For those of you who wanted to know about trace inductance at ~1.9GHz - I have a
friend (Nat Sokal) who is a genius at this stuff. I forwarded your inquiry to
him and here is is reply (contact him directly if you need more info):
Regards,
Bob Landman
-----Original Message-----
From: Nathan Sokal <[log in to unmask]>
To: Bob Landman <[log in to unmask]>
Date: April 17, 1999 10:52 PM
Subject: [DC] Trace Inductance
Bob & friends,
The one via, or two vias in parallel, connecting the ground-pad of a
decoupling capacitor to the PCB ground plane are one small inductance, or
two small inductances in parallel, that are in series with the decoupling
capacitor. Whether the one or two vias' inductance is important depends on
what that inductance is being compared with:
1. What is the reactance of that decoupling capacitor at the frequency of
interest? Use X(C) = -1/(2*pi*f*C), where C is the capacitor value, pi =
3.14159, f is the frequency of interest, and * indicates multiplication.
2. Find inductance L of each via, from equation (11) below. Two such
vias in parallel is approximately half the inductance (an accurate equation
is on p. 37 of the book referenced below).
3. Calculate reactance X(L) of the one via or the or the two vias in
parallel. Use X(L) = +2*pi*f*L .
4. Is X(L) appreciable for your purposes, compared with X(C)?
If this is an RF application where 1.9 GHz is the only frequency of
interest at that point in the circuit, you could exploit a useful trick and
*intentionally* make an inductive connection from the decoupling
capacitor's ground pad to the ground plane, where the positive reactance of
that intentional inductance is made equal to the negative reactance of the
decoupling capacitor. Then the sum of those two reactances in series will
be close to zero (but *at only that frequency f*), giving even better
decoupling action (*at that frequency f*). If you operate in a frequency
band in the vicinity of 1.9 GHz, you can make the above calculation at the
lower end, middle, and upper end of the frequency band.
Equation (11) below gives inductance L in microhenries, of a straight
tubular conductor (e.g., a PCB via) of length l, outer radius r1 and inner
radius r2; all dimensions in cm.
L = 0.002 l [ ln (2 l/r1) + ln (zeta) - 1 ] (11)
where ln (zeta) is given in Table 4 as a function of r2/r1.
TABLE 4
r2/r1 ln (zeta)
0 0.2500
0.10 0.2452
0.20 0.2320
0.30 0.2123
0.35 0.2007
0.40 0.1880
0.45 0.1745
0.50 0.1603
0.55 0.1456
0.60 0.1304
0.65 0.1148
0.70 0.0989
0.75 0.0827
0.80 0.0663
0.85 0.0499
0.90 0.0333
0.95 0.0167
1.00 0
A bonus: For the straight-line conductors on your PCB: Equation (9) below
gives inductance L (in microhenries) of an isolated straight conductor of
rectangular cross-section (e.g., a PC conductor) that is far away from the
conductor that carries the return current. [That's a bad way to lay-out
high-frequency circuits. The better way is to have the return conductor
(usually the ground plane, but not always) lie underneath the conductor
that is carrying current. In that case, the inductance is less, given by a
different equation that I can give you if anyone wants it.]
L = 0.002 l { ln [2 l/(B + C)] +1/2 - ln (E) } (9)
where
l is length of conductor
B and C are the thickness and the width of the conductor
all dimensions are in cm
ln (E) is given in Table 3 below, as a function of B/C or C/B, whichever is
smaller than 1.
TABLE 3
B/C or C/B ln (E)
0 0
0.025 0.00089
0.05 0.00146
0.10 0.00210
0.15 0.00239
0.20 0.00249
0.25 0.00249
0.30 0.00244
0.35 0.00236
0.40 0.00228
0.50 0.00211
0.55 0.00203
0.60 0.00197
0.70 0.00187
0.80 0.00181
0.90 0.00178
1.00 0.00177
Source: Frederick W. Grover, "Inductance Calculations: Working Formulas
and Tables," Dover Publications, Inc., New York, 1962 (unabridged and
corrected republication of book first published in 1946 by D. Van Nostrand
Co.), equation (11) on p. 36; Table 4 on p. 23; equation (9) on p. 35;
Table 3 on p. 22.
If anybody wants, I can send equations for inductance of a conductor above
another conductor that carries the return current. The second conductor
can be a ground plane or can be a conductor of same shape as the first
conductor (two different equations).
Nat
Nathan O. Sokal
Design Automation, Inc.
4 Tyler Road
Lexington, MA 02420-2404
U. S. A.
Tel. +1 (781) 862-8998
Fax +1 (781) 862-3769
E-mail [log in to unmask]
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