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 unguided wave propagation Transmission lines / waveguides guided wave propagation Transmission line examples
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Transmission Lines
Transmission lines and waveguides may be defined as devices usedto guide energy from one point to another (from a source to a load).Transmission lines can consist of a set of conductors, dielectrics orcombination thereof. As we have shown using Maxwell’s equations, wecan transmit energy in the form of an unguided wave (plane wave) throughspace. In a similar manner, Maxwell’s equations show that we cantransmit energy in the form of a guided wave on a transmission line.
Plane wave propagating in air � unguided wave propagation
Almost all transmission lines have a cross-sectional geometry whichis constant in the direction of wave propagation along the line. This typeof transmission line is called a uniform transmission line.
Uniform transmission line - conductors and dielectrics maintain thesame cross-sectional geometry along the transmission line inthe direction of wave propagation.
Given a particular conductor geometry for a transmission line, onlycertain patterns of electric and magnetic fields (modes) can exist forpropagating waves. These modes must be solutions to the governingdifferential equation (wave equation) while satisfying the appropriateboundary conditions for the fields.
Transmission line mode - a distinct pattern of electric and magneticfield induced on a transmission line under source excitation.
The propagating modes along the transmission line or waveguide may beclassified according to which field components are present or not presentin the wave. The field components in the direction of wave propagationare defined as longitudinal components while those perpendicular to thedirection of propagation are defined as transverse components.
Transmission Line Mode Classifications
Assuming the transmission line is oriented with its axis along the z-axis (direction of wave propagation), the modes may be classified as
1. Transverse electromagnetic (TEM) modes - the electric andmagnetic fields are transverse to the direction of wavepropagation with no longitudinal components [Ez = Hz = 0].TEM modes cannot exist on single conductor guiding
structures. TEM modes are sometimes called transmission linemodes since they are the dominant modes on transmissionlines. Plane waves can also be classified as TEM modes.
2. Quasi-TEM modes - modes which approximate true TEMmodes for sufficiently low frequencies.
3. Waveguide modes - either Ez, Hz or both are non-zero.Waveguide modes propagate only above certain cutofffrequencies. Waveguide modes are generally undesirable ontransmission lines such that we normally operate transmissionlines at frequencies below the cutoff frequency of the lowestwaveguide mode. In contrast to waveguide modes, TEMmodes have a cutoff frequency of zero.
Transmission Line Equations
Consider the electric and magnetic fields associated with the TEMmode on an arbitrary two-conductor transmission line (assume perfectconductors). According to the definition of the TEM mode, there are nolongitudinal fields associated with the guided wave traveling down thetransmission line in the z direction (Ez = Hz = 0).
From the integral form of Maxwell’s equations, performing the line integralof the electric and magnetic fields around any transverse contour C1 gives
where ds = dsaz. The surface integrals of E and H are zero-valued sincethere is no Ez or Hz inside or outside the conductors (PEC’s, TEM mode).
0
0
The line integrals of E and H for the TEM mode on the transmission linereduce to
These equations show that the transverse field distributions of the TEMmode on a transmission line are identical to the corresponding staticdistributions of fields. That is, the electric field of a TEM at any frequencyhas the same distribution as the electrostatic field of the capacitor formedby the two conductors charged to a DC voltage V and the TEM magneticfield at any frequency has the same distribution as the magnetostatic fieldof the two conductors carrying a DC current I. If we change the contourC1 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 atany point on a transmission line operating in the TEM mode. If a uniquevoltage and current can be defined at any point on the transmission line,then we may use circuit equations to describe its operation (as opposed towriting field equations).
Transmission lines are typically electrically long (severalwavelengths) such that we cannot accurately describe the voltages andcurrents along the transmission line using a simple lumped-elementequivalent circuit. We must use a distributed-element equivalent circuitwhich describes each short segment of the transmission line by a lumped-element equivalent circuit.
Consider a simple uniform two-wire transmission line with itsconductors parallel to the z-axis as shown below.
Uniform transmission line - conductors and insulating mediummaintain the same cross-sectional geometry along the entiretransmission line.
The equivalent circuit of a short segment �z of the two-wire transmissionline may be represented by simple lumped-element equivalent circuit.
