Cyclotron RF matching design rules

adjusting Cn for coincidence of maximum dee voltage and minimum plate current as the dee was tuned through resonance

Osterlund & Smythe, A Cyclotron Power-Amplifier RF System Using a 4CW50,000C/8350 Tetrode — COO-535-543 (1963) — p. 4

Editorial note, tabletop extrapolation: Neutralization is a triode/tetrode matter, and the coincidence test belongs to a neutralized tuned-plate PA: on that class of amplifier the dee-voltage peak and plate-current dip should line up through resonance, and a skew flags feedback. On a solid-state or matched-line chain there need be no input-current dip at resonance at all - verify resonance and match there with dee voltage, reflected power and the device's rated currents instead. [Note revised 2026-08-23: the earlier note generalised the test to 'any amplifier-dee chain' and changed the observable to PA input current.]

The following tuning procedure was found to be easy to follow: 1) Set the oscillator frequency at w0 and set the admittance meter to the correct reading for zero admittance at the junction to the system. 2) Tune the length of one of the lines for a null on the admittance meter. 3) Set the oscillator frequency to 3w and measure the admittance; note whether it is positive or negative. 4) Return to w0 and change a1, compensating with a change in a2 to keep the system tuned to w0. One can determine which way to tune by comparing the position of the resonances with numbers interpolated from Appendix III. 5) Remeasure admittance at 3w, and repeat the procedure until the admittance measures zero at 3w.

Goodman, A Square-Wave Cyclotron Oscillator — ORNL-2403 (1958) — p. 21

Editorial note, tabletop extrapolation: The written five-step procedure is a model for documenting any coupled-adjustment RF tune-up - a next machine's coupling loop and trimmer interact the same general way; precomputed knowing-which-way-to-tune tables are the transferable trick, though the dual-line tables themselves don't map onto a different topology.

if the coupling line is one-quarter of the fundamental wave length it is three-quarters of the third harmonic wave length, and the impedances are simply inverted by the line at both frequencies

Goodman, A Square-Wave Cyclotron Oscillator — ORNL-2403 (1958) — p. 22

Editorial note, tabletop extrapolation: A handy odd-harmonic identity - and a warning that a quarter-wave feeder presents TRANSFORMED impedances to your amplifier's harmonics even in a plain sine-wave system: evaluate the actual harmonic load with Z0(f) and measured S-parameters (connectors and loading shift the third-harmonic electrical length) before assuming either benefit or instability.

A deviation of an inch or two one way or the other will cause this response to rise or fall at either end of the range.

Anderson, Half-Scale Model Tests on the Three Quarter Wave R.F. System — UCRL-31 (1947) — p. 13

Editorial note, tabletop extrapolation: Even a fixed-frequency amateur system inherits this through the amp-to-dee coax and its stray resonances: sweep the ASSEMBLED feed system, not just the cavity, and choose line lengths from the measured input impedance and matching bandwidth at the operating frequency. The cited inch-scale sensitivity belongs to that swept resonant feedline - your system's sensitivity scale comes out of your sweep.

over about a 2 to 1 frequency shift, the voltage on the 1/4 wave-shorted line (which will be referred to as the "stub" line) is practically the same as the dee voltage.

MacKenzie, Preliminary Report on the “Three Quarter Wave” R.F. System for Frequency Modulated Cyclotrons — AECD-1850, University of California (1947) — p. 3

Editorial note, tabletop extrapolation: Even a fixed-frequency machine drifts with thermal expansion and plasma loading, and feeding at a voltage-stable point of the resonator helps - but a stable voltage RATIO is not constant drive impedance: detuning, Q and plasma loading still move what the amplifier sees, so measure the input impedance (or S11) across the expected drift and loading range before promising the amplifier anything.

start with the tap at the end of the stub line and then move toward the shorted end until the tube draws rated plate current at rated voltage.

