JLAB-TN-04-031
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 Q ext 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:
a1  5.375 0.0254
a1  0.137 (m)
Its height is 0.986 inch
b1  0.986 0.0254
b1  0.025 (m)
Standard waveguide coming out from klystron output is WR650 type. The waveguide width is 6.50 inch:
a2  6.50 0.0254
a2  0.165 (m)
WR650 waveguide height is 3.25 inch
b2  3.25 0.0254
b2  0.083 (m)
The speed of light:
c  299792458(m/s)
RF source frequency is:
6
f  1497 10 (Hz)
RF wave length is:
c
  0.20026 (m)
 
f
RF wave number is:
2 
(1/m)
k 

The waveguide material outside of cryomodule is aluminum. The permeability of aluminum is:
6
(H/m)
  1.256637061 10
The propagation constant for TE10 mode in port 1 of rectangular waveguide (reduced height waveguide
end) is:

 a1  k  

 a1 
2
2
 a1  21.328
(1/m)
The character impedance for TE10 mode in reduced height rectangular waveguide:
2  f  
 a1  554.204 ()
 a1 
 a1
The propagation constant for TE10 mode in WR650 waveguide:
 a2 
  
a 
 2
2
k 
2
 a2  24.945885
(1/m)
The phase delay of one-way path in 12-inch WR650 waveguide:
(rad)
d1   a2 12 0.0254 d1  7.604
 1  d1
180
 2  d2
180
(deg)
 1  435.649

The phase delay of two-way path in 12-inch WR650 waveguide:
d2  2 d1 d2  15.207 (rad)
(deg)
 2  871.298

The taper waveguide length is 12 inch:
L  12 0.0254 L  0.305 (m)
The taper tapping angle is:
a2  a1
(rad)
 ( l) 
2 l
The taper waveguide width at coordinate z:
az1( L 0)  0.137
az1( l z)  a1  2 z tan  ( l) 
(m)
az2( l z)  a2  2 z tan  ( l)  az2( L 0)  0.165
(m)
The propagation constant for TE10 mode in port 1 (reduced height waveguide end) at coordinate z:
2
 1( l  z) 
k 



 a ( l z) 
 z1

2
 1( L 0)  21.328
(1/m)
The propagation constant for TE10 mode in port 2 (WR650 waveguide end) at coordinate z:
  
 2( l z)  k  

 az2( l z) 
2
2
 2( L 0)  24.946
The fundamental power coupler external Q:
7
Q1ext  2.2 10
The field probe external Q:
12
Q2ext  1 10
The cavity's unloaded Q:
10
Q0  1 10
The field probe transformer ratio:
Q0
n 2 
n2  0.1
Q2ext
The fundamental power coupler transformer ratio:
Q0
n 1 
n1  21.32
Q1ext
The conductivity of room temperature aluminum:
7
  3.745 10 (1/(m)
The surface resistance of room temperature aluminum:
(1/m)
Rs 
 f  
() Rs  0.013 ()

Attenuation coefficient for TE10 mode, WR650 aluminum waveguide at room temperature:
 2 
Rs


 1  2
  

 2 a2 
120  b 2 1  
2
  

a2  2 a2 
b2

2
 (1/m)   6.944  10 4
2

(1/m)
Attenuation coefficient for TE10 mode, reduced height aluminum waveguide at room temperature:
 1 
Rs
  

 2 a1 
120  b 1 1  

b 1    2
3
(1/m)  1  2.344  10
 1  2  



a1  2 a1  
2 
(1/m)
Transmission matrix for l (l is a letter not number 1) mm long WR650 lossy waveguide:
 exp   i    l 

