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Lecture 3 Couplers and
Wakefields
Dr G Burt
Lancaster University
Engineering
Scattering Parameters
When making RF measurements, the most common measurement is the Sparameters.
Input signal
S1,1
Black Box
S2,1
forward transmission coefficient
input reflection coefficient
The S matrix is a m-by-m matrix (where m is the number of available
measurement ports). The elements are labelled S parameters of form Sab
where a is the measurement port and b is the input port.
S=
S11 S12
S21 S22
The meaning of an S parameter is the ratio of the voltage measured at the
measurement port to the voltage at the input port (assuming a CW input).
Sab =Va / Vb
Measuring S-Parameters
a1
Forward
S 21 =
b1
Incident
a2 = 0
b1
= a
1
b
a2 = 0
S 22 =
2
= a
1
a2 = 0
a1 = 0
Z0
DUT
Load
b1
Load
DUT
Reflected
Transmitted
b2
Transmitted
21
Z0
S 11
Reflected
Incident
S 11 =
S
Incident
Transmitted
S 12
S 12 =
Reflected
Incident
Transmitted
Incident
S 22
b2
= a
2
b
1
= a
2
a1 = 0
b2
Reverse
Reflected
a2
Incident
a1 = 0
S-Parameter Plot
As S-parameters are
measured using a
continuous wave signal
they are measured at a
single frequency.
S-Parameter Plot
0
1
1.1
1.2
1.3
1.4
1.5
-5
0 dBs 100% Transmission
Loss /dB
-10
A Frequency response
can be plotted.
-15
-3 dB
0 dB
-20
-25
-3 dBs 70.7%
reflection in amplitude
50% reflection in
power
-26 dB
-30
-35
Frequency /GHz
S1,1
-26 dBs 95% reflection
S (dB)  20. log 10 S 
S2,1
1.6
1.7
Cartesian and Polar Plots
S parameters are complex
quantities having both real and
imaginary parts. The real parts
represent a wave transmitted. The
imaginary parts represent a wave
transmitted with a 90 degree
phase shift.
Normally the magnitude of S is
referred to, but it is also often
useful to plot the S parameters in
polar form. For a cavity S11 will
trace a circle giving information
about the cavity coupling.
VSWR
It is also common to measure the VSWR, Voltage Standing Wave Ratio of a
transmission line. This is the ratio of the standing wave to travelling waves in
the device.
Voltage at the antinode
of a standing wave
Voltage of the
incident wave
Vmax Vi  Vr
  VSWR 

Vmin Vi  Vr
1  S11
VSWR 
1  S11
Voltage of the
reflected wave
Voltage at the node
of a standing wave
For 100% reflected wave, the resultant VSWR is infinity and for full
transmission it is unity.
Network Analyzer
Source
Switch
R1
Reference
Receiver
A
Reference
Receiver
R2
B
Measurement
Receivers
Port 1
A device which measures S
parameters is known as a
network analyser. They come
in two varieties
Scalar Analysers- Which
measure only magnitude
Vector analysers- Which
measure real and imaginary
parts of S
Port 2
Mode Coupling
• In RF we approximate the fields in the system by a
superposition of a number or ordinary waveguide modes.
• A cavity mode is said to coupled to a waveguide if the
fields at the intersection can be made to be continuous
by a superposition of a number of propagating modes.
Cavity Coupling
Probe coupling to E-field
Capacitive coupling
Higher penetration
higher coupling
Loop couples to the
B-field
Inductive coupling
Higher penetration
lower coupling
Couplers
The couplers can also
be represented in
equivalent circuits. The
RF source is
represented by a ideal
current source in
parallel to an
impedance and the
coupler is represented
as an n:1 turn
transformer.
External Q factor
Ohmic losses are not the only loss mechanism in cavities. We also
have to consider the loss from the couplers. We define this external
Q as,
P Q
U
Qe 
Pe

e
Pc

0
Qe
Where Pe is the power lost through the coupler when the RF sources
are turned off.
We can then define a loaded Q factor, QL, which is the ‘real’ Q of the
cavity
1
1
1


QL Qe Q0
U
QL 
Ptot
Cavity responses
A resonant cavity will reflect all power at frequencies outwith its bandwidth
hence S11=1 and S21=0.
The reflections are minimised (and transmission maximised) at the resonant
frequency.
If the coupler is matched to the cavity (they have the same impedance) the
reflections will go to zero and 100% of the power will get into the cavity
when in steady state (ie the cavity is filled).
S21
The reflected power in steady
state is given by
1.00
1  e
S11 
1  e
0.75
0.50
where
0.25
0.00
-10
-5
0
5
10
Q0
e 
Qe
Resonant Bandwidth
1.00
0.75
P
0.50
ω
D  1 = 0
tL QL
0.25
0.00
-10
-5
0
5
10
ω-ω0
SC cavities have much smaller resonant bandwidth and longer
time constants. Over the resonant bandwidth the phase of S21
also changes by 180 degrees.
Coupler Measurements
By measuring the S parameters of a cavity
we can determine all the Q factors. The
loaded Q is found from the resonant
bandwidth. Then we can find the coupling
factor, e, from a measurement of S11
Q0
1
 e  d2

