Enter the Paschen constants A and B, the pressure of the gas, the distance between the electrodes, and the secondary electron emission coefficient (γ) into the calculator to determine the breakdown voltage.
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Paschen's Law Formula
Paschen's Law estimates the voltage required to electrically break down a gas between two electrodes. It is most useful when you want to predict when a gas-filled gap will begin to conduct and form a discharge.
V_B = \frac{Bpd}{\ln(Apd) - \ln\!\left(\ln\!\left(1+\frac{1}{\gamma}\right)\right)}For this calculator, use gas constants that are consistent with SI units: constant A in 1/(Pa·m), constant B in V/(Pa·m), pressure in pascals, and distance in meters. The calculator's unit selectors help convert entered values into those base units before the final computation.
What the Inputs Mean
| Input | Description | Why It Matters |
|---|---|---|
| Constant A | A gas-specific ionization constant. | Controls how strongly the gas responds to the pressure-gap product. |
| Constant B | A gas-specific voltage scaling constant. | Sets the overall voltage level required for breakdown. |
| Pressure | The gas pressure inside the gap. | Higher or lower pressure changes collision frequency between electrons and gas molecules. |
| Distance | The electrode separation. | A larger gap usually requires more voltage, but the effect depends on pressure at the same time. |
| Secondary electron emission coefficient | The fraction of electrons released from the cathode by ion impact. | Influences how easily an avalanche can sustain itself after the first ionization events. |
| Breakdown Voltage | The predicted voltage at which the gas becomes conductive. | This is the threshold where a spark or discharge can begin. |
How to Use the Calculator
- Enter the correct Paschen constants for the gas you are analyzing.
- Select the pressure unit and enter the gas pressure.
- Select the distance unit and enter the electrode gap.
- Enter a positive value for the secondary electron emission coefficient.
- Calculate the result to obtain the predicted breakdown voltage.
If the output seems unrealistic, the most common cause is a mismatch between the gas constants and the unit system being used.
Why the Pressure-Distance Product Matters
Paschen's Law does not depend on pressure or gap length independently as much as it depends on their product. That product controls how often electrons collide with gas molecules while crossing the gap.
pd = p \cdot d
At very small values of the pressure-distance product, electrons may not collide often enough to sustain ionization. At very large values, electrons lose energy too quickly between collisions. Because of that, the breakdown voltage typically drops to a minimum over a certain range and rises again on either side.
Example Calculation
Using the sample values shown on this page:
- Constant A = 11.25 1/(Pa·m)
- Constant B = 273.7 V/(Pa·m)
- Pressure = 101325 Pa
- Distance = 0.001 m
- Secondary electron emission coefficient = 0.01
pd = 101325 \cdot 0.001 = 101.325
V_B = \frac{273.7 \cdot 101.325}{\ln(11.25 \cdot 101.325) - \ln\!\left(\ln\!\left(1+\frac{1}{0.01}\right)\right)}V_B \approx 5.03 \times 10^3 \ \text{V}So the predicted breakdown voltage is about 5.03 kV.
Interpreting the Result
- Lower breakdown voltage: the gas gap will ionize more easily and is more likely to spark under lower applied voltage.
- Higher breakdown voltage: the gas behaves as a stronger insulator for that combination of pressure and gap spacing.
- Very large output: this often means the chosen conditions are far from the minimum region of the Paschen curve, or the denominator is becoming very small.
Practical Notes
- Gas constants are not universal. Constant A and constant B depend on the gas being used.
- Gamma must be positive. Zero or negative values make the logarithmic term invalid.
- Unit consistency is critical. A and B must match the same unit basis as the pressure and distance used in the equation.
- Real systems can differ from the ideal model. Sharp electrode edges, contamination, rough surfaces, humidity, and non-uniform electric fields can change the actual breakdown voltage.
When This Calculator Is Useful
This calculation is commonly used when evaluating spark gaps, gas-insulated components, discharge tubes, plasma ignition, vacuum or low-pressure hardware, and compact high-voltage devices where gas breakdown is a limiting design condition.
