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Transmission Line Basics II - Class 6
Prerequisite Reading assignment: CH2
Acknowledgements: Intel Bus Boot Camp: Michael Leddige
Transmission Lines Class 6
2
Real Computer Issues
Dev a Dev b
Clk Switch Threshold
Signal Measured here
An engineer tells you the measured clock is non-monotonic and because of this the flip flop internally may double clock the data. The goal for this class is to by inspection determine the cause and suggest whether this is a problem or not.
data
Transmission Lines Class 6
3
AgendaThe Transmission Line ConceptTransmission line equivalent circuits
and relevant equationsReflection diagram & equationLoadingTermination methods and comparison Propagation delaySimple return path ( circuit theory,
network theory come later)
Transmission Lines Class 6
4
Two Transmission Line ViewpointsSteady state ( most historical view)
Frequency domainTransient
Time domainNot circuit element Why?
We mix metaphors all the timeWhy convenience and history
Transmission Lines Class 6
5
Transmission Line Concept
PowerPlant
ConsumerHome
Power Frequency (f) is @ 60 Hz Wavelength (λ) is 5× 106 m
( Over 3,100 Miles)
Transmission Lines Class 6
6
PC Transmission Lines
Integrated Circuit
Microstrip
Stripline
Via
Cross section view taken herePCB substrate
T
WCross Section of Above PCB
T
Signal (microstrip)
Ground/PowerSignal (stripline)Signal (stripline) Ground/Power
Signal (microstrip)
Copper Trace
Copper Plane
FR4 Dielectric
W
Signal Frequency (f) is approaching 10 GHz
Wavelength (λ) is 1.5 cm ( 0.6 inches)
Micro-Strip
Stripline
Transmission Lines Class 6
7
Key point about transmission line operation
The major deviation from circuit theory with transmission line, distributed networks is this positional dependence of voltage and current!
Must think in terms of position and time to understand transmission line behaviorThis positional dependence is added when the assumption of the size of the circuit being small compared to the signaling wavelength
( )( )tzfI
tzfV,,
==
V1 V2
dz
I2I1
Voltage and current on a transmission line is a function of both time and position.
Transmission Lines Class 6
8
Examples of Transmission Line Structures- I Cables and wires
(a) Coax cable(b) Wire over ground(c) Tri-lead wire (d) Twisted pair (two-wire line)
Long distance interconnects
(a) (b)
(c) (d)
+-
+
+ +-
- -
-
Transmission Lines Class 6
9
Segment 2: Transmission line equivalent circuits and relevant equations
Physics of transmission line structures
Basic transmission line equivalent circuit
?Equations for transmission line propagation
Transmission Lines Class 6
10
Remember fields are setup given an applied forcing function.
(Source)
How does the signal move from source to load?
E & H Fields – Microstrip Case
The signal is really the wave propagating between the
conductors
Electric field
Magnetic field
Ground return path
X
Y
Z (into the page)
Signal path
Electric field
Magnetic field
Ground return path
X
Y
Z (into the page)
Signal path
Transmission Lines Class 6
11Transmission Line “Definition” General transmission line: a closed system in which
power is transmitted from a source to a destination
Our class: only TEM mode transmission linesA two conductor wire system with the wires in close proximity, providing relative impedance, velocity and closed current return path to the source.Characteristic impedance is the ratio of the voltage and current waves at any one position on the transmission line
Propagation velocity is the speed with which signals are transmitted through the transmission line in its surrounding medium.
IVZ =0
r
cvε
=
Transmission Lines Class 6
12
Presence of Electric and Magnetic Fields
Both Electric and Magnetic fields are present in the transmission lines
These fields are perpendicular to each other and to the direction of wave propagation for TEM mode waves, which is the simplest mode, and assumed for most simulators(except for microstrip lines which assume “quasi-TEM”, which is an approximated equivalent for transient response calculations).
Electric field is established by a potential difference between two conductors.
