Introduction to RF · Harmonic time dependence and phasors Assuming sinusoidal electric field...
90
Title Introduction to RF Andrea Mostacci University of Rome “La Sapienza” and INFN, Italy 1
Transcript of Introduction to RF · Harmonic time dependence and phasors Assuming sinusoidal electric field...
Title
Introduction to RF
Andrea Mostacci University of Rome “La Sapienza” and INFN, Italy
1
Goal of the lecture
2
Goal of the
lectureShow principles behind the practice discussed in the RF
engineering module
Outline
Maxwell equations
General review
The lumped element limit
RF fields and particle accelerators
The wave equation
Energy conservation issue
Maxwell equations for time harmonic fields
Fields in media and complex permittivity
Boundary conditions and materials
Plane waves
The concept of mode
Maxwell equations and vector potentials
Cylindrical waveguides: TM, TE and TEM modes
Solving Maxwell Equations in metallic waveguides
Rectangular waveguide (detailed example)
Reading a simulation of a RF accelerating structure
Boundary value problems for metallic waveguides
3
Goal of the
lectureShow principles behind the practice
discussed in the RF engineering module
Outline
4
… The universe is written in the mathematical
language and the letters are triangles, circles
and other geometrical figures …
Maxwell equationsGeneral review
The lumped element limit
RF fields and particle accelerators
The wave equation
Energy conservation issue
Maxwell equations for time harmonic fields
Fields in media and complex permittivity
Boundary conditions and materials
Plane waves
The concept of mode
Maxwell equations and vector potentials
Cylindrical waveguides: TM, TE and TEM modes
Solving Maxwell Equations in metallic waveguides
Rectangular waveguide (detailed example)
Reading a simulation of a RF accelerating structure
Boundary value problems for metallic waveguides
Classical electromagnetic theory (Maxwell equations)
source
well
5
1.
Charges are the sources of
E-field.
2.
B-field has no sources.
3.
Time varying E-field and
B-field are chained.
4.
B-field is chained to
current.
Maxwell equations in vacuum
Electric Field
Magnetic Flux Density
Electric Charge Density
Electric Current Densitysources
fields
Magnetic constant
(permeability of free space)Electric constant
(permittivity of free space)
Speed of light
Divergence operator Curl operator
6
Some consequences of the IV equation
7
The current density has
closed lines.
Displacement
current Continuity
equation
At a given position the source of J
is the decrease of charge in time.
tFixed position
Maxwell equations: the static limit
Kirchhoff Laws
Ohm Law
Lumped elements
(electric networks)
The lumped elements model for electric networks is used also when
the field variation is negligible over the size of the network.
The E field is conservative.
The energy gain of a charge in closed circuit is zero.
No static, circular accelerators (RF instead!).
free space
Laplace
equation
8
Charged particle interaction with time varying fields
Beam manipulation
External sources acting on the beam
through EM fields.
