ECE 598 JS Lecture 04 Transmission Linesjsa.ece.illinois.edu/ece598js/Lect_04.pdfTitle Microsoft...

Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 1

ECE 598 JS Lecture ‐04Transmission Lines

Spring 2012

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]


Faraday’s Law of Induction

Ampère’s Law

Gauss’ Law for electric field

Gauss’ Law for magnetic field

Constitutive Relations

BEt

∂∇× = −

∂

H J∇× = Dt

∂+

∂

D ρ∇ ⋅ =

0B∇ ⋅ =

B Hμ= D Eε=

MAXWELL’S EQUATIONS


WAVE PROPAGATION

λ

λ

z

Wavelength : λ

λ = propagation velocity

frequency


Why Transmission Lines ?

In Free Space

At 10 KHz : λ= 30 km

At 10 GHz : λ = 3 cm

Transmission line behavior is prevalent when the structural dimensions of the circuits are comparable to the wavelength.


Let d be the largest dimension of a circuit

If d << λ, a lumped model for the circuit can be used

Transmission Line Model

circuit

z

λ


Transmission Line Model

circuit

z

λ

If d ≈ λ, or d > λ then use transmission line model


CT/2

LT

CT/2

CT

LT/2 LT/2

Zo

Short

LumpedReactive CKT

TransmissionLine

Low FrequencyMid-rangeFrequency High Frequency

or

Modeling Interconnections


TEM PROPAGATION

L

Δz

C

I

V

+

-

z

Zo βZ1 Z2

Vs


Telegrapher’s Equations

L: Inductance per unit length.

C: Capacitance per unit length.

L

Δz

C

I

V

+

-

V ILz t

∂ ∂− =

∂ ∂

I VCz t

∂ ∂− =

∂ ∂


Single wire near ground

+

h

d

o1 4for d << h, Z = ln

2h

dμ

π ε⎛ ⎞⎜ ⎟⎝ ⎠

-1oZ = 120 cosh (D/d)

2C = 4hlnd

πε⎛ ⎞⎜ ⎟⎝ ⎠

4L = ln2

hd

μπ


Single wire between grounded parallel planes

ground return

h/2

d+

h/2

o1 4Z = ln

2hd

μπ ε π

⎛ ⎞⎜ ⎟⎝ ⎠


Wires in parallel near ground

D

d++

h

for d << D, h

( ) ( ) ( ){ }21069 / log 4 / 1 2 /O rZ h d h Dε= +


Balanced, near ground

D

d+

h

for d << D, h

( ) ( )( )

10 2

2 /276 / log

1 / 2O r

D dZ

D hε

⎧ ⎫⎪ ⎪= ⎨ ⎬

+⎪ ⎪⎩ ⎭


D

d

Open 2‐wire line in air

-1oZ = 120 cosh (D/d)


Parallel-plate Transmission Line

μ, ε

w

a

L = μaw

C = εwa


εr

Coaxial line

Air

WaveguideCoplanar line

er

er

Microstrip

er

Stripline

er

Slot line

Types of Transmission Lines


Coaxial Transmission Line

abμ

ε

L = μ ln ba

C = 2πε

ln(b/a)

TEM Mode of Propagation


Coaxial Air Lines

abμ

ε

( )/ln /

2oZ b aμ επ

=

( ) ( ) ( )1/ 1//ln / 1 1

2 4 ln( / )o

a bZ b a j

f b aμ επ π μσ

⎡ ⎤+= + −⎢ ⎥

⎢ ⎥⎣ ⎦

Infinite Conductivity

Finite Conductivity


Coaxial Connector Standards

abμ

ε

Connector Frequency Range14 mm DC - 8.5 GHzGPC-7 DC - 18 GHzType NDC - 18 GHz3.5 mm DC - 33 GHz2.92 mm DC - 40 GHz2.4 mm DC - 50 GHz1.85 mm DC - 65 GHz1.0 mm DC - 110 GHz


ε

Microstrip


Microstrip

0 1 2 330

40

50

60

70

80

90

100

h = 21 milsh = 14 milsh = 7 mils

Microstrip Characteristic Impedance

W/h

Zo (o

hms)

dielectric constant : 4.3.


