Monday, November 30, 2020

An Anomalous Coupling Mode Supports Wide-Band RF Pulses in a High-Q, Narrow-Band Resonator

Parent Category: HFE Magazine

By Lewis Carroll


While preparing a conference paper on transmission line-driven inductive loop-coupling in cyclotron resonators (1) we discovered a heretofore unreported (as far as we are aware) ‘degenerate’, or ‘anomalous’ coupling mode where -- seemingly defying conventional wisdom and common sense -- the resonator’s power bandwidth expands by a factor of 20 or more, as if the system’s Q has collapsed -- though we know it hasn’t!

This mode is undesirable and should be avoided in a normal cyclotron application, but it is nonetheless interesting in its own right, and may have application in other technical fields.

In our conference paper we examined this anomalous mode using standard circuit analysis and computer-aided Q-factor analysis (2, 3). Here, we’ll illustrate the phenomenon with examples and data from actual measurements on a model resonator using standard RF lab instruments, including a Vector Network Analyzer (4).

To validate our analysis, we built and tested a coaxial resonator, as represented in Fig. 1. The inside of the cylinder is lined with copper foil. A is a coupling loop whose dimension is such that its un-coupled self-reactance is j50 ohms in accord with the example described in our conference paper. The loop is rotatable about its axis so that its orientation can be adjusted to vary the degree of inductive coupling, as required to establish a correct match to a 50 ohm transmission line. B is a copper support stem intended to simulate the corresponding structural element in a cyclotron resonator. C is a foil-covered disc representing an accelerating electrode (a ‘Dee’) in the cyclotron. D is a small capacitive RF pick-up probe -- loosely coupled to the resonator in order to sample RF signals without degrading the resonator’s Q. The un-coupled resonant frequency, f0, is ~80.7 MHz. Q0, the un-coupled Q, is ~3300 as determined by measurement of bandwidth at 3 dB below peak power.

1802 HFE coupling 01

Fig. 1 • Coaxial resonator.

Our guiding premise is that, for best efficiency, the output impedance of the RF generator -- ususally a high-power transmitter employing a triode or tetrode vacuum tube as the amplifying component -- should not be matched to Z0, the transmission feed-line’s characteristic impedance.

If the transmitter’s output impedance is matched to Z0, the frequency for correct Z0 match ‘automatically’ coincides with the frequency for maximum power in the resonator, but half of its power is then wasted -- absorbed in the transmitter’s output resistance. More power becomes available if the transmitter resembles either a nominal voltage (low resisteance) source, or a nominal current (high resistance) source. But when the generator output resistance is not matched to Z0, the frequency for maximum power may not necessarily coincide with the frequency for correct match.

Assuming the generator output is resistive (no reactance) the frequencies can be aligned by choosing an appropriate length of transmission line such that the resonator’s Q circle (the resonator’s S11 trace as recorded on a Vector Network Analyzer) is aligned symmetrically with respect to the horizontal (pure resistance) axis of a Smith Chart. In the case of our 50 ohm loop, the correct length is 45˚ for a voltage source, or 135˚ for a current source, plus addititional 1/2 wave sections if required to make the physical connection.

Bandwidth versus Line Length

Fig. 2 is a screen shot showing an S21 VNA trace sampled on our model resonator at terminal D of Fig. 1. The trace is taken with a 1000 ohm resistor connected in series with the amplified  (Mini - Circuits ZKL-1R5+) VNA signal in order to simulate a Thevenin-equivalent current source which, in turn, is connected to the coupling loop’s input through a 135˚ length of transmission line. The bandwidth is ~24.3 KHz, as shown by markers ‘3-2’ at the upper left corner of the figure. The inset shows the relevant S11 Q circle for the 135˚ line section -- here generated by computer simulation, since the VNA’s normal S11 function can’t be used when an amplifier and 1000 ohm resistor are in series with the feed-line.

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Fig. 2 • Screen shot of VNA trace.

Now, see what happens when we replace the 135˚ line with a 45˚ line (Fig. 3). The 3 dB bandwidth explodes to ~ 573 KHz! Why? The resonator Q hasn’t changed, and its input is still matched to Z0 at the same frequency as before, and the standing wave ratio (SWR) profile versus frequency is also unchanged, since that doesn’t vary with line length. But the Q circle is now congruent with an opposite iso-contour on the Smith Chart -- a circle of constant resistance which, like the Q circle itself, is tangent at the right edge of the chart, so that, when drivenby a nominal current source, the power (I2R) in the resonator is essentially constant over a correspondingly wide frequency range.

1802 HFE coupling 03

Fig. 3 • 3 dB bandwidth rises to ~ 573 KHz.

