Adapted from notes by ECE 5317-6351 Prof. Jeffery …courses.egr.uh.edu/ECE/ECE5317/Class...

of 29 /29
Prof. David R. Jackson Dept. of ECE ECE 5317 - 6351 Microwave Engineering Fall 2019 1 Notes 5 Smith Charts Adapted from notes by Prof. Jeffery T. Williams

Embed Size (px)

Transcript of Adapted from notes by ECE 5317-6351 Prof. Jeffery …courses.egr.uh.edu/ECE/ECE5317/Class...

  • Prof. David R. JacksonDept. of ECE

    ECE 5317-6351 Microwave Engineering

    Fall 2019

    1

    Notes 5Smith Charts

    Adapted from notes by Prof. Jeffery T. Williams

  • Recall: ( ) ( ) ( )( )

    ( ) ( ) ( )( )

    ( ) ( )( )( )( )

    20 0

    20 0

    0 0

    2

    0 02

    1 1

    1 1

    111 1

    z z zL

    z z zL

    zL

    zL

    V z V e e V e z

    V VI z e e e zZ Z

    V z zeZ z Z ZI z e z

    γ γ γ

    γ γ γ

    γ

    γ

    + − + + −

    + +− + −

    +

    +

    = + Γ = + Γ

    = − Γ = − Γ

    + Γ + Γ= = = − Γ − Γ

    Generalized reflection Coefficient: ( ) 2 zLz e γ+Γ = Γ

    Generalized Reflection Coefficient

    2

    0

    0

    LL

    L

    Z ZZ Z

    −Γ =

    +

    0,Z β ( )V z

    ( )I z

    0z =z

    LZ+

  • ( )

    ( ) ( )

    2

    2L

    zL

    j zL

    R I

    z e

    e e

    z j z

    γ

    φ γ

    +

    +

    Γ = Γ

    = Γ

    = Γ + Γ

    Lossless transmission line (α = 0)

    ( ) ( )2Lj zLz e φ β+Γ = Γ

    Generalized Reflection Coefficient (cont.)

    { }Re 0 1L LZ ≥ ⇒ Γ ≤

    ( )( )( )( )

    0

    0

    0

    0

    L LL

    L L

    L L

    L L

    R jX ZR jX Z

    R Z jXR Z jX

    + −Γ =

    + +

    − +=

    + +

    ( )( )

    2 22 0

    2 20

    1L LLL L

    R Z XR Z X

    − +⇒ Γ = ≤

    + +

    Proof:

    3

    Different forms for Γ(z) Magnitude property of Γ(z)

  • Complex Γ Plane( )

    ( ) ( )( )

    ( )

    ( )

    ( )

    2

    2

    2

    2

    L

    L

    R I

    j zL

    j zL

    j dL

    j dL

    z

    z j z

    e

    e

    e

    e

    β

    φ β

    φ β

    β

    +

    +

    Γ = Γ

    = Γ + Γ

    = Γ

    = Γ

    = Γ

    = Γ

    Increasing d(moving towards

    generator)

    4

    ReΓ

    Im Γ

    2L dφ β−

    Γ

    Lossless line

    z d= −d = distance from load

    1

    Note: Going λ/2 on the line

    corresponds to going all the way around the Smith chart.

    2 dβ

    Clockwise movement!

  • ( ) ( )( )011

    zZ z Z

    z + Γ

    = − Γ

    ( ) ( ) ( )( )011n

    Z z zZ z

    Z z + Γ

    ≡ = − Γ Define

    n n nZ R jX= +

    Hence we have:

    Z Chart

    ( )( )

    11

    R In n

    R I

    jR jX

    j + Γ + Γ

    + = − Γ + Γ

    Next, multiply both sides by the RHS denominator term and equate real and imaginary parts. Then solve the resulting equations for ΓR and ΓI in terms of Rn or Xn. This gives two equations.

    5

    Note: The z dependence is being

    suppressed here.

    The Z chart is the “usual” Smith chart.

