Transmission Lines
Transmission lines and waveguides may be defined as devices used
to guide energy from one point to another (from a source to a load).
Transmission lines can consist of a set of conductors, dielectrics or
combination thereof. As we have shown using Maxwell’s equations, we
can transmit energy in the form of an unguided wave (plane wave) through
space. In a similar manner, Maxwell’s equations show that we can
transmit energy in the form of a guided wave on a transmission line.
Plane wave propagating in air
Y
unguided wave propagation
Transmission lines / waveguides
Y
guided wave propagation
Transmission line examples
Transmission Line Definitions
Almost all transmission lines have a cross-sectional geometry which
is constant in the direction of wave propagation along the line. This type
of transmission line is called a uniform transmission line.
Uniform transmission line - conductors and dielectrics maintain the
same cross-sectional geometry along the transmission line in
the direction of wave propagation.
Given a particular conductor geometry for a transmission line, only
certain patterns of electric and magnetic fields (modes) can exist for
propagating waves. These modes must be solutions to the governing
differential equation (wave equation) while satisfying the appropriate
boundary conditions for the fields.
Transmission line mode - a distinct pattern of electric and magnetic
field induced on a transmission line under source excitation.
The propagating modes along the transmission line or waveguide may be
classified according to which field components are present or not present
in the wave. The field components in the direction of wave propagation
are defined as longitudinal components while those perpendicular to the
direction of propagation are defined as transverse components.
Transmission Line Mode Classifications
Assuming the transmission line is oriented with its axis along the zaxis (direction of wave propagation), the modes may be classified as
1.
Transverse electromagnetic (TEM) modes - the electric and
magnetic fields are transverse to the direction of wave
propagation with no longitudinal components [Ez = Hz = 0].
TEM modes cannot exist on single conductor guiding
2.
3.
structures. TEM modes are sometimes called transmission line
modes since they are the dominant modes on transmission
lines. Plane waves can also be classified as TEM modes.
Quasi-TEM modes - modes which approximate true TEM
modes for sufficiently low frequencies.
Waveguide modes - either Ez, Hz or both are non-zero.
Waveguide modes propagate only above certain cutoff
frequencies. Waveguide modes are generally undesirable on
transmission lines such that we normally operate transmission
lines at frequencies below the cutoff frequency of the lowest
waveguide mode. In contrast to waveguide modes, TEM
modes have a cutoff frequency of zero.
Transmission Line Equations
Consider the electric and magnetic fields associated with the TEM
mode on an arbitrary two-conductor transmission line (assume perfect
conductors). According to the definition of the TEM mode, there are no
longitudinal fields associated with the guided wave traveling down the
transmission line in the z direction (Ez = Hz = 0).
From the integral form of Maxwell’s equations, performing the line integral
of the electric and magnetic fields around any transverse contour C1 gives
0
0
where ds = dsaz. The surface integrals of E and H are zero-valued since
there is no Ez or Hz inside or outside the conductors (PEC’s, TEM mode).
The line integrals of E and H for the TEM mode on the transmission line
reduce to
These equations show that the transverse field distributions of the TEM
mode on a transmission line are identical to the corresponding static
distributions of fields. That is, the electric field of a TEM at any frequency
has the same distribution as the electrostatic field of the capacitor formed
by the two conductors charged to a DC voltage V and the TEM magnetic
field at any frequency has the same distribution as the magnetostatic field
of the two conductors carrying a DC current I. If we change the contour
C1 in the electric field line integral to a new path C2 from the “!”
conductor to the “+” conductor, then we find
These equations show that we may define a unique voltage and current at
any point on a transmission line operating in the TEM mode. If a unique
voltage and current can be defined at any point on the transmission line,
then we may use circuit equations to describe its operation (as opposed to
writing field equations).
Transmission lines are typically electrically long (several
wavelengths) such that we cannot accurately describe the voltages and
currents along the transmission line using a simple lumped-element
equivalent circuit. We must use a distributed-element equivalent circuit
which describes each short segment of the transmission line by a lumpedelement equivalent circuit.
