A Mathcad Program to Calculate the RF Waves Coupled from a WR650 Three-Stub Tuner to A

A MathCAD Program to Calculate the RF Waves Coupled from a WR650 Three-Stub Tuner to a CEBAF Superconducting Cavity

Haipeng Wang, SRF Institute, updated on August 05, 2004

Abstract

Three-stub WR650 waveguide tuners have been used on the CEBAF superconducting cavities for two changes of the external quality factors (Qext): increasing the Qext from 3.4~7.6×106 to 8×106 on 5-cell cavities to reduce klystron power at operating gradients and decreasing the Qext from 1.7~2.4×107 to 8×106 on 7-cell cavities to simplify control of Lorenz Force detuning. To understand the reactive tuning effects in the machine operations with beam current and mechanical tuning, a network analysis model was developed. The S parameters of the stub tuner were simulated by MAFIA and measured on the bench. We used this stub tuner model to study tuning range, sensitivity, and frequency pulling, as well as cold waveguide (WG) and window heating problems. This tech note is analytical part of the network model. I used the 7-cell cavity as an example and tune the stub tuner to decrease the Qext. The result of this analysis was used in my LINAC 2004’s paper [1]. In order to streamline my mathematic analytics and let readers easily copy or modify my work, this note is kept and written in MathCAD [2] format. Other MathCAD users can simply follow the math scripts in blue font, type in them for rework or just ask me for a copy of MathCAD (mcd) formatted file.

1. Characters of WR650 Waveguide 3-stub Tuner

1a. Constants Calculation

JLab Fundamental Couplers design uses special sized rectangular waveguide. It is called “reduced height waveguide”. Its width (long edge side) is 5.375 inch:

(m)

Its height is 0.986 inch

(m)

Standard waveguide coming out from klystron output is WR650 type. The waveguide width is 6.50 inch:

(m)

WR650 waveguide height is 3.25 inch

(m)

The speed of light:

(m/s)

RF source frequency is:

(Hz)

RF wave length is:

(m)

RF wave number is:

(1/m)

The waveguide material outside of cryomodule is aluminum. The permeability of aluminum is:

(H/m)

The propagation constant for TE10 mode in port 1 of rectangular waveguide (reduced height waveguide end) is:

(1/m)

The character impedance for TE10 mode in reduced height rectangular waveguide:

()

The propagation constant for TE10 mode in WR650 waveguide:

(1/m)

The phase delay of one-way path in 12-inch WR650 waveguide:

(rad)

(deg)

The phase delay of two-way path in 12-inch WR650 waveguide:

(rad)

(deg)

The taper waveguide length is 12 inch:

(m)

The taper tapping angle is:

(rad)

The taper waveguide width at coordinate z:

(m)

(m)

The propagation constant for TE10 mode in port 1 (reduced height waveguide end) at coordinate z:

(1/m)

The propagation constant for TE10 mode in port 2 (WR650 waveguide end) at coordinate z:

(1/m)

The fundamental power coupler external Q:

The field probe external Q:

The cavity's unloaded Q:

The field probe transformer ratio:

The fundamental power coupler transformer ratio:

The conductivity of room temperature aluminum:

(1/(m)

The surface resistance of room temperature aluminum:

() ()

Attenuation coefficient for TE10 mode, WR650 aluminum waveguide at room temperature:

(1/m) (1/m)

Attenuation coefficient for TE10 mode, reduced height aluminum waveguide at room temperature:

(1/m) (1/m)

Transmission matrix for l(l is a letter not number 1) mm long WR650 lossy waveguide:

Transmission matrix for l mm long reduced height lossy waveguide:

1b. MAFIA Simulation and Bench Measurement for One-stub Tuner

Both simulation and measurement were done at frequency=1.497GHz. Both data are agreed each other [1].

The S parameters of a single stub inside of WR650 with “zero” length of waveguide extension were fitted with MAFIA simulation data in the 5th order of polynomial:

The phase of S11 data has to be fitted into two sections due to the reference of “zero length”: one linear, one polynomial.

To check amplitude plots

First define the plot range: (mm)

Figure 1: Polynomial fitted S11 and S12 amplitudes plot for a single stub inside of WR650 waveguide with “zero” extra lengths.

To check phase plots

Figure 2: Polynomial fitted S11 and S12 phases plot for a single stub inside of WR650 waveguide with “zero” extra lengths.