0
a2

  2

1000
Twg2( l)  

l 


0
exp   2  i  a2 

1000 


Transmission matrix for l mm long reduced height lossy waveguide:
 exp   i    l 

0
a1

  1

1000
Twg1( l)  

l 


0
exp   1  i  a1 

1000 


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:
-5 2
-6 3
-7 4
-8 5
Sam11( d)  0.00188  0.00607 d  6.52086 10  d  1.91197 10  d  9.52745 10  d  1.79046 10  d
-4 2
-5 3
-6 4
-8 5
Sam12( d)  0.99981  0.00172 d  5.47489 10  d  5.10526 10  d  1.96439 10  d  2.07256 10  d
The phase of S11 data has to be fitted into two sections due to the reference of “zero length”: one linear, one
polynomial.
Sph11( d)  ( 9.4789 d  5.58783) if d  4
49.09987  2.02381d  0.18883d2  0.00896d3  1.4504510 4d4  7.3246110 7d5
2
-4 3
-5 4
-6 5
Sph12( d)  0.05372  0.15847 d  0.01259 d  7.47079 10  d  7.0572 10  d  1.07193 10  d
To check amplitude plots
First define the plot range: d  0 1 41 (mm)
if d  4
1
Sam11( d)
0.5
Sam12( d)
0
0
10
20
30
40
d
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
50
Sph11( d)
Sph12( d)
0
0
5
10
15
20
25
30
35
40
d
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:
 

Sre11( d)  Sam11( d)  cos  Sph11( d) 

180 

 

Sim11( d)  Sam11( d)  sin Sph11( d) 

180 

By geometry symmetry, it has relationship:
Sre22( d)  Sre11( d)
Sim22( d)  Sim11( d)
Transmissions:
 


180


 

Sim12( d)  Sam12( d)  sin Sph12( d) 

180 

Sre12( d)  Sam12( d)  cos  Sph12( d) 
By geometry symmetry, it has relationship:
Sre21( d)  Sre12( d)
Sim21( d)  Sim12( d)
Writing the S parameters into a complex format:
S11( d)  Sre11( d)  Sim11( d)  i
S12( d)  Sre12( d)  Sim12( d)  i
S21( d)  Sre21( d)  Sim21( d)  i
S22( d)  Sre22( d)  Sim22( d)  i
Writing the S parameters into a 2X2 complex matrix:
 S11( d) S12( d) 
S( d )  

 S21( d) S22( d) 
To check for d=20 mm:
 0.1884  0.1239i 0.9607  0.17i 
d  20 S( d)  

 0.9607  0.17i 0.1884  0.1239i 
For a reciprocal network, check matrix S=St [3]
T
 0.1884  0.1239i 0.9607  0.17i 

 0.9607  0.17i 0.1884  0.1239i 
S( d)  
For a lossless network check matrix StS*=U, here U is a unitary matrix [3]:
T  1.0027 0.404 
S( d )  S( d)  

 0.404 1.0027 
Converting the scattering matrix to a transmission matrix [4]:
S22( d )
 1


S12( d )
S21( d )
Tp ( d )  
 S11( d)
S22( d )
S12( d )  S11( d ) 

S21( d )
 S21( d)





To check for d=20 mm:
 1.009  0.179i 0.212  0.091i 
Tp ( d)  

 0.212  0.091i 0.932  0.127i 
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
T3st d1  d2  d3  Twg2( 77.4)  Tp d1 Twg2( 75.0)  Tp d2 Twg2( 75.0)  Tp d3 Twg2( 77.4)
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, with a 11.811" extra length; port2=reduced height (1”), with a
9.7795" extra length.
The S-parameter matrix calculated:
 7.84 10-2  2.34 10-2 i 6.01 10-1  7.95 10-1 i 

Stp  

-1
-1
-2
-2 
 6.01 10  7.95 10  i 4.39 10  6.91 10  i 
For a reciprocal network, to check S=St:
T  0.0784  0.0234i 0.601  0.795i 
Stp  

 0.601  0.795i 0.0439  0.0691i 
For a lossless network, to check StS*=U, here U is a unitary matrix:
T 
Stp  Stp 
5
5