Telegrapher’s Equations

L: Inductance per unit length.

C: Capacitance per unit length.

L

Δz

C

I

V

+

-

V ILz t

∂ ∂− =

∂ ∂

I VCz t

∂ ∂− =

∂ ∂


( ) j zV z Ae β−= j zBe β++

z

Zo β

forward wave

backward wave

1( ) j z

o

I z AeZ

β−⎡ ⎤= ⎣ ⎦j zBe β+−

LCβ ω=

Zo = LC

Transmission Line Solutions(Frequency Domain)


21 2

( ) 1

sj l

TVAe β

ω−=

− Γ Γ

22 j lB e Aβ−= Γ

1

o

o

ZTZ Z

=+

11

1

o

o

Z ZZ Z

=−

Γ+

22

2

= o

o

Z ZZ Z

−Γ

+

Transmission Line Equations

o

lvωβ =

z

Zo βZ1 Z2

Vsl

(Frequency Domain)


2 /1 1 2

0( ) oj kl vk k

kA TV e ωω

∞−

=

= Γ Γ∑ 2 ( 1) /11 1 2

0( ) oj k l vk k

kB TV e ωω

∞− ++

=

= Γ Γ∑

2 / /1 2 1

0( ) ( )o oj kl v j z vk k

fk

V z e e TVω ω ω∞

− −

=

= Γ Γ∑

2 ( 1) / /11 2 1

0( ) ( )o oj k l v j z vk k

bk

V z e e TVω ω ω∞

− + ++

=

= Γ Γ∑

( ) ( ) ( ) f bV z V z V z= +

Transmission Line Equations(Frequency Domain)


1 20

2( , ) k kf

k o

z klV z t T V tv

∞

=

⎛ ⎞+= Γ Γ −⎜ ⎟

⎝ ⎠∑

11 2

0

2( 1)( , ) k kb

k o

z k lV z t T V tv

∞+

=

⎛ ⎞− += Γ Γ −⎜ ⎟

⎝ ⎠∑

( , ) ( , ) ( , )f bV z t V z t V z t= +

Time‐Domain Solutions


Wave Shifting Solution*

1 1 2 1 1( ) ( ) ( )f b gv t v t TV tτ= Γ − +

2 2 1 2 2( ) ( ) ( )b f gv t v t T V tτ= Γ − +

2 1( ) ( )f fv t v t τ= −

1 2( ) ( )b bv t v t τ= −

11

1

o

o

Z ZZ Z

−Γ =

+

11

o

o

ZTZ Z

=+

22

2

o

o

Z ZZ Z

−Γ =

+

*Schutt-Aine & Mittra, Trans. Microwave Theory Tech., pp. 529-536, vol.36 March 1988.

1 1 1( ) ( ) ( )f bv t v t v t= +

2 2 2( ) ( ) ( )f bv t v t v t= +

line lengthdelay=

velocityτ =

Vg1

ZoZ1 Z2τ

vf1 vf2

vb2vb1

Vg2

22

o

o

ZTZ Z

=+


Frequency Dependence of Lumped Circuit Models

– At higher frequencies, a lumped circuit model is no longer accurate for interconnects and one must use a distributed model

– Transition frequency depends on the dimensions and relative magnitude of the interconnect parameters.

f ≈0.3 ×109

10d εrtr ≈

0.35f


Lumped Circuit or Transmission Line?

• Determine frequency or bandwidth of signal– RF/Microwave: f= operating frequency– Digital: f=0.35/tr

• Determine the propagation velocity and wavelength– Material medium v=c/(εr)1/2

– Obtain wavelength λ=v/f

• Compare wavelenth with feature size – If λ>> d, use lumped circuit: Ltot= L* length, Ctot= C* length– If λ ≈ 10d or λ<10d, use transmission-line model


PCB line 10 in > 55 MHz < 7nsPackage 1 in > 400 MHz < 0.9 nsVLSI int* 100 um > 8 GHz < 50 ps

Level Dimension Frequency Edge rate

* Using RC criterion for distributed effect

Frequency Dependence of Lumped Circuit Models

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