THE CERN ACCELERATOR SCHOOL

BASICS OF RF ELECTRONICS
( … OR GETTING STARTED WITH LLRF BUILDING BLOCKS)

Alessandro Gallo
INFN
Laboratori Nazionali di Frascati
Frascati (RM), Italy

CAS course on “RF for Accelerators”

Hotel Ebeltoft Strand, Ebeltoft, Denmark, June 8th to 17th 2010
1
SUMMARY  A. Gallo, Basics of RF Electronics

• Attenuators • Frequency Mixers

• (signal) Amplifiers • Phase Detectors (I&Q)
• RF Transformers • Bi-phase attenuators/
/I&Q modulators
• Power Splitters/
/Combiners • Peak Detectors
• Hybrid junctions/ • Step Recovery diodes
/Directional Couplers • PIN diode Switches/
• Circulators/Isolators /Attenuators
• Filters • Phase shifters
• Modulation Transfer • VCOs
Functions • PLLs
1 st Lecture 2nd Lecture 2
 FIXED ATTENUATORS
 A. Gallo, Basics of RF Electronics

Fixed attenuators are widely used in RF electronics to set the proper signal level in
the various circuit branches. Proper level setting is crucial to fully exploit the
instrumentation dynamic range and to avoid circuit overload and damaging.
Attenuators are also used as matching pads as they can be designed to connect lines
of different impedances. Also, the insertion of attenuators in front of mismatched
loads reduces the VSWR seen at the source side.
Fixed attenuators are mainly characterized by the following parameters:
• Attenuation ΔdB;
• Max average power rate;
• Max peak power rate;
• Frequency range;
• Attenuation flatness over the specified frequency range;
• VSWR, size and weight, performance over the given temperature range, …

3
 FIXED ATTENUATORS
 A. Gallo, Basics of RF Electronics

Fixed attenuators are passive, 2-ports devices generally made by a network of

resistors with a very broadband frequency response (dc ÷ many GHz, typical). They are
designed to provide both the required attenuation and matching of the input/output
lines, which might have different characteristic impedances. The attenuation ΔdB,
expressed in dB units, and the linear transmission coefficient α are defined as:
0 α
∆dB = 10 ⋅ log(Pin Pout ); α = Pout Pin = 10 − ( ∆dB 20 )
S =  
α 0 
Unbalanced T-type Attenuator
1 + α 2 − 2α Z 0out Z 0in
RA = Z 0in if Z 0in = Z 0out = Z 0
1−α 2

2α 1−α
RB = Z 0in ⋅ Z 0out RA = RC = Z0
1−α 2 1+ α

1 + α 2 − 2α Z 0in Z 0out 2α
RB = Z0
RC = Z 0out 1−α 2
1−α 2
Balanced
4
 FIXED ATTENUATORS
 A. Gallo, Basics of RF Electronics

Unbalanced
π-type Attenuator
1−α 2
RA = Z0
1 + α 2 − 2α Z 0in Z 0out in if Z 0in = Z 0out = Z 0

1−α 2 1+ α
RB = Z 0in ⋅ Z 0out RA = RC = Z0
2α 1−α

1−α 2 1−α 2
RC = Z 0out RB = Z0
1 + α − 2α Z 0out Z 0in
2
2α
Balanced

It is important to notice that in order to match unequal input/output line impedances

a minimum attenuation is required, according to (case Z0out ≥ Z0in ):

α max = Z 0 out
Z 0 in − Z 0out Z 0 in − 1 ⇒ ∆dBmin = 20 ⋅ log(1 α max )

Fixed attenuators are available in a huge variety of packages, power ratings (up to ≈1
kW), frequency ranges (to > 18 GHz), any attenuation value and all standard impedances
used in communication electronics. 5
 SIGNAL AMPLIFIERS
 A. Gallo, Basics of RF Electronics

Low-level RF amplifiers are used to increase the signal level whenever it is required
for proper treatment and/or manipulation. A very wide subject, can only be mentioned
here. Nowadays almost only solid state technology (silicon or GaAs semiconductors,
BJT and FET technology) is used for low/medium power (< 10 W) applications, up to
≈ 10GHz.
Construction techniques are MIC (Microwave Integrated Circuits) and MMIC
(Monolithic Microwave Integrated Circuits). In MIC realization the transistor and its
capacitance and resistors are soldered on microstrip lines laying on a proper substrate;
MMIC are completely integrated circuits where all components (the transistors and
their ancillaries) are fabricated on a common substrate.
Concerning small-signal amplifier, the operating class is generally “A” since power
efficiency is not an issue for these kind of applications.

