Program To find sum of five elements of an array

void main ()

{

int no[5],i,sum;

clrscr ();

for (i=0;i<=4;i++)

{

printf ("Enter Element: ");

scanf ("%d",&no[i]);

}

sum=no[0]+no[1]+no[2]+no[3]+no[4];

printf ("\nSum of five Elements: %d",sum);

getch ();

}

Program To find out Year is leap year or not.

void main ()

{

int a;

clrscr ();

printf ("Enter the Year: ");

scanf("%d",&a);

if (a%4==0)

{

printf ("\nYear is Leap");

}

else

{

printf("\nYear is not Leap");

}

getch ();

}

Output :

What is Corona?

Corona is caused by the electric field next to an object exceeding the breakdown value for air (or whatever it is immersed in). Since the magnitude of the field is inversely proportional to the radius of curvature, sharper edges break down sooner. The corona starting voltage is typically 30 kV/cm radius. Dust or water particles on the surface of the object reduce the corona starting voltage, probably by providing local areas of tighter curvature, and hence higher field stress.

The easiest case to analyze is that of a sphere. The magnitude of the electric field at the surface of a sphere in free space is simply the voltage/radius. Note that if the sphere is near another conductor, the field is no longer uniform, as the charge will redistribute itself towards an adjacent conductor, increasing the field.

Since corona is fundamentally a breakdown phenomenon, it follows Paschen's law: the voltage is a function of pd. Double all the dimensions and halve the gas pressure, and the corona voltage will be pretty much the same.

Corona Surface Factor

The following table gives empirically determined correction factors for various surface conditions. These factors are multiplied by the corona starting voltage (or field) to determine the corrected voltage.

Condition of Conductor	m0
New, unwashed	0.67-0.74
Washed with grease solvent	0.91-0.93
Scratch-brushed	0.88
Buffed	1.00
Dragged and dusty	0.72-0.75
Weathered (5 months)	0.95
Weathered at low humidity	0.92
For general design	0.87-0.90
7 strand concentric lay cable	0.83-0.87
19, 37, and 61 strand concentric lay cable	0.80-0.85

Eliminating or reducing corona

Smoothly radiusing the corners of objects at high voltages relative to nearby objects will reduce the local field strength.

Put the sharp corner in something with a higher breakdown strength than air. The trick here is to make sure that you have really got the replacement substance in contact with the conductor. By making the high field occur within a substance with a higher breakdown than the surrounding air, corona can be reduced.

Covering sharp corners with an insulating film increases the corona starting voltage at the points with high E-field stress. Generically known as "corona dope", this is an enamel or polystyrene paints or gels that you can apply. Glyptal is one example, and clear nail polish has also been used. Clear acrylic spray paint is another alternative, although the coating is quite thin.

Potting the entire assembly in an insulator (traditionally paraffin or sulfur were used, silicone RTV is a more popular modern alternative) achieves the same result. Immersing the assembly in oil or other insulating fluids will also work. All of the potting and immersion techniques depend on removing the air or gas bubbles to work. Commercial manufacturers pull a vacuum on the container while the assembly is being potted to facilitate the removal of the air bubbles. Experimenters building polyethylene and aluminum foil capacitors for tesla coils run them at low powers using the electrostatic forces between the plates to vibrate and pump the air bubbles out.

A popular approach to reducing corona on wires is to surrounding the conductor by a semiconducting film or layer of greater radius. This effectively increases the radius of the object, and hence lowers the field strength. You may not need a huge amount of copper to carry the required current (often micro or milliamps), but you want the diameter of the conductor large enough to reduce the corona. Wire of this type is manufactured by Belden, Rowe-Talley, and Caton, among others.

Field grading rings are often used on high voltage equipment to control the electric field distribution. Rather than rely the field that would exist in free space between two charged conductors, a series of other conductors are interposed at intermediate voltages. The intermediate voltages are derived from a capacitive or resistive divider. A capacitive divider may be a simple as the interelectrode capacitances of the grading rings themselves.

Running the system in a tank at high pressure, or in an insulating gas, will increase the corona starting voltage.

Kerr effect

The Kerr effect results from the impressed electric field causes the assymetric molecules of the liquid to align with the field. This causes the liquid to become anisotropic and birefringent. The change in index is given by:

n_o-n_e = K * E² * lambda

where:
lambda is the wavelength of the light
E is the electric field strength
K is the Kerr constant

A Kerr Cell is a cell containing the liquid (typically Nitrobenzene) between two flat parallel plates spaced several millimeters. A fairly high voltage (typically 10-20 kV) is placed on the plates..

If the field is such that the cell retards the extraordinary ray by a half wavelength, the polarization rotation will be 90 degrees. If a pair of polarizers is put around the cell, oriented at 45degrees, the assembly acts as a shutter. The voltage required to do this is called the "halfwavelength voltage".

At the half wave voltage, the following is true.

(n_o-n_e)*d = lambda / 2

where
d is length of cell
other variables as above.

