electromagnetic and electromechanical engineering ...2-1 electromagnetic and electromechanical...
TRANSCRIPT
![Page 1: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/1.jpg)
Electromechanics Magnetics and Energy Conversion2-1
Electromagnetic and Electromechanical Engineering Principles
Notes 02Magnetics and Energy Conversion
Marc T. Thompson, Ph.D.Thompson Consulting, Inc.
9 Jacob Gates Road Harvard, MA 01451
Phone: (978) 456-7722 Email: [email protected]
Website: http://www.thompsonrd.com
© Marc Thompson, 2006-2008Portions of these notes reprinted from Fitzgerald, Electric Motors, 6th edition
![Page 2: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/2.jpg)
Electromechanics Magnetics and Energy Conversion2-2
Course Overview --- Day 2
![Page 3: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/3.jpg)
Electromechanics Magnetics and Energy Conversion2-3
• Review of Maxwell’s equations • Ampere’s law, Gauss’ law, Faraday’s law• Magnetic circuits• Flux, flux linkage, inductance and energy
Overview of Magnetics
![Page 4: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/4.jpg)
Electromechanics Magnetics and Energy Conversion2-4
Review of Maxwell’s Equations• First published by James Clerk Maxwell in
1864• Maxwell’s equations couple electric fields
to magnetic fields, and describe:– Magnetic fields– Electric fields– Wave propagation (through the wave
equation)• There are 4 Maxwell’s equations, but in
magnetics we generally only need 3:– Ampere’s Law– Faraday’s Law– Gauss’ Magnetic Law James Clerk Maxwell
![Page 5: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/5.jpg)
Electromechanics Magnetics and Energy Conversion2-5
Review of Maxwell’s Equations• We’ll review Maxwell’s equations in words, followed by a
little bit of mathematics and some computer simulations showing the magnetic fields
![Page 6: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/6.jpg)
Electromechanics Magnetics and Energy Conversion2-6
Ampere’s Law• Flowing current creates a magnetic
field
• In magnetic systems, generally there is high current and low voltage (and hence low electric field) and we can approximate for low d/dt:
• In words: the magnetic flux density integrated around any closed contour equals the net current flowing through the surface bounded by the contour
AdEdtdAdJldH
So
SC
vrvvvr⋅+⋅=⋅ ∫∫∫ ε
∫∫ ⋅≈⋅SC
AdJldHvvvr
André-Marie Ampère
![Page 7: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/7.jpg)
Electromechanics Magnetics and Energy Conversion2-7
Finite-Element Analysis (FEA) • Very useful tool for visualizing and solving shapes and
magnitudes of magnetic fields• FEA is often used to simulate and predict the performance
of motors, etc.• Following we’ll see some 2-dimensional (2D) FEA results to
help explain Maxwell’s equations
![Page 8: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/8.jpg)
Electromechanics Magnetics and Energy Conversion2-8
Field From Current Loop, NI = 500 A-turns • Coil radius R = 1”; plot from 2D finite-element analysis
![Page 9: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/9.jpg)
Electromechanics Magnetics and Energy Conversion2-9
Faraday’s Law• A changing magnetic flux impinging on
a conductor creates an electric field and hence a current (eddy current)
• The electric field integrated around a closed contour equals the net time-varying magnetic flux density flowing through the surface bound by the contour
• In a conductor, this electric field creates a current by:
• Induction motors, brakes, etc.
∫∫ ⋅−=⋅SC
AdBdtdldE
vrvr
EJrr
σ=Michael Faraday
![Page 10: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/10.jpg)
Electromechanics Magnetics and Energy Conversion2-10
Circular Coil Above Conducting Aluminum Plate• Flux density plots at DC and 60 Hz• At 60 Hz, currents induced in plate via magnetic induction
create lift forceDC 60 Hz
![Page 11: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/11.jpg)
Electromechanics Magnetics and Energy Conversion2-11
Demonstration of Faraday’s Law: Electrodynamic Drag (NdFeB Magnet-in-Tube)
• Process:– Moving magnet creates changing magnetic field in
copper tube– Changing magnetic field creates induced voltage– Induced voltage creates current– By Lorentz force law, induced current and applied
magnetic field create drag force
![Page 12: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/12.jpg)
Electromechanics Magnetics and Energy Conversion2-12
Gauss’ Magnetic Law• Gauss' magnetic law says that the
integral of the magnetic flux density over any closed surface is zero, or:
• This law implies that magnetic fields are due to electric currents and that magnetic charges (“monopoles”) do not exist.
• Note: similar form to KCL in circuits.(We’ll use this analogy later…)
∫ =⋅S
dAB 0
B1, A1 B2, A2
B3, A3
221133 ABABAB +=
Carl Friedrich Gauss
![Page 13: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/13.jpg)
Electromechanics Magnetics and Energy Conversion2-13
Gauss’ Law --- Continuity of Flux Lines
1 2 3 0φ φ φ+ + =Reference: N. Mohan, et. al., Power Electronics Converters, Applications and Design, Wiley, 2003, pp. 48
![Page 14: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/14.jpg)
Electromechanics Magnetics and Energy Conversion2-14
Lorentz Force Law• Experimentally derived rule:
• For a wire of length l carrying current I perpendicular to a magnetic flux density B, this reduces to:
∫ ×= BdVJF
IBlF =
![Page 15: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/15.jpg)
Electromechanics Magnetics and Energy Conversion2-15
Lorentz Force Law and the Right Hand Rule
∫ ×= BdVJF
Reference: http://www.physics.brocku.ca/faculty/sternin/120/slides/rh-rule.html
![Page 16: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/16.jpg)
Electromechanics Magnetics and Energy Conversion2-16
Intuitive Thinking about Magnetics• By Ampere’s Law, the current J and the magnetic field H
are generally at right angles to one another• By Gauss’ law, magnetic field lines loop around on
themselves– No magnetic monopole
• You can think of high- μ magnetic materials such as steel as an easy conduit for magnetic flux…. i.e. the flux easily flows thru the high- μ material
![Page 17: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/17.jpg)
Electromechanics Magnetics and Energy Conversion2-17
Magnetic Field (H) and Magnetic Flux Density (B)• H is the magnetic field (A/m in SI units) and B is the
magnetic flux density (Weber/m2, or Tesla, in SI units)• B and H are related by the magnetic permeability μ by B =
μH• Magnetic permeability μ has units of Henry/meter• You can think of high- μ magnetic materials such as steel
as an easy conduit for magnetic flux…. i.e. the flux easily flows thru the high- μ material
• In free space μo = 4π×10-7 H/m• Note that B and H are vectors; they have both a magnitude
and a direction
![Page 18: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/18.jpg)
Electromechanics Magnetics and Energy Conversion2-18
Right Hand Rule and Direction of Magnetic Field
Reference: http://sol.sci.uop.edu/~jfalward/magneticforcesfields/magneticforcesfields.html
![Page 19: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/19.jpg)
Electromechanics Magnetics and Energy Conversion2-19
Forces Between Current Loops
Reference: http://sol.sci.uop.edu/~jfalward/magneticforcesfields/magneticforcesfields.html
![Page 20: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/20.jpg)
Electromechanics Magnetics and Energy Conversion2-20
Inductor Without Airgap
Constitutive relationships In free space:
HB oμ= Magnetic permeability of free space μo = 4π×10-7 Henry/meter. In magnetic material, magnetic permeability is higher than μo:
HB μ=
• Magnetic flux is constrained to flow within steel
![Page 21: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/21.jpg)
Electromechanics Magnetics and Energy Conversion2-21
Example: Flux in Inductor with Airgap
Ampere's law:
NIgHlHdAJdlH gcS
c =+⇒⋅=⋅∫ ∫
Let's use constitutive relationships:
NIgB
lB
o
gc
c
c =+μμ
![Page 22: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/22.jpg)
Electromechanics Magnetics and Energy Conversion2-22
cg
cc
AB
AB
Φ=
Φ=
Put this into previous expression:
⎟⎟⎠
⎞⎜⎜⎝
⎛+Φ=
goc
c
Ag
AlNI
μμ
Solve for flux:
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=Φ
goc
c
Ag
Al
NI
μμ
If g/μoAg >> lc/μAc,
go Ag
NI
μ
≈Φ
Example: Flux in Inductor with Airgap
![Page 23: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/23.jpg)
Electromechanics Magnetics and Energy Conversion2-23
Magnetic Circuits• Use Ohm’s law analogy to model magnetic circuits
• Use magnetic “reluctance” instead of resistance
• This is a very powerful method to get approximate answers in magnetic circuits
ℜ⇔Φ⇔
⇔
RI
NIV
Al
AlR
μσ=ℜ⇔=
![Page 24: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/24.jpg)
Electromechanics Magnetics and Energy Conversion2-24
Magnetic-Electric Circuit Analogy• In an electric circuit, voltage V forces current I to flow
through resistances R• In a magnetic circuit, MMF NI forces flux Φ to flow through
reluctances ℜ
![Page 25: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/25.jpg)
Electromechanics Magnetics and Energy Conversion2-25
C-Core with Gap --- Using Magnetic Circuits• Flux in the core is easily found by:
• Now, note what happens if g/μo >> lp/μc: The flux in the core is now approximately independent of the core permeability, as:
• Inductance:
N turns
Average magnetic path length lpinside core,
cross-sectional area A
Airgap, g
+-NI
Φ
ℜcoreℜgap
cocc
pgapcore
Ag
Al
NINI
μμ+
=ℜ+ℜ
=Φ
co
gap
AgNINI
μ
≈ℜ
≈Φ
co
gap
Ag
NNI
NL
μ
22
≈ℜ
≈Φ
=
![Page 26: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/26.jpg)
Electromechanics Magnetics and Energy Conversion2-26
C-Core with Gap --- FEA
![Page 27: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/27.jpg)
Electromechanics Magnetics and Energy Conversion2-27
Fringing Fields in Airgap• If fringing is negligible, Ac = Ag
![Page 28: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/28.jpg)
Electromechanics Magnetics and Energy Conversion2-28
Example: C-Core with Airgap• Fitzgerald, Example 1.1; with Bc = 1.0T, find reluctances,
flux and coil current
![Page 29: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/29.jpg)
Electromechanics Magnetics and Energy Conversion2-29
Example: C-Core with Airgap
![Page 30: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/30.jpg)
Electromechanics Magnetics and Energy Conversion2-30
Example: C-Core with Airgap --- FEA• NI = 400 A-turns
![Page 31: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/31.jpg)
Electromechanics Magnetics and Energy Conversion2-31
Example: C-Core with Airgap --- FEA• NI = 400 A-turns, close up near the core
![Page 32: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/32.jpg)
Electromechanics Magnetics and Energy Conversion2-32
Example: C-Core with Airgap --- FEA Result• Flux density in the core is approximately 1 Tesla
![Page 33: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/33.jpg)
Electromechanics Magnetics and Energy Conversion2-33
Example: C-Core with Airgap --- Gap Detail
![Page 34: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/34.jpg)
Electromechanics Magnetics and Energy Conversion2-34
Example: Simple Synchronous Machine• Fitzgerald, Example 1.2• Assuming μ ∞, find airgap flux Φ and flux density Bg
Assume I = 10A, N = 1000 turns, g = 1 cm and Ag = 2000 cm2
![Page 35: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/35.jpg)
Electromechanics Magnetics and Energy Conversion2-35
Aside: What Does Infinite Core μ Imply?