R = series resistance per unit length (�/m) of the transmission lineconductors.
L = series inductance per unit length (H/m) of the transmission lineconductors (internal plus external inductance).
G = shunt conductance per unit length (S/m) of the media betweenthe transmission line conductors (insulator leakage current).
C = shunt capacitance per unit length (F/m) of the transmission lineconductors.
We may relate the values of voltage and current at z and z+�z bywriting 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 � 0, the terms on the right hand side of theequations above become partial derivatives with respect to z which gives
Note the similarity in the functional form of the time-domain and thefrequency-domain transmission line equations to the respective source-freeMaxwell’s equations (curl equations). Even though these equations werederived without any consideration of the electromagnetic fields associatedwith the transmission line, remember that circuit theory is based onMaxwell’s equations.
Just as we manipulated the two Maxwell curl equations to derive thewave 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 takederivatives of both sides with respect to z.
We then insert the first derivatives of the voltage and current found in theoriginal 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 thetransmission 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 phaseconstant. The general equations for � and � in terms of the per-unit-lengthtransmission line parameters are
The general solutions to the voltage and current wave equations are
~~~~~ ~~~~~ +z-directed waves
� � �z-directed waves
The coefficients in the solutions for the transmission line voltage andcurrent are complex constants (phasors) which can be defined as
The instantaneous voltage and current as a function of position along thetransmission line are
Given the transmission line propagation constant, the wavelength andvelocity of propagation are found using the same equations as forunbounded waves.
The region through which a plane wave (unguided wave) travels ischaracterized by the intrinsic impedance (�) of the medium defined by theratio of the electric field to the magnetic field. The guiding structure overwhich the transmission line wave (guided wave) travels is characterized thecharacteristic impedance (Zo) of the transmission line defined by the ratioof 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 wavesdefines the characteristic impedance of the transmission line.
The transmission line characteristic impedance is, in general, complex andcan be defined by
The voltage and current wave equations can be written in terms of thevoltage coefficients and the characteristic impedance (rather than thevoltage and current coefficients) using the relationships
The voltage and current equations become
These equations have unknown coefficients for the forward and reversevoltage waves only since the characteristic impedance of the transmissionline is typically known.
Special Case #1 Lossless Transmission Line
A lossless transmission line is defined by perfect conductors and aperfect insulator between the conductors. Thus, the ideal transmission lineconductors have zero resistance (�=�, R=0) while the ideal transmissionline insulating medium has infinite resistance (�=0, G=0). Theequivalent 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 lineequations for the lossless line are
The characteristic impedance of the lossless transmission line is purely realand given by
The velocity of propagation and wavelength on the lossless line are
Transmission lines are designed with conductors of high conductivity andinsulators of low conductivity in order to minimize losses. The losslesstransmission line model is an accurate representation of an actualtransmission line under most conditions.
Special Case #2 Distortionless Transmission Line
On a lossless transmission line, the propagation constant is purelyimaginary and given by
The phase velocity on the lossless line is
Note that the phase velocity is a constant (independent of frequency) sothat all frequencies propagate along the lossless transmission line at thesame velocity. Many applications involving transmission lines require thata band of frequencies be transmitted (modulation, digital signals, etc.) asopposed to a single frequency. From Fourier theory, we know that anytime-domain signal may be represented as a weighted sum of sinusoids.A single rectangular pulse contains energy over a band of frequencies. Forthe pulse to be transmitted down the transmission line without distortion,all of the frequency components must propagate at the same velocity. Thisis the case on a lossless transmission line since the velocity of propagationis a constant. The velocity of propagation on the typical non-idealtransmission line is a function of frequency so that signals are distorted asdifferent components of the signal arrive at the load at different times.This effect is called dispersion. Dispersion is also encountered when anunguided wave propagates in a non-ideal medium. A plane wave pulsepropagating in a dispersive medium will suffer distortion. A dispersivemedium is characterized by a phase velocity which is a function offrequency.
For a low-loss transmission line, on which the velocity of propagationis near constant, dispersion may or may not be a problem, depending onthe length of the line. The small variations in the velocity of propagationon a low-loss line may produce significant distortion if the line is verylong. There is a special case of lossy line with the linear phase constantthat produces a distortionless line.