Note that the wavelength cancels out when rearranging and substituting to give:

E_halfwave = sqrt( 1/ (2 * d * K))
for calculating the halfwave E field

d = 1 / (2 * K * E^2)
for calculating required length of cell

Sample Kerr Constants

Nitrobenzene	2.4E-10 cm/V2
Glasses	3E-14 to 2E-23 cm/V2
Water	4.4E-12 cm/V2

For Nitrobenzene (K=2.4E-10 cm/V² ) and 30 kV/cm (breakdown of air), d = 2.3 cm

Pockels

(n_o-n_e) = pE

where E is the applied field, and p is a proportionality constant:

A similar calculation to that for Kerr cells can be made to determine the half wave voltage for the cell.

KDP (Potassium Dihydrogen Phosphate)	3.6E-11 meter/Volt
Deuterated KDP (KD*P)	8.0E-11 meter/Volt
Lithium Niobate (LiNbO3)	3.7E-10 meter/Volt

Faraday

Rotation = VBl

where
Rotation in radians
V is Verdet Constant
B is the magnetic field strength
l is the length

Verdet Constant (if l in millimeters, B in Tesla)

fused quartz	0.004
dense flint glass	0.11
Benzene	0.0087

A peculiarity of Faraday rotation is that it rotates the same direction (e.g. Clockwise or Counterclockwise) no matter which direction the light is travelling. This can be used to make a one way light valve with two polarizers set at 45 degrees to each other.

Electro-optical measurements

Kerr, Pockels, and Faraday

Conventional means of measuring high voltages and currents rely on the measurement consuming a (hopefully) small amount of the power from the system. For instance, a resistive divider does draw some small amount of current. In power engineering, this small power is called the "burden".

Instead of directly measuring the quantity of interest, you can measure the changes in properties of some material as a result of the surrounding electrical or magnetic field. The power required for making the measurement is provided by the measuring equipment itself. One family of these techniques relies on the changes in optical properties of certain materials in electrical and magnetic fields: the Kerr effect, the Pockels effect, and the Faraday effect.

All of these techniques rely on various mechanisms by which a material rotates the polarization of light passing through. The amount of rotation depends on the electric or magnetic field. The performace is determined largely by how well you can measure the change in polarization of the light. High quality polarizing film has a transmission ratio of 1000:1 between aligned and crossed.

If one wanted to measure the E field around an operating Tesla coil, as well as the waveforms, the electro-optic sensorl could be mounted on a long insulated rod with fiber optic cables to send the light to and from the measuring cell. An alternate scheme could be to use a laser and appropriate prisms or mirrors to send the light out to the cell along the support and to return it to a detector. In the latter case, the sensor itself could be mounted to the high voltage terminal, with the laser and detector mounted at some distance away.

Kerr Cells

Often used to create extremely high speed shutters, the Kerr effect is an anisotropic change in the index of refraction of a substance in response to an electric field. A practical implementation has the Kerr substance (often nitrobenzene, which has a very high Kerr Constant) between two crossed polarizers. The polarization of the light is rotated in proportion to the square of the E field, allowing some light to pass through the polarizers. As a shutter, the response time of the Kerr Cell is limited mainly by how fast the E field can be changed.

The problems with a Kerr Cell are: nitrobenzene is a volatile solvent which is remarkably toxic; the effect is proportional to the square of the E-field, which is no problem for a shutter application, but not as appropriate for a measurement application.

Pockels Cells

The Pockels effect is similar to that of the Kerr effect, except that the change in index is linearly proportional to the electric field. Substances such as KDP (Potassium Dihydrogen Phosphate), KD*P (Deuterated KDP) and LiNbO3 (Lithium Niobate) show large Pockels effects and are very popular as electro-optic modulators for laser work.

One problem with Pockels sensors is the cost of the crystals, particularly in large sizes. A small 1 cm diameter crystal suitable for turning on and off a laser beam isn't particularly expensive (several hundreds of dollars), but a larger one for use as a photographic shutter would be prohibitively expensive. For a HV measuring application, the cell could be on the scale of millimeters, particularly if fiberoptic cables are used.

Faraday Rotation

Faraday rotation is a magnetic effect. Notable in high density lead glass, the rotation is proportional to the magnetic field. A chunk of lead glass 1" thick and 2" in diameter would need a field of .5 Tesla (5000 Gauss) to rotate the polarization 90 degrees. (This is about 10,000 ampere turns for that physical size). The rotation is proportional to the length of the optical path and to the magnetic field, so a longer piece of glass makes a more sensitive detector.

Faraday rotation does provide a handy way to measure the current in EHV or UHV power lines. A piece of lead glass (which can be quite long) is placed near the power cable and a polarized laser is used to measure the rotation. In a Tesla coil application, lead glass sensors connected by fiber optic cables could be used to measure the current at various parts of the coil. For that matter, a glass fiber of the appropriate material could be used as the sensor itself.