They are agreed with both simulation and measurement. Then I convert them into real and image parts:

Reflections:

By geometry symmetry, it has relationship:

Transmissions:

By geometry symmetry, it has relationship:

Writing the S parameters into a complex format:

Writing the S parameters into a 2X2 complex matrix:

To check for d=20 mm:

For a reciprocal network, check matrix S=St [3]

For a lossless network check matrix StS*=U, here U is a unitary matrix [3]:

Converting the scattering matrix to a transmission matrix [4]:

To check for d=20 mm:

1c. Three-stub Tuner Transmission Matrix

From left to right is the direction of from source to cavity. Measured on a 12-inch long WR650 waveguide three-stub tuner: 77.4mm WG + stub #1 +75mm WG +stub #2 + 75mm WG +stub #3 +77.4mm WG

2. Character of the WR650 to JLab Reduced Height Waveguide Taper

2a. HFSS Simulation[5]

The drive frequency is at 1.5GHz

The simulated structure is: port1=WR650, witha 11.811" extra length; port2=reduced height (1”), with a 9.7795" extra length.

The S-parameter matrix calculated:

For a reciprocal network, to check S=St:

For a lossless network, to check StS*=U, here U is a unitary matrix:

Converting the scattering matrix to a transmission matrix:

The total transmission matrix is:

The transmission matrix for a straight section length of 11.811 inch of a lossless waveguide:

(m) (m)

The transmission matrix for a straight section length of 9.7795 inch of a lossless waveguide:

(m) (m)

Since T=T1T0T2, so T0=T1-1 T T2-1, to get the transmission matrix for taper only:

Converting the transmission matrix back to a scattering matrix:

Check for absolute values:

For a reciprocal network, to check S0=S0T

For a lossless network, to check StS*=U, here U is a unitary matrix

3. Characters of JLab Reduced Height Waveguide H-Bend

3a. HFSS Simulation [5]

The drive frequency is at 1.5GHz

The simulated structure is: port1=reduced height (1”), with a 350mm extra length; port2=reduced height (1”), with a 350mm extra length.

The S-parameter matrix calculated:

For a reciprocal network, to check S=St

For a lossless network, to check StS*=U, here U is a unitary matrix

Converting the scattering matrix to a transmission matrix:

The total transmission matrix is:

The transmission matrix for a straight section of 350 mm length of a lossless waveguide:

(m)

The transmission matrix for another straight section of 350 mm length of a lossless waveguide:

(m)

Since T=T1T0T2, so T0=T1-1 T T2-1, the H-bend waveguide only transmission matrix is:

Converting the transmission matrix back to a scattering matrix:

Check for absolute values:

For a reciprocal network, to check S0=S0T

For a lossless network, to check StS*=U, here U is a unitary matrix

4. Characters of CEBAF 7-cell Superconducting Cavity,

Fundamental Power Coupler and Field Probe Assembly

4a. Parallel LRC Circuit (Cavity) with Ideal Transformers (FPC+F.P.) without Beam Current Circuit Model (Sourceless)

The wave transmission matrix for a match load is a unitary matrix:

The field probe can be treated as a 1:n2's ideal transformer [4], here n2 is transformer ratio calculated in the 1a section.

The fundamental power coupler can be treated as a n1:1's ideal transformer [4], here n1 is transformer ratio calculated in the 1a section.

Normalized load conductance at the "detuned open" position of waveguide coupler as the function of detuned cavity frequency df is:

The cavity's shunt conductance transmission matrix is [4]:

The total FPC + cavity + FP transmission matrix is:

For df=0 or on the resonance:

5. Total Equivalent Circuit without Beam Loading Analysis

5a. Define waveguide length variables

Total WR650 waveguide length from 3-stub tuner to waveguide taper includes bends:

(mm)

This number is estimated, actual length should be surveyed from the drawings or installation site. This number also ignores the all WR650 bends effect (either H or E type). I just treat them as a straight section of WR650 waveguide here.

Total reduced height waveguide length from the H-bend Sweep to the superconducting cavity “detuned open" position on the FPC coupling waveguide:

(mm)

This number is estimated, actual length should be surveyed from drawings or installation site.

When these lengths change, the tuning result could be different. That is why each three-stub tuning varies cavity by cavity.

5b. Total wave transmission matrix from 3-stub tuner to field probe without beam loading

Converting the transmission matrix into a scattering matrix:

Calculate the S parameters in dB or degree like measured by a network analyzer, as a function of stub setting (d1, d2, d3) and frequency detuning (df) either by the cavity tuner or other sources (Lorenz force, microphonics etc.)

Plot S parameters as a frequency scan:

(start, start + incremental step… stop values) (Hz)

Figure 3: S21 amplitudescan plot with different stub settings. The red curve is “flush” stub setting with the original Q of 2e7, corresponding to FPC’s Qext. The peak not at df=0 is due to unmatched waveguide taper and H-bend. By inserting d3=31mm, the Q drops to 8e6 but also it causes a frequency pull in -26Hz (blue dot curve). When adding d2=18mm, the frequency pull goes back to zero (magenta dash curve).

