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What is a plasma toroid?

A plasma toroid is a glowing ring of ionized gas inside a sealed glass vessel filled with a noble gas at low pressure (a fraction of a Torr to a few Torr; normal air is 760 Torr). A coil of a few turns sits around or below the glass. A resonant RF driver (a single-MOSFET power oscillator, in the spirit of the ZVS / class-E drivers used in home builds) pushes megahertz current through the coil.

The oscillating magnetic field of the coil induces a circular electric field inside the gas (Faraday's law). Free electrons are accelerated around that circle, collide with gas atoms and knock more electrons loose, until the gas breaks down into a self-sustaining, ring-shaped discharge. This is an inductively coupled plasma: the coil is the primary of a transformer and the plasma ring acts as a one-turn secondary. Excited atoms emit light as they relax, which gives the ring its colour.

The name "tokamak in your home" is an analogy only. A tokamak also drives a ring-shaped (toroidal) plasma current by induction, but it confines a plasma at millions of degrees with strong magnetic fields. This plasma is cold, low-density and unconfined, so no fusion happens.

This is a simplified educational model. The equations below are tuned so the behaviour is plausible against the reference builds; they are not measurements and not engineering guidance.

Breakdown and ignition (Paschen's law)

A gas only conducts once the voltage is high enough for an avalanche of ionizing collisions. Paschen's law gives this breakdown voltage as a function of pressure × distance (p·d). The simulator uses the standard form

V_b = B·p·d / (ln(A·p·d) − ln(ln(1 + 1/γ)))

with the gas constants A and B from the table below and a secondary-emission coefficient γ = 0.01. Here p is the fill pressure in Torr and d is the discharge length in cm: the vessel diameter × a shape factor (see Vessel shape).

• The curve has a minimum. At high p·d the electrons collide too often to gain enough energy between collisions, so a higher voltage is needed. At low p·d there are too few collisions to build an avalanche.
• Far left of the minimum (A·p·d below about 4.85 in the model) breakdown is impossible: the readout shows ∞ and the reason "pressure too low to break down (left of Paschen minimum)".
• Example: xenon at 0.5 Torr in an 8 cm sphere needs about 449 V. The default settings give about 605 V of coil RF voltage, so the gas lights by itself.

The plasma ignites on its own when the coil RF voltage reaches V_b. A spark (the Ignite button, or 0.5 s of spark from Auto-ignite when you press Start) seeds free electrons, so only 60 % of V_b is needed. Once lit, the plasma keeps itself going down to 55 % of V_b. That hysteresis is why a bulb that will not self-start can run happily after a spark.

Each gas has its own A and B constants, so the curves differ. A lower breakdown voltage does not follow from a lower ionization energy: at the default 0.5 Torr in an 8 cm vessel, helium (24.59 eV) breaks down at about 142 V and xenon (12.13 eV) at about 449 V. What the heavy gases gain is reach to lower pressure. Xenon's larger A moves its Paschen cutoff down to p·d ≈ 0.19 Torr·cm (helium's is about 1.6), and its ring thresholds are far lower (0.08 Torr and 15 W versus 1.0 Torr and 60 W for helium).

The noble gases

Every element emits light at its own set of spectral lines, so each gas glows in its own colour: neon's strong red and orange lines, argon's violet, krypton's whitish lavender and xenon's blue-violet. Helium glows a pale peach-pink. The simulator draws each gas in one representative colour. Neon's main lines are red-orange, but low-pressure RF neon discharges often look and photograph pink, so neon is drawn pink-red.

A is in 1/(cm·Torr) and B in V/(cm·Torr). The ring and bean thresholds are empirical values tuned for this model.

Plasma states: ring, plasma beans, diffuse, unstable

The model picks one state, checked in this order:

1. Off: the driver is off, the MOSFET has failed, the pressure is left of the Paschen minimum, the coil voltage is below breakdown, or the gas has not been ignited.
2. Unstable: the plasma flickers and wobbles when it is barely sustained (coil voltage within about 7.5 % of the 55 % sustain level), badly detuned (resonance gain g below about 0.45, roughly 20–30 % off resonance), or overdriven (power demand more than 1.25 × what the supply current limit allows).
3. Diffuse glow: the pressure or the plasma power is below the gas's ring threshold, so the light fills the vessel instead of contracting into a ring.
4. Plasma beans: neon, argon, krypton or xenon (not helium) above the gas's bean pressure × the shape's bean shift, with 20–80 W of plasma power and the gas's ring thresholds met.
5. Ring: everything else, the classic rotating torus of light.

