Download III. Simulating Cross Talk in Matlab

Capacitive Discontinuity in Cross Talk Effect
Mihai Dărăban, Dan Pitică
Applied Electronics Department
Technical University of Cluj Napoca
Cluj Napoca, Romania
[email protected]
Abstract— There are a lot of studies that show the effect of
coupled lines, but most implementation in MatLab are based on
using an n-section lump circuit approximation model, which has
a limited bandwidth. Also on every PCB, on a trace it appears a
capacitive discontinuity, that influences the coupling of the
transmission lines. We propose another approach to describe the
coupled lines, and which will be capable to determine the effect of
capacitive discontinuity on the coupled lines.
Keywords-cross talk; pcb; transmision line; coupeld; noise
I.
INTRODUCTION
Cross talk is a phenomenon that appears more and more on
the pcb due to the small rise time of the logical gates and also
because the space between the traces is getting smaller,
following the direction of miniaturization.
All the signals on the pcb are affected by cross talk, but in
general bus traces are most affected by this signal-integrity,
and not only the buses from the on chip but also the ones on the
motherboards or other pcb board because of the faster and
faster logical families.
Because the traces are not uniform, it is possible to appear a
capacitive discontinuity that it can influence the propagation of
an unwanted signal from one net to an adjacent net (cross talk).
There it can appear and inductive discontinuities, but it is
complementary to the effect of capacitive discontinuity, so it is
enough to study one effect to understand the other one.
Capacitive discontinuity doesn’t mean an interrupt in the trace,
but a mismatch in the impedance of the trace, that the signal it
sees in high frequency.
A capacitive discontinuity it is mostly due to the receiver
input gate capacitance, corners, via or trace width change. It
will be shown the effect of capacitive discontinuity on the cross
talk between two adjacent traces.
II.
THEORETICAL BACKGROUND
A. Cross Talk Phenomenon
Cross talk appears between two or more adjacent nets
because of the electromagnetic field lines, there are electricfield lines between the signal and the return path and loops of
magnetic-field lines around the signal and return paths. The
electric and magnetic field lines are not confined to the
immediate space between the signal and the return paths, the
field lines are spread out into the surrounding volume (fringe
fields), [3].
Because of the proprieties of electromagnetic field, it is
possible that if a signal and his return path are passing together
in a region where there are still large fringe filed from another
net, the second trace may pick up noise from these field lines,
[3].
Cross talk can appear only when on a trace, the signal
voltage and current are changing, creating electric and
magnetic field lines. There are two types of scenarios in which
cross talk may occur: function cross talk noise and dynamic
crosstalk, [5]. Function cross talk noise appears when the
victim line experiences a voltage spike, because the aggressor
(active line) is switching. In the second scenario, cross talk is
experienced when aggressor and victim lines are switching
simultaneously.
On a passive line, cross talk appears at both ends. To
distinguish between the two ends of the line where the cross
talk can appear, the noise which appears near the source is
called “the near-end noise”, and the noise that appears at the
other end of the trace is called “far-end noise”, [3].
B. Models to Simulate Cross Talk
There are two methods by which cross talk between two
traces can be simulated. One method is based on describing the
traces using lump-circuit-model approximation. Each trace is
represented by n-section of LC cell network. The coupling
between the nets is described with the help of mutualcapacitance and mutual inductance elements.
The lump-circuit-model it is an approximation, because the
actual capacitance and the loop inductance that appears
between the trace and its return path, it is distributed uniformly
in the n-section of LC cell network. The error of the model it
gets lower, as the number of the LC cells which describe the
model, are increasing. The formula to compute the minimal
number of lumped sections for an accurate model, [3]:
n 10  BW  TD
where:
(1)
n – the minimum number of lumped sections; BW – the
required bandwidth of the model; TD – the time delay of each
transmission line.
Z odd 
Lodd
; TDodd  Lodd  Codd
Codd
Ceven  C11
(4)
(5)
Ceven – the total capacitance from the signal to the return of
one line when the pair is driven in the even mode
Leven  L11  L12
(6)
Figure 1. Structure of a transmision line described using LC cells
Another approach which can be used to simulate the cross
talk between two adjacent traces is by using odd- and evenmode impedance and the odd- and even-mode time delays,
which are able to describe all the transmission-line and
coupling effects. The biggest advantage of this model is that its
bandwidth is equal to that of an ideal lossless transmission line,
[3].
When the cross talk phenomenon is described using the odd
and the even mode, it is important how the transmission lines
are driven. For the odd mode a differential signal must be
applied, and for the even mode a common signal. This is
possible because any two waves can be described with the help
of a differential and common signal.
Leven – the total loop inductance from the signal to the return
path of one line when the pair is driven in the even mode
Z even 
Leven
; TDeven  Leven  Ceven
Ceven
(7)
C. Losses because of skin and dielectric effect
When the signal propagates down a trace on a PCB
structure, there are two kinds of losses that can appear, on is
because of the copper and the dielectric that insulates the traces
of the return plane. In copper the losses appear because every
trace it has a resistance, which at high frequency it increases
because the current it is no longer uniformly distributed in the
volume of the trace, appearing the skin effect.
The losses in the dielectric between the traces appear also at
high frequency when the conductivity of the dielectric
increases and because of this a AC – resistance leakage appears
between the trace and the return path.
Figure 2. EOdd and even propagation mode field lines
The transition from the model based on n-section LC cell
network to the model based on odd and even mode propagation
it is possible based on the following relations:
Codd  C11  2  C12
(2)
Codd – the total capacitance from the signal to the return
path of one line when the pair is driven in the odd mode; C11 –
the total capacitance from the signal to the return path for the
LC network model; C12 – the total mutual capacitance in LC
network model; Cload – the loaded capacitance of the signal line
(C11 + C12)
Lodd  L11  L12
(3)
Lodd – the total loop inductance from the signal to the return
path of one line when the pair is driven in the odd mode; L11 –
the total loop inductance from the signal to the return path, for
the LC network model; L12 – the total mutual inductance in LC
network model
In conclusion the losses are frequency-dependent, as can be
observed in the following formulas by which are computed the
losses in copper and dielectric, [3]:
R  0.8  
len
w  2.5 
1
f
(8)
G    tan( )  C
R – the resistance of the line, in Ohms; ρ – the bulk
resistivity of the conductor, in Ohm-inches; len – the length of
the line, in inches; w – the line width, in inches; f – the sine
wave frequency, in GHz, 0.8 – a factor due to the specific
current distribution in the signal and return paths
G – the shunt conductance from the dielectric, in Siemens;
tan(δ) – the dissipation factor; ω – the angular frequency, =
2∙π∙f; C – the capacitance between the signal and the return
path
By introducing this two parameters the compute of the
impedance of the trace it is changing, but also it is changing
and the formula by which the speed of the signal in the trace is
calculated:
Z
v
R  L
G  C

