Basics/Session Overview

– Relationship time and distance

– Signal propagation delay and propagation velocity
– Lumped systems versus distributed systems
– Mutual capacitance
– Capacitive crosstalk
– Mutual inductance
– Inductive crosstalk

08/10/2002 EE6471 (KR) 42

Basics/Time and Distance
• Electrical signals in conductors propagate at a finite
velocity (propagation velocity)
• Propagation velocity vp dependent on surrounding
medium
• Propagation delay Td (per unit length) is the inverse of vp

vp
x

08/10/2002 EE6471 (KR) 43

Basics/Time and Distance
c Permittivity εr
Propagation velocity vp =
Insulating Material Propagation velocity vp

εr
Air 1 300 mm/ns

Teflon 2 212 mm/ns

Polyimide 3 173 mm/ns

−1
Propagation delay Td = vp Silicon dioxide 3.9 152 mm/ns

FR4 (outer trace) 2.8-4.5 141…179 mm/ns

FR4 (inner trace) 4.5 141 mm/ns

Note:
Alumina (ceramic) 10 95 mm/ns
•Propagation delay increases with the
square root of the dielectric constant Silicon 11.7 88 mm/ns

•Equations assume that conductor is

surrounded by homogenuous medium
(if not, determine effective εr)
•Manufacturers of coax cable often use
foamed or ribbed material Signals on outer-layer PCB traces propagate
•Dielectric constant εr=f(T,f…) faster than those on inner-layer PCB traces!

08/10/2002 EE6471 (KR) 44

Basics/Time and Distance

Ldx Rdx Ldx Rdx Ldx Rdx

Cdx Gdx Cdx Gdx Cdx Gdx

• Often: Circuit elements of a component are distributed along its length and
not lumped in a single position
• The behaviour of distributed systems cannot be described by ordinary
differential equations (analysis requires partial differential equations)

• Question:
How small physically does a system need to be so that we can look at it as a
lumped system?
• Answer:
If the system is much smaller than the effective length of the fastest electric
feature in the signal.

08/10/2002 EE6471 (KR) 45

Basics/Time and Distance
•Series of snapshots of the
electric potential along a
trace.
lr t0
•Potential v(x) is not
x
uniform at all points
vp t1=t0+Tr
x
! Distributed System
•For systems physically
vp t2=t1+Tr small enough for all points
x
to react together: The
vp t3=t2+Tr voltage v(x) is uniform at
x
all points
vp t4=t3+Tr ! Lumped System
x

vp t5=t4+Tr
x

08/10/2002 EE6471 (KR) 46

Basics/Time and Distance

lr t0 Effective length of rising edge

x
lr = Tr ⋅ vp
vp t1=t0+Tr
x

t2=t1+Tr
•if l < lr/6
vp
x ! System behaves mostly
t3=t2+Tr
in a lumped fashion
vp
x

vp t4=t3+Tr
x

vp t5=t4+Tr
x

08/10/2002 EE6471 (KR) 47

Crosstalk/Mutual Capacitance
Circuit A
Vcc
•Wherever there is two circuit
va
nodes, there is mutual capacitance.
f
D Q •Circuits interact electrically
•Coefficient of electrical interaction
bitrate imutual due to electric fields is called their
mutual capacitance.
Cmutual

vb

Circuit B

{C mutual } = F = As
C mutual ∝
1
V distance
08/10/2002 EE6471 (KR) 48
Crosstalk/Mutual Capacitance
•Estimation of crosstalk. Assumptions:
Circuit A
Vcc •Capacitor Cmutual doesn’t load
circuit A significantly
va
f
D Q
•Coupled signal voltage (vb) is
small compared to signal voltage
bitrate imutual (va)
•Impedance of Cmutual is large
Cmutual compared to impedance to ground
of circuit B
vb
•Crosstalk is expressed as a fraction of
R the driving voltage
Circuit B
•Crosstalk is inversely proportional to
rise time Tr

∆Va R ⋅ imutual R ⋅ C mutual

imutual = C mutual dva = C mutual Crosstalk = =
dt Tr ∆Va Tr

08/10/2002 EE6471 (KR) 49

Crosstalk/Mutual Capacitance/Example

Linear Pulse Gaussian Pulse

6 5
5 5

va( t ) va( t)
2
V V
0
vb( t ) vb ( t)
10 ⋅ m ⋅V 0 10 ⋅ m ⋅V

− 2.5 4 − 4.502 5
0 50 100 150 0 50 100 150
0 t 150 0 t 150
n ⋅s n ⋅s
Example Parameter: R ⋅ C mutual
•Vcc=5V Crosstalk = = 0.5%
•Tr=10ns Tr
•Cmutual=0.5pF
•R=100Ω
Accuracy of crosstalk estimation depends strongly on pulse shape!
08/10/2002 EE6471 (KR) 50
Crosstalk/Mutual Capacitance
Estimation of the effect of
Cmutual_n
multiple interfering
Cmutual_2
sources
• Estimate the mutual
Cmutual_1 capacitances separately
• Sum the fractional
crosstalk figures
vb

