There are four distinct types of electrical breakdown of air, namely, Glow Discharges, Corona, Sparks, and Arcs. When the voltage across to conductors in air is very high, the air becomes ionized and begins to conduct electricity, or "breaks down." In fact, any gas that conducts electricity is called a plasma. Most people don't even realize that common fluorescent tubes and computer monitors use such a plasma to create light. An arc is a type of breakdown where the voltage is rather low and the current is relatively high. Some arc furnaces are used to melt metals and may require a 10 volt power supply . . . at 15,000 amps! The spark breakdown is nearly the opposite: High voltage and relatively low current. Static electricity is one such type of breakdown because the voltage can be as high as 500,000 volts, but at a current as low as maybe 0.00000001 amps. Sparks and arcs are generally difficult to catagorize when both the voltage and current are high, such as in lightning. Corona and glow discharges are somtimes confused, but they have one important distinction: Glow discharges are relatively "cold" breakdowns, whereas coronas are relatively hot breadowns. Glow discharges are used in neon signs and lamps, and fluorescent tubes, and are generally a desired effect or cause little harm. Coronas, on the other hand, have been a problem on power lines because coronas can destroy wire insulation and crack porcelan insulators. In high voltage power engineering, it is important to design the power transmission towers to prevent corona under all circumstances (rain, snow, fog, etc.).

    Coronas have two important voltage parameters: The Disruptive Critical Voltage, e₀, and the Visual Critical Voltage, e_v.^[1] The visual critical voltage is always higher than the disruptive critical voltage. Once e_v is reached, the corona can be seen and remains visible until the voltage reaches e₀. Calculations can be made on these voltage parameters as shown below (for parallel wires). In some cases, if the voltage is too high and the distance between the conductors is too small, a sparkover will occur. If the voltage is increasing, e_v will be reached. When this occurs, the air surrounding the conductor will ionize and begin to conduct. At this point, the wire's effective radius has been increased since the air is acting like a conductor. The equations below indicate that if this radius changes, the voltage gradient, g_v around the wire changes. If the g_v decreases when the effective radius increases, then corona forms. If, however, g_v increases when the effective radius increases, then sparkover will occur.

e₀ = m₀ g₀ delta r ( 1 + 0.301 / sqrt(r) )    in kv.

e_v = m_v g₀ delta r ( 1 + 0.301 / sqrt(r) ) ln (S/r) = m_v g_v ln (S/r)    in kv.

Where
• g_v = g₀ delta r ( 1 + 0.301 / sqrt(r) ) = Visual Critical Voltage Gradient = de / dr    in kv. / cm.
• g₀ = 29.8 kv. / cm. (for parallel wires) = Disruptive Critical Voltage Gradient
• m_v = m₀ = The wire roughness factor
  = 1 for polished wires
  = 0.98 to 0.93 for roughened, dirty or weathered wires
  = 0.87 to 0.83 for cables
  
• delta = air density factor = 1 for barometric pressure of 76 cm at 25 degrees C
  = 3.92 * (barometric pressure in cm) / (273 + temperature in C)
• S = distance between surfaces in cm.
  
• r = radius of wire in cm.

    Summary of calculations for parallel wires, concentric cylinders and equal spheres is shown below:^[2]

For Concentric Cylinders:
• Corona will not form when R / r < 2.718 = exp(1)
• Capacity C = ( 5.55x10^-13 k ) / (ln (R / r) )    farads / cm.
• Position Gradient g_x = e / (x ln (R / r) )    kv. / cm.
• Visual Critical Gradient g_v = 31 ( 1 + 0.308 / sqrt ( r ) )    kv. / cm.

For Parallel Wires:
• Corona will not form when S / r < 5.85
• Capacity C = ( 5.55x10^-13 k ) / (cosh^-1 (S / 2r) )    farads / cm.
• Position Gradient g_x = (e sqrt(S² - 4r²) ) / { [ ( r + x )( S - 2r ) - x² ] ln [ ( S / 2r ) + sqrt( (S / 2r)² - 1 ) ] }    kv. / cm.
• Visual Critical Gradient g_v = 30 ( 1 + 0.301 / sqrt ( r ) )    kv. / cm.

For Equal Spheres in Air:
• Corona will not form when X / R < 2.04
• Sparkover difficult to avoid when X / R < 8
• Capacity C = ( X 10^-11 ) / ( 36 ( f - 1 ) )    farads
• Position Gradient g_a = ( e / X ) { [2 X² ( X²( f + 1 ) + 4 ( X / 2 - a )²( f - 1 ) ) ] / [ X²( f - 1 ) - 4( X / 2 - a )²( f - 1 ) ]² }    kv. / cm.
• Visual Critical Gradient g_v = 27.2 ( 1 + 0.54 / sqrt ( R ) )    kv. / cm.

Where,
• e = voltage relative to ground in kilovolts.
• k = relative permittivity of the medium ( k_air = 1 )
• f = (1 / 4 ) [ ( X / R ) + 1 + sqrt ( ( X / R + 1 )² + 8 ) ]

    Corona discharge as used by existing ionic tweeters ^[3]. comes from a sharp point rather than a flat wire or sphere or cylinder. Using these basic geometric shapes, however, a rough model for such a corona-emitting sharp wire can be modelled. The figure shows a wire with radius R_w that carries the voltage (part #1). This voltage gets fed to the second shape, a cone: a wire with linearly decreasing radius, r(y), part #2. Finally, the bulk of the corona is emitted from the point of the wire, the sphere with radius r_c, part #3. Each of these shapes have their own voltage gradients and capacitances that have to be calculated and entered into the model.

