Transmission Line Basics
Acknowledgements: Intel Bus Boot Camp:
Michael Leddige
2
Real Computer Issues
data
Dev a
Dev b
Clk
Signal
Measured
here
Switch
Threshold
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.
Transmission Lines
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)
Transmission Lines
Two Transmission Line Viewpoints
Steady state ( most historical view)
Frequency domain
Transient
Time domain
Not circuit element Why?
Transmission Lines
4
5
Transmission Line Concept
Power Frequency (f) is @ 60 Hz
Wavelength (λ) is 5×
106 m
( Over 3,100 Miles)
Consumer
Home
Transmission Lines
Power
Plant
6
PC Transmission Lines
Signal Frequency (f) is
approaching 10 GHz
Wavelength (λ) is 1.5 cm
( 0.6 inches)
Microstrip
Integrated Circuit
Stripline
T
PCB substrate
Cross section view taken here
Stripline
W
Cross Section of Above PCB
Copper Trace
Via
FR4 Dielectric
MicroStrip
Signal (microstrip)
T
Ground/Power
Signal (stripline)
Signal (stripline)
Ground/Power
Copper Plane
Signal (microstrip)
W
Transmission Lines
7
Key point about transmission line operation
Voltage and current on a transmission line is
a function of both time and position.
V = f (z , t )
I = f (z , t )
I2
I1
V1
V2
dz
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
Transmission Lines
Examples of Transmission Line
Structures- I
Cables and wires
(a)
(b)
(c)
(d)
Coax cable
Wire over ground
Tri-lead wire
Twisted pair (two-wire line)
Long distance interconnects
+
-
+
(a)
-
-
+
(c)
-
(b)
+
(d)
Transmission Lines
-
8
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
E & H Fields – Microstrip Case
How does the signal move
from source to load?
Signal path
Y
Z (into the page)
X
Electric field
Magnetic field
Remember fields are setup given
an applied forcing function.
(Source)
Ground return path
The signal is really the wave
propagating between the
conductors
Transmission Lines
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
V
line
Z0 =
I
Propagation velocity is the speed with which signals are
transmitted through the transmission line in its
surrounding medium.
c
v=
εr
Transmission Lines
11
Presence of Electric and Magnetic Fields
I + ∆I
I
+
+
+
+
H
I
I + ∆I
E
V
I
-
-
-
-
V + ∆V
I + ∆I
V
I
H
V + ∆V
I + ∆I
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.
Transmission Lines
12
13
T-Line Equivalent Circuit
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
Ω/in]
Per-unit-length (Parallel) Conductance (G0) [S/in]
lR0
lL0
lG0
Transmission Lines
lC0
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.
lL0
lC0
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.
Transmission Lines
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.
Transmission Lines
15
T Line Rules of Thumb
So, what are the rules of thumb to use?
May treat as lumped Capacitance
Use this 10:1 ratio for accurate modeling
of transmission lines
Td < .1 Tx
May treat as RC on-chip, and treat as LC
for PC board interconnect
Td < .4 Tx
Transmission Lines
16
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
17
When does a T-line become a T-Line?
18
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?
Similarly, whether
Wavelength/edge rate
Transmission Lines
Tline
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
γ = ( R + jωL)(G + jωC ) = α + jβ
α is the attenuation (loss) factor
β is the phase (velocity) factor
Characteristic Impedance equation
( R + j ωL )
Z0 =
(G + jωC )
In class problem: Derive the high frequency, lossless
approximation for Z0
Transmission Lines
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 crosssectional 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
Z0 =
C0 =
v0 =
L0
;
C0
T0
;
Z0
1
µε
µ = µr µ0 ;
T d = L0 C0 ;
L0 = Z 0 T 0 ;
;
C0 L0 = µε;
ε = εr ε0 .
µ0= 4p X 10-7 H/m (free space)
µr is relative permeability
Don’t forget these relationships and what they mean!
Transmission Lines
21
Parallel Plate Approximation
Assumptions
TC
TEM conditions
ε
Uniform dielectric (ε )
between conductors
TC<< TD; WC>> TD
function of:
Material electric and
magnetic properties
Dielectric Thickness (TD)
Width of conductor (WC)
TD ; C0 , L0 , Z0 WC ; C0 , L0 , Z0 TD
WC
T-line characteristics are
Trade-off
22
ε * PlateArea Base
C=
d
C0
L0
Z0
WC  F 
ε⋅
⋅ 
TD  m 
TD
F
µ⋅
⋅ 
WC  m 
377 ⋅
TD
WC
⋅
µr
εr
equation
WC  pF 
8.85 ⋅ε r ⋅
⋅ 
TD  m 
T D  µH 
0.4 ⋅ π ⋅µ r ⋅
⋅

