T Line Basics

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    Chapter 2

    Part 1: Transmission Lines Basics

    Definition: A transmission line is a mechanical structure that guides EM energy in adesired direction.

    Basic properties of transmission lines: TEM mode transmission lines: E and H are in the transverse plane, a plane

    that is perpendicular to the direction of the EM wave propagation.

    Commonly used lines consisting a pair of parallel conductors usually

    belong to this class.

    Higher-order transmission lines: One of the field components is in the

    direction of the EM wave propagation. Waveguide types consisting a

    hollow conductor usually belong to this class (next semester).

    Dispersion in most commonly used transmission lines such as coaxial

    lines is negligible. However, the line loss is usually high especially at

    microwave frequencies.

    On the other hand, waveguides suffer much lower losses at microwavefrequencies but dispersion can be a problem.

    The most popular TEM lines are (Fig. 2-4, P. 38): Coaxial cable: Used mainlyin the high-frequency regime (MHz and up) at

    low- to medium -power levels.

    Parallel-wire: Used mainly for lower frequency transmission (AC power

    line) at high-power levels.

    Microstrip: Used mainly for microwave IC and PC board fabrications.

    The most popular higher order transmission lines are (Fig. 2-4, P. 38): Waveguide: Hollow conductors used at high frequencies and high-power.

    Optical fiber: Concentric dielectric layers used at optical frequencies. Differences and similarities between a transmission line and a waveguide: They both are mechanical structure used to carry EM energy.

    Transmission line has two parallel conductors; waveguide has only one,usually hollow, conductor.

    Waves inside transmission line are TEM; waveguide waves are TE or TM.

    Transmission line has higher loss, especially at high frequencies.

    Waveguide waves may be distorted due to dispersion.

    Transmission line has no low-frequency limit; waveguide acts like a high-

    pass filter.

    In general, the impact on the signal integrity is much severe from the effect ofdistortion than from the effect of losses.

    Beside guiding EM energy, short sections of transmission line can sometimes be usedas high-frequency discrete circuit components such as Z transformer, filter,

    Simplifications (Fig. 2-1, P. 35): Source: A voltage source and a series resistor are used to represent the

    generator (Thevenins theorem)

    Load: A resistance (impedance) is used to represent the load.

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    Transmission line: A two-port network is used to describe a general

    transmission line.

    The effect of short (transmission line effect): Guideline: l> 0.01 (1% of the wavelength)

    Phase shift: The voltage and current along a transmission line generally

    vary as a function of position.Short

    Long

    Reflection: If the line is not terminated properly (with a match load)

    portion of the energy will be reflected back to the generator. This isusually bad because:

    1) Reflected energy is wasted

    2) Generator may be damaged if the reflection is significant.

    Power loss: Energy is dissipated due to conduction loss and dielectric loss.

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    Distortion: A signal containing more than one frequency component may

    be distorted due to dispersiondifferent frequency components travelingat different speeds in a waveguide.

    Lumped-element model We will first consider the general properties of a TEM mode line.

    Similar to your calculus classes, we are going to cut a thin slice(differential section) of the line and examine its electrical properties.

    Transmission line parameters (Formula tableP. 41): SeriesL andR per unit length

    Parallel C and G per unit length

    Transmission line equations (Telegraphers equations): We can derive a set of circuitequations for a differential slice of the above model and from them a set differentialequations can be developed to describe the voltage and current in a transmission line.

    VCjGdz

    IddzCjGVdVIdII

    ILjRdz

    VddzLjRIVdVV

    ~)''(

    ~

    )'')(~~

    ()~~

    (~

    ~)''(

    ~

    )''(~

    )~~

    (~

    +=++=+

    +=+=+

    Wave equations: Combining the two 1st order/2-unknown equations we have a pair of2nd order/1-unknown equations.