1
3.52  10  3.82i  10 



5
5
1
 3.52  10  3.82i  10

Converting the scattering matrix to a transmission matrix:
Stp
 1
2 2

S
S
tp
 tp 1  2
2 1
Ttp  
Stp
 Stp 1  1
2 2
S

S

tp 1  2
tp 1  1
 Stp
Stp
2 1
 2 1







The total transmission matrix is:
 0.605  0.8i 0.029  0.077i 
Ttp  

 0.029  0.077i 0.605  0.8i 
The transmission matrix for a straight section length of 11.811 inch of a lossless waveguide:
(m)
l1  11.811 0.0254 (m) l1  0.3
 exp i  1( L 0)  l1
Ttp1  

0
 0.993  0.115i

 Ttp1  

0
0.993  0.115i 
exp i  1( L 0)  l1 

0
0

The transmission matrix for a straight section length of 9.7795 inch of a lossless waveguide:
l2  9.7795 0.0254 (m) l2  0.248 (m)
 exp i  2( L 0)  l2
Ttp2  

0

0
 0.996  0.087i

 Ttp2  

0
0.996  0.087i 
exp i  2( L 0)  l2 

0
Since T=T1T0T2, so T0=T1-1 T T2-1, to get the transmission matrix for taper only:
1
Ttp0  Ttp1
1
 0.628  0.783i 0.013  0.081i 

 0.013  0.081i 0.628  0.783i 
 Ttp  Ttp2
Ttp0  
Converting the transmission matrix back to a scattering matrix:
 Ttp02  1

1

 Ttp01  1 Ttp01  1
Stp0  
Ttp0
 1
1 2
 Ttp0
Ttp0
1 1
1 1



 0.0710  0.0407i 0.6234  0.7776i 
 Stp0  

 0.6234  0.7776i 0.0552  0.0605i 



Check for absolute values:
Stp0
 0.082
Stp0
1 1
 0.082
2 2
For a reciprocal network, to check S0=S0T
T
 0.071  0.0407i 0.6234  0.7776i 

 0.6234  0.7776i 0.0552  0.0605i 
Stp0  
For a lossless network, to check StS*=U, here U is a unitary matrix

T 
Stp0  Stp0  
1

5
5
 2.683  10  4.448i  10
5
2.683  10
5
 4.448i  10
1



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:
 1.20 10-2  2.10 10-2 i 7.75 10-1  6.32 10-1 i 

Shb  

-1
-1
-2
-2 
 7.75 10  6.32 10  i 1.82 10  1.60 10  i 
For a reciprocal network, to check S=St
T  0.012  0.021i 0.775  0.632i 
Shb  

 0.775  0.632i 0.0182  0.016i 
For a lossless network, to check StS*=U, here U is a unitary matrix

T 
Shb  Shb  
5
2.1  10
1.001

5
5
 2.1  10  4.34i  10
1.001
Converting the scattering matrix to a transmission matrix:
Shb
 1

2 2

Shb
 Shb1  2
2 1
Thb  
Shb
 Shb1  1
2 2
S

S

hb1  2
hb1  1
 Shb
Shb
2 1
 2 1
5
 4.34i  10









The total transmission matrix is:

3
0.775  0.632i
Thb  
3.993  10

3
 3.972  10  0.024i
 0.024i 

0.775  0.632i

The transmission matrix for a straight section of 350 mm length of a lossless waveguide:
l1  0.35 (m)
 exp i  a1 l1
Thb1  

0
 0.38  0.925i

 Thb1  

0
0.38  0.925i 
exp i  a1 l1 

0
0

The transmission matrix for another straight section of 350 mm length of a lossless waveguide:
l2  0.35 (m)
 exp i  a1 l2
Thb2  

0

0
 0.38  0.925i

 Thb2  

0
0.38  0.925i 
exp i  a1 l2 

0
Since T=T1T0T2, so T0=T1-1 T T2-1, the H-bend waveguide only transmission matrix is:
3
 0.996  0.094i
3.993  10  0.024i 