Qe
d 1
Polar plot of S11

1 
Qe  QL 1  
 e 
Cavity Filling
When filling, the impedance of a resonant cavity varies with time and hence so does
the match this means the reflections vary as the cavity fills.
Pref
Pfor
note:
No beam!
1
  0.1
0.8
0.6
 1
  10
0.4
0.2
0
0
1
2
3
4
5
0t / 2QL
As we vary the external Q
of a cavity the filling
behaves differently.
Initially all power is
reflected from the cavity,
as the cavities fill the
reflections reduce.
The cavity is only matched (reflections=0) if the external Q of the cavity is
equal to the ohmic Q (you may include beam losses in this).
A conceptual explanation for this as the reflected power from the coupler and
the emitted power from the cavity destructively interfere.
Coupling Strength
• Excited by a square pulse
critically coupled
under coupled
2
Pref
Pfor
over coupled
2
 1
1.5
2
  0.5
1.5
1
1
1
0.5
0.5
0.5
0
0
0
0
2
4
6
8
10
0
2
4
6
 2
1.5
8
10
0
2
4
6
8
10
0t / 2QL
Typical RF System
Low
Level
RF
Klystron
Feeder
System
Cavity
DC
Power
Supply
Low Level RF (LLRF) Control Tasks
• Provide low noise RF sources for all acceleration points in the
machine
• Maintain phase and amplitude of the RF system to accelerate the
beam
– Combat beam induced instabilities etc
• Provide diagnostics and be flexible
• Minimise environmental effects on the machine
– Temperature
– Vibration
Causes of Errors
•
•
•
•
•
•
•
Microphonics
Beam loading
Heating / Thermal Expansion
RF Source transient errors
Changes in the couplers
Multipactor or Field Emission
Excitation of other modes in passband
Phase Control Loop
Master
Oscillator
3dB
Electronic Phase
Shifter
Output
Phase
Shifter
Phase
Amplifier
Detector
Attenuator
Feeder
Waveguide
• The input phase is adjusted so that the output phase (and hence
cavity phase) is always equal to a preset value.
• The mechanical phase shifter is used so that the phase detector will
work in the correct range.
• Phase control loop can control the phase to less than 0.25 degree
Phase control performance
Locking Performance vs Gain with DSP Clock Speed of 50 MHz
0
-50
-40
-30
-20
0
-10
10
20
-10
Gain 0.2
-20
Gain 1.0
Gain 10.0
-30
dB
-40
-50
-60
-70
-80
-90
-100
Offset Hz
LLRF off
30
40
50
Frequency Control Loop
input
phase
detector
cavity
tuner
Change in cavity
resonant frequency
causes a change in
phase difference between
the cavity input and the
cavity output, S21.
motor
motor
control
 D 
Df

 tan 
f
 2Q 
• Cavity is tuned by squashing the shape using a stepper
motor and a pivot.
• The motor torque is varied to keep the phase constant.
Amplitude Control Loop
Gap voltage
setting
RF drive in
gap voltage
from cavity 1
gap voltage
from cavity 2
Attenuator
Attenuator
detector
detector
amplitude
loop board
voltage
controlled
attenuator
RF drive out
• Voltage controlled
attenuator controls
input power to the
source
• Loop keeps gap
voltages constant
(~1%) counteracting
beam loading.
Control Algorithms
These are 4 main types of control
•
Proportional – Varies the input proportionally to the output error.
– This is the simplest control loop
•
Integral - Varies the input proportionally to the integral of the output error.
– Combined with a proportional term we get a PI controller. The integral term
reduces will return the output to the set-point faster but can cause overshoot.
•
Differential - Varies the input proportionally to the differential of the output
error.
– The differential term reduces overshoot.
•
Feed-forward – Control based on anticipation of future cavity behaviour
– Useful for microphonics or Lorentz force detuning.
Loop Bandwidth and Gain
•
•
•
In a control loop it takes a set time (latency) for the measured input to
produce a change in the output.
This causes the loop to act as a low pass filter with a bandwidth such that it
can only correct for changes at a frequency below the loop bandwidth.
The open loop gain is the amount an error on the input affects the output. A
high gain will lead to faster corrections but can become unstable to noise.
Self Excited Loops
Digital LLRF
• For digital systems we use inphase
and quadrature (real and imaginary)
instead of phase and amplitude.
• This is because phase and amplitude
are coupled to each other, where I
and Q are independent.
• This can be used to calculate an
instantaneous phase and amplitude.
Vector controller