ECE 391 supplemental notes - #2 - Oregon State Universitytraylor/ece391/...Coax Microstrip...

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1 ECE 391 supplemental notes - #2 32 Oregon State University ECE391– Transmission Lines Spring Term 2014 Microstrip – Effective Dielectric Constant w h r r 10 1 1 2 1 2 1 eff + + + ε ε ε 10 -1 10 0 10 1 10 2 0 2 4 6 8 10 w/h ε eff ε r = 10 ε r = 4 ε r = 2 0 t

Transcript of ECE 391 supplemental notes - #2 - Oregon State Universitytraylor/ece391/...Coax Microstrip...

Page 1: ECE 391 supplemental notes - #2 - Oregon State Universitytraylor/ece391/...Coax Microstrip Characteristic Impedance of TLs . 3 35 Oregon State University ECE391– Transmission Lines

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ECE 391 supplemental notes - #2

32 Oregon State University ECE391– Transmission Lines Spring Term 2014

Microstrip – Effective Dielectric Constant

whrr

1011

21

21

eff +−++≈ εεε

10-1 100 101 1020

2

4

6

8

10

w/h

ε ef

f

εr = 10

εr = 4

εr = 2

0→t

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33 Oregon State University ECE391– Transmission Lines Spring Term 2014

Transmission Line Comparison

coaxial line

two-wire line (also twisted-pair)

microstrip

r s ( )rsZ 2cosh1 10

0−=

εµ

π

w t

h εr

ε0

⎟⎠⎞⎜

⎝⎛

++Ω=

twhZ

r 8.098.5ln

41.187

0 ε

a b

c

rε Z0 =µ

ε0εr

ln b a( )2π

34 Oregon State University ECE391– Transmission Lines Spring Term 2014

0 2 4 6 8 100

100

200

300

400

w/h or D/d or b/a

Z 0 ( Ω)

εr = 2.25 ___

εr = 1 - - -

Two-wire line

Coax

Microstrip

Characteristic Impedance of TLs

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35 Oregon State University ECE391– Transmission Lines Spring Term 2014

Launching a Wave on an Infinitely Long Transmission Line

VS(t)

RS )()( 0 tuvtvs = i+

i+

v+

+

_

v+ (t) = v0 u(t)Z0

Z0 + Rs

Z0, vp

i+ (t) = v+ (t)Z0

= v0 u(t)1

Z0 + Rs

RS

Z0

i+

i+

v+

+

_ VS(t)

36 Oregon State University ECE391– Transmission Lines Spring Term 2014

Wave Propagation on Transmission Line RS

VS

Z0, vp!

v1+ (z, t) = v0

Z0Rs + Z0

u(t − z / vp ) = v0Z0

Rs + Z0u(z − vpt)

i1+ (z, t) = v0

1Rs + Z0

u(t − z / vp ) = v01

Rs + Z0u(z − vpt)

First traveling wave

)()( 0 tuvtvs =),(),( 101 tziZtzv ++ =

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37 Oregon State University ECE391– Transmission Lines Spring Term 2014

Wave Propagation on Transmission Line RS

VS

Z0, vp!

)()( 0 tuvtvs =

zz1

v1+ (z, t = z1 vp ) = v0

Z0Rs + Z0

u(t1 − z / vp ) = v0Z0

Rs + Z0u((z1 − z) / vp )

i1+ (z, t = z1 vp ) = v0

1Rs + Z0

u(t1 − z / vp ) = v01

Rs + Z0u((z1 − z) / vp )

z20

v(z, t = z1 vp )vp

v0Z0

Rs + Z0t = z2 vp

38 Oregon State University ECE391– Transmission Lines Spring Term 2014

Transmission Line Circuit

Vr = ρL Vi

RS

RL VS

Z0, td!td = length/velocity

reflected wave

)/(1),(

)/(),(

001

0

001

ps

ps

vztuZR

vtzi

vztuZR

Zvtzv

−+

=

−+

=

+

+

First traveling wave

)()( 0 tuvtvs =

),(),( 101 tziZtzv ++ =

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Reflection Coefficient

Vr = ρL Vi

RS

RL VS reflected wave

{ })()(1)()(

)()(

110

11

11

dd

ddL

ddL

ttvttvZ

ttittiittvttvv

−−−=

−+−=

−+−=

−+

−+

−+

At load side at time t ≥ td"

)()( 0 tuvtvs =

0

0

ZRZR

VV

L

L

i

rL +

−==ρ

ρL

Solving for " LLL iRv =

Load-side reflection coefficient!