In core: Ccc AB=Φ
In airgap:
ggg AB=Φ In core, Bc is finite; this means that if μ ∞, then Hc 0 for finite Bc. Also, infinite μ implies zero reluctance
![Page 36: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/36.jpg)
Electromechanics Magnetics and Energy Conversion2-36
• Some initial thoughts, before doing any equations:– By symmetry, airgap
flux and flux density are the same in both gaps
– Since permeability is infinite, H inside steel is zero
Example: Simple Synchronous Machine
![Page 37: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/37.jpg)
Electromechanics Magnetics and Energy Conversion2-37
Reluctances:
WbturnsA
Ag
gogg
−=
×==ℜ=ℜ − 39789
)01.0)(2000)(104()01.0(
2721 πμ
Flux:
WbNI
gg
126.0)39789)(2()10)(1000(
21
==ℜ+ℜ
=Φ
Magnetic flux density:
TmWbWb
AB
g
63.063.0)01.0)(2000(
126.022 ===
Φ=
Example: Simple Synchronous Machine
![Page 38: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/38.jpg)
Electromechanics Magnetics and Energy Conversion2-38
Flux Linkage, Voltage and Inductance• By Faraday’s law, changing magnetic flux density creates
an electric field (and a voltage)
• Induced voltage:
∫∫ ⋅−=⋅SC
AdBdtdldE
vrvr
dtd
dtdNv λ
=Φ
=
λ = "flux linkage" = NΦ Inductance relates flux linkage to current
IL λ=
![Page 39: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/39.jpg)
Electromechanics Magnetics and Energy Conversion2-39
Finding Inductance Using Magnetic Circuits
gNA
IL
gINA
N
AgNI
go
go
go
2
2
μλ
μλ
μ
=≈
=Φ≈
≈Φ
• Let’s at first assume infinite core permeability; this means that the reluctance of the core is zero
• Note that inductance always scales as N2 (why is that?)
![Page 40: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/40.jpg)
Electromechanics Magnetics and Energy Conversion2-40
Magnetic Circuit with Two Airgaps
Total flux: 21 ℜℜ
=ΦNI
Reluctances: 1
11 A
g
oμ=ℜ and
2
22 A
g
oμ=ℜ
Inductance:
⎟⎟⎠
⎞⎜⎜⎝
⎛ℜℜℜ+ℜ
=Φ
==21
212NI
NI
L λ = ⎟⎟⎠
⎞⎜⎜⎝
⎛+
2
2
1
12
gA
gANoμ
![Page 41: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/41.jpg)
Electromechanics Magnetics and Energy Conversion2-41
Magnetic Circuit with Two Airgaps (cont.)
Flux in leg#1: 1
1
11 g
NIANI oμ=ℜ
=Φ
Flux in leg#2: 2
2
22 g
NIANI oμ=ℜ
=Φ
Flux density in leg#1: 11
11 g
NIA
B oμ=Φ
=
Flux density in leg#2: 22
22 g
NIA
B oμ=Φ
=
![Page 42: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/42.jpg)
Electromechanics Magnetics and Energy Conversion2-42
Inductance vs. Relative Permeability• What happens if we assume finite core permeability?• Reluctance of the core is now finite as well
Reluctances: c
cc A
lμ
=ℜ and A
g
og μ=ℜ
Let’s assume that Ac = Ag = A
Flux: gc
NIℜ+ℜ
=Φ
Flux linkage: gc
INNℜ+ℜ
=Φ=2
λ
Inductance:
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=
ℜ+ℜ==
μμ
μ
μμ
λ
oc
o
o
cgc lg
AN
Ag
Al
NNI
N222
Inductance is independent of core permeability if:
μμoclg >>
![Page 43: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/43.jpg)
Electromechanics Magnetics and Energy Conversion2-43
Inductance vs. Relative Permeability
![Page 44: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/44.jpg)
Electromechanics Magnetics and Energy Conversion2-44
Example: Effects of Finite Permeability• Fitzgerald, problem 1.5
![Page 45: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/45.jpg)
Electromechanics Magnetics and Energy Conversion2-45
Example: Effects of Finite Permeability• Relative μ vs. B
![Page 46: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/46.jpg)
Electromechanics Magnetics and Energy Conversion2-46
Example: Effects of Finite Permeability• B/H curve
![Page 47: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/47.jpg)
Electromechanics Magnetics and Energy Conversion2-47
Example: Effects of Finite Permeability• Current calculation
From Ampere’s law:
NIgHlH gcc =+ In core:
In gap (let’s assume Bc = Bg = B):
og
BH
μ=
Put this back into Ampere’s law:
AglN
BI
NIBgBl
r
c
o
o
c
8.65=⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=∴
=+
μμ
μμ
μc
cBH =
![Page 48: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/48.jpg)
Electromechanics Magnetics and Energy Conversion2-48
Example: Effects of Finite Permeability• Coil flux linkage λ as a function of coil current• Note that at low current, λ-I curve is linear, indicating
constant inductance
![Page 49: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/49.jpg)
Electromechanics Magnetics and Energy Conversion2-49
Example: MATLAB Script
![Page 50: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/50.jpg)
Electromechanics Magnetics and Energy Conversion2-50
Inductance and Energy• Magnetic stored energy (in Joules) is:
• This is a good thing to remember
2
21 LIW =
![Page 51: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/51.jpg)
Electromechanics Magnetics and Energy Conversion2-51
Magnetic Circuit with Two Windings• Note that flux is the sum of flux due to i1 and that due to i2
![Page 52: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/52.jpg)
Electromechanics Magnetics and Energy Conversion2-52
Magnetic Circuit with Two Windings• Note that by the right-hand rule the flux due to i1 and i2 are
additive given the current directions shown
![Page 53: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/53.jpg)
Electromechanics Magnetics and Energy Conversion2-53
Magnetic Circuit with Two Windings• Note that by the right-hand rule the flux due to i1 and i2
are additive given the current directions shownFlux: ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=Φ
gA
iNiN coμ)( 2211
Flux linkage for coil #1:
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=Φ=
gA
iNNgA
iNN coco μμλ 2211
2111
We can write this as:
2121111 iLiL +=λ L11 is “self inductance” of coil #1 L12 is “mutual inductance” between coils #1 and #2
![Page 54: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/54.jpg)
Electromechanics Magnetics and Energy Conversion2-54
Magnetic Circuit with Two WindingsFlux linkage for coil #2:
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=Φ=
gA
iNgA
iNNN coco μμλ 2
2212122
Or rewriting:
2221212 iLiL +=λ Or, in matrix form:
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
2221
1211
2
1
ii
LLLL
λλ
![Page 55: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/55.jpg)
Electromechanics Magnetics and Energy Conversion2-55
“Soft” Magnetic Materials• Materials with a small B/H curve, such as steels, etc.• Much of the previous analysis assumed that steel had
infinite permeability (μ ∞) or that permeability was constant and large.