Implies equivalent circuit model must contain capacitor. Magnetic field induced by current flowing on the line
Implies equivalent circuit model must contain inductor.
V
I
I
E
+
-
+
-
+
-
+
-
V + ∆ V
I + ∆ I
I + ∆ IV
IH
IH
V + ∆ V
I + ∆ I
I + ∆ I
Transmission Lines Class 6
13
General Characteristics of Transmission Line
Propagation delay per unit length (T0) time/distance [ps/in]Or Velocity (v0) distance/ time [in/ps]
Characteristic Impedance (Z0) Per-unit-length Capacitance (C0) [pf/in]Per-unit-length Inductance (L0) [nf/in]Per-unit-length (Series) Resistance (R0) [Ω/in]Per-unit-length (Parallel) Conductance (G0) [S/in]
T-Line Equivalent Circuit
lL0lR0
lC0lG0
Transmission Lines Class 6
14
Ideal T Line Ideal (lossless) Characteristics of
Transmission LineIdeal TL assumes:
Uniform linePerfect (lossless) conductor (R0→0)Perfect (lossless) dielectric (G0→0)
We only consider T0, Z0 , C0, and L0.
A transmission line can be represented by a cascaded network (subsections) of these equivalent models.
The smaller the subsection the more accurate the model
The delay for each subsection should be no larger than 1/10 th the signal rise time.
lL0
lC0
Transmission Lines Class 6
15 Signal Frequency and Edge Rate vs. Lumped or Tline ModelsIn theory, all circuits that deliver transient power from one point to another are transmission lines, but if the signal frequency(s) is low compared to the size of the circuit (small), a reasonable approximation can be used to simplify the circuit for calculation of the circuit transient (time vs. voltage or time vs. current) response.
Transmission Lines Class 6
16
T Line Rules of Thumb
Td < .1 Tx
Td < .4 Tx
May treat as lumped Capacitance Use this 10:1 ratio for accurate modeling of transmission lines
May treat as RC on-chip, and treat as LC for PC board interconnect
So, what are the rules of thumb to use?
Transmission Lines Class 6
17
Other “Rules of Thumb”
Frequency knee (Fknee) = 0.35/Tr (so if Tr is 1nS, Fknee is 350MHz)
This is the frequency at which most energy is below
Tr is the 10-90% edge rate of the signal Assignment: At what frequency can your thumb be
used to determine which elements are lumped?Assume 150 ps/in
Transmission Lines Class 6
18When does a T-line become a T-Line? Whether it is a
bump or a mountain depends on the ratio of its size (tline) to the size of the vehicle (signal wavelength)
When do we need to When do we need to use transmission line use transmission line
analysis techniques vs. analysis techniques vs. lumped circuit lumped circuit
analysis? analysis?
TlineWavelength/edge rate
Similarly, whether or not a line is to be considered as a transmission line depends on the ratio of length of the line (delay) to the wavelength of the applied frequency or the rise/fall edge of the signal
Equations & Formulas
How to model & explain transmission line behavior
Transmission Lines Class 6
20
Relevant Transmission Line Equations
Propagation equationβαωωγ jCjGLjR +=++= ))((
)()(0
CjGLjRZ
ωω
++=
Characteristic Impedance equation
In class problem: Derive the high frequency, lossless approximation for Z0
α is the attenuation (loss) factorβ is the phase (velocity) factor
Transmission Lines Class 6
21
Ideal Transmission Line Parameters Knowing any two out of Z0,
Td, C0, and L0, the other two can be calculated.
C0 and L0 are reciprocal functions of the line cross-sectional dimensions and are related by constant me.
ε is electric permittivity ε0= 8.85 X 10-12 F/m (free space) εri s relative dielectric constant
µ is magnetic permeability µ0= 4p X 10-7 H/m (free space) µr is relative permeability
.;
;;1
;;
;;
00
000
0000
00
00d0
00
εεεµµµ
µεµε
rr
LCv
TZLZTC
CLTCLZ
==
==
==
==
Don’t forget these relationships and what they mean!