Consider a simple uniform two-wire transmission line with its
conductors parallel to the z-axis as shown below.
Uniform transmission line - conductors and insulating medium
maintain the same cross-sectional geometry along the entire
transmission line.
The equivalent circuit of a short segment )z of the two-wire transmission
line may be represented by simple lumped-element equivalent circuit.
R = series resistance per unit length (S/m) of the transmission line
conductors.
L = series inductance per unit length (H/m) of the transmission line
conductors (internal plus external inductance).
G = shunt conductance per unit length (S/m) of the media between
the transmission line conductors (insulator leakage current).
C = shunt capacitance per unit length (F/m) of the transmission line
conductors.
We may relate the values of voltage and current at z and z+)z by
writing KVL and KCL equations for the equivalent circuit.
KVL
KCL
Grouping the voltage and current terms and dividing by )z gives
Taking the limit as )z 6 0, the terms on the right hand side of the
equations above become partial derivatives with respect to z which gives
Time-domain
transmission line
equations
(coupled PDE’s)
For time-harmonic signals, the instantaneous voltage and current may be
defined in terms of phasors such that
The derivatives of the voltage and current with respect to time yield jT
times the respective phasor which gives
Freq u en c y-domain
(phasor) transmission
line equations
(coupled DE’s)
Note the similarity in the functional form of the time-domain and the
frequency-domain transmission line equations to the respective source-free
Maxwell’s equations (curl equations). Even though these equations were
derived without any consideration of the electromagnetic fields associated
with the transmission line, remember that circuit theory is based on
Maxwell’s equations.
Just as we manipulated the two Maxwell curl equations to derive the
wave equations describing E and H associated with an unguided wave
(plane wave), we can do the same for a guided (transmission line TEM)
wave. Beginning with the phasor transmission line equations, we take
derivatives of both sides with respect to z.
We then insert the first derivatives of the voltage and current found in the
original phasor transmission line equations.
The phasor voltage and current wave equations may be written as
Voltage and
current wave
equations
where ( is the complex propagation constant of the wave on the
transmission line given by
Just as with unguided waves, the real part of the propagation constant (")
is the attenuation constant while the imaginary part ($) is the phase
constant. The general equations for " and $ in terms of the per-unit-length
transmission line parameters are
The general solutions to the voltage and current wave equations are
~~~~~
_
+z-directed waves
~~~~~
a
!z-directed waves
The coefficients in the solutions for the transmission line voltage and
current are complex constants (phasors) which can be defined as
The instantaneous voltage and current as a function of position along the
transmission line are
Given the transmission line propagation constant, the wavelength and
velocity of propagation are found using the same equations as for
unbounded waves.
The region through which a plane wave (unguided wave) travels is
characterized by the intrinsic impedance (0 ) of the medium defined by the
ratio of the electric field to the magnetic field. The guiding structure over
which the transmission line wave (guided wave) travels is characterized the
characteristic impedance (Zo ) of the transmission line defined by the ratio
of voltage to current.
If the voltage and current wave equations defined by
are inserted into the phasor transmission line equations given by
the following equations are obtained.
Equating the coefficients on e! ( z and e ( z gives
The ratio of voltage to current for the forward and reverse traveling waves
defines the characteristic impedance of the transmission line.
The transmission line characteristic impedance is, in general, complex and
can be defined by
The voltage and current wave equations can be written in terms of the
voltage coefficients and the characteristic impedance (rather than the
voltage and current coefficients) using the relationships
The voltage and current equations become
These equations have unknown coefficients for the forward and reverse
voltage waves only since the characteristic impedance of the transmission
line is typically known.
Special Case #1 Lossless Transmission Line
A lossless transmission line is defined by perfect conductors and a
perfect insulator between the conductors. Thus, the ideal transmission line
conductors have zero resistance (F = 4, R=0) while the ideal transmission
line insulating medium has infinite resistance (F =0, G=0). The
equivalent circuit for a segment of lossless transmission line reduces to
The propagation constant on the lossless transmission line reduces to
Given the purely imaginary propagation constant, the transmission line
equations for the lossless line are
The characteristic impedance of the lossless transmission line is purely real
and given by
The velocity of propagation and wavelength on the lossless line are
Transmission lines are designed with conductors of high conductivity and
insulators of low conductivity in order to minimize losses. The lossless
transmission line model is an accurate representation of an actual
transmission line under most conditions.