    ==+

    +===

    +=+=

    ++=+===

    +

    +

    ++++

    fuCjG

    LjRZ

    I

    V

    I

    V

    eIeIIeVeVV

    CjGLjRjIdz

    IdV

    dz

    Vd

    p

    zzzz

    ''

    ''

    ~~

    )'')(''(0~

    ~

    0~

    ~

    0

    0

    0

    0

    0

    0000

    2

    2

    22

    2

    2

    propagation constant

    attenuation constant = Re{}, phase constant = Im{},

    up phase velocity, Z0 characteristic impedance = v/i for a single wave

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    The time domain solution to the voltage wave equation due to sinusoidal excitation is:

    directionin theTravels

    )cos(

    directionin theTravels

    )cos(),( 00

    z

    zteV

    z

    zteVtzvzz

    +++

    +

    +=++++

    Electrical length: The electrical length of a line is the ratio of the physical length to the

    wavelength of the line signal. As a result, a physically long line may have

    a short electrical length if a low-frequency signal (long ) is beingtransmitted and vice versa.

    Electrically short lines: For example, AC power lines less than 50 mileslong are usually considered electrically short. For such cables we can

    consider them as discrete passive components.

    Medium-length lines: In general, power lines between 50 miles and 150miles long are considered to be medium-length lines. We can also consider

    them as discrete passive components with a more complex structure.

    Short and medium lines are important in low-frequency applications such

    as power transmission. They are handled with lumped-circuit theory. Electrically long lines: A line is considered to be electrically long if its

    physical length is comparable to the wavelength. For example, a 15 cm

    coaxial cable would be considered electrically long at 3 GHz (= c/f=

    3E8 / 3E9 = 0.1 m = 10 cm).

    Because at high frequencies the physical length of an electrically long line

    is often very short, the series resistance becomes negligible.

    Since low-loss materials are usually required to fabricate high-frequency

    cables, the shunt conductance can also be neglected.

    Lossless (ideal) line (R = G = 0 = 0):

    R + jX

    Z/2 Z/2

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    =====

    +=+=

    ==+=+

    +

    +

    ++++

    fu

    C

    LZ

    I

    V

    I

    V

    eIeIIeVeVV

    CLIdz

    IdV

    dz

    Vd

    p

    zjzjzjzj

    '

    '

    ~~

    ''0~

    ~

    0~

    ~

    0

    0

    0

    0

    0

    0000

    2

    2

    22

    2

    2

    For all TEM lines:

    h)wavelengtspace(freewhere

    1and

    '

    'and''

    00

    f

    c

    cu

    C

    GCL

    r

    r

    p

    ==

    ===

    ==

    Dispersion refers to the spreading of a signal due to the differences inspeed between different frequency components. It is often detrimental to

    digital communication since digital pulses contain multiple frequencycomponents.

    Since the phase velocity is independent of frequency, ideal TEM lines are

    nondispersive.

    Terminating an ideal line The characteristic impedanceZ0 of a transmission line is not a real

    resistance even though it has a unit of ohm.Z0 of a line merely represents

    the ratio of the voltage and current of one traveling wave along the line. If

    more than one wave (for example, incident and reflected waves) arepresent in the line, the ratio of the total voltage and total current would not

    equal toZ0.

    1

    1

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0000

    0

    0

    0

    0

    0000

    +

    =

    +

    ====

    =

    +=

    =+=

    +

    +

    +++

    ++

    Z

    Z

    Z

    Z

    ZZ

    ZZ

    I

    I

    V

    Ve

    Z

    VV

    Z

    VV

    Z

    V

    Z

    V

    Z

    V

    IIIVVV

    L

    L

    L

    Lj

    LL

    L

    LL

    r

    If a transmission line is terminated with a load resistanceZL equal to its

    characteristic impedanceZ0 there will be no energy reflected from theload. In other words, a matched load acts like an infinitely long line with

    the sameZ0.

    ZLZ0

    lz 0

    I-,V- I+,V+

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    For a mismatch load, part of the energy will be reflected back to the

    generator. The amplitude of the reflected wave is according to the aboveformula.