    Start with

  • 1) Equation #1:2 2

    2 11 1

    nR I

    n n

    RR R

    Γ − + Γ = + +

    Equation for a circle in the Γ plane

    ,011

    1

    n

    n

    n

    RR

    R

    = +

    =+

    Center

    Radius

    Z Chart (cont.)

    6

    1

    ΓR

    ΓI

  • ( )2 2

    2 1 11R In nX X

    Γ − + Γ − =

    Equation for a circle in the Γ plane

    11,

    1n

    n

    X

    X

    =

    =

    Center

    Radius

    2) Equation #2:

    7

    1

    ΓR

    ΓI

    Z Chart (cont.)

  • Short-hand version

    8

    Γ plane Γ plane

    Rn = 1

    Xn = 1

    Xn = -1

    Z Chart (cont.)

  • 9

    Open ckt. (Γ=1)

    Imag. (reactive)impedance

    Match pt. (Γ=0)

    Realimpedance

    Short ckt. (Γ= −1)

    Inductive (Xn > 0)

    Capacitive (Xn < 0)

    Γ planeRn = 1

    Xn = 1

    Xn = -1

    Γ plane

    Z Chart (cont.)

  • Note: ( ) ( )( )( )0

    11 11

    zY z

    Z z Z z − Γ

    = = + Γ

    ( )( )( )( )0

    11

    zY

    z + −Γ

    = − −Γ

    ( ) ( ) ( )( )( )( ) ( ) ( )011n n n

    zY zY z G z jB z

    Y z + −Γ

    ⇒ ≡ = = + − −Γ

    ( )0 01 /Y Z≡

    Admittance Calculations with the Z Chart

    Define:

    ( ) ( )z z′Γ ≡ − Γ( ) 1

    1nY z

    ′+ Γ = ′− Γ

    Same mathematical form as for Zn:

    Conclusion: The same Smith chart can be

    used as an admittance calculator.

    10

    ( ) 11n

    Z z + Γ = − Γ

  • 11

    Short ckt. (Γ′ =1)

    Imag. (reactive)admittance

    Match pt. (Γ′ =0)

    Realadmittance

    Open ckt. (Γ′ = −1)

    Capacitive (Bn > 0)

    Inductive (Bn < 0)

    Γ′ planeBn = 1

    Bn = -1

    Gn = 1

    Γ′ plane

    Admittance Calculations with the Z Chart (cont.)

  • Impedance or Admittance Calculations with the Z Chart

    12

    The Smith chart can be used for either impedance or

    admittance calculations, as long as we are consistent.

    The complex plane is either the Γ plane or the Γ′ plane.

    Normalized impedance or admittance coordinates

  • As an alternative way to do admittance calculations, we can continue to use the original Γ plane, and add admittancecurves to the chart.

    ( ) ( )( )( )( ) ( ) ( )11n n n

    zY z G z jB z

    z + −Γ

    = = + − −Γ

    Y Chart

    ( ) ( )( )( )( ) ( ) ( )11n n n

    zZ z R z jX z

    z + Γ

    = = + − Γ

    Compare with previous Smith chart derivation, which started with this equation:

    Side note: A 180o rotation on a Smith chart makes a normalized impedance become its reciprocal. 13

    Rn = 1 circle, rotated 180o, becomes Gn = 1 circle.Xn = 1 circle, rotated 180o, becomes Bn = 1 circle.

    ( ) ( )( ) ( )

    n n

    n n

    R z G z

    X z B z

    →( ) ( )z zΓ → −Γ (rotation of 180o)

    Examples:

  • Y Chart (cont.)

    14

    Open ckt.

    Match pt.

    Gn = 0

    Short ckt.

    Inductive (Bn < 0)

    Capacitive (Bn > 0)

    Gn = 1

    Bn = +1

    Bn = -1

    Bn = 0

    Γ planeΓ plane

    The Y chart is the “mirror image” of the usual Smith chart.

  • Short-hand version

    Γ plane

    15

    Gn = 1

    Bn = -1

    Bn = 1

    Y Chart (cont.)