Parasitic effects
Wakefields and coupling impedance
Extraction of beam energy
Particle acceleration, deflection …
Beam Instabilities
Diagnostics
RF devices
Particle acceleration by time varying fields
magnet
magnet X X X
cavity
Betatron or
”unbunched” acceleration
Resonant or
”bunched” acceleration
Linear accelerator (LINAC)
Synchrotron
𝐸
𝐵
𝐸
𝐵
Courtesy of P. Bryant
ChargeCharge
Parasitic effects: the wakefield
Particle in accelerators are charged, thus they are sources of EM fields …
Courtesy of Cho Ng, SLAC
Wakefields extract beam energy to EM field
The principle is used in general purpose RF sources (e.g. klystrons) as well as
in accelerators (e.g. particle wakefield accelerators)
Courtesy of Cho Ng, SLAC
Maxwell equations in matter: the physical approach
charges and currents IN VACUUM
Insulator
… the modelThe reality …
Superconductor,
Ferrite …
Magnetic medium:
13
Maxwell equations in matter: the mathematics
Electric insulators (dielectric)Magnetic materials
(ferrite, superconductor)
Polarization charges Magnetization currents
Magnetic Field Electric Flux Density
MagnetizationElectric Polarization
Constitutive relations
Equivalence Principles in Electromagnetics Theory14
Maxwell equations: general expression and solution
Magnetic Field
Electric Flux Density
Electric Field
Magnetic Flux Density
Electric Charge Density
Electric Current Densityso
urc
es
fie
lds
in vacuum
15
Maxwell Equations: free space, no sources
Wave
equation
Harmonic time dependence and phasors
Assuming sinusoidal electric field (Fourier)
Time dependence
Phasors are complex vectors
Power/Energy depend on time average of quadratic quantities
In the following we will use the same symbol for
Complex vectorsReal vectors
Note that, with phasors, a time animation is identical to phase rotation.16
2
Energy conservation
Energy of the e.m. field in the volume t
no
sources
bounding (closed) surface
Using Maxwell equations, vector identities and divergence theorem
Energy per unit time transferred from
electric field to moving charges in t
Dissipated power
(Joule effect) in t
E.M. energy flowing through surface
A per unit time (exiting from t)
Radiated power
through A
Rate of decrease of
e.m. energy in t
Energy conservation and Poynting theorem
no
sources
bounding (closed) surfacePoynting theorem (conservation of energy)
Poynting vector
Energy conservation and Poynting theorem
no
sources
bounding (closed) surfacePoynting theorem (conservation of energy)
Poynting vector
Phasor of the Poynting vector
Active power
Reactive power
Electromagnetic radiation spectrum: users point of view
300
MHz
30
MHz
3
MHz
300
kHz
300
GHz
30
GHz
3
GHz
300
THz30
THz
3
THz
3 1017
Hz
3 1016
Hz
3 1015
Hz
3 1018
Hz
The electromagnetic spectrum for RF engineers
22Source: Pozar, Microwave Engineering 4ed, 2012
near
future
today
The RF spectrum and particle accelerator devices
800MHz
110GHz3MHz
1.3GHz
3GHz 450GHz
dielectricfuture
The RF spectrum and particle accelerator electronics
1.3GHz
A. Gallo Lecture @ CAS RF engineering (2010)
Harmonic fields in media: constitutive relations
Hyp: Linear, Homogeneous, Isotropic and non Dispersive media
complex permittivity
complex permeability
Ohm Law conductivity
Losses (heat) due to damping of vibrating dipoles
Losses (heat) due to
moving charges
colliding with lattice
Source: Pozar, Microwave Engineering 4ed, 2012
Harmonic fields in media: Maxwell Equations
Hyp: Linear, Homogeneous, Isotropic and non Dispersive media
complex permittivity
complex permeability
Ohm Law conductivity
Losses (heat) due to damping of vibrating dipoles
Losses (heat) due to
moving charges
colliding with lattice
Dielectric constant
Loss tangent
27
Harmonic fields in media: Maxwell Equations
Hyp: Linear, Homogeneous, Isotropic and non Dispersive media
complex permittivity
complex permeability
Ohm Law conductivity
Losses (heat) due to damping of vibrating dipoles
Losses (heat) due to
moving charges
colliding with lattice
Dielectric constant
Loss tangent
Source: Pozar, Microwave Engineering 4ed, 2012
Boundary Conditions
Surface Charge Density
Surface Current Density
Fields at a lossless
dielectric interface
Perfect conductor
(electric wall)
Magnetic Wall
(dual of the E-wall)approx.
29
Helmotz equation and its simplest solution
Helmotz equation
Propagation/phase constant
Wave number
The simples solution:
the wave with the plane wave-front Uniform in x, y
Lossless medium
It is a wave, moving in the +z direction or –z direction
Velocity at which a fixed phase point on the wave travelsPhase velocity
Speed of light
30
Plane waves and Transverse Electro-Magnetic (TEM) waves
The ration of E and E component is an impedance called wave impedance
Distance between two consecutive maxima (or minima or …)Wave length
Intrinsic impedance of the medium
TEM waveE and H field are transverse to
the direction of propagation.