ρs

Z0, td!td = length/velocity

40 Oregon State University ECE391– Transmission Lines Spring Term 2014

Traveling Waves on TL

Outgoing waves:

Returning waves:

v1+ (z, t) = v1

+ (z − vpt) = v1+ (t − z vp )

v2+ (z, t) = ρsρLv1

+ (t − z vp − 2td )

1st

2nd

1st v1−(z, t) = v1

−(z+ vpt) = v1−(t + (z− zr ) vp )

v1− (z, t) = ρLv1

+ (t + (z − zr ) vp − td )zr is the length of the line

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Step Response of T-Line Circuit

Ls

L

Ls

L

s RRRVV

ZRZV

+=

−+

+=∞ 00

0

0

11

ρρρ

LsLs

L

s RRVV

ZRI

+=

−−

+=∞

1111

000 ρρ

ρ

V1+

42 Oregon State University ECE391– Transmission Lines Spring Term 2014

Reflection Diagram (also called Lattice or Bounce diagram)

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Load Reflection Coefficient ρL

V1+ +V1

− = RL I1+ + I1

−( )= RL

V1+

Z0+ V1

−Z0

⎛⎝⎜

⎞⎠⎟

ρL =VreflVinc

= V1−

V1+ = RL − Z0

RL + Z0

44 Oregon State University ECE391– Transmission Lines Spring Term 2014

VS − V1+ +V1

− +V2+( ) = RS I1

+ + I1− + I2

+( )

ρS =VreflVinc

= V2+

V1− = RS − Z0

RS + Z0

VS −V1+ = RS I1

+

− V1− +V2

+( ) = RS V1−

−Z0+ V2

+

Z0

⎛⎝⎜

⎞⎠⎟

For 1st outgoing wave:

For 2nd outgoing wave:

Source Reflection Coefficient ρS

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Example 25Ω

83.333Ω 50Ω 1V

ρL =RL − Z0RL + Z0

= 14

ρS =RS − Z0RS + Z0

= − 13

-1/3 1/4

V1+ = 50

751V= 2

3V

2/3 V

1/6 V

-1/18 V

-1/72 V

1/216 V

2/150 A

-1/300 A

-1/900 A

1/3600 A

1/10800 A

46 Oregon State University ECE391– Transmission Lines Spring Term 2014

Voltage and Current Step Response

lz 41=

0.667

1.111

0.963

0.8642 0.8971

667.0=Lρ334.0−=Sρ

2

0.667

0.222 0.0741 0.1728 0.2058

V0

0.9091 V∞

0.1818 I∞

20ZRs = 05ZRL =

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Single-Line Transient Responses

-1 0 1 2 3 4 5 6 7 8 9 1000.20.40.60.8

11.21.41.6

Time/TD

Volta

ge (v

olts

)

sourcenear endfar end

-1 0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

Time/TD

Volta

ge (v

olts

)

sourcenear endfar end

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

Time/TD

Volta

ge (v

olts

)

sourcenear endfar end

-1 0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

Time/TD

Volta

ge (v

olts

)

sourcenear endfar end

48 Oregon State University ECE391– Transmission Lines Spring Term 2014

Example RS

RL=0!VS

)()( 0 tuvtvs =

Z0, td!td = length/velocity

z

t/msec

z zr

ρL ρs

0

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Ideal Transmission Line – SPICE Model §  Ideal Model

§  Spice Implementation

Load VS

Z0,TD

+ -

RS

0

,

0

,

)(

)()()(

)(

)()()(

ZTDtiTDtVTDtVtV

ZTDtiTDtVTDtVtV

in

inrefinout

out

outrefoutin

−+

+−−−=

−+

+−−−=

Z0 =LC

TD = zr LC

50 Oregon State University ECE391– Transmission Lines Spring Term 2014

Equivalent Circuit for t < 3td iL

iL

vL Z0 RL +

_

iL

iL

vL RL +

_

vL (t) = 1+ ρL( ) Z0Rs + Z0

Vs  U t − td( )

vL (t) =RL

RL + Z02 Z0Rs + Z0

Vs  U t − td( )

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Reactive and Nonlinear Terminations

•  The equivalent circuit representation at the near or far end of a transmission line facilitates analysis of -  reactive terminations (capacitive, inductive) -  nonlinear terminations (see later).