  • All Four Possibilities for Smith Charts

    16

    Z chart, used for admittance

    Γ′ plane

    Z chart, used for impedance

    Γ plane

    The first two are the most common.

    The third is sometimes convenient.

    The fourth is almost never used.

    1 2

    Γ′ plane

    4Y chart, used for impedance

    Γ plane

    3Y chart, used for admittance

  • ZY Chart

    17

    This is convenient for doing matching problems that involve both series and shunt elements (done later).

    Short-hand version

    Γ plane

    Gn = 1 Rn = 1

    Xn = 1

    Xn = -1

    Bn = -1

    Bn = 1

    Inductive

    Capacitive

  • The SWR is given by the value of Rn on the positive real axis of

    the Smith chart (Rnmax).

    Standing Wave Ratio

    Proof:1

    SWR1

    L

    L

    + Γ=

    − Γ

    ( )( )

    2

    2

    2

    2

    11

    1111

    L

    L

    n

    j zL

    j zL

    j j zL

    j j zL

    zZ

    z

    eee ee e

    β

    β

    φ β

    φ β

    +

    +

    +

    +

    + Γ=

    − Γ

    + Γ=

    − Γ

    + Γ=

    − Γ

    18

    max 12 01

    LL n

    L

    z Rφ β+ Γ

    + = ⇒ =− Γ

    maxn nR R=

    2 0L zφ β+ =2L zφ β+

    Γ plane

    ( )zΓ

  • At this link:

    http://www.sss-mag.com/topten5.html

    Download the following zip file:smith_v191.zip

    Extract the following files:

    smith.exe mith.hlp smith.pdf

    This is the application file

    Electronic Smith Chart

    19

    http://www.sss-mag.com/topten5.html

  • 0 50100 50L

    ZZ j

    = Ω= + Ω

    ,0

    2 1LL nZZ jZ

    = = +

    ( )

    / 40.4 0.2

    / 4 20 10

    g

    n

    g

    dZ j

    Z j

    λ

    λ

    =

    ≈ −

    ⇒ − ≈ − Ω

    Example 1

    a

    20

    ( )I d−

    ( )V d−

    z d= −

    +

    0z =z

    LZ(- ) / 1 / 4, 3 / 8, 1 / 2gZ d d λ =Find at

    Use the Z chart.

    a

    b

    Z chart

    Γ plane

    ,L nZ

    1nR =

    1nX =

    1nX = −

  • ( )

    3 / 80.5 0.5

    3 / 8 25 25

    g

    n

    g

    dZ j

    Z j

    λ

    λ

    =

    ≈ +

    ⇒ − ≈ + Ω

    b

    ( )

    / 22 1

    / 2 100 50

    g

    n

    g

    dZ j

    Z j

    λ

    λ

    =

    ≈ +

    ⇒ − = + Ω

    c

    Example 1 (cont.)

    21

    3 / 8 0.212 0.5 0.087+ − =Note :

    ( )I d−

    ( )V d−

    z d= −

    +

    0z =z

    LZ

    a

    b

    0.087λg

    c0.5λg

    0.462λg

    0.212λgZ chart

    Γ plane

    0.0 gλ0.5 gλ

  • ( )0 050 20mS8mS 4 mSL

    Z YY j

    = Ω =

    = −

    ,0

    0.4 0.2LL nYY jY

    = = −

    ( )

    / 42 1

    / 4 40mS 20mS

    g

    n

    g

    dY j

    Y j

    λ

    λ

    =

    ≈ +

    ⇒ − ≈ +

    Example 2

    a

    22

    ( )I d−

    ( )V d−

    z d= −

    +

    0z =z

    LZ

    (- ) / 1 / 4, 3 / 8, 1 / 2gY d d λ =Find at

    Use the Y chart.

    a

    bc

    Y chart

    Γ plane

    ,L nY

    1nB = −

    1nB =

    1nG =

  • ( )

    3 / 81 1

    3 / 8 20mS 20mS

    g

    n

    g

    dY j

    Y j

    λ

    λ

    =

    ≈ −

    ⇒ − ≈ −

    ( )