Voltage Dividers

A general method for measuring high voltages is to use a voltage divider composed of two impedances in series. The ratio of impedance is such that the voltage across one of the elements is some convenient fraction (like 1/1000) of the voltage across the combination.

To make the power consumption of the divider as low as possible, the impedances are quite large: 10's of Gigaohm (1e9 ohms) might be used for measuring megavolt level signals (resulting in a current of a few tens of microamps). In an ideal world, the impedances would be pure resistors. The physically large size and the high impedances of high voltage equipment means that parasitic inductances and capacitances can be significant. Even at 60 Hz, a 10pF parasitic C has an impedance of 260 Megohm. 10 pF is roughly the capacitance of a 10 cm radius sphere (8" diameter). If the resistor string is 2 meters long, it's inductance is probably several microhenries, not particularly significant at power line frequencies, but a signficant concern at the higher frequencies encountered in fast impulse work. Measuring voltages or potentials with any AC component is greatly affected by these parasitic reactances, and much of high quality divider design goes to minimizing or compensating their effect.

For making AC measurements, purely capacitive dividers are popular. A fairly small capacitor forms the upper arm of the divider, and a larger, lower voltage capacitor forms the bottom. High pressure gas capacitors are popular for the high voltage arm. A high pressure gas capacitor can provide a reasonable capacitance with a high voltage rating in a physically small package, which is important for measurements on fast transients.

Thermal effects

Small as the current is through most high value resistive dividers, it may consititute a significant amount of power, which goes into heating yup the resistive elements. This heating will cause a change in the value of the resistor, changing the overall ratio of the divider.

Classic standards work, as reported in Craggs & Meek, used maganin resistors. Manganin has an extremely low temperature coefficient of resistance (1.5 ppm/deg C) (see Resistance Wire Table) compared to Nichrome ( 13 ppm) or Copper ( ppm).

Immersing the entire resistive divider in oil or rapidly circulating dielectric gas (e.g. SF6 or dry air) also ensures that all components are at the same temperature, so that, while the absolute values might change, the ratios will remain constant, for DC at least. Resistance value changes will change the parasitic RC time constants, changing the frequency response.

Voltage Coefficient of Resistance

Some resistive materials show a change in the resistivity as a function of the impressed electric field strength. This would manifest itself as a change in the resistance as the voltage changes. A long string of individual resistors, each run at a relatively low voltage, should not show this effect.

Safety Considerations

In the classic series resistor method for measuring voltage, the high value resistor string is in series with a sensistive current measuring meter (typically a d'Arsonval meter). If the resistor were to fail shorted, or flash over, the high voltage would appear across the meter, possibly producing a personnel safety hazard, as well as destroying the meter. A simple safety precaution is a spark gap across the meter, set for a kilovolt or so, that will arc over in case of a series resistor failure.

Another means is to measure current through the high value resistor by measuring the voltage across a resistor with a high impedance meter.

High Voltage Wire and Cable

There are two issues which need attention with high voltage wiring. The first is the level of insulation necessary to prevent arcs to adjacent components or wiring. The second is the diameter (or effective diameter) necessary to reduce corona losses. Reduction of corona is important because a common failure mode for insulation is the formation of small defects (i.e. pinholes) in the insulation due to corona discharges within the insulation.

With bare conductors, air is the insulator, and clearance distances can be calcuated using standard values for the breakdown of air. A common rule of thumb which is very conservative is 1 inch per 10 kV. Since the breakdown field for air is around 71 kV/inch, this provides a 7:1 safety factor.

Popular insulation materials for hookup type wire are polyethylene, PTFE, rubber, and silicone, particularly the latter. Neon signs are a cost sensitive application, so inexpensive wire ($.15/ft) rated at 15 kV with polyethylene insulation is widely available. Rubber is popular for test leads at the 5 kV level, although many rubbers degrade in the presence of ozone, which is often present in HV equipment. High quality high voltage wire has silicone insulation which is quite flexible and high temperature resistant.Typical prices for silicone insulated wire range from $.20/ft for 10kV rated to $2.00/ft for 50 kV rated.

Corona resistant wire is typically constructed with a central copper core surrounded by a semiconducting sheath, which in turn is surrounded by the insulation. The semiconducting sheath effectively increases the diameter of the wire, reducing the tendency for corona discharge. Suppliers of such wire include Belden, Caton, Tally, etc.

Coaxial cable of the RG-8 (RG-213) family is often used as high voltage cabling for several tens of kV. Grounding the outer shield makes the field distribution inside the cable very even, reducing the field concentrations that start corona. RG-8 is rated at 5 kV RMS, however, the polyethylene insulation is (.285-.01??) .120 inches thick which corresponds to 120 kV breakdown. I suspect that the 5kV rating (7 kV pk) allows for a substantial VSWR in transmission line use without breakdown. Certainly, many systems use RG-8 at 25 kV, and I have seen some at 50 kV using RG-8 as a conductor. Also, the field strength at the inner conductor is higher than that at the outer conductor

Equation here.

