EE564-Design of Electrical Machines

class: center, middle

# EE-564 Design of Electrical Machines

## Ozan Keysan

[ozan.keysan.me](http://ozan.keysan.me)

Office: C-113 <span class="meta">•</span> Tel: 210 7586

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# Electric Motors
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### How many electric motors are there in the classroom?

--
### Each one of you carry at least one of these:
--
<img src="http://upload.wikimedia.org/wikipedia/commons/e/ee/Vibramotor.jpg" alt="Drawing" style="width: 350px;"/>

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# Electric Machines

### But they can also be big:

Siemens Ship Propulsion Motors (Induction Motor)
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## Very Big

Enercon 7.5 MW, Direct-Drive Wind Turbine Generator (Synchronous Motor)

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## Very Very Big: [700 MW Synchronous Generator](https://www.itaipu.gov.br/en/energy/generating-units)
--

### Itaipu Dam, Brasil: 16 m diameter, 91 rpm, Rotor Mass: 2650 t
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## In Industry,
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 what percentage of energy used by electric motors?
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- # 65%
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## What is the percentage of energy used by electric motors overall?
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- # 45%

## What is the percentage of energy produced by electric machines?
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- # 97%
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# Richard Feynman: Electricity
--

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## So let's see how these machines work?
--
<img src="https://www.bbvaopenmind.com/wp-content/uploads/2016/06/bbva-openmind-maxwell-1-ppal.jpg" alt="Drawing" style="width: 750px;"/>
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## So let's see how these machines work?

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# Maxwell Equations

### $$ \nabla\cdot{\bf E} = \dfrac{\rho}{\epsilon_0} $$
### $$ \nabla\cdot{\bf B} = 0 $$
### $$ \nabla\times{\bf E} = - {{\partial{\bf B}}\over{\partial t}} $$
### $$ \nabla\times{\bf H} = {\bf J} + {\epsilon_0{\partial{\bf E}}\over{\partial t}} $$

#### [More info](http://www.maxwells-equations.com/), [Who's afraid of Maxwell equations?](http://majr.com/docs/Whos_Afraid_of_Maxwells_Equations_By_Elya_Joffe.pdf)
---

# Maxwell Equations:

### $$\textrm{Gauss' Law}\quad \nabla \cdot \vec{E}  =  \frac{\rho}{\varepsilon_0} $$
--

### $$\textrm{Faraday's Law} \quad \nabla \times \vec{E}  = - \frac{\partial \vec{B}}{\partial t}$$
--

## Need a reminder for vector calculus?

## [Curl](http://betterexplained.com/articles/vector-calculus-understanding-circulation-and-curl/), [Divergence](http://betterexplained.com/articles/divergence/), [Cross Product](http://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/)
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# Divergence: \$\nabla \cdot\$

--
## 1,2,3: Positive divergence, 4: Negative

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# Curl: \$\nabla \times \$

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## 1,2,3,4,5: High Curl, 6-7: No Curl

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### $$\textrm{Faraday's Law} \quad \nabla \times \vec{E}  = - \frac{\partial \vec{B}}{\partial t}$$

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# Gauss' Law:

# $$ \nabla \cdot \vec{E}  =  \frac{\rho}{\varepsilon_0} $$
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## Integral form:

# $$\oint_S {E dA} = {Q \over \varepsilon_0}$$

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# Gauss' Law (for Magnetic Field):

## $$\nabla \cdot \vec{B}  =  0 $$
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## $$\oint_S {B dA} = 0$$
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### Practical Meaning: 
### - There are no magnetic flux sources. 
### - No magnets with single pole!

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# Faraday's Law

### !! The most important equation for motor design
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### $$\nabla \times \vec{E}  = - \frac{\partial \vec{B}}{\partial t}$$
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### $$\oint_C \vec{E} dl  = - \frac{\partial \Phi}{\partial t} = -\frac{d}{dt}\iint B dA$$

### (-) Sign: Lenz Laz
--

### [Induction Melting](https://www.youtube.com/watch?v=8i2OVqWo9s0), [Levitation](https://www.youtube.com/watch?v=txmKr69jGBk)

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# Ampere's Law:
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### $$\nabla \times \vec{H}  =  \vec{J} + \varepsilon_0\frac{\partial \vec{E}}{\partial t}$$

### Neglecting the [displacement current](https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law) part (i.e. quasi-static state)
### and in integral form.

