Resistance of Traces
----------------------
Original Rev. 25-JUNE-1996
Most Recent Rev. 28-JULy-1998
Start with the resistance of Copper at 25 degrees C. A bar of 1 square
inch cross section and 1 foot long has a resistance of about 8.30 micro
Ohms at 25 degrees C. The coefficient of resistance is positive and is
about 0.38 % per degree C.
So for 1 oz coper (i.e. 0.0014 inch thickness) we have the following
table.
Trace Trace
Width Cross Section Approximate Resistance
mills. square inches Wire Gauge Ohms per ft
------ ------------- ----------- -----------
1 1.40 E-6 - 5.929 Ohm
7 9.80 E-6 39 0.85
10 1.40 E-5 37 0.59
15 2.10 E-5 36 0.40
20 2.80 E-5 35-34 0.30
25 3.50 E-5 33 0.24
50 7.00 E-5 31-30 0.12
75 1.05 E-4 29 0.079
100 1.40 E-4 28-27 0.059
So for 2 oz coper (i.e. 0.0028 inch thickness) we have the following
table.
Trace Trace
Width Cross Section Approximate Resistance
mills. square inches Wire Gauge Ohms per ft
------ ------------- ----------- -----------
1 2.80 E-6 - 2.964 Ohm
7 1.96 E-5 36 0.42
10 2.80 E-5 35-34 0.30
15 4.20 E-5 33 0.20
20 5.60 E-5 32-31 0.15
25 7.00 E-5 31-30 0.12
50 1.40 E-4 27-28 0.059
75 2.10 E-4 26 0.040
100 2.80 E-4 25-24 0.030
Self Inductance of a Trace
----------------------------
APPROXIMATED from F E Terman
For a 10 mill trace the self inductance is ON THE ORDER OF:
Reactance in Ohms
Trace Length Trace Inductance -----------------------
in inches micro Henrys at 10 MHz at 100 MHz
-------------- ------------------ --------- ----------
1 0.03 1.8 18
5 0.2 12.5 125
10 0.4 25.1 251
From Linear Technology Magazine May 1996 pg16
Approximate self inductance of various traces per inch:
Cu Thickness Trace Width Inductance per Inch
------------ ----------- -------------------
1 oz 10 mil 28.8 nH/in
2 oz 10 mil 28.3 nH/in
1 oz 100 mil 17.8 nH/in
2 oz 100 mil 17.7 nH/in
Inductance in nH
= 5.08 L [ 2.303 log ( (2L / (B+C)) + 0.5 + 0.2235 ((B+C) / L) ]
Where: L is trace length, B is width, and C is thickness.
Yes, I know that the parenthises are unbalanced. I just copied
it directly from the article; I will try to figure out what it
should be.
Bypass Capacitors
---------------------
Now look at the effectiveness of various types of bypass capacitors to
understand how low the resistance and reactance of the traces that connect
the bypass capacitors to ground and power planes need to be in order not to
degrade the operation of the bypass capacitors.
Effective Series Resistance of some Tantalum Capacitors:
10 V 4.7 uFd A case ESR is 6.0 Ohms
10 V 4.7 uFd B case ESR is 3.5 Ohms
10 V 15 uFd B case ESR is 3.5 Ohms
10 V 15 uFd C case ESR is 1.8 Ohms
10 V 22 uFd B case ESR is 1.8 Ohms
10 V 33 uFd B case ESR is 1.6 Ohms
10 V 47 uFd B case ESR is 1.2 Ohms
All of these ESR measurements are made at 100 kHz.
For tantalum capacitors in the range that we are likely to
use for bypassing (i.e. the values shown above), the Impedance
of the capacitor becomes equal to (i.e. becomes dominated by)
the ESR for frequencies above the range of 10 kHz to 100 kHz.
---> So connecting tantalum bypass capacitors via traces that
have a DC resistance of a fraction of an Ohm really does not
degrade the effectiveness of the bypassing that the capacitor
can perform.
The overall impedance of the tantalum capacitors, of values that we
are likely to use for bypassing (i.e. the values shown above) begins
to turn back up for frequencies above the range of 5 to 10 MHz.