WC  m 
⋅Ω
To a first order, t-line capacitance and inductance can
be approximated using the parallel plate approximation.
Transmission Lines
23
Improved Microstrip Formula
Parallel Plate Assumptions +
WC
Large ground plane with
zero thickness
To accurately predict
microstrip impedance, you
must calculate the effective
dielectric constant.
TC
ε
From Hall, Hall & McCall:
 5.98TD 
Z0 ≈
ln

εr + 1.41  0.8WC + TC 
87
εe =
F =
εr + 1
2
+
εr − 1
12TD
2 1+
WC
Valid when:
0.1 < WC/TD < 2.0 and 1 < r < 15
+ F − 0.217(εr − 1)
WC 

0 . 02 (ε r − 1 ) 1 −

T
D


0
TD
2
for
for
WC
TD
WC
TD
<1
>1
Transmission Lines
TC
WCTD
You can’t beat
a field solver
24
Improved Stripline Formulas
Same assumptions as
T D1
WC
used for microstrip
apply here
ε
TC
T D2
From Hall, Hall & McCall:
Symmetric (balanced) Stripline Case TD1 = TD2


4(TD1 + TD1)

Z 0 sym ≈
ln

+
0
.
67
π
(
0
.
8
W
C
T
C
)
εr 

60
Valid when WC/(TD1+TD2) < 0.35 and TC/(TD1+TD2) < 0.25
Offset (unbalanced) Stripline Case TD1 > TD2
You can’t beat a
field solver
Z 0 sym(2 A, WC , TC , εr ) ⋅ Z 0 sym(2 B,WC , TC , εr )
Z 0offset ≈ 2
Z 0 sym(2 A, WC , TC , εr ) + Z 0 sym(2 B,WC , TC , εr )
Transmission Lines
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 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
25
Reflection and Transmission
Incident
ρ
1+ρ
Transmitted
Reflected
Reflection Coeficient Transmission Coeffiecent
ρ
Zt − Z0
τ
(1 + ρ )
"" → ""
Zt + Z0
τ
Transmission Lines
2⋅ Zt
Zt + Z0
τ
1+
Zt − Z0
Zt + Z0
26
27
Derivation:
Transmission Lines Class 6
Time versus Frequency Domain
Reflection Coefficient in the Time
Domain is a REAL number that depends
on theTime and the Position
Reflection coefficient in Frequency
Domain is a COMPLEX number and
depends on the Frequency and the
Position
Transmission Lines Class 6
28
29
Complex Reflection Coefficient seen
through Length of transmission L.
Transmission Lines Class 6
Power Dissipated on a Load
Transmission Lines Class 6
30
Quarter Wave Length
Transmission Lines Class 6
31
32
Special Cases to Remember
A: Terminated in Zo
Zs
Zo
Vs
Zo
−
ρ = Zo Zo = 0
Zo + Zo
B: Short Circuit
Zs
Zo
Vs
−
ρ = 0 Zo = −1
0 + Zo
C: Open Circuit
Zs
Vs
Zo
Transmission Lines
ρ=
∞ − Zo
=1
∞ + Zo
33
Voltage Standing Wave Ratio Coefficient
Can be MEASURED
Transmission Lines Class 6
34
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, εr = 4
Symmetric 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
35
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
36
Review: Voltage Divider Circuit
Consider the
simple circuit that
contains source
voltage VS, source
resistance RS, and
resistive load RL.
RS
RL
VS
The output
