UNIVERSITY OF THE WITWATERSRAND
School of Electrical and Information Engineering

ELECTRICAL DISCHARGES IN GASES

TABLE OF CONTENTS

1. QUESTION 1 2 2. QUESTION 2 11 3. QUESTION 3 21 4. QUESTION 4 28 5. QUESTION 5 36 6. REFERENCES 45 7. APPENDIXTURE 48

1. QUESTION 1 a) Explain the processes which lead to electrical breakdown of air by the streamer mechanism. Your explanation should include a discussion of the influence of pressure, gap length and electric field uniformity The concept of streamer type of breakdown is based on the Townsend type breakdown, which requires a complete sequence of avalanches [2]. According to Reather, the streamer type of breakdown differs from the latter because it assumes that the growth of a single electron avalanche becomes unstable. The original criterion by Reather only addressed the primary ionization processes, i.e. electron collision. With increased understanding of electron processes the role of attachment became apparent and the criterion was modified to the following equation:

oxα-ηdx=18 (1)

Simultaneous with Raether’s work, Meek. identified the Streamer mechanism and formulated the following breakdown criterion:

According to Meek the instability is caused by the space charge field from the electrons and ions in the head of an avalanche. The breakdown occurs when the space charge field from the avalanche head becomes the same order of size as external applied field .The instability results in the formation of fast moving anode and cathode directed filamentary streamers from the avalanche head [3].A highly conducting plasma channel is then formed across the gap, which eventually causes the collapse of the voltage.

αx= e0xαdx=KEx (xp)12 (2)

According to Pedersen, the basic mechanism behind the formation of streamers is thought to be photo-ionisation in the gas [2][1]. Pedersen’s work was supported by Townsend’s early work which proposed the Townsend or Generation Mechanism (explain), given by:

i=ioeαd1-γ(eαd-1) (3)

Leading to the sparking threshold: γeαd (4)

Pedersen proposes that the Townsend and Streamer processes merge into a single criterion: αx e0xαdx=GX,ρ.Ex, μ.%H2O……. (5)

Due to the fact that gap length (x) and pressure (ρ) are the dominant influences, in practice: αx e0xαdx=GX,ρ (6)

It may seem that secondary effects have been ignored, but α is a substitute for the production of electrons and hence secondary process(i.e. photo-ionization, photo-electric emission, recombination process and ion bombardment) associated with electrons at the stage that the avalanche reaches its critical length. For a given air density or pressure this criterion reduces to:

ln⁡(αx)+oxαdx=g(X) (7)

Pedersen argues that for a given pressure, breakdown is only dependent on the critical avalanche length X. In the case of uniform field breakdown, the electrode spacing is always the critical avalanche length. Where (αx) is the numerical value of α at the avalanche head. In a uniform field x is equal to the gap length and α is constant. Hence, uniform field breakdown can be used to evaluate g(X). We thus have a breakdown criterion.

Though a satisfactory quantitative physical theory for streamer formation has not yet been formulated, it is generally accepted that the following conditions are necessary for streamer propagation [2].

* Sufficiently high energy photons must be produced in the head of the initial avalanche. * These photons must ionize gas molecules in the immediate vicinity of the avalanche head. * The space charge field at the head of the avalanche must be sufficient to produce. There are two main processes that lead to electrical breakdown in air, they are: * Primary and; * Secondary ionisation processes.
A neutral atom or particle has balanced charge, meaning there is an equal number of electrons and protons. Electrons occupy different orbits characterized by different permissible energy states and the electron orbit closest to the nucleus has the lowest energy and the one farthest has highest energy, according to Maruvada [1]. Hence, energy passed on to the atom will affect the electron that is in the outermost orbit.
In cases where adequate energy is passed on to the atom, the electron in the outermost orbit may jump to the next permissible energy orbit, and the atom will be excited. The excited electron may rapidly settle down to its original state, and subsequently release excess energy in the form of a photon. The frequency of the photon released in the process depends on the energy levels through which the electron jumps.
Should excessive energy be released to the atom, the electron may jump so far away from its orbit that it may not return to its orbit. The atom will then be ionised. The process of freeing an electron from an atom with the simultaneous production of a positive ion is called ionisation. * The primary process that leads to breakdown in air occurs through ionisation by collision.
Natural free electrons are available in air spaces. These free electrons created are by natural ionisation processes (i.e. Cosmic radiation, originating outside earth’s atmosphere and gamma rays produced by radioactive processes in the soil) attach quickly to oxygen atoms in air, forming negative ions. Hence, atmospheric air contains mainly positive and negative ions.

Complimentary, in air both the ions and electrons participate in the movement of molecules. The ionization of an air molecule with an electron means to knock off an electron creating a pair of positive and negative ions. Ionization by collision process is as a result of electrons colliding with an air molecule and gives rise to a new electron and positive air ion. In air, if two parallel conductors are subjected into an electric field, any electron starting from the cathode (negative) will be accelerated more and more between collisions with other gas molecules during its travel towards the anode (positive). This cumulative process leads to a number of electrons - ion pairs. These additional electrons make ionising collisions themselves. By this cumulative process, the number of electrons and ions grow rapidly and electron avalanches are formed.
Electron impact:
A + eA* + e (excitation) (8)
A + eA+ + e + e (ionisation) (9)
The positive ions are accelerated as well in the field, but they gain considerably inferior energy than the electrons because they lose excessive energy in each collision because of their bulky mass. The positive ions are unlikely to ionise in the gas [1]. As the positive ions accelerate towards the cathode and on bombardment on the cathode, they give rise to secondary electrons. This leads to secondary ionisation processes. * Photo-ionisation is one of the activities that contribute to the secondary process leads to breakdown in air.
The photo-ionisation process is caused by ionisation by radiation. This process occurs when the amount of radiation energy absorbed by the atom or molecule (conductor) exceeds its ionisation potential. Subsequently an atom is excited and when an electron that had been released earlier returns to its original lower state it emits radiation. Another important secondary process is photo-excitation or photoelectric emission. By Photon:
A + hfpA* (photo excitation or photo emission) (10)
A + hfpA*+ e (photo ionisation) (11) * Electron attachment and detachment
In some electronegative gasses like pure oxygen and sulfur, the outermost path is not completely filled in the neutral state, leaving some positions readily available to receive free electrons. As a result, they have the potential to attract and capture free electrons to form stable negative ions. The formation of negative ions is shown by: A + eA+ (attachment) (12)
In electron detachment a negative ion may give away its extra electron and revert to its neutral state. An amount of energy known as electron affinity is required to cause electron detachment. [1] * Recombination
The coexistence of positively and negatively charged particles in a gas leads to recombination in which charge neutralization takes place. The process may be represented symbolically by the equation,
A+ + B-AB + hfp (recombination) (13)
In this process A+ (positive ion) and B- (negative ion or electron). The process above may be considered as the reverse of photo-ionisation.

b) Consider a gap between the surfaces of two identical spheres. The diameter of the spheres is 1000mm and the gap length is 100mm. The published data for the breakdown of this gap, at a pressure of 101,3 kPa (1,0 atmosphere, 760 Torr) and 20◦C, is 266kV.1 (i) Determine, analytically, what you regard as the most accurate prediction of the breakdown voltage for this gap in air at 101,3kPa and 20◦C. Explain why you regard the method you use as the most accurate. Comment on the degree of agreement with the published data. (8 marks)

There are two analytical methods that can be used to predict the voltage breakdown at the above mentioned conditions, they are: * Pedersen method and; * Reather method.

