Magnetostatics EEE 161 1192020 EEE 161 Electromagnetics Magnetostatics

Magnetostatics EEE 161 11/9/2020 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 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 2

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Force on a conductor 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 5

Force on a conductor 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 6

Force on a conductor 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 7

Magnetic Force on a Current Element If the field was coming out of the page 11/9/2020 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 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 9

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Force between two conductors 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 11

Force between two conductors 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 12

Force between two conductors 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 13

Magnetic Force on a Current Element 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 14

Socrative Quiz #1 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 15

Magnetic Force on a Current Loop 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 16

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Socrative Quiz #2 11/9/2020 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? 11/9/2020 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 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 26

Mag field from a straight conductor 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 27

Magnetic Field due to Current Distributions Biot-Savart’s Law 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 28

Magnetic Field of a Current Loop This problem is too difficult for the final, but if you understand this problem, you will be able to solve all other problems like this. 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 29

http: //web. mit. edu/8. 02 t/www/802 TEAL 3 D/vi sualizations/magnetostatics/index. htm 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 30

Magnetic Field of a Current Loop r here is the distance vector, that is why we have r^3 in the denominator. 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 31

What is the distance vector between the current and the observation point? 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 32

Distance vector 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 33

What is dl? 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 34

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Final expression for the fields These integrals have to be solved numerically. 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 36

How does this look? 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 37

MS#2: Magnetic Field of Linear Conductor Example 5. 2 1. Show what dl is 2. Find components of R (find R through position vectors of dl and P) 3. Show mathematically x-product between R, dl 4. Confirm using RHR direction of H 5. Integrate over z not Theta (use Wolframalpha. com) 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 38 Cont

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MS#2: Magnetic Field of Linear Conductor 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 40

Ampère’s Law 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 41

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? 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 42

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) 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 43

Why can we take H in front of the integral? Why is H a constant? 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 44

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Radius of wire a 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 46

Example: Internal Magnetic Field of Long Conductor For r < a 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 47 Cont.

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MS#3 Find the magnetic field everywhere around an infinite hollow cylindrical conductor shown. J=const (current density). Given: I, a, b. Find H. 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 49

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 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 50

How can you calculate inductance? 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 51

How can you calculate inductance? Magnetostatic energy Electrostatic energy 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 52

Example: Using Magnetic Energy Density to find Inductance Magnetic field in the insulating material is Current is given 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 53

Example: Magnetic Energy Density Magnetic field in the insulating material is The magnetic energy stored in the coaxial cable is 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 54

Example: Magnetic Energy Density The magnetic energy stored in the coaxial cable is 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 55

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Graded Quiz MS#4 • 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. 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 57

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 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 58

Conceptual understanding of inductance 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 59

There are circular rings of magneticfield around all currents 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 60

Right Hand Rule Number of magnetic field lines unit is Weber http: //en. wikipedia. org/wiki/File: Manoderecha. svg 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 61

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 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 62

Resistance • Resistance is defined as • Do we have any such equations for inductance and capacitance? 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 63

Capacitance and Inductance do not relate directly current and voltage. 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 64

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What is inductance? ? ? 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 66

Definition of Inductance is the ratio of magnetic field flux around the conductor divided by 11/9/2020 current that produced the flux. EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 67

That brings the next question: What is flux? Weak Strong Number of vector lines “flowing” through (poking) a surface 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 68

Definition and Simplified Definition Inductance is the ratio of magnetic field flux around the conductor divided by current that produced the flux. 11/9/2020 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 69

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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 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 71

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 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 72

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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) • • • 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 74

Internal and External Self-Inductance 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 75

Self-Field Line Rings and Mutual-Field Line Rings 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 76

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 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 77

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. 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 78

Practice 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 79

Induced Voltage • When the number of magnetic field lines changes around one wire, the voltage is induced across the wire. 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 80

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 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 81

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. 82 11/9/2020 Dr. Markovic

Estimate the inductance of a via-hole through a substrate 64 mils thick. 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 83

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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. 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 86

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 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 87

What will happen here? Voltage droop 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 88

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 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 89

How many lines total around leg b? 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 90

Ground Bounce - Voltage between two points in the return path due to a changing current 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 91

Ground Bounce - Voltage between two points in the return path due to a changing current Lb smaller Lab larger 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 92

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) 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 93

Example: What if the wires carry current in the same direction? Total inductance is? ? 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 94

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Circuits 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 96

Loop Self and Mutual Inductance • Total number of field lines surrounding the entire loop, per amp of current in the loop. 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 97

Loop Inductance of a circular loop • A 30 gauge wire (10 mils=d) • Bent in a 1 inch diameter • Has 85 n. H inductance 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 98

Inductance formulas on web • http: //www. technick. net/public/code/cp_dpa ge. php? aiocp_dp=util_inductance_calculator 11/9/2020 EEE 161 Electromagnetics - Magnetostatics Dr. Markovic 99