Post on 30-Mar-2018
RF Systems I Erk Jensen, CERN BE-RF
Introduction to Accelerator Physics, Prague, Czech Republic, 31 Aug β 12 Sept 2014
Definitions & basic concepts dB
t-domain vs. Ο-domain
phasors
8th Sept, 2014 CAS Prague - EJ: RF Systems I 2
Decibel (dB)
β’ Convenient logarithmic measure of a power ratio.
β’ A βBelβ (= 10 dB) is defined as a power ratio of 101. Consequently, 1 dB is a power ratio of 100.1 β 1.259.
β’ If πππ denotes the measure in dB, we have:
πππ΅ = 10 dB logπ2
π1= 10 dB log
π΄22
π΄12 = 20 dB log
π΄2
π΄1
β’π2
π1=
π΄22
π΄12 = 10πππ 10 dB
π΄2
π΄1= 10πππ 20 dB
β’ Related: dBm (relative to 1 mW), dBc (relative to carrier) 8th Sept, 2014 CAS Prague - EJ: RF Systems I 3
πππ΅ β30 dB β20 dB β10 dB β6 dB β3 dB 0 dB 3 dB 6 dB 10 dB 20 dB 30 dB
π2
π1 0.001 0.01 0.1 0.25 .50 1 2 3.98 10 100 1000
π΄2
π΄1 0.0316 0.1 0.316 0.50 .71 1 1.41 2 3.16 10 31.6
Time domain β frequency domain (1)
β’ An arbitrary signal g(t) can be expressed in π-domain using the Fourier transform (FT).
π π‘ β πΊ π =1
2π π π‘ β β ππ‘
β
ββ
ππ‘
β’ The inverse transform (IFT) is also referred to as Fourier Integral.
πΊ π β π π‘ =1
2π πΊ π β ββ ππ‘
β
ββ
ππ
β’ The advantage of the π-domain description is that linear time-invariant (LTI) systems are much easier described.
β’ The mathematics of the FT requires the extension of the definition of a function to allow for infinite values and non-converging integrals.
β’ The FT of the signal can be understood at looking at βwhat frequency components itβs composed ofβ.
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Time domain β frequency domain (2)
β’ For π-periodic signals, the FT becomes the Fourier-Series, ππ becomes 2π π , β« becomes β.
β’ The cousin of the FT is the Laplace transform, which uses a complex variable (often π ) instead of β π; it has generally a better convergence behaviour.
β’ Numerical implementations of the FT require discretisation in π‘ (sampling) and in π. There exist very effective algorithms (FFT).
β’ In digital signal processing, one often uses the related z-Transform, which uses the variable π§ = β β ππ, where π is the sampling period. A delay of ππ becomes π§βπ.
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Time domain β frequency domain (3) β’ Time domain β’ Frequency domain
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π βπ
π
2 1 π
π 1
π
π 1
π
1
π
π
πΒ± π
1
π
π
1
π
π βπ
π
1
π
sampled oscillation sampled oscillation
modulated oscillation modulated oscillation
1
π
π βπ
1 π
π
Fixed frequency oscillation (steady state, CW) Definition of phasors
β’ General: π΄ cos ππ‘ β π = π΄ cos ππ‘ cos π + π΄ sin ππ‘ sin π
β’ This can be interpreted as the projection on the real axis of a rotation in the complex plane.
β π΄ cos π + β sin π β β ππ‘
β’ The complex amplitude π΄ is called βphasorβ;
π΄ = π΄ cos π + β sin π
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real part
imag
inar
y p
art Ο
Calculus with phasors
β’Why this seeming βcomplicationβ?: Because things become easier!
β’ Using π
ππ‘β‘ β π, one may now forget about the rotation with
π and the projection on the real axis, and do the complete analysis making use of complex algebra!
πΌ = π1
π + β ππΆ β
β
ππΏ
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Example:
Slowly varying amplitudes
β’ For band-limited signals, one may conveniently use βslowly varyingβ phasors and a fixed frequency RF oscillation.
β’ So-called in-phase (I) and quadrature (Q) βbaseband envelopesβ of a modulated RF carrier are the real and imaginary part of a slowly varying phasor.