* Pedersen method
Pedersen states that for air at atmospheric pressure the breakdown equation can be written as: (14)
Where is a numerical value of at the avalanche head. The breakdown voltage for air according to Pedersen’s method was determined by the use of the curve for g(x) that was derived from uniform field data as the benchmark. A voltage of 262.95 kV was chosen to determine g(x) of 29.8 which is the benchmark for the sphere-sphere gap. The curve intersects g(x) for the uniform field, and hence we have breakdown. Therefore the minimum applied voltage for which g(x) for the sphere-sphere gap intersects, or touches; g(x) for the uniform field is 262.95 kV as shown in appendix 1.

* Reather method.
According to Reather [2] streamer breakdown is the result of avalanche activity reaching a critical (15)
The above equation indicates that breakdown occurs when the total number of electrons in an avalanche attains a certain value. From experimental data a suitable value for K is given as 18. The breakdown data for air in uniform fields is available and hence the above equation proposed by Reather will be used.
The ionization and attachment coefficients for air was fitted to a 6th order polynomial and calculated excel (appendix 2) using the above.

According to appendix 2, the breakdown voltage of air is 256.3 kV, which occurs when K = 18

* Explain why you regard the method you use as the most accurate. * Comment on the degree of agreement with the published data.

The published data by Kuffels, for the breakdown of this gap, at a pressure of 101, 3 kPa (1, 0 atmosphere, 760 Torr) and 20◦C, is 266kV. Pedersen’s degree of variance is 1.147% from kuffels data, whilst Reather’s degree of variance is 3.64% from kuffels data, Therefore Pedersen is regarded as the most accurate method as it does not deviate by more than 3% from kuffels published data as reflected on Appendix 1 and 2. (ii) Repeat the prediction of breakdown for the gap with both pure oxygen and pure nitrogen instead of air. Comment on the reasons for differences/similarities between the three predictions.

* For oxygen
Only Reather’s method was considered, because a look up table for the breakdown in oxygen via Pedersen’s method is not available. The prediction of breakdown for the gap with pure oxygen was carried out in a similar manner as was in air. The ionization and attachment coefficients for oxygen was fitted to a 6th order polynomial and calculated in excel (appendix 3).

From the appendix 3, the breakdown voltage of air is 272.26 kV, which occurs when K = 18

Note: Experimental results prediction reveals that air is a better than our prediction yields

* For Nitrogen
Nitrogen is a non-attaching gas and is an electron retarding gas. In a non-attaching gas the electrons in an avalanche can proceed all the way to the anode as opposed to in a highly electronegative gas where breakdown will occur much earlier before reaching the anode. Nitrogen only experiences ionisation, the following ionisation coefficient was used on excel spreadsheet to calculate the breakdown of nitrogen. (15)
From the appendix 4, the breakdown voltage of air is 260.26kV, which occurs when K = 18

With regards to breakdown of air, Kuffels experimental data states that the two identical spheres with the gap length of 100 mm breakdown is 266 KV at a pressure of 1 bar, 20◦C. Pedersen’s degree of variance is 1.147% from kuffels data, whilst Reather’s degree of variance is 3.64% from kuffels data, Therefore Pedersen is regarded as the most accurate method as it does not deviate by more than 3% from kuffels published data as reflected on Appendix 1 and 2.

With regards to oxygen only Reather’s method was considered, because a look up table for the breakdown in oxygen via Pedersen’s method is not available. The prediction of breakdown for the gap with pure oxygen was carried out in a similar manner as was in air. The ionization and attachment coefficients for oxygen was fitted to a 6th order polynomial and calculated in excel (appendix 3).

From the appendix 3, the breakdown voltage of air is 272.26 kV, which occurs when K = 18

Also alpha –eta (α-η) for oxygen has a steeper gradient (111.1) such that when it breaks down it is vigorous. For air the gradient is 1000, meaning that once there is a spark breaks down, it will take some time before complete breakdown as compared to oxygen (111.1).Therefore even though oxygen (272.26KV) breaks at higher voltage than air (256.3KV), once oxygen has a spark breakdown, it takes longer to completely breakdown the gap. Based on appendix 4, the breakdown voltage of nitrogen is 260.26kV, which occurs when K = 18, from the above it seems as if oxygen is a better insulator of the three gases ,seconded by nitrogen if only the breakdown voltage magnitudes are considered.

2. QUESTION 2

a) Sulphurhexaflouride (SF6) is widely used as an insulating medium in high voltage substations. Explain why this is so considering the properties of the gas and the impact of such installations on the environment. What are the undesirable aspects associated with the use of SF6 as an insulating medium? The choice of insulation medium is the key consideration in High-voltage (HV) substation’s design and performance. Air Insulated Substations (AIS) necessitate relatively bigger areas, and their equipments are associated with high maintenance costs and operational risk to personnel, all these are unfavorable. On the other hand Gas Insulated Substations (GIS) necessitate relatively smaller areas, and they are favoured because they are associated with less maintenance costs and safer operations due to the use of SF6 . SF6 is a highly electronegative gas, meaning the gas removes free electrons from the discharge through the process of attachment.

SF6 is ideal for use in HV installations due to its excellent insulation properties. The main characteristics of SF6 are that it is chemically stable, odourless, non-toxic and non-flammable and its breakdown strength is approximately 2.5 to 3 times that of air.

SF6 is a notorious greenhouse gas. It demonstrates its electronegative properties by removing electrons from a discharge through the process of attachment, which can either be direct attachment (SF6 + e¯→ SF6¯) or dissociate attachment (SF6 + e¯→ SF5 + F¯). Through the above mentioned processes, the highly mobile electrons which could cause ionisation from collisions are replaced by heavy ions which are slow and unable of causing ionisation, and thus breakdown is prevented. Considering that its breakdown strength is about 2.5-3 times that of air, the size of the substation can be reduced by that order. In most applications SF6 is compressed thus increasing its dielectric properties three times more, consequently making its dielectric properties 9-10 times better than air. As a result SF6 installations are generally 10 times smaller than AIS systems. The environmental impact of SF6 installation may be significantly higher than that of air, but SF6 has the advantage of less space requirements, protection against pollution, and lower operating and maintenance costs. Considering that SF6 is a greenhouse gas it is imperative to ensure that the switchgear free from gas leaks. Detectors should be in place to sense any leaks, so that they can be addressed speedily. Greenhouse gases are notorious because aggravate global warming, which is harmful to the environment and to humans. In cases where SF6 is used as an interrupting medium, and exposed to spark discharges, partial discharges, switching arc and arcing failure such that the temperature increases above 1760C, the by-products formed are toxic to humans, and also corrosive. The by-products can be in the form of gases or powders. Therefore maintenance of SF6 installations must be carried out with caution by using protective equipment especially where arcing has taken place. The disposal of such by-products is an environmental concern because they are toxic to humans and bad for the environment.

