Published by Eric Bogatin on 25 May 2010
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Inductance is probably the most confusing topic in signal integrity and one of the most important. It plays a significant role in reflection noise, ground bounce, PDN noise and EMI.
Since this topic is not taught in school in a way that is at all useful to help solve signal integrity problems, practicing engineers are force to learn about this critical topic from professional development courses, books or articles.
Two inductance books are now available which should be on every engineer’s bookshelf.
The original definitive book on inductance, Inductance Calculations, by Fredrick Grover, first published in 1946, is now available as a reprint. The newest book, Inductance: Loop and Partial, by Clayton Paul, was just released in 2010. These two books cover calculating partial and loop inductance of conductors in a wide variety of geometries. The approximations offered are both immediately applicable to calculating inductance values for cable, connector, packages and circuit board interconnect structures.
Inductance Calculations was published in 1946 by D Van Nostrand CO. It was then reprinted by Dover in 1948 and was out of print by the mid 1980s. It was just re-printed this year and is now available from Amazon in paperback for a very low price.
Motors, generators and rf components experienced a period of high growth in the 1940s. At their core were inductors and being able to calculate their self and mutual inductance using pencil and paper was critical (before the use of electronic calculators). While many coil geometries had empirical formulas specific to their special conditions, Grover took on the task of developing a framework of calculation that could be applied to all general shapes and sizes of coils.
While he did not explicitly use the term, what he calculates in his book are really partial inductances, rather than loop inductances. There is a raging debate in the industry today about the value of this concept.
Proponents say it dramatically simplifies solving real world problems and is perfectly valid as a mathematical construct. You just have to be careful in translating partial inductances into loop inductances when applying the concept to calculate induced voltages.
Opponents say there is no such thing as partial inductance, it’s all about loop inductance and if you can’t measure it, you should not use the concept. There is too much danger of misapplying the term.
Inductance: Loop and Partial, goes the next step in putting partial and loop inductance in perspective. Both terms are clearly articulated and defined. Many of the formulas Grover introduced are presented in Paul’s book, with the added benefit of the details of the derivations shown. In particular, the methods of combining partial inductances in parallel and series is introduced to show the connection between loop and partial inductances.
As Clayton Paul points out, there are many cases where partial inductance is an easier approach to solving inductance problems. In a pin field connector, for example, the return path may not be defined until after the pins are connected up in the circuit. Using the partial inductance matrix values makes this an easy circuit to simulate, while using loop inductance matrix values is a more complicated solution.
I personally am a big fan of partial inductance, and use it extensively in my book, Signal and Power Integrity- Simplified. It makes understanding the concepts of inductance easier, and highlights the three physical design terms that reduce the loop inductance of a signal-return path: wider conductors, shorter conductors, and bringing the signal and return conductors closer together. Most importantly, partial inductance is a powerful concept to aid in calculating inductance for arbitrary shaped conductors.
Inductance is fundamentally the number of rings of magnetic field lines around a conductor, per amp of current through it. In this respect, it is a measure of the efficiency for which a conductor will generate rings of magnetic field lines. To calculate the inductance of a conductor, it is a matter of counting the number of rings of field lines and dividing this by the current through the conductor. Counting all the rings surrounding a conductor is really performing an integral of the magnetic field density on one side of the conductor.
Literally everything about the electrical effects of interconnects stems from Maxwell’s equations. Paul and Grover start from the basic Biot-Savart Law, which itself comes from Ampere’s Law and Gauss’s Law, each, one of Maxwell’s equations, and derives all the approximations for various geometries. The Biot-Savart Law describes the magnetic field at a point in space from a tiny current element.
Using this approach, the authors are able to calculate the magnetic field distribution around a wide variety of conductor geometries and integrate the field (count the field lines) to get the total number of rings per amp of current. Using clever techniques of calculus, they are able to derive analytical approximations for many of these geometries.
The advantage of Pauls’ book is that the hidden steps in many of the derivations are outlined. Luckily, we don’t have to do the derivations ourselves, but can rely on the work of experts and we are then in a position to use the results.
A commonly used approximation is for the partial self inductance of a long straight, rectangular conductor, such as a lead frame in a QFN package or a connector pin. It is calculated as:
L = the partial self inductance in nH
B, C are the thickness and width of the conductor cross section in inches
Len = the length of the conductor in inches.
For example, for a 1 inch long lead, 3 mils thick and 10 mils wide, the partial self inductance is 23 nH. This is roughly 25 nH per inch, or 1 nH/mm, which is a common rule of thumb for the partial self inductance of a wire.
If you deal with connectors, packages, vias, board discontinuities or odd shaped transmission lines, and need to estimate the loop inductances of non uniform sections, these two books will be essential resources. You will have a great collection of inductance approximations at your fingertips.