    The portion of power reflected back is 2. = 1 for o/c and = -1 for s/c || = 1 for purely reactive loads. Examples (P. 50-51)

    More examples:Z0 = 50 1. ZL = 55

    = (ZL Z0) / (ZL +Z0) = (55 50) / (55 + 50) = 5/105 = 0.0482 = 0.0482 =0.0023 = 0.23%

    2. ZL = 45 = (ZL Z0) / (ZL +Z0) = (45 50) / (45 + 50) = -5/95 = -0.0532 = 0.0532 =0.0028 = 0.28%

    3. ZL = 5 = (ZL Z0) / (ZL +Z0) = (5 50) / (5 + 50) = -45/55 = -0.822 = 0.822 =0.67 = 67%

    4. ZL = 5 K= (ZL Z0) / (ZL +Z0) = (5000 50) / (5000 + 50) =4950/5050 = 0.98

    2 = 0.982 =0.96 = 96%5. ZL = 75 + j25

    ZL Z0 25 25i+ =ZL Z0

    ZL Z0+:=

    ZL Z0+ 125 25i+ =

    ZL Z0 35.355=

    arg ZL Z0( ) 45deg=

    ZL Z0+ 127.475=

    arg ZL Z0+( ) 11.31deg=

    35.355

    127.4750.277= 45 11.31 33.69=

    0.231 0.154i+= arg ( ) 33.69deg=

    0.277= ( )2

    7.692%=

    Impedance transformation: The input impedance is the ratio of the total line voltage to the total line

    current. This ratio in general varies along the line and thus the inputimpedance is a function of the distance between the measurement point

    and the load.

    For an ideal line (lossless) terminated with a match load the inputimpedanceZin measured at any distance always equals toZ0.

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    On the other hand, ifZLZ0 thenZin will vary as a function of the distanceaway from the load.

    The input impedanceZin:

    +

    =

    +

    =

    =

    +

    =+

    ==

    +

    +

    +++

    +++

    lj

    lj

    ljlj

    ljlj

    in

    zjzj

    zjzj

    zjzj

    zjzj

    in

    e

    eZ

    ee

    eeZlZ

    lz

    ee

    eeZ

    Z

    eVeV

    eVeV

    zI

    zVzZ

    2

    2

    00

    0

    0

    00

    00

    1

    1)(

    Let

    )(~

    )(~

    )(

    The input impedanceZin measured at any distance lfrom the load is:

    ljZZ

    ljZZZ

    ljZlZ

    ljZlZZlZ

    L

    L

    L

    Lin

    tan

    tan

    sincos

    sincos)(

    0

    00

    0

    00 +

    +=++=

    As a result, the input impedance of an unmatched line can vary drasticallyas a function ofl.

    l l

    0 090 /4180 /2270 3/4

    360 Examples:Z0 = 50 ,f= 3 GHz

    1) ZL = 0 and = 3E8 / 3E9 = 10 cm Zin =jZ0 (sin l/ cos l)Zin = 0 forl= 0, 180, 360, Zin = forl= 90, 270, A shorted line 2.5 cm long sometimes looks like an open.

    2) ZL = and = 3E8 / 3E9 = 10 cm Zin =jZ0 (cos l/ sin l)Zin = forl= 0, 180, 360, Zin = 0 forl= 90, 270, An open line 2.5 cm long sometimes looks like a short.

    3) For ZL = 30 and v= 0.8 c, findZin for a 5.5 cm line

    ZLZin Z0

    l

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    arg Zin( ) 20.663deg=Zin 70.665=L 1.323 109 H=L

    24.935( )

    2 f( ):=

    Zin 66.119 24.935i+=arg

    ZL cos l( ) j Z0 sin l( )+

    Z0 cos l( ) j ZL sin l( )+

    20.663deg=

    ZL cos l( ) j Z0 sin l( )+

    Z0 cos l( ) j ZL sin l( )+1.413=

    arg 19.134 27.716i( ) 124.62 deg=arg 11.481 46.194i( ) 103.957 deg=

    19.134 27.716i 33.679=11.481 46.194i 47.599=

    Z0 cos l( ) j ZL sin l( )+ 19.134 27.716i=ZL cos l( ) j Z0 sin l( )+ 11.481 46.194i=

    j ZL sin l( ) 27.716i=Z0 cos l( ) 19.134=j Z0 sin l( ) 46.194i=ZL cos l( ) 11.481=

    Zin Z0

    ZL cos l( ) j Z0 sin l( )+

    Z0 cos l( )

    j ZL sin l( )

