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Chapter 4 Transmission Lines
General Considerations
• The family of transmission lines (TL) encompasses all
structures and media that serve to transfer energy or
information between two points:
- nerve fibers in the body for electrical waves,
- fluids for acoustic waves,
- solids for mechanical pressure waves
• Our study will be limited to TL used for guiding
electromagnetic signals:
- telephone wires,
- coaxial cables carrying audio and video information,
- optical fibers carrying light waves,
• A transmission line is a two-port network, with each
port consisting of two terminals.
Rg
A
B
+
Vg
-
Sending-end
port
A'
Generator circuit
Transmission line
Receiving-end
port
RL
B'
Load circuit
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The source is represented by an equivalent
generator circuit.
The load is represented by an equivalent
load resistance R, or impedance Z in the AC
(alternative current) case.
The role of Wavelength λ
• Consider a low-frequency electrical circuits. Wires are
used to connect the elements of the circuit.
A
+
i
B
+
+
R
VAA'
Vg
Transmission line
VBB'
-
C
-
-
A'
B'
l
This pair of wires between AA’ and BB’ is a TL.
The impact of this TL on the circuit depends on the length of
the line l and the frequency f of the signal from the generator.
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Example:
Consider the generator voltage Vg(t) = VAA' = Vo cos ωt
The voltage is propagating through the wire at the
speed of light, c=3x108 m/s, then the voltage VBB' will
have to be delayed in time;
VBB' (t) = VAA' ( t – l/c ) = VOcos [ω(t – l/c)]
Compare VBB' to VAA' at t=0 for a low-frequency
electronic circuit, f = 1 kHz and l = 5 cm
VAA' = VO and VBB' = VOcos (2π f l/c) = 0.9999999998VO
Thus, for all practical purposes, the length of the TL may
be ignored.
On the other hand, had the line been a 20-km long
telephone cable carrying a 1 kHz voice signal, then the
same calculation would have led to VBB' = 0.91VO
• The determining factor is the magnitude of ωl/c .
The velocity of propagation up = f λ = c in our case,
then
ωl/c = 2π l/λ
When l/λ is very small, transmission line effects may be
ignored, but when l/λ ≥ 0.01, we have to account for the
phase shift associated with the time-delay, and also for the
presence of reflected signals.
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Dispersive Transmission Line
• A dispersive transmission line is one on which the wave
velocity is not constant relative to the frequency f.
This means that a rectangular signal will be distorted through a
dispersive line:
Dispersionless line
Short dispersive line
Long dispersive line
The degree of distortion is proportional to the length of the
line.
Explanation: The Fourier analysis tells us that a periodic
wave is the sum of an infinity of sinusoidal waves.
Thus, every sinusoidal component of the rectangular
signal will propagate at different velocity, depending of
its frequency.
Preservation of pulse shape is very important in high-speed
data transmission.
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Propagation Modes
Transmission lines are classified into two basic types:
1)
Transverse electromagnetic (TEM) transmission
lines: The electric and magnetic fields of the waves
propagating on these lines are transverse to the
direction of propagation.
Example of the coaxial line;
metal
The electric field is in the radial
direction between the inner and outer
conductors, the magnetic field forms
circle around the inner conductor.
2b
2a
dielectric spacing
Other TEM transmission lines:
metal
w
2a
d
d
dielectric spacing
2)
High-order transmission lines: Waves propagating
along these lines have at least one field component
in the direction of propagation.
metal
Concentric
dielectric
layers
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Lumped-Element Model
• A transmission line is represented by a parallel-wire
symbol, regardless of its real shape.
(like the symbols used to represent resistors, capacitors…)
• To model the electrical properties of a transmission line,
which is just a pair of wire, we use an equivalent
electrical circuit for every differential length of the
circuit called lumped-element circuit.
Four basic elements are used:
- R':The combined resistance of both conductors per unit length, Ω/m.
- L':The combined inductance of both conductors per unit length, H/m.
- G':The combined conductance of both conductors per unit length, S/m.
- C':The combined capacitance of both conductors per unit length, F/m.
(a) Parallel-wire representation
z
z
z
z
(b) Differential sections each z long
R' z
L' z
G' z
z
R' z
C' z
L' z
G' z
z
R' z
C' z
L' z
G' z
R' z
C' z
z
L' z
G' z
C' z
z
(c) Each section is represented by an equivalent circuit
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• Expression of the parameters for a coaxial line:
It depends of the geometry of the line, and also of the
materials of the conductors and the insulating material
between them.
If
a = outer radius of inner conductor
b = inner radius of outer conductor
Then,
R' = (Rs/2π)(1/a + 1/b) with Rs = (π f µc /σc)1/2
L' = (µ/2π)ln(b/a)
G' = 2πσ/ln(b/a)
C' = 2πε/ln(b/a)
µc and σc characterize the conductor:
- µc
is the magnetic permeability
- σc
is the electrical conductivity
ε, µ, and σ characterize the insulation material:
- µ
is the magnetic permeability
- σ is the electrical conductivity
- ε is the electrical permittivity
Rs represent the surface resistance of the conductors.