    / 20.4 0.2

    / 2 8mS 4mS

    g

    n

    g

    dY j

    Y j

    λ

    λ

    =

    ≈ −

    ⇒ − = −

    Example 2 (cont.) b

    c

    23

    ( )I d−

    ( )V d−

    z d= −

    +

    0z =z

    LZ

    a

    bc

    Y chart

    Γ plane

    ,L nY

    1nB = −

    1nB =

    1nG =

  • Use a short-circuited section of air-filled TEM, 50 Ω transmission line(β = k0, λg = λd =λ0) to create an impedance of Zin = -j25 Ω at f = 10 GHz.

    ( ),25 0.550in n

    Z j j= − = −

    00.426 0 0.426 0.426g g gL λ λ λ λ= − = =

    Example 3

    24

    00 0 0

    2 2cf k

    π πλω µ ε

    = = =0 3.0 cmλ =

    25inZ j= − Ω

    050 , kΩ SC

    L

    1.28cmL =

    Z chart

    Γ plane

    SC

    -1/2

    50Ω

    Use the Z chart.

  • Use an open-circuited section of 75 Ω (Y0 = 1/75 S) air-filled transmission line at f = 10 GHz to create an admittance of Yin = j1/75 S:

    0.375cmL =

    Example 4

    25

    00.125 0.125gL λ λ= = 0 3.0 cmλ =

    ( ) ( ),1 / 75 S 1in in nY j Y j= ⇒ =

    075 , kΩ OC

    L

    Y chart

    Γ plane

    j1

    1/75 S

    OC

    L

    1nB = −

    1nB =

    Use the Y chart.

  • Example 5

    0 50Z = Ω

    100 100LZ j= + Ω

    ( ),1 0.25 .25

    2 2L nY j

    j= = −

    +

    /6 o0.62 0.62 30jL eπΓ = = ∠0

    0

    11

    L LnL

    L Ln

    Z Z ZZ Z Z

    − −Γ = =

    + +26

    In this example we will use the “usual”

    Smith chart (Z chart), but as an admittance

    calculator.

    Single-stub matching

    , 2 2L nZ j= +

    0Z

    0sZ

    LZ

    d

    slWe want Gin = Y0 (Gin,n = 1)

    0 0sZ Z=Choose

  • Example 5 (cont.)

    0.178 gλ

    , 0.25 0.25L nY j= −

    1 1.57j+

    1 1.57j−

    0.322 gλ

    0.041 gλ

    0.219- 1.57

    = 1.57 0.363gn

    gn

    dY

    Y j d

    j λ

    λ

    = =

    = +

    Add ator at

    Solution:0.170 0.2.0 981 14 gλ λλ =+0.320 0.3.0 321 64 gλ λλ =+

    wavelengths toward loadwavelengths toward generator

    Smith chart scale:

    Z chart

    27

    Γ′ plane

    (We’ll use the first choice.)

    SCOC

    1nG =

  • Example 5 (cont.)

    ( )( )

    ( )

    ( )

    0

    0

    ,

    1

    tan

    cot

    cot

    1.57 cot1cot 1.57; tan 0.637

    1.572 tan 0.637 0.567 [rad]

    scin s

    scin s

    s n s

    s

    s s

    s sg

    Z jZ l

    Y jY l

    B l

    l

    l l

    l l

    β

    β

    β

    β

    β βπβ

    λ−

    =

    = −

    = −

    − = −

    = = =

    = = =

    0.09s gl λ=From the Smith chart:

    0.0903s gl λ=

    Analytically:

    28

    S / C

    0 1.57j−

    0.09 gλ

    O / C

    Z chart

    Γ′ plane

    0.25 gλ

    0.34 gλ

    wavelengths toward generator

  • Example 5 (cont.)

    0 50Z = Ω

    100 100LZ j= + Ω

    29

    0Z

    0sZ

    LZ

    d

    sl

    Single-stub matching

    0.09s gl λ=

    0.219 gd λ=

    Summary:

    Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29