    +

    :=

    l360deg

    0.687=l 247.5deg=l l

    360deg

    :=

    0.08m=0.83 10

    s

    f:=l 5.5 10 2 m:=

    f 3 109

    Hz:=ZL 30:=Z0 50:=

    4) For ZL = 30 and v= 0.8 c, findZin for a 2.2 cm line

    C 4.031 1012=C 2 3 109 13.161( ) 1:=

    arg Zin( ) 9.359 deg=Zin 80.936=Zin 79.859 13.161i=

    argZL cos l( ) j Z0 sin l( )+

    Z0 cos l( ) j ZL sin l( )+

    9.359 deg=

    ZL cos l( ) j Z0 sin l( )+

    Z0 cos l( ) j ZL sin l( )+1.619=

    arg 19.134 27.716i( ) 124.62 deg=arg 11.481 46.194i( ) 103.957 deg=

    19.134 27.716i 33.679=11.481 46.194i 47.599=

    Z0 cos l( ) j ZL sin l( )+ 7.822 29.631i+=ZL cos l( ) j Z0 sin l( )+ 4.693 49.384i+=

    j ZL sin l( ) 29.631i=Z0 cos l( ) 7.822=j Z0 sin l( ) 49.384i=ZL cos l( ) 4.693=

    Zin Z0

    ZL cos l( ) j Z0 sin l( )+

    Z0 cos l( ) j ZL sin l( )+:=

    l 99 deg=ll

    360deg:=

    0.08=.

    3 109

    :=l 2.2 10 2:=ZL 30:=Z0 50:=

    From the examples we conclude that a short section of transmission linecan transform a short to an open and vice versa. It can also transform a

    capacitance to an inductance and vice versa.

    Properly terminating the output of microwave amplifiers and oscillators isimportant.

    Tune circuits at microwave frequencies usually require L and C withextremely small values. Using small segments of transmission line is one

    way to accomplish this requirement.

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    Complete solution of the wave equations (Example 2-5, P. 57-58):

    +=

    +=

    +ljlj

    ing

    ing

    eeZZ

    ZVgV

    1~

    0

    Standing waves:

    Transmission line waves (incident and reflected) are traveling waves inopposite directions. It is usually difficult to observe them separately.

    The combined wave displays an alternating pattern (repeats at every )of constructive and destructive interference between the incident and

    reflective waves.

    The envelop of this pattern is called the standing wave.

    The best way to visualize the standing wave pattern is to visit the

    following web site: http://graphics.cs.ucdavis.edu/~schussma/xl.htmlIn this transmission line simulation you can specify the magnitude and

    phase of the reflection coefficient. The program will then display the

    incident, reflective, and total wave along the line as well as the envelop of

    the total wave (standing wave pattern).!!! Everybody should try this !!!

    Match load: Since no reflection exists in the line, no standing wave patternis observed (0 V swing between maximum and minimum).

    Open-circuit (o/c) load: Since V+ and V- are the same atZL, we expect the

    load voltage to be doubled atZL and complete cancellation at away (2V swing). This pattern repeats every .

    http://graphics.cs.ucdavis.edu/~schussmaa/xl.htmlhttp://graphics.cs.ucdavis.edu/~schussmaa/xl.html
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    Short-circuited (s/c) load: Since V+ and V- are opposite atZL, we expect

    complete cancellation atZL and the voltage to be doubled at away (2V swing). This pattern repeats every .

    Larger load-100 : Similar to o/c with less voltage swing betweenmaximum (0.667V) and minimum (0.333 V).

    Smaller load-25 : Similar to s/c with less voltage swing betweenmaximum (0.667V) and minimum (0.333 V).

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    In summary, voltage maxima are located at n (1/2 , 1 , 1 1/2 ,) for resistive loads larger thanZ0 and at + n (1/4 , 3/4 , 11/4 , ) for resistive loads smaller thanZ0.

    Minima are located at away from maxima. The locations on the line corresponding to voltage maxima also

    correspond to current minima, and vice versa.

