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© H. Heck 2008 Section 5.4 1
Module 5: Advanced Transmission LinesTopic 4: Frequency Domain Analysis
OGI ECE564
Howard Heck
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Where Are We?
1. Introduction
2. Transmission Line Basics
3. Analysis Tools
4. Metrics & Methodology
5. Advanced Transmission Lines1. Losses
2. Intersymbol Interference
3. Crosstalk
4. Frequency Domain Analysis
5. 2 Port Networks & S-Parameters6. Multi-Gb/s Signaling
7. Special Topics
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Contents
Motivation Wave Equation Revisited Frequency Dependence Reflection Coefficient and Impedance Input Impedance Examples Summary References
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Motivation
At high frequencies, losses become significant. This makes time domain analysis difficult, as the properties are frequency dependent. Skin effect, dielectric loss & dispersion
We need to develop the means to understand those effects. Example: How would we measure R, L, G, C for a PCB trace?
Frequency domain analysis allows discrete characterization of a linear network at each frequency. Characterization at a single frequency is much easier
Frequency Analysis has advantages: Ease and accuracy of measurement at high frequencies Simplified mathematics Allows separation of electrical phenomena (loss, resonance …
etc).
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Key Concepts
The input impedance & the input reflection coefficient of a transmission line is dependent on: Termination and characteristic impedance Delay Frequency
S-Parameters are used to extract electrical parameters. Transmission line parameters (R,L,C,G, TD and Zo) Vias, connectors, socket … equivalent circuits
Periodic behavior of S-parameters can be used to gain intuition of signal integrity problems.
We’ll study S-parameters in section 5.5.
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Derive the lossy wave equation Add a sinusoidal stimulus
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Wave Equation Revisited
Goal: derive the frequency dependent impedance and reflection coefficients.
Method: Starting with the RLGC equivalent circuit, we derive the differential equations.
tjeVtv 0
dzzvLdzjziRdzzizv
dzLjRzizvdzzv
LjRzidz
dv
L
C
R
G+
v(z)
-
i(z)+
v(z+dz)
-
i(z+dz)
dzzidz
Cj
zv
dzG
zvzi
11
dzCjGzvzidzzi
CjGzvdz
di
KVL
Rearrange
Differentiate w.r.t. z
[5.4.1]
[5.4.2]
[5.4.3]
KCL
Rearrange
Differentiate w.r.t. z
[5.4.4]
[5.4.5]
[5.4.6]
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Wave Equation Revisited #2
Use the equations on the previous page to get:
tjeVtv 0
L
C
R
G+
v(z)
-
i(z)+
v(z+dz)
-
i(z+dz)
zvzvCjGLjRdz
vd 22
2
ziziCjGLjRdz
id 22
2
zR
zF eVeVzv
zR
zF eIeIzi
[5.4.7]
[5.4.8]
Which have solutions:
[5.4.9]
[5.4.10]
where jCjGLjR
[5.4.11]
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Wave Equation Solution
zRz
F eVeVLjR
CjGLjRzi
zRz
Fz
Rz
F eVeV
YZ
eVeVLjR
CjGzi
1
YZ
VI FF
YZ
VI RR
ziLjReVeVdz
dv zR
zF
Let’s work with [5.4.3] and [5.4.10] to relate the currents and voltages:
zRz
F eVeVLjR
zi
[5.4.12]
[5.4.13]
Differentiate w.r.t. z:
Substitute
[5.4.14]
[5.4.15]
[5.4.16b][5.4.16a]CjGY LjRZ
[5.4.17b][5.4.17a]
Algebra•••
where
note YZZo
so0Z
VI FF
0Z
VI RR
[5.4.19b][5.4.19a]
[5.4.18]
zRz
F eVeVZ
zi
0
1[5.4.20]
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Including Frequency Dependence
If a sinusoid is injected onto a transmission line, the resulting voltage can be expressed as a function of the distance from the load (z) and time.
tjzR
tjzF eeVeeVtzV ),(
Notice: The first term represents the forward traveling wave (toward
the load) The second term represents the backward traveling wave
reflected from the load (toward the source) The position dependent exponent is positive for the second
term because the wave is traveling back toward the source.
RS
v=V0ejwt
z=0
VF
VB
IF
IB
ZL
[5.4.21]
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Frequency Dependence #2 Note that and use to get:)sin()cos( je j j
tjzjR
tjzjF eeVeeVtzv )()(),(
)(sin)(cos)(sin)(cos),(
z
tjz
tVez
tjz
tVetzv Rz
Fz
[5.4.22]
[5.4.24]
[5.4.23] ztjzR
ztjzF eeVeeVtzv ),(
Separating the real and imaginary terms:
Expressing in terms of sine/cosine functions:
Where is the amplitude loss of the sinusoid
is the phase shift
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Frequency Dependence #3
zjR
zjF
tj
eVeVZ
etzi
0
,
Apply the sinusoid source to the expression for current:
ztjzR
ztjzF eeVeeV
Ztzi
0
1,
[5.4.25]
[5.4.26]
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Load Impedance
SSRF iRVVVzv 0
RS
VS
z=0
VF
VB
IF
IB
Lz
Rz
F
tj
zR
zF
tj
ZeVeV
Ze
eVeVe
tlzi
tlzv
0
,
,
zR
zF
zR
zF
L eVeV
eVeVZZ
0
[5.4.28]
[5.4.27]
[5.4.29]
Look at the boundary case.