Ring motion. In real toroids the ring drifts and rotates because the discharge is never perfectly uniform: ionization waves travel around it and it moves toward where coupling to the coil is strongest. The model animates this with a rotation speed of 0.4 + 2.5·B·√(f/10 MHz), where the brightness B = P_plasma / 60 W (capped at 1.5). More power and a higher frequency spin it faster. A wobble of 0.02 + 0.15 × instability adds flicker when the plasma is unstable. The ring gets thinner at higher pressure (thickness ∝ (0.5 / p)^0.35).

Plasma beans. At higher pressure a discharge column tends to contract and break up. Ionization instabilities (striations, standing or moving ionization waves) and filamentation split the uniform ring into a ring of discrete, round glowing balls of plasma, the "plasma beans" (also called a "string of beads" pattern), separated by dark gaps and drifting around the loop. The model shows this for Ne, Ar, Kr and Xe (not He). The beads are drawn slightly oval, stretched along the ring, with a dark gap of about a third of a bean length between neighbours. They drift at 0.6 × the ring speed, and their number is

beads = round(n0 · √((p / p_bean) · (D / 80 mm)) · (P_plasma / 40 W)^0.3), limited to 6–32,

where p_bean is the gas's bean pressure from the table multiplied by the shape's bean shift (sphere 1 ×, torus 0.9 ×, cylinder 1.1 ×). So higher pressure and bigger vessels give more beads, and more power gives slightly more. The Plasma Bean preset (xenon at 1.6 Torr) gives about 16 beads and the Neon Plasma Bean preset (neon at 3 Torr) about 17. The status card switches to Plasma beans exactly at p_bean; the beads start half-formed there and are fully formed by 1.1 × p_bean, while the ring fades out. See The science of plasma beans below for what the beads are and why their number is a whole number.

The science of plasma beans

Bead patterns in plasma toroids are seen in many photos and videos, but they have not been studied as much as the straight-tube discharges they resemble. What follows is the textbook picture of those related phenomena, hedged where the physics is still an active and largely qualitative research area.

What a bean is

Each bean is a small region where the gas is ionized and excited more strongly than around it. Excited atoms give off light when they drop back to lower levels, so this region glows brightly. Physically the beans belong to the same family as striations, also called ionization waves: the bright and dark bands that low-pressure noble-gas discharge tubes have shown since the 19th century. In a toroid the discharge is driven inductively by the RF coil and closes on itself, so the bands wrap around a ring instead of lining up along a straight tube. The dark gaps are places where electrons have just lost their energy in the previous bean and are being re-accelerated by the induced electric field, but are not yet energetic enough to excite or ionize atoms.

Why a smooth ring breaks up

A uniform ring can be unstable. If the electron density rises slightly in one spot, ionization there rises too, which raises the density further: a positive feedback called the ionization instability. It is especially strong when atoms are ionized in two steps through long-lived metastable states (stepwise ionization): each step needs less energy than direct ionization, so the ionization rate grows faster than linearly with electron density. Gas heating in the bright spots can add to this feedback.

The repeat pattern comes from electron kinetics. An electron that has spent its energy exciting atoms in a bright zone must travel a certain distance along the field, an energy relaxation length, before it again has enough energy to excite atoms. Excitation therefore repeats at roughly regular intervals. Because the electron energy distribution depends on where the electrons have been, not only on the local field, this is called nonlocal electron kinetics. Kolobov's review explains striations in rare gases in these kinetic terms, and recent kinetic simulations reproduce both moving and standing striations in noble gases.

Why the number of beans is a whole number

On a closed ring only a whole number of wavelengths fits around the loop. That number is the azimuthal mode number m = ring circumference / striation wavelength. Measured striation lengths in tubes follow similarity rules (Novák's law): the length, in units of the tube radius, depends on pressure × radius (p·R) and shrinks as p·R grows. Kinetic models give a length of roughly the distance over which an electron gains the excitation energy from the field.

The model turns this into a simple fit. If the wavelength scales like R · (p·R)^(−1/2) and the circumference grows with the vessel diameter D, then m grows like √(p·D). That is the √((p / p_bean) · (D / 80 mm)) factor in the bead formula. n0 sets the count at the reference point (p = p_bean, D = 80 mm, 40 W), and (P_plasma / 40 W)^0.3 is a weak empirical power term. Rounding and the 6–32 limit make the count discrete.

