Theory and Applications of Transmission...

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Theory and Applications ofTheory and Applications of

Transmission LinesTransmission Lines

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TopicsTopics

• Introduction

• Types

• General Transmission-Line Equations

• Wave Characteristics on Finite Transmission Lines

• Waveguides

• Optical Fiber

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Transmission LinesTransmission Lines

• Used for guiding electromagnetic (EM) waves

• Point-to-point “guided” transmission of power and information from “source” to “receiver”, e.g., data signal. (unguided=antenna)

• Transverse EM (TEM) waves applied to most transmission lines except waveguides.

• TEM waves -> uniform plane waves

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Types classified by materialsTypes classified by materials

• Metallic Transmission Lines (Conductor)

• Hollow or Dielectric-filled Waveguides (Conductor and dielectric)

• Optical Fiber (dielectric)

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Transmission LinesTransmission Lines

Two fundamental types

• Low Frequency

– used for power transmission

• High Frequency

– used for RF transmission

“wavelengths are shorter than or

comparable to the length of cable”

Note - transmission line = conductor - but only use “surface”

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Types of Metallic Transmission Types of Metallic Transmission

LinesLines

• Parallel Line

• Twisted Pair (Shielded & Unshielded)

• Coaxial

• Microstrips

• Strip Line

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Parallel PairParallel Pair

Low loss dielectric

Spacers

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Parallel Line Parallel Line ((akaaka Ribbon Cable)Ribbon Cable)

• Simple Construction

• Used primarily for power lines, rural telephone lines or TV antenna cable

• Freq up to 200MHz over short distances

• High Radiation Loss

– moving current = Ae

– need to be aware of other metallic conductors

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Twin Lead CableTwin Lead Cable

• Balanced

– 300 Ω

• Balun

– Balanced to unbalance transformer

)/log(2760 rDZ =

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Twisted PairTwisted Pair

Shielded

metal cladding

Unshielded

protective dielectric

coating is paper, rubber, PVC…can also have single pair, each wrapped individually

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Twisted PairTwisted Pair

• Twists tend to cancel radiation loss

• Helps reduce crosstalk

• Still fairly inexpensive

• Frequency < 100MHz

• Generally short distances

– analog ~5-6 km

– digital ~2-3 km

• Note - power line interference

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CAT5 CableCAT5 Cable

• UTP

• 4 pair

• terminating in RJ45

• 100MHz max frequency

• 1000 Mbps transmit rate

• Aside: Wire Gauge (smaller is bigger)

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Coaxial CableCoaxial Cable

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Coaxial CableCoaxial Cable

• Geometry creates a “shielded” system

– no EM energy outside the cable

• Can support frequencies > 100MHz

• Can support data rates > 1GHz

• Low self-inductance allows greater BW

• Used for long-distance telephone trunks, urban networks, TV cables

• Expensive + must keep dielectric dry

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StriplinesStriplines

• Micro Stripline• Embedded Stripline

• Coplanar Stripline

• Loss– Metallic

• Skin depth

• Localized current flow

– Dielectric• Loss tangent

– Surface roughness '

'''''

εε

δεεε =⇒−= Tanj

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MicrostripsMicrostrips

• Used for very high frequencies in semi-conductors

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Remember fields are setup given an applied forcing function.

(Source)

How does the signal move

from source to load?

E & H Fields E & H Fields –– Microstrip Microstrip

CaseCase

The signal is really the wave propagating between the conductors

Electric field

Magnetic field

Ground return path

X

Y

Z (into the page)

Signal path

Electric field

Magnetic field

Ground return path

X

Y

Z (into the page)

Signal path

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Transmission TheoryTransmission Theory

• Current and Voltage change with time along the line (the signal)

– superposition of waves in both directions

– but over short distances (<λ) are constant

• Energy is lost (heat - resistance) or stored (magnetic - inductance) / (capacitive - capacitance)

v = Ri v = Ldi

dti = C

dv

dt

= Attenuation Losses

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Transmission Line ConceptTransmission Line Concept

Power

Plant

Consumer

Home

Power Frequency (f) is @ 60 HzWavelength (λλλλ) is 5×××× 106 m ( Over 3,100 Miles)

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PC Transmission LinesPC Transmission Lines

Integrated Circuit

Microstrip

Stripline

Via

Cross section view taken here

PCB substrate

T

W

Cross Section of Above PCB

T

Signal (microstrip)

Ground/Power

Signal (stripline)

Signal (stripline)

Ground/Power

Signal (microstrip)

Copper Trace

Copper Plane

FR4 Dielectric

W

Signal Frequency (f) is approaching 10 GHzWavelength (λλλλ) is 1.5 cm ( 0.6 inches)

Micro-Strip

Stripline

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Key point about transmission line Key point about transmission line

operationoperation

The major deviation from circuit theory with transmission line, distributed networks is this positional dependence of voltage and current!

