A formula for the inductance PER INCH
of a finite-diameter wire floating in free
space parallel to an infinite ground plane
is:
L (henries) = 5.08E-9 * ln( 4 * h / d )
where L is in henries per inch
where ln() is the natural logarithm
where h is the height of the wire's centerline above the ground return plane
in inches
where d is the diameter of the wire in inches (Mike used the radius and got
2h/r)
This expression is widely distributed in the old
multi-wire literature.
It is also reproduced in "High-Speed Digital Design",
by this author, Prentice-Hall, 1993 (ISBN=0-13-395724-1).
This formula, along with other similar formulas for
the inductance, capacitance, impedance and delay of
various transmission line structures has been
translated into MathCad syntax and is available
on the Web at: http://www.sigcon.com
(MathCad is a software package for mathematical calculations
that runs on a variety of computing platforms. It is
similar in scope to Mathematica, MatLab, and probably
many other tools. It lets you place pictures, text,
and live equations on the page, and then solves
the equations.)
If you are interested in the loss effects of the
eddy currents, you will also need an approximation for the
current distribution under the wire:
j(x) = (1/(pi*h))*(1/(1 + (x/h)**2))
where all current in the ground plane flows parallel to the wire
where the ground plane is oriented parallel to the earth, and the sun is at
high noon
where j(x) is the current density (amps per perpindicular inch) at some point
located x inches away from the wire's shadow on the ground plane
where h is the height of the wire above ground, in inches
where pi is a constant equal to 3.1415926...
You can use j(x) to figure the total power loss in the ground plane.
This approximation is widely used for d/h ratios of about
1:1 or less.
If you are interested in wires very close to the surface, you
will additionally need to consider that the current flow on
the surface of the wire is not uniformly distributed. More
current flows on the side of the wire closest the ground plane.
This is called the proximity effect, and is documented (among
other places) on 158 of "High-Speed Digital Design". At a
separation ratio of 1:1 (d=h) the proximity effect has increased
the resustance of the wire by about 15% more than would have been
predicted by the skin effect alone. The proximity effect will
also distort the ground current pattern, probably by a similar
amount (although I've never seen this documented).
Best regards,
Dr. Howard Johnson
>>[ your message below ]
>>Return-Path: <alterra@adnc.com>
>>Errors-To: si-admin@silab.Eng.Sun.COM
>>Errors-To: si-admin@silab.Eng.Sun.COM
>>X-Sender: alterra@adnc.com
>>Date: Thu, 13 Feb 1997 16:37:07 -0800
>>To: si-list@silab.Eng.Sun.COM
>>From: Eric Wheatley <alterra@adnc.com>
>>Subject: Inductance calculation
>>
>>Hello all,
>>
>>I am looking for an analytic expression for the partial self-inductance of a
>>cylindrical wire of finite length that is parallel to and some distance away
>>from an infinite conducting, floating plane. I am primarily interested in
>>the limit where all conductors are perfect conductors so the source current
>>is entirely on the surface of the cylinder and the eddy currents in the
>>plane are also on the surface. (so the thickness of the plane is not
>>important). I realize that this situation cannot be realized physically in
>>that current magically appears at a source at one end of the cylinder and
>>disappears in a sink at the other end (which is why it is a partial
>>inductance calculation).
>>
>>Grover gives an expression for the inductance of a tubular conductor of
>>finite length in the limit to zero wall thickness which is my starting
>>point. I want to determine the effects of the eddy currents in the floating
>>plane on the self-inductance.
>>
>>Does anyone know if this calculation has been published anywhere? Thanks
>>for your help,
>>
>>Eric Wheatley
>>Alterra Technology Co.
>>
>>
>[ Mike's reply ]
>Return-Path: <jenkins@lsil.com>
>Errors-To: si-admin@silab.Eng.Sun.COM
>Errors-To: si-admin@silab.Eng.Sun.COM
>Sender: jenkins@lsil.com
>Date: Thu, 13 Feb 1997 18:11:09 -0800
>From: Mike Jenkins <jenkins@lsil.com>
>Organization: LSI Logic
>To: Signal Integrity Reflector <si-list@silab.Eng.Sun.COM>
>Subject: Re: Inductance calculation
>References: <2.2.32.19970214003707.002e3174@adnc.com>
>
>Eric,
>
>If you consider Grover's expression as bounding your desired
>inductance from above, I think I can bound it from below --
>by the case where the plane carries the return current. The
>expression for this inductance comes from the following trick:
>the fields are the same as for two identical wires (carrying
>equal and opposite currents) spaced twice as far apart as the
>wire is from the plane. For this case, the formula is:
>
> L(wire-wire) = 4e-7 * ln(separation/radius) Henries/Meter
>
>So for the wire "height" above ground plane, the current is
>the same but the voltage (and so the inductance) is halved:
>
> L(wire-plane) = 2e-7 * ln(2*height/radius) Henries/Meter
>
>Hope that helps you (and that I haven't lost a factor of two
>somewhere). I suspect the actual answer is an ugly finite
>element problem.
>
>Regards,
>Mike Jenkins
>--
>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
> Mike Jenkins Phone: 408.433.7901 _____
> LSI Logic Corp, ms/G750 Fax: 408.433.2840 LSI|LOGIC| (R)
> 1525 McCarthy Blvd. mailto:jenkins@lsil.com | |
> Milpitas, CA 95035 http://www.lsilogic.com |_____|
>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>
>
_________________________________________________
Dr. Howard Johnson, Signal Consulting, Inc.
16541 Redmond Way, Suite 264, Redmond, WA 98052
U.S. tel (206) 556 0800 // fax 206 881 6149 // email howiej@sigcon.com
http://www.sigcon.com