b) Explain the mechanism of electrical breakdown of a gap in SF6 and the criterion for the threshold of electrical breakdown, on the basis of the mechanism you have described. (5 marks)

Electrical breakdown in gases can occur by streamer or leader breakdown. The streamer breakdown theory assumes a uniform or near uniform fields, with relatively short gaps, and the low pressure, although in HV applications where SF6 is used, the fields are non-uniform and the pressure is high, therefore the mechanism for breakdown is the leader mechanism. Nevertheless, the streamer breakdown mechanism can be used to explain the breakdown in SF6. The following conditions are necessary for streamer formation: 1. Sufficient high energy photons must be produced in the head of the initial avalanche 2. These photons must ionise gas molecules in the immediate vicinity of the avalanche head. 3. The space charge field at the head of the avalanche must be sufficient to produce adequate to produce secondary avalanches in the enhanced field.

Although breakdown in SF6 is by leader mechanism, it is assumed that all three conditions for streamer formation are satisfied, if condition (1) which relates to the production of high-energy photons is fulfilled.

The leader breakdown mechanism in SF6 can be explained in the following way in line with the Figure 2.1 below.

Figure 2-1: Leader breakdown mechanisms in SF6 [9]

Stage a: Illustrates the formation of the streamer. The applied voltage must be sufficient such that a critical volume a critical value Vcr is formed next to the high-field electrode, and this results in the electric field exceeding a critical value Ecr. Stage b: Once streamers are formed, they cannot propagate unless the excitation and recombination processes results in photoionisation within the first avalanches triggers streamers and they form the first corona. Stage c: In this stage, two processes occur in the corona remnants. The first process is thermal expansion of the former streamer channels, on receipt of the current from the corona propagation stage and forms stem (S) mechanism. The second process is the formation of space charge filaments as the positive and negative ions drift apart in the electrical field and forms space charge filaments and forms the precursor (P) mechanism. Stage d: The stem mechanism provides a conducting filament propagating from the electrode to the corona margin and the precursor starts at the corona and propagates towards the electrode. These processes/mechanisms bridge the corona zone and serve as an electrode where a second corona can be launched into the gas volume. Stage e: This process is repeated until the conditions of the stem or precursor formation are no longer fulfilled or a corona streamer touches the opposite electrode and causes a breakdown [9].This process occurs due to the second corona supplying a current pulse to the stem or the precursor filament leading to additional ohmic energy and subsequently resulting in more heating and expansion of the channel. Resulting in the gas reduced to molecular fragments compromising its attachment properties, and as the density drops the critical field is reduced even further.
The criterion for voltage breakdown in SF6 is derived by assuming all conditions for streamer development are satisfied when sufficient high energy photos are produced at the head of the avalanche. The rate of production of photons is reflected in the equation below: (16)
Where β is the radiative recombination coefficient, Ne is the total number of electrons present in the avalanche, and N+ is the volume density of positive ions in the avalanche head. N+ is given by: (17)
Assuming that streamers are produced when the rate of photon production reaches a certain value k, the breakdown equation becomes: (18)
In the absence of data to determine the coefficients of radiative recombination for SF6, but with the understanding that the role it plays is genuine then the breakdown criterion can be written in the form: (19)
When simplifying the criterion by writing it in the logarithmic form, be criterion becomes: (20)
Where f(x,p) for SF6 requires reliable data for impulse breakdown, but in the unavailability of such data, the empirical data available indicates that f(x,p) is almost constant, and the combination of the logarithmic term is constant, such that (21)
The formula means that the integral in (20) should give a constant value irrespective of the avalanche length (x) or the gas density (ρ).
According to experimental uniform field breakdown data, the value of K is 18. Therefore the breakdown criteria for SF6 can be written as (22) c) Consider a coaxial duct, with the inner conductor having a radius of 50mm and the outer conductor an inner radius of 100mm:

(i) Determine the breakdown voltage of the duct in pure SF6 at a pressure of 3,0 bar; (5 marks) SF6 is a highly electronegative gas. The outer orbit of the SF6 molecule is not completely filled with electrons. That means that the molecule of SF6 possesses attractive properties in attaching low energy electrons. [8]. Figure 2-1 below shows that the breakdown voltage of SF6 is 710.3 kV and it breaks down at gap length of 1.813mm. The Matlab program was used for the plot (see code on appendix 5).

Figure 2-1: Breakdown voltage of SF6 at 3.0 bar. (ii) Determine the breakdown voltage of the duct in a mixture of 80% N2 and 20% SF6 at a total pressure of 3,0 bar; (5 marks). Figure 2-2 below shows the breakdown voltage of 80%/20% mixture of nitrogen and SF6 at 3.0 bars. The breakdown voltage is 574.8 kV at 3.155 mm. SF6 is a highly electronegative gas and it attaches free electrons in the low energy levels. Nitrogen on the other hand, although it does not have the attachment properties, it is an electron retarding gas. When mixed with SF6, nitrogen slows down the electrons that are in higher energy levels so that they can be attached by the SF6. According to [8] an experiment was conducted to determine the effects of basic physical properties such as electron attachment and scattering on the breakdown strength of gas mixtures of an electronegative and electron retarding gases. Based on the results it is shown that SF6 has a much higher breakdown voltage than N2. It is also shown that the combination of both gases gives a better dielectric characteristic than the sum of the individual gases. This mixture does have synergistic effects. Based on [8] it is shown that the electron capturing capability of SF6 is only strong for low energy electrons. The lifetime of an SF6 ion formed by capturing the slow electron is much longer than the one formed by capturing the fast electron. This is where N2 comes in; it slows fast electrons into the low energy range very effectively.

Figure 2-2: Breakdown voltage of 80%/20% mixture of nitrogen and SF6 at 3.0 bar.

(iii) Determine the breakdown voltage of the duct in pure N2; (5 marks)

Figure 2-3: Breakdown voltage of pure nitrogen at 3.0 bar.

Based on the Matlab simulation shown in Figure 2-.3 above, the curve for N2 does reach the value of 18 and only changes slowly with further penetration of the avalanche into the gas. It is because N2 in non-attaching. Based on the above, it may be concluded that pure N2 gas simply saturates once the applied voltage exceeds 331.5 k V, because N2 in non-attaching.

(iv) Comment on the three predictions that you have made, give reasons for the differences and explain which arrangement you regard as most suitable for a 4km long transmission link. (6 marks) SF6 is a highly electronegative gas. The outer orbit of the SF6 molecule is not completely filled with electrons. That means that the molecule of SF6 possesses attractive properties in attaching low energy electrons [8]. Figure 2-1 above shows that the breakdown voltage of SF6 is 710.3 kV and it breaks down at gap length of 1.813mm. The Matlab program was used for the plot (see code on appendix 5).