• However, soft magnetic materials exhibit both saturation and losses
![Page 56: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/56.jpg)
Electromechanics Magnetics and Energy Conversion2-56
B-H Curve and Saturation• Definition of magnetic permeability: slope of B-H curve
Reference: N. Mohan, et. al., Power Electronics Converters, Applications and Design, Wiley, 2003, pp. 48
![Page 57: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/57.jpg)
Electromechanics Magnetics and Energy Conversion2-57
BH Curve for M-5 Steel• Note horizontal scale is logarithmic
![Page 58: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/58.jpg)
Electromechanics Magnetics and Energy Conversion2-58
Hysteresis Loop• Real-world magnetic materials have a “hysteresis loop”• Hysteresis loss is proportional to shaded area
![Page 59: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/59.jpg)
Electromechanics Magnetics and Energy Conversion2-59
B-H Loop for M-5 Grain-Oriented Steel• Only the top half of the loops shown for steel 0.012” thick
![Page 60: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/60.jpg)
Electromechanics Magnetics and Energy Conversion2-60
BH Curves for Various Soft Magnetic Materials
Reference: E. Furlani, Permanent Magnet and Electromechanical Devices, Academic Press, 2001, pp. 41
![Page 61: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/61.jpg)
Electromechanics Magnetics and Energy Conversion2-61
Relationship Between Voltage, Flux and Current
![Page 62: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/62.jpg)
Electromechanics Magnetics and Energy Conversion2-62
Exciting RMS VA per kg at 60 Hz
![Page 63: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/63.jpg)
Electromechanics Magnetics and Energy Conversion2-63
Hysteresis Loop Size Increases with Frequency• Hysteresis loss increases as frequency increases
Reference: Siemens, Soft Magnetic Materials (Vacuumschmelze Handbook), pp. 30
![Page 64: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/64.jpg)
Electromechanics Magnetics and Energy Conversion2-64
Total Core Loss• Total core loss is the sum of:
– Hysteresis loss– Eddy current losses
• Eddy current losses are due to induced currents (via Faraday’s law)
• Eddy current losses are minimized by laminating magnetic cores
![Page 65: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/65.jpg)
Electromechanics Magnetics and Energy Conversion2-65
Magnetic steel lamination
Insulator
0.5 t
t (typically 0.3 mm)
• Cores made from conductive magnetic materials must be made of many thin laminations. Lamination thickness < skin depth.
Laminated Core
![Page 66: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/66.jpg)
Electromechanics Magnetics and Energy Conversion2-66
x
dx
-x
L
d
w
B sin( ωt)
x
y
z
Eddy current flow path
Eddy Current Loss in Lamination
![Page 67: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/67.jpg)
Electromechanics Magnetics and Energy Conversion2-67
Total Core Loss• M-5 steel at 60 Hz
![Page 68: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/68.jpg)
Electromechanics Magnetics and Energy Conversion2-68
Total Core Loss vs. Frequency and Bmax• Core loss depends on peak flux density and excitation
frequency• This is the curve for a high frequency core material
Reference: http://www.jfe-steel.co.jp/en/products/electrical/jnhf/02.html
![Page 69: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/69.jpg)
Electromechanics Magnetics and Energy Conversion2-69
Process to Find Core Loss• Find maximum B• From this B and
switching frequency, find core loss per kg
• Total loss is power density × mass of core
![Page 70: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/70.jpg)
Electromechanics Magnetics and Energy Conversion2-70
Example: C-Core with Airgap --- Current• Fitzgerald, Example 1.7; find the current necessary to
produce Bc = 1T
![Page 71: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/71.jpg)
Electromechanics Magnetics and Energy Conversion2-71
Example: C-Core with Airgap --- SolutionThe value of Hc needed for Bc = 1 Tesla is read from the chart:
meterturnsAH c /11 −= The MMF drop in the core is:
turnsAlH cc −== 33)3.0)(11( The MMF drop in the airgap is:
turnsAgB
gHo
gg −=
××
== −
−
396104
)105)(0.1(7
4
πμ
The winding current is:
ANMMF
I 8.0500
39633=
+== ∑
![Page 72: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/72.jpg)
Electromechanics Magnetics and Energy Conversion2-72
Example: Inductor• Fitzgerald, Example 1.8 Material: M-5 steel
f = 60 Hz N=200 Bc = 1.5 sinωt Tesla Steel is 94% of cross section Density of steel = 7.65 g/cm3
Find: (a) Applied voltage (b) Peak current (c) RMS current (d) Core loss
![Page 73: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/73.jpg)
Electromechanics Magnetics and Energy Conversion2-73
Example: Excitation VoltageFrom Faraday’s law:
dtdB
NAdtdN
dtde c
c=Φ
==λ
232 104.276.394.022 minininAc−×==××=
)cos(565)cos()5.1( ttdt
dBc ωωω ==
)cos(274))cos(565)(104.2)(200( 3 tte ωω =×= −
![Page 74: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/74.jpg)
Electromechanics Magnetics and Energy Conversion2-74
Example: Peak Current in Winding
From Figure 1.10, B = 1.5T requires H = 36 A-turns/m
From Ampere’s law, NIHlc =
lc = 2×(8”+6”) = 28” = 0.71m
AN
HlI c 13.0
200)71.0)(36(===
![Page 75: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/75.jpg)
Electromechanics Magnetics and Energy Conversion2-75
Example: RMS Winding CurrentFrom Figure 1.12, at Bmax = 1.5T, Pa = 1.5 VA/kg
Core volume: 3332 107.15.105)28)(94.0)(4( minininVc
−×===
Core mass:
kgmkgmVM cc 2.13)7650)(107.1( 3
33 =×== −ρ
Core Volt-
Amperes: VAkgkgVAPc 8.192.135.1 =×=
Current: AEVAI
RMSRMS 10.0
2274
7.19=
⎟⎠
⎞⎜⎝
⎛==
![Page 76: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/76.jpg)
Electromechanics Magnetics and Energy Conversion2-76
Example: Core Loss
From Figure 1.14, core loss density = 1.5 W/kg at Bmax = 1.5 Tesla. Total core loss is:
WkgWMPc 20)5.1)(2.13(5.1 ==×=
![Page 77: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/77.jpg)
Electromechanics Magnetics and Energy Conversion2-77
Permanent Magnets• “Soft” magnetic materials such as magnetic steel can
behave as very weak permanent magnets• Permanent magnets, or “hard” magnetic materials, have a
high coercive force Hc and can produce significant flux in an airgap; they also have a “wide” hysteresis loop
Reference: E. Furlani, Permanent Magnet and Electromechanical Devices, Academic Press, 2001, pp. 39
![Page 78: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/78.jpg)
Electromechanics Magnetics and Energy Conversion2-78
Brief History of Permanent Magnets• c. 1000 BC: Chinese compasses using lodestone
– Later used to cross the Gobi desert
Reference: K. Overshott, “Magnetism: it is permanent,” IEE Proceedings-A, vol. 138, no. 1, Jan. 1991, pp. 22-31
![Page 79: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/79.jpg)
Electromechanics Magnetics and Energy Conversion2-79
Brief History of Permanent Magnets
Reference: R. Parker, Advances in Permanent Magnetism, John Wiley, 1990, pp. 3
![Page 80: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/80.jpg)
Electromechanics Magnetics and Energy Conversion2-80
Brief History of Permanent Magnets (cont.)• c. 200 BC: Lodestone (magnetite) known to the Greeks
– Touching iron needles to magnetite magnetized them• 1200 AD: French troubadour de Provins describes use of a
primitive compass to magnetize needles• 1600: William Gilbert publishes first journal article on
permanent magnets• 1819: Oersted reports that an electric current moves compass
needle
References: 1. K. Overshott, “Magnetism: it is permanent,” IEE Proceedings-A, vol. 138, no. 1, Jan. 1991, pp. 22-312. R. Petrie, “Permanent Magnet Material from Loadstone to Rare Earth Cobalt,” Proc. 1995 Electronics Insulation and Electrical Manufacturing and Coil Winding Conf., pp. 63-643. Rollin Parker, Advances in Permanent Magnetism, John Wiley, 19904. E. Hoppe, “Geshichte des Physik,” Vieweg, Braunshweig, 1926, pp. 3395. W. Gilbert, “De Magnete 1600,” translation by S. P. Thompson, 1900, republished by Basic Books, Inc., New York, 1958
![Page 81: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/81.jpg)
Electromechanics Magnetics and Energy Conversion2-81
Brief History of Permanent Magnets (cont.)