Transmission Lines Class 6
22Parallel Plate Approximation Assumptions
TEM conditionsUniform dielectric (ε ) between conductorsTC<< TD; WC>> TD
T-line characteristics are function of:
Material electric and magnetic propertiesDielectric Thickness (TD)Width of conductor (WC)
Trade-offTD ; C0 , L0 , Z0 WC ; C0 , L0 , Z0
TD
TC
WC
ε
To a first order, t-line capacitance and inductance can be approximated using the parallel plate approximation.
dPlateAreaC *ε= Base
equationC0 ε
WCTD
⋅Fm
⋅ 8.85 ε r⋅WCTD
⋅pFm
⋅
L0 µTDWC
⋅Fm
⋅ 0.4 π⋅ µ r⋅TDWC
⋅µ Hm
⋅
Z0 377TDWC
⋅µ rε r
⋅ Ω⋅
Transmission Lines Class 6
23
Improved Microstrip Formula Parallel Plate Assumptions +
Large ground plane with zero thickness
To accurately predict microstrip impedance, you must calculate the effectiveeffective dielectric constant.
TD
TC
ε
WC
From Hall, Hall & McCall:
++≈
CC
D
r TWTZ
8.098.5ln
41.187
0ε
( )DC
Cr
C
D
rre
TWTF
WT
1217.01212
12
1 −−++
−++= εεεε
( )2
1102.0
−−
D
Cr
TWε
=F1<
D
C
TW
for
0 1>D
C
TW
for
Valid when: 0.1 < WC/TD < 2.0 and 1 < r < 15
You can’t beat a field solver
Transmission Lines Class 6
24
Improved Stripline Formulas Same assumptions as
used for microstrip apply here TD2
TCε
WC TD1
From Hall, Hall & McCall:
+
+≈)8.0(67.0
)(4ln60 110
CC
DD
rsym
TWTTZ
πε
Symmetric (balanced) Stripline Case TD1 = TD2
),,,2(),,,2(),,,2(),,,2(2
00
000
rCCsymrCCsym
rCCsymrCCsymoffset
TWBZTWAZTWBZTWAZZ
εεεε
+⋅≈
Offset (unbalanced) Stripline Case TD1 > TD2
Valid when WC/(TD1+TD2) < 0.35 and TC/(TD1+TD2) < 0.25
You can’t beat a field solver
Transmission Lines Class 6
25
Refection coefficient Signal on a transmission line can be analyzed by
keeping track of and adding reflections and transmissions from the “bumps” (discontinuities)
Refection coefficientAmount of signal reflected from the “bump”Frequency domain ρ=sign(S11)*|S11|If at load or source the reflection may be called gamma (ΓL or Γs)Time domain ρ is only defined a location
The “bump”Time domain analysis is causal.Frequency domain is for all time.We use similar terms – be careful
Reflection diagrams – more later
Transmission Lines Class 6
26Reflection and Transmission
ρ
1+ρIncident
Reflected
Transmitted
Reflection Coeficient Transmission Coeffiecentτ 1 ρ+( ) "" ""→ τ 1
Zt Z0−Zt Z0+
+ρZt Z0−Zt Z0+
τ2 Zt⋅
Zt Z0+
Transmission Lines Class 6
27
Special Cases to Remember
1=+∞−∞=
ZoZoρ
0=+−=
ZoZoZoZoρ
100 −=
+−=
ZoZoρ
VsZs
Zo Zo
A: Terminated in Zo
VsZs
Zo
B: Short Circuit
VsZs
Zo
C: Open Circuit
Transmission Lines Class 6
28
Assignment – Building the SI Tool BoxCompare the parallel plate approximation to the improved microstrip and stripline formulas for the following cases:Microstrip:
WC = 6 mils, TD = 4 mils, TC = 1 mil, εr = 4Symmetric Stripline:
WC = 6 mils, TD1 = TD2 = 4 mils, TC = 1 mil, εr = 4
Write Math Cad Program to calculate Z0, Td, L & C for each case.
