Non-Sinusoidal Waves on
(Mostly Lossless)Transmission Lines
Don Estreich
Salazar 2010C
Adjunct Professor
Engineering Science
October 2012
1
https://www.iol.unh.edu/services/testing/sas/tools.php
Outline of Presentation
1. Motivation: Signal Integrity
2. Discontinuities in PCB – microstrip and stripline in PCB
3. Some examples of PCB microwave circuits
4. Dispersion in Transmission Lines
5. Time Domain Reflectometry (unit step echoes)
6. Probing PCB and electronic assemblies
7. Brief review of transmission lines (with Smith Chart)
8. Surge impedance terminology
9. Step response for resistive loads
10.Step response for opens and shorts
11.Step response for and capacitive and inductive loads
12.Discontinuities along transmission lines
13.Using “lattice diagrams’ for transmission lines
14.Wrap-up
2
Signal Integrity in Packages and PCBs (PWBs)
PCB Interconnect
Carrier or
Package
Edge
connectors
Model with
Parasitics
3
Signal Integrity in a backplane
Fast IC
> 2 GHz signals
4
Sources of discontinuities in Printed Circuit Boards
Most high speed signals are differential
Some particular
origins of
discontinuities
in PCB/PWB
Agilent Signal Integrity Analysis Series
Part 1: Single-Port TDR, TDR/TDT, and 2-Port TDR
5
Microstrip and Stripline Transmission Lines
Microstrip
Stripline
Air
Dielectric (Alumina)
Ground plane
6
Printed Circuit Board (PCB) layouts
http://www.designacircuit.net/printed-circuit/
When is a PCB trace a transmission line?
What about crosstalk (i.e., coupled lines)?
7
Stripline
PCB Module (~ DC to 26.5 GHz) with Shielding
IC
IC
IC
RF Connectors
SMA type
Microstrip
8
Dispersion on Transmission Lines (1)
Remember: pulse-like waveforms
are made up of many frequencies.
Dispersion is the result of frequency-dependent group velocity – the separate
frequency components spread out and arrive at differing times.
http://ricksturdivant.com/dispersioneffects/
9
Dispersion on Transmission Lines (2)
Example of “10111” on an optical fiber shown as the start (0 km)
& at distances of 50 km and 100 km.
10
Dispersion on Transmission Lines (3)
Lossy Line
Model of
Transmission
Line
R+jωL
Z0 =
G+jωC
Early telegraph operators experienced the merging of dots and dashes from
dispersion over long transmission lines between stations.
Heaviside’s Condition: No dispersion resulted when
G
R
=
C
L

Group velocity 
Im  (R  j L)(G  jC) 
11
Lossy Transmission Lines
Ohmic loss in the metallization
Skin Effect
(frequency-dependent R and L)
Radiation losses (unshielded TL)
Insulating substrate loss
Dielectric losses and leakage
(frequency-dependent G
Semiconductor substrates
Si has lossy substrate from residual resistance
GaAs and InP have semi-insulating substrates
12
Time Domain Reflectometry (1)
Voltage decay
transient generated
by a fault in a cable.
Example
Fast step excitation
Pulse
Generator
Reference (t = 0)
Oscilloscope
Transmission Line under test
13
Time Domain Reflectometry Example (2)
Multi-section transmission line – TDR can locate where
discontinuities are present and characteristic impedance
of different sections of transmission line.
5 nH
5 nH
Voltage
60W/0.5ns
1 pF
45W/0.5ns
50W term
Time (ns)
14
Time Domain Reflectometry (3)
Practical waveforms
Ideal waveforms
http://pe2bz.philpem.me.uk/Comm01/-%20TestEquip/-%20TDR/Info-907-Theory/P2/pico.html
15
Reflection at discontinuity along transmission line
Vi
Vr
Z 01
Ii
-I r
Vt
z
Z 02
C
It
reflection coefficient

Z02  Z01
Z02  Z01
transmission coefficient T 
Pulse echo
2Z02
Z02  Z01
16
Probing an RF Printed Circuit Board
17
Optimize Probe Performance by Minimizing Tip Length
5 cm
Courtesy of Mike McTigue and Dave Dascher,
Agilent (Colorado Springs, CO)
1 cm
There is still
some LC
ringing from
the tip!
18
State-of-the-art in high frequency probing today (to  30 GHz)
Infiniium 90000 X-series Oscilloscope
InfiniiMax III probing system
19
Review: Model of transmission lines
Single incremental
section of line
  (R  jL)(G  jC) per meter
When R and G are small (our favorite conditions of course)