MacKenzie, Preliminary Report on the “Three Quarter Wave” R.F. System for Frequency Modulated Cyclotrons — AECD-1850, University of California (1947) — p. 11

Editorial note, tabletop extrapolation: The walk-the-tap idea transfers as a method of converging on coupling empirically rather than committing to a computed position - executed safely: find the initial setting at low power or with a VNA, move taps only de-energized, approach the operating point with current limiting, and watch plate current AND dissipation AND reflected power together - rated plate current alone is one indicator, not proof of match, and the line's high-impedance end carries hazardous RF voltage.

the assembly was supported by a 1.3 cm thick copper rod that also connected the dee to the RF matching transformer

Cressman, The Design and Construction of a Small Cyclotron — Houghton College thesis (2006) — p. 17

Editorial note, tabletop extrapolation: Making the RF feed a structural member gives a rigid connection and can save a penetration - count your own: an internal support needn't pierce the wall at all, and low loop inductance comes from the LENGTH and return-path geometry, not rod thickness. The 1.3 cm is Rutgers' as-built datum; size a new rod from RF current, mechanical load and the actual loop.

The capacitance of the dee and chamber of the Houghton College cyclotron has been determined to be 79 pf. If the maximum magnetic field of 1.127 T is used then the frequency of orbit for singly ionized helium is 4.32 MHz, and thus the inductance, using (23), must be 17.2 uH. In this system the maximum energy for singly ionized helium is 77.2 keV. For doubly ionized helium, the frequency in the same magnetic field is 8.63 MHz, so the inductance is 4.29 uH, and the maximum energy is 309 keV.

Cressman, The Design and Construction of a Small Cyclotron — Houghton College thesis (2006) — p. 46

Editorial note, tabletop extrapolation: A rare published electrode-system capacitance anchor for microhenry-scale resonator sizing at this machine class - noting the energy quadrupling with charge state at fixed field (the implied orbit radius is ~7.1 cm), that 79 pF is build-specific, and that the installed resonance still needs the parasitics-and-trim treatment (dg-1462).

With an improved matchbox, two cyclotron frequencies of 2.82 MHz and 5.64 MHz are available.

Wolf & Prechtl, COLUMBUS — A Small Cyclotron for School and Teaching Purposes — THPO001, Proceedings of Cyclotrons2022 (2022) — p. 1

Editorial note, tabletop extrapolation: Two-frequency matching lets a small machine serve a light ion and its molecular ion, or run one ion at half field: at the fundamental, 2.82 MHz pairs with H+ at 185 mT or H2+ at 370 mT, 5.64 MHz with H+ at 370 mT (f = qB/2*pi*m). The paper states availability; which pairings were demonstrated as beam operating points, and the switching mechanics (COLUMBUS's own book documents a vacuum-relay inductor switch, dg-1393), need their own evidence per machine.

A directional-coupler in the input-circuit of the matchbox makes it possible to control and minimize the reflections back into the RF-source and a RF pick-up, i.e. Fig. 6, in the output-circuit allows to check whether the cyclotron is tuned to the cyclotron-frequency of 5.63 MHz

Wolf, Frank & Held, COLUMBUS — A Small Cyclotron for School and Teaching Purposes — WE1PB03, Proceedings of Cyclotrons2013 (2013) — p. 2

Editorial note, tabletop extrapolation: Two independent indications, reflected power at the input and detected RF at the dee side, help separate matching problems from resonance problems during tune-up - though both respond to coupling and resonance, so neither is unambiguous alone, and the pick-up reads amplitude: the drive frequency itself should be known independently (a counter is cheap) and compared against qB/2πm.

they are injected axially using a spiral electrostatic inflector, Figure 3. The inflector geometry has been optimized with the computer codes CASINO, RELAX3D and Poisson so that the emittance of the ion beam coming out of the inflector matches the acceptance of the cyclotron

Clark, Halbach, Kunkel, Leung, Li & Young, A Compact Permanent Magnet Cyclotron for Accelerator Mass Spectrometry — Proceedings of Cyclotrons'95, Cape Town (1995) — p. 2

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: emittance matching at the inflector exit — the output phase-space distribution oriented so it lies within the cyclotron acceptance, not mere geometric survival — is the design criterion; the code roles (electrode field solve plus 3-D trajectory integration in the real fields) map onto modern open tools.