A transmission line can be made distortionless (linear phase constant) bydesigning the line such that the per-unit-length parameters satisfy
Inserting the per-unit-length parameter relationship into the generalequation for the propagation constant on a lossy line gives
Although the shape of the signal is not distorted, the signal will sufferattenuation as the wave propagates along the line since the distortionlessline is a lossy transmission line. Note that the attenuation constant for adistortionless transmission line is independent of frequency. If this werenot true, the signal would suffer distortion due to different frequenciesbeing attenuated by different amounts.
In the previous derivation, we have assumed that the per-unit-lengthparameters of the transmission line are independent of frequency. This isalso an approximation that depends on the spectral content of thepropagating signal. For very wideband signals, the attenuation and phaseconstants will, in general, both be functions of frequency.
For most practical transmission lines, we find that RC > GL. In orderto satisfy the distortionless line requirement, series loading coils aretypically placed periodically along the line to increase L.
Transmission Line Circuit(Generator/Transmission line/Load)
The most commonly encountered transmission line configuration isthe simple connection of a source (or generator) to a load (ZL ) through thetransmission line. The generator is characterized by a voltage Vg andimpedance Zg while the transmission line is characterized by a propagationconstant � and characteristic impedance Zo.
The general transmission line equations for the voltage and current as afunction 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 termsof the current and voltage at the load.
Just as a plane wave is partially reflected at a media interface when theintrinsic impedances on either side of the interface are dissimilar (�1��2),the guided wave on a transmission line is partially reflected at the loadwhen the load impedance is different from the characteristic impedance ofthe transmission line (ZL�Zo). The transmission line reflection coefficientas a function of position along the line [(z)] is defined as the ratio of thereflected wave voltage to the transmitted wave voltage.
Inserting the expressions for the voltage coefficients in terms of the loadvoltage 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 loadmeans that all of the energy associated with the forward traveling wave isdelivered to the load). The reflection coefficient at the load is zero whenZL = Zo. If ZL = Zo, the transmission line is said to be matched to the loadand no reflected waves are present. If ZL � Zo, a mismatch exists andreflected waves are present on the transmission line. Just as plane wavesreflected from a dielectric interface produce standing waves in the regioncontaining the incident and reflected waves, guided waves on atransmission line reflected from the load produce standing waves on thetransmission line (the sum of forward and reverse traveling waves).
The transmission line equations can be expressed in terms of thereflection coefficients as
The last term on the right hand side of the above equations is the reflectioncoefficient (z). The transmission line equations become
Thus, the transmission line equations can be written in terms of voltagecoefficients (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 givenby the ratio of voltage to current at that point. Inserting the expressions forthe 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 voltageand current as
Inserting these equations for the voltage coefficients into the impedanceequation gives
We may use the following hyperbolic function identities to simplify thisequation:
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 tangentfunction reduces to
which yields
Input impedance at any point along a lossless transmission line
Special Case #1 Open-circuited lossless line (�ZL��� , L = 1)
Special Case #2 Short-circuited lossless line (�ZL��0, L = �1)
The impedance characteristics of a short-circuited or open-circuitedtransmission line are related to the positions of the voltage and currentnulls along the transmission line. On a lossless transmission line, themagnitude of the voltage and current are given by
The equations for the voltage and current magnitude follow the crankdiagram form that was encountered for the plane wave reflection exampleand the minimum and maximum values for voltage and current are
For a lossless line, the magnitude of the reflection coefficient is constantalong 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 ofmaximum to minimum voltage magnitudes (or maximum to minimumcurrent magnitudes).
The standing wave ratio on a lossless transmission line ranges between 1
and �.
We may apply the previous equations to the special cases of open-circuited and short-circuited lossless transmission lines to determine thepositions of the voltage and current nulls.
Open-Circuited Transmission Line (L=1)
Current null (Zin=�) at the load and every /2 from that point.Voltage null (Zin= 0) at /4 from the load and every /2 from that point.
Short-Circuited Transmission Line (L=�1)
Voltage null (Zin= 0) at the load and every /2 from that point.Current null (Zin=�) at /4 from the load and every /2 from that point.