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Amplitude modulation
1 + π cos π β cos πππ‘ = β 1 +π
2β β π +
π
2β ββ π β β πππ‘
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50 100 150 200 250 300
1.5
1.0
0.5
0.5
1.0
1.5
green: carrier black: sidebands at Β±ππ blue: sum
example: π = πππ‘ = 0.05 πππ‘π = 0.5
π: modulation index or modulation depth
Phase modulation
β β β πππ‘+π sin π = β π½π π β β ππ+πππ‘
β
π=ββ
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50 100 150 200 250 300
1.0
0.5
0.5
1.0
Green: π = 0 (carrier) black: π = 1 sidebands red: π = 2 sidebands blue: sum
where
π: modulation index (= max. phase deviation)
example: π = πππ‘ = 0.05 πππ‘π = 4
π = 1
Spectrum of phase modulation
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4 2 0 2 41.0
0.5
0.0
0.5
1.0
4 2 0 2 41.0
0.5
0.0
0.5
1.0
4 2 0 2 41.0
0.5
0.0
0.5
1.0
4 2 0 2 41.0
0.5
0.0
0.5
1.0
4 2 0 2 41.0
0.5
0.0
0.5
1.0Phase modulation with π = π: red: real phase modulation blue: sum of sidebands π β€ 3
M=1
M=4
M=3
M=2
Plotted: spectral lines for sinusoidal PM at ππ Abscissa: π β ππ ππ
M=0 (no modulation)
Spectrum of a beam with synchrotron oscillation, π = 1 = 57 Β°
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carrier synchrotron sidelines
f
Vector (I-Q) modulation
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More generally, a modulation can have both amplitude and phase modulating components. They can be described as the in-phase (I) and quadrature (Q) components in a chosen reference, cos πππ‘ . In complex notation, the modulated RF is:
β πΌ π‘ + β π π‘ β β πππ‘ =
β πΌ π‘ + β π π‘ cos πππ‘ + β sin πππ‘ =
πΌ π‘ cos πππ‘ β π π‘ sin πππ‘
So I and Q are the Cartesian coordinates in the complex βPhasorβ plane, where amplitude and phase are the corresponding polar coordinates.
πΌ π‘ = π΄ π‘ cos π π π‘ = π΄ π‘ sin π
I-Q modulation: green: I component red: Q component blue: vector-sum
1 2 3 4 5 6
1.5
1.0
0.5
0.5
1.0
1.5
Vector modulator/demodulator
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mixer
0Β°
90Β° combiner splitter
3-dB hybrid
low-pass
low-pass
tI
tI
tQ tQ
1 2 3 4 5 6
1.5
1.0
0.5
0.5
1.0
1.5
1 2 3 4 5 6
1.5
1.0
0.5
0.5
1.0
1.5
1 2 3 4 5 6
1.5
1.0
0.5
0.5
1.0
1.5
1 2 3 4 5 6
1.5
1.0
0.5
0.5
1.0
1.5
1 2 3 4 5 6
2
1
1
2
3-dB hybrid 0Β°
90Β°
ππ ππ β
mixer
mixer
mixer
Sampling and quantization β’ Digital Signal Processing is very powerful β note recent progress in digital
audio, video and communication!
β’ Concepts and modules developed for a huge market; highly sophisticated modules available βoff the shelfβ.
β’ The βslowly varyingβ phasors are ideal to be sampled and quantized as needed for digital signal processing.
β’ Sampling (at 1 ππ ) and quantization (π bit data words β here 4 bit):
8th Sept, 2014 CAS Prague - EJ: RF Systems I 18
1 2 3 4 5 6
1.5
1.0
0.5
0.5
1.0
1.5
The βbasebandβ is limited to half the sampling rate!
ADC
DAC Original signal Sampled/digitized
Anti-aliasing filter Spectrum
The βbasebandβ is limited to half the sampling rate!
Digital filters (1) β’ Once in the digital realm, signal processing becomes βcomputingβ!
β’ In a βfinite impulse responseβ (FIR) filter, you directly program the coefficients of the impulse response.
8th Sept, 2014 CAS Prague - EJ: RF Systems I 19
sf1
Transfer function: π0 + π1π§
β1 + π2π§β2 + π3π§β3 + π4π§β4
π§ = β β πππ
Digital filters (2)
β’ An βinfinite impulse responseβ (IIR) filter has built-in recursion, e.g. like
8th Sept, 2014 CAS Prague - EJ: RF Systems I 20
Transfer function: π0 + π1π§β1 + π2π§β2
1 + π1π§β1 + π2π§β2
Example: π0
1 + πππ§βπ
0 1 2 3 4 5
2
4
6
8
10
β¦ is a comb filter.
