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Microwave theory, March 14, 2011
Anders Karlsson, anders.karlsson@eit.lth.se
Electrical and information technology
Microwave theory, March 14, 2011
Homepage
All information is on the course home page (www.eit.lth.se/)
Anders Karlsson, Electrical and information technology
Examination
I Hand in problems. Three assignments.
I Project.
I Oral exam for grades 4 and 5.
Anders Karlsson, Electrical and information technology
Assignments
The hand in problems should be solved individually. You areallowed to
I discuss the problem with others
I ask the teacher (Anders) if you get stuck
You are not allowed to copy solutions!
Anders Karlsson, Electrical and information technology
Project
I Groups of two students (or, if needed three or one).
I All students should contribute to the project.
I All students should take part in the oral presentation of theproject.
I Discussions with the teacher (Anders) are recommended.
I You are not allowed to copy results from other groups.
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Comsol
The finite element method program Comsol is used frequently inthe course. It is installed on most of the computers in thebasement of the E-building. You can also install it on your owncomputer. You find information on the home page.On Tuesday 22/3 there is a problem session devoted for Comsol.
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Microwave theory
I Maxwell equations
I Transmission lines
I Waveguides
I Resonance cavitiesI Dielectric waveguides
I Optical fibersI Dielectric resonators
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Example
reflector
hornrectangular
waveguide
rectangular waveguide circular waveguide
coaxial
cable
reflector
horn circular
waveguide
coaxial
cable
coaxial cable
optical fiber
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electron
gun
input
input
resonator
output
resonator
output
collector
electron
beam
rectangular waveguides
cavity for particle acceleration
coaxial
waveguides
particle beam
tube
z
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The Maxwell equations
Time domain!
∇× ~E(~r, t) = −∂~B(~r, t)∂t
∇× ~H(~r, t) = ~J(~r, t) +∂ ~D(~r, t)∂t
~E=electric field~D=electric flux density~H=magnetic field~B=magnetic flux density~J=current density
Anders Karlsson, Electrical and information technology
The Maxwell equations
What else is needed in order to find ~E and ~H in a typical problem?
I Sources
I Constitutive relations: ~D = ε0ε ~E, ~B = µ0µ ~H, ~J = σ ~E
I Boundary conditions
Anders Karlsson, Electrical and information technology
Example
reflector
hornrectangular
waveguide
coaxial
cable
Anders Karlsson, Electrical and information technology
The Maxwell equations
Frequency domain!
∇× ~E(~r) = −jω ~B(~r)
∇× ~H(~r) = ~J(~r) + jω ~D(~r)
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Phasors: examples
Transformation between time and frequency domain:
~E(~r, t) = ~E0(~r) cos(ωt+ φ)←→ ~E(~r) = ~E0(~r)ejφ
v(t) = V0 cos(ωt+ π/4) ←→ V = V0ejπ/4
E(~r) =complex electric field. V =complex voltage.
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Helmholz equation
From Maxwell equations in the frequency domain we get
∇2 ~E(~r) + k2 ~E(~r) = ~0
∇2 ~H(~r) + k2 ~H(~r) = ~0
k = ω√µ0µε0ε =wave number in the material.
Helmholtz equations for ~E and ~H are crucial for waveguides andcavities!
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Boundary conditions
Boundary between two materialsTangential component of ~E is the same on both sides!Tangential component of ~H is the same on both sides!
Anders Karlsson, Electrical and information technology
Boundary conditions
Boundary between two materials
Anders Karlsson, Electrical and information technology
Boundary conditions
Boundary between two materials
Anders Karlsson, Electrical and information technology
Boundary conditions
Boundary to a perfect conductorTangential component of ~E is zero!
PEC=perfect electric conductor
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Circuits
Discrete circuits!
LZ+-
Z i
v
d
λ
d� λ ⇒discrete circuit ⇒ Circuit theory.
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Discrete?
Which of the following systems ca be considered to be discrete?
I Power system in a house. d ≈10 m,λ = c/f = 3 · 108/50 = 6000000 m ⇒ d� Yes!
I Mobile phone. d ≈ 10 cm. λ = c/f = 3 · 108/2 · 109 = 15cm. No!
I Accelerators. f = 100 MHz in Maxlab (λ = 3 m). f = 700MHz in ESS (λ = 0.4 m.) No!
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Transmission lines
If the system is not discrete then we need transmission linetheory, or Maxwell equations.Examples of transmission lines:
I Parallel wires (e.g. twisted pairs for LAN)
I Coaxial cables
I Micro strips in integrated circuits and on PCB.
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Transmission lines
In transmission line theory we use voltage, current and lineparameters (R, L, G, C) to describe signals. We use Maxwellequations to determine the line parameters.
Anders Karlsson, Electrical and information technology
Transmission line parameters
zzzz d+
R/2 /2dz
R/2 dz
L dz
/2L dzC dz G dz
i(z, t)
i(z, t)
i(z +
+
dz, t)
i(z + dz, t)
v(z, t) v(z + dz, t)
-
+
-1
2 3
4
A
R =resistance per unit lengthL =inductance per unit lengthG =conductance per unit lengthC =capacitance per unit length
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Voltage and current
Voltage and current depend on both time and position!
i(z, t)
i(z, t)v(z
z
, t) +
-
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Waves along transmission line
z
v v
v+(z − vt) v−(z + vt)
The voltage and current are superpositions of waves traveling tothe left and to the right! They travel with the speed of light.
Anders Karlsson, Electrical and information technology
Important quantities
Time domain:
I Wave speed vpI Characteristic impedance Z0
I Reflection coefficient Γ
Anders Karlsson, Electrical and information technology
Important quantities
Frequency domain:
I Wave speed vpI Phase constant β = ω/vp = 2π/λI Propagation constant γ
I Characteristic impedance Z0
I Input impedance Z(0)I Reflection coefficient Γ
Anders Karlsson, Electrical and information technology
Motespunkter
1. Val av justeringsman.2. Dagordning.3. Paminnelse: Uppdatering av aktivitet och avklarad del av
forskarstudierna.4. Web-sidorna.
I Thomas Johansson: Kort presentation av Web-sidorna forforskning. (www.eit.lth.se/)
I Diskussion.I Diskussion kring lista med utvalda publikationer.
5. Storlek pa forskargrupper. Viktor inleder.6. Antal forskarutbildningsamnen. Skall vi reducera antalet?
Racker det med ett (electrical engineering)?7. Extra tilldelning av fakultetsmedel. Riktade satsningar?8. Situationen i Helsingborg. Viktor informerar.9. Biblioteket. Nagot nytt? ESS. Nagot nytt?
10. Ovriga fragor.11. Nasta mote
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