From the equations for the maximum and minimum transmission linevoltage and current, we also find
It will be shown that the voltage maximum occurs at the same location asthe current minimum on a lossless transmission line and vice versa. Usingthe definition of the standing wave ratio, the maximum and minimumimpedance values along the lossless transmission line may be written as
Thus, the impedance along the lossless transmission line must lie withinthe range of Zo /s to sZo.
Example
A source [Vsg=100�0oV, Zg= Rg = 50 �, f = 100 MHz] is connectedto a lossless transmission line [L = 0.25�H/m, C=100pF/m, l=10m]. Forloads of ZL = RL = 0, 25, 50, 100 and � �, determine (a.) the reflectioncoefficient at the load (b.) the standing wave ratio (c.) the input impedanceat the transmission line input terminals (d.) voltage and current plots alongthe transmission line.
RL (�) (a.) Zin (�) (b.) L (c.) s
0 0 �1 �
25 25 �0.333 2
50 50 0 1
100 100 +0.333 2
� � +1 �
RL (�) �Vs(z)�max (V) �Vs(z)�min (V) �Vs( l)� (V)
0 100 0 0
25 66.7 33.3 33.3
50 50 50 50
100 66.7 33.3 66.7
� 100 0 100
RL (�) �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
� 2 0 0
Smith Chart
The Smith chart is a useful graphical tool used to calculate thereflection coefficient and impedance at various points on a (lossless)transmission line system. The Smith chart is actually a polar plot of thecomplex reflection coefficient (z) [ratio of the reflected wave voltage tothe 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 definedas
where L is the reflection coefficient at the load (z = l) given by
where zL defines is the normalized load impedance. The magnitude of thecomplex-valued reflection coefficient ranges from 0 to 1 for any value ofload impedance. Thus, the reflection coefficient (and correspondingimpedance) can always be mapped on the unit circle in the complex plane.
(1)
The corresponding standing waveratio s is
The magnitude of L is constanton any circle in the complexplane so that the standing waveratio (VSWR) is also constant onthe same circle.
Smith chart center � �L� = 0 Constant VSWR (no reflection - matched, s = 1) circle
Outer circle � �L� = 1(total reflection, s = �)
Once the position of L is located on the Smith chart, the location of thereflection coefficient as a function of position [(z)] is determined usingthe reflection coefficient formula.
This equation shows that to locate (z), we start at L and rotate throughan angle of �z=2�(z�l) on the constant VSWR circle. With the loadlocated at z= l, moving from the load toward the generator (z < l) definesa negative angle �z (counterclockwise rotation on the constant VSWRcircle). Note that if �z=�2 , we rotate back to the same point. Thedistance traveled along the transmission line is then
Thus, one complete rotation around the Smith chart (360o) is equal to onehalf wavelength.
On the Smith chart ...
• CW rotation � toward the generator
• CCW rotation � toward the load
• /2=360o =720o
• �� and s are constant on a lossless transmission line. Movingfrom point to point on a lossless transmission line is equivalentto rotation along the constant VSWR circle.
• All impedances on the Smith chart are normalized to thecharacteristic impedance of the transmission line (when usinga normalized Smith chart).
The points along the constant VSWR circle represent the complexreflection coefficient at points along the transmission line. The reflectioncoefficient at any given point on the transmission line corresponds directlyto the impedance at that point. To determine this relationship between L
and ZL, we first solve (1) for zL.
where rL and xL are the normalized load resistance and reactance,respectively. Solving (2) for the resistance and reactance gives equationsfor the “resistance” and “reactance” circles.
(2)
In a similar fashion, the reflection coefficient as a function ofposition (z) along the transmission line can be related to the impedanceas a function of position Z(z). The general impedance at any point alongthe length of the transmission line is defined by the ratio of the phasorvoltage 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 plottingthe complex reflection coefficient (rotating around the constant VSWRcircle), we are also plotting the corresponding impedance.
Once a normalized impedance is located on the Smith chart for aparticular point on the transmission line, the normalized admittance at thatpoint is found by rotating 180o from the impedance point on the constant
reflection coefficient circle.