The same occurs when a nominal voltage source drives the resonator through 135˚. The Q circle then being situated on the left side of the Smith Chart, now congruent with a circle of constant conductance. But note that a voltage source becomes a current source -- and vice versa -- by adding (or subtracting) a 90˚ section of transmission line in series with the transmitter’s output. Rules of thumb: Voltage source at the right side of the chart = narrow-band; voltage source at the left side of the chart = wide-band, and conversely for a current source.

Wide-Band RF pulses

To further demonstrate the effect of our anomalous coupling mode, we inserted an RF switch (Analog Devices/Hittite #108436-2) driven by a pulse generator (B&K Precision #4030) between the VNA and amplifier to modulate the CW signal at frequency f0 behind 1000 ohms in order to simulate a Thevenin equivalent current source, as shown previously.

Fig. 4 shows 40 :sec wide pulses at 20 :sec/cm. The vertical amplitudes are not all to scale. The oscilloscope is a vintage Tektronix 2235A. Fig. 4a is the RF waveform sampled directly -- not through the resonator. Fig. 4b is the signal driven through the narrow-bandwidth (135˚) line section, sampled at terminal D. Fig. 4c is the same, but driven through the wide-bandwidth (45˚) line section.

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In Fig. 5 the upper trace is also sampled at terminal D via the 45˚ wide-band connection, showing 2 :sec wide pulses recorded at 2 :sec/cm. The lower trace is sampled at the junction connecting the 1K resistor with the 45˚ transmission line. Note the ringing and reflections due to high-order sidebands.

Cautions, Limitations, and Conclusion

High-order sidebands inherent with wide-bandwidth RF pulses will cause substantial reflections in the transmission line, as shown in Fig. 5. The transmitter must be able to tolerate same.

1802 HFE coupling 05

Fig. 5. Note the ringing and reflections due to high-order sidebands.

Our treatment thus far has assumed that the generator’s output impedance is purely resistive. If the generator’s output impedance includes reactance, then either an additional (opposite) reactive ‘trim’ element at the generator end, or an additional line length is required to align the Q circles for both the resonator and that of the generator,

A potential limitation to bandwidth in the wide-band case is due to line losses for an extended transmission line totaling more than a few wavelengths when additional 1/2 wave sections are needed to make a physical connection. Such losses cause the Q circle to shrink away from the perimeter of the Smith Chart, thereby reducing the degree of congruence with the corresponding iso-contour.

Aligning best match with max power in the narrow-band case, as in a normal cyclotron application, is relatively non-critical. Simulation studies show that as much as plus or minus 22.5˚ feed-line length relative to ‘optimum’ results in a negligible frequency displacement of the peak for maximum power relative to that for best match.

However, in the anomalous wide-band case, achieving congruence between the Q circle and the corresponding iso-contour requires care and precision in trimming the transmission line to the prescribed length. An error of as little one degree, or ~1 mm for a 45˚ length of RG-type coaxial cable at 80.7 MHz, results in a substantial offset of the peak frequency in our high-Q example. However, this offset error can be compensated by means of a reactive ‘trim’ adjustment at the generator end, as mentioned above.

In any case, the required degree of precision may actually be an asset for certain metrology or RF range-finding applications. Preliminary computer-aided simulation studies and initial bench tests suggest that, with the appropriate circuit configuration, the anomalous mode can be applied to other, more compact resonator types, such as a quartz crystal. Suggestions are welcome!


(1) Carroll, L. ‘Characterization of Inductive Loop Coupling in a Cyclotron Dee Structure’ Physics Procedia 90 (2017) pp 126-135, Posted on line (free download) at <>

(2) Kajfez ‘Q factor Measurements, Analog and Digital’. Posted on line at <>

(3) Harriman, W, Simsmith RF Simulation software; Description and Download at <>

(4) SDR Kits, VNWA3 Computer-aided Vector Network Analyzer, described at <>

About the Author

1802 HFE coupling 06Lewis Carroll

Carroll & Ramsey Associates provides service and technical support to the community of accelerator builders and users, including leading cyclotron manufacturers, U.S. Department of Energy National Laboratories, University Research and Clinical PET/Nuclear Medicine Imaging Centers, and commercial isotope producers.

Principals Lewis Carroll and Fred Ramsey are veteran contributors to the accelerator and Positron Emission Tomography (PET) fields. Their experience spans almost 50 years, first as members of the technical staff of The Cyclotron Corporation (TCC), then as founding members of CTI Cyclotron Systems (now part of Siemens Medical Systems), where they were lead members of the team that developed the RDS- series of automated, self-shielded PET cyclotrons and, finally, as partners in an independent consulting firm.

Lewis Carroll has worked in the accelerator and radiation safety and radiation science fields since 1969. He also led the early development of innovative Positron Emission Tomography (PET) Systems, including the first commercially-produced multi-slice Neuro-Tomograph. Today, Carroll & Ramsey Associates’ specialized radiation detectors are used in commercial and academic research labs and radio-pharmacies around the world.

Contact: Lewis Carroll <>;

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