Printed Circuit Design & Fab - May 2008 - (Page 17)

lapping. For a simple isolated current loop, where the wire radius r0 is much smaller than the loop’s radius a, then the loop’s self inductance is approximated as in 3,4: L ≈ µ0 a ln () 8a −2 r0 EQ. 3 If multiple turns of the wire loop are used, then the inductance is simply multiplied by the number of turns to find the total inductance of the number of loops. It is important to note that Equation 3 clearly shows that the value of the inductance will change quickly with the loop area, and much more slowly with the conductor size. For example, the loop area will dominate the amount of inductance, not the trace width or thickness on a PCB. where w = the width of the rectangle (wide dimension) h = the height of the rectangle (short dimension), and a = the wire radius. Once again, note that EQUATION 7 clearly shows that the value of the inductance will change quickly with the loop area, and much more slowly with the conductor size. Inductance of a Via There is no such thing as inductance of a via by itself. Somehow, the return path must be considered. In the case of a PCB via, the intentional signal current flows on the via, and then must return. If there is a return via close to the signal via, the amount of loop area, and therefore the inductance, will be small. However, as the return via moves further away from the signal via, then the loop area (and inductance) grows quickly. Eventually, the return via is no longer used for any significant amount of return current, and all the return current flows though the dielectric displacement current (and the return via has no impact in the inductance the signal experiences). As the return via moves further away the return current spreads out in all directions in order to use the dielectric displacement current. This current spread provides many more opportunities to couple onto other signal vias and cause noise problems for other signals. Isolated Square Loop For an isolated square loop in free space, the self-inductance can be found to be: 2µ a L = π 0 ln ( p + 1 + p2 1+ 2 + 1 1 − 1+ 2 − p p 1 + p2 ) EQ. 4 where p = a/r0 a = length of side, and r0 = wire radius. Once again, note that Equation 4 clearly shows that the value of the inductance will change quickly with the loop area, and much more slowly with the conductor size. For the case where the wire radius is much smaller than the loop radius (r0 << a), Equation 4 reduces to: L= 2µ 0a Summary The concept of inductance only has meaning when the entire loop is considered. A wire that does not form a loop will have no inductance since there can be no current flow. When calculating inductance, the loop area will dominate the value of inductance. Conductor thickness plays a minor role. So PCB designers can have a greater influence on their signal path (including the return current path) by minimizing this loop area. PCD&F π ln ( 2p − 2+ 2 1+ 2 ) EQ. 5 Note that for these calculations of self-inductance the contribution by the internal flux within the conductor has been neglected. This term is most important at low frequencies when skin depth is not important and the current is uniformly distributed across the cross section of the conductor. For high frequencies, this term can be ignored. This self-inductance from the internal flux is given as a per-unit-length parameter as: µ L'int = 0 8π EQ. 6 This term is then multiplied by the loop length to find the total contribution of the internal flux. REFERENCES 1. Oliver Heavyside first coined the term ‘inductance’ in 1886. He also created the terms ‘capacitance’ and ‘impedance’. Source: D. Johnson, J. Johnson and J. Hilborn, Electric Circuit Analysis, Prentice-Hall, 1989. 2. The basic definition of voltage is the electric field multiplied by the distance. 3. J.D. Kraus and K.R. Carver, Electromagnetics, 2nd Edition, McGrawHill, 1973. 4. A.E. Ruehli, “Inductance Calculations in a Complex Integrated Circuit Environment, IBM J. Research and Development, 16, pp 470-481, ” 1972. 5. F .W. Gover, Inductance Calculations, Dover Publications, NY, 1946. Isolated Rectangular Loop For a single turn rectangular loop in free space5, the selfinductance can be found from: h + h 2 + w2 A = h ln w B = wln C = h ln L= ( ) ( ) () () w + h 2+ w 2 h 2h 2w + w ln a a EQ. 7 µ0 − 2(w + h ) + 2 h 2 + w2 − A − B + C π ( ) DR. BRUCE ARCHAMBEAULT is a IBM Distinguished Engineer and IEEE Fellow. He can be reached at barch@us.ibm.com. PRINTED CIRCUIT DESIGN & FAB 17 MAY 2008

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Printed Circuit Design & Fab - May 2008 Contents Our Line Market Watch Around the World Happenings ROI EMC For the Real World PCB East Conference Brochure Positive Plating Don't Let your Signals Stub Their Toes Improve PCB Layout With Skill Utility Programs The Next Generation Design Tool Challenge Thermally Conductive Microwave Materials PCB Dielectric Degradation in Lead-Free Assembly Applications A Tale of Two Trade Shows Eliminating Board Defects Off the Shelf Marketplace Ad Index BGA Bulletin

Printed Circuit Design & Fab - May 2008

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