    In general, voltage maxima occur at locations where the incident andreflective waves are in phase (constructive interference). For a complex

    load impedance, the locations of voltage maxima are given by:

    =

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    ( )

    ( )

    ( )( )

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    +

    =

    +=

    +

    ==

    1

    1

    1

    1

    1

    1~

    ~

    0

    min

    max

    ZZ

    S

    S

    V

    VS

    L

    Example 2-5 (Slotted-line), P.54-55

    Special cases: Short-circuited line: A short-circuited line behaves like a discrete reactive

    component with:

    ljZZSCin tan0=

    For length between 0 and it is inductive with equivalent L:

    ==

    0

    10 tan1tan

    Z

    Ll

    lZL

    eq

    eq

    For length between and it is capacitive with equivalent C:

    ==

    0

    1

    0

    1tan

    1

    tan

    1

    ZCl

    lZC

    eq

    eq

    The reactive type repeats beyond l > following the same pattern (Fig.2-15d, P. 59).

    Open-circuited line: An open-circuited line behaves like a discrete reactive

    component with:

    l

    jZljZZOCin

    tancot 00

    ==

    For length between 0 and it is capacitive with equivalent C:

    ( )01

    0

    tan1tan

    ZClZ

    lC eqeq

    ==

    For length between and it is with inductive equivalent L:

    ==

    eq

    eqL

    Zl

    l

    ZL

    010 tan

    2

    1

    tan

    The reactive type repeats beyond l > following the same pattern (Fig.2-17d, P. 61).

    In microwave and high-speed circuits small C and L are often needed. It is

    much easier to fabricate them using transmission lines segments.

    Transmission parameters (Z0 and ) can be determined from the results ofOC and SC measurements (Example 2-8, P. 61).

    ==

    OC

    in

    SC

    inOC

    in

    SC

    in

    Z

    Z

    lZZZ

    1

    0tan

    1

    1/2 wave window: A 1/2 section of transmission line, regardless ofcharacteristic impedance, can be inserted into an existing transmission line

    system without disturbing the electrical properties of the system.

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    L

    L

    L

    L

    Lin Z

    jZZ

    jZZZ

    ljZlZ

    ljZlZZZ =

    +

    +=

    +

    +=

    )0()1(

    )0()1(

    sincos

    sincos

    1

    11

    1

    11

    1/4 wave transformer: A 1/4 section of transmission line (Z02) can beinserted into used to match any Z01 toZL.

    L

    LL

    L

    L

    Lin

    ZZZ

    ZjZ

    jZZ

    jZZ

    jZZZ

    ljZlZ

    ljZlZZZ

    =

    ==++

    =++

    =

    0102

    0102

    02

    02

    0202

    02

    0202

    )1(

    )1(

    )1()0(

    )1()0(

    sincos

    sincos

    Example: match a 50- line to a 200- load at 100 MHz (v= c/8)./4 = (c/8) / f / 4 = 0.094 m

    Z1 = (50 200)1/2 = 100 Summary of lossless transmission line properties (Table 2-3, P. 63):

    Zin is real at maxima and minima.

    At voltage maxima:Zin =Z0 [(1 + ||)/(1 ||)] At voltage minima:Zin =Z0 [(1 ||)/(1 + ||)]

    Power flow in ideal lines Instantaneous power:

    Average power:

    ZL

    Zin = Z01

    Z01

    l /4

    Z02

    ZLZ0

    l /2

    Z1

    Zin = ZL

    )(cos)(

    )(cos)(

    2

    0

    2

    2

    2

    0

    2

    r

    r

    i

    tZ

    VtP

    tZ

    VtP

    ++=

    +=

    +

    +

    +

    +

    [ ]20

    2

    2

    0

    2

    2

    0

    2

    12

    2

    2

    =

    ==

    =

    +

    +

    +

    Z

    V

    P

    PZ

    V

    P

    Z

    V

    P

    av

    i

    av

    r

    av

    i

    av

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    Chapter 2

    Part 2: Smith chart

    Introduction The primary objective of the Smith chart is to graphically compute the

    input impedance at any position along a transmission line terminated withan arbitrarily load.

    Powerful graphical tool for computing the complex impedance of

    transmission line circuits (Originally called transmission line calculator).

    Useful CAD presentation tool for displaying the performance (noise, gain,stability, ) of microwave circuits.

    Still a good way to design matching networks.

    Useful for lossless or lossy lines.

    Components of the chart Polar plot for the reflection coefficient at the load (Fig. 2-20, P. 67).

    Lines of a pair of parametric equations, one for the real (resistive) part andthe other for the imaginary (reactive) part, are plotted on the chart to

    indicate the normalized load impedance (zL = ZL /Z0)(Fig. 2-21, P. 69).