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Reflection Coefficients & Impedance
l
F
Rl
Ftj
lR
tj
V eV
V
eVe
eVe
zv
zv
2
Vl
F
Rl
Ftj
lR
tj
I eV
V
eIe
eIe
2
F
lRF
VV V
eVV
2
1
F
lRF
II V
eVV
2
1
z
F
R
z
F
R
zR
zF
zR
zF
eVV
eVV
ZeVeV
eVeVZ
tzi
tzvzZ
2
2
00
1
1
,
,
zlv
zlv
ee
eeZzZ
22
22
0 1
1
Define the reflection coefficients:
[5.4.30]
[5.4.33]
[5.4.34]
[5.4.32]
[5.4.31]
Define the impedance in terms of reflection coefficients:
Note: most microwave texts use the gamma () symbol to represent the reflection coefficient. I have chosen to continue to use in order to remain consistent with our definition from module 2.
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Input Impedance
02
02
0 1
10
ee
eeZzZZ
lv
lv
in
v
vll
v
llv
L Zee
eeZlzZZ
1
1
1
1022
22
0
Define the input impedance:
[5.4.35]
[5.4.37]
The impedance at the load is:
lv
lv
in e
eZZ
2
2
0 1
1
Solving [5.4.36] for v, we get the familiar equation for the reflection coefficient at the load:
[5.4.38]
Substituting [5.4.37] into [5.4.30], we get the equation reflection coefficient as a function of position along the line:
[5.4.36]
zl
L
Lv e
ZZ
ZZz
2
0
0
0
0
ZZ
ZZlz
L
Lv
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Input Impedance #2
lLL
lLL
l
L
L
l
L
L
in eZZZZ
eZZZZZ
eZZZZ
eZZZZ
ZZ
200
200
02
0
0
2
0
0
0
1
1
Substituting [5.4.38] into [5.4.35] and doing the algebra:
[5.4.39]
02
2
2
2
0
020
2
20
2
0
11
11
11
11
Zee
Z
ee
ZZ
ZeZeZ
eZeZZZ
l
l
L
l
l
L
llL
llL
in
x
x
e
ex
2
2
1
1tanh
0
00 tanh
tanh
ZlZ
lZZZZ
L
Lin
Use the following relationship:
To get:
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Input Impedance #3 Alternate expression (for lossless lines):
[5.4.40]
jxjx
jxjx
eexj
eex
sin
cos Use the following relationships:
To get:
0
00
10
0Z
ee
ee
eeZV
eeV
eVeVZ
eVeV
zi
zvZ
ljV
lj
ljV
lj
ljV
ljF
ljV
ljF
lR
lF
lR
lF
in
lj
Llj
L
ljL
ljL
lj
L
Llj
lj
L
Llj
in eZZeZZ
eZZeZZZ
eZZ
ZZe
eZZ
ZZe
Z
00
000
0
0
0
0
ljljljljL
ljljljljL
in eeZeeZ
eeZeeZZ
0
0
lZljZ
ljZlZZ
L
Lin
cossin
sincos
0
0
ljZZ
ljZZZZ
L
Lin
tan
tan
0
00
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Example
118
110
566102
108.12
mradj
sm
sradj
vjj
p
LosslessLossless0
l1 = 1.5 mm
Z01 = 20 ZL = 20Z02 = 30
l2 = 2 mm
vpl = vp2 = 2 x 108 m/s
f = 18 GHz
0
00 tanh
tanh
ZlZ
lZZZZ
L
Lin
radjmmmmmmradjljl 132.1102566 13122
Looking into Z02:
j11.778+36.686
30132.1tanh20
132.1tanh3020302 j
jZ in
Use Zin2 as the load impedance to get the input impedance looking into Z01:
j14.617-18.897
20848.0tanhj11.778+36.686
848.0tanh20j11.778+36.686201 j
jZ in
radjmmmmmmradjljl 848.0105.1566 13111
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Example #2 What is v as measured at z = 0 for
the lossless transmission line system as a function of frequency?
Start with [5.4.38]:
LClj
L
Llj
L
Ll
L
Lv e
ZZ
ZZe
ZZ
ZZe
ZZ
ZZ 2
0
02
0
02
0
0
Which can be rewritten:
LCfljLCflZZ
ZZ
L
Lv 4sin4cos
0
0
Notice that the real part is zero when . Solving for f:2
4 nLCfl
LCl
nf8
where 5,3,1n
The imaginary part is zero when . Solving for f: nLCfl 4
LCl
nf
4 where 5,3,1n
ZL = 50Z0= 75
l=5 in
vp= 205 ps/ft
z=0 z=l
fjfv sec10341.5sinsec10341.5cos2.0 99
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Example #2 (2)
ZL = 50Z0= 75
l=5 in
vp= 205 ps/ft
z=0 z=l
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.0E
+00
5.0E
+08
1.0E
+09
1.5E
+09
2.0E
+09
2.5E
+09
3.0E
+09
3.5E
+09
4.0E
+09
4.5E
+09
5.0E
+09
frequency [Hz]
Ref
lect
ion
Co
effi
cien
t
rho(real)rho(imag)
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Summary
We now have the basis for using measurement equipment to characterize interconnect in the frequency domain.
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References R.E. Matick, Transmission Lines for Digital and
Communication Networks, IEEE Press, 1995. D.M. Posar, Microwave Engineering, John Wiley
& Sons, Inc. (Wiley Interscience), 1998, 2nd edition. B. Young, Digital Signal Integrity, Prentice-Hall
PTR, 2001, 1st edition. W. Dally and J. Poulton, Digital Systems
Engineering, Cambridge University Press, 1998. Ramo, Whinnery, and Van Duzer, Fields and
Waves in Communication Electronics, 1985. U. Inan, A. Inan, Engineering Electromagnetics,
Addison Wesley, 1999, 1st edition. Ramo, Whinnery, and Van Duzer, Fields and
Waves in Communication Electronics, 1985.