Impedance of Z5U Ceramic Chip Capacitors:
The impedance of a 1206 case 0.1 uFd Z5U chip capacitor reaches
a minimum in the range of 10 MHz to 30 MHz. In the 10 MHz to 30 MHz
range impedance of such a capacitor is about 0.1 Ohm. In the range
of 2 MHz to 200 MHz the impedance is less than 1.0 Ohm.
---> So if the traces connecting the ceramic capacitor to the ground
and power planes have a reactance of less then 0.1 Ohm then
they do not seriously degrade the effectiveness of the capacitor
in doing its bypass job.
Z of Traces in Our Designs
-----------------------------
Get the terminology:
---- ---- ---------------------------
---- ----
--------------------------- ---------------------------
two Microstrips two Striplines
In both cases: The lines are of thinkness "T" and of width "W" and are
spaced from each other a distance "S" and the total thickness
of the board is "H".
The Microstrips are on top of a board of thickness "H" and
the Striplines are in the middle of a board of thickness "H".
The dielectric constant of FR4 / G10 "Er" is 4.3
So now lets look at some interesting examples that reflect our boards:
MICROSTRIP
Trace Trace Board Construction Trace
Width Thickness Layers, Mat thickness Z
in Mills in oz and Mills Mills WRT Gnd
-------- --------------- --------------------------- --------
7 1 oz 1.4 mil GND, 5 mil mat, Trace 54 Ohm
7 1 oz 1.4 mil GND, 6 mil mat, Trace 59 Ohm
7 1 oz 1.4 mil GND, 7.5 mil mat, Trace 66 Ohm
7 1 oz 1.4 mil GND, 8 mil mat, Trace 70 Ohm
7 1 oz 1.4 mil GND, 10 mil mat, Trace 77 Ohm
7 1 oz 1.4 mil GND, 12 mil mat, Trace 84 Ohm
7 1 oz 1.4 mil GND, 15 mil mat, Trace 93 Ohm
7 1 oz 1.4 mil GND, 16 mil mat, Trace 94 Ohm
10 1 oz 1.4 mil GND, 5 mil mat, Trace 44 Ohm
10 1 oz 1.4 mil GND, 7.5 mil mat, Trace 56 Ohm
10 1 oz 1.4 mil GND, 10 mil mat, Trace 68 Ohm
10 1 oz 1.4 mil GND, 15 mil mat, Trace 82 Ohm
STRIPLINE
Trace Trace Board Construction Trace
Width Thickness Layers, Mat thickness Z
in Mills in oz and Mills Mills WRT Gnd
-------- --------------- ----------------------------- --------
7 1 oz 1.4 mil GND, 5 Mil, TRC, 5 Mil, GND 30 Ohm
7 1 oz 1.4 mil GND, 6 Mil, TRC, 6 Mil, GND 36 Ohm
7 1 oz 1.4 mil GND, 7 Mil, TRC, 7 Mil, GND 43 Ohm
7 1 oz 1.4 mil GND, 8 Mil, TRC, 8 Mil, GND 44 Ohm
7 1 oz 1.4 mil GND, 9 Mil, TRC, 9 Mil, GND 47 Ohm
7 1 oz 1.4 mil GND, 10 Mil, TRC, 10 Mil, GND 50 Ohm
7 1 oz 1.4 mil GND, 12 Mil, TRC, 12 Mil, GND 53 Ohm
7 1 oz 1.4 mil GND, 15 Mil, TRC, 15 Mil, GND 61 Ohm
7 1 oz 1.4 mil GND, 16 Mil, TRC, 16 Mil, GND 61 Ohm
10 1 oz 1.4 mil GND, 5 Mil, TRC, 5 Mil, GND 21 Ohm
10 1 oz 1.4 mil GND, 7 Mil, TRC, 7 Mil, GND 34 Ohm
10 1 oz 1.4 mil GND, 10 Mil, TRC, 10 Mil, GND 42 Ohm
10 1 oz 1.4 mil GND, 12 Mil, TRC, 12 Mil, GND 45 Ohm
10 1 oz 1.4 mil GND, 15 Mil, TRC, 15 Mil, GND 53 Ohm
Now for Stripline NOT in the CENTER
between two ground planes we can estimate:
STRIPLINE
Trace Trace Board Construction Trace
Width Thickness Layers, Mat thickness Z
in Mills in oz and Mills Mills WRT Gnd
-------- --------------- ----------------------------- --------
7 1 oz 1.4 mil GND, 5 Mil, TRC, 10 Mil, GND 41 Ohm
7 1 oz 1.4 mil GND, 6 Mil, TRC, 12 Mil, GND 46 Ohm
7 1 oz 1.4 mil GND, 7 Mil, TRC, 15 Mil, GND 52 Ohm
7 1 oz 1.4 mil GND, 8 Mil, TRC, 16 Mil, GND 53 Ohm
Series Termination
----------------------
In order to design a series termination setup you need to know both the
characteristic impedance of the circuit board trace (see above) and the
output impedance of the integrated circuited that is driving the trace.