Figure 2-2 above shows the breakdown voltage of 80%/20% mixture of nitrogen and SF6 at 3.0 bars. The breakdown voltage is 574.8 kV at 3.155 mm. SF6 is a highly electronegative gas and it attaches free electrons in the low energy levels. Nitrogen on the other hand, although it does not have the attachment properties, it is an electron retarding gas. When mixed with SF6, nitrogen slows down the electrons that are in higher energy levels so that they can be attached by the SF6. According to [8] an experiment was conducted to determine the effects of basic physical properties such as electron attachment and scattering on the breakdown strength of gas mixtures of an electronegative and electron retarding gases. Based on the results it is shown that SF6 has a much higher breakdown voltage than N2. It is also shown that the combination of both gases gives a better dielectric characteristic than the sum of the individual gases. This mixture does have synergistic effects. Based on [8] it is shown that the electron capturing capability of SF6 is only strong for low energy electrons. The lifetime of an SF6 ion formed by capturing the slow electron is much longer than the one formed by capturing the fast electron. This is where N2 comes in; it slows fast electrons into the low energy range very effectively.
Based on the Matlab simulation shown in Figure 2-.3 above, the curve for N2 does reach the value of 18 and only changes slowly with further penetration of the avalanche into the gas. It is because N2 in non-attaching. Based on the above, it may be concluded that pure N2 gas simply saturates once the applied voltage exceeds 331.5 k V, because N2 in non-attaching.

When comparing the results of the 3 simulations/predictions above it noted that SF6 breakdown at 710.3 kV which is the highest of the 3 simulations, but the breakdown occurs at the shortest of the three distances, 1.8 mm. This points out that an extremely high field would be required for breakdown to occur, but once the critical fields is reached, the breakdown will take place quickly even before the streamer propagates an big distance through the gap. This indicates that the breakdown will take place almost on the electrode surface.

But for the mixture, with the breakdown at 575 kV (at 3.15 mm), unlike the SF6 case where the breakdown will take place very close to the electrode, in this case the breakdown will take place at a noticeable distance from the electrode, at a lower voltage though.
It is critical to maintain the breakdown voltage high for a 4km transmission line; hence a duct filled with nitrogen only would not be sensible. In South African there are 220, 275, 400 and 765 kV transmission voltages. Assuming that the 4 km is operated at 400 kV pure SF6 would be an ideal insulating gas.. But considering that SF6 is costly, it would be ideal so strike a compromise and choose the mixture of 80% N2 and 20% SF6, bearing in mind that experimental measurements for the mixture prove that there is synergy in mixing N2 and SF6, therefore the breakdown will be even greater than predicted 575 kV.

3. QUESTION 3 a) Describe the mechanism of leader breakdown in long, nonuniform field gaps. What are the main characteristics of this breakdown mechanism? In what ways does it differ from streamer breakdown? Comment, briefly, on your view of the influence of gap length and pressure When a slow switching surge is applied to a gap, refer to figure 3.1 from [4] below: Stage a: Absence of activity. At the voltage Ui, there is no phenomenon. Stage c: Corona formulation, primary dark periods. With increasing voltage, more explosions of corona are interspersed with dark periods caused by a space charge formed as a result of streamers requiring to drift away from the electron tip before ionisation can begin. [6] Stage i: Leader stem inception at Ul, secondary dark periods. The leader stem and the corona to the electrode combine. Dark periods are caused by the reduction of charge in the leader channel. Stage l: Leader continues to propagate at Ucl. The leader temperature ensures continuous current flow to the tip of the corona explosions, owing to high level of corona activity; Final stage is the final jump at voltage Ub. The leader breakdown mechanism differs from the streamer breakdown because the latter is composed of electron avalanches and while in the former the electron avalanches are continuous because of the collision along the electric field and leader inception happens at a voltage Uc, after the secondary dark periods.

Figure 3-1: The stages in leader breakdown from [6] b) A live worker is required to perform maintenance on a conductor which is 6,0 m above ground. The worker, wearing a conducting suite, will be raised on an insulated arm from ground level to the conductor. Explain the mechanisms which lead to breakdown in the gap, with the worker between the conductor and ground,

(i) when the worker is close to ground and, When the worker is close to the ground, it means the primary gap (d1) (i.e. between the live worker and the conductor) is larger than the secondary gap (i.e. between the live worker and the ground) , this is according to Rizk [5], for positive leader initiation from the protrusion, the required voltage of the floating object in this instance, the live line worker, is: (23) For large values of d1, leader inception from the high voltage will take place below streamer breakdown, so the mechanism that will lead to breakdown of the gap is the leader breakdown. This is the case because of the natural capacitance between the live worker and the ground; the live worker will have enough self capacitance such that the leader takes place as soon as primary gap is bridged by streamers. If this is the case the potential of the live worker will rise. Figure 3-2: Shows configuration of live line worker (ii) When the worker is close to the conductor. This means the primary (d1) gap is small. From Rizk [5], when a switching impulse is applied to the conductor, it will induce a potential in the floating object (live worker). The streamer breakdown of the gap between the conductor and the live worker will take place when the potential for the streamer breakdown is less than the leader inception voltage Ulc [3]. Streamer breakdown will take place when the applied voltage is = Us, which is given by: (24) where Ug the potential of the floating object and Es1 is the minimum streamer gradient of the primary gap which is 400 kV. Therefore: . (25) c) Assuming that the worker can be represented by a sphere 1,0 m in diameter, estimate the minimum breakdown voltage to ground when the gap between the worker’s suit and the conductor is 1,5 m and when it is 2,5 m. Explain the reasons for the way in which you have done the estimations. Assume the voltage coefficient (k◦) for the sphere 1,5m from the conductor is 0,088 and 0,045 when the sphere is 2,5 m from the conductor. Part 1 Given: d = 6 m, d1= 1.5 m, d0= 1m, l0= 1.1m, ko = 0.088 Assumptions: V = 400 kV, lp (protrusion ) = 10%, to eliminate trapped charge in and provide local field enhancement necessary to initiate leader inception from the live worker. Rod – plane geometry. * = Kv (26) * kV (27)
(The capacitance of the primary gap will ensure that sufficient charge can transfer along the streamer channel to convert it to a leader channel at a much lower gradient of El = 50kV/m.)

Therefore the potential of the floating object becomes:

* kV (28) * The voltage of the floating object for leader breakdown of the secondary gap is given by the following equation: * (29) kV
This is less than the potential of the floating object (543-716kV) or equal to Rizk’s value of 637kV and leader breakdown of the gap will take place. * The 50% breakdown gap is kV (30) Part 2
The second scenario that Rizk considers is where the potential for streamer breakdown of the primary gap is higher than the leader breakdown potential. He considers d=6,00m, d1=2,5m, d0 =1,00m, l◦=1,10m, k◦=0,045. * The potential for breakdown of the primary gap is: = kV (31)

This is greater than the continuous leader inception of the full 6,00m gap of
944kV. Breakdown is, therefore, likely by leader in both gaps.

