chapter 3 part 1.ppt

EM Waves and Guide Structures
ECEg 4291
Chapter 3 Part 1
Transmission Lines

Transmission Lines
Introduction
 So far we have seen wave propagation in unbounded media,
media of infinite extent.
 Such wave propagation is said to be unguided in that the
uniform plane wave exists throughout all space
 And EM energy associated with the wave spreads over a wide
area
 Wave propagation in unbounded media is used in radio or TV
broadcasting, where the information being transmitted is
meant for everyone who may be interested.
 Such means of wave propagation will not help in a situation
like telephone conversation, where the information is received
privately by one person.

Transmission Lines
Introduction
 Another means of transmitting power or information is by
guided structures.
 Guided structures serve to guide (or direct) the propagation
of energy from the source to the load.
 Typical examples of such structures are transmission lines
and waveguides.
 Transmission lines are commonly used in power
distribution at lower frequency and in communications at
higher frequency
 A transmission line basically consists of two or more
parallel conductors used to connect a source to a load.

Transmission Lines
Introduction
 The source may be a hydroelectric generator, a transmitter, or
an oscillator; the load may be a factory, an antenna, or an
oscilloscope, respectively
 Examples of transmission lines are shown in next slide
 Coaxial cables are routinely used in electrical laboratories and
in connecting TV sets to TV antennas.
 Micro strip lines are particularly important in integrated
circuits where metallic strips connecting electronic elements
are deposited on dielectric substrates.

Transmission Lines
Introduction
 Examples: (a) coaxial cable (b) two-wire line
(c) planar lines (d) wire above conducting plane
(e) microstrip

Transmission Lines
Transmission Line Parameters
Line Parameters:
1. R – resistance per unit length (series)
- opposition to current flow
2. L – inductance per unit length (series)
- self inductance
3. C – capacitance per unit length (shunt)
- two conductors separated by an insulator
4. G – conductance per unit length (shunt)
- due to dielectric medium separating the conductors

Transmission Lines
Line Parameters:
Distributed parameters of a two-conductor transmission line

Transmission Lines
Formulas for calculating the values of R, L, C, and G:

Transmission Lines
Dimensions: (a) coaxial line (b) two-wire line (c) planar line

Transmission Lines
Notes:
1. The line parameters are not discrete or lumped but
distributed. By this we mean that the parameters are
uniformly distributed along the entire length of the line.
2. For each line, the conductors are characterized by σc, μc,
εc=εo and the homogenous dielectric separating the
conductors is characterized by σ, μ, ε.
3. G ≠ 1/R. R is the ac resistance per unit length of the
conductors comprising the line and G is the conductance
per unit length due to the dielectric medium separating the
conductors.

Transmission Lines
Notes:
4. The value of L shown in the table is the external
inductance per unit length; that is L = Lext . The effect of
internal inductance Lin (= R/ω) are negligible as high
frequencies at which most communication system operate.
Self inductance (internal) – inductance measured with
the current flowing in the conductor
Mutual inductance (external) – flux linkage due to
nearby current carrying conductor to the conductor current
5. For each line,
LC = με and G/C = σ/ε

Transmission Lines
 Let us consider how an EM wave propagates through a two-
conductor transmission line.
 For example, consider the coaxial line connecting the
generator or source to the load as in Figure in the next slide.
 When switch S is closed, the inner conductor is made positive
with respect to the outer one so that the E field is radially
outward
 According to Ampere's law, the H field encircles the current
carrying conductor
 The Poynting vector (E X H) points along the transmission
line.

Transmission Lines
 Thus, closing the switch simply establishes a disturbance,
which appears as a transverse electromagnetic (TEM) wave
propagating along the line.
 This wave is a non-uniform plane wave and by means of it
power is transmitted through the line.

Transmission Lines
Transmission Line Equations
 As mentioned a two-conductor transmission line supports a
TEM wave
 An important property of TEM waves is that the fields E
and H are uniquely related to voltage V and current I,
respectively:
 Let us examine an incremental portion of length Δz of a
two-conductor transmission line.
 We intend to find an equivalent circuit for this line and
derive the line equations.

Transmission Lines
 From Figure in slide 7, we expect the equivalent circuit of
a portion of the line to be as in Figure in next slide
 The model in Figure in the next slide is in terms of the line
parameters R, L, G, and C, and may represent any of the
two-conductor lines of Figure in slide 7
 The model is called the L-type equivalent circuit
 In this model, we assume that the wave propagates along
the +z-direction, from the generator to the load.