• c. 1825: Sturgeon invents the electromagnet, resulting in a way to artificially magnetize materials
• 7-ounce magnet was able to lift 9 pounds
References: 1. W. Sturgeon, Mem. Manchester Lit. Phil. Soc., 1846, vol. 7, pp. 6252. Britannica Online
![Page 82: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/82.jpg)
Electromechanics Magnetics and Energy Conversion2-82
Brief History of Permanent Magnets (cont.)• c. 1830: Joseph Henry
(U.S.) constructs electromagnets
Photo reference: Smithsonian Institute archives
Joseph Henry
![Page 83: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/83.jpg)
Electromechanics Magnetics and Energy Conversion2-83
Brief History of Permanent Magnets (cont.)• 1917: Cobalt magnet steels developed by Honda and
Takagi in Japan• 1940: Alnico --- first “modern” material still in common use
– Good for high temperatures• 1960: SmCo (samarium cobalt) rare earth magnets
– Good thermal stability• 1983: GE and Sumitomo develop neodymium iron boron
(NdFeB) rare earth magnet– Highest energy product, but limited temperature range
References: 1. K. Overshott, “Magnetism: it is permanent,” IEE Proceedings-A, vol. 138, no. 1, Jan. 1991, pp. 22-312. R. Petrie, “Permanent Magnet Material from Loadstone to Rare Earth Cobalt,” Proc. 1995 Electronics Insulation and Electrical Manufacturing and Coil Winding Conf., pp. 63-64
![Page 84: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/84.jpg)
Electromechanics Magnetics and Energy Conversion2-84
Magnetizing Permanent Magnets• Material is placed inside magnetizing fixture• Magnetizing coil is energized with a current producing
sufficient field to magnetize the PM material
Reference: E. Furlani, Permanent Magnet and Electromechanical Devices, Academic Press, 2001, pp. 57
![Page 85: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/85.jpg)
Electromechanics Magnetics and Energy Conversion2-85
Pictorial View of Magnetization Process
Reference: R. Parker, Advances in Permanent Magnetism, John Wiley, 1990, pp. 49
![Page 86: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/86.jpg)
Electromechanics Magnetics and Energy Conversion2-86
Permanent Magnets• External effects of permanent magnets can be modeled as
surface current
Reference: P. Campbell, Permanent Magnet Materials and their Applications, Cambridge University Press, 1994, pp. 7
![Page 87: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/87.jpg)
Electromechanics Magnetics and Energy Conversion2-87
Permanent Magnets• After magnetization, M has values of either +Msat or -Msat
Reference: P. Campbell, Permanent Magnet Materials and their Applications, Cambridge University Press, 1994, pp. 14-15
![Page 88: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/88.jpg)
Electromechanics Magnetics and Energy Conversion2-88
Permanent Magnets• By a constitutive relationship, B = μo(H+M)• Since M has values of either +Msat or -Msat, it follows that
the slope of the BH curve for the permanent magnet is μo
Reference: P. Campbell, Permanent Magnet Materials and their Applications, Cambridge University Press, 1994, pp. 15, 23
![Page 89: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/89.jpg)
Electromechanics Magnetics and Energy Conversion2-89
Demagnetization Curves of Ceramic 8• Typical sintered ceramic magnet
Reference: P. Campbell, Permanent Magnet Materials and their Applications, Cambridge University Press, 1994, pp. 62
![Page 90: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/90.jpg)
Electromechanics Magnetics and Energy Conversion2-90
Demagnetization Curves for NdFeB• Strong neodymium-iron-boron
Reference: P. Campbell, Permanent Magnet Materials and their Applications, Cambridge University Press, 1994, pp. 74
![Page 91: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/91.jpg)
Electromechanics Magnetics and Energy Conversion2-91
Permanent Magnets vs. Steel• Note that PM has much higher coercive force
Permanent magnet: Alnico 5 M-5 steel
![Page 92: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/92.jpg)
Electromechanics Magnetics and Energy Conversion2-92
Lines of Force• Iron filings follow magnetic field lines
![Page 93: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/93.jpg)
Electromechanics Magnetics and Energy Conversion2-93
Cast Alnico
Reference: R. Parker, Advances in Permanent Magnetism, John Wiley, 1990, pp. 65
![Page 94: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/94.jpg)
Electromechanics Magnetics and Energy Conversion2-94
Example: PM in Magnetic Circuits• Fitzgerald, Example 1.9
g = 0.2 cm
lm = 1.0 cm
Am = Ag = 4 cm2
Find flux in airgap Bg for magnetic materials
(a) Alnico 5
(b) M-5 steel
![Page 95: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/95.jpg)
Electromechanics Magnetics and Energy Conversion2-95
Example: Solution with Alnico PMNI = 0, so by Ampere’s law:
0=+ gHlH gmm
Solve for Hg: ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
gl
HH mmg
Continuity of flux (Gauss’ law):
⎟⎟⎠
⎞⎜⎜⎝
⎛=⇒=
g
mmgmmgg A
ABBlABA
Next, solve for Bm as a function of Hm:
mm
gmmom
m
ggo
m
ggm
HAA
gl
HB
AA
HAA
BB
61028.6 −×−=⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−=⇒
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
μ
μ
Plot this load line on Alnico BH curve
![Page 96: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/96.jpg)
Electromechanics Magnetics and Energy Conversion2-96
Example: Solution with Alnico• Result: Bg = 0.3 Tesla
Reference: P. Campbell, Permanent Magnet Materials and their Applications, Cambridge University Press, 1994, pp. 89
![Page 97: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/97.jpg)
Electromechanics Magnetics and Energy Conversion2-97
Example: Load Line Solution with M-5 Steel • Use same load line; Bg = 0.38 Gauss (much lower than
with Alnico)• Note: Earth’s magnetic field ~ 0.5 Gauss
![Page 98: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/98.jpg)
Electromechanics Magnetics and Energy Conversion2-98
Some Common Permanent Magnet Materials • Other tradeoffs not shown here include: mechanical
strength, temperature effects, etc.
![Page 99: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/99.jpg)
Electromechanics Magnetics and Energy Conversion2-99
Typical NdFeB B-H Curve• Neodymium-iron-boron (NdFeB) is the highest strength
permanent magnet material in common use today• Good material for applications with temperature less than
approximately 80 - 150C• Cost per pound has reduced greatly in the past few years• B/H curve below for “grade 35” or 35 MGOe material
Bm, Tesla
Hm, kA/m
1.2Br
Hc
-915
0.6
-457.5
(BH)max
![Page 100: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/100.jpg)
Electromechanics Magnetics and Energy Conversion2-100
NdFeB B-H Curves for Different Grades
Reference: Dexter Magnetics, Inc. http://www.dextermag.com
![Page 101: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/101.jpg)
Electromechanics Magnetics and Energy Conversion2-101
Maximum Energy Product• BH has units of Joules per unit volume
![Page 102: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/102.jpg)
Electromechanics Magnetics and Energy Conversion2-102
Maximum Energy Product
Why is maximum energy product important?
(1) ⎟⎟⎠
⎞⎜⎜⎝
⎛=
g
mmg A
ABB
(2) 1−=gHlH
g
mm
Let’s find Bg2
( )
( )mmgap
mago
mmo
g
mm
gog
mmg
HBVolVol
glH
AA
B
HAA
BB
−⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−=
×⎟⎟⎠
⎞⎜⎜⎝
⎛=
μ
μ
μ2
Solve for magnet volume Volmag
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=mm
gapgmag HB
VolBVol 2
To use minimum volume of magnet for a given Bg, operate magnet at (BH)max point
![Page 103: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/103.jpg)
Electromechanics Magnetics and Energy Conversion2-103
Progress in PM Specs
Reference: J. Evetts, Concise Encyclopedia of Magnetic and Superconducting Materials, Pergamon Press, Oxford, 1992
• One figure of merit is (BH)max product
![Page 104: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/104.jpg)
Electromechanics Magnetics and Energy Conversion2-104
Progress in PM Specs
Reference: K. Overshott, “Magnetism: it is permanent,” IEE Proceedings-A, vol. 138, no. 1, Jan. 1991, pp. 22-31
![Page 105: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/105.jpg)
Electromechanics Magnetics and Energy Conversion2-105
Applications for Permanent Magnets• Disk drives• Speakers• Motors
– Rotary motors (Toyota Prius)– Linear motors (Maglev, people moving)
• Refrigerator magnets• Proximity sensors and switches• Compasses• Magnetic bearings and magnetic suspensions (Maglev)• Water filtration• Plasma fusion research, NMR• Eddy current brakes (ECBs)• Etc.References: 1. R. Parker, Advances in Permanent Magnetism, John Wiley, 19902. P. Campbell, Permanent Magnet Materials and their Applications, Cambridge University Press, 1994
![Page 106: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/106.jpg)
Electromechanics Magnetics and Energy Conversion2-106
Example: Maximum Energy Product• Fitzgerald, Example 1.10: Find magnet dimensions for
desired airgap flux density Bg = 0.8 TeslaAt maximum (BH), Bm=1.0 T and Hm = -40 kA/m
22 6.10.18.0)2( cmcm
BB
AAm
ggm =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
cmcmH
Bg
HH
glmo
g
m
gm 18.3
)40000)(104(8.0)2.0( 7 =⎟⎟
⎠
⎞⎜⎜⎝
⎛−×
−=⎟⎟⎠
⎞⎜⎜⎝
⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−= −πμ
So, magnet is 3.18 cm long and 1.6 cm2 in area
![Page 107: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/107.jpg)
Electromechanics Magnetics and Energy Conversion2-107
Open-Circuited Permanent Magnet
Reference: P. Campbell, Permanent Magnet Materials and their Applications, Cambridge University Press, 1994, pp. 89
![Page 108: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/108.jpg)
Electromechanics Magnetics and Energy Conversion2-108
Open Circuited Permanent Magnet --- FEA
![Page 109: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/109.jpg)
Electromechanics Magnetics and Energy Conversion2-109
Short-Circuited Permanent Magnet• Find B inside core, ignoring any leakage and assuming
infinite permeability in core