Having the outer surface of the cable at ground potential also confers some safety advantages. Don't forget though, that in systems with sufficient stored energy, the coax can literally explode in the event of a dielectric failure. If you have several tens of kJoules stored up, the energy has to go somewhere. At least you won't get shocked, just burned.

Coaxial cable using foamed dielectrics (e.g. RG-8X) are not useful, since the nitrogen used to make the foam has a much lower breakdown than the PE. The same goes for RG-59 cable TV remnants, because they are usually foamed insulation (cheaper and lower loss).

Coaxial cable also has the advantage of low series impedance in pulsed circuits, as does other types of transmission lines such as twinlead and quadroline.

The so-called UHF connector (SO-279, PL-259) can be modified as a high voltage connector for use with RG-8 family coax by drilling out the center and extending the center conductor (of the plug) into a tube with a banana jack at the end. The jack can be modified by mounting the threaded outer housing (drilled out) on a block of insulator (acrylic, G10 glass epoxy).

Photo here

Another ubiquitous source of high voltage hookup wire is spark plug cable for automotive use. The more common variety has a resistive core (used to slow the rise time reducing EMI) of a few kOhms per foot. A less common variety, called solid core or copper core, the conductor is normal wire. Spark plug cable typically has a very rugged silicone or hypalon jacket, as well as a fibrous armor layer. Spark plug cable costs about $1/ft

High Voltage Fuses

A fuse is a circuit element designed to melt when the current exceeds some limit, thereby opening the circuit. In high voltage and high power applications, some additional design considerations come into play. For instance, if the length of the fuse wire or strip is short enough, an arc will form between the ends maintaining the circuit as long as there is current to supply it. In systems with high peak current capability (i.e. with capacitors and low impedance circuitry), the fuse can be melted and vaporized so fast that an explosion occurs. This phenomenon is actually used in Exploding Bridge Wire detonators to create a shockwave that detonates high explosive.

The basic design equation for fuses is the Preece equation (W.H. Preece, Royal Soc. Proc., London, 36, p464, 1884) for wires in free air:

i = A * D^1.5

where A is a constant depending on the metal and D is the diameter of the wire.

Fuse Wire	A (d in inches)	A (d in mm)	Melting temp (deg C)	Boiling temp (deg C)
Copper	10244	80.0	1083	2300
Aluminum	7585	59.3	660	1800
German Silver	5230	40.9
Platinum	5172	40.4	1774	4300
Silver*	3200	49.8	960	1950
Iron	3148	24.6	1535	3000
Tin	1642	12.8	232	2260
Lead	1379	11.8	327	1620
Tungsten*	105	1.5	3370	5900

*Exponent in the equation should be adjusted to 1.287 for silver and 1.32 for tungsten.

Table taken from Standard Handbook for Electrical Engineers, 6th ed., Sec 15, p153.

Some fuses put the wire inside an insulating tube so that the tube walls can contain the gases created by the vaporized wire. The tube walls also cool the gases extinguishing the arc. In exploding wire type fuses, the tube walls reflect a shock wave back to the center of the arc channel increasing the pressure to raise the breakdown voltage. Expulsion protector tube type fuses use the expanding gases to actually blow the arc out the end of the tube. One author comments: "Care should be taken in locating the vents of the expulsion gaps, for flaming gas is blown for a considerable distance upon operation.... [Flames may range] from 5 ft for a 1,000 amp crest current to 12 ft for a 10 kA crest current." (Cobine, p411)

Fuses may be filled with a refractory material: silica (sand), alumina, or zirconia. The arc energy is used up in fusing the filler. Silica can absorb about 2 kJ/g.

In many circuits, the creation of an arc is actually necessary, since it provides a gradually increasing voltage drop as it cools to "gradually" interrupt the circuit. The energy dissipated in the arc also absorbs the inductive energy stored in the circuit. A sudden total interruption may cause very high terminal voltages due to series L. A similar problem crops up in the design of circuit breakers or other interrupters.

Here is another design equation due to I.M. Onderdonk:

Ifuse = Area * SQRT( LOG((Tmelt-Tambient)/(234-Tambient)+1)/ (Time * 33))

where

Tmelt = melting temp of wire in deg C
Tambient = ambient temp in deg C
Time = melting time in seconds
Ifuse = fusing current in amps
Area = wire area in circular mils

Practical example :

16 gauge copper wire: Tmelt = 1083, Area = 2581 circ mil, Time = 5 sec,diam = .0524 inches

Using Preece equation:

= 10244*.0524^1.5 = 123 Amps

Using Onderdonk equation:

Ifuse = 2581 * SQRT( LOG((1083-25)/(234-25)+1)/(5*33))

= 2581 * sqrt(log(1058/209+1)/165)
= 2581 * sqrt(.0047)
= 178 Amps

Wheeler Formulas for Inductance

These formulas, developed by Wheeler at the (then) NBS, give approximate inductances for various coil configurations. They are primarily based on empirical measurements, and are accurate to a few percent.

