Transmission Line Theory - ITESO - Home

Signal Integrity and High-Speed InterconnectsJanuary-May 2006

Dr. J. E. Rayas Sánchezhttp://iteso.mx/~erayas [email protected]

1

Transmission Line Theory(Part 3)

Dr. José Ernesto Rayas Sánchez

2Dr. J.E. Rayas Sánchez

Outline

Input impedance in lossless transmission lines (TL)

Special cases of lossless terminated TL

Insertion loss

Transmission coefficient

Smith Chart interpretation

Basic Smith Chart applications




Input Impedance in Lossless TL

Zo , β

0l

ZL

)()( zjzjo eeVzV ββ Γ +−+ +=

)()( zjzj

o

o eeZVzI ββ Γ +−

+

−=

lj

lj

oljlj

ljlj

oin eeZ

eeeeZ

lIlVlZ β

β

ββ

ββ

ΓΓ

ΓΓ

2

2

11

)()()( −

−

−

−

−+=

−+==

l

loin ZlZ

ΓΓ

−+=

11)(

oin

oinl ZlZ

ZlZ+−=

)()(Γ


Input Impedance in Lossless TL

)tan()tan(

)()()(

ljZZljZZZ

lIlVlZ

Lo

oLoin β

β++==

Lin ZlZ == ),2

3,,2

,0( Lλλλ

λπβ 2 Since =

L

oin Z

ZlZ2

),4

5,4

3,4

( == Lλλλ

(Period = λ/2)

Zo , β

0l

ZL

)()( zjzjo eeVzV ββ Γ +−+ +=

)()( zjzj

o

o eeZVzI ββ Γ +−

+

−=

βl is the electrical length




Short-Circuited Lossless TL

Zo , β

0l

)()()( zjzjo

zjzjo eeVeeVzV ββββ Γ +−++−+ −=+=

)()()( zjzj

o

ozjzj

o

o eeZVee

ZVzI ββββ Γ +−

++−

+

+=−=

1−=+−=

oL

oL

ZZZZΓ

zjVzV o βsin2)( +−= zZVzI

o

o βcos2)(+

=


Short-Circuited Lossless TL (cont)

)tan()tan()(

ljZZljZZZlZ

Lo

oLoin β

β++=Zo , β

0l )tan()( ljZlZ oin β=

Zin is purely reactive

C

LLL

C




Open-Circuited Lossless TL

Zo , β

0l

1+=+−=

oL

oL

ZZZZΓ

)()( zjzjo eeVzV ββ +−+ +=

)()( zjzj

o

o eeZVzI ββ +−

+

−=

zVzV o βcos2)( += zZVjzIo

o βsin2)(+

=


Open-Circuited Lossless TL (cont)

)tan()tan()(

ljZZljZZZlZ

Lo

oLoin β

β++=

)cot()tan(

)( ljZlj

ZlZ oo

in ββ

−==

Zin is purely reactive

Zo , β

0l

LL

C CC




Transmission Coefficient, T

It is a measure of the capacity of a TL to transmit an incoming wave

0for )()( ≤+= +−+ zeeVzV zjzjo

ββ Γo

o

ZZZZ

+−=

1

1Γinfinite) TL2 (assuming 0for )( ≥= −+ zTeVzV zj

oβ

TL2TL1

++ =+= oo TVVV ]1[)0( Γ Γ+=1T


Insertion Loss, IL

It is a measure of the reflections caused by an inserted TL

TL2TL1

(dB) ||log20 TIL −=

dB)0|(| ∞==TIL dB0)1|(| ==TIL




Lightwave Analogy to RF Energy

RF

Incident

Reflected

Transmitted

Lightwave

(Hewlett-Packard's RF Design and Measurement Seminar, 2000)


Reflection Parameters

∞ dB

No reflection(ZL = Zo)

ρRL

VSWR

0 1

Full reflection(ZL = open, short)

0 dB

1 ∞

=ZL − ZO

ZL + OZReflectionCoefficient =

Vreflected

Vincident= ρ ΦΓ

=ρ ΓReturn loss = -20 log(ρ),

VSWR = Emax

Emin=

1 + ρ1 - ρ

Voltage Standing Wave RatioEmaxEmin





Transmission Parameters


V TransmittedV Incident

Transmission Coefficient = Τ =VTransmitted

VIncident= τ∠φ

DUT

Gain (dB) = 20 Log V Trans

V Inc = 20 log τ

Insertion Loss (dB) = - 20 Log V Trans

V Inc = - 20 log τ

Insertion Phase (deg) = V Trans

V Inc = φ


The Smith Chart

Developed in 1939 by P. H. Smith at Bell LabsVery useful for visualizing transmission line phenomena and impedance matching problemsIt is part of most current CAD tools and modern measurement equipmentsIt is essentially a plot of Γ in polar coordinatesAny impedance can be mapped in the Γ plane




The Smith Chart (cont)

-90 o

0o180 o+-.2

.4.6

.8

1.0

90 o

Polar plane θΓΓ j

oL

oL eZZZZ ||=

+−=

(R. Ludwig and P. Bretchko, RF Circuit Design, Prentice Hall, 2000)