In our Run II boards we have two types of IC's that drive long series
terminated traces: the TTL outputs from 100325QC's and outputs from
XC4013L-5PQ240C FPGA's.
100325QC: The 1992 National "F100k ECL 300 Series Databook and Design
Guide" says that the 100325 has a "standard FAST" output
circuit and it gives the following DC characteristics:
The output pulling towards Voltage HIGH with 2.0 ma of
current, the output voltage will be at least 2.5 Volts.
---> output impedance < 1250 Ohms
The output trying to go Voltage HIGH but the output held
at 0.0 Volts it will pull up with at least 60 ma but not
more than 150 ma
---> 33 Ohms < output impedance < 83 Ohms
The output pulling towards Voltage LOW with 20 ma of
current, the output voltage will be under 0.5 Volts.
---> output impedance < 25 Ohms
XC4013L: This is the modern 3.3V part and I do not have any specific
information about its output impedance.
There is information on page 9-28 of the 1994 Xilinx
"Programmable Logic Data Book" that says, "Typically, an LCA
device with an output impedance of 60 Ohms drives a 3.5 Volt
step into a 100 Ohm line".
In the "1993 IDT High-Speed CMOS Logic Design Guide" on page
54 it gives the output impedance of IDT FCT devices as
"6 Ohms in the LOW state and 25 Ohms in the HIGH state".
It makes sense that the small scale IDT devices could be
designed to have a lower output impedance than the large I/O
count FPGA devices (to control "ground bounce").
Determination of Impedance for Stacked Paddle Cards (28-July-1998)
---------------------------------------------------
Geometry of the connectors on the paddle cards:
spacing s between pins 0.100 in or 2.54 E-3 m
"diameter" d of pins 0.025 in or 6.35 E-4 m
total length of pins 1.125 in or 0.0286 m
length of pin in air 0.675 in or 60%
length in plastic 0.450 in or 40%
Treating the pins as round parallel transmission lines, the
capacitance C is determined by:
Pi * E
---------- (1)
acosh(s/d)
where E is the dielectric constant [1/(36 Pi) E-9 in air, 1/(12 Pi)
E-9 in plastic]. Thus the capacitance in air is 1.35 E-11 F/m and the
capacitance in the plastic is 4.039 E-11 F/m.
The input impedance Z is given by:
sqrt(L/C) = sqrt(uE/C^2) (2)
assuming lossless lines so that L = uE/C. Here u is the permeability
constant in air (4 Pi E-7). This gives an input impedance for the
connector in air of 247 Ohms and an input impedance in plastic of 143
Ohms; the weighted average for the connector is 205 Ohms.
The measured capacitance of the connector is 1 pF or 3.5 E-11 F/m.
The capacitance of the paddle card itself was found to be 5.5 pF for
pins with long traces and 2.5 pf for pins with short traces.
Substituting 1 pF into Equation 2, the inductance L can be determined.
This value for L can then be used along with a total capacitance of
either 6.5 or 3.5 pF per 0.0286 m (so 2.27 E-10 or 1.22 E-10 F/m) to
obtain a value for the overall input impedance. The input impedances
corresponding to the long and short traces respectively were
calculated to be 80 Ohms and 110 Ohms.
The same results can be obtained by correcting the original impedance
of 205 Ohms according to:
Z' = Z
------------
sqrt(1+Cd/Co)
Here Co is the measured capacitance of the connector per meter and Cd
is the capacitance of the paddle card per meter.
As a starting point, 110 Ohm resistors will be used for termination.