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    CHAPTER 5TRANSMISSION LINE

    By

    Nur Diyana Kamarudin

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    un amen a s o

    Transmission Lines Transmission lines are considered to be

    impedance matching circuits designed to deliverpower (RF) from the transmitter to the antennaand maximum signal from the antenna to thereceiver.

    Nowadays, the transmission line is made ofparallel-wire (unbalanced) line and coaxial(unbalanced) line.

    It consists of 2-wire line since transmission line

    fortransverse electromagneticTEM wavepropagation always have 2 conductors.

    The characteristic of Transmission Line aredetermined by its electrical properties :Conductivity & Insulator Dielectric Constant

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    Fundamentals The parallel-wire line is employed where balanced

    properties are required: for instance: inconnecting a folded-dipole antenna to a TVreceiver.

    The coaxial line is used when unbalanced

    properties are needed as in the interconnection ofa broadcast transmitter to its grounded antenna.

    It is also employed at UHF and microwavefrequencies to avoid the risk of radiation from thetransmission line itself.

    Parallel lines are never used in microwaveswhereas coaxial lines may be employed for

    frequencies up to at least 18GHz.16 March 2010 3

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    Types of Transmission Lines

    Parallel line

    Coaxial Cable

    Open-Wire

    Twin-Lead

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    Types of Transmission lines

    Open-wire Twin lead

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    Types of transmission lines

    Unshielded twisted-pair

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    Types of transmission lines

    Coaxial cable

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    Ideal Transmission Line

    No losses conductors have zero resistance dielectric has zero conductance possible only with superconductors approximated by a short line

    No capacitance or inductance possible with a real line only at dc with low frequencies and short lines this can

    be approximated

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    The electrical properties determine the PRIMARYelectricalconstant:

    Series Resistance, R

    Series Inductance, L

    Shunt Capacitance, C

    Shunt Conductance, G

    Th

    e combined above parameter is called LUMPED

    PARAMETERS.

    The transmission line characteristic is called SECONDARYCONSTANT.

    The secondary constants are:1. Characteristic Impedance2. Propagation constant

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    Two-wire parallel transmission lineelectrical equivalent circuit

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    The characteristic Impedance, Zo is defined as :

    A transmission line must be terminated at purelyresistive load for maximum power transfer.

    Zo = Zo = for Low Frequencies Zo = for high Frequencies

    The conductance between 2 wires are determinedby the shunt leakage resistance, Rs

    The load impedance, ZL must match withcharacteristic impedance, ZO

    GR /

    )(/)( jwCGjwLR

    CL /

    Characteristic Impedance

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    The characteristic impedance can be calculatedby using Ohms Law:

    Zo = Eo/ Io

    where Eo

    is source voltage and Io

    is transmission linecurrent

    The characteristic impedance also can becalculated by its physical dimension:

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    Two wire parallel Transmission lines

    Two-wireparallel transmission line

    0 276logD

    Zr

    !

    Z0= the characteristic impedance (ohms)

    D = thedistancebetween the centers= 2s

    r = theradiusof the conductor

    I0 = thepermittivity offreespace (F/m)Ir = therelativepermittivity ordielectric constant of the

    medium (unitless)

    Q0 = thepermeability offreespace (H/m)

    1

    p

    o

    v

    Q I

    0rI I I!

    0

    1

    o

    c

    Q I!

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    Coaxial Cable transmission lines

    Coaxial cable

    0

    138log

    r

    DZ

    dI

    Z0= the characteristic impedance (ohms)

    D = thediameterof theouterconductor

    d =thediameter

    oftheinnerc

    onduct

    orI = thepermittivity of thematerial

    Ir = therelativepermittivity ordielectric constant of

    themediumorinsulators=k

    Q0 = thepermeability offreespace

    1

    p

    o

    v

    Q I

    0rI I I!

    0

    1

    o

    c

    Q I!

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    EXAMPLE 1 :

    Determine the characteristic impedance of thecoaxial cable with L= 0.118 uH/ft and C = 21 pF/ft.

    Solution :

    Zo = = 75;

    -seealsoexample7.1, 7.2 and7.3 giveninhandouts.

    CL /

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    Losses in Transmission Lines ConductorLosses

    Increaseswithfrequency duetoskineffect

    Conductorheating or I2R

    loss-proportional to current

    andinversely proportional to

    characteristic impedance. Dielectric Heating Losses

    Alsoincreaseswithfrequency

    RadiationLosses Not significant with good

    quality coaxproperly installed

    Canbeaproblemwithopen-

    wire cable

    Coupling Losses

    Corona

    Skineffect

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    Transmission Lines Losses

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    Velocity factor The velocity of light and all other electromagnetic

    waves depends on the medium through whichthey travel.

    It is nearly 3 x 108 m/s in a vacuum and slower inall other media.

    The velocity of light in a medium is given by

    v = vc/ k

    where v = velocity in the medium

    vc = velocity of light in a vacuum

    k = dielectric constant of the medium ( 1for vacuum and very nearly 1 for air)

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    Velocity factor The velocity factor of a dielectric substance and

    thus of a cable is the velocity reduction ratio andis therefore given by

    vf= 1/k

    The dielectric constants of materials commonlyused in transmission lines range from about 1.2 to2.8, giving corresponding velocity factors from 0.9to 0.6.