dia1 (Vag  Z ag ia1  Va 2 ) L11dx  (V pg  Z pg i p1  V p 2 ) L12dx

dt
( L11dx  L22dx  L12dx  L21dx )
(9)
1
( R 2   2 L2 )(G 2   2C 2 )   2 LC  RG 
2 

By using R and G we can compute the losses in cross talk
using the first model, and for the second model based on odd
and even propagation mode, the effect of losses can be observe
by calculating the Z and TD using equation (9).
III.
SIMULATING CROSS TALK IN MATLAB
As mentioned above in the paper, the cross talk
phenomenon can be simulated by two methods. One method is
using n-section of LC cell network to describe the two traces,
[1], which are coupled by mutual inductance and capacitance.
This model is described mathematical by using ordinary
differential equations, which are applied for each LC cell.
di p1
dt

(V pg  Z pg i p1  V p 2 ) L22dx  (Vag  Z ag ia1  Va 2 ) L21dx
( L11dx  L22dx  L12dx  L21dx )
dva1

dt
(C11dx  C12dx )(ia1  ia 2 )  C12dx (i p1  i p 2 )
dv p1
(C11dx  C12dx )(i p1  i p 2 )  C12dx (ia1  ia 2 )
dt

(10)
(C11dx  C 22dx  2  C11dx  C 21dx )
(C11dx  C 22dx  2  C11dx  C 21dx )
Electrical behavior of each cell that makes up the
transmission line is computed using a second set of differential
equations, where k it is the number of the cell.
The second method to describe the coupling between two
nets is by using odd and even propagation method. From the
mathematical point of view this method is simple because it
doesn’t require the use of ordinary differential equations (ode).
This method only uses the propagation delay and the reflection
coefficient when a mismatch occurs.
Figure 4. The structure of the k cell in the model
A. Decribing the N-Sectiont of LC Cell Network Using
Ordinary Differential Equations
There are three distinct cases to be treated if nets are
defined by small infinitesimal, discrete lumped elements placed
periodically down the length of the traces. These three
scenarios will be described using different sets of ordinary
differential equations. The first system of differential
equations, it is describing the LC cell electrical behavior at the
end of the trace near the voltage source.
diak (Va ( k 1)  Vak ) L11dx  (V p ( k 1)  V pk ) L12dx