)Γ( o=Γ


The Smith Chart (cont)

θΓΓ j

o

o eZZZZ ||=

+−=

Normalizing Z w.r.t Zo

θΓ jezz ||

11 =

+− where

oZZz =

θ

θ

ΓΓ

j

j

eez

||1||1

−+=

ir

ir

jjjxrΓΓΓΓ

−−++=+

)1()1(

22

22

)1(1

ir

irrΓΓ

ΓΓ+−

−−=

22

2

)1(2

ir

ixΓΓ

Γ+−

=

22

2

11

1

+=+

+−

rrr

ir ΓΓ

( )22

2 111

=

−+−

xxir ΓΓSolving for r and x

Rearranging




The Smith Chart – Circles of Constant r


22

2

11

1

+=+

+−

rrr

ir ΓΓ


The Smith Chart – Circles of Constant x


( )22

2 111

=

−+−

xxir ΓΓ




The Smith Chart – Combining both Circles



The Smith Chart – Interpretation

0=LZ

0.2

0.5

1.0

2.0

5.0

+0.2

-0.2

+0.5

-0.5

+1.0

-1.0

+2.0

-2.0

+5.0

-5.0

0.0 ∞ ∞=LZ

oL ZZ =

oL jZZ −=

oL jZZ +=

(open circuit)(short circuit)

(pure inductive)

(pure capacitive)

(matched)Circles of constant XL

Circles of constant RL

0=Γ

o1801∠=Γo01∠=Γ

o901∠=Γ

o901 −∠=Γ




Practical Smith Chart


Practical Smith Chart (cont)




Smith Chart – Calculating Γ, RL, SWR

Example:Ω50=oZ

Ω 7525 jZL +=

5.15.0 jzL +=

Γ = ?RL = ?SWR = ?


Smith Chart – Calculating Γ, RL, SWR (cont)

Example:Ω50=oZ

Ω 7525 jZL +=

63.4o

5.15.0 jzL +=o4.63|| ∠= ΓΓ





Example:Ω50=oZ

Ω 7525 jZL +=

5.15.0 jzL +=o4.63|| ∠= ΓΓ



Example:Ω50=oZ

Ω 7525 jZL +=

5.15.0 jzL +=o4.6375.0 ∠≈Γ

0.75

7

2.57≈SWR

dB 5.2≈RL




Smith Chart – Circles of SWR

Example:Ω50=oZ

Ω 7525 jZL +=

5.15.0 jzL +=o4.6375.0 ∠≈Γ

7≈SWRdB 5.2≈RL

63.4o

SWR = 7


Smith Chart – Circles of SWR (cont)

Examples:Ω50=oZ

Ω 2575 c) jZL +=

Ω 50 a) =LZΩ 5.48 b) =LZ

Ω 510 d) jZL −=




Smith Chart – Calculating Γl

ljlj

o

olj

o

ljo

l eeVV

eVeVl ββ

β

β

ΓΓ 22)( −−+

−

+

−−

===

Zo , β

0l

ZL

λπβ 2 Since =

(Period = λ/2)lj

l el λπ

ΓΓ22

)(−

=

thsin wavelnglength theis λl


Smith Chart – Calculating Zin

Zo , β

0l

ZL

oin

oinl ZlZ

ZlZ+−=

)()(Γ

lj

l el λπ

ΓΓ22

)(−

=

thsin wavelnglength theis λl




Smith Chart – Calculating Γl and Zin

Example:Ω50=oZ

Ω 7525 jZL +=cm1=lGHz1=f4=rε?=lΓ?=inZ

5.15.0 jzL +=o4.6375.0 ∠≈Γ

63.4o

0.75


63.4o

0.75

Smith Chart – Calculating Γl and Zin (cont)

Example:Ω50=oZ


o4.6375.0 ∠≈Γ

r

p

fc

fv

ελ ==

cm154GHz1

Gm/s3.0 ==λ




63.4o

0.75


Example:Ω50=oZ


o4.6375.0 ∠≈Γ

067.0cm15

cm1 ==λl

229.0067.0162.0 =+


63.4o

0.75

15o


Example:Ω50=oZ


o4.6375.0 ∠≈Γ

0667.0cm15

cm1 ==λl

229.0067.0162.0 =+

o1575.0 ∠≈lΓ




63.4o

0.75

15o


Example:Ω50=oZ


o4.6375.0 ∠≈Γo1575.0 ∠≈lΓ

4.38.3 jzin +=Ω )4.38.3(50 jZin +=Ω 170190 jZin +≈


Smith Chart – Circles of SWR, Vmax and Vmin

Example:Ω50=oZ

Ω 4060 jZL +=Ω 8.02.1 jzL +=

1.2≈SWR

θΓΓ je|| where =|||1||||)(| Since )2( lj

o eVlV βθΓ −++ +=

We can locate the positions of Vmax and Vmin along the line

56o

Circle of SWR = 2.1

Γl for VmaxΓl for Vmin

|Γl| = 0.35

0.078 wavelengths

172.025.0max −=WL078.0max =WL

o5635.0 ∠≈Γ

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