– Must think in terms of position and time to understand transmission line behavior

– This positional dependence is added when the assumption of the size of the circuit being small compared to the signaling wavelength

( )( )tzfI

tzfV

,

,

=

=V1 V2

dz

I2I1

Voltage and current on a transmission line is

a function of both time and position.

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Transmission Line ModelTransmission Line Model

• Distributed circuit concept

R is the resistance in both conductors per unit length in W /m

L is the inductance in both conductors per unit length in H/m

G is the conductance of the dielectric media per unit length in S/m

C is the capacitance between the conductors per unit length in F/m

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Transmission Line Model (contTransmission Line Model (cont’’d)d)

Using Kirchhoff´s voltage law on the circuit in the figure

letting ∆z → 0 we get

(1)

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Transmission Line Model (contTransmission Line Model (cont’’d)d)

To get another equation relating G and C we apply Kirchhoff´s

current law on the circuit and get:

letting ∆ z → 0 in this equation also we get:

(2)

(1),(2) : General Transmission-line Equations

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Transmission Line Model (contTransmission Line Model (cont’’d)d)

These equations can be simplified if the voltage v(z,t) and

the current i(z,t) are time-harmonic cosine functions

the general transmission line equations become:

(3)

(4)

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Wave equations & solutionsWave equations & solutionsBy combining (3) and (4):

where γ is the propagation constant:

The general solution of (5), (6)

(5) (6)

Characteristic

Impedance

(7)

(8)

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Special CasesSpecial Cases

CLZLCu

LCjj

p /;/1/

;

0 ===

=+=

βω

ωβαγ

Lossless Line (R=0,G=0)

Distortionless Line (R/L,G/C)

CLZLCu

LCLCR

LjRLC

CjLRCLjRj

p /;/1/

;/

);(/

)/)((

0 ===

==

+=

++=+=

βω

ωβα

ω

ωωβαγ

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In an infinitely long line there are only forward travelling waves and

no reflected waves. The second term in (7) and (8) will be zero.

This is however also true for a line terminated with its characteristic

impedance. A line is called a matched line when the load impedance

is equal to the characteristic impedance. If we consider a line with

the characteristic impedance Z0, a propagation constant γ and with

the length l terminated with a load impedance ZL connected to

a sinusoidal voltage source, and then the voltage and current

distribution on the line can be calculated as:

Finite Transmission LinesFinite Transmission Lines

zzzzeIeIzIeVeVzV

γγγγ −−+−−+ +=+= 0000 )(;)(

−

−

+

+

−==0

0

0

00

I

V

I

VZ

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Finite Transmission Lines (2)Finite Transmission Lines (2)

0000

00

00

00

// ZeVZeV

eVeV

eIeI

eVeV

I

V

I

VZ

ll

ll

ll

ll

L

L

lz

L

γγ

γγ

γγ

γγ

−−+

−−+

−−+

−−+

=

−+

=

++

==

=

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Finite Transmission Lines (3)Finite Transmission Lines (3)

( ) ( )

( ) ( ) l

LLl

LL

l

LLl

LL

eZZI

eZIVV

eZZI

eZIVV

γγ

γγ

−−−

+

−=−=

+=+=

000

000

22

1

;22

1

( ) ( )[ ]

( ) ( )[ ])(

0

)(

0

0

)(

0

)(

0

2)(

;2

)(

zl

L

zl

LL

zl

L

zl

LL

eZZeZZZ

IzI

eZZeZZI

zV

−−−

−−−

−−+=

−++=

γγ

γγ

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Finite Transmission Lines (4)Finite Transmission Lines (4)

( ) ( )[ ]

( ) ( )[ ] zlzeZZeZZZ

IzI

eZZeZZI

zV

z

L

z

LL

z

L

z

LL

−=−−+=

−++=

−

−

';2

)'(

;2

)'(

'

0

'

0

0

'

0

'

0

γγ

γγ

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Finite Transmission Lines (5)Finite Transmission Lines (5)

'tanh

'tanh)'(

0

00

zZZ

zZZZzZ

L

L

γγ

++

=

lZZ

lZZZlzZZ

L

Li γ

γtanh

tanh)'(

0

00 +

+===

Input Impedance:

Matched load if ZL=Z0

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Reflection CoefficientReflection Coefficient

( ) ( )[ ]

( )

( ) [ ]'2'

0

'2

0

0'

0

'

0

'

0

12

12

2)'(

zz

LL

z

L

Lz

LL

z

L

z

LL

eeZZI

eZZ

ZZeZZ

I

eZZeZZI

zV

γγ

γγ

γγ

−

−

−

Γ++=

+−

++=

−++=

ΓΓ=+−

=Γ θj

L

L eZZ

ZZ||

0

0

||1

||1

||

||

min

max

Γ−Γ+

==V

VS

Reflection Coefficient Standing Wave Ratio (SWR)

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Reflection and TransmissionReflection and Transmission

Γ

1+ΓIncident

Reflected

Transmitted

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Special Cases to RememberSpecial Cases to Remember

