This document provides a summary of key concepts in radio frequency (RF) acceleration for particle accelerators. It discusses early electrostatic accelerators like the Van de Graaff and their limitations. It then introduces the concept of RF acceleration using oscillating electric fields to continuously accelerate particles. Key RF acceleration techniques are summarized, including drift tube linear accelerators, cavity linacs, and the resonant TM010 mode used in most RF cavities. Important cavity concepts like the quality factor, resonant frequencies, and accelerating voltage are defined. Peak surface fields, power dissipation, and the equivalent circuit model of cavities including their shunt impedance are also covered at a high level.
This presentation contains basics of RLC circuit. Also there is a model developed for a given RLC circuit problem. The solution of this model is also included with MATLAB code.
Types of Transducers
Analog and Digital Transducer
Characteristic of Transducer
Selection factor of Transducer
Measurement of Displacement
LVDT and RVDT
Different types of strain Gauges
Manometers
Pressure Measuring Elements
Hall Effect
Thermocouple
Today, communications is the largest sector of the electronics field. In addition, wireless, networking or other communications technologies are now contained in almost every electronic product. This makes a knowledge and understanding of communication a must rather than an option for every student. Without at least one course in communications, the student may graduate with an incomplete view of the products and systems so common today.
This presentation contains basics of RLC circuit. Also there is a model developed for a given RLC circuit problem. The solution of this model is also included with MATLAB code.
Types of Transducers
Analog and Digital Transducer
Characteristic of Transducer
Selection factor of Transducer
Measurement of Displacement
LVDT and RVDT
Different types of strain Gauges
Manometers
Pressure Measuring Elements
Hall Effect
Thermocouple
Today, communications is the largest sector of the electronics field. In addition, wireless, networking or other communications technologies are now contained in almost every electronic product. This makes a knowledge and understanding of communication a must rather than an option for every student. Without at least one course in communications, the student may graduate with an incomplete view of the products and systems so common today.
This is one of a type of Analog to Digital Converter (ADC).
Through this presentation, you will have a clear view of how an ADC works. This one specifies one of the types of Analog to Digital Convertor.
An Auto Transformer is a transformer with only one winding wound on a laminated core. An auto transformer is similar to a two winding transformer but differ in the way the primary and secondary winding are interrelated. A part of the winding is common to both primary and secondary sides.
Its looks like a letter Z due to symmetrical switching characteristics for each polarity of the applied voltage. ... This means that, unlike the triac and the SCR, the disc cannot be estimated to maintain a low voltage drop until its current falls below the level of holding current. A triac is a 4-layer semiconductor device with two power terminals (MT1 and MT2) and a gate terminal. It is used as a power control device for 50/60Hz AC mains applications. It is placed in series with the load connected across the mains. ... A disc is a similar 4-layer device but does not have a gate terminal.
http://bit.ly/2PIOIQM
Introduction
Band Pass Amplifiers
Series & Parallel Resonant Circuits & their Bandwidth
Analysis of Single Tuned Amplifiers
Analysis of Double Tuned Amplifiers
Primary & Secondary Tuned Amplifiers with BJT & FET
Merits and de-merits of Tuned Amplifiers
This is a pdf file on the topic Gamow theory of alpha decay which gives description about how the scientist Gamow had solved the theory of the alpha decay via tunneling .
This is one of a type of Analog to Digital Converter (ADC).
Through this presentation, you will have a clear view of how an ADC works. This one specifies one of the types of Analog to Digital Convertor.
An Auto Transformer is a transformer with only one winding wound on a laminated core. An auto transformer is similar to a two winding transformer but differ in the way the primary and secondary winding are interrelated. A part of the winding is common to both primary and secondary sides.
Its looks like a letter Z due to symmetrical switching characteristics for each polarity of the applied voltage. ... This means that, unlike the triac and the SCR, the disc cannot be estimated to maintain a low voltage drop until its current falls below the level of holding current. A triac is a 4-layer semiconductor device with two power terminals (MT1 and MT2) and a gate terminal. It is used as a power control device for 50/60Hz AC mains applications. It is placed in series with the load connected across the mains. ... A disc is a similar 4-layer device but does not have a gate terminal.
http://bit.ly/2PIOIQM
Introduction
Band Pass Amplifiers
Series & Parallel Resonant Circuits & their Bandwidth
Analysis of Single Tuned Amplifiers
Analysis of Double Tuned Amplifiers
Primary & Secondary Tuned Amplifiers with BJT & FET
Merits and de-merits of Tuned Amplifiers
This is a pdf file on the topic Gamow theory of alpha decay which gives description about how the scientist Gamow had solved the theory of the alpha decay via tunneling .