52 Oregon State University ECE391– Transmission Lines Spring Term 2014

Reactive Terminations Reactive terminations occur e.g. as

-  capacitive input of devices (buffers) -  shunt capacitance of pads -  inductance of bond wire -  discontinuities in PCB traces (step,

bend, via …) -  parasitics of packaging leads

Source: Google Images

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Capacitive Termination

RS = Z0

VS

Z0, td!td = length/velocity

C

vcap (t) = ?

)()( 0 tuvtvs =

54 Oregon State University ECE391– Transmission Lines Spring Term 2014

Equivalent Circuit at Termination

Rth = Z0

Vth=2Vinc(t-td)! !

=V0 u(t-td) C

{ } )(1)( /)(0cap d

tt ttueVtv d −−= −− τ

τ = ?

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Capacitive Termination

0 2 4 6 80

0.2

0.4

0.6

0.8

1

t/ td

vL(

t)/ V0

load-end source-end

RS = Z0

VS

Z0, td!td = length/velocity

C

vcap (t) =V0 1− e−(t−td )/τ{ } u(t − td )

CZ0=τ

s )()( 0 tuvtvs =

56 Oregon State University ECE391– Transmission Lines Spring Term 2014

RLC Series Termination (source matched)

pF0132.1nF10050 ==Ω= LLL CLR nsec22 ≈= LLCLT π

nsec1=TD

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

t/ td

Vol

tage

(vol

ts)

VsourceVloadVcap

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Effect of Source Resistance

driver too small!-> long delay

driver too large!-> ringing!-> long settling time

58 Oregon State University ECE391– Transmission Lines Spring Term 2014

Example – Underdriven Line

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

t/ td

vL(

t)/ V0

50% threshold

∞→LR0ZRs >>

05ZRs =V1

+ =

)()( 0 tuVtvs =

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Example – Overdriven Line

0 5 10 15 20 25 300

0.5

1

1.5

2

t/ td

vL(

t)/ V0

∞→LR0ZRs <<

5/0ZRs =V1

+ =

)()( 0 tuVtvs =

60 Oregon State University ECE391– Transmission Lines Spring Term 2014

Example

+!- vs

Ω=10sourceR

Ω= K50LoadR

ns2,500 =Ω= DtZ

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Example

+!- vs

Ω= 250sourceR

Ω= K50LoadR

ns2,500 =Ω= DtZ

RsCtotal = 10 nsec!

62 Oregon State University ECE391– Transmission Lines Spring Term 2014

Basic TDR Principle Waveforms for resistive termination

ρL

measured load resistance totalinc

total

L

LL VV

VZZR−

=−+=

211

00 ρρ

TDR = Time-Domain Reflectometry

1+ ρL

1− ρL

=1+

VreflVinc

1−VreflVinc

=Vinc +VreflVinc −Vrefl

= VtotalVinc −Vrefl + Vinc −Vinc( ) =

Vtotal2Vinc −Vtotal

TDR Equipment

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TDR Waveforms for Resistive Terminations

V1

2t

d

V1

(1+ρL)V

1

ρLV

1

2td

V1

(1+ρ

L)V

1

-ρLV

1

2td

VTDR

VTDR

VTDR

64 Oregon State University ECE391– Transmission Lines Spring Term 2014

TDR Waveforms for Reactive Terminations

V1

(1+ρL)V

1

V1

2td

V1

(1+ρL)V

1

2td

[ ]( ){ } )'(1)'( /'initialfinalinitialload tueVVVtttv t

dτ−−−+=−=

LL

L CZRZR

0

0

+=τ

0ZRL

L

L

+=τ

VTDR

VTDR