Figure 4: S21 phasescan plot with different stub settings. The curve color represents same condition as in Figure 3.

Figure 5: S11 amplitudescan plot with different stub settings. The curve color represents same condition as in Figure 3.

Figure 6: S11 phasescan plot with different stub settings. The curve color represents same condition as in Figure 3.

5c. Equivalent External Q Calculation Using Power Transmission Method

Following “for loop” tries to find resonance peak on the amplitude of S21 curve and approximately calculate the equivalent external Q by the peak value. A simple derivation is from when at resonance:

Here and

Now the port1 is FPC and three stub tuner plus anything in between, the port 2 is the Field Probe. In CEBAF case, we have Q2ext >Q0>Q1ext (or Q1eqext). That is1>1>2. Then:

We can check the equivalent external Q at different stub settings.

When all stubs are in “flush” position:

When the third stub in 31 mm. the coupling Q changes into:

When the second stub in 18 mm, the frequency pull draws back to zero but the Q drops further into:

Now we can map out the equivalent external Q vs. stub tuner setup:

The d3 is changing from 0 to 40 mm (full range) in an incremental step of 1 mm.

Figure 7: Equivalent external change on a CEBAF 7-cell superconducting cavity by moving second and third stubs. The tuning range and sensitivity due to these changes can be read out from this graph.

The d1 is changing from 0 to 40 mm (full range) in an incremental step of 1 mm.

Figure 8: Equivalent external change on a CEBAF 7-cell superconducting cavity by moving first and second stubs. The tuning range and sensitivity due to these changes can be read out from this graph.

d2 is changing from 0 to 40 mm (full range) in an incremental step of 1 mm.

Figure 9: Equivalent external change on a CEBAF 7-cell superconducting cavity by moving second and third stubs. The tuning range and sensitivity due to these changes can be read out from this graph.

6. Klystron Incident Powerand SW Waveform on FPC Waveguide for Required Eacc without Beam Loading

6a. Constants Calculation

The 7-cell cavity's acceleration length:

(m)

The 7-cell Old Cornell ("OC") shape cavity's R/Q per unit length (r/Q) calculated by SuperFish:

(/m)

Required cavity's acceleration gradient:

(MV/m)

Transmitted power through the Field Probe for a given Eacc=12MV/m:

(W) (W)

Field Probe voltage on the 50  terminated transmission line (power meter cable is matched to the power meter’s input impedance):

(V) (V)

6b. Klystron Incident Power Required

We can check klystron incident powers at resonance peak for different stub setups:

(W) at

(W) at

Klystron incident powers at -3dB points for different stub setups:

(W) at

(W) at

(W) at

(W) at

6c. Transmission Line Voltage Calculation to Examine Standing Wave Amplitude on the FPC Reduced Height Waveguide and at the Location of Warm Window

The waveguide voltage on the WR650 input waveguide of the 3-stub tuner for different stub setups and different frequency pulling can be expressed as:

The input voltage will be expressed directionally with the first row as the incident and the second row as the reflected. With a “flushed” stub setting and df=15 Hz, the input waveguide voltage is:

With the third stub in 30 mm and df=-13Hz, the input waveguide voltage is:

Assume the warm window flange's "hot spot" is at a L3 mm away from the superconducting cavity's "detuned open" position upstream of the reduced height waveguide, the partial transmission matrix from the input waveguide of the 3-stub tuner to the "hot spot" is:

Then the incident and reflected voltages at the "hot spot" of the warm window is:

The Standing Wave Voltage amplitude on reduced height waveguide is:

Please pay attention to the voltage de-normalization and re-normalization from 50  to waveguide impedance

Now we can plot the Standing Wave Voltage waveform along the reduced height waveguide length:

(mm) Plot the waveguide distance from the cavity’s “detuned open” position to 3 meters away with the increment of 1 mm at each data point.

Figure 10: Standing Wave Voltage Amplitude (SWVA) waveform at a constant gradient of Eacc=12 MV/m along the FPC’s reduced height waveguide. Red curve corresponds to “flushed” stub setting with original Qext but detuned or de-Qed by the waveguides components between the stub tuner and the FPC. The blue curve is when the cavity tuner tunes to df=32Hz at -3dB point. The dash-green curve, when the third stub in 31mm, it de-Qs the system into 8.03e6 but also detuned the system by df=-26Hz. The dash-dot-magenta curve indicates when the second stub in 18mm additionally to d3=31mm, the detuning return to zero. The SWVA will be about same as original (red) one. This is an important conclusion that the minimization of the frequency detune will minimize the SWVA and heating on the window or cold waveguide components. Because the minimum in frequency detune reduces the klystron incident power.