Thb0 


3
0.996  0.094i
 3.972  10  0.024i

1
1
Thb0  Thb1  Thb Thb2
Converting the transmission matrix back to a scattering matrix:
 Thb02  1

1

 Thb01  1 Thb01  1
Shb0  
Thb0
 1
1 2
 Thb0
Thb0
1 1
1 1

Check for absolute values:
Shb0
 0.024
Shb0
1 1

 6.2062  10 3  0.0234i


0.9956  0.0944i


S

 hb0 

3

0.9956  0.0944i
1.7190  10  0.0242i 



2 2
 0.024
For a reciprocal network, to check S0=S0T
T
Shb0 
 6.2062  10 3  0.0234i

0.9956  0.0944i




3
0.9956  0.0944i
1.719  10  0.0242i 

For a lossless network, to check StS*=U, here U is a unitary matrix

T 
Shb0  Shb0  
1.001
5
2.1  10

5
5
 2.1  10  4.34i  10
5
 4.34i  10
1.001



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:
1 0
TRL  

0 1
The field probe can be treated as a 1:n2's ideal transformer [4], here n2 is transformer ratio calculated in the
1a section.



TFP  



2



 5.05 4.95 

 TFP  
 4.95 5.05 



2
1  n2
1  n2
2 n 2
2 n 2
2
2
1  n2
1  n2
2 n 2
2 n 2
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.



TFPC  



2
2
1  n1
n1  1
2 n 1
2 n 1
2
n1  1
2
1  n1



 10.683 10.637 

 TFPC  
 10.637 10.683 



2 n 1
2 n 1
Normalized load conductance at the "detuned open" position of waveguide coupler as the function of
detuned cavity frequency df is:
f  df
f 
Yca( df )  1  i Q0 


f
f

df 

The cavity's shunt conductance transmission matrix is [4]:
Yca( df )
Yca( df ) 

1

2
2


Tca( df ) 
 Yca( df )
Yca( df ) 
1


2
2


The total FPC + cavity + FP transmission matrix is:
Tc( df )  TFPC Tca( df )  TFP TRL
For df=0 or on the resonance:
 106.837 106.833 
Tc( 0)  

 106.363 106.368 
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:
L1  8000 (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:
L2  300 (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
Ttotald1  d2  d3  df   T3st d1  d2  d3 Twg2L1 Ttp0 Thb0 Twg1L2 Tc( df )
Converting the transmission matrix into a scattering matrix:
 Ttotal d1  d2  d3  df  2  1
1


Ttotal d 1  d 2  d 3  df  1  1 Ttotal d 1  d 2  d 3  df  1  1
Stotal d 1  d 2  d 3  df   

Ttotal d 1  d 2  d 3  df  1  2
1

 Ttotal d1  d2  d3  df  1  1 Ttotal d1  d2  d3  df  1  1





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.)
SdB12d1  d2  d3  df   20 log Stotald1  d2  d3  df 1  2