    Three concentric scales are on the perimeter of the chart (Fig. 2-22, P. 70).

    The innermost scale indicates the phase angle of the reflection coefficient.

    The other two scales indicate the distance, in terms of electrical length (),

    of the movement of the measurement plane.

    o The outermost scale WTG (toward generator) indicates the

    movement of the measurement plane toward the generator (away fromthe load).

    o The middle scale WTL (toward load) indicates the movement of the

    measurement plane toward the load (away from the generator). Graphical calculations

    The reflection coefficient along the line: The magnitude ofis constantalong the line. The only other unknown is the phase angle r.

    1) Plot the normalized load impedance (zL = ZL /Z0) on the chart.

    2) Draw a radial line through zL until it intersects the r scale.

    3) r is at the position where the radial line intersects the r scale.

    VSWR:1) Plot the normalized load impedance (zL = ZL /Z0) on the chart.

    2) Starting atzL, draw a complete circle centering at the origin. This is

    called the constant-|| circle, or more commonly known as the SWRcircle since S is a function of || only.3) Sequals to the value where the SWR circle intersects the = 0

    radial line.

    The first voltage minimum equals to the angular displacement betweenzL

    and the = 180 radial line using the WTG scale (Fig. 2-24, P. 74).

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    The first voltage maximum equals to the angular displacement between zL

    and the = 0 radial line using the WTG scale. You can also use lmin to obtain lmax.

    The normalized input impedance (zin = Zin /Z0)along the line:

    1) Plot the normalized load impedance (zL = ZL /Z0) on the chart.

    2) Draw a radial line through zL until it intersects the WTG scale. This isthe starting position WTGstart.

    3) From the physical distance between the measurement plane and the

    load, compute the equivalent electrical length (in ).4) From the WTG scale compute the equivalent angular movement.

    WTGfinal = WTGstart + 5) Draw a radial line throughWTGfinal.

    6) Starting atzL, draw a complete circle centering at the origin. This is

    called the constant-|| circle, or more commonly known as the SWRcircle since S is a function of || only.

    7) zin is at the position where the WTGfinal radial line intersects the theSWR circle.

    Zin is real at = 0 and is maximum (=Z0 S);Zin is real at = 180 andis minimum (=Z0 / S)

    Z-Yconversion: It is often more convenient to work with Ysince parallelcomponents are easier to insert into a transmission line circuit (???) and

    the result can be obtained by simply adding the Ys.

    In a Smith chartz(=Z/Z0) andy(= Y/Y0 = Y Z0) are 180 apart. Example 2-10, P. 76-78 and 2-11, P. 78.

    Examples 1-5

    Impedance matching Whenever a transmission line is not terminated with a matched load (ZL =

    Z0, portion of the incident energy will be reflected back. This usually

    causes two problems: 1) energy is wasted; 2) the reflected energy couldupset/damage the transmitter.

    Components are added between the generator and the load to achieve

    matching. This is called an impedance-matching network.

    The match point in a Smith chart is at the origin where zin=yin = 1 +j 0

    The single-stub matching technique:

    1) Since the stub, which is a section of transmission line with a slidingshort at the end, is connected in parallel to the line, it is more

    convenient to work with admittance than impedance. First transformzLintoyL by rotating thezL 180 along the SWR circle.

    2) Determine where to attach the stub by rotatingyL in the CW (WTG)direction along the SWR circle until it intersects the real circle = 1. At

    this location theyin has a real part = 1.

    3) There will be two points on the chart that satisfy the above condition(real{yin} = 1).

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    4) Measure the electrical length (in ) for the above rotation and convert

    it into physical distance d. This should be the distance between the

    load and the stub.5) Determine the imaginary part (susceptance) ofyin (=j B) from the

    chart.

    6) Starting from the short circuit point (where and why), rotate in the CW(WTG) direction along the SWR = circle (outer rim) until itintersects the imaginary line = -j B. At this location theyin of the stub

    has a susceptance = -j B.

    7) Measure the electrical length (in ) for the above rotation and convert

    it into physical distance l. This should be the length of the stub.

    Example 2-12, P. 80-84.