Assumptions:
• #1: The wire carrying the voltage will largely be out of the Faraday shield's domain and near the grounded chassis of the device, so perhaps a fair approximation would be the parallel wire scenario.
• #2: The cone will be completely inside the Faraday shield, so it will be modelled as a wire with varying radius in a cylinder.
• #3: The point is a roughly a sphere inside of another sphere, but can be roughly approximated by the concentric cylinder case.

Position Gradients:
1. Let R_w = 0.3 cm., S = 10 cm., x be the distance from the wire, then,
   g_w(x) = (e sqrt(10² - 4(0.3)²) ) / { [ ( 0.3 + x )( 10 - 2(0.3) ) - x² ] ln [ ( 10 / (2*0.3) ) + sqrt( (10 / (2*0.3) )² - 1 ) ] } = ( 28.4 e ) / (9.4 x - x² + 2.82 )   kv / cm.
2. Let the length of the tapering zone be 1 cm. Let R_g = 7 cm. Then r(y) = R_w ( 1 - y ) = 0.3 ( 1 - y ). If R_g is large compared to 1 cm, then R_cyl = R_g. The gradient a distance x from the cone is approximately:
   g₁(x,y) = e / { x ln [R_g / r(y) ] } = e / { x ln [ 7 / ( 0.3 ( 1 - y ) ) ] } = e / [ x ln ( 3.15 - ln (1 - y ) ]   kv / cm.
3. Let r_c = 0.05 cm. R_g = 7 , as above. The gradient a distance x from the point is then:
   g_c(x) = e / (x ln (R_g / r_c ) ) = e / ( x ln ( 7 / 0.05) ) = e / ( 4.94 x )   kv / cm.

Capacitances:
1. Let the length of the wire, L_w be 20 cm, and the permittivity of air, k, be unity. Let the imaginary wire be an effective distance of 5 cm away, then,
   C_w = ( 5.55x10^-13 k ) / (cosh^-1 [S / r] ) = ( 5.55x10^-13 ) / (cosh^-1 [5 / (2*0.3) ] ) = 0.2   picofarads per cm.
   C_w (total) = L_w C_w = 20 * 0.2 = 4   picofarads
2. Let k=1, as above, then,
   C₁(y) = ( 5.55x10^-13 k ) / (ln [R_g / r(y) ] ) = ( 5.55x10^-13 ) / { ln [0.07 / ( ( 1 - y ) R_w ) ] } = 0.555 / ( 3.15 - ln [ 1 - y ] )   picofarads
   C₁ (total) = integral ( C₁(y) ), from y = 0 to y = (1-r_c) = 0.14   picofarads
3. Let k=1, as above, then,
   C_c = ( 5.55x10^-13 k ) / (ln [R_g / r_c] ) = ( 5.55x10^-13 ) / (ln ( 7 / 0.05) ) = 0.11   picofarads
   Since half of the sphere is part of the cone, only half of the capacitance will actually be present,
   C_c (total) = ~0.5 C_c = 0.06   picofarads

Required Visual Corona Potentials:
1. e_vw = g₀ ( 1 + 0.301 / sqrt ( r ) ) r ln ( S / r )
   = 29.8 ( 1 + 0.301 / sqrt ( 0.3 ) ) 0.3 ln ( 10 / 0.3 ) = 48.6   kv
2. e_v1 (min) = g₀ ( 1 + 0.308 / sqrt ( r ) ) r ln ( S / r )
   = 31 ( 1 + 0.308 / sqrt ( 0.05 ) ) 0.05 ln ( 10 / 0.05 ) = 19.5   kv
   e_v1 (max) = g₀ ( 1 + 0.308 / sqrt ( r ) ) r ln ( S / r )
   = 31 ( 1 + 0.308 / sqrt ( 0.3 ) ) 0.3 ln ( 10 / 0.3 ) = 50.9   kv
3. e_vc = e_v1 (min)
   = 31 ( 1 + 0.308 / sqrt ( 0.05 ) ) 0.05 ln ( 10 / 0.05 ) = 19.5   kv

Conclusions:
• Total Capacitance of the emitter: 4.2 pf
• From the voltage calculations, it is obvious that the sharp point will form a corona first since the required visual critical voltage is the lowest at the sharp points, at 19.5 kv. Since the equations are approximations, the actual corona voltages should be higher in the case that the emitter doesn't accurately represent the model or is worn down by the corona, or that the atmospheric conditions cause e_v to increase. Experimentation has shown that the visual critical voltage must be at least ~30 kv (peak value) at normal atmospheric conditions to produce corona. The diameter of the corona on a parallel wire of R_w = 0.23 cm becomes a linear function of voltage above ~34 kv (approximately 1mm of corona per 2 kv) and is ~0.6 cm in diameter. Between 32.5 kv and 34 kv, the size of the corona is a non-linear function of the voltage. For ion tweeter purposes, the voltage should be in the linear range from ~34 kv to 44 kv, which keeps the sharp point in corona range and the wire out of corona range.
• The equations do not cover the possibility of the frequency of the voltage affecting the critical voltages, since they were intended to be used for 60Hz power transmission. Experiments have shown that 1kHz has little has a very similar effect to the 60Hz, but 30kHz seemed to lower the visual critical voltage very slightly.

• Peek, F.W., Dielectric Phenomena in High Voltage Engineering, McGraw-Hill, pp.48-108
• Peek, F.W., Dielectric Phenomena in High Voltage Engineering, McGraw-Hill, pp.30-31
• Existing Ion Tweeters by several hobbiests.