* As the leader penetrates the primary gap, the potential of the floating object will rise. The floating object potential for continuous leader inception in the secondary gap can be readily calculated.

kV (32)

The minimum leader breakdown voltage for the secondary gap is:

(33)

These are much less than the 944kV required for continuous leader inception in the full gap, confirming leader breakdown in both gaps. Rizk has used a numerical field plotting programme to determine the length of the primary leader at the stage of continuous leader inception in the secondary gap. This length is 1,13m.

* kV (34) * The 50% breakdown gap is kV (35)

4. QUESTION 4 a) Briefly explain what corona is and how it manifests in each half cycle of a power frequency wave. Simple sketches should be used to aid your explanation. Your explanation should include discussion on the influence of air density, conductor diameter and conductor surface condition. No more than 600 words are required. (12 marks)

Negative DC Corona Modes
When a non uniform field of a cylindrical conductor-plane air gap is considered, which has a high voltage of negative polarity applied to the conductor and the plane being at ground potential. The field is the highest at the conductor surface and reduces gradually towards the plane.
When the voltage is amplified, an electron avalanche is initiated at the conductor surface and ionisation occurs. Subsequently electron-ion pairs are formed and at some point along the gap all the electrons are attached by oxygen molecules and negative ions are formed. The impact of returning positive ions and photons created in the avalanche, on the conductor surface produce secondary electrons which are required for corona onset [1].
Subsequent to the initial avalanche, negative and positive space charge clouds are formed, with the positive moving towards the conductor and the negative towards the plane. The consequence of the space charges is the increase in the electric field towards the conductor and a decrease of electric field away from the conductor. This has direct influence on the discharge development and gives rise to three modes of negative corona, namely: trichel streamer, negative glow and negative streamer.
The trichel streamer occurs slightly above corona onset. As the voltage is increased, the trichel pulse frequency increases until it reaches a critical value. beyond this voltage streamers merge and form negative glow which is evident by change in visual appearance and steady corona current. The glow continues over a certain voltage, above which negative streamer is formed. The discharge process of the negative streamer never completely stops and an increase in voltage leads to a breakdown, according to Maruvada [1].

Positive DC Corona Modes
When a positive voltage is applied in the conductor-plane gap, the resulting discharge phenomenon is positive corona. The electron avalanche is initiated along the field where the effective ionisation coefficient is bigger than zero. The highest field intensified ionisation occurs in close proximity to the conductor. The secondary electrons necessary for producing a self sustained discharge of positive corona is generated by photo-ionisation. The modes of positive corona are:
Burst corona which occurs at the onset and is caused by spread of ionisation on the conductor surface. The onset streamers are caused by radial development of the discharge. This mode of corona is pulsative in nature and it is the key source of radio interference and audible noise. Under special conditions bust corona may progress to a stable glow of non pulsative called positive glow, which occurs due to a particular combination of rate of creation and removal of positive ions near the conductor. As voltage is increased, breakdown streamer occurs.

AC Corona Modes According to Maruvada [1], with regards to an AC source, corona first appears in the negative half cycle in the form of Trichel streamers. In the positive half cycle, breakdown streamers follow glow corona. Although corona first appears in the negative half cycle the gap breakdown always occurs in the positive half cycle. Figure 4-1 below is an extract taken from Maruvada [1] showing the laboratory results gathered by Trinh and Jordan 1968. It shows the formation of Trichel streamers and glow on the negative half cycle for various voltage levels as well as the breakdown streamers (strong electric field) on the positive half cycle. Onset streamers are suppressed in favor of glow corona. As the voltage is increased the results show that the gap breakdown occurs in the positive half cycle at 106 kV Maruvada [1]. Figure 4-1: AC Corona modes [1]

b) Give and explain the basis for the most commonly used criterion for corona inception on overhead conductors. (5 marks) The surface conductor irregularity factor is m and for an ideal conductor m is 1. The surface of a practical transmission line conductor is not ideal, due to stranding and other irregularities. All these irregularities tend to enhance the electric field in the immediate vicinity of the conductor surface and consequently reduce the corona inception voltage and this could result in very high corona losses. [1] The approximations are as follows * For a clean stranded conductor, depending on the radii of the outer strand and overall conductor m varies between 0.75 and 0.85. * Nick and scratches may reduce this value to between 0.6 and 0.8. * Insects, vegetable matter, water drops, snow and ice – 0.3 to 0.6 * Extreme conditions may reduce m to 0.2. When designing the transmission line, knowledge of corona onset characteristics is essential. For bundled conductors: * Increase the conductor threshold and adjust surface roughness. * Do not design for thick conductors as they are difficult to handle and transport. * Thin conductors as they are easier to manufacture, handle and transport, there is improved mechanical handling, improved heat dissipation, minimised skin effect and lower manufacturing costs.

c) A two-conductor bundle is to be used at an altitude of 2 000m (relative air density _ 0:79) at a three phase system voltage of 275kV line to line . The normal working voltage to ground is, therefore, 275/√3kV. The average height of the conductor bundle above round is 12,0m. To accommodate the anticipated current flow, the overall diameter of each conductor has been chosen as 30,0mm. Arbitrarily, the separation between conductors in the bundle has been chosen at 8 times the conductor diameter, i.e. 240mm.Estimate the corona inception voltage, explaining any assumptions that you make. Comment on whether this is acceptable for the operating voltage of the line. (15 marks) Input Data | r(mm) | 15.00 | θ deg) | 45.00 | h (mm) | 12000.00 | V(kV) | 158.77 | m | 0.70 | δ | 0.79 | B(mm) | 240 | Assume the surface condition irregularity factor m=0.7
Corona inception on cylindrical conductors is determined from Peek’s empirical relationship [4]: (36) = =kV/mm

Figure 5: 2 – Conductor Bundles
Assuming that the electric field will be maximum at point P and weakest at point a. Now we need to find the voltage that is going to give the above corona field q = v.c = V.4π.εOln 2hbr (37) q4π.εO = Vln 2hbr (38) = Vln 2×120.24 ×0.015 (39) = Vln 400 = V5,99 r1=(r sin45)2+(b+r cos45)2 (40) r1=( 10.61)2+(250.612) r1=250.83 β=90-tan-1(10.61250.61) (41) β=90- 2.42=87.58o α=87.58-45 =42.580 (42) εTOT=ε1 +ε2 =q,22πϵo1r+ 1r1 cosα (43) εTOT =q,2πϵo10.015+ 10.25061 cos42.58 εTOT =v5.9966.67+ 2.938 εTOT=11.62vm= 0.01162 v/mm 0.01162v= 2.09kv Therefore, the corona inception voltage (C.I.V), v =2.09kv0.01162 v=179.86kv peak
Comment on whether this is acceptable for voltage of the line
The service voltage is 224.54 kV. The corona inception voltage is 179.86 kV. This is not acceptable as it is far higher than the service voltage of 224.54 kV

d) How could the corona inception voltage be increased? Explore increasing the conductor separation in the bundle, without increasing the conductor diameter, to give an increase in the corona inception voltage of 10%

Table 4-1: Separation distance increase in order to get increase in the corona inception voltage of 10%.