SdB11d1  d2  d3  df   20 log Stotald1  d2  d3  df 1  1 
SdB21d1  d2  d3  df   20 log Stotald1  d2  d3  df 2  1 
SdB22d1  d2  d3  df   20 log Stotald1  d2  d3  df 2  2 
Sph12d1  d2  d3  df   argStotald1  d2  d3  df 1  2
Sph11d1  d2  d3  df   argStotald1  d2  d3  df 1  1
Sph21d1  d2  d3  df   argStotald1  d2  d3  df 2  1
Sph22d1  d2  d3  df   argStotald1  d2  d3  df 2  2
Plot S parameters as a frequency scan:
df  500 499 500 (start, start + incremental step… stop values) (Hz)
3-Stub Tuner Changes S.C. Cavity's Qext
Amplitude of S21 from FPC toFP (dB)
35
40
45
50
55
60
65
500
400
300
200
100
0
100
200
frequency pulling of system {Hz}
300
400
500
d1=0, d2=0, d3=0
d1=0, d2=0, d3=31mm
d1=0, d2=18, d3=31mm
Figure 3: S21 amplitude scan 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).
3-Stub Tuner Changes S.C. Cavity's Qext
Phase of S21 from FPC to FP (deg)
100
50
0
50
100
150
500
400
300
200
100
0
100
200
frequency pulling of system {Hz}
300
400
500
d1=0, d2=0, d3=0
d1=0, d2=0, d3=31mm
d1=0, d2=18, d3=31mm
Figure 4: S21 phase scan plot with different stub settings. The curve color represents same condition as
in Figure 3.
3-Stub Tuner Changes S.C. Cavity's Qext
Amplitude of S11 from FPC (dB)
10
0
10
20
30
500
400
300
200
100
0
100
200
frequency pulling of system {Hz}
300
400
500
d1=0, d2=0, d3=0; 400 times scale
d1=0, d2=0, d3=31mm; original scale
d1=0, d2=18, d3=31mm, original scale
Figure 5: S11 amplitude scan plot with different stub settings. The curve color represents same
condition as in Figure 3.
3-Stub Tuner Changes S.C. Cavity's Qext
Phase of S11 from FPC (deg)
200
100
0
100
200
500
400
300
200
100
0
100
200
frequency pulling of system {Hz}
300
400
500
d1=0, d2=0, d3=0
d1=0, d2=0, d3=31mm
d1=0, d2=18, d3=31mm
Figure 6: S11 phase scan 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:
S21 
2
4 1  2
(1  1   2 ) 2
Here
1 
Q0
Q1eqext
and
2 
Q0
Q 2 ext
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
2. Then:
2
Q1eqext 
S21
Q2ext
4
Q1eqext  d 1  d 2  d 3 
for i  1  1000
df  500  i
i


A  Stotal d 1  d 2  d 3  df 1  2
i
i
peak  max( A )
2 Q2ext
peak 
4
We can check the equivalent external Q at different stub settings.
When all stubs are in “flush” position:
7
Q1eqext ( 0 0 0)  2.548  10
When the third stub in 31 mm. the coupling Q changes into:
6
Q1eqext ( 0 0 31)  8.029  10
When the second stub in 18 mm, the frequency pull draws back to zero but the Q drops further into:
6
Q1eqext ( 0 18 31)  6.371  10
Now we can map out the equivalent external Q vs. stub tuner setup:
d3  0 1 40 The d3 is changing from 0 to 40 mm (full range) in an incremental step of 1 mm.
Equavlent External Q with 3-Stubs
1 10
9
1 10
8
1 10
7
1 10
6
1 10
5
Equavlent Qext Change vs 3-Stub Setups
0
5
10
15
20
25
Third Stub Height d3 (mm)
30
35
40
d1=0, d2=0
d1=0, d2=10mm
d1=0, d2=20mm
d1=0, d2=30mm
d1=0, d2=40mm
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.
d1  0 1 40 The d1 is changing from 0 to 40 mm (full range) in an incremental step of 1 mm.
Equavlent External Q with 3-Stubs
1 10
9
1 10
8
1 10
7
1 10
6
Equavlent Qext Change vs 3-Stub Setups
0
5
10
15
20
25
First Stub Height d1 (mm)
30
35
40
d2=0, d3=0
d2=10mm, d3=0
d2=20mm, d3=0
d2=30mm, d3=0
d2=40mm, d3=0
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  0 1 40
d2 is changing from 0 to 40 mm (full range) in an incremental step of 1 mm.
Equavlent External Q with 3-Stubs
1 10
9
1 10
8
1 10
7
1 10
6
Equavlent Qext Change vs 3-Stub Setups
0
5
10
15
20
25
Second Stub Height d2 (mm)
30
35
40
d1=0, d3=0
d1=0, d3=10mm
d1=0, d3=20mm
d1=0, d3=30mm
d1=0, d3=40mm
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 Power and SW Waveform on FPC Waveguide for Required
Eacc without Beam Loading
6a. Constants Calculation
The 7-cell cavity's acceleration length:
Lacc  0.7 (m)
The 7-cell Old Cornell ("OC") shape cavity's R/Q per unit length (r/Q) calculated by SuperFish:
roQ  960 (/
Required cavity's acceleration gradient:
Eacc  12 (MV/m)
Transmitted power through the Field Probe for a given E acc=12MV/m:
2
Ptr 
12
Lacc Eacc  10
roQ Q2ext
(W)
Ptr  0.105
(W)
Field Probe voltage on the 50  terminated transmission line (power meter cable is matched to the power
meter’s input impedance):
 Ptr 50 
 2.291  (V)
VFP  
 (V) VFP  