For a perfect conductor, σc is infinite, then Rs approaches zero,
and so does R'.
G' accounts for current flow between the outer and inner
conductors, made possible by the material conductivity, σ.
A perfect dielectric is such that σ=0.
C' accounts for the voltage created by the charges of the two
conductors.
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Transmission-Line Equations
Before considering a complete circuit (Source + TL +
Load), we need to define the relations of the voltage and
current between both ends of the TL.
N i(z, t)
+
N+1
R' z
v(z , t)
i(z+ z, t)
+
L' z
G' z
C' z
-
v(z + z, t)
z
• Kirchhoff's voltage law in the outer loop:
v(z,t) – R'∆z i(z,t) – L'∆z ∂i(z,t)/∂t – v(z+∆z,t) = 0
Then, after rearranging terms
- [(v(z+∆z,t) – v(z,t)) / ∆z] = R' i(z,t) + L' ∂i(z,t)/∂t
At the limit ∆z Æ 0, we obtain a differential equation:
− ∂v(z,t)/∂z = R' i(z,t) + L' ∂i(z,t)/∂t
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• Kirchhoff's current law at the node N+1:
i(z,t) – G'∆z v(z+∆z,t) – C'∆z ∂v(z+∆z,t)/∂t – i(z+∆z,t) = 0
By taking the limit as ∆z Æ 0, it provides the second
differential equation,
− ∂i(z,t)/∂z = G' v(z,t) + C' ∂v(z,t)/∂t
• These two equations are the time-domain form of the
transmission line equations (telegrapher's equation).
− ∂v(z,t)/∂z = R' i(z,t) + L' ∂i(z,t)/∂t
− ∂i(z,t)/∂z = G' v(z,t) + C' ∂v(z,t)/∂t
• Phasors representation:
We define
v(z,t) = Re[ V(z) ejωt ]
i(z,t) = Re[ I(z) ejωt ]
After substitution in the two differential equations, we
have the phasor form of the telegrapher's equations:
-dV(z)/dz = (R'+jωL')I(z)
-dI(z)/dz = (G'+jωC')V(z)
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Wave Propagation on a Transmission Line
• These two first-order coupled equations can be
combined to give two second-order uncoupled wave
equations.
If you differentiate both sides of the first equation,
-d2V(z)/dz2 = (R'+jωL') dI(z)/dz
And substitute the value of dI(z)/dz,
d2V(z)/dz2 -
γ2V(z) = 0
Where γ = [(R'+jωL')(G'+jωC')]1/2 is the complex propagation
constant of the TL.
The real part α of γ is called the attenuation constant, and the
imaginary part β is called the phase constant.
In the same way we find,
d2I(z)/dz2 -
γ2I(z) = 0
The solutions of these wave equations are
V(z) = Vo+ e-γz + Vo- eγz
I(z) = Io+ e-γz + Io- eγz
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The term e-γz represents wave propagation in the +zdirection and eγz term represents propagation in the –zdirection.
The characteristic impedance of the line is
Z0 = (R'+jωL') / γ
( = Vo+ / Io+ = - Vo-/ Io-)
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The Lossless Transmission Line
• We have seen that a TL is characterized by two
fundamental properties, its propagation constant γ and
characteristic impedance Z0. They are specified by the
angular frequency ω and the line parameters R', L', G',
and C'.
• Usually a TL is designed to minimize ohmic losses by
selecting conductors with high conductivities and
dielectric materials with negligible conductivities.
That's why we assume R' = G' = 0 for a lossless line.
Therefore,
α=0
β = ω(L'C')1/2
Z0 = (L'/C')1/2
(Lossless line)
For all TEM line L’C’ = µε and G’/C’ = σ/ε
Then also have the following relations,
β = ω(µε)1/2
uP = 1/(µε)1/2
(rad/m)
(m/s)
(Lossless TEM line)
µ is the magnetic permeability
ε is the electric permittivity
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• Voltage Reflection Coefficient
For the lossless line γ = jβ, and the solutions of the wave
equations become,
V(z) = Vo+ e-jβz + Vo- ejβz
I(z) = (Vo+/Z0) e-jβz - (Vo-/Z0) ejβz
To determine the two unknowns, Vo+ and Vo- , the voltage
amplitudes of the incident and reflected waves, we need to
consider the complete circuit: Generator + TL + Load.
- For convenience, the reference of the z-axis is
chosen such that z = 0 corresponds to the location
of the load.
- The load impedance is ZL=VL / IL
- At the sending end (at z = - l), the line is connected
to a sinusoidal voltage source with phasor Vg and
internal impedance Zg.
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• The voltage VL is equal to the total voltage on the line
V(z) evaluated at z = 0, and IL is equal to I(z) evaluated
at z = 0.
VL = V(z=0) = Vo+ + VoIL = I(z=0) = (Vo+/Z0) - (Vo-/Z0)
Then using these equations, we obtain:
ZL =
Solving for Vo- gives
• The ratio of the amplitude of the reflected voltage wave to
the amplitude of the incident voltage wave at the load is
known as the voltage reflection coefficient Γ.