For an increase in the corona inception voltage of 10%, the separation distance will have to be increased from 240 mm to 263 mm.

5. QUESTION 5
A new power station is to be built in the north-west area of Limpopo to tap into the Waterberg coalfields. A steel refinery is also to be built in the vicinity. Both the power station and the refinery are scheduled to commence operation in 2017. In the first phase of operation, the generated voltage will be stepped up and transmitted via 3-phase, 765kV lines, to the main transmission grid.

You have been tasked to select the outdoor insulation to be used for the station and transmission lines.

6.1 Describe what steps you would take to determine the most appropriate insulation for this substation.

In open wire transmission and distribution systems, the performance of the insulators is the key factor in the reliability of these systems [7]. Factors which influence the performance of the insulators for the site in question need to be identified and then steps need to be taken to select insulators which will perform satisfactorily. The first 2 years of the planning stage this project would be dedicated to insulator specification (selection) prior to procurement by assessing the pollution level according to IEC 60815 documents, the following steps practical would be taken:

a) Evaluation of the type and severity of the pollution at site.
When the pollution flashover process of insulators is analysed it can be concluded that ambient pollution severity, the occurrence of fog or mist, wind, natural washing by rain, insulator profile and insulator surface properties are the key factors affect the flashover performance. According to Reynders [7] the IEC 815 standard document has identified 4 levels of pollution, which are classified according to various environmental factors, as shown in Table 5.1 below:

POLLUTION LEVEL | EXAMPLES OF TYPICAL ENVIRONMENTS | 1. Light | * Areas without industries and with low intensity of houses equipped with heating plants. * Areas with low density of industries or houses but subjected to frequent winds and/or rainfalls * Agricultural areas (1) * Mountainous areas All these areas must be situate far from the sea (10 to 20 km) and must not in any case be exposed to winds from the sea. | II - Medium | * Areas with industries not producing particularly polluting smokes and/or with average density of houses equipped with heating plants. * Areas with density of houses and/or industries but subjected to frequent clean winds and/or rainfalls. * Areas exposed to wind from the sea but not too close to the coast (at least a few kilometers) (2). | III - Heavy | * Areas with high density of industries and suburbs of large cities with high density of heating plants producing pollution. * Areas close to sea or in any case exposed to relatively strong winds from the sea. | IV - Very Heavy | * Areas generally of moderate extension, subjected to conductive dusts and to industrial smokes producing particularly thick conductive deposits. * Areas generally of moderate extension, very close to the coast and exposed to sea sprays or to very strong and polluting winds from the sea. * Those desert areas, characterised by no rain for long periods, exposed to strong winds carrying sand and salt, and subjected to regular condensation. |
1. Use of fertilisers sprayed by gun or by plane, or where crop burning has taken place can lead to a higher pollution level due to dispersal by wind.

2. These distances from sea coast depend on the topography of the coastal area and on the wind conditions.

Table 5-1: Pollution levels, with rough description of some typical environments which correspond to each pollution level [7].

In order to evaluate and classify the pollution level of the site in question tests would have to be conducted. A range of tests are available on the market, they are:

* The Dust deposit Gauge * Equivalent Salt Deposit Density (ESDD). * Surface conductance * Leakage current

The pollution tests to be performed at the site to determine the pollution performance and environmental severity are based on the pollution flashover process as mentioned in [8]. These are:

* The Dust deposit Gauge:- The advantages of this technique is that it is the simplest and probably the cheapest because it can be carried out at an unenergised site without insulators or any facilities other than those required for the mounting of the gauges. Its major disadvantages relate to the fact that actual insulators are not used and therefore it is not possible to assess the self cleaning properties of insulators and the effect of the shed profile on the deposition process on the insulator surfaces. In areas of high rainfall, a higher Index can be tolerated whereas in areas of low rainfall but high fog occurrence, the actual severity is higher than that indicated by the gauges. This measurement relates to Phase 1 of the pollution flashover process.

* Due to the techniques limitations, it should be used as the initial preforma test which will give a rough idea of the pollution level in the area, it than needs to be supplemented by tests/techniques that are carried out on actual insulators which assess the self cleaning properties and shed profile performance. Table 5.2 below gives a guide for the relationship between the Pollution Index and the Pollution Level as defined in table 5-1 [8]. POLLUTION LEVEL | POLLUTION INDEX | I – Light | | | | 0 – 75 | | II - Medium | | | | 76 – 250 | | II – Heavy | | | | 251 – 500 | | IV – Very Heavy | | | | 500 | | Table 5-2: Pollution Levels and Pollution Index deduced from the Dust Gauge Measurements.

* Equivalent Salt Deposit Density: - This would compliment and support the previous test. The test provides an equivalent deposit in mg NaCl/cm2 of the surface area of an insulator. It also relates to phase 1 of the pollution process. The test can be carried out on actual insulators and self cleaning properties and shed profile performance can be assessed. Table 5-3 below gives a range of ESDD levels [7]

POLLUTION LEVEL | ESDD | I – Light | 0.03 – 0.06 | II – Medium | 0.10 – 0.20 | III – Heavy | 0.30 – 0.60 | IV –Very Heavy | >0.80 | Table 5-3: Pollution level and ESDD [7] * Surface conductance: - This is the ratio of the power frequency current flowing over a sample insulator to the applied voltage. The conductance will aid indicate the overall state of the insulator surface. The shed profile of the insulator can be assessed in terms of the contamination collection.

The advantage of this method is that actual insulators are used and thus the self-cleaning properties, the performance of the shed profile and the creepage length of the insulator can be assessed. This test involves phases 1 and 2 of the pollution flashover process. Table 5-4 gives a range of surface conductivity and pollution level.

POLLUTION LEVEL | SURFACE CONDUCTIVITYµS | I – Light | 15-20 | II – Medium | 24-35 | III – Heavy | 36 | Table 5-4: Pollution level and Surface Conductivity [7]. * Leakage current: - This test could be carried out on a similar existing substation which has similar characteristics (voltage, etc) as the proposed new station in the north-west area of Limpopo close to the Waterberg coalfields. This test requires a fully energized site. The measurement of the leakage current incorporates phase 1 to phase 5 in the pollution flashover process. Since the layer of pollution allows for the passage of current pulses over the surface of an insulator. This test will help assess the severity of the site as well as provide information on the self cleaning properties, the shed profile performance and the suitability of the creepage length.