A presentation on Coulomb-Blockade Oscillations in Semiconductor Nanostructures made by Deepak Rajput. It was presented as a course requirement at the University of Tennessee Space Institute in Fall 2008.
Laboratory session in Physics II subject for September 2016-January 2017 semester in Yachay Tech University (Ecuador). Topic covered: electricity, magnetism
Based on Bruna Regalado's work
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Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
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3. Van-de Graaff - 1930s
A standard
electrostatic
accelerator is
a Van de
Graaf
These devices are limited to
about 30 MV by the voltage hold
off across ceramic insulators
used to generate the high
voltages (dielectric breakdown).
4. RF Acceleration
- + - +
- + - +
By switching the charge on the plates in phase with the particle
motion we can cause the particles to always see an
acceleration
You only need to hold off the voltage between two plates not the full
accelerating voltage of the accelerator.
We cannot use smooth wall waveguide to contain rf in order to accelerate a
beam as the phase velocity is faster than the speed of light, hence we
cannot keep a bunch in phase with the wave.
5. Early Linear Accelerators (Drift
Tube)
• Proposed by Ising (1925)
• First built by Wideröe (1928)
• Alvarez version (1955)
Replace static fields by time-varying fields by only exposing the
bunch to the wave at certain selected points. Long drift tubes shield
the electric field for at least half the RF cycle. The gaps increase
length with distance.
6. Cavity Linacs
• These devices store large amounts of
energy at a specific frequency allowing low
power sources to reach high fields.
7. Cavity Quality Factor
• An important definition is the cavity Q factor, given
by
Where U is the stored energy given by,
The Q factor is 2p times the number of rf cycles it
takes to dissipate the energy stored in the cavity.
• The Q factor determines the maximum energy the
cavity can fill to with a given input power.
c
P
U
Q
0
dV
H
U
2
0
2
1
0
0
exp
t
U U
Q
8. Cavities
• If we place metal walls at
each end of the waveguide
we create a cavity.
• The waves are reflected at
both walls creating a standing
wave.
• If we superimpose a number
of plane waves by reflection
inside a cavities surface we
can get cancellation of E|| and
BT at the cavity walls.
• The boundary conditions
must also be met on these
walls. These are met at
discrete frequencies only
when there is an integer
number of half wavelengths in
all directions.
(/c)2=(mp/a)2+ (np/b)2+ (pp/L)2
L
The resonant frequency of a rectangular
cavity can be given by
Where a, b and L are the width, height
and length of the cavity and m, n and p
are integers
a
9. Pillbox Cavities
• Transverse Electric (TE) modes
• Transverse Magnetic (TM) modes
im
n
m
m
z e
a
r
J
A
r
E
,
1
, z
t
n
m
z
t E
a
ik
E
2
,
2
z
t
n
m
t E
z
a
i
H
ˆ
2
,
2
im
n
m
m
z e
a
r
J
A
r
H
,
1
'
, z
t
n
m
z
t H
a
ik
H
2
,
2
'
z
t
n
m
t H
z
a
i
E
ˆ
'2
,
2
0
1
1 2
2
2
2
z
k
r
r
r
r
r
im
t
m e
r
k
J
A
)
(
1
Wave equation in cylindrical co-ordinates
Solution to the wave equation
10. Bessel Function
• Ez (TM) and Hz (TE)
vary as Bessel
functions in pill box
cavities.
• All functions have
zero at the centre
except the 0th order
Bessel functions.
-0.5
-0.3
0.0
0.3
0.5
0.8
1.0
0 2 4 6 8 10
m=0
m=1
m=2
m=3
Jm(kTr)
kTr
First four Bessel functions.
One of the transverse fields varies with the differential of the
Bessel function J’
All J’ are zero in the centre except the 1st order Bessel
functions
11. Cavity Modes
TE1,1
TE0,1 TM0,1
TE2,1
TEr,θ
r θ
Cylindrical (or pillbox) cavities are more common than rectangular cavities.
The indices here are
m = number of full wave variations around theta
n = number of half wave variations along the diameter
P = number of half wave variations along the length
The frequencies of these cavities are given by f = c/(2p * (z/r)
Where z is the nth root of the mth bessel function for TE modes or the nth root of
the derivative of the mth bessel function for TE modes or
12. TM010 Accelerating mode
Electric Fields
Magnetic Fields
Almost every RF cavity operates
using the TM010 accelerating mode.
This mode has a longitudinal electric
field in the centre of the cavity which
accelerates the electrons.
The magnetic field loops around this
and caused ohmic heating.
