependence on waveshape—although did not use short duration pulse waves as here—and slight dependence on coating material. Short rise-time turn-to-turn-like waveforms used here and as seen in other studies (e.g. [10, 16]), have peak voltages of much shorter duration than square waves (as little as 50 ns). In supplementary experiments, the frequency of 50 ns pulse waves was also observed to have little influence on inception voltage, supporting the conclusions of other studies (e.g. [7]) for short duration pulses.

Figure 3 shows the empirically observed likelihood of partial discharge occurring as a function of applied voltage follows a sigmoid function, consistent with the cumulative Weibull distribution function used to model its behaviour in Volume-Time theory. Whilst the probability of partial discharge is an interplay between the applied electric field strength and duration over which it is present, this data confirms the theoretical expectation that with sufficiently high electric field strength, partial discharge cannot be avoided, even for very short duration pulses associated with transient turn-to-turn potential differences within machine windings. The differences in discharge probability curves also indicates that machine insulation systems should be tested with the specific waveform profiles they will experience in operation rather than ones with merely similar rise times for specific equipment under test, echoing similar existing sentiments [47, 48]. Knowing the partial discharge probability curves for a given system can provide more information about its discharge response than simply measuring the repetitive partial discharge inception voltage alone (that is, when 50% of voltage pulses produce a discharge event), particularly if the operating voltage may vary from nominal.

Low pressure environments exhibited a relatively rapid onset of partial discharge activity for both square and pulse waves, reaching 100% discharge occurrence with much smaller voltage increases above inception (figure 3). This is likely due to the much stronger initial response of the effective ionisation coefficient with increasing electric field strength at low pressure (equation (4)) which more rapidly increases the critical volume. However, rapid onset was also observed at ambient pressure for square waves—particularly at positive polarity. This is likely accounted for by the increased integration time from (equation (1)) relative to pulse waves, increasing the duration of the critical volume and making partial discharge more likely. This similarly explains why square waves were observed to have a reduced 50% discharge occurrence voltage relative to shorter duration pulse waves at all pressures. It is not clear why at low pressure, negative polarity favours lower inception voltages for both square and pulse waveforms, yet the trend is reversed at higher pressure. While there are known differences in electron avalanche development processes with polarity (e.g. [26, 27]), this does not appear to fully account for the observations. Nevertheless, voltage polarity does not exhibit as significant a difference in partial discharge likelihood under given conditions, relative to pressure or waveform.

When electric fields are sufficient for partial discharge to occur and are sustained during peak voltage in square and pulse waveforms, discharge occurrence is temporally stochastic (case 3, section 4.2). It is known from theoretical and experimental studies of gas breakdown (e.g. [32, 49]) that the magnitude and spatial distribution of gas ion density affects breakdown statistical time lag. Free electron production rate is determined by considering their sources and sinks from gas molecular ionisation, attachment, and detachment processes, and must account for ion mobility, diffusion coefficient, and proximity of a free electron to the cathode. This is accounted for in Volume-Time theory previously outlined. The probability of electron detachment is electric field-dependent. Higher voltages—and the associated increase in local electric fields—increase the production of free electrons from ions—whose concentration can also be increased by space charge presence, as discussed earlier. The electrons are sourced from spontaneous detachment from negative ions or initiated by cosmic radiation and natural radioactivity [49], supplementing sources driven by collisional detachment. The increased free electron production rate ultimately increases the probability of critical avalanche formation within the critical volume, reducing the average partial discharge time delay. Higher electric field strength also increases the critical volume itself, making critical avalanche formation more likely. The effects of these factors are observed empirically in figure 4, with increasing voltage reducing the modal delay time between voltage change and subsequent partial discharge.