6
SIGNAL AMPLIFIERS: BASIC SPECIFICATIONS
 A. Gallo, Basics of RF Electronics
• Frequency range;
from DC to > 10 GHz, multi-decades covered by a single device.
• Output level;
Maximum power at the amp output.
• Gain and gain flatness;
Ratio between the output (non-saturated) and input levels, typically expressed in dB.
The flatness is defined as half of the gain variation over the entire specified frequency band.
• 1 dB compression point;
Output level corresponding to a 1 dB reduced gain because of the incipient device saturation.
• Noise figure;
Ratio between the input and output signal-to–noise ratios assuming an input unilateral spectral
noise power density dPin/df=kT, (k=Boltzmann constant, T=290 K). Being G its power gain, the
amp generates an extra output spectral noise d(ΔPnout)/df= =G(NF- 1 )kT .
• Dynamic range;
Potential excursion of the output level, upper-limited by compression/saturation and lower-
limited by the noise power integrated over the application frequency band.
•Two-Tone Third-order intercept point ;
Measures the amp linearity. If two-tone fed (two equal-amplitude signals of frequencies f1
and f2) the amp generates intermodulation products at m1 f1 ± m2f2 frequencies. The
amplitudes of 3rd order products (2f1 - f2, 2f2- f1 ) grow with the 3rd power of the input signals
so that an input level corresponding to equal fundamental and 3rd order products amplitudes
can be extrapolated (laying usually beyond the amp dynamic range).
• Input/Output VSWR or Return Loss;
Measures of the input/output matching characteristics of the amplifier.
7
 RF TRANSFORMERS
 A. Gallo, Basics of RF Electronics

Transformers are widely used in RF electronics. They are very

effective to:
• Match lines of different impedance with negligible insertion loss;
• De-couple ground while transmitting RF signals;
• Connect balanced and unbalanced circuits (balun) Autotransformer (n ≥ 1)
RF transformers are also embedded in a number of
other devices (splitters/combiners, mixers, amplifiers, …)

Ideal Transformer transfer functions

Vs = n V p Center-tap transformer

Is = − I p n
Z p = Vp I p = −Z s n2

8
 RF TRANSFORMERS
 A. Gallo, Basics of RF Electronics

Circuital model of a
real RF Transformer
Together with transform ratio n and connection topology, real transformers are characterized by
operating bandwidth, insertion loss, max power rating, … The lower cutoff frequency is
determined by the windings active inductance Lact, while the high frequency cutoff is dominated
by the inter-windings and intra-windings capacitances Cp- p, Cs- s and Cp- s .
In-band insertion loss is due to the magnetic core dissipation and to the windings ohmic losses,
accounted by the resistances Rloss , Rp and Rs .

MCL TMO -1-1+ MCL TMO -1-1+

Insertion Loss [dB]

Return Loss [dB]

Frequency [MHz] Frequency [MHz]

9
 POWER SPLITTERS/COMBINERS
 A. Gallo, Basics of RF Electronics

Power splitters/combiners are used • Max power ratings;

to divide a signal into N equal copies
• Splitting technique (reactive or
(N = any number, preferably a power
resistive);
of 2), or to make a vector sum of N
different signals. • Insertion loss (over the nominal
Ideally, the power into any output 10 Log N );
channel is Pout = Pin/N. • Isolation between channels;
Basic characteristics are: • Phase and amplitude unbalance
• Number of channels N; among output channels;

• Operating frequency range; • …

A 2-way reactive splitter/combiner

based on a double tapped auto-
transformer provides impedance
matching at all ports and isolation
between A and B channels.

10
 POWER SPLITTERS/COMBINERS
 A. Gallo, Basics of RF Electronics

Isolation between ports A and B is obtained by a proper choice of the Rint value
(Rint =2Z0) in the 2nd auto-transformer, while the 1st transformer is needed to match
the characteristic input impedance.

AT2 transformer equations

V p2 = Vs2 2; I p2 = 2 I s2

Kirchhoff current law at node B

I B = Vs2 Rint − I s2 =
= I s2 (2 Z 0 Rint − 1) =if 0
 
Rint = 2 Z 0

Micro-strip design of a 2-
way splitter/combiner with
similar characteristics of a
transformer based one.