Gas and pressure

Higher pressure means more collisions and a shorter energy relaxation length, so the discharge contracts and stratifies more readily. The heavier gases argon, krypton and xenon have lower ionization energies and low-lying metastable levels, which favours stepwise ionization, so in the model they form a ring, and then beans, at lower power than neon; krypton and xenon also bead at lower pressure (1.8 and 1.2 Torr), while argon needs 3.0 Torr. Neon also stratifies readily (it is the classic gas for striation experiments), so the model gives it a bean mode from 2.5 Torr, but only above its 45 W ring threshold. Striations are observed in helium discharges too, but as a simplification the model gives helium no bean mode.

Why the beans drift

Striations are often moving ionization waves rather than fixed patterns, and their speed and direction depend on the discharge conditions. In a toroid, gas flow and heating and small asymmetries in the coil also push the beans around. The model drifts the beans at 0.6 × the ring rotation speed, so more power and a higher frequency make them drift faster.

How the controls change the beans in this simulator

• Pressure: must reach the shifted p_bean (table value × the shape's bean shift); above that the count grows with √p.
• Vessel diameter: the count grows with √D, and a larger p·d also changes the breakdown voltage.
• Gas: sets p_bean, n0 and the colour (see the gas table); helium has no bean mode.
• Plasma power, set by supply voltage, drive and current limit: must be 20–80 W and above the gas's ring power; it has a weak ^0.3 effect on the count.
• Frequency and coil turns: act through the tank resonance. Detuning lowers the power and can make the plasma unstable, which overrides the beans. A higher frequency also makes them drift faster.
• Shape: changes p_bean (torus 0.9 ×, cylinder 1.1 ×) and the coupling. A torus gains beans from both: its lower shifted p_bean raises √(p / p_bean), and its better coupling gives more power. A cylinder loses beans for the opposite reasons.

With the Plasma Bean settings this gives about 16 beads in the sphere, 17 in the torus, 15 in the cylinder and 14 with argon at 3.5 Torr; the Neon Plasma Bean preset gives about 17.

What real devices show

In photos and videos of commercial plasma toroid lamps and of home builds like those in the references, the glowing ring can break into a string of bright beans that drift around the loop or hold still, often switching back and forth as the pressure or power changes. Counts of roughly a dozen to twenty beans are common in such pictures.

Caveats

The bead formula and its thresholds are a phenomenological fit chosen to look plausible, not a first-principles model. Real bean formation depends on gas purity, wall temperature, electron kinetics and the shape of the RF field. Predicting it from first principles needs kinetic simulations like those in the references.

Vessel shape

Because the donut's discharge length is its narrow tube, it needs a higher pressure for the same p·d and breakdown. Its ring path matches the induced electric field best, so it couples the most power into the plasma and forms beans at slightly lower pressure. The cylinder couples the least.

The driver circuit

The driver is a single power MOSFET switching current into a resonant LC tank formed by the coil and a capacitor. The tank resonates at f0 = 12 MHz × 4 / N for N coil turns (inductance L = 0.125 µH·N², so 2 turns → 24 MHz, 4 → 12 MHz, 8 → 6 MHz). The tank has a quality factor Q = 4, and driving it off resonance reduces the gain:

g = 1 / √(1 + (Q·(f/f0 − f0/f))²)

• Coil RF voltage: V_coil = 36 · V_supply · drive · g · √(N/4). Coil current: I_coil = V_coil / (2π·f·L).
• Power demand: (V_supply · drive)² / 6 Ω · (0.3 + 0.7·g), × 1.3 while the plasma is lit (it loads the tank).
• The bench supply limits current, so input power is P_in = min(demand, V_supply × current limit) and supply current is I_in = P_in / V_supply. When the cap is hit the HUD says "current limited".
• Efficiency falls with frequency because switching losses grow: η = 0.8 − 0.025·(f − 8 MHz), kept between 0.55 and 0.85. At 12 MHz η = 0.70.
• Plasma power: P_plasma = P_in · η · coupling (only while lit).

Where the losses go: the MOSFET dissipates P_in · (1 − η) (conduction plus switching loss), the coil heats by I_coil² · R_ac (with R_ac = 0.05 Ω · (N/4) · √(f/10 MHz), rising with frequency from skin effect) plus 4 % of the plasma power, and the glass absorbs 60 % of the plasma power.