This test can be divided into surge counting and Ihighest. For this site the Ihighest is recommended. This is the highest peak current recorded during a given period, over a sample or actual insulator continuously energised at its working voltage. This current can be related to the maximum peak current which the insulator surface will support without flashover occurring. The following relationship can be used. (44) where S is the specific creepage in mm/kV.
For satisfactory insulator performance:- (45)

b) Selection of suitable insulators on the basis of:

(i) Creepage distance
(ii) Shed profile (iii) Shed material

* Creepage distance: - The dry band arc is controlled by the series resistance of the pollution layer. The greater the resistance is the smaller the arc current. This resistance is related to the length of the current path along the insulator surface and is described as the creepage distance. If hydrophobic materials are used, a shorter creepage distance can be adopted. It is necessary to determine this for the installation. For the site in question, the minimum creepage distance should be 31 kV/mm as shown in table 5-5 below [7].

POLLUTION LEVEL | MINIMUM SPECIFIC NOMINAL CREEPAGE DISTANCE BETWEEN PHASE AND GROUND | I – Light | 16 | II – Medium | 20 | III – Heavy | 25 | IV – Very Heavy | 31 | Table 5-5: Pollution level and Creepage Distance [7] * Shed profile: - The shed profile needs to be designed so that, the insulator surfaces are sufficiently aerodynamic to minimise the deposition of pollution. Rain washing also needs to be effective, and it has to be ensured that the bridging of adjacent sheds by the rain will not take place during heavy rainfall conditions [7]. * Shed material and composite insulators: - The importance of choosing the correct material for insulator use is crucial. Glass and porcelain have been used extensively in the past. In recent years, non ceramic materials that are carbon or silicon based have been used due to their hydrophobic nature and inhibit the formation of a conducting path along the insulator surface. It is also possible to get away with shorter creepage distances than required for glass and porcelain insulators. Certain non-ceramic insulators need not have an axial length as long as that of porcelain or glass. These are the relative axial lengths found for the same performance in a heavily polluted environment [7].

INSULATOR MATERIAL | RELATIVE LENGTH | Silicone Rubber | 0.7 | Ethylene Propylene Rubber | 0.9 | Porcelain | 1.0 | Epoxy-resin | 1.1 | Table 5-6: Relative axial lengths of insulators for equal flashover performance [7].

c) Site testing of prototypes prior to specification of final installation if possible.

6.2 Assume what you regard as reasonable outcomes from these steps and select insulators for both the substation suspension and the line insulation. A justification for this selection must be provided. The procedures listed above need to be followed in order to make necessary recommendations for the correct selection of the insulators for the site under discussion. Based on the pollution severity and environmental tests, one can make the following assumptions: * The Dust deposit Gauge: Based on the site of the new power station and steel refinery, this test is important as it will show the equivalent deposit on the insulators in a simple and cheap manner. * Equivalent Salt Deposit Density: Based on the site of the new power station and steel refinery, this test is important as it will show the equivalent deposit on the insulators which will have electrical conductivity equal to that which will be on site. * Surface Conductance test: This will help us choose the correct insulator type with the corresponding form factor. * Leakage current: The technique provides information on the severity of the site and the ability of the insulator to cope with the pollution. * Table 5-6: Relative axial lengths of insulators for equal flashover performance [7].
Assumptions for reasonable outcomes for the site under discussion
Since the site is situated in mining environment in the vicinity of the Waterburg coalfields and considering the proposed steel refinery in close proximity, pollution level 3 (heavy) has been chosen. The main concern on this site is a combination of deposits which it will be prone to like industrial smokes, and coal dust particles. Measurement | Outcome | Dust gauge | 470 | ESDD | 0.6 | Surface conductivity(µS) | 35 | Creepage distance(S) | 25 | Imax | (25/15.36)2=2.65 | Ihighest | 0.66 | Table 5-7: Assumptions for reasonable outcomes for the site under discussion Based on the above assessments, the insulators that will be suitable for the power lines and substation are the polymeric insulators, specifically the Silicon Rubber. * Silicon rubber is said to be hydrophobic and thus inhibit the formation of the conducting path along the insulator surface. * They can perform satisfactorily with shorter minimum specific creepage distance. * They are less prone to vandalism due to their elasticity. * Silicon rubber contains low molecular weight components which are fundamental in the diffusion process and they are the main agent for surface recovery after ageing has led to loss of hydrophobicity. This is proves that normal ageing of silicon rubber does not lead to loss of performance. [9] * They are light weight, easy and economical to transport and install. * The only disadvantage is the main common cause of failure is mechanical.

6.3 Do you foresee the need for monitoring and/or maintenance to be carried out on the insulators? Give your reasons and explain the likely outcomes of any procedures that you recommend.

Constant monitoring and maintenance will have to be carried out on the Silicon rubber insulators because early detection or warning signs for any possible defects on the line, substation equipment and the power station would improve the performance of the electrical network. Silicon rubber insulators are known to be prone to corona and their hydrophobic properties exacerbates corrosion failures. In order to monitor the performance of the insulators the leakage current flowing over the surface of the insulators will have to be measured continuously and stored [7]. Due to the expected heavy pollution, water greasing or the use of insulation coating should be considered as a control measure [7]. Washing also comes at very high costs but a good investment.The proposed maintenance plan is showed below.

DURATION (Stn / Lines) yrs | | SUBSTATION | | LINES | | | | | | 1,5 / ≥ 2 -3 | | WASHING | | WASHING | | | | | | 1,5 / ≥ 2 -3 | | GREASING | | GREASING | | | | | | 0,5 / ≥ 2 -3 | | RADIATION | | Not Needed | | | | | |
Table 5-8. Shows the proposed maintenance plan for this station

6. REFERENCES

P.S. Maruvada, Chapter 8: Corona and Gap Discharge Phenomena, EPRI AC Transmission Line Reference Book – 200 kV and above.

[2] Pedersen A. Calculation of Spark Breakdown or Corona Starting Voltages in Nonuniform Fields,.IEEE Transaction on Power Apparatus and Systems. Vol. PAS- . 86, number 2, February. 1967, pp 200-206

[3] Pedersen A. Criteria For Spark Breakdown In Sulfur Hexafluoride .I Department of Physics. The Technical University. Lyngby. Denmark.EEE Transaction on Power Apparatus and Systems. Vol. PAS- . 89, number 8, November/December. 1970, pp 2043-2048

[4] Reynders J P, Electrical Discharges in gases – Lecture notes, 2011.

[5] Rizk F A M, Effect of conducting objects on critical switching impulse breakdown of long airgaps, Cigr´e Session 1994, Paper 33-301.

[6] Rizk F A M, A model for switching leader impulse inception and breakdown of long air-gaps. IEEE Transactions on Power Delivery, V4, N◦1, January 1989, pp596-606.

[7] Reynders J P, Guide for the choice of outdoor insulators for ac systems under polluted conditions. National Energy Council Report, 1990.
[8] Zhou M, Reynders J P, Synergy between SF6 and other gases to enhance dielectric strength. 7th ISH, Montreal, August, 1997, [1] Niemeyer L, Ulrich L, Wiegart N, The mechanism of leader breakdown in electronegative gases IEEE transactions on Electrical Insulation, V 24, N◦2, April 1989,pp309-324.