Compute H …
31
Plane waves and Transverse Electro-Magnetic (TEM) waves
The ratio of E and H component is an impedance called wave impedance
Distance between two consecutive maxima (or minima or …)Wave length
Intrinsic impedance of the medium
TEM waveE and H field are transverse to
the direction of propagation.32
Plane waves and Transverse Electro-Magnetic (TEM) waves
z
x
y
l
Note that E and B fields oscillate with the same phase.
Current density, Poynting vector and transport phenomena
probability
Probability current density
chargeCurrent density
e.m.
energy Poynting vector
Continuity equation
Poynting theorem
massMass flux density
Continuity equation
Plane wave in lossy media
Definition:
Uniform in x, y
Positive z direction
Phase constantAttenuation constant
time
complex
Attenuating
TEM “wave” …
37
Plane waves in good conductors
Good conductorConduction current >> displacement current
Characteristic depth of penetration: skin depth
vacuum conductor
z
38
vacuum conductor
z
Plane waves in good conductors
Good conductorConduction current >> displacement current
Characteristic depth of penetration: skin depth
vacuum conductor
z
@ 10 GHz
Al
Cu
Au
Ag
? Copper @ 100 MHz
39
impedance of
the medium
Surface Impedance
Incident
plane wave
Reflected
plane wave
Good conductor
The power that is transmitted into the conductor
is dissipated as heat within a very short
distance from the surface.
No transmitted
field
Skin
depth
Goal: account for an imperfect conductor
S
when
ApproximationReplace the exponentially decaying volume
current volume with a uniform current
extending a distance of one skin depth
Power loss
Surface resistance
computed as if the metal
were a perfect conductor
Being
valid if
Reflection of plane waves (a first boundary value problem)
Courtesy of
M. Ferrario, INFN-LNF
x
z
𝒌
𝜽
Simulations by
L. Ficcadenti, INFN
Reflection of plane waves (a first boundary value problem)
Courtesy of
M. Ferrario, INFN-LNF
z = z cosq - x sinq
z' = z cosq ' + x sinq '
E x,z,t( ) = E+ xo,zo, to( )eiwt-ikz +E- xo,zo, to( )eiwt-ikz '
x
z
Plane wave reflected by a perfectly conducting plane
In the plane xz the field is given by the superposition of the incident and reflected
wave:
s =¥
And it has to fulfill the boundary conditions (no tangential E-field)
Reflection of plane waves (a first boundary value problem)
Ez x,z, t( ) = E+ sinq( )eiwt- ik z cosq -x sinq( )
- E+ sinq( )eiwt- ik z cosq +x sinq( )
= 2iE+ sinq sin kx sinq( )eiwt- ikz cosq
Standing Wave
pattern (along x)
Guided wave
pattern (along z)
Taking into account the boundary conditions the
longitudinal component of the field becomes:
vfz =w
kz=
w
k cosq=
c
cosq> c
The phase velocity is given by
Courtesy of
M. Ferrario, INFN-LNF
x
z
𝒌
𝜽
From reflections to waveguides
vfz =w
kz=
w
k cosq=
c
cosq> c
x
z
𝒌
𝜽
Courtesy of
M. Ferrario, INFN-LNF
Put a metallic boundary parallel to the first
wall (the E-field is normal).
Between the two walls there must be an
integer number of half wavelengths (at
least one).
For a given distance, there is a
maximum wavelength, i.e. there is cut-
off frequency.
It can not be used as it is for
particle acceleration/deflection
Simulations by
L. Ficcadenti, INFN
From reflections to waveguides
Courtesy of
M. Ferrario, INFN-LNFSimulations by
L. Ficcadenti, INFN
Incident Electric Field
Electric Field
between the plates
Maxwell equations and boundary value problem
Maxwell equation with sources + boundary conditions = boundary value problem
Homogeneous mediumSources
46
Do you see asymmetries?