 0 
 0 
6b. Klystron Incident Power Required
Pinc d1  d 2  d 3  df  
2
Lacc Eacc
roQ Q2ext
12

1 10
Stotal d1  d 2  d 3  df  1  2 
We can check klystron incident powers at resonance peak for different stub setups:
3
Pinc( 0 0 0 4)  1.031  10

2
(W) at
7
Q1eqext ( 0 0 0)  2.548  10
3
6
(W) at Q1eqext ( 0 0 31)  8.029  10
Pinc( 0 0 31 26)  3.269  10
Klystron incident powers at -3dB points for different stub setups:
3
Pinc( 0 0 0 25.5)  2.051  10
3
Pinc( 0 0 31 64.5)  6.504  10
3
Pinc( 0 0 0 32)  2.047  10
Q1eqext ( 0 0 0)  2.548  10
(W) at
Q1eqext ( 0 0 31)  8.029  10
(W) at
3
Pinc( 0 0 31 118)  6.523  10
7
(W) at
6
7
Q1eqext ( 0 0 0)  2.548  10
6
(W) at Q1eqext ( 0 0 31)  8.029  10
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:
Vinput d1  d2  d3  df   Ttotald1  d2  d3  df  VFP
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:
 49.164  239.906i 
Vinput ( 0 0 0 15)  

 49.081  238.798i 
With the third stub in 30 mm and df=-13Hz, the input waveguide voltage is:
 354.572  152.476i 
Vinput ( 0 0 30 13)  

 154.542  192.079i 
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:
Tpartiald1  d2  d3  df  L3  T3st d1  d2  d3 Twg2L1 Ttp0 Thb0 Twg1L3
Then the incident and reflected voltages at the "hot spot" of the warm window is:
1
Vwindow d1  d2  d3  df  L3  Tpartial d1  d2  d3  df  L3  Vinput  d1  d2  d3  df 
The Standing Wave Voltage amplitude on reduced height waveguide is:
 a1
Uwindow d1  d2  d3  df  L3 
 Vwindow d1  d2  d3  df  L3 1  1  Vwindow d1  d2  d3  df  L3 2  1
50
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:
L3  0 1 300 (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.
Standing Wave Voltages for Eacc=12MV/m
2500
voltage on reduced height WG (V)
2000
1500
1000
500
0
0
50
100
150
200
dist. from cavity's "detuned open" (mm)
250
300
d1=0, d2=0, d3=0, peak at df=4Hz, Qext=2.55E7
d1=0, d2=0, d3=0, -3dB point at df=32Hz, Qext=2.55E7
d1=0, d2=0, d3=31mm, peak at df=-26Hz, Qeqext=8.03E6
d1=0, d2=18, d3=31mm, peak at df=0Hz, Qeqext=8.03E6
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 Power and 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:
Vacc  Eacc Lacc (MV) Vacc  8.4 (MV)
To confirm the Vacc from the transmission matrix:
Vac( df )  Tca( df )  TFP VFP
roQ Lacc Q0
(V)
50
At resonance df=0, and pay attention to the impedance re-normalization here
 8.442  106 


4 
 4.2  10 
Vac( 0)  
(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:
 I0 10 3 roQ Q0 
  exp i b   (1/)

