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) though
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
Transmission lines / waveguides Y
Transmission line examples
unguided wave propagation
guided wave propagation
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.
2.
3.
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
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, integrating 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 mode 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 Äz of the two-wire transmission
line may be represented by simple lumped-element equivalent circuit.
RN = series resistance per unit length (Ù/m) of the transmission line
conductors.
LN = series inductance per unit length (H/m) of the transmission line
conductors (internal plus external inductance).
GN = shunt conductance per unit length (S/m) of the media between
the transmission line conductors (insulator leakage current).
CN = 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+Äz by
writing KVL and KCL equations for the equivalent circuit.
KVL
KCL
Grouping the voltage and current terms and dividing by Äz gives
Taking the limit as Äz 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 jù
times the respective phasor which gives
Freq u en cy-d o ma in
(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 the electric and magnetic fields of 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 ã 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 (á)
is the attenuation constant while the imaginary part (â) is the phase
constant. The general equations for á and â 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 (ç) 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! ã z and e ã 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. These coefficients must be determined based on the
knowledge of voltages and currents at the transmission line connections.
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 (ó=4, R=0) while the ideal transmission
line insulating medium has infinite resistance (ó=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 RNCN > GNLN. In
order to satisfy the distortionless line requirement, series loading coils are
typically placed periodically along the line to increase LN.
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 ã 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.
Thus, the voltage coefficients in the transmission line equations can be
determined given the voltage and current at the load. Once these
coefficients are obtained, the voltage and current at any location on the line
can be determined using the transmission line equations.
Just as a plane wave is partially reflected at a media interface when
the intrinsic impedances on either side of the interface are dissimilar
(ç1 ç2), 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 Zo). The transmission line
reflection coefficient as a function of position along the line [Ã(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 ÃL = 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 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 Ã(z). The transmission line equations become
Thus, the transmission line equations can be written in terms of voltage
coefficients (Vo+,Vo! ) or in terms of one voltage coefficient and the reflection
coefficient (Vo+,Ã).
The input impedance at any point on the transmission line is given by
the ratio of voltage to current at that point. Taking a ratio of the phasor
voltage to the phasor current using the original form of the transmission
line equations gives
The voltage coefficients have been previously 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 [ã(l!z)] gives
Input impedance at any
point along a general
(lossy) transmission line
For a lossless line, ã=jâ 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, ÃL = 1)
Special Case #2 Short-circuited lossless line (*ZL*60, ÃL = !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.
The minimum and maximum values for the 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 à 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 ( ÃL =1)
Current null (Zin = 4) at the load and every ë/2 from that point.
Voltage null (Zin = 0) at ë/4 from the load and every ë/2 from that point.
Short-Circuited Transmission Line ( ÃL =!1)
Voltage null (Zin = 0) at the load and every ë/2 from that point.
Current null (Zin = 4) at ë/4 from the load and every ë/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 [Vg =100p0o V, Zg = Rg = 50 Ù, f = 100 MHz] is connected
to a lossless transmission line [L = 0.25ìH/m, C=100pF/m, l=10m]. For
loads of ZL = RL = 0, 25, 50, 100 and 4 Ù, 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.
Zin (0) = RL because the transmission line
in this example is an integer multiple of
half-wavelengths long (l = 5ë).
RL (Ù)
(a.)
ÃL
(b.)
s
(c.)
Zin (Ù)
0
!1
4
0
25
!1/3
2
25
50
0
1
50
100
1/3
2
100
4
1
4
4
Writing the general case transmission line equations in terms of a
lossless line (ã = jâ) gives
The voltage coefficient Vo+ must be determined in order to plot the
voltage and current along the transmission line. Since Vo+ is the
voltage coefficient of the forward wave, it is independent of the value
of the load impedance. For simplicity, we may determine Vo+ for the
matched case (RL = Zo = 50 Ù, ÃL = 0). The input impedance seen
looking into the input terminals of the transmission line, under
matched conditions, is 50 Ù. The equivalent circuit at the input to the
transmission line under matched conditions is shown below.
~
~
~
RL (Ù)
*V(z)*max (V)
*V(z)*min (V)
*V(l)* (V)
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
~
~
~
RL (Ù)
*Is(z)*max (A)
*Is(z)*min (A)
*Is (l)* (A)
0
2
0
2
25
1.33
0.67
1.33
50
1
1
1
100
1.33
0.67
0.67
4
2
0
0
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 Ã(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 phasor voltage at any point on the lossless transmission line is
The reflection coefficient at any point on the transmission line is defined
as
where ÃL is the reflection coefficient at the load (z = l) given by
(1)
where zL defines is the normalized load impedance. The magnitude of the
complex-valued reflection coefficient ranges from 0 to 1 for any value of
load impedance. Thus, the reflection coefficient (and corresponding
impedance) can always be mapped on the unit circle in the complex plane.
The corresponding standing wave
ratio s is
The magnitude of the reflection
coefficient 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 *ÃL* = 0
(no reflection - matched, s = 1)
Outer circle Y *ÃL* = 1
(total reflection, s = 4)
Once the position of ÃL is located on the Smith chart, the location of the
reflection coefficient as a function of position [Ã(z)] is determined using
the reflection coefficient formula.
This equation shows that to locate Ã(z), we start at ÃL and rotate through an
angle of èz =2â(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 growing
negative angle èz (clockwise rotation on the constant VSWR circle). Note
that if èz =!2ð, 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
•
ë/2=360o
ë=720o
•
*Ã* 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 ÃL
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
Ã(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. 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.
