[hsdd] High-Speed Digital Design Newsletter
- From: "Dr. Howard Johnson" <howiej@xxxxxxxxxx>
- To: <hsdd@xxxxxxxxxxxxx>
- Date: Tue, 18 Mar 2003 16:03:11 -0800
SHORT-TERM IMPEDANCE OF PLANES
HIGH-SPEED DIGITAL DESIGN ? online newsletter ?
Vol. 6 Issue 05
It's a gorgeous day here in the Methow Valley region
of Washington state. The snow is nearly all melted
and the grass is turning a healthy green color. I
hope you have enjoyed a pleasant winter season.
The last newsletter left open a basic question about
the efficacy of parallel planes at extremely high
speeds, a question which I hope to fully answer in
this issue.
______________________________________________________
SHORT-TERM IMPEDANCE OF PLANES
By Dr. Howard Johnson
In a four-layer board, assume you have a trace that
jumps through a via from layer 1 to layer 4. Before
the via, the trace lies adjacent to plane 2. After
the jump, the trace lies adjacent to a different
plane, layer 3.
This configuration is illustrated in Figure 1
(copied from newsletter v6-04, "Via Inductance").
The figure shows how at every point where high-
frequency signal current flows, an equal and
opposite current flows on the solid reference plane
adjacent to the signal conductor. The principle of
equal and opposite current on the plane implies
that, somewhere near the signal via, the returning
signal current associated with this trace must hop
between layers 2 and 3.
-sig trace-->-----via-- layer 1
--<--return------ || ------------------ layer 2
||
----------------- || ---<--return------ layer 3
--via-----sig trace-->- layer 4
Figure 1--Near the via, returning signal current
originating on layer 2 must somehow leap to layer 3
("High-Speed Signal Propagation", Fig. 5.33, p. 353)
[Ed. note--A much improved version of the figure
appears in the web-based version of this article
located at www.sigcon.com, under "archives". ]
Doesn't the returning signal current just pop
between the planes through the parasitic capacitance
of the planes themselves, you might ask?
The short answer is that in typical cases where the
planes are widely spaced (as in a stripline cavity)
the current seeks out the nearest metallic
connection, such as a via or bypass capacitor, and
follows that path.
To investigate that question I have set up a
simulation that calculates the impedance between two
parallel planes, as measured at the location the
signal via in Figure 1. To keep the math simple,
I will assume the planes are circular (as opposed to
the square geometry shown in the figure).
I will try various plane radiuses including 1, 2, 3,
5, and 8 cm. The inter-plane separation in each case
will be h = 0.030 in. (0.76 mm), filled with a
material having a dielectric constant of 4.3,
corresponding to the situation normally used when
implementing a very thick board with striplines
between the planes.
In microwave parlance, the "coupling port" that
gives access to the cavity between the planes is the
via clearance hole, which has an assumed radius of
rc = 0.010 in. (0.25 mm).
The simulation pumps a 20-mA step of current from
one plane to the other at the clearance hole
location. The size of this step corresponds to the
jolt of current that would prevail when sending a 1-
V p-p signal from a 50-ohm trace through the signal
via. The 10-90% signal risetime is tr = 100 ps
(pretty fast).
If you have a calculator handy, you should check the
length of this rising edge as it slides down the
transmission structure, as measured from its 10% to
its 90% values. The length is calculated as
trc/sqrt(Er) , where c is the speed of light in a
vacuum, and Er is the dielectric constant of the
material between the planes. Your answer should come
out to 0.57 in. (1.4 cm). Half a risetime
corresponds to 0.7 cm. Hold onto this bit of data;
we will need it in a moment.
To set up this experiment I first broke the planes
into a set of concentric rings. The thickness (in
the radial direction) of each ring was 0.04 cm
(about 1/35th of the length of the rising edge). The
number of rings required to model a disk of radius 8
cm was 200.
-------Extra for experts---------
The technical setup for this experiment is best
comprehended by first cutting the whole arrangement
into eight wedges (like a pie). A 1/8th slice of the
pie forms a horn-shaped transmission line, broken
into chunks at radial increments of 0.04 cm. Each
little chunk acts like a short, fat segment of
transmission line. As you proceed out from the
center, the chunks get wider and wider, which means
the characteristic impedance of each chunk must be
getting progressively smaller.
The experiment calculates the effective impedance
and length of each of the little chunks and then
concatenates all 200 of these chunks together into
one big cascade of two-port models. I used MathCad
to compute the input impedance of this whole
structure. By symmetry, I then take the impedance of
this one single pie slice and divide it by eight to
obtain the impedance of the whole pie.
I checked my solution against the standard Bessel-
function solution for the impedance of a circular
plate and it matched perfectly for the case of no
losses within the planes. Then I modified the
MathCad model to incorporated both skin effect and
dielectric losses using the approach outlined in
High-Speed Signal Propagation. The losses
substantially damp resonances in the frequency
response.
-------End extra section ---------
Figure 2 illustrates the impedance of the various-
sized disks used in this experiment (all having a
dielectric thickness of 0.030 in.). At frequencies
below 100 MHz, all five disks exhibit a clearly
capacitive behavior (impedance trending down with
increasing frequency). The total capacitances of the
disks, listed in order of decreasing size, is 1005,
392, 141, 63, and 16 pF.