dt
( L11dx  L22dx  L12dx  L21dx )
di pk
dt

(V p ( k 1)  V pk ) L22dx  (Va ( k 1)  Vak ) L21dx
( L11dx  L22dx  L12dx  L21dx )
dvak (C11dx  C12dx )(iak  ia ( k 1) )  C12dx (i pk  i p ( k 1) )

dt
( L11dx  L22dx  L12dx  L21dx )
dv pk
dt

(11)
(C11dx  C12dx )(i pk  i p ( k 1) )  C12dx (iak  ia ( k 1) )
( L11dx  L22dx  L12dx  L21dx )
The last system of ordinary differential equations, it
describes the cell from the far end of the line, where the lode is
placed.
Figure 3. The structure of the first cell in the model
Figure 5. The structre of the last cell in the model
diaN (Va ( N 1)  VaL) ) L11dx  (V p ( N 1)  V pL ) L12dx

dt
( L11dx  L22dx  L12dx  L21dx )
di pN
dt

(V p ( N 1)  V pL ) L22dx  (Va ( N 1)  VaL ) L21dx
( L11dx  L22dx  L12dx  L21dx )
dvaL (C11dx  C12dx )(iaN  iaL )  C12dx (i pN  i pL )

dt
( L11dx  L22dx  L12dx  L21dx )
dv pL
dt

(12)
(C11dx  C12dx )(i pN  i pL )  C12dx (iaN  iaL )
( L11dx  L22dx  L12dx  L21dx )
Figure 7. The signals that apears in odd even propagation mode
B. Describing the Coupling Between Two Adjacent Lines
Using Odd and Even Propagation Mode
Analyzing the cross talk phenomenon using n section LC
cell network it needs a lot of computation resources and time,
because each cell it is described by a system of four ordinary
differential equations. A method to reduce the time need to
compute is using odd and even mode propagation method.
For this method it is needed only two transmission lines,
one it simulates the odd-mode and the second one the even
mode. The propagation time and impedance for each mode are
computed using relations described in section II.B.
Because of the difference between the two propagation
modes it is important that the value of time step used to resolve
the equations in MatLab, must be the lowest common divisor
between todd and teven. The odd and even propagation mode will
be resolved using this time step so that there will be an integer
number of steps for each mode of propagation and also the
results of calculation to have the same time base, to be able to
add and subtract the results from the odd and even propagation
mode to obtain the signals on the active and passive line.
The signals from the active line are obtained by adding the
results from the odd and even propagation modes. On the other
hand the signals from the passive line are obtained by
subtracting the odd propagation mode results from the results
obtained in the even propagation mode.
Figure 6. The model to simulate odd even propagation mode
Because any wave can be described with the help of an odd
and even wave, the odd and even wave which are applied on
the two transmission lines are obtained as follows:
Vodd 
Veven 
Vactive  V pasive
2
Vactive  V pasive
(13)
Figure 8. The signals on the active and passive line
The two transmission lines that simulate the odd and even
propagation modes are implemented in MatLab using
reflection coefficient. The two pair of lines are geometrically
symmetric with the same line widths and dielectric spacing. If
the conductors are not symmetric the odd and even mode
voltage are no longer so simple, and a field solver is needed to
determine the specific odd and even mode voltage patterns
2
Let’s presume that there are two transmission lines coupled,
each line has a total inductance of 89,6nH and a total
capacitance of 38.61pF, the value of total mutual inductance is
of 20.03nH, and the total mutual capacitance is of 4.48pF.
Computing the parameters of odd and even propagation
mode, using the values above, we obtain Zodd=38.24Ω,
Zeven=53.28Ω, todd=1.82ns and teven=2.06ns. The small
difference of 0.24ns between the two types of propagation it’s
responsible for the near and noise and far end noise on the
passive line.
C. Comparison Between the Two Models
The time step which we showed is very important how it
selected for the odd and even mode propagation model, it is
also a problem in the first method which has been described.
Every cell in the n section LC cell network has its time delay
which must be higher than the time between two samples of the
signal applied to the line, to obtain accurate results.
In the first model the time step it can get very small
depending on the value of n, which determines the number of
LC cell sections. As can be observed in figures below by
increasing the number of sections the model error it gets
smaller but the time delay per cell it gets smaller also,
increasing a lot the time needed to finish the simulation.
Figure 9. Near end noise on the passive line when ranging the number of
cells in the model
Figure 12. Spectrum if the near and far end noise on the passive line
Each of the methods presented it is capable of determine
the losses that appear in the transmission line because of skin
effect and losses in dielectric. This is done by breaking the
signals applied to the lines in their spectral components using
fast fourier transformation. To the transmission lines are
applied only the sine waves components with their
corresponding amplitudes and phases. The number of sine
waves which are applied it is determined by the minimal
bandwidth required to reproduce the rise time of the original
signal, [3]:
Figure 10. Far end noise on the passive line when ranging the number of cells