![Page 110: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/110.jpg)
Electromechanics Magnetics and Energy Conversion2-110
Short Circuited Permanent Magnet• For infinite permeability, load line is vertical• Intersection of load lines occurs at B ≈ Br
Reference: P. Campbell, Permanent Magnet Materials and their Applications, Cambridge University Press, 1994, pp. 89
![Page 111: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/111.jpg)
Electromechanics Magnetics and Energy Conversion2-111
Short Circuited Permanent Magnet --- FEA
![Page 112: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/112.jpg)
Electromechanics Magnetics and Energy Conversion2-112
Circuit Modeling of Permanent Magnets
Reference: E. P. Furlani, Permanent Magnet and Electromechanical Devices, Academic Press, 2001
![Page 113: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/113.jpg)
Electromechanics Magnetics and Energy Conversion2-113
Circuit Modeling of Permanent Magnets
![Page 114: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/114.jpg)
Electromechanics Magnetics and Energy Conversion2-114
Circuit Modeling of Permanent Magnets
![Page 115: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/115.jpg)
Electromechanics Magnetics and Energy Conversion2-115
Circuit Modeling of Permanent Magnets
![Page 116: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/116.jpg)
Electromechanics Magnetics and Energy Conversion2-116
Example: Circuit Modeling of Permanent Magnets
![Page 117: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/117.jpg)
Electromechanics Magnetics and Energy Conversion2-117
Example: Circuit Modeling of Permanent Magnets
![Page 118: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/118.jpg)
Electromechanics Magnetics and Energy Conversion2-118
Example: Circuit Modeling of Permanent Magnets---FEA
![Page 119: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/119.jpg)
Electromechanics Magnetics and Energy Conversion2-119
Example: Magnetic Circuit With Steel• Estimate airgap field Bg assuming grade 37 NdFeB• Airgap g, magnet thickness tm
![Page 120: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/120.jpg)
Electromechanics Magnetics and Energy Conversion2-120
Example: Magnetic Circuit With Steel• This analysis also ignores leakage
![Page 121: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/121.jpg)
Electromechanics Magnetics and Energy Conversion2-121
Example: Magnetic Circuit
With Steel ---Operating Point
vs. Magnet Thickness tm
![Page 122: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/122.jpg)
Electromechanics Magnetics and Energy Conversion2-122
Example: FEA with Magnet Thickness tm = g/2
![Page 123: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/123.jpg)
Electromechanics Magnetics and Energy Conversion2-123
Example: FEA with Magnet Thickness tm = g
![Page 124: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/124.jpg)
Electromechanics Magnetics and Energy Conversion2-124
Example: FEA with Magnet Thickness tm = 2g
![Page 125: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/125.jpg)
Electromechanics Magnetics and Energy Conversion2-125
Example: Comparison of Different Magnet Thicknesses
-1.00E+00
-8.00E-01
-6.00E-01
-4.00E-01
-2.00E-01
0.00E+00
2.00E-01
4.00E-01
6.00E-01
8.00E-01
1.00E+00
0.00E+001.00E+002.00E+003.00E+004.00E+005.00E+006.00E+007.00E+008.00E+009.00E+00
![Page 126: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/126.jpg)
Electromechanics Magnetics and Energy Conversion2-126
PM and a Winding• Many motors have permanent magnets, steel and windings
Analysis of PM in closed core, with excitation:
mm
mm
lNIH
NIlH
=∴
=
![Page 127: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/127.jpg)
Electromechanics Magnetics and Energy Conversion2-127
PM and a Winding --- Load Line• Note that demagnetization can occur if current is
sufficiently high
![Page 128: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/128.jpg)
Electromechanics Magnetics and Energy Conversion2-128
Another Example --- Excitation and Airgap(1) Ampere’s law:
NIgHlH gmm =+
(2) Constitutive law: gog HB μ=
(3) Gauss’ law: ggmm ABAB =
Solve for Bm-Hm load line:
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
mm
m
mmom l
NIHgAlA
B μ
![Page 129: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/129.jpg)
Electromechanics Magnetics and Energy Conversion2-129
Another Example --- Excitation and Airgap ---Load Line
![Page 130: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/130.jpg)
Electromechanics Magnetics and Energy Conversion2-130
Interesting Calculation Tool
Reference: www.magnetsales.com
![Page 131: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/131.jpg)
Electromechanics Magnetics and Energy Conversion2-131
What Can You Buy?
Reference: www.dextermag.com
![Page 132: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/132.jpg)
Electromechanics Magnetics and Energy Conversion2-132
What Can You Buy?
Reference: www.dextermag.com
![Page 133: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/133.jpg)
Electromechanics Magnetics and Energy Conversion2-133
What Can You Buy?
Reference: www.magnetsales.com
![Page 134: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/134.jpg)
Electromechanics Magnetics and Energy Conversion2-134
Magnetization Patterns
Reference: www.magnetsales.com
![Page 135: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/135.jpg)
Electromechanics Magnetics and Energy Conversion2-135
Comparison
Reference: www.magnetsales.com
![Page 136: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/136.jpg)
Electromechanics Magnetics and Energy Conversion2-136
Comparison of Maximum Operating Temperatures
Reference: http://www.electronenergy.com/media/Magnetics%202005.pdf
![Page 137: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/137.jpg)
Electromechanics Magnetics and Energy Conversion2-137
Comparison of Maximum Operating Temperatures
Reference: http://www.electronenergy.com/media/Magnetics%202005.pdf
![Page 138: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/138.jpg)
Electromechanics Magnetics and Energy Conversion2-138
Magnet Comparisons
Reference: www.magnetsales.com
![Page 139: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/139.jpg)
Electromechanics Magnetics and Energy Conversion2-139
PM Online Resources• http://members.aol.com/marctt/• www.dextermag.com• www.magnetsales.com• http://www.grouparnold.com/• Magnetic Materials Producers Association (MMPA
standard)
![Page 140: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/140.jpg)
Electromechanics Magnetics and Energy Conversion2-140
Some Very Brief Comments on Superconductors• Superconductors have zero resistance if the temperature is
low enough, the field acting on the superconductor is low enough, and the current through the superconductor is low enough
• Superconductors are classified as “low temperature” (NbTi, NbSn) or “high temperature” (YBCO, BSCCO)
• Low-Tc superconductors are usually chilled with liquid helium (4.2K)
• High-Tc superconductors are usually used in the 20K-77K range
Reference: Y. Iwasa “Case Studies in Superconducting Magnets,” Plenum Press, 1994
![Page 141: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/141.jpg)
Electromechanics Magnetics and Energy Conversion2-141
Some Data on Low-Tc Material• Shown for niobium titanium• This type of superconductor is used in the Japanese MLX
500 km/hr Maglev
![Page 142: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/142.jpg)
Electromechanics Magnetics and Energy Conversion2-142
Some Data on High-Tc Material• Some superconductors are anisotropic; i.e. superconducting
tapes
Reference: American Superconductor, www.amsuper.com
![Page 143: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/143.jpg)
Electromechanics Magnetics and Energy Conversion2-143
Quotes
It is well to observe the force and virtue and consequence of discoveries, and these are to be seen nowhere more conspicuously than in printing, gunpowder, and the magnet.--- Sir Francis Bacon
The mystery of magnetism, explain that to me! No greater mystery, except love and hate.---John Wolfgang von Goethe
![Page 144: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/144.jpg)
Electromechanics Magnetics and Energy Conversion2-144
• Selected history• Types of transformers• Voltages and currents• Equivalent circuits• Voltage and current transformers• Per-unit system
Transformers --- Overview
![Page 145: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/145.jpg)
Electromechanics Magnetics and Energy Conversion2-145
Selected History• 1831 --- Transformer action
demonstrated by Michael Faraday
• 1880s: modern transformer invented
Reference: J. W. Coltman, “The Transformer (historical overview),” IEEE Industry Applications Magazine, vol. 8, no. 1, Jan.-Feb. 2002, pp. 8-15
![Page 146: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/146.jpg)
Electromechanics Magnetics and Energy Conversion2-146
Early Transformer (Stanley, c. 1880)
William Stanley
![Page 147: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/147.jpg)
Electromechanics Magnetics and Energy Conversion2-147
Faraday’s Law• A changing magnetic flux impinging on
a conductor creates an electric field and hence a current (eddy current)
• The electric field integrated around a closed contour equals the net time-varying magnetic flux density flowing through the surface bound by the contour
• In a conductor, this electric field creates a current by:
• Induction motors, brakes, etc.
∫∫ ⋅−=⋅SC
dABdtddlE
EJ σ=
![Page 148: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/148.jpg)
Electromechanics Magnetics and Energy Conversion2-148
Magnetic Circuit with Two Windings• Note that flux is the sum of flux due to i1 and that due to i2
![Page 149: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/149.jpg)
Electromechanics Magnetics and Energy Conversion2-149
Types of Transformers• There are many different types and power ratings of
transformers: single phase and multi-phase, signal transformers, current transformers, etc.