Special Case #2 Distortionless Transmission Line
On a lossless transmission line, the propagation constant is purely
imaginary and given by
The phase velocity on the lossless line is
Note that the phase velocity is a constant (independent of frequency) so
that all frequencies propagate along the lossless transmission line at the
same velocity. Many applications involving transmission lines require that
a band of frequencies be transmitted (modulation, digital signals, etc.) as
opposed to a single frequency. From Fourier theory, we know that any
time-domain signal may be represented as a weighted sum of sinusoids.
A single rectangular pulse contains energy over a band of frequencies. For
the pulse to be transmitted down the transmission line without distortion,
all of the frequency components must propagate at the same velocity. This
is the case on a lossless transmission line since the velocity of propagation
is a constant. The velocity of propagation on the typical non-ideal
transmission line is a function of frequency so that signals are distorted as
different components of the signal arrive at the load at different times.
This effect is called dispersion. Dispersion is also encountered when an
unguided wave propagates in a non-ideal medium. A plane wave pulse
propagating in a dispersive medium will suffer distortion. A dispersive
medium is characterized by a phase velocity which is a function of
frequency.
For a low-loss transmission line, on which the velocity of propagation
is near constant, dispersion may or may not be a problem, depending on
the length of the line. The small variations in the velocity of propagation
on a low-loss line may produce significant distortion if the line is very
long. There is a special case of lossy line with the linear phase constant
that produces a distortionless line.
A transmission line can be made distortionless (linear phase constant) by
designing the line such that the per-unit-length parameters satisfy
Inserting the per-unit-length parameter relationship into the general
equation for the propagation constant on a lossy line gives
Although the shape of the signal is not distorted, the signal will suffer
attenuation as the wave propagates along the line since the distortionless
line is a lossy transmission line. Note that the attenuation constant for a
distortionless transmission line is independent of frequency. If this were
not true, the signal would suffer distortion due to different frequencies
being attenuated by different amounts.
In the previous derivation, we have assumed that the per-unit-length
parameters of the transmission line are independent of frequency. This is
also an approximation that depends on the spectral content of the
propagating signal. For very wideband signals, the attenuation and phase
constants will, in general, both be functions of frequency.
For most practical transmission lines, we find that RC > GL. In order
to satisfy the distortionless line requirement, series loading coils are
typically placed periodically along the line to increase L.
Transmission Line Circuit
(Generator/Transmission line/Load)
The most commonly encountered transmission line configuration is
the simple connection of a source (or generator) to a load (ZL ) through the
transmission line. The generator is characterized by a voltage Vg and
impedance Zg while the transmission line is characterized by a propagation
constant ( and characteristic impedance Zo.
The general transmission line equations for the voltage and current as a
function of position along the line are
The voltage and current at the load (z = l ) is
We may solve the two equations above for the voltage coefficients in terms
of the current and voltage at the load.
Just as a plane wave is partially reflected at a media interface when the
intrinsic impedances on either side of the interface are dissimilar (01 ú 02),
the guided wave on a transmission line is partially reflected at the load
when the load impedance is different from the characteristic impedance of
the transmission line (ZL ú Zo). The transmission line reflection coefficient
as a function of position along the line [ '(z)] is defined as the ratio of the
reflected wave voltage to the transmitted wave voltage.
Inserting the expressions for the voltage coefficients in terms of the load
voltage and current gives
The reflection coefficient at the load (z=l) is
and the reflection coefficient as a function of position can be written as
Note that the ideal case is to have 'L = 0 (no reflections from the load
means that all of the energy associated with the forward traveling wave is
delivered to the load). The reflection coefficient at the load is zero when
ZL = Zo. If ZL = Zo, the transmission line is said to be matched to the load
and no reflected waves are present. If ZL ú Zo, a mismatch exists and
reflected waves are present on the transmission line. Just as plane waves
reflected from a dielectric interface produce standing waves in the region
containing the incident and reflected waves, guided waves on a
transmission line reflected from the load produce standing waves on the
transmission line (the sum of forward and reverse traveling waves).