Heat and time to danger

The MOSFET, coil and glass are each modelled as one thermal node with a heat capacity C (J/K) and a thermal resistance to ambient R (K/W): C · dT/dt = P − (T − T_ambient) / R. Each node heats with time constant τ = R·C toward a steady state of T_ambient + P·R. The cooling option changes R and C:

• The MOSFET junction is hotter than its case: T_j = T_case + P_mos × 0.5 K/W.
• A MOSFET's on-resistance R_DS(on) rises as it heats, so the model raises its loss by 0.4 % per °C above 25 °C. A hotter MOSFET loses more and heats faster. If that feedback outruns the cooling there is no safe steady state: thermal runaway.
• Danger limits: the MOSFET uses the "MOSFET danger limit" slider (default 125 °C), the coil 150 °C (enamel wire rating) and the glass 150 °C (burn and thermal-stress hazard). Past 175 °C (the IRFP260N's maximum junction temperature) the MOSFET fails whatever the settings, and only Reset replaces it.
• Labels: Safe below 75 % of a limit, Warm up to the limit, Danger at or above it.
• Auto-shutdown stops the driver when the MOSFET or coil reaches its limit. The glass is not monitored by auto-shutdown, but it is part of the countdown.

Time to danger assumes the current operating point stays fixed and solves each node exactly: t = R·C · ln((T_ss − T) / (T_ss − T_limit)). The MOSFET node includes the R_DS(on) feedback in closed form. The shortest time of the three is shown. If every steady state stays below its limit the readout says "Safe — steady ≈ X °C" instead. Changing any control changes the estimate.

Parameter reference

Presets

Load a known setup. Xenon Toroid classic: 80 mm sphere, xenon 0.5 Torr, 24 V, 12 MHz, 4 turns, 70 % drive, 3.5 A, fan; it self-ignites into a rotating ring. Plasma Bean: xenon 1.6 Torr, 28 V, 75 % drive; needs a spark, then forms plasma beans (about 16). Neon Plasma Bean: neon 3 Torr, 30 V, 75 % drive, 4 A; self-ignites into a ring of about 17 pink beans. Neon Glow: neon 2 Torr, 60 % drive; gives a diffuse pink-red glow.

Glass shape

Donut (torus), sphere or cylinder; default sphere. Sets the discharge length, the coupling of power into the plasma, the bean pressure and where the ring sits (see Vessel shape).

Diameter

Outer size of the vessel, 60–130 mm in 5 mm steps, default 80 mm. A larger vessel raises p·d, which lowers breakdown voltage at low pressure and raises it at high pressure, and gives more beads in bean mode.

Noble gas

He, Ne, Ar, Kr or Xe, default Xe. Sets the Paschen constants, glow colour and the ring and bean thresholds (see the gas table). Heavier gases can light and form a ring at lower pressure and lower plasma power, though at a given pressure they need a higher breakdown voltage than helium or neon.

Pressure

Gas fill pressure, 0.01–10 Torr on a logarithmic slider, default 0.5 Torr. Too low and the gas cannot break down; too high and breakdown needs more voltage than the coil provides. Higher pressure gives a thinner ring and, for Ne/Ar/Kr/Xe, plasma beans.

Supply voltage

DC supply to the driver, 6–36 V in 0.5 V steps, default 24 V. Coil RF voltage scales with it and power demand with its square, so it raises brightness and heat together.

Drive frequency

Switching frequency, 5–20 MHz in 0.1 MHz steps, default 12 MHz. Best at the tank resonance (12 MHz for 4 turns); detuning lowers the gain g, the coil voltage and the power, and can make the plasma unstable. Higher frequency also lowers efficiency, so the MOSFET runs hotter, and spins the ring faster.

Coil turns

Turns on the RF coil, 2–8, default 4. More turns lower the resonance (f0 = 48 / N MHz), raise the coil voltage by √(N/4) and raise the coil's AC resistance. Re-tune the frequency after changing it.

Drive (potentiometer)

Drive level, 0–100 %, default 70 %. Coil voltage scales with it and power demand with its square.

Supply current limit

Bench-supply current cap, 1–5 A in 0.1 A steps, default 3.5 A. Input power cannot exceed supply voltage × limit; hitting it shows "current limited", and far past it the plasma becomes unstable.

MOSFET cooling

None, Heatsink or Heatsink + fan; default Heatsink + fan. Changes the thermal resistances and capacities (see the thermal table). With no cooling the MOSFET reaches 125 °C in about a minute at the default settings.

