Editing Tesla coil (section)

===Output voltage===
[[File:Teslacoil.jpg|thumb|upright=1.5|Large coil producing 3.5 meter (10 foot) streamer arcs, indicating a potential of millions of volts]]

In a resonant transformer the high voltage is produced by resonance; the output voltage is not proportional to the turns ratio, as in an ordinary transformer.<ref name="BurnettOperation"/><ref name="Gerekos3"/> It can be calculated approximately from [[conservation of energy]]. At the beginning of the cycle, when the spark starts, all of the energy in the primary circuit <math>W_1</math> is stored in the primary capacitor <math>C_1</math>. If <math>V_1</math> is the voltage at which the spark gap breaks down, which is usually close to the peak output voltage of the supply transformer ''T'', this energy is
:<math>W_1 = {1 \over 2}C_1V_1^2\,</math>

During the "ring up" this energy is transferred to the secondary circuit. Although some is lost as heat in the spark and other resistances, in modern coils, over 85% of the energy ends up in the secondary.<ref name="Anderson"/> At the peak (<math>V_2</math>) of the secondary sinusoidal voltage waveform, all the energy in the secondary <math>W_2</math> is stored in the capacitance <math>C_2</math> between the ends of the secondary coil
:<math>W_2 = {1 \over 2}C_2V_2^2\,</math>
Assuming no energy losses, <math>W_2\;=\;W_1</math>. Substituting into this equation and simplifying, the peak secondary voltage is<ref name="Sprott"/><ref name="Anderson"/><ref name="BurnettOperation"/>
{{Equation box 1 |indent =: |cellpadding=0 |border=1 |border colour=black |background colour=transparent |equation=<math>V_2 = V_1\sqrt{C_1 \over C_2} = V_1\sqrt{L_2 \over L_1}.</math>
}}
The second formula above is derived from the first using the resonance condition <math>L_1 C_1\;=\;L_2 C_2</math>.<ref name="BurnettOperation"/> Since the capacitance of the secondary coil is very small compared to the primary capacitor, the primary voltage is stepped up to a high value.<ref name="Anderson"/>

The above peak voltage is only achieved in coils in which air discharges do not occur; in coils which produce sparks, like entertainment coils, the peak voltage on the terminal is limited to the voltage at which the air [[electrical breakdown|breaks down]] and becomes conductive.<ref name="Anderson"/><ref name="BurnettOperation"/><ref name="BurnettParts"/> As the output voltage increases during each voltage pulse, it reaches the point where the air next to the high-voltage terminal [[ionization|ionizes]] and [[corona discharge|corona]], [[brush discharge]]s, and [[streamer discharge|streamer arcs]] break out from the terminal. This happens when the [[electric field]] strength exceeds the [[dielectric strength]] of the air, about 30&nbsp;kV per centimeter. Since the electric field is greatest at sharp points and edges, air discharges start at these points on the high-voltage terminal. The voltage on the high-voltage terminal cannot increase above the air breakdown voltage, because additional electric charge pumped into the terminal from the secondary winding just escapes into the air. The output voltage of open-air Tesla coils is limited to a few million volts by air breakdown,<ref name=":0"/> but higher voltages can be achieved by coils immersed in pressurized tanks of [[transformer oil|insulating oil]].