6
 
180 
 Eacc 10

Yb Eacc  I0  b  


When Eacc=12MV/m, I0=0.2mA, on crest. They are extreme CEBAF operation parameters.
(1/)
Yb( 12 0.2 0)  160
Thus the beam's shunt conductance transmission matrix is [4]:
Yb Eacc  I0  b
Yb Eacc  I0  b 

1

2
2


Tb Eacc  I0  b 
 Yb Eacc  I0  b
Yb Eacc  I0  b 
1


2
2


7b. Total Wave Transmission Matrix from 3-Stub Tuner to Field Probe with Beam Loading




Ttot d1  d2  d3  df  Eacc  I0  b  T3st d1  d2  d3 Twg2L1 Ttp0 Thb0 Twg1L2 TFPC Tca( df )  Tb Eacc  I0  b  TFP TRL
The waveguide voltage on the WR650 input waveguide of the 3-stub tuner:
Vin d1  d2  d3  df  Eacc  I0  b  Ttot d1  d2  d3  df  Eacc  I0  b  VFP
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:
 173.188  270.301i  (V)
Vin( 0 0 0 4 12 0.2 0)  

 68.965  113.004i 
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:
 437.866  367.895i  (V)
Vin( 0 0 31 32 12 0.4 0)  

 233.909  86.557i 
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:
Tpartiald1  d2  d3  df  L3  T3st d1  d2  d3 Twg2L1 Ttp0 Thb0 Twg1L3
Then the incident and reflected voltages at the "hot spot" of the warm window is:






Vwin d1  d2  d3  df  Eacc  I0  b  L3  Tpartial d1  d2  d3  df  L3

6

1


 Vin d1  d2  d3  df  Eacc  I0  b
 216.945  256.504i 
Vwin 0 0 31 32 12 10  0.0001 0 0  

 215.866  256.379i 
The Standing Wave Voltage amplitude on reduced height waveguide is:
 a1
Uwin d1  d2  d3  df  Eacc  I0  b  L3 
 Vwin d1  d2  d3  df  Eacc  I0  b  L3 1  1  Vwin d1  d2  d 3  df  Eacc  I0  b  L3 2  1
50
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:
L3  0 1 300 (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.
Standing Wave Voltages for Eacc=12MV/m
2500
voltage on reduced height WG (V)
2000
1500
1000
500
0
0
50
100
150
200
dist. from cavity's "detuned open" (mm)
250
300
d1=0, d2=0, d3=0, peak at df=4Hz, Qext=2.55E7
d1=0, d2=0, d3=0, -3dB point at df=32Hz, Qext=2.55E7
d1=0, d2=0, d3=31mm, peak at df=-26Hz, Qeqext=8.03E6
d1=0, d2=18, d3=31mm, peak at df=0Hz, Qeqext=6.37E6
same as curve 4, plus beam I0=0.3mA, phib=0 (on-crest).
Figure 11: Standing Wave Voltage Amplitude (SWVA) waveform at a constant gradient of E acc=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]
[2]
[3]
[4]
[5]
[6]
H. Wang, M. Tiefenback, “Waveguide Stub Tuner Analysis for CEBAF Application”, Proceedings
of LINAC 2004, Lubeck, Germany, Aug. 16-20, 2004, p836.
Web page: http://www.mathcad.com/.
David Pozar, Microwave Engineering, second edition, John Wiley and Sons, Inc., chapt 4:
Microwave Network Analysis, p200.
Keqian Zhang, Dejie, Li, Electromagnetic Theory for Microwaves and Optoelectronics, Chinese
edition 2001, Publishing House of Electronics Industry, Beijing, or English edition 1998, Springer,
p142.
Private communication with Genfa Wu.
L. Doolittle, Waveguide Surface Currents, JLab Tech Note in 1999, JLAB-TN-99-012. (Haipeng
Wang notes that there are some errors in this tech note as well as other publications refer to it. He
would like to correct and comment them in a separated tech note).