Ambient temperature

Room temperature, 0–40 °C, default 25 °C. Every part starts and cools toward it, so a hot room shortens the time to danger.

MOSFET danger limit

Junction temperature you treat as dangerous, 80–175 °C, default 125 °C. Used for the MOSFET label, the countdown, the chart line and auto-shutdown. The MOSFET still fails at 175 °C.

Auto-shutdown at danger limit

On or off, default on. Stops the driver when the MOSFET junction reaches its danger limit or the coil reaches 150 °C. Off lets you watch it run on into failure.

Auto-ignite (spark on Start)

On or off, default on. Fires a 0.5 s spark when you press Start, so breakdown needs only 60 % of V_b.

Start / Stop

Switches the driver on or off. Start is refused while the MOSFET is failed.

Ignite (spark)

Fires a 0.5 s spark at any time, lowering the ignition threshold to 60 % of V_b. Useful at higher pressure (the Plasma Bean preset needs it when Auto-ignite is off).

Reset / cool down

Stops the driver, returns every part to ambient temperature, clears the chart, run time and auto-shutdown, and replaces a failed MOSFET.

Time speed

1×, 2×, 5×, 10×, 30× or 60×, default 1× (real time). Speeds up the simulated clock so you can watch minutes of heating quickly.

Run time (simulated)

Simulated time the driver has been on, in minutes:seconds; it follows the time speed. Reset sets it to 00:00.

3D view

Drag to orbit, scroll or pinch to zoom.

Readouts

Plasma status card

Driver

Running, Stopped, AUTO-SHUTDOWN (stopped by a temperature limit) or MOSFET FAILED.

Ignition

Lit, Sparking… (a spark is active) or Not lit.

Plasma

Off, Diffuse glow, Ring, Plasma beans (with bead count) or Unstable.

Coil RF / breakdown

Coil RF voltage V_coil and the Paschen breakdown voltage V_b. ∞ means left of the Paschen minimum.

Input power

Power drawn from the supply, P_in, with "current limited" when the supply cap is reached.

Plasma power

Power delivered into the plasma, P_in · η · coupling; zero when not lit.

Supply current

DC current from the supply, P_in / V_supply.

Coil current

RF current in the coil, V_coil / (2π·f·L).

Resonance (detune)

Tank resonance f0 and how far the drive frequency is from it, in %.

Status note

Why the plasma is off, unstable or diffuse, e.g. "below breakdown: need 1047 V, have 756 V".

Circuit temperature card

MOSFET junction

Junction temperature T_j, with a Safe / Warm / Danger label against the MOSFET danger limit.

Coil

Coil temperature with its label against 150 °C.

Glass

Glass temperature with its label against 150 °C.

Limits

The danger limits in use: MOSFET (from the slider), coil and glass.

Time to danger

A countdown such as "2 m 10 s until MOSFET reaches 125 °C"; "Safe — steady ≈ X °C" when nothing will reach its limit; "Cooling down" when off and still hot; "DANGER" at a limit; "MOSFET failed — press Reset" after a failure.

Temperature chart

The last 300 s of MOSFET, coil and glass temperatures, with dashed lines at the MOSFET and coil limits.

Other messages

Danger banner

Red alert at the top when a part is at its limit, auto-shutdown has tripped or the MOSFET has failed.

Status line

Short announcements below the simulation buttons (also read by screen readers), e.g. state changes and preset loads.

Safety

A real plasma toroid is a high-voltage RF device. Its hazards include electric shock, RF burns from touching the coil or glass (RF burns can be deep and painless at first), interference with nearby electronics, UV light and ozone, hot MOSFETs and glass, fire, and the implosion of an evacuated glass vessel that is cracked or overheated. Build one only with proper training, shielding and supervision.

Simplified educational simulation — not engineering guidance.

References

• Tokamak in Your Home: a Xenon Plasma Toroid (Instructables)
• BagelGen
• Plasma Toroid Generator (Make:)
• V. I. Kolobov, "Striations in rare gas plasmas", J. Phys. D: Appl. Phys. 39, R487 (2006)
• Ionization waves (striations) in low-current DC discharges in noble gases obtained with a hybrid kinetic-fluid model (arXiv:2208.14321)
• Plasma stratification in AC discharges in noble gases at low currents (arXiv:2406.05926)
• Inductively coupled plasma (Wikipedia), background on RF inductive discharges
• three.js, used for the 3D view