    Example 6Discrete conjugate matching: Handout

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    Chapter 2

    Part 3: Transient (Pulse) Transmission Line Responses

    Introduction So far we have studied the high-frequency behavior of transmission lines

    under harmonic excitations, in this part we are going to examine the pulseresponses of transmission lines.

    Because of reflections due to mismatch, an electrically long transmission

    line may respond irregularly to pulsed signals.

    Transient response of a line is a time record of the pulses bouncingbetween the transmitting and receiving end.

    Transmission line responses The steady-state response can be calculated using ordinary dc circuit

    analysis techniques.

    The transient response calculations must include the reflected waves

    between the generator and load.

    The reflections may cause severe distortion to the wave shape of the

    signal.

    They may even damage the receiver or transmitter because of EOS(electric overstress) or polarity reversal.

    Computing the transient response The transient response is usually difficult to compute. To simplify the

    analysis, a rectangular pulse is decomposed into two step functions. Thestep response is then computed and the pulse response is computed by

    adding the two step responses (Fig. 2-32, P. 84).

    Bounce diagram (Fig. 2-35, P. 89): The bounce diagram is used to keep

    track of the multiple reflections along the line. TDR: Measures the round-trip time as well as the shape of the reflected

    pulse to determine the distance and the type of discontinuity. Used forfault detection, location, and identification. It can also be used to measure

    the length of long transmission lines (Example 2-13, P. 89).

    Computer simulation results:

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    RG = RL = 10

    RG = RL = 100

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    RG = 10 RL = 100

    RG = 100 RL = 10

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    Chapter 2

    Appendix

    I. Transient Response

    The response of a circuit can be extremely complex. To simplify the analysis we

    often separate the response into two parts: Steady-state response: This response represents the behaviors of the circuit

    after a long period of time (until the effect of initial conditions has

    diminished to a negligible level). The steady-state response can usually becomputed using basic AC/DC circuit laws.

    Transient response: This response represents the behaviors of the circuit

    immediately after the input is applied. It is usually more difficult to solve

    since it involves both the input and the initial conditions.

    There are other classifications of the circuit responses such as frequency

    response vs. time response, force response vs. natural response, and zero-

    state response vs. zero-input response.

    Properties of transient response: Preshoot: A change of amplitude in the opposite direction before the pulse.

    Overshoot: The overshooting of the amplitude after the initial transition.

    Rounding: The rounding of the amplitude after the initial transition.

    Ringing: The peak-to-peak amplitude following the overshooting.

    Note: The above parameters are expressed as % of the steady-state amplitude.

    Period: The duration required for the waveform to repeat itself.

    Pulse width: The time duration between the 50% point of the leading and

    trailing edges of the pulse.

    Rise time: The duration between the 10 and 90% of the steady-state value.

    Settling time: The duration required to reach the steady-state amplitudeafter the pulse has reached the 100% point.

    Droop: The decrease of amplitude with time.

    Nonlinearity: The deviation from the line between the 10 and 90% points.

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    II. Electrical Component Parameters

    Conductance (G) / resistance (R): The basic electrical property of a resistor is howwell (poorly) it conducts electricity (AC and DC).

    R The resistance of a resistive component.

    G The conductance of a resistive component.

    R = 1 / G Reactance (X) / susceptance (B): The basic electrical property of a capacitor or an

    inductor (reactive component) is how much it shifts the phase angle of an AC

    signal at a particular frequency.

    X The reactance of a reactive component.

    XC = -1/2fC

    XL = 2fL

    B The susceptance of a reactive component.

    BC = 2fC

    BL = -1/2fL

    X= 1 /B Inductance (L) / capacitance (C):

    L: XL > 0, BL < 0

    C: XC < 0, BC > 0

    Impedance (Z) / admittance (Y): The combined property of a component or circuitwith both resistance and reactance.

    Z=R +jX

    Y= G +jB

    Z= 1 / Y

    ZYConversion:

    Zto Y:

    jBGY

    YBYG

    RR

    X

    RR

    X

    XRZY

    jXRZ

    YY

    ZY

    +===

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    jXRZ

    ZXZR

    GG

    B

    GG

    B

    BGYZ

    jBGY

    ZZ

    YZ

    +=

    ==

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