Dee Capacitance & RF Matching Calculator
The dee and the grounded liner around it form a capacitor; resonating that capacitance with an inductor turns the dee into a tank circuit, and the RF problem reduces to matching that tank to a 50 Ω amplifier. This calculator estimates the dee capacitance from a parallel-plate model, the inductance that resonates it at your drive frequency, and the values of a simple L-network match. Every number here is a starting point for measurement, not a final design — real dee capacitance depends on stem, posts, and liner details that no parallel-plate formula captures.
Results
| Dee face area (one side) | — |
| Estimated dee capacitance Cdee | — |
| Inductance to resonate at f | — |
| Reactance of Cdee at f | — |
| L-network Q = √(Rp/50 − 1) | — |
| Series capacitor Cs (source side) | — |
| Shunt inductor Lp (dee side) | — |
| Matched −3 dB bandwidth ≈ f/Q | — |
The math
Capacitance estimate
A dee of radius r presents a half-disc face of area A = πr²/2 toward the liner above it and again below it — two parallel-plate capacitors in parallel:
The fringe factor (default 20%) crudely accounts for edge fields; the stem allowance covers the dee stem and feed structure. This is deliberately a crude model — expect the real value to differ by tens of percent, and measure it (grid-dip a known inductor against the dee, or use a VNA) before winding a final resonator coil.
Resonance
At resonance the dee tank looks, from its feed point, like a pure resistance Rp — the parallel loss resistance set by conductor losses (Rp = Qtank · XC). Amateur-scale copper dee assemblies commonly land in the several-kΩ to tens-of-kΩ range; measure yours.
Matching 50 Ω to Rp
With Rp > 50 Ω, the canonical L-network places the series element on the low-resistance (source) side and the shunt element across the high-resistance (dee) side. This calculator uses the high-pass form — series capacitor, shunt inductor — which also blocks DC and drains static charge off the dee:
In practice Lp and the resonating inductance L can be merged into one tapped coil (the common amateur arrangement); the two-element form above keeps the algebra transparent. Component values are exact only at f; the match degrades over a fractional bandwidth of roughly 1/Q.
Assumptions and limits
- Parallel-plate capacitance with a flat fringe factor — no account of dee thickness, aperture, pillars, or the accelerating-gap capacitance to the dummy dee.
- Rp is an input, not a prediction; it varies with construction quality, plating, and joint resistance, and it sets the drive power for a given dee voltage (P = Vdee²/2Rp).
- Ideal lossless matching components; at tens of kΩ transformation ratios, real component Q and stray reactance matter. Beam loading is ignored (fine at microamp beams).
Worked check
Dee radius 13 cm, 25 mm gaps top and bottom, 20% fringe, 5 pF stem: face area 265 cm², Cdee ≈ 27.6 pF. At 8.87 MHz (protons at 0.582 T) the resonating inductance is 11.7 µH. Matching 50 Ω to Rp = 20 kΩ: Q = 20.0, Cs = 18.0 pF, Lp = 18.0 µH, bandwidth ≈ 440 kHz.
Sources
- Parallel-plate capacitance: any electromagnetics text, e.g. D. J. Griffiths, Introduction to Electrodynamics, §2.5.
- L-network design equations: C. Bowick, RF Circuit Design, 2nd ed., Newnes, 2008, ch. 4; ARRL Handbook, impedance matching chapter.
- Cyclotron RF systems: J. J. Livingood, Principles of Cyclic Particle Accelerators, Van Nostrand, 1961 — ch. 12.