# Re: [SI-LIST] : Expression for inductance of a flat conductor

[email protected]
Tue, 28 Jul 1998 17:13:22 -0400

For a straight PCB land or rectangular strap of:
* Width W meters.
* Thickness T meters <= W.
* Length L meters.
* Metal of resistivity rho ohm-meters and relative permeability mur.
* pi = 3.14159...
* muv = 1.2566 microhenries/meter

For low frequencies, f <= fb = (4 * rho * (W+T)^2 ) / (pi * uv * ur * W^2 *
T^2) Hertz:
* Resistance = (rho * L) / (W * T) ohms.
* Inductance = ((muv * L) / (2 * pi)) * [ ln ((2 * L) / (W + T)) + 0.5 +
(0.2235 * (W + T) / L)] Henries.

For high frequencies, f > fb Hertz:
* Skindepth = SQRT(rho / (pi * muv * mur * f)) meters.
* Resistance = (rho * L)/(2 * (W + T) * skindepth) ohms.
* Geometric mean distance GMD = (T + W) * [0.2929 - (0.0698 * (W - T)^2 /
(W + T)^2)] meters.
* Inductance = ((muv * L) / (2 * pi)) * [ ln ((2 * L) / (GMD)) -1 + (GMD
/ L)] Henries.

These formulas from pages 166, 168, and 169 of my book "Electronic System
Design: Interference and Noise Control Techniques" (copyright 1987). Or
you can go back to Frederick Grover's book "Inductance Calculations"
(copyright 1973), where I got them.

The basic principle is "keep them short and wide". A common guideline for
bonding straps is L < 5 * W or L < 3 * W.

If you have multiple parallel lands or straps spaced fairly-well apart (by
over 3 times their width), as on a gridded circuit board, you will get
approximately the same inductance as if you had one very wide land/strap
spanning all the lands/straps. Thus if a cross-section of your ground-net
looks like:

--- --- --- -
--- - ---
<------------------------------------ W
--------------------------------->
you will have about the same inductance as this cross-section:

---------------------------------------------------------------------------

```--
<------------------------------------ W
--------------------------------->

John R. Barnes
Lexmark
International

on 07/28/98 03:44:05 PM

To:   si-list%[email protected]
cc:    (bcc: John Barnes/Lex/Lexmark)
bcc:  John Barnes/Lex/Lexmark
Subject:  [SI-LIST] : Expression for inductance of a flat conductor

Hello All,

Has anyone come across an expression describing the inductance of a flat
conductor?  The are of my pcb is 57.2 square inches.

We are working on a spice model for a power distribution system.

Thank you in advance.

Sal Aguinaga
Hardware Design EE
Carrier Systems
```