Circuit B Crosstalk total = ∑ Crosstalk n

n

Conservative estimation!
08/10/2002 EE6471 (KR) 51
Crosstalk/Mutual Inductance
•Wherever there is two current
ia loops, there is mutual inductance.
•Circuits interact electrically
•Coefficient of electrical interaction
due to magnetic fields is called their
H-field
Circuit A mutual inductance.

•Magnetic field is a vector quantity:

•Sensitivity to loop orientation
(induced noise voltage reverses
Vb polarity)
•If loop B is in parallel to H-
field, no noise coupling
Circuit B

{Lmutual } = H = Vs
Lmutual ∝
1
n
with n = 2..3
A distance
08/10/2002 EE6471 (KR) 52
Crosstalk/Mutual Inductance
•Estimation of crosstalk. Assumptions:
•Inductor Lmutual doesn’t load circuit A
va R ia
significantly
•Coupled signal current ib is much
smaller than signal current ia
•Secondary impedance of Lmutual is
Circuit A small compared to impedance to ground
Lmutual of circuit B
•Crosstalk is expressed as a fraction of
the driving voltage
•Crosstalk is inversely proportional to
Vb
rise time Tr
•Assess multiple interfering sources
Circuit B
separately. Sum the fractional crosstalks.

Lmutual ⋅ ∆Va Lmutual

vb = Lmutual
dia
= Lmutual
1 dv a
= Crosstalk =
dt R dt R ⋅ Tr R ⋅ Tr

08/10/2002 EE6471 (KR) 53

Crosstalk/Mutual Inductance/Example
Linear Pulse Gaussian Pulse
10 10
5 9.004

5 5

va( t) va( t)
V V
0 0
vb ( t) vb( t)
10 ⋅ m ⋅V 10 ⋅ m ⋅V

5 5

− 5 10 − 9.004 10
0 20 40 60 80 100 0 20 40 60 80 100

0 t 100 0 t 100
n ⋅s n ⋅s

Example Parameter: Lmutual

•Vcc=5V Crosstalk est = = 1%
•Tr=10ns R ⋅ Tr
•Lmutual=5nH
•R=50Ω
Accuracy of crosstalk estimation depends strongly on pulse shape!
08/10/2002 EE6471 (KR) 54
Crosstalk/Mutual Inductance
Example Parameter:
Vcc
Vcc
•Vcc=5V
•Tr=10ns
v1 R C i1 Lmutual •Lmutual=3nH
v2
•C=100pF
•R=30Ω

Circuit A Circuit B
Lmutual
Crosstalk est = = 1%
10
6.33

R ⋅ Tr
5
v1 j Note:
V
•Estimation of crosstalk is based
i1 j
10 ⋅ m ⋅A
0 on linear approximation
v2 j •Use with care!
10 ⋅ m ⋅V

5
•Highly non-linear situations:
Analyse numerically! (Spice,
Mathcad)
− 6.33 10
0 10 20 30 40 50 60 70 80 90 100
0 t( j ) 99.994
n ⋅s
08/10/2002 EE6471 (KR) 55
Crosstalk/Mutual Inductance/Estimations
Estimation of mutual inductance Lmutual :
A2
d
h
d

1
Lmutual ≈ Lindividual ⋅
( h)
A1 2
1+ d
nH A1 ⋅ A2
Lmutual ≈ 200 ⋅
meter d3 Note:
•Mutual Inductance for transmission lines
valid for d > A1 and d > A2 •good estimation for stripline, microstrips,
and twisted pair
Note:
•Mutual Inductance for well separated loops

08/10/2002 EE6471 (KR) 56

Crosstalk/Capacitive vs Inductive Crosstalk

• In today’s high speed digital designs inductive crosstalk is typically a

more serious problem than capacitive crosstalk. Multiple reasons:
– low-impedance gate/driver output stages
– small and uncontrolled rise and fall times
– transmission lines directly driven by silicon without driver-side termination
– inadequate grounding
» insufficient or sectioned ground planes in PCB
» not enough ground pins in high pin-count connectors
» ground loops

08/10/2002 EE6471 (KR) 57