Single layer air core solenoid

L (uH) = r^2 * n^2 / (9 * r + 10 * l)

where

r = coil radius in inches
l = coil length in inches
n = number of turns

Multi layer air core solenoid

L {uH} = 31.6 * N^2 * r1^2 / (6*r1 + 9*L + 10*(r2-r1))

L{uH} = Inductance in microHenries
N^2 = Total Number of turns on coil Squared
r1 = Radius of the inside of the coil {meters}
r2 = Radius of the outside of the coil {meters}
L = Length of the coil {meters}

Note the similarity to the formula for the single layer
air core coil. Hope this helps everyone.

L (uH) = 0.8 * a^2 * n^2 / (6*a + 9*b + 10*c )

where
a = average radius of windings
b = length of the coil
c = difference between the outer and inner radii of the coil.
all dimensions in inches.

It states that it is accurate to 1% when the terms in the denominator are
about equal. This is also an equation by Wheeler. It applies as long as the
coil has a rectangular cross section.

Flat "pancake" coil

L (uH) = r^2 * n^2 / (8 * r + 11 * w)

where

r = radius to center of windings in inches
w = width of windings (in inches)
n = number of turns

High Voltage Resistors

Even such a mundane component as a resistor has special requirements when it comes to high voltage applications. The usual little 1/4 watt carbon film resistor used in most other electronics is only rated to 250 or 500 volts, a far cry from the kilovolt levels needed. The voltage limitation is usually set by power dissipation issues: a 10K resistor with 1 kV across it dissipates 100 watts! And, of course, the physical length of the device of around a centimeter means that around 5-10 kV, arcing around the resistor body is a significant problem. There are a number of manufacturers of resistors intended specifically for high voltage applications, and of course, you can construct a resistor suitable for your application.

Commercial products

• High resistance, low power - These are typically used for measuring high voltages as part of a voltage divider (or as a meter series multiplier). Generally constructed by thin film techniques on a ceramic substrate of appropriate size for the voltage rating. Suppliers include Caddock, Gigohm, ....
• High precision - Used in precision voltage dividers for measurement applications.
• High power - Typified by the products of the Cesewid corporation (formerly Carborundum), these units are designed to take large peak or average powers and are often constructed so they are non-inductive. A typical application would be as a current limiting device in a capacitor discharge circuit, components in pulse forming/shaping networks, or as a energy dump load. Suppliers include Cesewid and Maxwell Labs..

Insulating gases

Electronegative gases make good insulators since the ions rapidly combine with the ions produced in the spark. However, they tend to be corrosive. Some gases though, dissociate only where the discharge is (or wants to be), making them particularly good insulators.

Gases with electronegative species (i.e. halogens such as chlorine) make good insulators, hence the popularity of SF6, which is not only dense (breakdown voltage is roughly proportional to density) but is mostly Fluorine, a highly electronegative element. The halogenated hydrocarbon refrigerants are also a popular insulator. CCl₄, CCl₂F₂, CCl₃F, and C₂Cl₂F₄

Unfortunately, the cost of insulating gases has greatly increased in the last few years largely due to the various treaties regulating halocarbon refrigerants. The traditional Freons (R-12, R-22) are not being produced any more, and are quite expensive. Since the regulatory thrust eliminated chlorinated alkanes, modern refrigerants are relying more on fluorinated or per-fluoro hydrocarbons (e.g.HC-134a) . Unfortunately, plant capacity is limited, and plants that used to make SF6 are now making fluorinated hydrocarbons resulting in much higher prices for SF6. In the mid 1980's SF6 was about $3-4/lb. Now, in the mid 90's, it is about $100/lb. Since a pound is only about 10 liters, filling up a large insulating tank with SF6 has become a very expensive proposition.

The breakdown voltage of most gases can be increased by increasing the absolute pressure. In the case of some gases, there is a limit imposed by the liquefaction point at normal operating temperatures (i.e. Freon 12 liquifies at 5 atmospheres). Mixtures of gases can overcome some of these issues and a mixture of Freon 12 and Nitrogen was popular.

One disadvantage of the halogenated compounds is that the dissociation products are highly corrosive, so it is important that operating voltages remain well below corona starting voltages. Even air forms highly reactive nitrogen oxides and other corrosive compounds, particularly if there is any water vapor present. High pressure air can also support combustion due to the oxygen content.Pure Nitrogen seems to not have these disadvantages, although its breakdown is only about 15 % better than air.

Air - approximate breakdown is 30 kV/cm at 1 atm. = 30 + 1.53d where d in cm. The breakdown of air is very well researched, to the point where the breakdown voltage of a calibrated gap is used to measure high voltages.

Freons- The vapor pressure of CCl₂F₂ (R-12) is 90 psi at 23C, where the breakdown is some 17 times that of air at 1 atm. An even higher insulating strength can be obtained by adding nitrogen to the saturated CCl₂F₂ to bring the total pressuire to the desired value. The saturated vapor pressure of C₂Cl₂F₄ at 23C is 2 atm abs, at which condition it has a relative dielectric strength of 5.6 times N2 at 1 atm

Sulfur Hexafluoride (SF6) - Sulfur Hexafluoride is probably the most popular insulating gas, although its cost has risen dramatically recently.