13. TM010 Monopole Mode
0 0
0 1
0
2.405
0
0
2.405
0
0
i t
z
z
r
i t
r
r
E E J e
R
H
H
i r
H E J e
Z R
E
E
E
H
Beam
Z0=377 Ohms
15. Accelerating Voltage
Position, z
Ez, at t=0 Normally voltage is the
potential difference between
two points but an electron can
never “see” this voltage as it
has a finite velocity (ie the field
varies in the time it takes the
electron to cross the cavity
Position, z
Ez, at t=z/v
The voltage now depends
on what phase the electron
enters the cavity at.
If we calculate the voltage at
two phases 90 degrees
apart we get real and
imaginary components
16. Accelerating voltage
• An electron travelling close to the speed of light traverses through a
cavity. During its transit it sees a time varying electric field. If we use
the voltage as complex, the maximum possible energy gain is given
by the magnitude,
• To receive the maximum kick with multiple cells the particle should
traverse the cavity in a half RF period (see end of lecture).
2
c
L
f
/2
/
/2
,
L
i z c
b z
L
E eV e E z t e dz
17. Transit time factor
• An electron travelling close to the
speed of light traverses through a
cavity. During its transit it sees a time
varying electric field. If we use the
voltage as complex, the maximum
possible energy gain is given by the
magnitude,
• Where T is the transit time factor given
by
• For a gap length, g.
• For a given Voltage (=E0L) it is clear
that we get maximum energy gain for a
small gap.
/2
/
0
/2
,
L
i z c
z
L
E eV e E z t e dz E LT
/ 2
/
/ 2
/ 2
/ 2
, sin
,
L
i z c
z
L
L
z
L
g
E z t e dz
T
g
E z t dz
p
p
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5
Transit
time
factor,
T
g/
18. f
V
fs
Vp
Phase stability is given by
off-crest acceleration
Stable
region
Overvoltage
• To provide a stable bunch you often will accelerate off
crest. This means the particles do not experience the
maximum beam energy.
• Vb=Vc cos(fs) = Vc q
• Where Vc is the cavity voltage and Vb is the voltage
experienced by the particle, f is the phase shift and q is
known as the overvoltage.
19. For TM010 mode
/2
/
/2
/2
0
/2
/2
0
/2
0
,
cos /
sin /
/
2sin / 2
/
L
i z c
z
L
L
L
L
L
V E z t e dz
E z c dz
z c
E
c
L c
E
c
0
0
cos
2
cos
z
V E LT
V E L
p
This is often approximated as
Where L=c/2f, T=2/p
Hence voltage is maximised when L=c/2f
Position, z
Ez, at t=z/v
20. Peak Surface Fields
• The accelerating gradient is the average gradient seen by an
electron bunch,
• The limit to the energy in the cavity is often given by the peak
surface electric and magnetic fields. Thus, it is useful to
introduce the ratio between the peak surface electric field and
the accelerating gradient, and the ratio between the peak
surface magnetic field and the accelerating gradient.
max
2
acc
E
E
p
c
acc
V
E
L
max /
2430
/
acc
H A m
E MV m
Electric Field Magnitude
For a pillbox
21. Surface Resistance
As we have seen when a time
varying magnetic field impinges
on a conducting surface current
flows in the conductor to shield
the fields inside the conductor.
However if the conductivity is
finite the fields will not be
completely shielded at the
surface and the field will
penetrate into the surface.
2
r
This causes currents to flow
and hence power is absorbed
in the surface which is
converted to heat.
Skin depth is the distance in
the surface that the current has
reduced to 1/e of the value at
the surface, denoted by
Current
Density, J.
x
.
The surface resistance is defined as
1
surf
R
For copper 1/ = 1.7 x 10-8 Wm
22. Power Dissipation
• The power lost in the cavity walls due to ohmic heating is given by,
Rsurface is the surface resistance
• This is important as all power lost in the cavity must be replaced by
an rf source.
• A significant amount of power is dissipated in cavity walls and hence
the cavities are heated, this must be water cooled in warm cavities
and cooled by liquid helium in superconducting cavities.
2
1
2
c surface
P R H dS
23. Capacitor
–
E-Field
The electric field of the
TM010 mode is contained
between two metal
plates
This is identical to a capacitor.
This means the end plates
accumulate charge and a
current will flow around the
edges
Surface
Current
25. Resistor
Surface
Current
Finally, if the cavity has
a finite conductivity, the
surface current will flow
in the skin depth
causing ohmic heating
and hence power loss.
This can be accounted for by
placing a resistor in the circuit.
In this model we assume the
voltage across the resistor is the
cavity voltage. Hence R takes the
value of the cavity shunt
impedance (not Rsurface).