The development includes a broadband solid state RF power amplifier up to 2 kW CW, Impedance matching network (IMN) and GDR based Digital LLRF Controller. The aim of the RF system is to generate and maintain high RF voltage across Dee-Dummy Dee to accelerate the particles from the ion source of Cyclotron.

IUAC, Annual Report 2024–25, Chapter 3 — Research Support Facilities (table-top cyclotron RF system) — p. 18

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: the amplifier-IMN-dee chain with a digital feedback controller is the modern minimal RF architecture for a small cyclotron, and the dee/dummy-dee (single-dee) geometry matches common amateur practice. The 2 kW CW is this amplifier's rated maximum, not a derived drive requirement — the power a given machine needs follows from its dee voltage, shunt impedance, coupling and losses, so treat the rating as one professional team's headroom choice for an MeV-class teaching machine.

An HP8165 digital programmable RF signal source was used to drive an ENI350L 100 watt solid state amplifier. This method was much more convenient as fine tuning was easily achieved at the signal source rather than by manually tuning the tank circuit. The smallest adjustment capable of the HP8165 is 10kHz, which proved to be sufficiently sensitive. The output of the ENI350L amplifier was then passed through a Bird wattmeter (model 4410) and on to the RF cabinet.

Koeth, The Construction and Operation of a Nine Inch Cyclotron (undated scan; the machine ran 1995–1999) — p. 3

Editorial note, tabletop extrapolation: Calibration data from one resonator, plus one broadly good idea. The data: 100 W of solid-state drive bought ~1700 V peak dee here (16 W forward on the beam run of record), and 10 kHz source steps proved finer than the ~90 kHz loaded bandwidth — comfortable for THIS tank. The idea: fine-tune at the SIGNAL SOURCE, not the tank — it removes mechanical tuning from the operator's inner loop. Size your own amplifier from your dee capacitance, loaded Q, coupling and target voltage, with headroom for mismatch and discharge transients.

It utilizes the lumped capacitance of the DEE, which is approximately 70pF, to create a tank circuit out of the chamber itself. Using the resonance equation for an inductor in parallel with a capacitor: fr=1/2(pi)sqrt(LC) L, the inductance, was chosen to bring the fr to resonance at 13.56 MHz. Initially, coarse tuning was to create an 8 turn coil of length 5 inches, with a cross sectional area of 2.14 inches^2, out of 1/4-inch copper refrigeration tubing.

Koeth, The Construction and Operation of a Nine Inch Cyclotron (undated scan; the machine ran 1995–1999) — p. 3

Editorial note, tabletop extrapolation: The topology is the copyable part: use the dee-to-lid capacitance itself (~70 pF here) as the tank C, add an air-core tubing inductor, and couple through a coaxial few-turn primary with movable taps to find 50 Ω — no quarter-wave stem, no vacuum variable. Two numbers to reconcile on your bench: resonance at 13.56 MHz with 70 pF wants ≈2.0 µH (computed from the source's own equation), while Wheeler's formula on the printed coil geometry (8 turns, 5 in long, 2.14 in² area) yields only ≈0.8 µH — leads, strays and the actual in-situ capacitance evidently make up the difference, which is precisely why you measure fr in place and provide fine tuning rather than copying dimensions. (The 3-turn coaxial primary, adjustable taps and 50-ohm loading are printed on p.4.)

Cooling became a necessity when the RF power began to heat the secondary coil such that thermal expansion changed the tank fr. General Electric Dielectrol transformer oil is pumped through the 1/4 inch tubing of the secondary. The oil was then passed through a small heat exchanger that is cooled by flowing water. The oil is then returned to the pump reservoir of approximately two gallons volume. No effort was made to measure the cooling rate of the oil.

Koeth, The Construction and Operation of a Nine Inch Cyclotron (undated scan; the machine ran 1995–1999) — p. 4

Editorial note, tabletop extrapolation: A concrete failure mode plus fix at tabletop RF power levels (tens of watts to ~100 W into a high-Q tank): the tank drifts off tune as it warms, and the fix is to circulate a dielectric coolant inside the hollow tubing that already forms the inductor. Using transformer oil rather than water keeps the coolant non-conductive at the high-voltage end. The author notes no calorimetry was done, so no efficiency number can be taken from this.