Transmission Lines
L-type equivalent circuit model of a differential length Δz of a two-
conductor transmission line.

Transmission Lines
 By applying Kirchhoff's voltage law to the outer loop of
the circuit in Figure in previous slide, we obtain
 Taking the limit of eq. (11.3) as Δz -> 0 leads to
 Similarly, applying Kirchoff's current law to the main node
of the circuit in Figure in previous gives

Transmission Lines
As Δz —> 0, eq. (11.5) becomes
 If we assume harmonic time dependence so that

Transmission Lines
 where Vs(z) and Is(z) are the phasor forms of V(z, t) and
I(z, t), respectively, eqs. (11.4) and (11.6) become

Transmission Lines
 In the differential eqs. (11.8) and (11.9), Vs and Is are
coupled.
 To separate them, we take the second derivative of Vs in eq.
(11.8) and employ eq. (11.9) so that we obtain
 Where

Transmission Lines
 By taking the second derivative of Is in eq. (11.9) and
employing eq. (11.8), we get
 We notice that eqs. (11.10) and (11.12) are, respectively,
the wave equations for voltage and current
 The wavelength and wave velocity u are, respectively,
given by



f
u 




2


Transmission Lines
 The solutions of the linear homogeneous differential
equations (11.10) and (11.12) are
 Thus, we obtain the instantaneous expression for voltage
as

Transmission Lines
 The characteristic impedance Zo of the line is the ratio of
positively traveling voltage wave to current wave at any
point on the line.
 By substituting eqs. (11.15) and (11.16) into eqs. (11.8)
and (11.9) and equating coefficients of terms and ,
we obtain

Transmission Lines
 Ro should not be mistaken for R— while R is in ohms per
meter; Ro is in ohms.
 The propagation constant and the characteristic impedance Zo
are important properties of the line because they both depend
on the line parameters R, L, G, and C and the frequency of
operation.
 The reciprocal of Zo is the characteristic admittance Yo, that
is, Yo = 1/Zo.
 The transmission line considered so far is the lossy type in
that the conductors comprising the line are imperfect and the
dielectric in which the conductors are embedded is lossy

Transmission Lines
 We may now consider two special cases of lossless
transmission line and distortionless line.
Case for Lossless Transmission Line
 A transmission line is said to be lossless if the conductors
of the line are perfect (σc ≈ ∞) and the dielectric
medium separating them is lossless (σ ≈ 0).
 For such a line, it is evident that when:
σc ≈ ∞ and σ ≈ 0,
This is necessary for a line to be lossless.
G
R 
 0

Transmission Lines
Case for Lossless Transmission Line
Thus, for such a line,
,
0

 LC
j
j 

 




f
LC
u 


1
,
0

O
X
C
L
R
Z O
O 


Transmission Lines
Case for Distortionless Transmission Line
 A signal consists of band of frequencies; wave amplitudes
of different frequency components will be attenuated
differently in a lossy line as α is frequency dependent.
This results in distortion.
 A distortionless line is one in which the attenuation
constant α is frequency independent while the phase
constant β is linearly dependent on frequency.
 From the general expression for α and β, a distortion line
results if the line parameters are such that
C
G
L
R


Transmission Lines
Thus, for a distortionless line,
or
This shows that α does not depend on frequency
whereas β is a linear function of frequency















G
C
j
R
L
j
RG


 1
1



j
G
C
j
RG 








 1
,
RG

 LC

 

Transmission Lines
Also,
or
and
 
  O
O
O jX
R
C
L
G
R
G
C
j
G
R
L
j
R
Z 








1
1
,
C
L
G
R
RO 
 0

O
X



f
LC
u 


1

Transmission Lines
Summary

Transmission Lines
Note:
1. The phase velocity is independent of frequency because
the phase constant β linearly depends on frequency. We
have shape distortion of signals unless α and u are
independent of frequency.
2. u and ZO remain the same as for the lossless lines.
3. A lossless line is also distortionless line, but a
distortionless line is not necessarily lossless. Although
lossless lines are desirable in power transmission,
telephone lines are required to be distortionless.

Transmission Lines
Example 1:
An air line has characteristic impedance of 70 Ω and phase
constant of 3 rad/m at 100 MHz. Calculate the inductance per
meter and the capacitance per meter of the line. Note: air line
can be regarded as a lossless line.