26. Equivalent circuits
2
2
c
CV
U
2
2
c
c
V
P
R
1
LC
The stored energy is just the stored energy in the capacitor.
The voltage given by the equivalent circuit does not contain the transit
time factor, T. So remember
Vc=V0 T
To increase the
frequency the
inductance and
capacitance has to
be increased.
27. Shunt Impedance
• Another useful definition is the shunt impedance,
• This quantity is useful for equivalent circuits as it
relates the voltage in the circuit (cavity) to the
power dissipated in the resistor (cavity walls).
• Shunt Impedance is also important as it is
related to the power induced in the mode by the
beam (important for unwanted cavity modes)
2
1
2
c
s
c
V
R
P
28. TM010 Shunt Impedance
0
0 1
0
2
2.405
c
E L
V
i r
H E J
Z R
p
2
2
2
0
, 1
2
0
2
2
0
, 1
2
0
2
2
0
1
2
0
1
2
2.405
2
2.405
2.405
c surface
c ends surface
c walls surface
c surface
P R H dS
E r
P R r J dr
Z R
E
P RL R J
Z
E
P R R L R J
Z
p
p
p
2 4
0
2
3
1
2 5 10
2.405
s
surface
surface
Z L x
R
R
R R L R J
p
29. Geometric shunt impedance,
R/Q
• If we divide the shunt impedance by the
Q factor we obtain,
• This is very useful as it relates the
accelerating voltage to the stored
energy.
• Also like the geometry constant this
parameter is independent of frequency
and cavity material.
2
2
c
V
R
Q U
30. TM010 R/Q
0
0 1
0
2
2.405
E L
V
i r
H E J
Z R
p
2
0
2
2
0
0 1
2
0
2
2
2
0 0
1
1
2
2.405
2.405
2
U H dV
E r
U L r J dr
Z R
E
U R L J
p
p
2
0
2
0 1
8
150 196
2.405 2.405
Z
R L L
Ohms
Q R R
c J
p
31. Geometry Constant
• It is also useful to use the geometry constant
• This allows different cavities to be compared
independent of size (frequency) or material, as it
depends only on the cavity shape.
• The Q factor is frequency dependant as Rs is
frequency dependant.
0
surface
G R Q
32. Q factor Pillbox
2
2
2
0 0
1 2.405
2
E
U R L J
p
2
2
0
1
2
0
2.405
c surface
E
P R R L R J
Z
p
0
0
453 /
2 1 /
453 /
260
1 /
surface surface
RL L R
Q
R L R R L R
L R
G
L R
33. Equivalent circuits
0
c
U C
Q R
P L
2
0
1
2
R V L
Q U C C
These simple circuit equations
can now be used to calculate
the cavity parameters such as Q
and R/Q.
In fact equivalent circuits have been proven to accurately
model couplers, cavity coupling, microphonics, beam loading
and field amplitudes in multicell cavities.
34. Cavity geometry
• The shunt impedance is
strongly dependant on
aperture
Figures borrowed from Sami
Tantawi
Similarly larger
apertures lead to higher
peak fields.
Using thicker walls has
a similar effect.
Higher frequencies
need smaller apertures
as well
35. Frequency Scaling
• Rsurf ~ f0.5 normal conducting
• Rsurf ~ f2 superconducting
• Qo ~ f-0.5 normal conducting
• Qo ~ f-2 superconducting
• Rs ~ f-0.5 normal conducting
• Rs ~ f-2 superconducting
• R/Q ~ f0 normal conducting
• R/Q ~ f0 superconducting
36. Multicell
• It takes x4 power to double the voltage in one cavity but only x2 to
use two cavities/cells to achieve the same voltage (Rs ~number of
cells).
• To make it more efficient we can add either more cavities or more
cells. This unfortunately makes it worse for wakefields (see later
lectures) and you get less gradient per unit power.
• In order to make our accelerator more compact and cheaper we can
add more cells. We have lots of cavities coupled together so that we
only need one coupler. For N cells the shunt impedance is given by
This however adds
complexity in tuning,
wakefields and the gradient of
all cells is limited by the worst
cell.
sin
total gle
R NR
37. Synchronous particle
• Imagine we have a series of gaps. The phase change
between two gaps when the beam arrives is given by
• Where a is the phase advance, (the phase difference
between adjacent coupled cavities)
• Hence the distance between cells should be
• In a linac we choose a synchronous phase fs and design the
lengths so that the synchronous particle sees the desired
phase (not always constant)
• For a standing wave structure the synchronous phase occurs
when the cavity is half a free space wavelength long.
1
1
1
n
n n a
n
l
c
f f
1
a n c
d