2ππ
ππ
Digital LLRF building blocks β examples
β’General D-LLRF board: β’ modular!
FPGA: Field-programmable gate array
DSP: Digital Signal Processor
β’DDC (Digital Down Converter) β’ Digital version of the
I-Q demodulator CIC: cascaded integrator-comb
(a special low-pass filter)
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RF system & control loops
e.g.: β¦ for a synchrotron:
Cavity control loops
Beam control loops
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Minimal RF system (of a synchrotron)
8th Sept, 2014 CAS Prague - EJ: RF Systems I 23
β’ The frequency has to be controlled to follow the magnetic field such that the beam remains in the centre of the vacuum chamber.
β’ The voltage has to be controlled to allow for capture at injection, a correct bucket area during acceleration, matching before ejection; phase may have to be controlled for transition crossing and for synchronisation before ejection.
Low-level RF High-Power RF
Fast RF Feed-back loop
8th Sept, 2014 CAS Prague - EJ: RF Systems I 24
β’ Compares actual RF voltage and phase with desired and corrects. β’ Rapidity limited by total group delay (path lengths) (some 100 ns). β’ Unstable if loop gain = 1 with total phase shift 180 Β° β design requires to stay
away from this point (stability margin) β’ The group delay limits the gainΒ·bandwidth product. β’ Works also to keep voltage at zero for strong beam loading, i.e. it reduces the
beam impedance.
β ββ ππ
Fast feedback loop at work
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Without feedback, ππππ = πΌπΊ0 + πΌπ΅ β π π , where
π π =π 1 + π½
1 + β πππ0
βπ0π
Detect the gap voltage, feed it back to πΌπΊ0 such that πΌπΊ0 = πΌππππ£π β πΊ β ππππ, where πΊ is the total loop gain (pick-up, cable, amplifier chain β¦) Result:
ππππ = πΌππππ£π + πΌπ΅ βπ π
1 + πΊ β π π
β’ Gap voltage is stabilised! β’ Impedance seen by the beam is reduced by
the loop gain!
Plot on the right: 1+π½
π
π π
1+πΊβπ π vs. π, with
the loop gain varying from 0 dB to 50 dB.
1-turn delay feed-back loop
β’ The speed of the βfast RF feedbackβ is limited by the group delay β this is typically a significant fraction of the revolution period.
β’ How to lower the impedance over many harmonics of the revolution frequency?
β’ Remember: the beam spectrum is limited to relatively narrow bands around the multiples of the revolution frequency!
β’ Only in these narrow bands the loop gain must be high!
β’ Install a comb filter! β¦ and extend the group delay to exactly 1 turn β in this case the loop will have the desired effect and remain stable!
8th Sept, 2014 CAS Prague - EJ: RF Systems I 26
0.5 1.0 1.5 2.0
2
4
6
8
10
Field amplitude control loop (AVC)
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β’ Compares the detected cavity voltage to the voltage program. The error signal serves to correct the amplitude
Tuning loop
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β’ Tunes the resonance frequency of the cavity ππ to minimize the mismatch of the PA. β’ In the presence of beam loading, the optimum ππ may be ππ β π. β’ In an ion ring accelerator, the tuning range might be > octave! β’ For fixed π systems, tuners are needed to compensate for slow drifts. β’ Examples for tuners:
β’ controlled power supply driving ferrite bias (varying Β΅), β’ stepping motor driven plunger, β’ motorized variable capacitor, β¦
Beam phase loop
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β’ Longitudinal motion: π2 Ξπ
ππ‘2 + Ξ©π 2 Ξπ 2 = 0.
β’ Loop amplifier transfer function designed to damp synchrotron oscillation.
Modified equation: π2 Ξπ
ππ‘2 + πΌπ Ξπ
ππ‘+ Ξ©π
2 Ξπ 2 = 0
Other loops
β’ Radial loop: β’ Detect average radial position of the beam, β’ Compare to a programmed radial position, β’ Error signal controls the frequency.
β’ Synchronisation loop (e.g. before ejection): β’ 1st step: Synchronize π to an external frequency (will also act
on radial position!). β’ 2nd step: phase loop brings bunches to correct position.
β’ β¦
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Wave vector π:
the direction of π is the direction of propagation,
the length of π is the phase shift per unit length.
π behaves like a vector.