11
 POWER SPLITTERS/COMBINERS
 A. Gallo, Basics of RF Electronics

Splitter/combiners can be also resistive, consisting in a “star” connection of equal

resistors. The frequency response can be much wider (extending from DC) and
flatter in this case, at the expense of a larger insertion loss and no isolation
(all ports equally coupled).
N −2
Matching ⇒ R = Z0
N
2
P V  1
Transmission ⇒ out =  out  =
Pin  Vin  (N − 1)2

As the insertion loss

grows linearly with the
number of ports, practical
use is restricted to 3-
ports devices.

12
 HYBRID JUNCTIONS/
DIRECTIONAL COUPLERS
 A. Gallo, Basics of RF Electronics

Hybrid junctions and Directional

couplers are 4-ports passive devices
based on the same operational principles
but with different coupling levels 90° Hybrid
between ports. The two class of devices
 0 0 −j 1 
are used for different purposes.  
1  0 0 1 − j
S =
 

2 − j 1 0 0 
 0 0 − j 1− c2 c  
 
 1 −j 0
 0 
 0 0 c − j 1− c2 
S =  
 
 − j 1− c
2
c 0 0 
 
 − j 1− c2 
 c 0 0 

Directional Coupler 1 80° Hybrid

 0 0 1 − 1
 
−j  0 0 1 1
S = 
2  1 1 0 0

 −1
 1 0 0 

13
 HYBRID JUNCTIONS/DIRECTIONAL COUPLERS
- COUPLED LINE
 A. Gallo, Basics of RF Electronics

Distributed coupling between 2 lines

travelling close each other is one of the
possible hybrid/coupler configuration.
The lines have a characteristic
impedance Z0 when travelling separately,
while the 3-conductors system of the 2
coupled lines has even and odd excitation
impedances Z0+ and Z0-, respectively.
The scattering matrix can be worked out by exploiting the 4-fold symmetry of the
network. Being β± the propagation constants of the even and odd modes, we get:

β + = β − ≡ β ; Z 0+ Z 0− = Z 02 

c ma x sin (2βd )   − j 1 − c 2 e jφ c c e jφ c 
c=   0 0 
1 − c ma x cos (2 β d )  
2 2

  0 0 c e jφ c − j 1 − c 2 e jφ c 
Z 0+ − Z 0− ⇒ S =  
with:: c max = +   − j 1 − c 2 e jφ c c e jφ c 0 0 
Z 0 + Z 0−   
  c e jφ c − j 1 − c 2 e jφ c 
1− c 2
  0 0 
tan φ c = ma x

tan (2β d ) 
14
 HYBRID JUNCTIONS/DIRECTIONAL COUPLERS
- COUPLED LINE
 A. Gallo, Basics of RF Electronics

If the length of the coupled line is exactly 2d=λ/4=π/2β, the scattering matrix has its
simplest form:
 − j 1 − cma
2 
2βd = π / 2   0 0 x cma x 
⇓   
  0 0 cma x − j 1 − cma x 
2
Z 0+ − Z 0−  ⇒ S =  
c = cmax = + 
Z0 + Z0  −
 − j 1− c 2
cma x 0 0 
  ma x

φc = 0   
 c ma x − j 1 − c 2
ma x 0 0 
P1 P3
If coupling factors lower than 10 dB (cmax > 0.3) are required,
+
broadside coupled strips can be used. At cmax = .707 (i.e. Z0 =5.8
-
Z0 ) the device scattering matrix is that of a 90° hybrid junction.
Coupled wound coils on ferrite cores are used at low frequencies.
P4 P2

Another possible directional coupler structure is

represented by 2 parallel lines connected through
P2 P4 coupling holes. Two equal holes separated by λ/4 are
sufficient to produce the proper scattering matrix at a
P1 P3 given frequency. Device bandwidth can be enlarged with
multiple holes with different optimal dimensions.
15
 HYBRID JUNCTIONS/DIRECTIONAL COUPLERS
- BRANCH LINE
 A. Gallo, Basics of RF Electronics

The branch line coupler is another geometry

giving the desired port-to-port coupling.
The scattering matrix can be still worked out
by exploiting the 4-fold symmetry of the
network. Under the assumptions:
I) βd1 = βd2=π/2;
II) (1/Z1)2 -(1/Z2)2= (1/Z0)2
the scattering matrix has the following form:

Directional
90° Hybrid
coupler

 0 0 − jZ1 Z 0 − Z1 Z 2  0 0 j 1
   
 0 0 − Z1 Z 2 − jZ1 Z 0  Z2 = Z0 1 0 0 1 j

S = = =−  
 − jZ1 Z 0 − Z1 Z 2 0 0  Z1 = Z 0 2 2  j 1 0 0
   
 −Z Z − jZ1 Z 0   1 j 0 0 
 1 2 0 0 

16
 HYBRID JUNCTIONS/DIRECTIONAL COUPLERS
- HYBRID RING
 A. Gallo, Basics of RF Electronics

The hybrid ring (also called “rat-race”) coupler

is a suitable geometry to get 180° hybrids.
Under the assumptions:
d1,2=d2,3=d3,4=π/2β=λ/4; d1,4=3π/2β=3λ/4
the scattering matrix has the following form:
Unmatched
Isolated

180° Hybrid
 Z12 − 2 Z 02 − 2 jZ 1Z 0 2 jZ 1Z 0 
 2 0 
 Z1 + 2 Z 0 Z12 + 2 Z 02 Z12 + 2 Z 02 
2

  0 1 0 − 1
 − 2 jZ 1Z 0 Z12 − 2 Z 02 − 2 jZ 1Z 0
0   
 Z12 + 2 Z 02 Z12 + 2 Z 02 Z12 + 2 Z 02  −j 1 0 1 0
S = = Z1 = 2 Z 0 =  
 − 2 jZ 1Z 0 Z12 − 2 Z 02 − 2 jZ 1Z 0  2 0 1 0 1
   −1 0 1 0 
0
 Z12 + 2 Z 02 Z12 + 2 Z 02 Z12 + 2 Z 02  
 2 jZ Z
 1 0 − 2 jZ 1Z 0 Z12 − 2 Z 02  V1   0 
0
 Z 2 + 2Z 2
 1 Z12 + 2 Z 02 Z12 + 2 Z 02  0  
− j V3 + V1 
0
S ⊗   =
V3  2  0 
    17
0 V3 − V1 
 HYBRID JUNCTIONS/
DIRECTIONAL COUPLERS
 A. Gallo, Basics of RF Electronics
Directional couplers are used to sample Basic characteristics are:
or to unequal split/sum RF signals .
• Coupling coefficient (3 dB for
Hybrids are used whenever splitting/ hybrids);
combination of RF signals out-of-phase
• Directivity / Isolation between un-
(90°) or counter-phase (180°) are
coupled ports;
required (I&Q modulators/ detectors,
differential combination, …). • Operating frequency range;
• Max power ratings;
• Coupling type (holes, distributed,
rings, …);
• Insertion loss (over the nominal
(
VP4 = c VP1 + d VP3 ) with d << 1 coupling factor);

 0 c⋅d − j 1− c2 c 

• Phase and amplitude unbalance

 c⋅d 0 c − j 1− c 
2
 among output channels;
S = 
  • Phase and amplitude flatness over
 − j 1− c c⋅d 
2
c 0



 frequency;
 c − j 1− c2 c⋅d 0 
• …
Coupler Directivity: -20 Log|d|
measurement of the imperfect isolation between ideally uncoupled port.
18
 CIRCULATORS / ISOLATORS
 A. Gallo, Basics of RF Electronics
Circulators are non-reciprocal 3-ports (typically) devices whose scattering
matrix is ideally given by:
0 0 1
 
S = 1 0 0
0 1 0
 
Their basic structure is a symmetrical, 120° Y junction
with a ferrite disk placed at the center biased by an axial magnetic field.
Biased ferrites show a tensor magnetic permeability, i.e. an anisotropic behavior. The
incident wave on one port excites 2 unbalanced (because of the anisotropy) waves rotating in
the 2 opposite directions, so that the coupling to the output ports is also unbalanced. By
proper design the junction it is possible to have almost 100% transmission in one port and no
transmission in the other in a given frequency band (> 1 octave).

Isolators are circulators with one port

internally terminated. Circulators and
isolators are used for a number of tasks, such
us matching sources and loads, protecting
sources against backward power, capturing
and draining reflected power from a device to
another device.

