Magnetostatics EEE 161 6152021 EEE 161 Electromagnetics Magnetostatics

Magnetostatics EEE 161 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 1

Learning Objectives • Students will be able to – Apply cross product (right-hand rule) – Visualize magnetic field from a straight conductor – Calculate the magnetic force • On a conductor in an external magnetic field • Between two current-carrying conductors – Explain operation of a simple DC-motor 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 2

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Force on a conductor 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 5

Force on a conductor 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 6

Force on a conductor 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 7

Magnetic Force on a Current Element If the field was coming out of the page 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 8

Right Hand Rule Number of magnetic field lines unit is Weber http: //en. wikipedia. org/wiki/File: Manoderecha. svg 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 9

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Force between two conductors 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 11

Force between two conductors 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 12

Force between two conductors 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 13

Magnetic Force on a Current Element 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 14

Socrative Quiz #1 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 15

Magnetic Force on a Current Loop 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 16

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Socrative Quiz #2 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 20

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MS#1 Loop of current is given. It is fixed in the middle of top and bottom wires. Magnetic flux density B. Find the direction of the forces on the loop and CW, CCW? 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 25

How do we find H? Ampere’s law Bio-Savart’s Law How do we find E? Gauss’ Law Coulomb’s Law 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 26

Ampère’s Law 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 27

Guided Example: Magnetic field of an infinite line of current 1. What is the direction of magnetic field? 2. Then we have to pick an appropriate contour (we pick circle because the magnetic field is constant on this circle – all points are the same distance away from the wire) 3. What is the contour piece dl? 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 28

What is the direction of magnetic field? Magnetic field on this circle will be constant (all points are the same distance away from the source (current) 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 29

Why can we take H in front of the integral? Why is H a constant? 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 30

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Radius of wire a 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 32

Example: Internal Magnetic Field of Long Conductor For r < a 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 33 Cont.

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MS#2 Find the magnetic field everywhere around an infinite hollow cylindrical conductor shown. J=const (current density). Given: I, a, b. Find H. 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 35

How is B different than H? H – magnetic field B- magnetic flux density Magnetic permeability How is D different than E? E – electric field D- electric flux density electric permittivity 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 36

How can you calculate inductance? 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 37

How can you calculate inductance? Magnetostatic energy Electrostatic energy 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 38

Example: Using Magnetic Energy Density to find Inductance Magnetic field in the insulating material is Current is given 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 39

Example: Magnetic Energy Density Magnetic field in the insulating material is The magnetic energy stored in the coaxial cable is 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 40

Example: Magnetic Energy Density The magnetic energy stored in the coaxial cable is 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 41

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Graded Quiz MS#3 • Using magnetic energy density equation, find the internal inductance of a wire of circular cross section, with radius of a=1 mm. The current is constant throughout the wire. 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 43

Inductance recall • Inductor creates magnetic field • Current in one conductor will cause current in another conductor that’s mutual inductance • J omega L • Inductor resists current changes • Voltage over inductor is inductance x di/dt • Positive VAR 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 44

Conceptual understanding of inductance 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 45

There are circular rings of magneticfield around all currents 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 46

Right Hand Rule Number of magnetic field lines unit is Weber http: //en. wikipedia. org/wiki/File: Manoderecha. svg 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 47

Learning Outcomes • Students will be able to – Describe inductance through flux of magnetic field – Conceptually explain what is inductance – Describe what affects inductance – Estimate inductance 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 48

Resistance • Resistance is defined as • Do we have any such equations for inductance and capacitance? 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 49

Capacitance and Inductance do not relate directly current and voltage. 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 50

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What is inductance? ? ? 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 52

Definition of Inductance is the ratio of magnetic field flux around the conductor divided by 6/15/2021 current that produced the flux. EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 53

That brings the next question: What is flux? Weak Strong Number of vector lines “flowing” through (poking) a surface 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 54

Definition and Simplified Definition Inductance is the ratio of magnetic field flux around the conductor divided by current that produced the flux. 6/15/2021 Inductance is the ratio of number of magnetic field lines around the conductor divided by the current though the wire. EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 55

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What affects the # of magnetic field lines? • Current. (double I -> double # lines) • Length of wire ( len 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 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 57

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 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 58

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Types of Inductances Self-Inductance Mutual Inductance Loop inductance Partial Inductance Total Inductance / Net Inductance / Effective inductance • Internal and External inductance (due to magnetic fields inside and outside of metal) • • • 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 60

Internal and External Self-Inductance 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 61

Self-Field Line Rings and Mutual-Field Line Rings 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 62

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 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 63

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. 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 64

Practice 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 65

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 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 66

Partial Self-Inductance of a Rod • 30 -gauge wire -> 25 n. H/inch or 1 n. H/mm EEE 161 Electromagnetics - Magnetostatics The more spread out the current distribution, the lower the partial self-inductance. 67 6/15/2021 Dr. Markovic

Estimate the inductance of a via-hole through a substrate 64 mils thick. 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 68

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Partial Mutual Inductance between two round wires • In general partial mutual inductance between two wires is much smaller than partial selfinductance of either wire. 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 71

Rule of thumb for partial mutual inductance • If the spacing between two conductors is farther apart than their length: • For example two 20 mil via holes spaced 20 mils apart have no coupling between them 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 72

Induced Voltage • When the number of magnetic field lines changes around one wire, the voltage is induced across the wire. 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 73

What will happen here? Voltage droop 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 74

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 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 75

How many lines total around leg b? 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 76

Ground Bounce - Voltage between two points in the return path due to a changing current 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 77

Ground Bounce - Voltage between two points in the return path due to a changing current Lb smaller Lab larger 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 78

Ground Bounce • To decrease groundbounce: – Decrease the partial selfinductance of the return path (short lengths and wider interconnects) – Increase the mutual inductance of the two legs (by brining them closer together) 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 79

Example: What if the wires carry current in the same direction? Total inductance is? ? 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 80

Example: Calculate Ground Bounce • Two wire bonds of a chip are 1 mil in diameter and 100 mils long. • One carries power current and the other carries ground current (the same I but opposite direction), ΔI=100 m. A that switches in 1 nsec. • Find the following: – – – Partial self-inductance of each wire Mutual inductance if the wires are more than 100 mils apart Mutual inductance if the wires are 5 mils apart Total inductance in both cases (when 100 mils and 5 mils apart) Ground bounce voltage in both cases

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

Placement for decoupling capacitor pads between Vss and Vcc planes • Conventional placement • Optimized for lowest voltage-collapse noise

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Circuits 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 87

Loop Self and Mutual Inductance • Total number of field lines surrounding the entire loop, per amp of current in the loop. 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 88

Loop Inductance of a circular loop • A 30 gauge wire (10 mils=d) • Bent in a 1 inch diameter • Has 85 n. H inductance 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 89

Inductance formulas on web • http: //www. technick. net/public/code/cp_dpa ge. php? aiocp_dp=util_inductance_calculator 6/15/2021 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 90