Hydrogen - Hydrogen gas is not a particularly good insulator (65% of air) from a breakdown voltage standpoint. Its very low viscosity and high thermal capacity make it an insulating gas of choice for high speed, high voltage machinery such as turbogenerators. There isn't an explosion hazard, provided that the oxygen content in the hydrogen tank is kept below the flammable limit (around 5%). Of course, hydrogen has lots of other handling problems, including hydrogen embrittlement, it leaks through very tiny holes (even the pores in the metal tanks), and perfectly colorless, but very hot, flames.

Relative spark breakdown strength of gases

Gas	N2	Air	NH3	CO2	H2S	O2	Cl2	H2	SO2	C2Cl2F4	CCl2F2
V/Vair	1.15	1	1	0.95	0.9	0.85	0.85	0.65	0.30	3.2	2.9

Power Factor Correction

Many loads are highly inductive, such a lightly loaded motors and illumination transformers and ballasts. You may want to correct the power factor by adding parallel capacitors. You can also add series capacitors to "remove" the effect of leakage inductance that limits the output current.

Why correct the power factor?

The current flow through the circuit is increased by the reactive component. Normally, loads are represented by a series combination of a resistance and a purely imaginary reactance. For this explanation, it is easier to contemplate it as an equivalent parallel combination. The diagram below illustrates a partially reactive load being fed from a real system with some finite resistance in the conductors, etc.

The current through the reactive component itself dissipates no power, and neither does it register on the watt hour meter. However, the reactive current does dissipate power when flowing through other resistive components in the system, like the wires, the switches, and the lossy part of a transformer. Switches have to interrupt the total current, not just the active component. Wires have to be big enough to carry the entire current, etc. Correcting the power factor reduces the amount of oversizing necessary.

Correcting power factor

Given the reactive load component (Xload), you can calculate the capacitance to exactly match it using the equation:

Xc = 2*pi * 60 / C = 377/C

or, rearranging: C = 377/Xc

Power factor correction capacitors are often rated in kVar, instead of uF, because that is how the power company works. Say a factory has several thousand horsepower worth of motors at .85 power factor. They might have a reactive component of several hundred kVar. At a distribution voltage of 14,400 volts, this would require a capacitor with an impedance of about 1037 ohms, or about 2.5 microfarads, a reasonable sized and priced package. However, if you were crazy enough to try to compensate this at 230 volts, you would need about .01 Farads (i.e. 10,000 uF), a sizeable package.

For very large systems, even capacitors get unwieldy. One approach is to use large over excited synchronous motors which look like capacitors, electrically. Another approach is clever systems of thyristors and inductors which simulate the capactive reactance by drawing "displacement current".

Loads that draw non-sinusoidal current

Classic reactive loads, like transformers, lighting ballasts, and AC motors still have a sinusoidal current flow. The phase of the current is just shifted from that of the supply voltage. However, there are some loads which draw distinctly non-sinusoidal currents. The most recently notorious is the switching power supply in a PC. These power supplies start with a bridge rectifier feeding a capacitor, and so, particularly at part load, draw their current in little peaks, when the instantaneous line voltage is above the capacitor voltage, forward biasing the rectifier. Another notorious non-sinusoidal current draw is the popular phase controlled light dimmer, which uses a TRIAC or SCR to reduce the RMS voltage to the load by turning on partway through the half cycle. Not only is the current waveform highly non-sinusoidal, but it is also out of phase with the voltage supply. Hence, these loads have a non-unity power factor, and draw reactive power.

However, to compensate these loads, you have to come up with a means to supply the reactive current at the appropriate times. A simple capacitor doesn't do this. A capacitor only compensates nice sinusoidal power factor lags, like those from linear (non-saturating) inductors.

Example of Power Factor Correction

Let's take an example. A 3/4 HP electric motor has a power factor of .85. The nameplate current is 10 Amps at 115 Volts, or 1150 Volt Amps.

• Apparent power = 1150 Volt Amps
• Active power = .85 * 1150 = 977.5 Watts
• Reactive Power = sqrt(1150^2 - 977.5^2) = 605 VAR

So, we need 605 var of power factor correction. Calculating the required impedance from Q = E^2/X

• 605 = 115^2/X => X = 115^2/605 = 21 ohms
• C = 1 /( 2 * pi * f *X) = 1/ (377 * 21) = 126 uF

which is a fairly large capacitor. Furthermore, it will have a current of about 5.2 amps flowing through it, so its series resistance should be pretty low, or it will dissipate a fair amount of energy.

If the line voltage were higher, the correction impedance would be increased as the square of the line voltage. The capacitance would be reduced as the square of the line voltage. That is, if the same motor were run off 230 Volts, the capacitor would only need to be 31.5 uF. And if we were to do power factor compensation at the distribution voltage of 4160 volts (for example), you would only need about .1 uF. This is why power factor correction is usually done in the distribution network at MV or HV, and not at the end voltage.