Maxwell equations and boundary value problem
Maxwell equation with sources + boundary conditions = boundary value problem
Homogeneous mediumSources
Actual or equivalent
equivalent
Vector Helmotz Equation
Step 1 Source free region
So
luti
on Homogeneous problem
Step 2
47
Method of solution of Helmotz equations
Sources Radiated Fields
Vector
Potentials
Integration of 6 Helmotz equations
Solution of the
homogeneous equation
Shape of
radiated fieldMODES
48
Solution of Helmotz equations using potentials
Sources Radiated Fields
Vector
Potentials
Integration of 6 Helmotz equations
Solution of the
homogeneous equationsMODES
Why/when is
it convenient?
49
Modes of cylindrical waveguides: propagating field
Field propagating in the
positive z direction
2 Helmotz equations
(transverse coordinates)
Only E field along z
E-mode
Transverse Magnetic (TM)
Only H field along z
H-mode
Transverse Electric (TE)
50
Modes of cylindrical waveguides: propagating field
Field propagating in the
positive z direction
Only E field along z
E-mode
Transverse Magnetic (TM)
Only H field along z
H-mode
Transverse Electric (TE)
TM
modes
TE
modes+
51
Transverse Electric Magnetic mode
Look for a Transverse Electric Magnetic mode
Hint 1
For a given
Start from a TM mode (vector potential A)
Example
Hint 2
Solution
52
The transverse E field is “electrostatic”1.
2. As plane waves:
Transverse Electric Magnetic mode in waveguides
TEM waves are possible only if there are at least two conductors.
For a given Solution
3.
4. The plane wave is a TEM wave of two infinitely large plates separated to infinity
5. Electrostatic problem
with boundary conditions
54
Example
Common TEM waveguides: stripline with dielectrics
Animations by G. Castorina59
metal strip
metal plane
dielectric
layer
General solution for fields in cylindrical waveguide
Write the Helmotz equations for potentials1.
TM waves
TE waves
Cartesian coordinates Cylindrical coordinates
2.
Separation of variables60
General solution for fields in cylindrical waveguide
Eigenvalue problem: Eigenvalues + Eigen-function 3.
TM
TE
Mode (m,n)
Compute the fields and apply the boundary conditions4.
5. It can be complex
It depends on the sources
61
Eigenvalues and cut-off frequencies (TE mode, rect. WG)
constraint
condition
Cut-off frequencies fc such that
mode m, n is attenuated exponentially (evanescent mode)
mode m, n is propagating with no attenuation
65
Waveguide dispersion curve
Courtesy of S. Pisa
Cut-off
Unimodal
propagation
Same curve for TE and TM mode, but n=0 or m=0 is possible only for TE modes.
In any metallic waveguide the fundamental mode is TE.
Fundamental
mode
66
High
dispersion
Single mode operation of a rectangular waveguide
Find the smallest ratio a/b allowing the largest
bandwidth of single mode operation
Exercise
1.
2. State the largest bandwidth of single mode operation
3. Defining the single mode bandwidth (BW) as
Find the single mode BW for WR-90 waveguide (a=22.86mm and b=10.16 mm)
I I
67
Hint:
Place the possible cut-off frequencies for different
modes on the frequency axis
I IFrequency
axis
Cut-off frequencies
Single mode operation of a rectangular waveguide
Find the smallest ratio a/b allowing the largest
bandwidth of single mode operation
Exercise
1.
2. State the largest bandwidth of single mode operation
3. Defining the single mode bandwidth (BW) as
Find the single mode BW for WR-90 waveguide (a=22.86mm and b=10.16 mm)
I I
69
a=0.9 inches b=0.4 inches
Single mode BW
Eigenfunctions and mode pattern (TE mode, rect. WG)
74
Exercise
Draw the field pattern in
the xz plane for TE10
E field
H field
Field pattern (TE10 mode, rect. WG)
m (n) is the number of half periods (or
maxima/minima) along the x (y) axis in the cross-
section.
