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1
Theory and Applications ofTheory and Applications of
Transmission LinesTransmission Lines
2
TopicsTopics
• Introduction
• Types
• General Transmission-Line Equations
• Wave Characteristics on Finite Transmission Lines
• Waveguides
• Optical Fiber
3
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
4
Types classified by materialsTypes classified by materials
• Metallic Transmission Lines (Conductor)
• Hollow or Dielectric-filled Waveguides (Conductor and dielectric)
• Optical Fiber (dielectric)
5
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
12
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.
22
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
23
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
25
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)
26
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)
27
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
28
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
32
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
33
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)
34
Reflection and TransmissionReflection and Transmission
Γ
1+ΓIncident
Reflected
Transmitted
35
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
36
Waveguides Waveguides akaaka plumbingplumbing
• width is ~ wavelength
37
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)
38
Optical FiberOptical Fiber
Can be considered “circular waveguides”
39
History of Fiber OpticsHistory of Fiber Optics
Total Internal reflection is the basic idea of fiber optic
John Tyndall demonstration in 1870
40
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.
41
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?
42
How Does fiber optic transmit How Does fiber optic transmit
light?light?
43
Source and transmittersSource and transmitters
• A basic fiber optic communications system consists of three basic elements:
– Fiber media
– Light sources
– Light detector
44
A Light SourcesA Light Sources
LED (Light emitting diode) ILD (injection laser diode)
45
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
46
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
47
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.
48
Fiber TechnologyFiber Technology
49
Fiber TechnologyFiber Technology
50
Total internal ReflectionTotal internal Reflection
51
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
52
Fiber TypesFiber Types
53
Fiber TypesFiber Types
54
The loss of fiber opticThe loss of fiber optic
• Material absorption
• Material Scattering
• Waveguide scattering
• Fiber bending
• Fiber coupling loss
55
Fiber AttenuationFiber Attenuation
56
Fiber BandwidthFiber Bandwidth
57
Fiber BandwidthFiber Bandwidth
58
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
59
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
60
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
