[hsdd] High-Speed Digital Design Newsletter
- From: "Dr. Howard Johnson" <howiej@xxxxxxxxxx>
- To: <hsdd@xxxxxxxxxxxxx>
- Date: Thu, 24 Apr 2003 15:53:37 -0700
*****RESONANCE IN SHORT TRANSMISSION LINE*****
HIGH-SPEED DIGITAL DESIGN ? online newsletter ?
Vol. 6 Issue 06
Announcing: An all-new advanced seminar based on my
latest book, High-Speed Signal Propagation: Advanced
Black Magic. This course will appear in the U.S.
starting in the fall of 2003. If you'd like to bring
this new seminar to your site please write to
Jennifer (jennifer@xxxxxxxxxx) or call her at (U.S.)
509.997.0505.
This will be an advanced-level course for
experienced digital designers who want to press
their designs to the upper limits of speed and
distance. Focusing on long transmission environments
like backplanes and cables, this course teaches a
unified theory of transmission impairments that
apply to any transmission media. Topics include:
lossy media, single-ended and differential
signaling, frequency-domain modeling, and clock
jitter. This course is an advanced sequel to my
original course High-Speed Digital Design. For more
information please see www.sigcon.com.
The first public presentation of the new course will
take place at Oxford University May 22-23, 2003. If
you want to preview to course to see if it would
benefit your organization, this is a good time to
catch it. Register now at
http://www.conted.ox.ac.uk/electronics/courses/hspp.html.
This month's article includes a lot of pictures. The
pictures are located in the web version of this
newsletter, at www.sigcon.com under "archives". Some
day, if you ever have grandchildren, you can tell
them about the good old days when it wasn't
considered polite to email pictures on a
distribution list, so everybody had to go to the a
web site if they wanted to see the color graphics.
Anyway, the picture aren't essential to your
understanding of this article, but they do highlight
the text in a visually interesting way.
______________________________________________________
RESONANCE IN SHORT TRANSMISSION LINE
by Dr. Howard Johnson
Many engineers use rules of thumb to determine when
terminations will be necessary. One popular rule
compares the risetime of the driver to the delay
(length) of a transmission line. If the risetime is
six times, or three times, or sometimes even only
two times the raw, one-way unloaded line delay then
you might be tempted to conclude that a termination
is not necessary. The efficacy of such a rule hinges
on a crucial hidden assumption: that the driver
source impedance is not too much less than the line
impedance. Violate that assumption and the rule
fails miserably under certain conditions of loading.
For example, start with a simple linear driver
having a 10-ohm output impedance (RS = 10 ohms) and
a risetime tr = 1 ns. Connect this driver to a
transmission line having a characteristic impedance
of 50 ohms and a physical length of 1 inch (176 ps
raw unloaded trace delay) (see Figure 1). The driver
risetime in this case is about six times the line
delay, indicating at first glance no need for
termination, but at the same time the driver output
impedance (10 ohms) lies far below the transmission
line impedance (50 ohms), raising the possibility of
severe resonance. Will this line ring or not?
(Figure 1.)
______________________________________________________
Driver,
RS=10ohm ---(50 ohms, 176 ps/in.)----(CL to ground)
Where the driver has a 10-ohm source impedance, the
line has a characteristic impedance of 50 ohms and a
delay of 176 ps (~ 1 in. of FR-4), and the
capacitive load CL varies from 0-200 pF.
Figure 1--A short transmission line with a capacitive
load sometimes rings horribly.
______________________________________________________
Those of you familiar with analog circuits will find
the pi model helpful in visualizing how this circuit
functions (Figure 2). The pi model for a short
section of transmission line is composed of a
capacitor C1 shunting the signal to ground, followed
by a series inductance L, followed by another
capacitor C2 shunting to ground.
______________________________________________________
-------- L --------------
| |
C1 C2
| |
-------------------------
Figure 2--Under ordinary conditions the pi model is
accurate to about 2% when the transmission-line
delay T is less than 1/6th the rise or fall time.
______________________________________________________
The value of the components in the pi model are
calculated from the transmission line parameters,
where if td equals the raw, unloaded one-way delay
of the transmission line and Z0 its characteristic
impedance, then
L = td*Z0 [1]
C1 = C2 = (1/2)(td/Z0) [2]
The pi-model applies in an accurate way only to
transmission lines whose delay td is short compared
to the signal rise of fall time. When td is less
than 1/6th the rise or fall time you can expect the
accuracy of the simple pi-model approximation to be
better than about 2% under ordinary conditions of
loading as used in digital applications. When td is
enlarged to 1/3rd the rise or fall time the accuracy
deteriorates to about 20%.
In the present example td is about 1/6th the
risetime tr, so the pi model applies. Note that the
ratio td/tr doesn't actually determine whether a
termination is needed, only whether the pi method of
analysis is useful to predict the circuit behavior.