Example (Smith chart)
A 50Ù lossless transmission line of length 5.25ë is terminated by a
load impedance of (100 + j75) Ù. The line is driven by a source of 24V
(rms) with an impedance of 50Ù. Using the Smith chart, determine (a.) ÃL
(b.) s (c.) the transmission line input impedance and (b.) the distance in
wavelengths from the load to the first voltage minimum (dmin).
(1.) Locate the normalized load impedance zL and draw constant
VSWR circle through this point.
Draw a vertical line intersecting the leftmost point on the
VSWR circle downward through the scales located below the
Smith chart. The values of |ÃL| and s are found using these
scales (s - left side, top scale, |ÃL| - left side third scale from
top). The phase angle for ÃL is determined using the angle scale
(degrees) surrounding the Smith chart.
(2.) Move 5.25ë (10.5 revolutions) toward generator (CW) to find
normalized input impedance zin. Denormalize to find Zin.
(3.) Rotate from the load toward the generator (CW) on the VSWR
circle to the location of the voltage minimum (the leftmost point
on the VSWR circle) using either the wavelength or degree
scales. If degree scale is used, convert degrees to wavelengths.
Example (Smith chart)
A 60Ù lossless line has a maximum impedance Zin = (180 + j0) Ù at
a distance of ë/24 from the load. If the line is 0.3ë, 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 VSWR circle. From
this point, move ë/24 toward the load (CCW) to find zL.
(c.) From zL, move 0.3ë toward the generator (CW) to find zin.
Quarter Wave Transformer
When mismatches between the transmission line and load cannot be
avoided, there are matching techniques that may be employed to eliminate
reflections on the feeder transmission line. One such technique is the
quarter wave transformer.
If Zo ZL, *Ã*> 0
(mismatch)
Insert a ë/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 300Ù transmission line and a 75 Ù load, determine the
characteristic impedance of the quarter wave transformer necessary to
match the transmission line. Verify the input impedance using the Smith
chart.
Stub Tuner
A quarter-wave transformer is effective at matching a resistive load
RL to a transmission line of characteristic impedance Zo when RL 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) Ù to a 50 Ù transmission line.
From yL, move toward the generator (CW) to the first intersection
with the g = 1 circle (ytl = 1 + j0.75 at 69.8o) to find the distance d.
Shunt stub reactance must be ys = !j0.75 (!106.3o)
To find the stub length l, rotate from the short circuit admittance (ysc,
rightmost point on Smith chart, 0o) toward the generator (CW) to the
stub reactance (ys,!106.3o)
Power Flow on a Lossless Transmission Line
The phasor voltage and current on a lossless transmission line can be
written as
The time-average power at any location on the transmission line can be
expressed in terms of the voltage and current as
The first two terms in the equation above are real-valued while the last two
terms combine to yield a purely imaginary result. This yields a timeaverage power which is independent of position on the line.
The first term in the average power equation is the power associated with
the forward wave on the transmission line while the second term is the
power associated with the reflected wave.
Transients on Transmission Lines
The frequency domain solutions for the voltage and current along a
transmission line considered thus far are valid only for single frequency
time-harmonic signals under steady-state conditions. In many applications
containing digital or wideband signals, the transient response of the
transmission line is important.
As previously shown using the transmission line equivalent circuit,
the instantaneous voltage and current on a lossless transmission line
(RN=GN=0) must satisfy the following coupled time domain PDE’s.
These equations can be decoupled by differentiating one equation with
respect to time (t), differentiating the opposite equation with respect to
position (z), and combining the results. The resulting decoupled time
domain equations for the voltage and current are time domain wave
equations.
The general solutions to the time domain wave equations are
where the individual terms in the general solutions represent forward and
reverse traveling waves on the transmission line and u is the velocity of
propagation on the lossless transmission line given by
Inserting the general solutions for voltage and current into the original time
domain PDE’s gives the following relationships for the forward and reverse
waves.
Restating the voltage and current equations in terms of only voltage
coefficients gives
Consider the standard transmission line connection with a resistive
termination on the end of the lossless transmission line as shown below.
Let the instantaneous generator voltage be a step of amplitude Vo at t = 0.
A forward wave is launched down the transmission line at time t = 0,
traveling at a velocity u. Given only a forward wave on the transmission
line, the ratio of voltage to current at the transmission line input is equal to
Zo. Thus, the equivalent circuit seen by the
generator is a resistance of Zo. The voltage at
the transmission line input, by voltage
division, is
The voltage and current as a function of time and position, prior to the
arrival of the wavefront at the load, is then
At the load (z = l), if RL Zo, a reflected wave will be launched in the
reverse direction. The voltage and current at the load are
The ratio of the voltage to current at the load connection must equal the
load resistance RL, so that
Solving this equation for the reflection coefficient at the load gives
The reflected wave travels back toward the transmission line input. If the
generator impedance Rg is not equal to Zo, another wave is reflected in the
forward direction. The reflection coefficient at the generator is
This reflection process continues at both ends of the transmission line
[unless one or both ends of the transmission line is (are) matched]. Each
subsequent reflection is smaller than the previous such that the reflections
eventually become negligible.
If we denote the nth forward and reverse wave amplitudes as
the wave amplitudes in terms of the reflection coefficients are:
A convenient way of representing the transient voltage and current along
a transmission line as a function of time and position is the voltage bounce
diagram and current bounce diagram.
Example (transmission line transients)
Consider the lossless transmission line below with Zo = 50 Ù, Rg =
100 Ù, RL = 200 Ù, u = 108 m/s and l = 100 m. Assume the generator
waveform is a 12 V step voltage at t = 0. Draw (a.) the voltage bounce
diagram, (b.) the voltage waveform at z = 0 for (0 # t # 4 ìs), (c.) the
voltage waveform at z = l for (0 # t # 4 ìs).
(a.)
(b.)