introduction to rf - indico · 2018. 11. 20. · propagation/phase constant wave number the simples...
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Title
Introduction to RF
Andrea Mostacci University of Rome “La Sapienza” and INFN, Italy
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Outline
Maxwell equations
General review
The lumped element limit
The wave equation
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)
Boundary value problems for metallic waveguides
2
Goal of the
lectureShow principles behind the practice
discussed in the RF engineering module
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Outline
Maxwell equations
General review
The lumped element limit
The wave equation
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)
Boundary value problems for metallic waveguides
3
Goal of the
lectureShow principles behind the applications
discussed in the RF engineering module
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Maxwell equations in vacuum
Electric Field
Magnetic Flux Density
Electric Charge Density
Electric Current Densitysources
fields
Magnetic constant Electric constant Speed of light
Divergence operator Curl operator
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Maxwell equations in matter: the physical approach
charges and currents IN VACUUM
Insulator
… the modelThe reality …
Superconductor,
Ferrite …
Magnetic medium:
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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 Theory7
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Maxwell equations: general expression
Continuity equation is included
Magnetic Field
Electric Flux Density
Electric Field
Magnetic Flux Density
Electric Charge Density
Electric Current Densityso
urc
es
fie
lds
in vacuum
tFixed position
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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
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Solution of Maxwell Equations: the EM waves
xI II
Wave equation (1D)
Maxwell Equations: free space, no sources
Wave equation (3D)
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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.11
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The electromagnetic spectrum for RF engineers
14Source: Pozar, Microwave Engineering 4ed, 2012
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The RF spectrum and particle accelerators
800MHz
110GHz50MHz
1.3GHz
3GHz
600 (mm)3000
200 (mm)1000
Animations by G. Castorina
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The RF spectrum and particle accelerators
1.3GHz
A. Gallo Lecture @ CAS RF engineering (2010)
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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
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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
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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
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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.
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Helmotz equation and its simplest solution
Helmotz equation
Propagation/phase constant
Wave number
The simples solution: the plane wave 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
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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 …
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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.
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Plane wave in lossy media
Definition:
Uniform in x, y
Positive z direction
Phase constantAttenuation constant
time
complex
Attenuating
TEM “wave” …
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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
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impedance of
the medium
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Maxwell equations and boundary value problem
Maxwell equation with sources + boundary conditions = boundary value problem
Homogeneous mediumSources
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Do you see asymmetries?
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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
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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
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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?
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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)
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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+
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Transverse Electric Magnetic mode
Look for a Transverse Electric Magnetic mode
Hint 1
For a given
Start from a TM mode ( vector potential A)
Exercise
Hint 2
Solution
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Transverse Electric Magnetic mode
Look for a Transverse Electric Magnetic mode
Hint 1
The transverse E field is “electrostatic”
For a given
Start from a TM mode ( vector potential A)
Exercise
Hint 2
Solution
1.
2. As plane waves:
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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
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General solution for fields in cylindrical waveguide
Write the Helmotz equations for potentials1.
TM waves
TE waves
Cartesian coordinates Cylindrical coordinates
2.
Separation of variables37
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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
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Rectangular waveguides: TE modeExample
constraint
condition
Write the Helmotz equation
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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
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Waveguide dispersion curve
Courtesy of S. PisaCut-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
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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 as
Find the single mode BW for WR-90 waveguide (a=22.86mm and b=10.16 mm)
I I
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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 as
Find the single mode BW for WR-90 waveguide (a=22.86mm and b=10.16 mm)
I I
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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 as
Find the single mode BW for WR-90 waveguide (a=22.86mm and b=10.16 mm)
I I
47
a=0.9 inches b=0.4 inches
Single mode BW
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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
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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
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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??
53
Simulations by L. Ficcadenti
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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
54
Simulations by L. Ficcadenti
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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.
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Conclusions
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… The universe is written in the mathematical
language and the letters are triangles, circles
and other geometrical figures …