Class06_Transmission_line_basics.ppt

Transmission Line Basics II - Class 6
Prerequisite Reading assignment: CH2
Acknowledgements: Intel Bus Boot Camp:
Michael Leddige

2
Transmission Lines Class 6
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

3
Agenda
The Transmission Line Concept
Transmission line equivalent circuits
and relevant equations
Reflection diagram & equation
Loading
Termination methods and comparison
Propagation delay
Simple return path ( circuit theory,
network theory come later)

4
Two Transmission Line Viewpoints
Steady state ( most historical view)
Frequency domain
Transient
Time domain
Not circuit element Why?
We mix metaphors all the time
Why convenience and history

5
Transmission Line Concept
Power
Plant
Consumer
Home
Power Frequency (f) is @ 60 Hz
Wavelength (l) is 5 106 m
( Over 3,100 Miles)

6
PC Transmission Lines
Integrated Circuit
Microstrip
Stripline
Via
Cross section view taken here
PCB substrate
T
W
Cross Section of Above PCB
T
Signal (microstrip)
Ground/Power
Signal (stripline)
Signal (stripline)
Ground/Power
Signal (microstrip)
Copper Trace
Copper Plane
FR4 Dielectric
W
Signal Frequency (f) is
approaching 10 GHz
Wavelength (l) is 1.5 cm
( 0.6 inches)
Micro-
Strip
Stripline

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 behavior
This positional dependence is added when the
assumption of the size of the circuit being
small compared to the signaling wavelength
 
 
t
z
f
I
t
z
f
V
,
,


V1 V2
dz
I2
I1
Voltage and current on a transmission line is
a function of both time and position.

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)
+
-
+
+ +
-
- -
-

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

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

11
Transmission 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 lines
A 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.
I
V
Z 
0
r
c
v



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 + DV
I + DI
I + DI
V
I
H
I
H
V + DV
I + DI
I + DI

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) [W/in]
Per-unit-length (Parallel) Conductance (G0) [S/in]
T-Line Equivalent Circuit
lL0
lR0
lC0
lG0

14
Ideal T Line
 Ideal (lossless) Characteristics of
Transmission Line
Ideal TL assumes:
Uniform line
Perfect (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 1/10th the signal rise time.
lL0
lC0

15
Signal Frequency and Edge Rate
vs.
Lumped or Tline Models
In 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.

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?

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

18
When 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
use transmission line
analysis techniques vs.
lumped circuit
analysis?
Tline
Wavelength/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

20
Relevant Transmission Line Equations
Propagation equation




 j
C
j
G
L
j
R 



 )
)(
(
)
(
)
(
0
C
j
G
L
j
R
Z





Characteristic Impedance equation
In class problem: Derive the high frequency, lossless
approximation for Z0
 is the attenuation (loss) factor
 is the phase (velocity) factor

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.
  is electric permittivity
0= 8.85 X 10-12 F/m (free space)
ri s relative dielectric constant
 m is magnetic permeability
m0= 4p X 10-7 H/m (free space)
mr is relative permeability
.
;
;
;
1
;
;
;
;
0
0
0
0
0
0
0
0
0
0
0
0
0
d
0
0
0



m
m
m
m
m
r
r
L
C
v
T
Z
L
Z
T
C
C
L
T
C
L
Z








Don’t forget these relationships and what they mean!

22
Parallel Plate Approximation
 Assumptions
TEM conditions
Uniform dielectric ( )
between conductors
TC<< TD; WC>> TD
 T-line characteristics are
function of:
Material electric and
magnetic properties
Dielectric Thickness (TD)
Width of conductor (WC)
 Trade-off
TD ; C0 , L0 , Z0 
WC ; C0 , L0 , Z0 
TD
TC
WC

To a first order, t-line capacitance and inductance can
be approximated using the parallel plate approximation.
d
PlateArea
C
*

 Base
equation
C0 
WC
TD

F
m






 8.85 r

WC
TD

pF
m







L0 m
TD
WC

F
m






 0.4 
 mr

TD
WC

mH
m







Z0 377
TD
WC

mr
r
 W


23
Improved Microstrip Formula
 Parallel Plate Assumptions +
Large ground plane with
zero thickness
 To accurately predict
microstrip impedance, you
must calculate the effective
dielectric constant.
TD
TC

WC
From Hall, Hall & McCall:









C
C
D
r T
W
T
Z
8
.
0
98
.
5
ln
41
.
1
87
0

 
D
C
C
r
C
D
r
r
e
T
W
T
F
W
T
1
217
.
0
12
1
2
1
2
1







 



 
2
1
1
02
.
0 







D
C
r
T
W


F
1

D
C
T
W
for
0 1

D
C
T
W
for
Valid when:
0.1 < WC/TD < 2.0 and 1 < er < 15
You can’t beat
a field solver