19
 FILTERS
 A. Gallo, Basics of RF Electronics
Filters are 2-ports devices “tailored” to obtain a specific required frequency
response (the s21 of the network). Typically the response is maximized at some bands
of interest, and minimized at other frequency bands that have to be rejected.
Filters are classified on the base of their cells, DR cavities, µ-strip cells, …);
nature, topology, dissipation, … • Reflective/Absorbing depending on
• Analog/Digital, depending on the nature the path of the stopbands (reflected
(continuous or sampled & digitized) of or internally dissipated);
the input signal; • LowPass/HighPass/BandPass/Notch/
• Lumped/Distributed, depending on the Comb, depending on the profile of
nature of the internal components (L-C the frequency response.

20
 FILTERS
 A. Gallo, Basics of RF Electronics
Filters with different response around transition between pass and stop bands
are available for different applications. They implements different rational
complex polynomials in their transfer functions, the most popular ones being:
• Bessel, for a maximally flat group delay;
• Butterworth, for a maximally flat frequency response in the pass-band;
• Gaussian, providing a gaussian response to a Dirac pulse and no overshoot for
an input step function. The Gaussian filter also minimizes the group delay;
• Chebyshev, providing a steep transition with some passband (type I) or
stopband (type II) ripples. They provide the closest possible response w.r.t. an
ideal rectangular filter;
• Elliptic, providing the steeper possible transition between the pass-band and
the stop-band by equalizing the ripple in both.

Butterworth Chebyshev I Chebyshev II Elliptic 21

 FILTERS
 A. Gallo, Basics of RF Electronics
Design and construction of filters is becoming more and more a specialized activity,
so that “home made” devices are seldom used, and mainly for very specific task.
Design phase make use of dedicated software packages (such as Touchstone)
through various iterative steps. In addition to the typologies already listed, other
important characteristics defining filter performances are:

• Insertion Loss, defined as the

in-band signal attenuation;
• Phase linearity/Group delay: figures of the quality
of the filter phase response across the pass-band,
that should present a constant negative slope to
avoid distortion of time-profile of in-band pulses.
The slope of the response phase is the filter group
delay, equal to the signal latency while travelling
across the device.
• Input/output impedance and VSWR: characteristic
impedance of the device and reflectivity of signals
transmitted (within pass-band) and rejected (within
stop-band). 22
 DIGITAL FILTERING
 A. Gallo, Basics of RF Electronics
Digital filters act on sampled and digitized input signals. There are 2 basic filter architectures:
• Finite Impulsive Response (FIR), where the output y is a linear combination of the last N sampled
values of the input x. The coefficients hi of the expansion represent the discretization of the filter
Green’s function.
• Infinite Impulsive Response (IIR), where the output y is a linear combination of the last N and M
sampled values of the input x and output y, respectively. IIR filters directly implement a feedback
architecture, which may generate sharp frequency responses with a limited number of samples.
t

H(τ) =filter Green’s function y (t ) = ∫ H (τ ) x(t − τ ) dτ

−∞
N
y (t n ) = h0 x(t n ) + h1 x(t n −1 ) + h2 x(t n − 2 ) + .... = ∑ hi xn −i
i =0

FIR
N
Y ( z)
H ( z) = = h0 + h1 z + h2 z + ... = ∑ hi z −i
−1 −2

X ( z) i =0

y (t n ) = h0 x(t n ) + h1 x(t n −1 ) + h2 x(t n − 2 ) + ....

N M
... − p1 y (t n −1 ) − p2 x(t n − 2 ) + .... = ∑ hi xn −i − ∑ pk yn − k
i =0 k =1
N
−1
Y ( z ) h0 + h1 z + h2 z + ...
−2 ∑h z i
−i

H ( z) = = = i =0
X ( z ) 1 + p1 z −1 + p2 z − 2 + ...
IIR
M
1 + ∑ pk z − k
k =1
23
 DIGITAL FILTERING
 A. Gallo, Basics of RF Electronics
The z-domain transfer function Hz (z) gives direct information on the filter frequency response
being related to the Laplace HL (s) and Fourier HF (jω) transfer functions by the mathematical
expressions:
H L ( s ) = H z ( z ) z =e sT H F ( j ω ) = H z ( z ) z = e j ωT
Differences between analog and digital filtering are quite evident.
Digital filtering is a complex operation requiring many steps such
as down-conversion (necessary in most cases), A-to-D conversion,
digital data manipulation, D-to-A conversion and final frequency
up-conversion.
However, powerful ICs nowadays available (DSP, FPGA, …) are
capable to perform various tasks in a single chip.
On the other hand, digital filtering provide incomparable
flexibility and operational adaptivity, since the transfer function
can be modified and optimized in real time by simply changing the
weighting coefficients. Beam feedbacks can greatly benefit this
feature.