The transmission line equations can be expressed in terms of the
reflection coefficients as
The last term on the right hand side of the above equations is the reflection
coefficient '(z). The transmission line equations become
Thus, the transmission line equations can be written in terms of voltage
coefficients (Vso+,Vso! ) or in terms of one voltage coefficient and the
reflection coefficient (Vso+, ' ).
The input impedance at any point on the transmission line is given
by the ratio of voltage to current at that point. Inserting the expressions for
the phasor voltage and current [Vs (z) and Is (z)] from the original form of
the transmission line equations gives
The voltage coefficients have been determined in terms of the load voltage
and current as
Inserting these equations for the voltage coefficients into the impedance
equation gives
We may use the following hyperbolic function identities to simplify this
equation:
Dividing the numerator and denominator by cosh [((l!z)] gives
Input impedance at any
point along a general
transmission line
For a lossless line, ( =j$ and Zo is purely real. The hyperbolic tangent
function reduces to
which yields
Input impedance at any
point along a lossless
transmission line
Special Case #1 Open-circuited lossless line (*ZL*64 , 'L = 1)
Special Case #2 Short-circuited lossless line (*ZL*60, 'L = !1)
The impedance characteristics of a short-circuited or open-circuited
transmission line are related to the positions of the voltage and current
nulls along the transmission line. On a lossless transmission line, the
magnitude of the voltage and current are given by
The equations for the voltage and current magnitude follow the crank
diagram form that was encountered for the plane wave reflection example
and the minimum and maximum values for voltage and current are
For a lossless line, the magnitude of the reflection coefficient is constant
along the entire line and thus equal to the magnitude of ' at the load.
The standing wave ratio (s) on the lossless line is defined as the ratio of
maximum to minimum voltage magnitudes (or maximum to minimum
current magnitudes).
The standing wave ratio on a lossless transmission line ranges between 1
and 4.
We may apply the previous equations to the special cases of opencircuited and short-circuited lossless transmission lines to determine the
positions of the voltage and current nulls.
Open-Circuited Transmission Line ( 'L =1)
Current null (Zin = 4 ) at the load and every 8/2 from that point.
Voltage null (Zin = 0) at 8/4 from the load and every 8/2 from that point.
Short-Circuited Transmission Line ( 'L = !1)
Voltage null (Zin = 0) at the load and every 8/2 from that point.
Current null (Zin = 4 ) at 8/4 from the load and every 8/2 from that point.
From the equations for the maximum and minimum transmission line
voltage and current, we also find
It will be shown that the voltage maximum occurs at the same location as
the current minimum on a lossless transmission line and vice versa. Using
the definition of the standing wave ratio, the maximum and minimum
impedance values along the lossless transmission line may be written as
Thus, the impedance along the lossless transmission line must lie within
the range of Zo /s to sZo.
Example
A source [Vsg =100 p0o V, Zg = Rg = 50 S, f = 100 MHz] is connected
to a lossless transmission line [L = 0.25 :H/m, C=100pF/m, l=10m]. For
loads of ZL = RL = 0, 25, 50, 100 and 4 S, determine (a.) the reflection
coefficient at the load (b.) the standing wave ratio (c.) the input impedance
at the transmission line input terminals (d.) voltage and current plots along
the transmission line.
RL (S)
(a.) Zin (S)
(b.) 'L
(c.) s
0
0
25
25
!0.333
2
50
50
0
1
100
100
+0.333
2
4
4
+1
4
!1
4
RL (S)
*V (z)*
s
max
(V)
*V (z)*
s
min
(V)
*V (l)* (V)
s
0
100
0
0
25
66.7
33.3
33.3
50
50
50
50
100
66.7
33.3
66.7
4
100
0
100
Smith Chart
The Smith chart is a useful graphical tool used to calculate the reflection
coefficient and impedance at various points on a (lossless) transmission line
system. The Smith chart is actually a polar plot of the complex reflection
coefficient (z) [ratio of the reflected wave voltage to the forward wave
voltage] overlaid with the corresponding impedance Z(z) [ratio of overall
voltage to overall current].