What factors cause the errors with the parallel plate approximation?
Transmission Lines Class 6
29
Transmission line equivalent circuits and relevant equations
Basic pulse launching onto transmission lines
Calculation of near and far end waveforms for classic load conditions
Transmission Lines Class 6
30Review: Voltage Divider Circuit Consider the
simple circuit that contains source voltage VS, source resistance RS, and resistive load RL.
The output voltage, VL is easily calculated from the source amplitude and the values of the two series resistors.
RS
RLVS VL
RSRL
RLVSVL +
=
Why do we care for? Next page….
Transmission Lines Class 6
31
Solving Transmission Line ProblemsThe next slides will establish a procedure that
will allow you to solve transmission line problems without the aid of a simulator. Here are the steps that will be presented:
3.Determination of launch voltage & final “DC” or “t =0” voltage
4.Calculation of load reflection coefficient and voltage delivered to the load
5.Calculation of source reflection coefficient and resultant source voltage
These are the steps for solving all t-line problems.
Transmission Lines Class 6
32
Determining Launch Voltage
Step 1 in calculating transmission line waveforms is to determine the launch voltage in the circuit.
The behavior of transmission lines makes it easy to calculate the launch & final voltages – it is simply a voltage divider!
Vs ZoRsVs
0
TD
Rt
A B
t=0, V=Vi(initial voltage)
RSZ0
Z0VSVi +
=RSRt
RtVSVf +=
Transmission Lines Class 6
33
Voltage Delivered to the Load
Vs ZoRsVs
0
TD
Rt
A B
t=0, V=Vi
t=TD, V=Vi + ρB(Vi )t=2TD, V=Vi + ρB(Vi) + ρA(ρB)(Vi )
(signal is reflected)
(initial voltage)
Step 2: Determine VB in the circuit at time t = TD The transient behavior of transmission line delays the
arrival of launched voltage until time t = TD. VB at time 0 < t < TD is at quiescent voltage (0 in this case)
Voltage wavefront will be reflected at the end of the t-line VB = Vincident + Vreflected at time t = TD
ZoRtZoRt
+−=ρΒ
Vreflected = ρΒ (Vincident)
VB = Vincident + Vreflected
Transmission Lines Class 6
34
Voltage Reflected Back to the Source
VsZo
RsVs
0
TD
Rt
A B
t=0, V=Vi
t=TD, V=Vi + ρB (Vi )t=2TD, V=Vi + ρB (Vi) + ρA
(ρB)(Vi )
(signal is reflected)
(initial voltage)
ρA ρB
Transmission Lines Class 6
35Voltage Reflected Back to the Source
Step 3: Determine VA in the circuit at time t = 2TD The transient behavior of transmission line delays the
arrival of voltage reflected from the load until time t = 2TD. VA at time 0 < t < 2TD is at launch voltage
Voltage wavefront will be reflected at the source VA = Vlaunch + Vincident + Vreflected at time t = 2TD
In the steady state, the solution converges to VB = VS[Rt / (Rt + Rs)]
ZoRsZoRs
+−=ρΑ
Vreflected = ρΑ (Vincident)
VA = Vlaunch + Vincident + Vreflected
Transmission Lines Class 6
36
Problems Consider the circuit