The beauty of the pi model is how well it
illustrates the resonant properties of a
transmission circuit. The pi-model in conjunction
with a capacitive load forms a circuit virtually
bristling with L's and C's. That should raise in
your mind immediate concerns about the possibility
of terrible resonance, which is exactly what can
happen.
Let's begin the analysis assuming no capacitive
loading (CL = 0). Without worrying too much about
math, let me just tell you that the resonant
frequency fr of an unloaded line driven by a low-
impedance source works out to (1/4)(1/td). Given
that in this case td = (tr/6), a little algebraic re-
arrangement shows that:
fr = (1/4)(1/td) = (1/4)(6/tr) = (3/2)(1/tr) [3]
The expression for fr shows that the resonant
frequency of an unloaded line lies at a point three
times higher than the knee frequency fknee =
(1/2)(1/tr) associated with the rise and fall time
of the driver. This relationship means that the
resonance, even though technically present, has no
practical effect because the resonance occurs at
frequencies outside the bandwidth of the driving
signal. This relationship is also the basis for the
popular myth that lines shorter than a certain
amount (roughly 1/6th or perhaps 1/3rd the rise or
fall time) never require termination. What goes
wrong with that rule of thumb is easily seen in a
frequency-domain plot showing the gain of the
transmission circuit (Figure 3).
The figure shows for CL = 0 a towering resonance at
about 1.4 GHz. This is the unloaded resonance
associated with the transmission line delay.
Increasing the load CL pulls the resonant frequency
lower. In this circuit, any load greater than 12 pF
creates a resonance below the knee frequency
associated with the driver. That's the essence of
the problem. If capacitive loading decreases the
resonant frequency associated with your transmission
line to a point near (or below) the knee frequency
associated with your driver you will see ringing and
overshoot in the time-domain (Figure 4).
Going back to the pi-model analogy, those of you
with analog design experience may immediately
recognize that increasing the capacitance of the
load in this circuit reduces the resonant frequency.
I like very much how the pi-model circuit
illustrates this effect. A few years back, one of my
more astute students noticed that as the resonant
frequency decreases, so does the Q. The Q of a
resonant circuit is a measure of the severity of the
resonance: bigger values of Q imply more ringing and
overshoot, provided that the resonance falls near
(or below) the knee frequency of your driver. Figure
3 shows how increasing CL reduces the Q. Above 200
pF the Q is reduced to the point where the resonance
disappears?producing a critically damped circuit.
In the time-domain waveforms (Figure 4) you can see
that for very large values of CL the circuit no
longer rings, but merely produces a long, sluggish
response to a crisp step input.
The various behaviors described in this note suggest
two ways to combat ringing on short, unterminated
transmission lines.
1. Reduce the load capacitance, raising the resonant
frequency well above the knee frequency of your
driver. To make this work you need
CL << C1 = (1/2)(td/Z0) [4]
2. Increase the load capacitance to the point where
the line reacts in a critically-damped fashion. This
approach sacrifices speed for monotonicity, a good
trade in some cases (especially for relatively slow
clocks produced by overly speedy drivers). To make
this work you need
(CL + 2*C1) > (L/(RS*RS)) = (tdZ0/(RS*RS)) [5]
From the critical-damping expression [5] you can see
the effect of increasing RS. If RS is increased to
the point where it equals the characteristic
impedance of the line (RS = sqrt(L/(2*C1)) then the
line damps itself even with CL = 0. That's the
benefit of raising RS. The higher you make RS the
more natural damping you get. The lower you make RS
the harder you have to work to damp the line
elsewhere (i.e., by making CL larger or by adding
other termination components).
Capacitances in the middle range between these two
extremes [4] and [5] cause the worst problems with
ringing and overshoot.
______________________________________________________
The following color pictures appear in the web
version of this article at www.sigcon.com, under
"archives":
Figure 1?A short transmission line with a capacitive
load sometimes rings horribly.
Figure 2?Under ordinary conditions the pi model is
accurate to about 2% when the transmission-line
delay T is less than 1/6th the rise or fall time.
Figure 3?Increasing the load CL pulls the resonant
frequency lower. In this circuit loads greater than
12 pF resonate at frequencies below the knee
frequency associated with the driver.
Figure 4?The resonance associated with capacitive
loading manifests itself as overshoot and ringing in
the time domain.
______________________________________________________
Join us at our upcoming seminars at Oxford
University May 19-20 (regular High-Speed Digital
Design), and May 22-23 (Advanced High-Speed Signal
Propagation). A full schedule of cities and dates
appears at: www.sigcon.com .
High-Speed Signal Propagation: Advanced Black Magic
is here! This book is an all-new sequel to (not just
an update of) my earlier book High-Speed Digital
Design: A Handbook of Black Magic. See preface and
table of contents at www.sigcon.com , under "books".
Check for it at www.barnesandnoble.com , or
www.amazon.com , ISBN 013084408X.
If you have an idea that would make a good topic for
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