The corresponding standing wave
ratio s is
The magnitude of 'L is constant
on any circle in the complex
plane so that the standing wave
ratio (VSWR) is also constant on
the same circle.
Smith chart center Y *'L* = 0
(no reflection - matched, s = 1)
Outer circle Y *'L* = 1
(total reflection, s = 4)
Constant VSWR
circle
Once the position of 'L is located on the Smith chart, the location of the
reflection coefficient as a function of position [ '(z)] is determined using
the reflection coefficient formula.
This equation shows that to locate '(z), we start at 'L and rotate through
an angle of 2z =2$(z !l) on the constant VSWR circle. With the load
located at z=l, moving from the load toward the generator (z < l) defines
a negative angle 2z (counterclockwise rotation on the constant VSWR
circle). Note that if 2z = !2B, we rotate back to the same point. The
distance traveled along the transmission line is then
Thus, one complete rotation around the Smith chart (360o) is equal to one
half wavelength.
On the Smith chart ...
•
CW rotation
Y
toward the generator
•
CCW rotation
Y
toward the load
•
8 /2=360o
8 =720o
•
*'* and s are constant on a lossless transmission line. Moving
from point to point on a lossless transmission line is equivalent
to rotation along the constant VSWR circle.
•
All impedances on the Smith chart are normalized to the
characteristic impedance of the transmission line (when using
a normalized Smith chart).
The points along the constant VSWR circle represent the complex
reflection coefficient at points along the transmission line. The reflection
coefficient at any given point on the transmission line corresponds directly
to the impedance at that point. To determine this relationship between 'L
and ZL, we first solve (1) for zL.
(2)
where rL and xL are the normalized load resistance and reactance,
respectively. Solving (2) for the resistance and reactance gives equations
for the “resistance” and “reactance” circles.
In a similar fashion, the reflection coefficient as a function of
position '(z) along the transmission line can be related to the impedance
as a function of position Z(z). The general impedance at any point along
the length of the transmission line is defined by the ratio of the phasor
voltage to the phasor current.
The normalized value of the impedance zn(z) is
(3)
Note that Equation (2) is simply Equation (3) evaluated at z =l.
(2)
(3)
Thus, as we move from point to point along the transmission line plotting
the complex reflection coefficient (rotating around the constant VSWR
circle), we are also plotting the corresponding impedance.
Once a normalized impedance is located on the Smith chart for a
particular point on the transmission line, the normalized admittance at that
point is found by rotating 180o from the impedance point on the constant
reflection coefficient circle.
The locations of maxima and minima for voltages and currents along
the transmission line can be located using the Smith chart given that these
values correspond to specific impedance characteristics.
Voltage maximum, Current minimum
Y
Impedance maximum
Voltage minimum, Current maximum
Y
Impedance minimum
Example (Smith chart)
A 75 S lossless transmission line of length 1.25 8 is terminated by a
load impedance of 120 S. The line is energized by a source of 100V (rms)
with an internal impedance of 50 S. Determine (a.) the transmission line
input impedance and (b.) the magnitude of the load voltage.
Using the Smith chart software,
(1.) Define normalization factor for the Smith chart (Zo =75 S).
(2.) Define load impedance (ZL =120 S).
(3.) Insert series t-line (Zo =75 S).
(4.) Move 1.25 8 (2.5 revolutions) on the constant VSWR circle
to find Zin.
(a.)
DP#1 (120.0 + j0.0) S
DP#2 (46.9 + j0.0) S
Zin = (46.9 + j0) S
(b.)
Using a pencil and paper Smith chart,
(1.) Find the normalized load impedance.
(2.) Move 1.25 8 (2.5 revolutions) toward generator (CW) to find
normalized input impedance.
(3.) De-normalize the input impedance.