shown to the right with a resistive load, assume propagation delay = T, RS= Z0 . Calculate and show the wave forms of V1(t),I1(t),V2(t), and I2(t) for (a) RL= ∞ and (b) RL= 3Z0
Z0 ,Τ 0
V1 V2
l
I2I1
VS RL
RS
Solved Homework
Transmission Lines Class 6
37
Step-Function into T-Line: Relationships
Source matched case: RS= Z0
V1(0) = 0.5VA, I1(0) = 0.5IA
ΓS = 0, V(x,∞) = 0.5VA(1+ ΓL) Uncharged line
V2(0) = 0, I2(0) = 0 Open circuit means RL= ∞
ΓL = ∞ /∞ = 1V1(∞) = V2(∞) = 0.5VA(1+1) = VA
I1(∞) = I2 (∞) = 0.5IA(1-1) = 0 Solution
Transmission Lines Class 6
38
Step-Function into T-Line with Open Ckt
At t = T, the voltage wave reaches load end and doubled wave travels back to source end
V1(T) = 0.5VA, I1(T) = 0.5VA/Z0
V2(T) = VA, I2 (T) = 0 At t = 2T, the doubled wave reaches the
source end and is not reflectedV1(2T) = VA, I1(2T) = 0V2(2T) = VA, I2(2T) = 0
Solution
Transmission Lines Class 6
39
Waveshape:Step-Function into T-Line with Open Ckt
Z0 ,Τ 0
V1 V2
l
I2I1
VSOpen
RS
Cur
rent
(A)
2Τ Time (ns)3ΤΤ 4Τ0
0.5IA
0.25IA
IA
0.75IA
I1I2
Vol
tage
(V)
2Τ Time (ns)3ΤΤ 4Τ0
0.5VA
0.25VA
VA
0.75VA
V1V2
This is called This is called “reflected wave “reflected wave switching”switching”
Solution
Transmission Lines Class 6
40
Problem 1b: Relationships Source matched case: RS= Z0
V1(0) = 0.5VA, I1(0) = 0.5IA
ΓS = 0, V(x,∞) = 0.5VA(1+ ΓL) Uncharged line
V2(0) = 0, I2(0) = 0 RL= 3Z0
ΓL = (3Z0 -Z0) / (3Z0 +Z0) = 0.5V1(∞) = V2(∞) = 0.5VA(1+0.5) = 0.75VA
I1(∞) = I2(∞) = 0.5IA(1-0.5) = 0.25IA Solution
Transmission Lines Class 6
41
Problem 1b: Solution At t = T, the voltage wave reaches load end
and positive wave travels back to the sourceV1(T) = 0.5VA, I1(T) = 0.5IA
V2(T) = 0.75VA , I2(T) = 0.25IA
At t = 2T, the reflected wave reaches the source end and absorbed
V1(2T) = 0.75VA , I1(2T) = 0.25IA
V2(2T) = 0.75VA , I2(2T) = 0.25IA
Solution
Transmission Lines Class 6
42Waveshapes for Problem 1b
Z0 ,Τ 0
V1 V2
l
I2I1
VS RL
RS
Curr
ent (
A)
2Τ Time (ns)3ΤΤ 4Τ0
0.5IA
0.25IA
IA
0.75IA
I1I2
Volta
ge (V
)
2Τ Time (ns)3ΤΤ 4Τ0
0.5VA
0.25VA
VA
0.75VA
I1I2
Note that a Note that a properly terminated properly terminated wave settle out at wave settle out at 0.5 V0.5 V
SolutionSolution
Transmission Lines Class 6
43Transmission line step response
Introduction to lattice diagram analysis
Calculation of near and far end waveforms for classic load impedances
Solving multiple reflection problems
Complex signal reflections at different types of transmission line “discontinuities” will be analyzed in this chapter. Lattice diagrams will be introduced as a solution tool.
Transmission Lines Class 6
44
Lattice Diagram Analysis – Key Concepts
Diagram shows the boundaries (x =0 and x=l) and the reflection coefficients (GL and GL )
Time (in T) axis shown vertically
Slope of the line should indicate flight time of signal
Particularly important for multiple reflection problems using both microstrip and stripline mediums.