[Ed. note--Figure 2 appears in the web-based version
of this article located at www.sigcon.com, under
"archives". -- Sorry, I can't send pictures through
an email reflector. ]
Somewhere above 100 MHz, depending on the size of
the disk, the impedance reaches its nadir and begins
to climb. This minimum-impedance point is called a
point of series resonance. The resonance has to do
with the speed of round-trip wave propagation back
and forth from side to side (or center to edge,
depending on how you look at it) across the disk.
The bigger disks have a lower point of series
resonance.
Above the point of first series resonance, the
impedance trends generally upward amongst a
multitude of horrid resonances. Figure 2 shows a
black line corresponding to the impedance of a 1-nH
inductor. Notice that, regardless of the size of the
disk, all the curves trend upward in a pattern
reminiscent of the 1-nH inductance.
In this example, you may model the region
surrounding the first series resonance as simply a
capacitor equal to the total capacitance of the disk
placed in series with an effective series inductance
of 1 nH. If you can "see through" the gyrations
induced by the upper-level resonances this simple L-
C series model serves pretty well at all
frequencies.
I should like to thank Minjia Zu, Yun Ji, Todd
Hubing, Thomas P. Van Doren, and James L. Drewniak
for making clear the theoretical basis for the
existence of this effective series inductance in
their paper titled, "Development of a Closed=Form
Expression for the Input Impedance of Power-Ground
Plane Structures", published in the proceeding of
the IEEE EMC Symposium 2000.
Figure 3 shows the step-response waveforms
corresponding to the six impedance plots in figure
2. Amazingly, the response is the same during the
rising edge in each case, regardless of the size of
the disk. This happens because the outer reaches of
the disk (more than 1/2 of a rise-time from the
center) are located too far away to possibly
influence the signal before the conclusion of the
rising edge. In order the influence the response
during the initial risetime, an object must be
located close enough that electromagnetic energy can
propagate to the object, interact with it, and
return, before the risetime is complete. From this
constraint derives the idea that no areas of the
disk further away than 1/2 of a risetime can
possible influence the initial response.
[Ed. note--Figure 3 appears in the web-based version
of this article located at www.sigcon.com, under
"archives". It's worth a look. ]
Your earlier calculation showed that half a risetime
corresponds to 0.7 cm. Since all the disks have a
radius at least that big, you should expect no
difference in the initial response in each case.
Additional plane material located further away than
0.7 cm simply doesn't make any difference on a scale
of time as small as 100 ps. All that matters during
the initial rising edge is the effective inductance
of the disk, which is the same in each case. In this
example with the planes separated by 0.030 in.,
Er=4.3, and the access port radius equal to 0.010
in. the effective inductance is roughly 1 nH.
Reflections from the edge of the disk arrive at
successively later intervals in the larger disks.
After the initial response, the each disk begins to
charge up on a ramp slope corresponding to the
amount of that disk's total capacitance.
The impact of the plane's effective inductance on
your signals is precisely the same as if you had
connected an inductance of that size in series with
your trace. A single-ended signal traversing an
inter-plane cavity of height 0.030 in. through a
clearance hole of 0.010-in. radius will therefore
generate a short reflected pulse whose amplitude, as
a fraction of the incoming step height, equals
(1/2)(L/Z0)(1/tr), where L is the effective
inductance, Z0 the characteristic impedance of the
line, and tr the signal risetime. Plugging in 1 nH,
50 ohms, and 100 ps gives a reflected pulse height
of 10% of the incoming step amplitude. At a
risetime of 1 ns (1000 ps) you get only a 1%
reflection, which is why we digital designers have
not worried about this effect up until now. When you
get down to sub-nanosecond risetimes, however, it
really starts to matter.
The last (lowest) plot in Figure 2 shows the step
response of a 1 nH inductance. This is the effective
inductance you would get if you connect the planes
together with a single via having a hole radius of
0.005 in. (0.010-in. diameter), separated 0.140 in.
from the signal via. During the first rising edge a
via of this configuration presents the same
impedance as the planes, so it takes half the
current. This reduces the magnitude of the reflected
signal by a factor of two as well reducing both EMI
and crosstalk.
That brings me back to the subject of this article,
which has to do with the flow of returning signal
current in a multi-layer board. If you interconnect
the planes with vias or low-inductance bypass
capacitors, a substantial fraction of the returning
signal current flows through your intentional
interconnections.
As you scale down the height h between the planes
the impedance of the planes shrinks in direct
proportion to h. This scaling principle is
enormously helpful. It is the reason I like to see
your power and ground planes as close together as
possible. If you get the planes close enough (for
example a power and ground pair with 2-mil height),
the impedance of the planes themselves can become
sufficiently low that you do not need additional
interconnections to help your return current pop
from one plane to the other. On the other hand, when
your signals traverse a relatively tall stripline
cavity, additional interconnection vias play an
important role in containing signal reflections,
crosstalk, and EMI.
Best Regards,
Dr. Howard Johnson
______________________________________________________
Join us at our upcoming seminar in Dallas, TX, Mar.
24-25, 2003. A few seats are still available. A full
schedule of cities and dates appears at:
www.sigcon.com.
Questions & Comments: all students who attend our
High-Speed Digital Design seminars have the
opportunity to talk directly with Dr. Johnson about
signal integrity issues.
High-Speed Signal Propagation: Advanced Black Magic
is here! This 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.
Related articles:
newsletter v2-14 "Measuring Power-to-Ground Impedance"
newsletter v2-27 "Measuring Power Plane Resonance"
newsletter v3-21 "Interplane Capacitance"
newsletter v6-04, "Via Inductance").
If you have an idea that would make a good topic for
a future newsletter, please send it to hsdd@xxxxxxxxxxx
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