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    Reflection Coefficients

    It is a vector quantity that represents the ratioof reflected voltages to incident voltage orreflected current incident current.Refer to figure 12.19 for the figure ofincidence wave and reflected wave.

    It is defined as :+ = (z 1) / (z + 1)

    z is a normalized impedance

    It canalsobewrittenin termofreflectedvoltageorcurrent andincident voltage/current.

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    Reflection Coefficient

    r r

    i i

    V Ior

    V I+

    + =reflection coefficientV

    i=incident voltage

    Vr=reflectedvoltage

    Ii =incident currentIr=reflected current

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    ZL

    TransmissionLine

    Z0

    Reflection of Pulses

    r r

    i i

    Ior

    I+ @

    total voltagei r

    V V!

    total currenti r

    I I!

    i r

    L

    i r

    ZI I

    !

    0i r

    i r

    V VZ

    I I

    ! !

    0

    0

    L

    L

    Z Z

    Z Z

    + !

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    Reflection Coefficient

    More complex situation: Load has an arbitraryimpedance

    not equal to Z0 not shorted or open

    impedance may be complex (either capacitiveor inductive as well as resistive)

    Method needed to calculate reflected voltage inthese cases

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    Wave Propagation on Lines

    Start by assuming a matched line

    Waves move down the line at propagationvelocity

    Waves are the same at all points except for phase

    Phase changes 360 degrees in the distance awave travels in one period

    This distance is called the wavelength

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    Incidentand ReflectedWaves

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    StandingWaves

    When an incident wave reflects from a

    mismatched load (Z0 ZL), an interference patterndevelops

    Both incident and reflected waves move at thepropagation velocity, but the interference pattern

    is stationary

    The interference pattern is called a set ofstanding waves

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    StandingWaves

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    Standing-Wave Ratio When line is mismatched but neither open nor

    shorted, voltage varies along line without everfalling to zero Greater mismatch leads to greater variation Voltage standing-wave ratio (VSWR or SWR) is

    defined as a ratio of the maximum voltage/current

    to the minimum voltage/current of a standingwave of the transmission line

    often called Voltage Standing Wave Ratio (VSWR)

    min

    max

    V

    VSWR !

    http://www.bessernet.com/Ereflecto/tutorialFrameset.htm

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    Standing waves

    max

    min

    VSWR

    V!

    max i r i iV E E E E ! ! +

    min i r i iV E E E E ! ! +

    0

    0

    1or

    1

    L

    L

    Z ZSWR

    Z Z

    +! !

    +

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    StandingWave Ratio (SWR)

    If there is no reflected wave or signals, then SWRwill be equal to 1 where Zo = ZL

    If the load is purely resistive, then the SWR willbe:SWR = Zo/ ZL

    1

    1

    SWR

    SWR

    + !

    0

    0

    L

    L

    Z Z

    Z Z

    + !

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    Effects of High SWR

    High SWR causes voltage peaks on the line thatcan damage the line or connected equipmentsuch as a transmitter

    Current peaks due to high SWR cause losses to

    increase

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    Quarterand half wavelength

    lines Impedance inversion by quarter-wavelength

    Zs s/D

    Zo ZL

    Consider figure above which shows a load of

    impedance ZL connected to a piece oftransmission line of length s and having Zo as itscharacteristic impedance.

    Zs = Zo2/ ZL

    (reflective impedance)

    If the impedances are normalized with respect toZo, we have

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    Zs/Zo = Zo/ZL

    Where

    Zs/Zo = zsand

    ZL/Zo = zL

    we know that zs = 1/ zL = yL

    where yL is the normalized admittance of the load.

    this equation states that if a quarter-wavelength line is connected to an impedance, then thenormalized input impedance of this line is equal tothe normalized load admittance.

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    Quarter-Wave Transformerand

    impedancematching

    It used to match transmission line to the purelyresistive load in order to find the shortestdistance to the load (in term ofP).

    Refer to figure 12-31 for the diagram of theQuarter-wave transformer.

    At quarter-wavelength (P/4), the characteristicimpedance of the transmission line will be

    defined as:Zo

    = LZZo

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    Quarter-Wave Transformerand

    impedancematching

    Condition of Quarter-Wave Transformer:1. RL = Zo Acts as Transformer for ratio of 1:1

    2. RL > Zo Acts as Step down Transformer

    3. RL < Zo Acts as Step up Transformer

    A quarter-wave Transformer is simply onequarter (P/4) long of the transmission line length.

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    Quarter-Wave Transformerand

    impedancematching

    EXAMPLE:

    Determine the physical length and newcharacteristic impedance for the transmissionline of 50; RG-8A/U. Use Quarter-Wave

    Transformer to match the 150; load impedancewhere the source frequency is 150 MHz andvelocity factor of 1.

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    SOLUTION :

    1. Physical Length:

    P = (c/f) = (3 x108/ 150 x 106)

    =0.5m

    2. The characteristic impedance is:

    Zo = sqrt [(50)(150)] = 86.6 ;

    - Seealsohandout givenasanotherexample.