voltage, VL is
easily calculated
from the source
amplitude and the
values of the two
series resistors.
VL = VS
Why do we care for?
Next page….
Transmission Lines
RL
RL + RS
VL
37
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.
Transmission Lines
38
Determining Launch Voltage
TD
Vs
0
Rs A
B
Zo
Vs
Rt
(initial voltage)
t=0, V=Vi
Vi = VS
Z0
Z0 + RS
Vf = VS
Rt
Rt + RS
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!
Transmission Lines
39
Voltage Delivered to the Load
TD
Vs
Rs A
Zo
Vs
0
B
Rt
(initial voltage)
t=0, V=Vi
ρΒ =
t=2TD,
ρA(ρB)(Vi )
V=Vi
− +Zo
Rt+ ρB(Vi)
Rt + Zo
(signal is reflected)
t=TD, V=Vi +ρB(Vi )
Vreflected = ρΒ (Vincident)
VB = Vincident + Vreflected
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
Transmission Lines
Voltage Reflected Back to the Source
Vs
0
Rs A
Vs
B
Zo
ρA
ρB
Rt
TD
(initial voltage)
t=0, V=Vi
(signal is reflected)
t=2TD,
V=Vi + ρB (Vi) + ρA(ρ B )(Vi )
Transmission Lines
t=TD, V=Vi + ρB (Vi )
40
Voltage Reflected Back to the Source
ρΑ
− Zo
Rs
=
Rs + Zo
Vreflected = ρΑ (Vincident)
VA = Vlaunch + Vincident + Vreflected
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)]
Transmission Lines
41
42
Problems
Solved Homework
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
RS
VS
Transmission Lines
I1
I2
Z 0 ,Τ 0
V1
l
V2
RL
43
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 = ∞ /∞
∞ =1
V1(∞
∞) = V2(∞
∞) = 0.5VA(1+1) = VA
I1(∞
∞) = I2 (∞
∞) = 0.5IA(1-1) = 0
Solution
Transmission Lines
44
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
Transmission Lines
45
Waveshape:
Step-Function into T-Line with Open Ckt
I1
I2
Current (A)
IA
RS
0.75IA
0.5I A
VS
I1
I2
Z 0 ,Τ 0
V1
l
V2
0.25IA
0
Τ
2Τ
3Τ
V1
V2
VA
Voltage (V)
4Τ Time (ns)
0.75VA
This is called
“reflected wave
switching”
0.5V A
0.25VA
Solution
0
Τ
2Τ
3Τ
4Τ Time (ns)
Transmission Lines
Open
46
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) = 0.5
V1(∞
∞) = V2(∞
∞) = 0.5VA(1+0.5) = 0.75VA
I1(∞
∞) = I2(∞
∞) = 0.5IA(1-0.5) = 0.25IA
Solution
Transmission Lines
47
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
Transmission Lines
Waveshapes for Problem 1b
I1
I2
C ur ren t (A )
IA
RS
0.75IA
0.5I A
VS
48
I1
I2
Z 0 ,Τ 0
V1
l
V2
0.25IA
0
Τ
2Τ
3Τ
I1
I2
VA
V olta ge (V )
4Τ Time (ns)
Note that a
properly terminated
wave settle out at
0.5 V
0.75VA
0.5V A
0.25VA
Solution
0
Τ
2Τ
3Τ
4Τ Time (ns)
Transmission Lines
Solution
RL
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.
Transmission Lines
49
50
Lattice Diagram Analysis – Key Concepts
The lattice diagram is a
tool/technique to simplify
the accounting of
reflections and waveforms
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.
Vs