Measuring active and reactive power with just a VOM

If you just measure the RMS voltage and RMS current and multiply them, you get the apparent power. As long as the load is purely resistive, the apparent power is equal to the active power. For a reactive load, though, where the voltage and current are out of phase, the apparent power will be greater than the active power. Measuring active power is a bit tricky and is traditionally done by a watt meter (or more commonly a watt hour meter, like that measuring the electrical energy consumption of your house). Watt meters (and watt hour meters) instantaneously multiply the voltage and current and integrate the result, so they measures true active power.

Commercial power monitors actually have an a/d that samples the voltage and current waveform and do the math internally, which makes measurements for three phase power systems much easier. If you have a enough a/d channels on your data acquisition system, and are handy with software, you can do the same. These units will correctly calculate power for non-sinusoidal waveforms, as well.

You can use a dual trace oscilloscope to measure the phase shift between the current and voltage and use that to calculate active power using the equation

active power = cos(theta) * apparent power

where theta is the phase difference between voltage and current. The term cos(theta) is the power factor, typically in the range .80-.95 for motors, fluorescent light ballasts, and the like.

The load is represented by a resistance (Rload) and a reactance (Xload) in series. The series resistor shouldn't be too big, say 2-10 ohms. Make sure it can dissipate the power. If your load is going to draw 10 amps, and you have a 10 ohm resistor, it is going to dissipate 1000 Watts. In use, you set the output of the variac to get the load voltage to be whatever its rated input voltage is, e.g. 115Volts. To do the calculation, you'll need the following measurements:

• The RMS voltage at the load (call that V1)
• The RMS voltage out of the variac (call that V2)
• The RMS current through the load (call that I)
• The resistance of the series resistor (call that Rseries)

Now do the calculation:

Resistive component of load

• Rload = ((V2/I)^2 - (V1/I)^2 - Rseries^2)/(2 * Rseries)

Reactive Component of Load

• Xload = sqrt( (V2/I)^2 - Rload^2)

(Of course, you don't know the sign of the reactive component from this measurement.)

Active power = Rload * I^2

Reactive Power = Xload * I^2

If you want to determine if the reactive load is capacitive or inductive, you can add a small capacitor or inductor (the reactive impedance must be less than the existing circuit's) to the circuit, make the measurements again, and see if Xload got bigger or smaller. For instance, if you had measured a reactive impedance of 5 Ohms, and then you added a capacitor and got a reactive impedance of 6 Ohms, then the original reactive impedance was capacitive. If the reactive impedance decreased, then the original reactive impedance was inductive.

LIMITATION

The above technique does not necessarily work for non-sinusoidal waveforms. A good example of a non-sinusoidal waveform is the current drawn by a capacitor input filter (e.g. the input of a switching power supply) or the output of a phase controlled light dimmer.

Basic Circuit Theory

Circuit theory for high voltage systems is essentially the same as for any other circuit. Ohm's law and Kirchoff's Voltage and Current laws still apply. The actual circuits for most high voltage equipment are actually quite simple, so not much analytical work is necessary to get an understanding of the expected behavior. In fact, slavish use of mathematical circuit modelling may not be the best approach for high voltage circuits, because the characteristics of the components are not known accurately, and the effects of dielectric breakdown (e.g. corona) are unpredictable. In much of high voltage engineering, empiricism still rules the day.

RC circuits

discharge: V(t) = Vinitial * EXP ( -t / (R*C))

The product R*C is referred to as the time constant.

LC circuits

RLC circuits

Nonlinear elements

The most common non-linear circuit encountered in high voltage circuits is a spark or arc. The voltage drop across a high pressure arc (e.g. in a xenon flash tube) is proportional to the square root of the current. This is known as the Goncz relation.

E = K0 * SQRT(I)

Another common non-linear element is the essentially constant voltage glow discharge, typified by a neon lamp. In fact, glow discharges can have a negative resistance characteristic, in that the voltage drop across the discharge decreases as the current increases.

In some simple cases (like a xenon flash tube discharging in a simple RLC loop), a fairly accurate analytical solution can be determined. In more complex cases, numerical integration is the best approach.

Paschen's Law reflects the Townsend breakdown mechanism in gases, that is, a cascading of secondary electrons emitted by collisions in the gap. The significant parameter is pd, the product of the gap distance and the pressure. Typically, the Townsend mechanism (and by extension Paschen's law) apply at pd products less than 1000 torr cm, or gaps around a centimeter at one atmosphere. Furthermore, some modifications are necessary for highly electronegative gases because they recombine the secondary electrons very quickly.

In general, an equation for breakdown is derived, and suitable parameters chosen by fitting to empirical data.

Here are three equations:

Breakdown voltage:
Vbreakdown = B * p * d / (C + ln( p * d))

Breakdown field strength:
Ebreakdown = p * ( B / ( C + ln ( p * d)))

where:
C = A / ln ( 1 + 1 / gamma)

where:
gamma is the (poorly known) secondary ionization coefficient.