Example (Smith chart)
A 60 S lossless line has a maximum impedance Zin = (180 + j0) S at
a distance of 8/24 from the load. If the line is 0.38, determine (a.) s (b.)
ZL and (c.) the transmission line input impedance.
(a.)
(b.) [Zin]max occurs at the rightmost point on the s=3 circle. From this
point, move 8/24 toward the load (CCW) to find ZL.
(c.) Insert series transmission line section and from ZL, move 0.3 8 toward
the generator (CW) to find Zin.
DP#1 (117.2 + j78.1) S
DP#2 (20.0 + j2.8) S
Quarter Wave Transformer
When mismatches between the transmission line and load cannot be
avoided, there are matching techniques that we may use to eliminate
reflections on the feeder transmission line. One such technique is the
quarter wave transformer.
If Zo ú ZL, *'*> 0
(mismatch)
Insert a 8/4 length section of different transmission line (characteristic
impedance = ZNo ) between the original transmission line and the load.
The input impedance seen looking into the quarter wave transformer is
Solving for the required characteristic impedance of the quarter wave
transformer yields
Example (Quarter wave transformer)
Given a 300 S transmission line and a 75 S load, determine the
characteristic impedance of the quarter wave transformer necessary to
match the transmission line. Verify the input impedance using the Smith
chart.
DP#1 (75.0 + j0.0) S
DP#2 (300.0 - j0.0) S
Stub Tuner
A quarter-wave transformer is effective at matching a resistive load
RL to a transmission line of characteristic impedance Zo when RL ú Zo.
However, a complex load impedance cannot be matched by a quarter wave
transformer. The stub tuner is a transmission line matching technique that
can used to match a complex load. If a point can be located on the
transmission line where the real part of the input admittance is equal to the
characteristic admittance (Yo =Zo!1) of the line (Yin =Yo ±jB ), the
susceptance B can be eliminated by adding the proper reactive component
in parallel at this point. Theoretically, we could add inductors or
capacitors (lumped elements) in parallel with the transmission line.
However, these lumped elements usually are too lossy at the frequencies
of interest.
Rather than using lumped elements, we can use a short-circuited or
open-circuited segment of transmission line to achieve any required
reactance. Because we are using parallel components, the use of
admittances (as opposed to impedances) simplifies the mathematics.
l ! length of the shunt stub
d !distance from the load to
the stub connection
Ys ! input admittance of the stub
Ytl ! input admittance of the terminated
transmission line segment of length d
Yin ! input admittance of the stub in parallel
with the transmission line segment
Ytl = Yo + jB
[Admittance of the terminated t-line section]
Ys = !jB
[Admittance of the stub (short or open circuit)]
Yin = Ytl + Ys = Yo
[Overall input admittance]
[Transmission line characteristic admittance]
In terms of normalized admittances (divide by Yo), we have
ytl = 1 + jb
ys = !jb
yin = 1
Note that the normalized conductance of the transmission line segment
admittance (ytl = g + jb) is unity (g = 1).
Single Stub Tuner Design Using the Smith Chart
1.
2.
3.
Locate the normalized load impedance zL (rotate 180o to find
yL). Draw the constant VSWR circle [Note that all points on
the Smith chart now represent admittances].
From yL, rotate toward the generator (CW) on the constant
VSWR circle until it intersects the g = 1. The rotation distance
is the distance d while the admittance at this intersection point
is ytl = 1 + jb.
Beginning at the stub end (short circuit admittance is the
rightmost point on the Smith chart, open circuit admittance is
the leftmost point on the Smith chart), rotate toward the
generator (CW) until the point at ys = !jb is reached. This
rotation distance is the stub length l.
Short circuited stub tuners are most commonly used because a
shorted segment of transmission line radiates less than an open-circuited
section. The stub tuner matching technique also works for tuners in series
with the transmission line. However, series tuners are more difficult to
connect since the transmission line conductors must be physically
separated in order to make the series connection.
Example
Design a short-circuited shunt stub tuner to match a load of
ZL = (60 ! j40) S to a 50S transmission line.
Short circuit admittance - rightmost point on Smith chart (0o)
Shunt stub reactance