Homogeneous plane wave
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33
z
x
Ey
πΈ β π’π¦ cos ππ‘ β π β π
π΅ β π’π₯ cos ππ‘ β π β π
π β π =π
ππ§ cosπ + π₯ sinπ
πβ₯ =ππ
π
ππ§ =π
π1 β
ππ
π
2
π =π
π
π
Wave length, phase velocity
β’ The components of π are related to the wavelength in the direction of that
component as ππ§ =2π
ππ§ etc. , to the phase velocity as π£π,π§ =
π
ππ§= πππ§.
8th Sept, 2014 CAS Prague - EJ: RF Systems I 34
z
x
Ey
πβ₯ =ππ
π π =
π
π
πβ₯ =ππ
π
ππ§ =π
π1 β
ππ
π
2
π =π
π
Superposition of 2 homogeneous plane waves
8th Sept, 2014 CAS Prague - EJ: RF Systems I 35
+ =
Metallic walls may be inserted where without perturbing the fields. Note the standing wave in x-direction!
z
x
Ey
This way one gets a hollow rectangular waveguide!
Rectangular waveguide
CAS Prague - EJ: RF Systems I 36
power flow: 1
2Re πΈ Γ π»βππ΄
Fundamental (TE10 or H10) mode in a standard rectangular waveguide.
E.g. forward wave
electric field
magnetic field
power flow
power flow
Waveguide dispersion
8th Sept, 2014 CAS Prague - EJ: RF Systems I 37
What happens with different waveguide dimensions (different width π)? The βguided wavelengthβ ππ varies
from β at ππ to π at very high frequencies.
cutoff: ππ =π
2 π
2
1
cz
ck
ck
1
2
3
1: π = 52 mm π
ππ= 1.04
π = 3 GHz
2: π = 72.14 mm π
ππ= 1.44
3: π = 144.3 mm π
ππ= 2.88
π ππ
ππ§
Phase velocity π£π,π§
8th Sept, 2014 CAS Prague - EJ: RF Systems I 38
2
1
cz
ck
ck
1
2
3
The phase velocity is the speed with which the crest or a zero-crossing travels in π§-direction. Note in the animations on the right that, at constant π, it is π£π,π§ β ππ.
Note that at π = ππ , π£π,π§ = β!
With π β β, π£π,π§ β π!
z
z
v
k
,
1
z
z
v
k
,
1
z
z
v
k
,
1
1: π = 52 mm π
ππ= 1.04
π = 3 GHz
2: π = 72.14 mm π
ππ= 1.44
3: π = 144.3 mm π
ππ= 2.88
cutoff: ππ =π
2 π π ππ
ππ§
Radial waves
β’ Also radial waves may be interpreted as superpositions of plane waves.
β’ The superposition of an outward and an inward radial wave can result in the field of a round hollow waveguide.
8th Sept, 2014 CAS Prague - EJ: RF Systems I 39
πΈπ§ β π»π2
πππ cos ππ πΈπ§ β π»π1
πππ cos ππ πΈπ§ β π½π πππ cos ππ
Round waveguide modes
8th Sept, 2014 CAS Prague - EJ: RF Systems I 40
TE11 β fundamental
mm/
9.87
GHz a
fc mm/
8.114
GHz a
fc mm/
9.182
GHz a
fc
TM01 β axial field TE01 β low loss
πΈ
π»
Waveguide perturbed by discontinuities (notches)
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βnotchesβ
Reflections from notches lead to a superimposed standing wave pattern. βTrapped modeβ
Signal flow chart
Short-circuited waveguide
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TM010 (no axial dependence) TM011 TM012
πΈ
π»
Single waveguide mode between two shorts
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short circuit
short circuit
Eigenvalue equation for field amplitude π:
π = π β ββ ππ§2π Non-vanishing solutions exist for 2ππ§π = 2 ππ.
With ππ§ =π
π1 β
π
ππ
2, this becomes π0
2 = ππ2 + π
π
2π
2.
Signal flow chart
β ββ ππ§π
β ββ ππ§π
β1 β1
π
Simple pillbox cavity
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electric field (purely axial) magnetic field (purely azimuthal)
(only 1/2 shown)
TM010-mode
Pillbox with beam pipe
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electric field magnetic field
(only 1/4 shown) TM010-mode
One needs a hole for the beam pipe β circular waveguide below cutoff
A more practical pillbox cavity
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electric field magnetic field
(only 1/4 shown) TM010-mode
Round of sharp edges (field enhancement!)
Some real βpillboxβ cavities
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CERN PS 200 MHz cavities