Magnetostatics

Recall: Magnetostatics Current in an inductor Bio-Savart’s Law Right-hand rule Not conservative Magnetic Flux Density Magnetic Field Induction machines Transformers? Magnets Inductance.. Emf

Magnetic Force on a Current Element

Magnetic Force on a Current Loop

Ampère’s Law

Internal Magnetic Field of Long Conductor For r < a Cont.

External Magnetic Field of Long Conductor For r > a

Magnetic Field of Toroid Applying Ampere’s law over contour C: Ampere’s law states that the line integral of H around a closed contour C is equal to the current traversing the surface bounded by the contour. The magnetic field outside the toroid is zero. Why?

Recall: Inductance Types? Where did you hear about inductance? What is it? Units Any other info?

Coils of wire Stores electromagnetic field Oppose the change in current Electromagnetic waves Mutual inductance Units Henry Using transformers to do something to voltage Motors or generators Step up or step down Isolation

There are circular rings of magnetic-field around all currents

Right Hand Rule Number of magnetic field lines unit is Weber http://en.wikipedia.org/wiki/File:Manoderecha.svg

What affects the # of magnetic field lines? Current. (double I -> double # lines) Length of wire ( l increases -> #lines increases) Cross section of the wire (CS increases -> # lines decreases) Other currents in the vicinity (can increase or decrease # of lines) Metal that the wire is made of if ferromagnetic

Inductance is # of H lines around the conductor per amp of current through it L=N/I Unit Henry Inductance is related to the geometry of conductors, and magnetic material properties

Types of Inductances Self-Inductance Mutual Inductance Loop inductance Partial Inductance Total Inductance / Net Inductance / Effective inductance

Self-Field Line Rings and Mutual-Field Line Rings

Self Inductance Self-Field line rings – magnetic field rings due to current in the wire Number of field-line rings around a wire per amp of current in its own wire

Mutual Inductance Mutual-Field line rings- magnetic field rings around a wire due to a current from another wire. Number of field-line rings around one wire, per Amp of current in another wire.

Practice

Induced Voltage When the number of magnetic field lines changes around one wire, the voltage is induced across the wire.

Partial Inductance We can’t have a partial current Currents always flow in loops However, sometimes we don’t know how the rest of the loop looks like so: We define inductance of a small part of the loop as partial inductance Two types of partial inductance Partial self-inductance Partial mutual-inductance

Partial Self-Inductance of a Rod 30-gauge wire, 1mm long -> 25nH/inch or 1nH/mm The more spread out the current distribution, the lower the partial self-inductance.

Estimate the inductance of a via-hole through a substrate 64mils thick.

Partial Mutual Inductance between two round wires In general partial mutual inductance between two wires is much smaller than partial self-inductance of either wire.

Rule of thumb for partial inductance If the spacing between two conductors is farther apart than their length: For example two 20mil via holes spaced 20 mils apart have no coupling between them

What will happen here?

Effective, total or Net Inductance The total number of field lines around just this section per amp of current in the loop It included the contribution of field line rings from all the current segments in the loop

Ground Bounce - Voltage between two points in the return path due to a changing current

Ground Bounce To decrease ground-bounce: Decrease the partial self-inductance of the return path (short lengths and wider interconnects) Increase the mutual inductance of the two legs (by brining them closer together)

Example: What if the wires carry current in the same direction? Total inductance is??

Loop Self and Mutual Inductance Total number of field lines surrounding the entire loop, per amp of current in the loop.

Loop Inductance of a circular loop A 30 gauge wire (10mils=d) Bent in a 1 inch diameter Has 85nH inductance

Inductance formulas on web http://www.technick.net/public/code/cp_dpage.php?aiocp_dp=util_inductance_calculator

Loop inductance of two adjacent straight and round wires

Loop inductance per square of planes Parallel-plate waveguide capacitance

Spreading Inductance The planes are not connected like this But through vias

Internal and External Self-Inductance

Current Distributions and Skin Depth

Skin depth vs. ½ oz copper trace

How can you calculate inductance?

How do we find H? How do we find E? Ampere’s law Bio-Savart’s Law Gauss’ Law Coulomb’s Law

Maxwell’s Equations

What do we know about motors/generators Generator is a motor in reverse Stator and rotor typically We use coupled coils to transfer magnetic energy