Transmission Lines
Dividing eq. (11.1.1) by eq. (11.1.2) yields

Transmission Lines
From eq. (11.1.1),
Example 2: A distortionless line has Zo = 60 Ω, α = 20 mNp/m,
u = 0.6c (c = speed of light). Find R, L, G, C and λ at 100 MHz.
Hence,

Transmission Lines
 Transmission Line Equations
But
From eq. (11.2.2b),
Dividing eq. (11.2.1) by eq. (11.2.3) results in

Transmission Lines
From eq. (11.2.2a),
Multiplying eqs. (11.2.1) and (11.2.3) together gives

Transmission Lines
Input impedance, SWR and power
 Consider a transmission line of
length l, characterized by γ and
Zo, connected to a load ZL as
shown below.
 Looking into the line, the
generator sees the line with the
load as input impedance Zin.

Transmission Lines
 Let the transmission line extend from z = 0 at the generator
to z = l at the load.
 First of all, we need the voltage and current waves, that is
 To find V0
+ and V0
- , the terminal conditions must be given.
For example, if we are given the conditions at the input, say

Transmission Lines
 Substituting these into the wave and current waves
equations results in
 If the input impedance at the input terminals is Zin, the input
voltage V0 and the input current I0 are easily obtained from
the figure.

Transmission Lines
 On the other hand, if we are given the conditions at the load,
say
 Substituting these into the wave and current wave’s
equations gives

Transmission Lines
 Next, we determine the input impedance Zin = Vs(z)/Is(z) at
any point on the line.
 At the generator, for example, voltage and current wave’s
equations yield
 Substituting the equations of Vo
+ and Vo
- into Zin and
utilizing some useful identities and properties
(Lossy Line)

Transmission Lines
Useful Identities:

Transmission Lines
 For a lossless line, γ = jβ, tanh j βl = jtan βl , and Zo = Ro,
so Zin becomes
(Lossless line)
 Showing that the input impedance varies periodically with
distance from the load.
 The quantity βl is usually referred to as the electrical length
of the line and can be expressed in degrees or radians.

Transmission Lines
 The standing wave ratio s (otherwise denoted by SWR) as
 As mentioned at the beginning of this chapter, a transmission
is used in transferring power from the source to the load
 The average input power at a distance l from the load is given
by
 Since the last two terms are purely imaginary, we have

Transmission Lines
 Since the last two terms are purely imaginary, we have
 The first term is the incident power Pi while the second term is
the reflected power Pr. Thus eq. above may be written as
Pt = Pi – Pr
 where Pt is the input or transmitted power and the negative
sign is due to the negative going wave since we take the
reference direction as that of the voltage/current traveling
toward the right.

Transmission Lines
We now consider special cases when the line is connected to
the load , ZL = 0, ZL = ∞, ZL = Zo
A. Shorted Line (ZL = 0)
From the equation of input impedance of lossless line, at ZL
= 0, it becomes:
Also,
Г = -1, SWR = ∞

Transmission Lines
A. Shorted Line (ZL = 0) .....cont...
We notice that Zin is a pure reactance, which could be
capacitive or inductive depending on the value of l.
B. Open-Circuited Line (ZL = ∞)
In this case, Zin becomes
Also,
Г = 1, SWR = ∞

Transmission Lines
C. Matched Line (ZL = Zo)
 This is the most desired case from the practical point of
view. For this case, equation of Zin reduces to
Also,
Г = 0, SWR = 1
 The incident power is fully absorbed by the load.
 Thus, maximum power transfer is possible when a
transmission line is matched to the load.

Transmission Lines
Input Impedance of a lossless line when Shorted (ZL = 0)

Transmission Lines
Input Impedance of a lossless line when Shorted (ZL = ∞)

Transmission Lines
Example 1: A certain transmission line operating at ω = 106
rad/s has α = 8 dB/m, β = 1 rad/m, and ZO = 60 + j40 Ω and
is 2 m long. If the line is connected to a source of 10 /0o V,
Zg = 40 Ω and terminated by a load of 20 + j50 Ω,
determine: (a) Zin (b) the sending-end current (c) the
current at the middle of the line
Solution

Transmission Lines
Example 3: A 50-Ω coaxial cable feeds a 75 + j20-Ω dipole
antenna. Find Γ and s.
Solution

Transmission Lines
Example 4 :

Transmission Lines
Example 5: Refer to the lossless transmission line shown in
Figure in example 4 slide. (a) Find Γ and s. (b) Determine Zin at
the generator.

chapter 3 part 1.ppt

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