24
Improved Stripline Formulas
 Same assumptions as
used for microstrip
apply here TD2
TC

WC
TD1
From Hall, Hall & McCall:











)
8
.
0
(
67
.
0
)
(
4
ln
60 1
1
0
C
C
D
D
r
sym
T
W
T
T
Z


Symmetric (balanced) Stripline Case TD1 = TD2
)
,
,
,
2
(
)
,
,
,
2
(
)
,
,
,
2
(
)
,
,
,
2
(
2
0
0
0
0
0
r
C
C
sym
r
C
C
sym
r
C
C
sym
r
C
C
sym
offset
T
W
B
Z
T
W
A
Z
T
W
B
Z
T
W
A
Z
Z







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

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 coefficient
Amount of signal reflected from the “bump”
Frequency domain r=sign(S11)*|S11|
If at load or source the reflection may be called gamma (GL
or Gs)
Time domain r 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

26
Reflection and Transmission
r
1r
Incident
Reflected
Transmitted
Reflection Coeficient Transmission Coeffiecent
 1 r

  "" ""
  1
Zt Z0

Zt Z0


r
Zt Z0

Zt Z0


2 Zt

Zt Z0


27
Special Cases to Remember
1






Zo
Zo
r
0




Zo
Zo
Zo
Zo
r
1
0
0 




Zo
Zo
r
Vs
Zs
Zo Zo
A: Terminated in Zo
Vs
Zs
Zo
B: Short Circuit
Vs
Zs
Zo
C: Open Circuit

28
Assignment – Building the SI Tool Box
Compare 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, r = 4
Symmetric Stripline:
WC = 6 mils, TD1 = TD2 = 4 mils, TC = 1 mil, r = 4
Write Math Cad Program to calculate Z0, Td, L
& C for each case.
What factors cause the errors with the parallel
plate approximation?

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

30
Review: 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
RL
VS VL
RS
RL
RL
VS
VL
+
=
Why do we care for?
Next page….

31
Solving Transmission Line Problems
The 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:
1.Determination of launch voltage &
final “DC” or “t =0” voltage
2.Calculation of load reflection coefficient and
voltage delivered to the load
3.Calculation of source reflection coefficient
and resultant source voltage
These are the steps for solving
all t-line problems.

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
Zo
Rs
Vs
0
TD
Rt
A B
t=0, V=Vi
(initial voltage)
RS
Z0
Z0
VS
Vi
+
=
RS
Rt
Rt
VS
Vf
+
=

33
Voltage Delivered to the Load
Vs Zo
Rs
Vs
0
TD
Rt
A B
t=0, V=Vi
t=TD, V=Vi +r
B(Vi )
t=2TD,
V=Vi +r
B(Vi) +r
ArB)(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
Zo
Rt
Zo
Rt



rB
Vreflected = rB (Vincident)
VB = Vincident + Vreflected

34
Voltage Reflected Back to the Source
Vs
Zo
Rs
Vs
0
TD
Rt
A B
t=0, V=Vi
t=TD, V=Vi + r
B
(Vi )
t=2TD,
V=Vi + r
B
(Vi) + r
A
r
B
)(Vi )
(signal is reflected)
(initial voltage)
rA rB

35
Voltage 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)]
Zo
Rs
Zo
Rs



rA
Vreflected = rA (Vincident)
VA = Vlaunch + Vincident + Vreflected

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 ,T0
V1 V2
l
I2
I1
VS RL
RS
Solved Homework

37
Step-Function into T-Line: Relationships
 Source matched case: RS= Z0
V1(0) = 0.5VA, I1(0) = 0.5IA
GS = 0, V(x,) = 0.5VA(1+ GL)
 Uncharged line
V2(0) = 0, I2(0) = 0
 Open circuit means RL= 
GL =  / = 1
V1() = V2() = 0.5VA(1+1) = VA
I1() = I2 () = 0.5IA(1-1) = 0
Solution

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 reflected
V1(2T) = VA, I1(2T) = 0
V2(2T) = VA, I2(2T) = 0
Solution

39
Waveshape:
Step-Function into T-Line with Open Ckt
Z0 ,T0
V1 V2
l
I2
I1
VS
Open
RS
Current
(A)
2T Time (ns)
3T
T 4T
0
0.5IA
0.25IA
IA
0.75IA
I1
I2
Voltage
(V)
2T Time (ns)
3T
T 4T
0
0.5VA
0.25VA
VA
0.75VA
V1
V2
This is called
“reflected wave
switching”
Solution

40
Problem 1b: Relationships
 Source matched case: RS= Z0
V1(0) = 0.5VA, I1(0) = 0.5IA
GS = 0, V(x,) = 0.5VA(1+ GL)
 Uncharged line
V2(0) = 0, I2(0) = 0
 RL= 3Z0
GL = (3Z0 -Z0) / (3Z0 +Z0) = 0.5
V1() = V2() = 0.5VA(1+0.5) = 0.75VA
I1() = I2() = 0.5IA(1-0.5) = 0.25IA
Solution