   LC
R
2
C G L

L 2 C
nepers / meter
radians / meter for R

c
 group velocity 

r
L & G
C
where c  300, 000 km / sec
20
Reflection coefficients on transmission lines
Z0
z=0
+
z = zL
Z0
+
VS(t)
ZL
V1
_
+
V2
_
_

vr (at z  z L ) ZL  Zo

 L   L

vi (at z  z L ) ZL  Zo
Smith
Chart
ZL
L
21
Reflection gives standing waves with sinusoidal
excitation
22
Remember the Smith chart – A way to visualize Z
=
50 W
Z0 normalized to 50 ohms
23
Special cases to remember
Terminated in Zo
Zs
Vs
Zo
Zo

  Zo Zo  0
Zo  Zo
Short Circuit
Zs
Vs
Zo

  0 Zo  - 1
0  Zo
Open Circuit
Zs
Vs
Zo

  Zo
 1
  Zo
24
Concept of Surge Impedance (or Surge Admittance)
Transmission line model
with loss
Z0 =
R+jωL
L
=
G+jωC
C
(if R = 0 and G = 0)
Characteristic Impedance Zo is defined by equation above.
A surge of energy on a transmission line will see an impedance of Z0 prior
to any reflections arriving; hence, Surge Impedance is an alternative name
for characteristic impedance.
Input Impedance is looking into loaded transmission in steady state.
25
Transmission Line Bandwidth
  
  
Questions:
What is the bandwidth of an ideal lossless transmission line?
What are practical limitations for bandwidth?
26
Step response for Z0 loaded TL (i.e., matched case)
Z0
z=l
z=0
+
Z0 ; T
+
VS(t)
Z0
VS
V1
_
+
VS(t) = VS·u(t)
0
_
V2
_
t=0
No
Reflection
VS
2.00
1.75
Z0 = 50 W
voltage [V]
1.50
1.25
1.00
V1
0.75
V2
0.50
0.25
0.00
0
T
2T
time
3T
4T
27
Step response for resistively-loaded transmission line
VS
Z0
z=0
+
+
VS
V1
_
voltage [V]
Z0 ; T
+
VS(t)
VS(t) = VS·u(t)
0
_
Rt
V2
_
t=0
Z0 = 50 W
VS
2.00
z=l
1.75
Rt =  ( = 1)
1.50
Rt > Z0
1.25
Rt = Z0 ( = 0)
open
½VS
1.00
Rt < Z0
0.75
V2
0.50
Rt = 0 ( = -1)
0.25
short
0.00
0
T
2T
time
3T
4T
This is just a simple
resistive voltage divider
but with time delay.
28
Step response for open circuited transmission line
I1
Z0
I2
z=0
z=l
Z0 ; T
+
VS(t)
V1
+
V2
VS
_
VS(t) = VS·u(t)
0
_
t=0
Current (A)
IS
0.75IS
I1
0.5I S
0.25IS
I2
T
0
Voltage (V)
VS
2T
3T
4T Time
2T
3T
4T Time
V2
0.75VS
V1
0.5VS
0.25VS
0
T
29
Voltage and current components for open circuit load
Voltage
Vs
2I
½Vs
I
0
T
c
Current
l
z
0
Vs
2I
½Vs
I
0
l
z
0
Vs
2I
½Vs
I
0
0
l
z
0 < t1
l
z
t1 < t2 < T
l
z
T < t3
l
z
-I
30
Step response for short circuited TL
I1
Z0
I2
z=0
z=l
Z0 ; T
+
VS(t)
V1
+
V2 = 0
VS
_
VS(t) = VS·u(t)
0
_
t=0
Current (A)
IS
I2
0.75IS
I1
0.5I S
0.25I
S
0
T
2T
3T
4T Time
2T
3T
4T Time
Voltage (V)
IS
0.75IS
V1
0.5I S
0.25IS
V2
0
T
31
BTW – This gives us a way to generate fast pulses
32
Summary for reflections (up to this point)
 Reflected voltage and current waves are generated
when incident waves encounter a discontinuity in a
transmission line
 The magnitude of the reflection is determined by the
impedances of the lines and by the amplitude of the
incident signal
 Special cases:
o Open circuits fully reflect the voltage signal
o Short circuits reflect the incident signal with equal magnitude
but opposite sign
o Matched circuits do not generate reflections
o Resistive loads generate reflections determined by voltage
division of resistances RL and Re[Z0]
33
Step response for capacitively-loaded TL
Z0
z=0
z=l
Z0 ; T
+
VS(t)
+
CL
VS
V1
VS(t) = VS·u(t)
0
_
V2
_
t=0
Z0 = 50 W
VS
2.00
1.75
voltage [V]
1.50
V2
1.25
V1
1.00
0.75
0.50
0.25
0.00
0
T
2T
time
3T
4T
34
Step response for inductively-loaded TL
Z0
z=0
z=l
Z0 ; T
+
VS(t)
+
LL
VS
V1
VS(t) = VS·u(t)
0
_
V2
_
t=0
Z0 = 50 W
VS
2.00
1.75
voltage [V]
1.50
1.25
1.00
0.75
V2
0.50
V1
0.25
0.00
0
T
2T
time
3T
4T
35
Step response for series RL and parallel RC loading

