L1-1

ELECTROMAGNETICS AND APPLICATIONS

Lecture 8TEM Transmission Lines

Luca Daniel

L8-2

Course Outline and Motivations

• Electromagnetics:– How to transport signals and power on “guided systems”

• Applications– digital electronics: e.g. analyze transients when you send a

signal from the CPU chip to the GPU chip, or from your keyboard to your iPad

CPU

RAM

GPU A/DD/A

keyboard

iPad

L8-3

Course Outline and Motivations

• Electromagnetics:– How to transport signals and power on “guided systems”

• Applications– digital electronics: e.g. analyze transients when you send a

signal from the CPU chip to the GPU chip, or from your keyboard to your iPad

– analog and biomedical electronics: e.g. match load of RF cables bringing signal from power amplifier to MRI coil antennas to avoid reflections

CPU

RAM

GPU A/DD/A PA

L8-4

• Review of Fundamental Electromagnetic Laws

• Electromagnetic Waves in Media and Interfaces

• Digital & Analog Communicationso TEM transmission lines (cables and IC/PCB traces)

o Digital communications (transients)

o RF communications (matching loads to amplifiers)

o RLC and TEM resonators

• Microwave Communications

• Optical Communications

• Wireless Communications

• Acoustics

Course Outline

L8-5

• Review of Fundamental Electromagnetic Laws

• Electromagnetic Waves in Media and Interfaces

• Digital & Analog Communications

o TEM transmission lines (cables and IC/PCB traces) Parallel plate TEM transmission line

Characteristic Impedance

From Maxwell eqn. to Telegrapher eqn.

Finite difference - equivalent circuit interpretation

Examples of Practical TEM transmission lines (cables, traces…)

o Digital communications (transients)

o RF communications (matching loads to amplifiers)

o RLC and TEM resonators

Today’s Outline

TodayToday

L8-6

Parallel-Plate Transmission Line

Boundary Conditions:

Uniform Plane Wave

E H 0 at perfect conductor plates

zE tzˆ ˆE(t,z) xE t , H(t,z) y

x polarized wave propagating in+z direction in free space

H

z

x

y

E

k

between perfect conductors plates

E produces surface charges on plates (and voltage)

H produces surface currents on plates

L8-7

Currents and Voltages Corresponding to the Fields

Currents in Plates:

I(z) = H(z) W [A]

C AH ds J da I(z)

Surface current density Js [A/m]:

sJ (z) H(z) [A/m]

H

z

W

I(z)

C

Voltages between plates:

2

1 21 E ds V(z)

z cSince H = 0 E ds 0 at any fixed z,

V(z) is uniquely defined at fixed z

Surface charge density s(z) [C/m2]:

(Boundary condition; from ) s(z) E(z) D

1

2

h

V(z)-

+ E

ds

c

x

y z

I(z)

V(z) = E(z) h [V], independent of path

L8-8

0

zv t

Z

Characteristic Impedance of the Transmission Line

z zv t h E t -

o

hZ [ohms] "Characteristic impedance"W

zE tzi t W

zˆE(t,z) xE t

zE tˆH(t,z) y

W

z

xy

z z

cE t

k

-

+ z

cv t zci t

ElectromagneticFields in the mediumbetween the plates

Corresponding Voltagesand Currents on the Plates(induced by boundary conditions)

zc

H t

h

L8-9

Maxwell Equations vs. Telegrapher’s Equations

dHEdt

yx dHdEdz dt

1 dv 1 dih dz W dt

dv h didz W dt

dEHdt

y xdH dEdz dt 1 di 1 dv

W dz h dt di W dv

dz h dt

ElectromagneticFields in the mediumbetween the plates

Corresponding Voltagesand Currents on the Plates(induced by boundary conditions)

2 2x x

2 2

E E

z t

2 2

2 2v h W v

W hz t

totz zˆ ˆE (t,z) xE t xE t

tot

z zE t E tˆ ˆH (t,z) y y

totz zv (t,z) v t - v t +

tot

0 0

z zv t v ti (t,z)

Z Z

Solutions:

tottot

0

v (t,z)i (t,z)

ZNote:

tot

totE (t,z)

H (t,z)Note:

L8-10

Telegrapher’s Equations – Equivalent Circuit Interpretation

Equivalent Circuit:

i(z) i(z+z)Lz

v(z) v(z+z)

z

Lz Lz

Cz Cz Cz

Finite Difference Approximation:

di(z)hv(z z) v(z) zW dt

S

dv h didz W dt

di W dvdz h dt

z

hL inductance per unit lenght [H/m]W

0

h h h LZW W W C

h W

W h

1 1 1LC

dHEdt

dEHdt

dH(z)h E(z z) h E(z) h z

dt

c sdHE ds n̂dadt

c s

dEH ds ndaˆdt

dE(z)W H(z z) W H(z) W z

dt

E(z) E(z z)

WC capacitance per unit lenght [F/m]h

dv(z)Wi(z z) i(z) zh dt

h

L8-11

Examples of Practical TEM Transmission Lines

microstrip

Parallel wires Coaxial cableParallelplates

z

Arbitrary cross-section

2) Compute the capacitance per unit length C (OR the inductance per unit length L)

1) Compute velocity from medium properties: 1

3) Compute Characteristic Impedance: 0 01Z = OR Z = LC

General procedure to compute the Characteristic Impedance:

0

1LC

LZC