Γ = Vo- / Vo+ = - Io- / Io+ = (ZL – Z0) / (ZL + Z0)
For a lossless line Z0 is a real number. However, ZL is in
general a complex quantity: for a series RL circuit ZL = R
+ jωL. Hence, Γ may also be complex.
jθ
Γ = |Γ| e
• A load is said to be matched to the line if ZL=Z0
because then there will be no reflection by the load
(Γ = 0, and Vo- = 0).
On the other hand, when the load is an open circuit
(ZL=∞), Γ = 1, and Vo- = Vo+ .
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Example 1:
A 100 Ohm TL is connected to aload consisting of
a 50 Ohm resistor in series with a 10-pF capacitor.
Q1: Draw the figure for this circuit.
Q2: Find the reflection coefficient at the load for a
100-MHz signal.
Example 2:
Q: Show that |Γ| = 1 for a purely reactive load.
Answer: A purely reactive load means that ZL = jXL
(There is no resistor in the load)
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• Standing Waves
Using the relation Vo- = Γ Vo+ in the solutions of the
phasors equations
V(z) = Vo+ (e-jβz + Γ ejβz)
I(z) = (Vo+/Z0)(e-jβz - Γ ejβz)
which now contain only one unknown, Vo+ .
• Let us first examine the physical meaning represented
by these expressions.
For that we determine the magnitude of V(z):
|V(z)| = [V(z)V*(z)]1/2 = ?
Use the fact that Γ = |Γ| ejθ and ejx + e-jx = 2 cosx
The variation of |V(z)|
and |I(z)| as a function
of z are illustrated
besides:
~
|V(z)|
~
|V |max
1.4 V
1.2
1.0
0.8
0.6
0.4
0.2
~
|V |min
-
lmin -3
4
4
2
~
(a) |V(z)| versus z
z
lmax 0
~
|I(z)|
30 mA
25
20
15
10
5
~
|I |max
~
|I |min
-
-3
2
4
4
~
(b) |I(z)| versus z
z
0
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• Distance where the voltage magnitude is maximum.
If we denote lmax = -z this distance, |V(z)| is maximum
when the cosine is equal to one, therefore when
2βz + θr = -2βlmax + θr = -2nπ
with n=0, -1, 1, -2, 2, -3, 3,…
Solving for lmax , we have
lmax = - (θr + 2nπ)/2β = - (θrλ/4π) + (nλ/2)
If θr ≥ 0, the first voltage maximum occurs at
lmax= -(θrλ/4π) corresponding to n=0.
• Distance where the voltage magnitude is minimum.
Similarly, if lmin = -z, |V(z)| is minimum when the cosine
is equal to -1, therefore when
2βz + θr = -2βlmin + θr = -(2n+1)π
The voltage standing-wave ratio S, is the ratio of |V|max to
|V|min ,
S = (1 + |Γ|) / (1 - |Γ|)
This quantity (VSWR) provide a measure of the mismatch
between the load and the TL; for a matched load with
Γ=0, we get S=1, and for a line with |Γ|=1, S=∞ .
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Example 3:
A 50 Ohm TL
ZL=(100+j50).
is
terminated
in
a
load
with
Find the voltage standing-wave ratio (SWR).
Example 4:
A slotted-line is an instrument used to measure the
unknown impedance of a load. It's a special TL that
allows us to measure |V|max and |V|min , and the
distances from the load at which they occur.
The slotted-line impedance is 50 Ohm, and the SWR
has been found equal to 3.
The first voltage minimum is located at 12 cm, and
the distance between successive minima is 30 cm.
Q: Determine the load impedance ZL.
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Input Impedance of the Lossless Line
• The input impedance Zin of the TL is given by
Zin(z)= V(z)/I(z) = Vo+(e-jβz + Γ ejβz)/[(Vo+/Z0)(e-jβz - Γ ejβz)]
Zin(z)= Z0(1+ Γ ej2βz)/(1- Γ ej2βz)
Zin is the ratio of the total voltage to the total current at
any point z on the line, in contrast with the characteristic
impedance Z0 which relates the voltage and current of the
incident and reflected waves individually.
The input impedance at the input of the line (at z = -l) is
Zin (-l) = Z0(1+ Γ e-j2βl )/(1- Γ e-j2βl )
• From the standpoint of the generator circuit, the TL can
be replaced with an impedance Zin(-l), and then,
Vi = Ii Zin(-l) = Vg Zin(-l) / (Zg + Zin(-l))
but from the standpoint of the TL, the voltage Vi is also
equal to
Vi = V(-l) = Vo+(ejβl + Γ e-jβl )
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Then solving these two equation for Vo+ leads to the
result
Vo+ = [Vg Zin(-l) / (Zg + Zin(-l))] [1 / (ejβl + Γ e-jβl )]
This completes the solution of the TL wave equations for
the special case of the lossless TL.
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