41
Problem 1b: Solution
 At t = T, the voltage wave reaches load end
and positive wave travels back to the source
V1(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

42
Waveshapes for Problem 1b
Z0 ,T0
V1 V2
l
I2
I1
VS RL
RS
Current
(A)
2T Time (ns)
3T
T 4T
0
0.5IA
0.25IA
IA
0.75IA
I1
I2
Voltage
(V)
2T Time (ns)
3T
T 4T
0
0.5VA
0.25VA
VA
0.75VA
I1
I2
Note that a
properly terminated
wave settle out at
0.5 V
Solution
Solution

43
Transmission 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.

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
Vs
Rs
Zo
V(source) V(load)
TD = N ps
0
Vs
Rt
The lattice diagram is a
tool/technique to simplify
the accounting of
reflections and waveforms
Time V(source) V(load)
a
source
r load
r
b
A
c
A’
B’
C’
d
B
e
0
N ps
2N ps
3N ps
4N ps
5N ps

45
Lattice Diagram Analysis – Detail
V(source) V(load)
Vlaunch
source
r load
r
Vlaunch rload
Vlaunch
0
Vlaunch(1+rload)
Vlaunch(1+rload +rload rsource)
Time
0
2N ps
4N ps
Vlaunch rloadrsource
Vlaunch r2
loadrsource
Vlaunch r2
loadr2
source
Vlaunch(1+rload+r2
loadrsource+ r2
loadr2
source)
Time
N ps
3N ps
5N ps
Vs
Rs
Zo
V(source) V(load)
TD = Nps
0
Vs
Rt

46
Transient Analysis – Over Damped
Assume Zs=75 ohms
Zo=50ohms
Vs=0-2 volts
Vs
Zs
Zo
V(source) V(load)
1
50
50
2
.
0
50
75
50
75
8
.
0
50
75
50
)
2
(



























Zo
Zl
Zo
Zl
Zo
Zs
Zo
Zs
Zo
Zs
Zo
Vs
V
load
source
initial
r
r
0.8v
2
.
0

source
r 1

load
r
0.8v
0.8v
0.16v
0v
1.6v
1.92v
0.16v
1.76v
0.032v
TD = 250 ps
0
500 ps
1000 ps
1500 ps
2000 ps
2500 ps
0
2 v
Response from lattice diagram
0
0.5
1
1.5
2
2.5
0 250 500 750 1000 1250
Time, ps
V
olt
s
Source
Load

47
Transient Analysis – Under Damped
1
50
50
33333
.
0
50
25
50
25
3333
.
1
50
25
50
)
2
(




























Zo
Zl
Zo
Zl
Zo
Zs
Zo
Zs
Zo
Zs
Zo
Vs
V
load
source
initial
r
r
Assume Zs=25 ohms
Zo=50ohms
Vs=0-2 volts
Vs
Zs
Zo
V(source) V(load)
TD = 250 ps
0
2 v
1.33v
3333
.
0


source
r 1

load
r
1.33v
1.33v
-0.443v
0v
2.66v
1.77v
-0.443v
2.22v
0.148v
0
500 ps
1000 ps
1500 ps
2000 ps
2500 ps 1.92
0.148v
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
Source
Load

48
Two Segment Transmission Line Structures
Vs
Rs
Zo1 Rt
Zo2
X X
a
b
c
d e
f g
h i
j k
l
3
3
2
2
2
2
4
2
1
2
1
3
1
2
1
2
2
1
1
1
1
1
1
1
r
r
r
r
r
r


















T
T
Z
Rt
Z
Rt
Z
Z
Z
Z
Z
Z
Z
Z
Z
Rs
Z
Rs
Z
Rs
Z
V
v
o
o
o
o
o
o
o
o
o
o
o
o
o
o
s
i
2
3
3
2
4
1
2
3
3
2
4
1
2
2
hT
i
k
iT
h
j
g
i
f
h
dT
e
g
eT
d
f
b
e
c
d
a
c
aT
b
v
a i















r
r
r
r
r
r
r
r
r
h
f
d
c
A
C
d
c
a
B
a
A









l
k
i
g
e
b
C
i
g
e
b
B
e
b
A












'
'
'
A
B
C
A’
B’
C’
1
r 2
r 3
r 4
r
3
T 2
T
TD TD
TD
3TD
2TD
4TD
5TD

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 ,T01
V1 V2
l1
I2
I1
VS
RS
Z02 ,T02
V3
l2
I3
Short
Previous examples are the
preparation

Class06_Transmission_line_basics.ppt

Recommended

Recommended

More Related Content

Similar to Class06_Transmission_line_basics.ppt

Similar to Class06_Transmission_line_basics.ppt (20)

Recently uploaded

Recently uploaded (20)

Class06_Transmission_line_basics.ppt