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[ §3 · calculators · coupled poles ]

Tesla coil coupling and pole frequencies

calculators

Both poles for the general case, where primary and secondary are tuned to different frequencies, as they are in practice. The f/√(1±k) shortcut is only right at equal tuning, and the self-check line below shows the difference. It also tells you which pole a stock driver will sit on, and whether that changes during the ramp.

JavaTC gives the secondary's frequency with no arc and the coupling k from your coil's dimensions; the primary's comes from the primary tank page. Under an arc the streamer adds capacitance to the top and pulls the secondary down. That frequency cannot be calculated exactly, so the page estimates it from the arc's length and shows how much the estimate depends on the arc's capacitance per metre. If you have measured it, enter the measurement instead.

under an arc
what you have
secondary under arc 355 kHz (150 cm at 4.14 pF/m) at 8 pF/m (Mikkelsen) 313 kHzAt 8 pF/m the ramp ends on the UPPER pole instead: thearc's capacitance decides it, so check on the scope. no arc upper 519 kHz lower 317 kHzunder arc upper 463 kHz lower 296 kHzpole split 202 → 168 kHz A stock driver will settle on: burst start → LOWER pole (f_pri 349.9 vs f_sec 426.5) ramp end → EITHER (too close to call) pole (f_pri 349.9 vs f_sec 355) Just below equal tuning the rule is not reliable: asimulated ideal driver sat on the UPPER pole there. Check onthe scope which one yours picks. → The phase lead page (ramp start) and the tank page (running frequency) want the pole it runs on: LOWER here, 317 kHz. A driver started above the upper pole holds the UPPER one (519), as the reference coil's modified UD2 does: then use that.An 8 % error in f_pri moves the upper pole 2.5 % and the lower 6.0 %: the upper is the robust one here. self-check at equal tuning 426/426: general 560.5 / 357.8 simplified f/√(1∓k) 560.5 / 357.8
formula
from (1 - ω₁²/ω²)(1 - ω₂²/ω²) = k²   [lossless coupled circuits]
ω±² = [ω₁²+ω₂² ± √((ω₁²-ω₂²)² + 4k²ω₁²ω₂²)] / [2(1-k²)]
at ω₁=ω₂ this reduces to ω₀/√(1∓k)
from a sweep (dips f₋, f₊, peak f_s):  P = f₊²f₋², A = f₊² + f₋²
  f_pri² = P·f_s²/(A·f_s² - P)     k² = 1 - f_pri²·f_s²/P
under arc: f = f_sec·√(C_es/(C_es + c·l)), c = 4.14 pF/m (this site's fit
  on the reference coil's 1.5 m arc) or 8 (Mikkelsen's arc model)
pole rule (davekni, Uspring): f_pri < f_sec → lower;  f_pri > f_sec → upper
  1 - k²/2 < f_pri/f_sec < 1: too close to call (simulated: upper)
limits

The formula is lossless. Under arc load the ZCS points (the zeros of phase) no longer sit at these poles. Uspring showed that under heavy load there may be only one zero left, and otherwise there are three, the middle one unstable. Our circuit simulation (01.11.2026) puts numbers on it: three zeros appear whenever k²·Q₂/2 exceeds |1 - (f_pri/f_sec)²|, so at a coupling of 0.2 even with the two tuned 20 % apart. The zero nearer the primary's tuning moved under 0.5 % with load and the other up to about 3 %, and a zero-crossing driver never locked on the middle one.

It needs a tuned primary: DRSSTC or QCW. An SSTC with no primary capacitor has no poles in this sense.

The pole rule comes from two practitioners’ experience. It is not a theorem.