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    Reactance properties of TL:

    Stubs

    It is used to remove the reactive parts of thetransmission line in order to obtain themaximum energy transferred to the loadsince purely inductive or capacitive loadabsorbs no energy.

    In order to implement it, a piece oftransmission line is placed across theprimary line as close to the load as possible.

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    Reactance properties of TL :

    Stubs

    Procedure for Stub Matching is definedbelow:1. Calculate the load admittance

    2. Calculate stub susceptance

    3. Connect stub to the load, the resultingadmittance being the load conductance G.4. Transform conductance to resistance, andcalculate Z

    o of the quarter-wave transformer

    as before.

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    EXERCISE :

    A (200 + j75) load is to be matched to a 300 line to give SWR = 1. Calculate:

    1. Reactance of the stub

    2. the characteristic impedance of the quarter-wave transformer,

    both connected directly to the load.

    - See the solution as given in the handouts.

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    We need to model the transmission line due to

    the following reasons:

    Propagation Line (Phase Shift)

    Reflected Signals or Waves

    Power Loss Dispersion

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    Smith Chart Smith Chart is a tool used to determine the

    matched IMPEDANCEorADMITTANCEforthe lossless transmission line.

    Smith Chart consists of the following:

    The outer section of the Smith Chart is referto the distance of wavelength towardsgenerator (source) and load.

    Clockwise rotation movement towardsgenerator

    Counterclockwise rotation movementtowards load The scale for the wavelength is from 0 => 0.5P

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    The body of Smith Chart consists of the following:

    Body of smith chart is made of families of orthogonalcircles (intersect at right angles).

    The impedance or admittance at any point on the line

    can be plotted by find the intersection of the realcomponent (resistance or conductance) indicated inthe along horizontal axis with the imaginarycomponent (reactance or susceptance).

    Counterclockwise rotation movement towards load

    The scale for the wavelength is from 0 => 0.5P

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    The top half of the smith chart is referring to thepositive Inductive Reactance (jX/Zo) & CapacitiveSusceptance (jB/Yo).

    The bottom half of the smith chart is referring tothe negative Capacitive reactance (-jX/Zo) &

    Inductive Susceptance (-jB/Yo)

    The centerof theSmithChart isreferring to therealcomponent, Resistance (R/Zo)andConductance(G/Yo).

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    Smith Chart (Simplediagram)

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    The impedance & admittance value in the SmithChart is often expressed in term of normalized,

    = Z /Zo EXAMPLE 2 :

    Plot the impedance of 100 + j25 ; on a 50 ; line.

    Solution :

    1. Normalized the impedance by:z = (100 + j25) / 50 = 2 + j0.5

    2. Use Smith Chart to plot the normalizeimpedance.

    3. Since the value of impedance is positive, find the

    point of the plot on the upper side of smith chart andfind the value of resistance (2) and reactance (0.5)and then draw the circle.4. The radius of the circle is called SWR (StandingWave

    Ratio) of the line.

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    Plot the following impedance in the smith chart ifthe characteristic impedance is 50 ;.

    i) 50 + j75 ;

    ii) 150 + j75 ;

    EXERCISE 1:

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    EXAMPLE3 :

    Determine the input impedance and SWR forthe 1.25P long transmission line where thecharacteristic impedance, Zo = 50 ; and loadimpedance, ZL= 30 + j40;.

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    SOLUTION FOR EXAMPLE3:

    1. Find the normalized impedance,z = (30 + j40)/50 = 0.6 + j0.8

    2. Plot the normalized impedance anddetermine the SWR.SWR = 2.9

    3. Find the wavelength from normalizedimpedance and rotate it by 1.25Pthe distance required is 0.37P.draw the intersection line to determine the input

    impedance,Zin= (0.63 j0.77)x50=31.5 j38.5

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    EXAMPLE 4 :

    Quaterwave TransformerMatching

    A load impedance, ZL= 75 + j50; to matchthe 50; source with a quarter-wavetransformer.

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    SOLUTION FOR EXAMPLE 4:

    1. Find the normalized impedance,z = (75 + j50)/50 =1.5 + j1

    2. Plot the normalized impedance and

    determine the SWR.

    SWR = 2.4 labeled as point C

    3. Label it as point A and then extend the line to

    determine the distance from the load:

    0.192P labeled as point B

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    4.At point C, thedistanceis0.250P

    thus, thedistancefro

    mpo

    intC

    to

    B is0.250P - 0.192P =>0.058P

    It isa Quarter-waveTranformer

    5.Thus, theinput impedanceisdeterminedfrom

    itsSWR,Zin =SWR = (2.4)x50= 120;

    6.TheCharacteristic ImpedancefromQuarter-Wave Transformer :

    Zo

    =sqrt(

    ZoZ

    in)=

    sqrt (50

    x120

    )=

    77.5

    ;

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    EXERCISE 2:

    Determine the SWR, Characteristic ofQuarter-Wave Transformer and the distanceto match the 75 ; transmission line to theload impedance of 25-j50 ;.