![Page 150: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/150.jpg)
Electromechanics Magnetics and Energy Conversion2-150
Instrument Transformer• Instrument transformers (voltage and current) provide line
current and line voltage information to protective relays and control systems
• Current transformer shown below
Reference: L. Faulkenberry and W. Coffer, Electrical Power Distribution and Transmission, Prentice Hall, 1996, pp. 131
![Page 151: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/151.jpg)
Electromechanics Magnetics and Energy Conversion2-151
200A Current Transformer (CT)
Reference: http://rocky.digikey.com/WebLib/Amveco-Talema/Web%20Data/AC1200.pdf
![Page 152: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/152.jpg)
Electromechanics Magnetics and Energy Conversion2-152
Voltage Instrument Transformer
Reference: http://www.geindustrial.com/products/brochures/ITI.pdf
![Page 153: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/153.jpg)
Electromechanics Magnetics and Energy Conversion2-153
Audio Transformer
Reference: http://rocky.digikey.com/WebLib/Hammond/Web%20Data/560,%20800-844,%20850%20Series.pdf
![Page 154: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/154.jpg)
Electromechanics Magnetics and Energy Conversion2-154
Pulse Transformer
Reference: http://rocky.digikey.com/WebLib/Tamura-Microtran/Web%20Data/STT-107.pdf
• Used for triggering SCRs, etc. where isolation is needed
![Page 155: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/155.jpg)
Electromechanics Magnetics and Energy Conversion2-155
Power Distribution Transformer
Reference: http://www.geindustrial.com/products/brochures/DEA-271-English.pdf
• Provides voltage for the customer• Typical voltages are 2.3-34.5kV primary, and 480Y/277V or
208Y/120V 3-phase or 240/120V single phase• Pole-top transformers typically 15-100 kVA
![Page 156: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/156.jpg)
Electromechanics Magnetics and Energy Conversion2-156
E Core Transformer
![Page 157: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/157.jpg)
Electromechanics Magnetics and Energy Conversion2-157
Offline Flyback Power Supply
P. Maige, “A universal power supply integrated circuit for TV and monitor applications,” IEEE Transactions on Consumer Electronics, vol. 36, no. 1, Feb. 1990, pp. 10-17
![Page 158: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/158.jpg)
Electromechanics Magnetics and Energy Conversion2-158
Transcutaneous Energy Transmission
H. Matsuki, Y. Yamakata, N. Chubachi, S.-I. Nitta and H. Hashimoto, “Transcutaneous DC-DC converter for totally implantable artificial heart using synchronous rectifier,” IEEE Transactions on Magnetics, vol. 32 , no. 5 , Sept. 1996, pp. 5118 - 5120
![Page 159: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/159.jpg)
Electromechanics Magnetics and Energy Conversion2-159
Power Transformer
Reference: J. W. Coltman, “The Transformer (historical overview),” IEEE Industry Applications Magazine, vol. 8, no. 1, Jan.-Feb. 2002, pp. 8-15
![Page 160: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/160.jpg)
Electromechanics Magnetics and Energy Conversion2-160
Superconducting Transformer
Reference: W. Hassenzahl et. al., “Electric Power Applications of Superconductivity,” Proceedings of the IEEE, vol. 92, no. 10, October 2004, pp. 1655-1674
![Page 161: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/161.jpg)
Electromechanics Magnetics and Energy Conversion2-161
Transformer Under No-Load ConditionBy Faraday’s law:
)cos(max
11
tNdtdN
dtd
e
ωω
λ
Φ=
Φ==
RMS value of e1:
maxmax
,1 22
2Φ=
Φ= fN
fNE rms π
π
If resistive drop in winding is negligible:
fNE rms
π2,1
max =Φ
![Page 162: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/162.jpg)
Electromechanics Magnetics and Energy Conversion2-162
No-Load Phasor Diagram• Winding current has harmonics,
and fundamental is generally out of phase with respect to flux
• In-phase component is from core losses
• Magnetizing current is 90 degrees out of phase
cc IEP θϕ cos1=
![Page 163: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/163.jpg)
Electromechanics Magnetics and Energy Conversion2-163
Example: Transformer Calculations• Fitzgerald, Example 2.1. In Example 1.8, the core loss
and VA at Bmax = 1.5T and 60 Hz were found to be: Pc = 16W and VI = 20 VA with induced voltage 194V. Find power factor, core loss current Ic and magnetizing current Im.
![Page 164: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/164.jpg)
Electromechanics Magnetics and Energy Conversion2-164
Example: Transformer Calculations --- SolutionPower factor:
8.02016)cos( === cPF θ
Exciting current
AVVAI 1.0
19420
===ϕ
Core loss component
AIc 082.019416
==
Magnetizing component
AII cm 060.0sin == θϕ
![Page 165: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/165.jpg)
Electromechanics Magnetics and Energy Conversion2-165
Ideal Transformer with LoadIdeal transformer:
1
2
1
2
22
11
NN
vv
dtdNv
dtdNv
=
Φ=
Φ=
By Ampere’s law:
1
2
2
12211 0
NN
ii
iNiN =⇒=−
(Also, by power balance,
2211 iviv = )
![Page 166: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/166.jpg)
Electromechanics Magnetics and Energy Conversion2-166
Impedance Transformation
22
11
ˆˆ VNN
V =
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛===
21
1
2
1
2
2
2
1
22
1
21
1ˆˆ
ˆˆZ
VNN
NN
ZV
NNI
NNI
Therefore, impedance at input terminals is:
2
2
12
1
1
ˆˆ
⎟⎟⎠
⎞⎜⎜⎝
⎛=
NN
ZIV
![Page 167: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/167.jpg)
Electromechanics Magnetics and Energy Conversion2-167
Equivalent Circuits• These 3 circuits have the same impedance as seen from
the a-b terminals
![Page 168: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/168.jpg)
Electromechanics Magnetics and Energy Conversion2-168
Example: Use of Equivalent Circuits• Fitzgerald, Example 2.2• (a) Draw equivalent circuit with series impedance
referred to primary• (b) For a primary voltage of 120VAC and a short at the
output, find the primary current and the short circuit current at the output
![Page 169: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/169.jpg)
Electromechanics Magnetics and Energy Conversion2-169
Example: Impedance Transformation
( )
10025
22
2
2
1'2
'2
j
jXRNNjXR
+=
+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
![Page 170: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/170.jpg)
Electromechanics Magnetics and Energy Conversion2-170
Example: Input Current with Output Short Circuit
AI
jII
AI
jI
jj
jjXRVI
rms
rms
8.565.54.1
)65.5(4.1ˆ5ˆ16.113.128.0
)13.1(28.0ˆ1002510025
10025120ˆˆ
22,2
12
22,1
1
'2
'2
11
=+=
−==
=+=
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
+=
+=
![Page 171: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/171.jpg)
Electromechanics Magnetics and Energy Conversion2-171
Non-Ideal Effects in Transformers --- Magnetizing Inductance
• A real-world transformer doesn’t pass DC• From either set of terminals, the impedance looks like an
inductor if the other set of terminals is open-circuited• Can model this as an ideal transformer with a magnetizing
inductance added.• The magnetizing current im produces the mutual flux which
couples to the secondary
![Page 172: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/172.jpg)
Electromechanics Magnetics and Energy Conversion2-172
Mutual and Leakage Flux• Not all of the flux created by winding #1 links with winding #2.
![Page 173: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/173.jpg)
Electromechanics Magnetics and Energy Conversion2-173
Mutual and Leakage Flux
![Page 174: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/174.jpg)
Electromechanics Magnetics and Energy Conversion2-174
Non-Ideal Effects in Transformers --- Leakage• Not all of the flux created by winding #1 links with winding #2.• Therefore, real-world voltage transformation is not exactly
equal to the turns ratio, due to the voltage drops on Lk1 and Lk2
![Page 175: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/175.jpg)
Electromechanics Magnetics and Energy Conversion2-175
Transformer Equivalent Circuits
![Page 176: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/176.jpg)
Electromechanics Magnetics and Energy Conversion2-176
Example: Use of Equivalent Circuits• Fitzgerald, Example 2.3: A 50-kVA 2400:240V 60 Hz distribution transformer has a leakage impedance of 0.72+j0.92Ω in the high voltage winding and 0.0070 + j0.0090 Ω in the low-voltage winding. The impedance Zϕof the shunt branch (equal to Rc +jXm in parallel) is 6.32 + j43.7 Ω when viewed from the low voltage side.
Draw the equivalent circuits referred to the high voltage side and the low voltage sides.
SOLUTION:Note that N1:N2 is 1:10, so impedances step up and down by 100
![Page 177: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/177.jpg)
Electromechanics Magnetics and Energy Conversion2-177
Example: Solution--- Leakage impedance of 0.72+j0.92Ω in the high voltage winding and 0.0070 + j0.0090 Ω in the low-voltage winding. --- The impedance Zϕ of the shunt branch (equal to Rc +jXm in parallel) is 6.32 + j43.7 Ω when viewed from the low voltage side.