1====++++∞∞∞∞−−−−∞∞∞∞

====Zo

Zoρρρρ

0====++++−−−−

====ZoZo

ZoZoρρρρ

10

0 −−−−====++++−−−−

====Zo

Zoρρρρ

Vs

ZsZo Zo

A: Terminated in Zo

Vs

ZsZo

B: Short Circuit

Vs

ZsZo

C: Open Circuit

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Waveguides Waveguides akaaka plumbingplumbing

• width is ~ wavelength

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WaveguidesWaveguides

• Uses a different transmission method

• “Ducting” not “conducting”

• >1GHz

• Expensive

• May need to be filled

• Cannot turn sharp corners

• Any defects will cause significant attenuation (sparking)

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Optical FiberOptical Fiber

Can be considered “circular waveguides”

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History of Fiber OpticsHistory of Fiber Optics

Total Internal reflection is the basic idea of fiber optic

John Tyndall demonstration in 1870

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History of Fiber opticsHistory of Fiber optics

• During 1930, other ideas were developed with this fiber optic such as transmitting images through a fiber.

• During the 1960s, Lasers were introduced as efficient light sources• In 1970s All glass fibers experienced excessive optical loss, the loss

of the light signal as it traveled the fiber limiting transmission distance.

• This motivated the scientists to develop glass fibers that include a separating glass coating. The innermost region was used to transmit the light, while the glass coating prevented the light from leaking out of the core by reflecting the light within the boundaries of the core.

• Today, you can find fiber optics used in variety of applications such as medical environment to the broadcasting industry. It is used to transmit voice, television, images and data signals through small flexible threads of glass or plastic.

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Optical fiber transmits light. But, what prevents the light froOptical fiber transmits light. But, what prevents the light from m

escaping from the fiber?escaping from the fiber?

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How Does fiber optic transmit How Does fiber optic transmit

light?light?

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Source and transmittersSource and transmitters

• A basic fiber optic communications system consists of three basic elements:

– Fiber media

– Light sources

– Light detector

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A Light SourcesA Light Sources

LED (Light emitting diode) ILD (injection laser diode)

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DetectorsDetectors

•Detector is the receiving end of a fiber optic link.

There are two kinds of Detectors

1. PIN (Positive Intrinsic Negative)

2. APD (Avalanche photo diodes)

PIN

APD

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The advantages of fiber optic over The advantages of fiber optic over

wire cablewire cable

• Thinner

• Higher carrying capacity

• Less signal degradation

• Light signal

• Low power

• Flexible

• Non-flammable

• Lightweight

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Disadvantage of fiber optic over Disadvantage of fiber optic over

copper wire cablecopper wire cable

• Optical fiber is more expensive per meter than copper

• Optical fiber can not be join together as easily as copper cable. It requires training and expensive splicing and measurement equipment.

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Fiber TechnologyFiber Technology

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Fiber TechnologyFiber Technology

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Total internal ReflectionTotal internal Reflection

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There are three types of fiber optic cable commonly usedThere are three types of fiber optic cable commonly used

Single Mode

Step-index Multimode fiber

Plastic optic fiber

Fiber media

Optical fibers are the actual media that guides the light

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Fiber TypesFiber Types

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Fiber TypesFiber Types

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The loss of fiber opticThe loss of fiber optic

• Material absorption

• Material Scattering

• Waveguide scattering

• Fiber bending

• Fiber coupling loss

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Fiber AttenuationFiber Attenuation

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Fiber BandwidthFiber Bandwidth

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Fiber BandwidthFiber Bandwidth

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Pulse Propagation through Pulse Propagation through

FibersFibers

Fundamentals of Photonics - Saleh and Teich

Response of a multi-mode fiber to a single short pulse

Broadening of a short pulse after transmission through different types of fibers

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Fiber Attenuation and Fiber Attenuation and

Chromatic DispersionChromatic Dispersion

0.1

0.2

0.3

0.4

0.5

0.6A

tte

nu

ati

on

(d

B/k

m)

1600 1700140013001200 15001100

Wavelength (nm)

Dispersion-unshiftedFiber

Dispersion-shifted Fiber

20

10

0

-10

-20 Dis

pe

rsio

n (

ps

/nm

×k

m)

EDFAband

TrueWaveFiber

Attenuation(all Fiber types)

+ Dispersion

Input Pulse Output Pulse

Slide Courtesy of Stan Lumish

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Four Wave Mixing (FWM)Four Wave Mixing (FWM)

λλλλ0

Dispersion-Shifted Fiber (25 km)

D ≈≈≈≈ 0 ps/nm-km

Wavelength (1 nm/division)

1546.55

1 nm2 nm 1.5 nm

Optical Launch Power = 3 dBm/channel

10 d

B/d

ivis

ion

TrueWave Fiber (50 km)

D ≈≈≈≈ 2.52.52.52.5 ps/nm-km

Wavelength (1 nm/division)

1546.55

1 nm2 nm 1.5 nm

Slide Courtesy of Stan Lumish