0
Vs
Zo
V(source)
Rs
TD = N ps
N ps
V(load)
V(source)
0
a
A’
A
b
B’
2N ps
3N ps
c
B
d
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
Rt
ρload
ρsource
Time
V(load)
4N ps
5N ps
Transmission Lines
C’
e
Lattice Diagram Analysis – Detail
ρ
ρ
source
load
V(load)
V(source)
0
51
Vlaunch
0
Time
N ps
Vlaunch
Vlaunch ρload
Vlaunch(1+ρload)
2N ps
Time
Vlaunch ρloadρsource
Vlaunch(1+ρload +ρload ρsource)
3N ps
Vlaunch ρ2loadρsource
Vlaunch(1+ρload+ρ2loadρsource+ ρ2loadρ2source)
4N ps
Vlaunch ρ2loadρ2source
0
V(load)
V(source) Zo
Vs
Rs
TD = N ps
Vs
Rt
5N ps
Transmission Lines
Transient Analysis – Over Damped
2v
0
Vs
Zo
V(source)
Zs
TD = 250 ps
ρ source = 0 . 2
0
Assume Zs=75 ohms
Zo=50ohms
Vs=0-2 volts
V(load)
ρ load = 1
Vinitial = Vs
V(load)
Time V(source)
ρ source =
0.8v
0v
500 ps
ρload =
0.8v
0.8v
1000 ps
52
Zo
 50 
= (2)
 = 0.8
Zs + Zo
 75 + 50 
Zs − Zo 75 − 50
=
= 0.2
Zs + Zo 75 + 50
Zl − Zo ∞ − 50
=
=1
Zl + Zo ∞ + 50
1.6v
Response fr om lattice diagram
0.16v
2.5
1500 ps 1.76v
2
2000 ps
2500 ps
1.92v
0.032v
V olt s
0.16v
1.5
Sour ce
1
Load
0.5
0
0
2 50
500
750
Tim e , ps
Transmission Lines
1000
1250
Transient Analysis – Under Damped
V(source)
2v
0
Zo
Zs
TD = 250 ps
Vs
ρsource = −0 . 3333
Time
Assume Zs=25 ohms
Zo =50ohms
Vs=0-2 volts
V(load)
V(load)
V(source)
0
ρload = 1
1.33v
Vinitial =Vs
−
−
ρsource = Zs Zo = 25 50 = −0.33333
Zs + Zo
0v
500 ps 1.33v
 50 
Zo
= (2) 
 =1.3333
+
+
Zs Zo
 25 50 
−
∞ −50
ρload = Zl Zo =
=1
Zl + Zo
1.33v
2.66v
1000 ps
∞ + 50
Response from lattice diagram
-0.443v
3
1500 ps 2.22v
-0.443v
0.148v
Volts
2.5
1.77v
2000 ps
25 + 50
2
1.5
Source
1
0.5
2500 ps
Load
0
1.92
0.148v
0
250
500
750 1000 1250 1500 1750 2000 2250
Time, ps
2.07
Transmission Lines
53
Two Segment Transmission Line Structures
X
Rs
X
Zo2
TD
Zo1
TD
Vs
T3 T2
ρ 2 ρ3
ρ1
Rt
ρ4
A=a
A' = b + e
B = a+c+d
B' = b + e + g + i
C = A+c+ d + f + h
C' = b + e + g + i + k + l
a
TD A
2TD
3TD B
4TD
5TD C
vi = Vs
c
b
d
e
f
g
h
i
j
k
ρ1 =
A’
ρ2 =
B’
l
C’
Z o1
Rs + Z o1
Rs − Z o1
Rs + Z o1
Z o 2 − Z o1
Z o 2 + Z o1
Z −Z
ρ 3 = o1 o 2
Z o1 + Z o 2
a = vi
b = aT2
c = aρ 2
d = cρ1
e = bρ 4
f = dρ 2 + eT3
g = eρ 3 + dT2
h = fρ1
Rt − Z o 2
ρ4 =
Rt + Z o 2
i = gρ 4
T2 = 1 + ρ 2
j = hρ 2 + iT3
T3 = 1 + ρ 3
k = iρ 3 + hT2
Transmission Lines Class 6
54
55
Assignment
Previous examples are the
preparation
Consider the two segment
transmission line shown to
the right. Assume RS=
3Z01 and Z02= 3Z01 . Use
I
I
R I
Lattice diagram and
Z ,Τ
Z ,Τ
l
l
calculate reflection
V
V
V
V
coefficients at the
interfaces and show the
wave forms of V1(t), V2(t),
and V3(t).
S
1
01
02
01
Check results with PSPICE
Transmission Lines
1
02
2
1
S
3
2
2
3
Short