(3)
(2)
The locations of maxima and minima for voltages and currents alongthe transmission line can be located using the Smith chart given that thesevalues correspond to specific impedance characteristics.
Voltage maximum, Current minimum � Impedance maximum
Voltage minimum, Current maximum � Impedance minimum
Example (Smith chart)
A 75� lossless transmission line of length 1.25� is terminated by aload impedance of 120�. The line is energized by a source of 100V (rms)with an internal impedance of 50�. Determine (a.) the transmission lineinput impedance and (b.) the magnitude of the load voltage.
Using the Smith chart software,(1.) Define normalization factor for the Smith chart (Zo=75�).(2.) Define load impedance (ZL=120�).(3.) Insert series t-line (Zo=75�).(4.) Move 1.25� (2.5 revolutions) on the constant VSWR circle
to find Zin.
(a.)
DP#1 (120.0 + j0.0)� DP#2 (46.9 + j0.0)�
Zin = (46.9 + j0)�
(b.)
Using a pencil and paper Smith chart,(1.) Find the normalized load impedance.
A 60� lossless line has a maximum impedance Zin = (180 + j0) � ata distance of �/24 from the load. If the line is 0.3�, 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 thispoint, move �/24 toward the load (CCW) to find ZL.
(c.) Insert series transmission line section and from ZL, move 0.3� towardthe generator (CW) to find Zin.
DP#1 (117.2 + j78.1) �DP#2 (20.0 + j2.8) �
Quarter Wave Transformer
When mismatches between the transmission line and load cannot beavoided, there are matching techniques that we may use to eliminatereflections on the feeder transmission line. One such technique is thequarter wave transformer.
If Zo � ZL, ���> 0 (mismatch)
Insert a �/4 length section of different transmission line (characteristicimpedance = Zo� ) 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 wavetransformer yields
Example (Quarter wave transformer)
Given a 300� transmission line and a 75� load, determine thecharacteristic impedance of the quarter wave transformer necessary tomatch the transmission line. Verify the input impedance using the Smithchart.
DP#1 (75.0 + j0.0) �DP#2 (300.0 - j0.0) �
Stub Tuner
A quarter-wave transformer is effective at matching a resistive loadRL to a transmission line of characteristic impedance Zo when RL�Zo.However, a complex load impedance cannot be matched by a quarter wavetransformer. The stub tuner is a transmission line matching technique thatcan used to match a complex load. If a point can be located on thetransmission line where the real part of the input admittance is equal to thecharacteristic admittance (Yo=Zo
�1) of the line (Yin=Yo±jB ), thesusceptance B can be eliminated by adding the proper reactive componentin parallel at this point. Theoretically, we could add inductors orcapacitors (lumped elements) in parallel with the transmission line.However, these lumped elements usually are too lossy at the frequenciesof interest.
Rather than using lumped elements, we can use a short-circuited oropen-circuited segment of transmission line to achieve any requiredreactance. Because we are using parallel components, the use ofadmittances (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 segmentadmittance (ytl = g + jb) is unity (g = 1).
Single Stub Tuner Design Using the Smith Chart
1. Locate the normalized load impedance zL (rotate 180o to findyL). Draw the constant VSWR circle [Note that all points onthe Smith chart now represent admittances].
2. From yL, rotate toward the generator (CW) on the constantVSWR circle until it intersects the g = 1. The rotation distanceis the distance d while the admittance at this intersection pointis ytl = 1 + jb.
3. Beginning at the stub end (short circuit admittance is therightmost point on the Smith chart, open circuit admittance isthe leftmost point on the Smith chart), rotate toward thegenerator (CW) until the point at ys = �jb is reached. Thisrotation distance is the stub length l.
Short circuited stub tuners are most commonly used because ashorted segment of transmission line radiates less than an open-circuitedsection. The stub tuner matching technique also works for tuners in serieswith the transmission line. However, series tuners are more difficult toconnect since the transmission line conductors must be physicallyseparated in order to make the series connection.
Example
Design a short-circuited shunt stub tuner to match a load ofZL = (60 � j40)� to a 50� transmission line.
Short circuit admittance - rightmost point on Smith chart (0o)