The probability of electron detachment is also pressure dependent. At lower pressure, the density of negative ions is reduced for a given temperature, diminishing detachment rate within the critical volume (equation (5)), and reducing partial discharge probability. This is reflected in the larger range of delay times in figure 5 at lower pressure. During square waves with a relatively long 50 ms pulse duration, partial discharge was observed to occasionally occur considerably after voltage onset, particularly at low pressure, where delays beyond 1 ms could be observed. This observation suggests that testing based on IEC 61934 [50] may need to have a sufficiently long detection time to ensure a delayed partial discharge event is accurately detected. Under environmental conditions relevant to aerospace with the presence of fast rise-time voltage waveforms that allow a wider temporal distribution of partial discharge activity following voltage change, if the detection window is too short in duration, misdetection or under-detection of partial discharge may occur.

The temporal distribution of partial discharge delay times after voltage rise is generally within the period that the high voltage is present. However, examination of the influence of pulse duration more closely, under conditions where partial discharge likelihood was consistently 15%–20% in each case (figure 6), revealed the surprising observation that while longer pulse durations allow for longer delay times between voltage increase and partial discharge, the distributions were not simply broadened. Longer duration pulses did not exhibit the presence of shorter discharge delay times after voltage rise. While the shorter duration pulses required a greater applied voltage (electric field strength) to initiate the same level of partial discharge activity—owing to the reduced integration time of the shorter pulse from (equation (1))—this does not fully account for the observation, since pulse durations of 5 ms, 500 µs, and 50 µs were all measured at the same applied voltage. It is also unlikely that space charge could explain this either; greater homocharge injected during the longer duration pulses would increase local electric field strength and reduce inception voltage. The applied voltage for these three longest pulse durations was unchanged and the partial discharge likelihood was consistently 15%–20%, indicating homocharge build-up was not significant and thus cannot account for the observation. It is not clear from known models of partial discharge how this behaviour could be explained.

A corollary of the conceptual model proposed by Fabiani et al [22], is that for unipolar square waves, partial discharge events are expected after applied voltage returns to zero. Such discharge events reflect a field restoration process, should a prior discharge have led to conducive field strengths. This was observed here (figure 6), supporting the model; however, the distribution in time delays between discharges occurring after voltage rise compared with voltage fall is very different. Partial discharge occurring after voltage fall typically occurs sooner and stops altogether for pulse durations of 5 µs and lower. Fabiani et al [22] also showed a clear difference in the distribution of discharge delay times between voltage rise and voltage fall; however, their subsequent study (Cavallini et al [51]) did not observe this and found symmetric partial discharge distributions. It is not clear what causes this marked difference in partial discharge behaviour, and we are not able to propose an explanation from known theory, but nevertheless demonstrates additional distinction between square waves and shorter duration pulse waves.

Partial discharge magnitudes are generally observed to be greater at low pressure than at ambient pressure. This likely relates to increased area of the surface discharged at lower pressures during a partial discharge event due to occurrence of glow discharges that form nearer to Paschen's minimum [52]. There may also be greater charge transferred in longer channel avalanches that are possible at lower pressures where the critical volume and surface are larger; however, this is likely secondary to the increased contribution from greater surface area involved in the discharge.

The magnitude of observed partial discharge also reveals differences between waveforms and pulse duration. Square waves typically have lower partial discharge voltage magnitudes, and this broadens to larger magnitudes with decreasing pulse duration (figure 7). The shortest pulse durations tested of 500 ns showed a transition to more distinct discharge magnitude behaviour and a stronger polarity dependence. For positive voltage polarity, discharges were exclusively high magnitude, and while true for a negative applied voltage, the magnitudes were not as large. It is not clear what drives the onset of polarity asymmetry. However, the increasing discharge magnitude is likely explained by the need for a greater applied voltage to initiate partial discharge for shorter pulse durations—which integrate over a shorter time period in (equation (1)), thereby reducing likelihood of inception, as discussed above in section 5.2—and that the partial discharge magnitude is also dependent on electric field strength [22]. This means that shorter duration pulses require a larger electric field strength to incept partial discharge, but their magnitude is greater as a result, which has implications for system degradation and lifetime.