Simulations by L. Ficcadenti
75
3l/2
3l/2
Exercise
Field pattern (TE10 mode, rect. WG)
m (n) is the number of half periods (or
maxima/minima) along the x (y) axis in the cross-
section.
Animations by L. Ficcadenti
76
Exercise
Power Flow pattern (TE10 mode, rect. WG)
Instantaneous power density
E-field
The power flowing through a surface
is the flux of the is the Poynting Vector
Animations by F. Bosco, Sapienza
Power Flow pattern (TE10 mode, rect. WG)
Average of the power density over one period
Animations by F. Bosco, Sapienza
Power Flow pattern (TE10 mode, rect. WG)
E-field
Time varying part of the instantaneous power density
Animations by F. Bosco, Sapienza
Power Flow pattern (TE10 mode, rect. WG)
E-field
Time varying part of the instantaneous power density
Animations by F. Bosco, Sapienza
Power Flow pattern (TE10 mode, rect. WG)
Reactive power density
There is energy stored in the e.m. field
No net flux through the cross section
Animations by F. Bosco, Sapienza
Field pattern at the cross section
m (n) is the number of half periods (or
maxima/minima) along the x (y) axis in the cross-
section.TE?? TE?? TE??
TM??
TM?? TM??
82
Simulations by L. Ficcadenti
Field pattern at the cross section
m (n) is the number of half periods (or
maxima/minima) along the x (y) axis in the cross-
section.TE11 TE21 TE31
TM21
TM11 TM31
83
Simulations by L. Ficcadenti
Field pattern (TE mode, rect. WG)
m (n) is the number of half periods (or
maxima/minima) along the x (y) axis in the cross-
section.
84
Full EM simulation of a RF accelerating structureExercise
With phasors, a time animation is identical to phase rotation.
X-band (12GHz) accelerating structure for high brightness LINAC
Power
INPower
OUT
Particle axis
Vacuum
port
Vacuum
port
Iris loaded waveguide (phase velocity < c)
E-field along particle axis, i.e. z-axis (log-scale)
Cut-off
(why?)
Cut-off
(why?)
Waveguide TE10 mode
(phase velocity > c)
Simulation assuming perfect conductor
Courtesy of J. Arnesano, Sapienza
ExerciseFull EM simulation of a RF accelerating structure
Which field is this one? E or H field?
Hint: always zero on the metallic
lateral surface …
z x
y
Simulation assuming perfect conductor
Courtesy of J. Arnesano, Sapienza
ExerciseFull EM simulation of a RF accelerating structure
z x
y
Which field? E or H?
Which component? Along x, y or z?
Simulation assuming perfect conductor
Courtesy of J. Arnesano, Sapienza
H
Exercise
3 cell periodicity 2p/3 phase advance
Full EM simulation of a RF accelerating structure
Simulation assuming perfect conductor
Particle
axis
Accelerating
E-field
Courtesy of J. Arnesano, Sapienza
Which is the
periodicity of the
accelerating field?
Exercise
3 cell periodicity 2p/3 phase advance
Full EM simulation of a RF accelerating structure
Simulation assuming perfect conductor
Particle
axis
Accelerating
E-field
Courtesy of J. Arnesano, Sapienza
ExerciseFull EM simulation of a RF accelerating structure
Temperature breakdown: seek
for maximum power loss
Simulation with
perfect conductor
Courtesy of J. Arnesano, Sapienza
ExerciseFull EM simulation of a RF accelerating structure
Courtesy of J. Arnesano, Sapienza
Comparison between a tuned coupler and a detuned coupler
Input coupler
detuned
Simulation with perfect conductor
Input coupler
tuned
ExerciseFull EM simulation of a RF accelerating structure
Courtesy of J. Arnesano, Sapienza
Input and output
coupler detuned
Simulation with perfect conductor
Input and output
coupler tuned
Comparison between a tuned coupler and a detuned coupler