![Page 178: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/178.jpg)
Electromechanics Magnetics and Energy Conversion2-178
Approximate Transformer Equivalent Circuits• “Cantilever circuits”• Ignoring voltage drop in primary or secondary leakage
impedances
![Page 179: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/179.jpg)
Electromechanics Magnetics and Energy Conversion2-179
Approximate Transformer Equivalent Circuits• Circuit if we ignore the magnetizing inductance and core
resistance
• Circuit if we further ignore the winding resistance
![Page 180: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/180.jpg)
Electromechanics Magnetics and Energy Conversion2-180
Example: Use of Cantilever Circuit• Fitzgerald, Example 2.4
Using T-model:
( ) 0315.09.2391012400ˆ
12 j
ZZZ
Vl
+=⎟⎠⎞
⎜⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
+=
ϕ
ϕ
Using cantilever model:
( ) 2401012400ˆ
2 =⎟⎠⎞
⎜⎝⎛=V
![Page 181: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/181.jpg)
Electromechanics Magnetics and Energy Conversion2-181
Example: Find V2
• Fitzgerald, Example 2.5
![Page 182: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/182.jpg)
Electromechanics Magnetics and Energy Conversion2-182
Example: Find V2
From node equations: LLs IjXRIVV ˆˆˆˆ
2 ++=
![Page 183: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/183.jpg)
Electromechanics Magnetics and Energy Conversion2-183
Example: Find V2
-We need length of vector Oa (which is V2) -We know length of vector Oc (which is 2400V)
4.71)6.0)(42.3)(8.20()8.0)(72.1)(8.20(sincos =+=+= θθ IXIRab 5.35sincos =−= θθ IRIXbc
Solve for V2: VVVbcabV s 233)()( 2
2222 =⇒=++
![Page 184: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/184.jpg)
Electromechanics Magnetics and Energy Conversion2-184
Transformer Testing to Determine Parameters• By doing various tests on a transformer, we can
determine the equivalent circuit parameters• Testing includes open-circuit and short-circuit testing
![Page 185: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/185.jpg)
Electromechanics Magnetics and Energy Conversion2-185
Short-Circuit Test
![Page 186: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/186.jpg)
Electromechanics Magnetics and Energy Conversion2-186
Short-Circuit Testing
Normally: Lm >> Lk1, Lk2 and Rc >> Rw1, Rw2
Using the simplified circuit, we can approximate:
TEST
TESTeq I
VZ ≈
2'
21TEST
SCWWSC I
PRRR =+=
22
SCeqSC RZX −≈
![Page 187: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/187.jpg)
Electromechanics Magnetics and Energy Conversion2-187
Open-Circuit Testing
![Page 188: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/188.jpg)
Electromechanics Magnetics and Energy Conversion2-188
Open-Circuit Testing• There is no secondary current
TEST
TESTm
OC
TESTc
IV
X
PV
R
≈
≈2
![Page 189: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/189.jpg)
Electromechanics Magnetics and Energy Conversion2-189
AutotransformerConventional transformer
Connection as autotransformer
Autotransformer redrawn
![Page 190: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/190.jpg)
Electromechanics Magnetics and Energy Conversion2-190
Staco Autotransformer
Reference: Staco
![Page 191: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/191.jpg)
Electromechanics Magnetics and Energy Conversion2-191
Autotransformer Drawing
Reference: Staco
![Page 192: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/192.jpg)
Electromechanics Magnetics and Energy Conversion2-192
Autotransformer Brush
![Page 193: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/193.jpg)
Electromechanics Magnetics and Energy Conversion2-193
Damaged Autotransformer
![Page 194: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/194.jpg)
Electromechanics Magnetics and Energy Conversion2-194
Example: Autotransformer• Fitzgerald, Example 2.7• 2400:240V 50-kVA transformer is connected as an
autotransformer with ab being the 240V winding and bc is the 2400V winding.
(a) Compute the kVA rating(b) Find currents at rated power
![Page 195: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/195.jpg)
Electromechanics Magnetics and Energy Conversion2-195
Example: Autotransformer Example --- Solution
For 50 kVA, rating of 240V winding is: A208240/50000 =
The autotransformer VA rating is:
kVAAkVIV HH 549)208)(64.2( == Rated current at low-voltage winding:
AII HL 22924002640
=⎟⎠⎞
⎜⎝⎛=
![Page 196: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/196.jpg)
Electromechanics Magnetics and Energy Conversion2-196
Some Comments on Autotransformers• Autotransformers differ from isolation transformers in that
there is no isolation between primary and secondary• However, this lack of isolation allows some of the
transferred power to be conducted from primary to secondary instead of magnetic induction
• Autotransformers in general require less core material per kVA rating
• Autotransformers used where lack of isolation doesn’t pose a safety issue
![Page 197: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/197.jpg)
Electromechanics Magnetics and Energy Conversion2-197
Example: Magnetic Circuit Problem• Fitzgerald, Problem 2.2• A magnetic circuit with a cross-sectional area of 15 cm2
is to be operated from a 120V RMS supply. Calculate the number of turns required to achieve a peak magnetic flux density of 1.8 Tesla in the core
![Page 198: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/198.jpg)
Electromechanics Magnetics and Energy Conversion2-198
Example: Magnetic
Circuit Problem --- Solution
Flux is: )sin(max tωΦ=Φ The time rate of change of flux is:
)cos(max tdtd ωωΦ=Φ
The time rate of change of flux linkage is:
VtNdtdN
dtd
=Φ=Φ
= )cos(max ωωλ
Let’s relate flux density to flux:
maxmax BA
=Φ
So, we can solve for N
7.166)105.1)(8.1)(602(
)120)(2(23
max
=××
== −πω ABVN
Round up to N = 167
![Page 199: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/199.jpg)
Electromechanics Magnetics and Energy Conversion2-199
Example: Transformer Problem• Fitzgerald, Problem 2.4• A 100-Ohm resistor is connected to the secondary of an
ideal transformer with a turns ratio of 1:4 (primary to secondary). A 10V RMS, 1-kHz voltage source is connected to the primary. Calculate the primary current and the voltage across the 100-Ohm resistor
![Page 200: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/200.jpg)
Electromechanics Magnetics and Energy Conversion2-200
Example: Transformer Problem --- Solution
We know that this is a step-up transformer, so V1 = 10V and V2 = 40V. We next find the secondary current I2
AVI 4.010040
2 =Ω
=
The voltage steps up, so the current steps down; hence I1 = 4I2 = 1.6A
![Page 201: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/201.jpg)
Electromechanics Magnetics and Energy Conversion2-201
3-Phase Connections of Transformers
![Page 202: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/202.jpg)
Electromechanics Magnetics and Energy Conversion2-202
Example: Finding 3-Phase Currents• Problem: A three phase, 208V (line-line) Y connected
load has:– Zan = 3 + j4– Zbn = 5– Zcn = -5j
• Find– (a) Phase voltages– (b) Line and phase currents– (c) Neutral currents
![Page 203: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/203.jpg)
Electromechanics Magnetics and Energy Conversion2-203
Example: Finding 3-Phase Currents --- Solution
Line-neutral voltages are found by taking the line-line voltage and dividing by 3
1203
208==−NLV
![Page 204: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/204.jpg)
Electromechanics Magnetics and Energy Conversion2-204
Example: 3-Phase Currents --- Solution
Line (also called phase) currents are found by taking the line-line voltages and dividing by impedance
oo
c
oo
b
oo
a
jI
I
jI
3024)5(
240120
12024)5(120120
13.5324)43(
0120
−∠=−∠
=
∠=∠
=
−∠=+∠
=
![Page 205: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/205.jpg)
Electromechanics Magnetics and Energy Conversion2-205
Example: 3-Phase Currents --- Solution• Note that loads are unbalanced so there is a net neutral
current
ocban IIII 9.1554.25 ∠=++=
![Page 206: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/206.jpg)
Electromechanics Magnetics and Energy Conversion2-206
480V Y System• Line-line = 480V; line-neutral = 277V
Reference: Ralph Fehr, Industrial Power Distribution, Prentice Hall, 2002
![Page 207: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/207.jpg)
Electromechanics Magnetics and Energy Conversion2-207
Delta System• Line-line = 480V
Reference: Ralph Fehr, Industrial Power Distribution, Prentice Hall, 2002
![Page 208: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/208.jpg)
Electromechanics Magnetics and Energy Conversion2-208
Instrument Transformer• Used for protection, relaying, voltage or current monitoring,
etc.
![Page 209: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/209.jpg)
Electromechanics Magnetics and Energy Conversion2-209
Per-Unit System• Computations in electric machines and transformers are
often done using the “per-unit” system• Actual circuit quantities (Watts, VArs, etc.) are scaled to
the per-unit system• This method allows removal of transformers from
diagrams• To convert to per-unit, 4 base quantities are established
– Base power VAbase– Base voltage Vbase– Base current Ibase– Base impedance Zbase
![Page 210: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/210.jpg)
Electromechanics Magnetics and Energy Conversion2-210
Example: Per-Unit System• Example: A system has Zbase = 10 Ω and Vbase = 400V.
Find base VA and Ibase
• Solution:– Ibase = Vbase/Zbase = 400/10 = 40A– (VA)base = VbaseIbase = (400)(40) = 16 kVA
![Page 211: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/211.jpg)
Electromechanics Magnetics and Energy Conversion2-211
Example: Per-Unit System• A 5 kVA, 400/200V transformer has 2 Ω reactance
referred to the 200V side. Express the transformer reactance in p.u.
• Solution:– (VA)base = 5000– Vbase = 200– Ibase = (VA)base/Vbase = 5000/200 = 25A– Zbase =Vbase/Ibase = 200/25 = 8 Ω– Z = 2.0/8.0 = 0.25 p.u.