7. APPENDIXTURE
Appendix 1: Breakdown voltage prediction in air using Pedersen

Appendix 2: Breakdown voltage prediction in air using Reather

Appendix 3: Breakdown voltage prediction in oxygen using Reather

Appendix 4: Breakdown voltage prediction in pure nitrogen using Reather

Appendix 5: Matlab code for pure sulpher breakdown voltage at pressure of 3 Bars
% Mixcyl.m clear delR = ((0:.001:10)*1e-3)'; % Plotting range & interval, m px=mean(delR)*0.9*1e3; R1 = 0.05; % Inner conductor radius, m
R2 = 0.1; % Outer conductor, inner radius, m pn2 = 0.0; %Partial pressure of Nitrogen psf6 = 3.0;%0.8;%0.3; %Partial pressure of SF6 ruf = 66/89; %Pederson's roughness coefficient for SF6 pn = pn2/(pn2+psf6); ps = psf6/(pn2+psf6); ks = 0.028; %Coefficient for alpha bar in SF6 ka = ks*ps; kc = 8.89e6; %Critical field at 1 bar in SF6 kn = 5.32e5; %Coefficient for alpha in N2 at 1 bar ki = 2.08e7; %Index coefficient for alpha in N2 at 1 bar kb = kn*pn;
R = R1 + delR; pt = pn2+psf6; % Total pressure, bar V=input('Voltage estimate in kv '); V1=V*1000; ints = ka*V1*log(R/R1)/log(R2/R1) - ka*kc*pt*ruf*delR; %Streamer integral for SF6 intn = kb*V1/(ki*log(R2/R1))*(exp(-(ki*pt*log(R2/R1)*R1/V1))-exp(-(ki*pt*log(R2/R1)*R/V1))); %Streamer integral for N2 int = ints+intn; Intmax=max(int)
Distance=delR(find(int>=max(int)))*1000

plot(delR*1000,int,'linewidth',2); xlabel('{\fontsize{12}Distance from inner conductor, mm}') ylabel('{\fontsize{12}Integral}') title(['Breakdown in SF_6/N_2 mixture: pSF_6 = ',num2str(psf6),' bar',', pN_2 = ',num2str(pn2),' bar',', Roughness coeff = ',sprintf('%.2f',ruf),', R_1 = ',num2str(R1*1000),'mm',', R_2 = ',num2str(R2*1000),'mm']) clear grid grid text(px,0.75*max(int),['Max integral ',sprintf('%.1f',max(int)),' at ',num2str(Distance),' mm']) text(px,0.82*max(int),[' V = ',num2str(V),'kV'])

Appendix 6: Matlab code for 20% sulpher and 80% nitrogen breakdown voltage at pressure of 3 Bars
% Mixcyl.m clear delR = ((0:.001:10)*1e-3)'; % Plotting range & interval, m px=mean(delR)*0.9*1e3; R1 = 0.05; % Inner conductor radius, m
R2 = 0.1; % Outer conductor, inner radius, m pn2 = 0.0; %Partial pressure of Nitrogen psf6 = 3.0;%0.8;%0.3; %Partial pressure of SF6 ruf = 66/89; %Pederson's roughness coefficient for SF6 pn = pn2/(pn2+psf6); ps = psf6/(pn2+psf6); ks = 0.028; %Coefficient for alpha bar in SF6 ka = ks*ps; kc = 8.89e6; %Critical field at 1 bar in SF6 kn = 5.32e5; %Coefficient for alpha in N2 at 1 bar ki = 2.08e7; %Index coefficient for alpha in N2 at 1 bar kb = kn*pn;
R = R1 + delR; pt = pn2+psf6; % Total pressure, bar V=input('Voltage estimate in kv '); V1=V*1000; ints = ka*V1*log(R/R1)/log(R2/R1) - ka*kc*pt*ruf*delR; %Streamer integral for SF6 intn = kb*V1/(ki*log(R2/R1))*(exp(-(ki*pt*log(R2/R1)*R1/V1))-exp(-(ki*pt*log(R2/R1)*R/V1))); %Streamer integral for N2 int = ints+intn; Intmax=max(int)
Distance=delR(find(int>=max(int)))*1000

plot(delR*1000,int,'linewidth',2); xlabel('{\fontsize{12}Distance from inner conductor, mm}') ylabel('{\fontsize{12}Integral}') title(['Breakdown in SF_6/N_2 mixture: pSF_6 = ',num2str(psf6),' bar',', pN_2 = ',num2str(pn2),' bar',', Roughness coeff = ',sprintf('%.2f',ruf),', R_1 = ',num2str(R1*1000),'mm',', R_2 = ',num2str(R2*1000),'mm']) clear grid grid text(px,0.75*max(int),['Max integral ',sprintf('%.1f',max(int)),' at ',num2str(Distance),' mm']) text(px,0.82*max(int),[' V = ',num2str(V),'kV'])

Appendix 7: Matlab code for pure nitrogen breakdown voltage at pressure of 3 Bars
% Mixcyl.m clear delR = ((0:.01:100)*1e-3)'; % Plotting range & interval, m px=mean(delR)*0.9*1e3; R1 = 0.05; % Inner conductor radius, m
R2 = 0.1; % Outer conductor, inner radius, m pn2 = 3.0; %Partial pressure of Nitrogen psf6 = 0.0;%0.8;%0.3; %Partial pressure of SF6 ruf = 66/89; %Pederson's roughness coefficient for SF6 pn = pn2/(pn2+psf6); ps = psf6/(pn2+psf6); ks = 0.028; %Coefficient for alpha bar in SF6 ka = ks*ps; kc = 8.89e6; %Critical field at 1 bar in SF6 kn = 5.32e5; %Coefficient for alpha in N2 at 1 bar ki = 2.08e7; %Index coefficient for alpha in N2 at 1 bar kb = kn*pn;
R = R1 + delR; pt = pn2+psf6; % Total pressure, bar V=input('Voltage estimate in kv '); V1=V*1000; ints = ka*V1*log(R/R1)/log(R2/R1) - ka*kc*pt*ruf*delR; %Streamer integral for SF6 intn = kb*V1/(ki*log(R2/R1))*(exp(-(ki*pt*log(R2/R1)*R1/V1))-exp(-(ki*pt*log(R2/R1)*R/V1))); %Streamer integral for N2 int = ints+intn; Intmax=max(int)
Distance=delR(find(int>=max(int)))*1000

plot(delR*1000,int,'linewidth',2); xlabel('{\fontsize{12}Distance from inner conductor, mm}') ylabel('{\fontsize{12}Integral}') title(['Breakdown in SF_6/N_2 mixture: pSF_6 = ',num2str(psf6),' bar',', pN_2 = ',num2str(pn2),' bar',', Roughness coeff = ',sprintf('%.2f',ruf),', R_1 = ',num2str(R1*1000),'mm',', R_2 = ',num2str(R2*1000),'mm']) clear grid grid text(px,0.75*max(int),['Max integral ',sprintf('%.1f',max(int)),' at ',num2str(Distance),' mm']) text(px,0.82*max(int),[' V = ',num2str(V),'kV'])