For air:
A = 15 cm^-1Torr ^-1
B = 365 Vcm^-1 Torr^-1
and gamma = 10^-2
so
C = 1.18

Paschen's Law

In 1889, F. Pashchen published a paper set out what has become known as Paschen's Law. The law essentially states that the breakdown characteristics of a gap are a function (generally not linear) of the product of the gas pressure and the gap length, usually written as V= f( pd ), where p is the pressure and d is the gap distance. In actuality, the pressure should be replaced by the gas density.

For air, and gaps on the order of a millimeter, the breakdown is roughly a linear function of the gap length: V = 30pd + 1.35 kV, where d is in centimeters, and p is in atmospheres.

Much research has been done since then to provide a theoretical basis for the law and to develop a greater understanding of the mechanisms of breakdown. Some of this will be described in the rest of this section, but it should be realized that there are many, many factors which have an effect on the breakdown of a gap, such as radiation, dust, surface irregularities. Excessive theoretical analysis might help understanding why a gap breaks down, but won't necessarily provide a more accurate value for the breakdown voltage in any given situation.

Calculating the magnitude of the Electric Field

A lot of practical high voltage design requires knowing what the maximum E-field is, for insulation design, corona reduction, etc. The exact field can, of course, be calculated numerically by solving Laplace's equation over a suitable field with appropriate boundary conditions. As complicated and time consuming as this is, it is necessary when performance is critical, in integrated circuit design, designs for absolute minimum cost, and so forth. However, for more run of the mill experimentation and use, where a little overdesign can be tolerated, approximations to the field are just as useful.

The mean or average field is just the voltage difference divided by the distance between the conductors. For the proverbial infinite flat plates, this makes the calculation simple.

Emax = Eaverage = V / Distance

For two concentric cylinders (i.e. like coaxial cable) the maximum field is:

Emax = V / ( Rinner * LN( Router / Rinner))

where:
Rinner is radius of the inner electrode
Router is the radius of the outer electrode
LN() is the log base e of the argument
V is the voltage between the electrodes

For concentric spheres, using the same variables, the maximum field is:

Emax = V * Router / (Rinner *(Router - Rinner))

For two parallel cylinders of equal radius:

Emax = V * SQRT(D^2 - 4 * R^2)/ (2 * R*(D-R)*INVCOSH(D/(2*R)))

approx equal to: V / (2*R) * LN(D/R) if D>>R

where:
D is distance between the centers of the conductors
R is the radius of the conductors

For two spheres:

Emax = approximately V/S * F

where:
S is spacing between spheres = D - 2*R
F is a field enhancement factor =

F = (S/R+1) * sqrt( (S/R+1)^2+8)/4

For spheres, if S>>R then Emax = approx V/ (2*R)

For other configurations:

Emax = Eaverage * F

For sphere/plane: F = 0.94*Spacing/Radius + 0.8

For cylinder/plane: F = 0.25 * Spacing / Radius + 1.0

Basic Electrostatics

Electrostatics deals with charges, potentials, and the like where things aren't changing, i.e. they're static. Basic principles of electrostatics are used all the time in high voltage work for a lot of reasons. Popular high voltage generators like the Van de Graaf are based on electrostatic principles (even though a current is flowing). The burning question in a lot of high voltage work is whether the system will electrically breakdown as the voltage is raised. This is generally a question of quasi-static potential gradients which can be answered by simple electrostatics.

Coulomb's Law

The force on a charged point exerted by a second charge is proportional to the product of charges, and inversely proportional to the square of the distance between the charges, and acts either directly towards each other (opposite charges) or away from each other (same sign of charge).

F(vector) = k * q1 * q2 / r12^2 * direction(r12)

where

k = 1 / (4 * pi * epsilon) = 8.99E9 Newton Meter^2/Coulomb^2

where epsilon is the dielectric constant of the medium ( = 8.85E-12 for vacuum)

r12 is the scalar distance from point 1 to point 2

direction (r12) is a unit vector from point 1 to point 2

q1,q2 are charges on each point

Capacitance of two parallel plates

C = epsilon * Area / DistanceBetweenPlates

this neglects fringing effects, which for plates that are smaller than, say, 10 times the spacing, are pretty significant.

Capacitance of two concentric cylinders (e.g. coaxial cable)

C = 2 * pi * epsilon * length / ln( rOuter/rInner)

this assumes length >> r

Capacitance of two concentric spheres

C = 4 * pi * epsilon * rInner * rOuter/ (rOuter - rInner)

as rOuter goes to infinity, the fraction rOuter/(rOuter-rInner) goes to one, leading to the following handy equation:

Capacitance of isolated sphere

C = 4*pi*epsilon*radius = approx 111.2 pF/meter

this equation is derived from the equation for two concentric (nested) spheres, and letting the radius of the outer sphere go to infinity.