R - Z0
V(t)= Vs  1+
R+ Z0

a
Vs
  R - Z0  -t / t 
 +  1e 
R+
Z
0 
 


R - Z0 
Vs  1+

R+ Z0 

where τ =
L
R+ Z0
R
ZL
L
Vs
t=0
b
Vs
RZ0
where τ =
C
R+ Z0
-Vs
t=0

R - Z0 
Vs  1+

R + Z0 


R - Z0
V(t)= Vs  1+
R+ Z0

ZL
R
C


-t / t
+
1e






36
Step response for parallel RL and series RC loading
c

R - Z0
V(t)= Vs  1+
R+ Z0

R - Z0
Vs
R + Z0
 -t / t 
e 


R - Z0
where τ =
L
RZ0
Vs
ZL
L
R
0
t=0
d
R - Z0
Vs
R + Z0
Vs
 R - Z0  -t / t  2 Vs

V(t)= Vs  2  
e
R+
Z
0 
 


ZL
R
C
where τ = ( R+ Z0 ) L
t=0
37
Discontinuity at point z’ along transmission line
Z 01  Z 02
Vi
Vr
Z 01
Ii
-I r
Vr
Ir
ρV = ; ρI =
Vi
- Ii
Vt
Z’
Z 02
Z02
It
Vr Z02  Z01

 V
Vi Z02  Z01
38
Inductive discontinuity between Z0 transmission lines
Z0
Z0
+
VS
Z0
Z0
_
V
V
V
V
x
39
Voltage echoes from several embedded components
TDR Signatures
40
TDR signatures in transmission lines
41
Basics of a “lattice diagram” for transmission line (1)
V1
Vs
0
Z0
V2
TD
RL
RS
Vs
load
source
V2
V1
Time = 0
a
TD
1
b
2TD
c
2
3 TD
d
4TD
e
5TD
42
Terms for “lattice diagram” for transmission line (2)
load
 source
V2
Time
V1
0
Vs
Vlaunch
0
Vlaunch
2TD
V1
0
Z0
V2
TD
RL
RS
Vs
TD
Vlaunch load
Vlaunch(1+load)
Vlaunch loadsource
Vlaunch(1+load +load source)
3TD
Vlaunch 2loadsource
4TD
Vlaunch(1+load+2loadsource+ 2load2source)
Vlaunch 2load2source
5TD
43
Example using the “lattice diagram”
0
Z0
V1
2V
V2
Zs
TD = 250 ps
VS
 load  1
source  0 . 2
Time =
500 ps
V2
V1
0
open
0.8V
Vinitial  Vs
 50 
Zo
 (2)
 0.8


Zs Zo
 75 50 


source  Zs Zo  75 50  0.2
Zs  Zo
0v
0.8V
75  50

  50
load  Zl Zo 
1
Zl  Zo
0.8v
1000 ps
Assume: Zs = 75 ohms
Z0 = 50 ohms & RL is infinite (open circuit)
Vs= 0 to 2 volts
  50
1.6v
0.16V
1500 ps 1.76v
0.16v
2000 ps
1.92v
0.032v
2500 ps
0.032v
44
The “lattice diagram” is very useful for multiple lines
Z0
+
VS
0
Z0
_
S
1 ’1
2
’2
3 ’3
4 ’4
5
Easy to analyze multiple sections
45
Ringing behavior from reflections
Can you explain this?
46