7. Klystron Incident Powerand SW Waveform on FPC Waveguide for Required Eacc with Beam Loading

7a. Constants Calculation

The cavity's acceleration length, r/Q, acceleration gradient Eacc=12MV/m, transmitted power and field probe voltage are all same as in section 6 (without bean loading condition).

But we need calculate the cavity voltage (includes cavity’s Transit Time Factor or TTF here).

When Eacc=12 MV/m:

(MV) (MV)

To confirm the Vacc from the transmission matrix:

(V)

At resonance df=0, and pay attention to the impedance re-normalization here

(V)

It agrees with simple calculation above.

When the RF cavity has a beam load on it, its normalized shunt beam conductance with beam current I0 (mA), acceleration gradient Eacc (MV/m) and off-crest angle b (deg) is:

(1/)

When Eacc=12MV/m, I0=0.2mA, on crest. They are extreme CEBAF operation parameters.

(1/)

Thus the beam's shunt conductance transmission matrix is [4]:

7b. Total Wave Transmission Matrix from 3-Stub Tuner to Field Probe with Beam Loading

The waveguide voltage on the WR650 input waveguide of the 3-stub tuner:

For a “flushed” stub tuner setup with a detune of df=4Hz, at Eacc=12MV/m, I0=0.2mA, and beam on crest, the input waveguide voltage is:

(V)

When the third stub in 31mm (Q drops to 8e6) with a detune of df=32Hz, at Eacc=12MV/m, I0=0.4mA, and beam on crest, the input waveguide voltage is:

(V)

Assume the warm window flange's "hot spot" is at a L3 mm away from the superconducting cavity's "detuned open" position upstream of the reduced height waveguide, the partial transmission matrix from the input waveguide of the 3-stub tuner to the "hot spot" is:

Then the incident and reflected voltages at the "hot spot" of the warm window is:

The Standing Wave Voltage amplitude on reduced height waveguide is:

Please pay attention to the voltage de-normalization and re-normalization from 50  to waveguide impedance

Now we can plot the Standing Wave Voltage waveform along the reduced height waveguide length:

(mm) Plot the waveguide distance from the cavity’s “detuned open” position to 3 meters away with the increment of 1 mm at each data point.

Figure 11: Standing Wave Voltage Amplitude (SWVA) waveform at a constant gradient of Eacc=12 MV/m along the FPC reduced height waveguide. All first four curves’ conditions are as same as in Figure 10 except the last cyan color curve is the curve 4’s condition plus a beam current of 0.3mA and beam on crest operation. As seen similar to the surface current waveform in the Figure 1 of Reference [6], The voltage nodes will be floating up when a beam current loads up With an extreme CEBAF beam current load, the FPC waveguide never sees a “critical coupling” in the SWVA waveform of a straight line. The FPC Qext is always over-coupled with such light beam loading.

8. Conclusion

Based on this model analysis, I have concluded that three-stub tuner can modify (increase or decrease) the external coupling Q of a superconducting cavity over a range of 2 orders of magnitude. Stub position could be sensitive to the Q and phase change. Minimizing the frequency pulling away from the matched system is the key step to properly set up the stubs to avoid extra RF heating on the waveguide components. Based on the experience and result of this program, I judge that the phase drifting problem as the tunnel’s temperature variation is related to the reactance change on the waveguide components. To relief this problem, I recommend installing the stub tuner close to the cavity inside accelerator tunnel with a stepper-motor remote control. I can use this program or the network model to study this problem further. This model can be also modified to improve the reactive tuning compensation technique for other application.

The experiment (or test plan) on SL21 (new 7-cell “OC” shape cavity) cryomodule has confirm this analysis. No extra heating on both cold waveguide and warm window has been observed when the frequency pull was minimized [1].

Some of the figures and parameters in this note have been used in my LINAC2004’s publication [1].

9. Acknowledgements

I am grateful to M. Tiefenback for his clue on the frequency pulling effect on the waveguide component heating and his idea on the reactive tuning on the superconducting cavity. Thanks also go to Genfa Wu for his help on the HFSS simulations and Jay Benesch and Robert Rimmer for their constructive discussions. I would like also to acknowledge S. Chattopadhyay, A. Hutton and W. Funk for their encouragements and supports during the course of this analysis and experiment.

10. References

[1]H. Wang, M. Tiefenback, “Waveguide Stub Tuner Analysis for CEBAF Application”, Proceedings of LINAC 2004, Lubeck, Germany, Aug. 16-20, 2004, p836.

[2]Web page:

[3]David Pozar, Microwave Engineering, second edition, John Wiley and Sons, Inc., chapt 4: Microwave Network Analysis, p200.