in the model
In cross talk the value of n computed using equation (1), it
is no longer correct because the spectrum of cross talk is larger
than that of the signal that appears on a transmission line which
is not coupled. By increasing the number of cells the
components of high frequency that appear on the far signal on
the passive line are not cut off, and so we obtain an accurate
value of the amplitude. Also the higher number of cells is
reducing the ringing effect that is a falls noise, which appears
because of the limited bandwidth of the model.
Figure 11. Spectrum of the signals on the active line
Odd and even propagation model it has a bandwidth which
is equal to that of an ideal lossless transmission line, and so the
time step it only depends on the lowest common divisor
between todd and teven.
BW 
0.35
rise time
(14)
BW – the required bandwidth to reproduce the rise time of
the signal; rise time – the rise time of the signal o which FFT
was applied
In the first model the cell it is changing by introducing R,
responsible for losses in cooper, and G responsible for losses in
dielectric. Each of these two parameters is computed each time
a sine component it is applied, by using the formulas from
section II.C. The structure of each cell is that from the figure
13.
Figure 13. The cell structure when introducing skin and dielectric losses
Figure 14. The signals on the active and passive line when introducing skin
and dielectric losses (simulated using 30 cells, BW=1.5GHz)
By introducing the skin and dielectric losses the amplitude
of the noise on the passive line decreases, because of the
attenuation introduced by the increased resistance of the traces
and the leakage resistance of the dielectric.
IV.
HOW DOES CAPACITIV DISCONTINUITY INFLUENCE THE
CROSS TALK BETWEEN TWO TRACES
Coupling between two traces can be influence by the
distance between them or by their dimensions, but another
source which can affect the coupling between the two traces, is
a mismatch in impedance. A mismatch on one of the traces
coupled can occur because of a capacitive discontinuity, which
appears because of a via, or a change in the width of the trace
or because of the input gate capacitance.
The capacitive discontinuity it appears for the signal like a
lump capacitor, which produces a mismatch in the impedance
the signal it sees when traveling down the line, resulting in
reflections. The values of capacitive discontinuity it varies
between 0.2-1pF for a via, [4], and between 2pF to 10pF for
the input gate capacitance.
For the first model it is very simple to observe the effects of
capacitive discontinuity, it is only necessary to specify the
position along the line of the capacitive discontinuity, by point
the LC cell in which appears. In that cell the equations that
describe the propagation of signals are modified, according to
the new lump capacitance that appears on the line.
A solution to reduce the crosstalk when a capacitive
discontinuity appears is to place it as much as possible to the
near end of the transmission line but that affects the signal on
the active line, [2], so a compromise must be made.
Figure 16. How the possition of capacitiv discontinuity can influence the
noise levels
V.
The second model is better to be used in simple coupled
transmission lines, and that lines should be symmetrical as
much as possible to obtain accurate results. The model it has an
infinite bandwidth and the time needed to compute it is smaller
than that of the ordinary differential equation model.
In both models time step it is crucial but especially in the
first model, in the second one the precision whit which the
results are computed it is more important, because the reflected
wave after a number of propagation tend to get very small, and
so we need a lot of digits after the decimal point to obtain good
results.
REFERENCES
[1]
[2]
[3]
[4]
[5]
Figure 15. Simulating capacitiv discontinuity using LC network cell
CONCLUSION
As we showed the both methods have their ups and downs.
The first model it is easier to be used in complex coupled
transmission lines because it only requires some minor
modification in the equation system to describe the
discontinuity that can appear. The accuracy of the results
depends of a lot of parameters. If we want a good accuracy
then the time needed to compute must increase, and also we
must pay attention to the bandwidth of the model by selecting
the correct number of the LC network cells, which describes
the model.
M. Rangu and C. Negrea, “An educational tool for transmission lines
animations”, ISSE Conference, 12-16 May 2010 Warsaw, Poland, in
press
M. Daraban and D. Pitica, “The Influence of Capacitive Discontinuity in
Signal Inegrity on PCB Traces”, ISSE Conference, 12-16 May 2010
Warsaw, Poland, in press
E. Bogatin, “Signal and power integrity - simplified”, Second edition,
Pretince Hall, pp. 337–553, July 2009
N. Codreanu, “Metode avansate de investigatie a structurilor PCB”,
Cavallioti, pp. 175-237, 2009
B. K. Kausik and S. Sarkar, “Crosstalk analysis for a CMOS gate driven
inductively and capacitively coupeld interconects”, Microelectronics
Journal, vol. 39, pp. 1834-1842, 2008