![Page 212: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/212.jpg)
Electromechanics Magnetics and Energy Conversion2-212
Example: Per-Unit Applied to Transformer• Fitzgerald, Example 2.12• Convert this circuit showing a 100 MVA transformer to the
per-unit system
![Page 213: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/213.jpg)
Electromechanics Magnetics and Energy Conversion2-213
Example: Low-Voltage Side
Ω=⎟⎟⎠
⎞⎜⎜⎝
⎛××
=
===
==
− 638.010100
)1099.7(
)(
99.7100)(
6
23
2
BASE
BASEBASEBASEBASE
BASE
BASE
VAV
ZXR
kVVMVAVA
In per-unit system:
..7.178638.0
114
..063.0638.004.0
..0012.0638.0
1076.0 3
upX
upX
upR
m
L
L
==
==
=×
=−
![Page 214: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/214.jpg)
Electromechanics Magnetics and Energy Conversion2-214
Example: High-Voltage Side
Ω=⎟⎟⎠
⎞⎜⎜⎝
⎛××
=
===
==
− 5.6310100
)107.79(
)(
7.79100)(
6
23
2
BASE
BASEBASEBASEBASE
BASE
BASE
VAV
ZXR
kVVMVAVA
In per-unit system:
...059.05.63
75.3
..00133.05.63
085.0
upX
upR
H
H
==
==
![Page 215: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/215.jpg)
Electromechanics Magnetics and Energy Conversion2-215
Example: Model• Turns ratio is 1:1, so we can remove transformer
![Page 216: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/216.jpg)
Electromechanics Magnetics and Energy Conversion2-216
Example: Get Rid of 1:1 Transformer
![Page 217: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/217.jpg)
Electromechanics Magnetics and Energy Conversion2-217
• Forces and torques• Energy balance• Determining magnetic forces and torques from energy• Multiply-excited systems• Forces and torques in systems with permanent magnets
Energy Conversion --- Overview
![Page 218: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/218.jpg)
Electromechanics Magnetics and Energy Conversion2-218
• For determining the direction magnetic-field component of the Lorentz force F=q(v x B) = JxB.
Right-Hand Rule
![Page 219: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/219.jpg)
Electromechanics Magnetics and Energy Conversion2-219
• Fitzgerald, Example 3.1• Find θ-directed torque as a function of α
Example: Single-Coil Rotor
![Page 220: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/220.jpg)
Electromechanics Magnetics and Energy Conversion2-220
Example: Single-Coil RotorFor wire #1:
αθ sin1 oIlBF −= For wire #2:
αθ sin2 oIlBF −= Total torque (T = force × distance):
RIlBT o αθ sin2−= What happens if B points left-right instead of up-down?
RIlBT o αθ cos2−=
![Page 221: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/221.jpg)
Electromechanics Magnetics and Energy Conversion2-221
• This box can be used to model motors, actuators, lift magnets, etc.• Note 2 electrical terminals (voltage and current) and 2 mechanical terminals (force ffld and position x)• The lossless magnetic energy storage system converts electrical energy to mechanical energy
Electromechanical Energy Conversion Device
![Page 222: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/222.jpg)
Electromechanics Magnetics and Energy Conversion2-222
Interaction Between Electrical and Mechanical Terminals
WFLD = stored magnetic energy In words, the rate of change of magnetic energy equals the power in minus the mechanical work out
dtdxfei
dtdW
fldFLD −=
By Faraday’s law, e = dλ/dt, so let’s rework:
dtdxf
dtdi
dtdW
fldFLD −=
λ
Multiply through everywhere by dt:
dxfiddW fldFLD −= λ
![Page 223: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/223.jpg)
Electromechanics Magnetics and Energy Conversion2-223
Interaction Between Electrical and Mechanical Terminals
In a lossless system, we can rewrite the energy balance:
FLDMECHELEC dWdWdW += Differential energy in: λiddWELEC = Differential work out: dxfdW fldMECH = Change in magnetic energy: FLDdW
![Page 224: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/224.jpg)
Electromechanics Magnetics and Energy Conversion2-224
• This solenoid is an example of a force-producing device
Force-Producing Device
![Page 225: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/225.jpg)
Electromechanics Magnetics and Energy Conversion2-225
EnergyThinking about energy, we start out with:
dxfiddW fldFLD −= λ If magnetic energy storage is lossless, this is a conservative system and Wfld is determinedby state variables λ and x In a conservative system, the path you take todo this integration doesn’t matter
![Page 226: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/226.jpg)
Electromechanics Magnetics and Energy Conversion2-226
Another Conservative System• Roller coaster, ignoring friction, the path doesn’t matter. Speed of both coasters is the same at the bottom of the hill
![Page 227: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/227.jpg)
Electromechanics Magnetics and Energy Conversion2-227
• This illustrates lossless magnetic structure with external losses due to resistance
Magnetic Relay
![Page 228: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/228.jpg)
Electromechanics Magnetics and Energy Conversion2-228
Integration Path for Finding Magnetic Stored Energy
∫∫ +=bpath
FLDapath
FLDooFLD dWdWxW22
),(λ
On path 2a, dλ = 0 and ffld = 0 since zero λ means zero magnetic force, therefore:
λλλλ
dxixW oooFLD ),(),(0∫=
![Page 229: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/229.jpg)
Electromechanics Magnetics and Energy Conversion2-229
Special Case --- Linear SystemIn the special case of a linear system, the flux linkage is proportional to current, or λ ∝ i.
)(21
)(
),(),(
2
0
''
0
''
xLd
xL
dxixWFLD
λλλ
λλλ
λ
λ
==
=
∫
∫
![Page 230: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/230.jpg)
Electromechanics Magnetics and Energy Conversion2-230
Example: Relay with Movable Plunger• Fitzgerald, Example 3.2• Find magnetic stored energy WFLD as a function of x with I = 10A
![Page 231: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/231.jpg)
Electromechanics Magnetics and Energy Conversion2-231
Example: Relay with Movable Plunger
)(21 2
xLWFLD
λ= with I constant.
We know that IxL )(=λ , so
222
)(21
)()(
21 IxL
xLIxLWFLD ==
I’ll give you that )1(2
)(2
dxld
gN
xL o −=μ ,
so magnetic stored energy is: 2
2
)1(4
Idxld
gN
W oFLD −=
μ
![Page 232: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/232.jpg)
Electromechanics Magnetics and Energy Conversion2-232
Determining Magnetic Force from Stored Energy• Next, if we go to all this trouble to find stored energy, let’s figure out how to find forces from the energy (very important!)
Remember dxfiddW fldFLD −= λ Next, remember the “total differential” from calculus:
2.2
1.1
21
12
),( dxxFdx
xFxxdF
constxconstx ==∂∂
+∂∂
=
Let’s rewrite the stored energy expression:
λ
λλ dx
Wd
dW
dW FLD
x
FLDFLD
∂+
∂=
From this, we see that
x
FLD
dW
iλ
∂= and
λxWf FLD
fld ∂∂
−=
![Page 233: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/233.jpg)
Electromechanics Magnetics and Energy Conversion2-233
Determining Magnetic Force from EnergyFor linear systems with λ = L(x)I
Energy )(2
1 2
xLWFLD
λ=
Force:
dxxdL
xL
xLxf
tcons
fld
)()(2
)(21
2
2tan
2
λ
λ
λ
=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−==
With λ = L(x)I
dxxdLIf fld)(
2
2
=
![Page 234: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/234.jpg)
Electromechanics Magnetics and Energy Conversion2-234
Determining Magnetic Force from Energy• The bottom line: if your system is linear, and if you can calculate inductance as a function of position, then finding the force is pretty easy
![Page 235: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/235.jpg)
Electromechanics Magnetics and Energy Conversion2-235
Example: Curve Fit for Inductance of Solenoid with Plunger
• Fitzgerald, Example 3.3• Assume that the following inductance vs. plunger position was measured. We then run the solenoid with 0.75A current
![Page 236: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/236.jpg)
Electromechanics Magnetics and Energy Conversion2-236
• We can fit a polynomial to the inductance, and use energy methods to find the plunger force as a function of position
Example: Force as a Function of Position
![Page 237: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/237.jpg)
Electromechanics Magnetics and Energy Conversion2-237
• Can solve this as before, by analogy
Torque in Magnetic Circuit
By analogy:
θθ
ddLIT fld
)(2
2
=
![Page 238: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/238.jpg)
Electromechanics Magnetics and Energy Conversion2-238
Example: Finding Torque• Fitzgerald, Example 3.4
Assume L(θ) = Lo + L2cos(2θ) with Lo = 10.6 mH and L2 = 2.7 mH. Find the torque with I = 2A. Solution:
( )
mN
LId
dLIT
−×−=
−+==
− )2sin(1008.1
)2sin(22
)(2
)(
2
2
22
θ
θθθθ
![Page 239: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/239.jpg)
Electromechanics Magnetics and Energy Conversion2-239
Example: Finding Torque• Fitzgerald, Practice Problem 3.4
Assume L(θ) = Lo + L2cos(2θ) + L4sin(4θ) with Lo = 25.4 mH, L2 = 8.3 mH and L4 = 1.8 mH. Find the torque with I = 3.5A. Solution:
mNT −+−= )4cos(044.0)2sin(1017.0)( θθθ
![Page 240: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/240.jpg)
Electromechanics Magnetics and Energy Conversion2-240
Another Example --- Torque vs. Rotor Angle
0 50 100 150 200 250 300 350 400-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Theta [deg]
Torq
ue [N
-m]
![Page 241: Electromagnetic and Electromechanical Engineering ...2-1 Electromagnetic and Electromechanical Engineering Principles Notes 02 Magnetics and Energy Conversion Marc T. Thompson, Ph.D](https://reader035.vdocuments.net/reader035/viewer/2022070218/6127764aaf04827aa3142f58/html5/thumbnails/241.jpg)
Electromechanics Magnetics and Energy Conversion2-241
Today’s Summary• Today we’ve covered:
• Maxwell’s equations: Ampere’s, Faraday’s and Gauss’ laws• Soft magnetic materials (steels, etc.)• Hard magnetic materials (permanent magnets)• Basic transformers• The per-unit system• We started electromechanical conversion basics