anders karlsson, [email protected] · i thomas johansson: kort presentation av web-sidorna f...

Microwave theory, March 14, 2011

Anders Karlsson, [email protected]

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.


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!


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.


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.


Microwave theory

I Maxwell equations

I Transmission lines

I Waveguides

I Resonance cavitiesI Dielectric waveguides

I Optical fibersI Dielectric resonators


Example

reflector

hornrectangular

waveguide

rectangular waveguide circular waveguide

coaxial

cable

reflector

horn circular

waveguide

coaxial

cable

coaxial cable

optical fiber


electron

gun

input

input

resonator

output

resonator

output

collector

electron

beam

rectangular waveguides

cavity for particle acceleration

coaxial

waveguides

particle beam

tube

z


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



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


Example

reflector

hornrectangular

waveguide

coaxial

cable



Frequency domain!

∇× ~E(~r) = −jω ~B(~r)

∇× ~H(~r) = ~J(~r) + jω ~D(~r)


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.


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!


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!


Boundary conditions

Boundary between two materials


Boundary conditions

Boundary to a perfect conductorTangential component of ~E is zero!

PEC=perfect electric conductor


Circuits

Discrete circuits!

LZ+-

Z i

v

d

λ

d� λ ⇒discrete circuit ⇒ Circuit theory.


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!


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.


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.


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


Voltage and current

Voltage and current depend on both time and position!

i(z, t)

i(z, t)v(z

z

, t) +

-


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.


Important quantities

Time domain:

I Wave speed vpI Characteristic impedance Z0

I Reflection coefficient Γ


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


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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