Calculate voltage amplitude for each successive reflected wave
Total voltage at any point is the sum of all the waves that have reached that point
VsRs
ZoV(source) V(load)
TD = N ps0Vs
RtThe lattice diagram is a tool/technique to simplify the accounting of reflections and waveforms
Time V(source) V(load)
a
sourceρ loadρ
bA
c
A’
B’
C’
dB
e
0
N ps
2N ps
3N ps
4N ps
5N ps
Transmission Lines Class 6
45Lattice Diagram Analysis – DetailV(source) V(load)
Vlaunch
sourceρ load
ρ
Vlaunch ρload
Vlaunch
0
Vlaunch(1+ρload)
Vlaunch(1+ρload +ρload ρsource)
Time
0
2N ps
4N ps
Vlaunch ρloadρsource
Vlaunch ρ2loadρsource
Vlaunch ρ2loadρ2
source
Vlaunch(1+ρload+ρ2loadρsource+ ρ2
loadρ2source)
Time
N ps
3N ps
5N psVs
RsZoV(source) V(load)
TD = N ps0Vs
Rt
Transmission Lines Class 6
46Transient Analysis – Over DampedAssume Zs=75 ohms Zo=50ohmsVs=0-2 voltsVs
ZsZoV(source) V(load)
Time V(source) V(load)
15050
2.050755075
8.05075
50)2(
=+∞−∞=
+−=
=+−=
+−=
=
+=
+=
ZoZlZoZl
ZoZsZoZs
ZoZsZoVsV
load
source
initial
ρ
ρ0.8v
2.0=sourceρ 1=loadρ
0.8v0.8v
0.16v
0v
1.6v
1.92v
0.16v1.76v
0.032v
TD = 250 ps
0
500 ps
1000 ps
1500 ps
2000 ps
2500 ps
02 v
Response from lattice diagram
0
0.5
1
1.5
2
2.5
0 250 500 750 1000 1250
Tim e , ps
Vol
tsSource
Load
Transmission Lines Class 6
47Transient Analysis – Under Damped
15050
33333.050255025
3333.15025
50)2(
=+∞−∞=
+−=
−=+−=
+−=
=
+=
+=
ZoZlZoZl
ZoZsZoZs
ZoZsZoVsV
load
source
initial
ρ
ρ
Assume Zs=25 ohms Zo =50ohmsVs=0-2 voltsVs
ZsZoV(source) V(load)
TD = 250 ps02 v
Time V(source) V(load)
1.33v
3333.0−=sourceρ 1=loadρ
1.33v1.33v
-0.443v
0v
2.66v
1.77v
-0.443v2.22v
0.148v
0
500 ps
1000 ps
1500 ps
2000 ps
2500 ps 1.920.148v
2.07
Response from lattice diagram
0
0.5
1
1.5
2
2.5
3
0 250 500 750 1000 1250 1500 1750 2000 2250
Time, ps
Volts
SourceLoad
Transmission Lines Class 6
48Two Segment Transmission Line Structures
VsRs
Zo1 RtZo2
X X
a
bc
d e
f g
h i
j k
l
33
22
2
24
21
213
12
122
1
11
1
1
1
1
ρ
ρ
ρ
ρ
ρ
ρ
+=
+=
+−=
+−=
+−=
+−=
+=
T
T
ZRtZRt
ZZZZ
ZZZZ
ZRsZRs
ZRsZVv
o
o
oo
oo
oo
oo
o
o
o
osi
23
32
4
1
23
32
4
1
2
2
hTik
iThj
gi
fh
dTeg
eTdf
be
cd
ac
aTb
va i
+=
+=
=
=
+=
+=
=
=
=
=
=
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
hfdcACdcaB
aA
++++=++=
=
lkigebCigebB
ebA
+++++=+++=
+=
'''
A
B
C
A’
B’
C’
1ρ 2ρ 3ρ 4ρ3T 2T
TD TD
TD
3TD
2TD
4TD
5TD
Transmission Lines Class 6
49
Assignment Consider the two segment
transmission line shown to the right. Assume RS= 3Z01
and Z02= 3Z01 . Use Lattice diagram and calculate reflection coefficients at the interfaces and show the wave forms of V1(t), V2(t), and V3(t).
Check results with PSPICE
Z01 ,Τ 01
V1 V2
l1
I2I1
VS
RSZ02 ,Τ 02
V3l2
I3
Short
Previous examples are the preparation