k = 80 1
H z ( z) =
pk = 1 − 2 − 4 1 − p k z −k
Comb filter transfer function

24
MODULATION TRANSFER FUNCTIONS
 A. Gallo, Basics of RF Electronics

LLRF servo-loops and feedback loops often need to apply AM and PM

modulation to the RF drive signal. The response of a resonant cavity to AM
and PM excitations depends on its bandwidth and tuning relative to the
carrier:
ω =ω
vi (t ) = Ai [1 + ai (t )]cos (ωc t ) vo (t ) = Ao [1 + ao (t )]cos (ωc t )
r c

vi (t ) = Ai cos [ωc t + φi (t )] vo (t ) = Ao cos [ωc t + ∆φo + φo (t )]


L−transform
aˆo ( s ) φˆo ( s ) 1 ωr
x(t ) ⇒ xˆ ( s ) G(s) = = =
aˆi ( s ) φˆi ( s ) 1 + s / σ
with σ =
2QL

vo (t ) = Ao [1 + ao ,a (t )] cos [ωc t + ∆φo + φo ,a (t )]

ω r ≠ ωc
vi (t ) = Ai [1 + ai (t )]cos (ωc t )

vi (t ) = Ai cos [ωc t + φi (t )] [ ] [
vo (t ) = Ao 1 + ao , p (t ) cos ωc t + ∆φo + φo , p (t ) ]

aˆo, a ( s ) φˆo, p ( s ) φˆo, a ( s ) aˆo, p ( s )

Gaa ( s ) = ; G pp ( s ) = ; Gap ( s ) = ; G pa ( s ) =
aˆi ( s ) φˆi ( s ) aˆi ( s ) φˆi ( s )
25
 MODULATION TRANSFER FUNCTION
 A. Gallo, Basics of RF Electronics

It may be demonstrated that direct and cross modulation transfer functions

are given by:

1  A(s + jωc ) A(s − jωc )  1  A(s + jωc ) A(s − jωc ) 

G pp ( s ) = Gaa ( s ) =  +  ; G ( s ) = −G ( s ) =  − 
2  A( jωc ) A(− jωc )  2 j  A( jωc ) A(− jωc ) 
ap pa

with A(s) = transfer function in Laplace domain of the filter applied to the
modulated signal. If the signal is filtered by a resonant cavity, one has to
consider A(s)=Acav(s) given by:
2σ s
Acav ( s ) = A0 2 with ωr ≈ ωc + σ tan φ z
s + 2σ s + ωr2
where φz is the cavity tuning angle, i.e. the phase of the cavity transfer
function at the carrier frequency ωc. Finally one gets:

σ s + σ 2 (1 + tan 2 φ z ) σ tan φ z s
G pp ( s ) = Gaa ( s ) = ; Gap ( s ) = −G pa ( s ) = −
(
s 2 + 2σ s + σ 2 1 + tan 2 φ z ) (
s 2 + 2σ s + σ 2 1 + tan 2 φ z )

26
 MODULATION TRANSFER FUNCTION
 A. Gallo, Basics of RF Electronics

The general form of the modulation transfer functions features 2 poles

(possibly a complex conjugate pair) and 1 zero, and degenerates to a single
pole LPF response if the cavity is perfectly tuned (cross modulation terms
vanish in this case).

27
MODULATION TRANSFER FUNCTION:
THE PEDERSEN MODEL
 A. Gallo, Basics of RF Electronics
In circular accelerators the beam phase depends on the cavity RF phase through the
beam transfer function, while the cavity RF amplitude and phase depend on the beam
phase through the beam loading mechanism. The whole generator-cavity-beam linear
system can be graphically represented in a diagram called Pedersen Model.
The modulation transfer functions vary with the stored current and definitely couple
the servo-loops and the beam loops implemented around the system.

Cavity + Beam
Generator B (s )

p p
p g
G pp (s ) + G bpp (s )

g
G pa (s ) G bpa (s )

g b
Gap (s ) Gap (s )
a
a g
Gaa (s ) + b
Gaa (s )
a
1
tan φs
28
END OF THE 1 ST PART
 A. Gallo, Basics of RF Electronics

Thank you for the moment

and …

See you tomorrow at 1 5. 30

29