### $$\oint_C {\vec{H}.d\vec{\ell}} = \iint \vec{J}dA = \sum I_n $$
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# Ampere's Law:

### Problem of neglecting displacement current:
 
<img src="./images/ee564/displacement_current.png" alt="Drawing" style="width: 350px;"/>

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# Ampere's Law:

### $$\oint_C {\vec{H}.d\vec{\ell}} = \iint \vec{J}dA = \sum I_n $$

### Magnetic field intensity(H) is independent of the material properties!

![](http://farside.ph.utexas.edu/teaching/302l/lectures/img740.png)

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## Material Properties:
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## \$\sigma\$
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: Conductivity
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## \$\rho\$
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: Resistivity
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# \$\mathbf{J}=\sigma \mathbf{E}\$

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## Resistivity and Conductivity of Copper?
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## \$\sigma = 5.96\,10^{7}\$
--
S/m, or 1/Ωm

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## \$\rho = 1.68\,10^{-8}\$ Ωm

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## Material Properties:
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## \$\varepsilon\$
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: Electric Permittivity
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# \$\mathbf{D}=\varepsilon \mathbf{E}\$
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## \$\varepsilon_0\$: Permittivity of free space
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## \$\varepsilon_0= 8.85\,10^{-12}\$
--
 F/m, or
--
 As/Vm

### Materials with high permittivity (dielectric constant) used in capacitors, (mica, ceramis, metal oxides etc)

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## Material Properties:
--

## \$\mu\$
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: Magnetic Permeability
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# \$\mathbf{B}=\mu \mathbf{H}\$
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## \$\mu_0\$: Permeability of free space
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## \$\mu_0= 4\pi\,10^{-7}\$
--
 H/m, or
--
 Vs/Am

### Materials with high permeability (ferromagnets) used in inductors

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# Material Properties
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## Other Important Parameters for Motor Design
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# Thermal:

- ## Thermal Conductivity, Heat Capacity...

# Mechanical:

- ## Density, Strength, Young's Modulus...

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# Material Properties:
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## Isotropic:
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Parameter is constant for all directions

## e.g.: \$\mu_x=\mu_y=\mu_z\$
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## Otherwise it is anisotropic

## Can be constant (linear material)
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## Can be non-linear (for example magnetic saturation)

## Can be depedent to temperature (e.g. condutivity)...
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## Electric Materials
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## Conductors:
--
 Copper, Aluminium, Iron...
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## Insulators: Plastic, Air...

## How conductive copper is compared to air?
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### Copper is \$10^{22}\$ times more conductive than air.
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### Copper is \$10^{7}\$ times more conductive than sea water.

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# Electric Materials

## What is the best conductor?

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- ## **Copper:** \$1.68 \; 10^{-8} \; \Omega.m \$

--
- ## **Aluminium:** \$2.82\; 10^{-8} \; \Omega.m \$
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- ## **Silver:** \$1.59 \; 10^{-8} \; \Omega.m \$

--
- ## **Gold:** \$2.44 \; 10^{-8} \; \Omega.m \$
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## Why some connectors are gold plated?

![](http://www.safestchina.com/products_image/hdmi-cable-1-3v-1080p-gold-plated-41946.jpg)

### Gold Plated High Speed HDMI cable [1199 TL](http://www.hepsiburada.com/monster-black-platinum-hdmi-kablo-15-metre-p-HBV000006CXNE): SCAM!

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# What about the price?
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# Copper vs. Aluminium
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<img src="./aluminium_price.png" alt="Drawing" style="width: 600px;"/>

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# What about the price?

# Copper vs. Aluminium

<img src="./copper_price.png" alt="Drawing" style="width: 600px;"/>
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# Electric vs. Magnetic Materials
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## Conductors --> 
--
Ferromagnets
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 (e.g. Iron, Cobalt)

## Large permeability ( `$\mu_r >> 1$`)
<img src="https://c1.staticflickr.com/3/2716/4468925382_6c3ceecbfe.jpg" alt="Drawing" style="width: 450px;"/>

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# Non-magnetic Materials
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## Paramagnetic 
### ( `$\mu_r \gtrsim 1$`) (very slightly attracted)
### E.g.:Aluminium
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## Diamagnetic 
### ( `$\mu_r \lesssim 1$`) (very slightly repulsed)

### E.g.: Copper, Water, [Frogs](https://www.youtube.com/watch?v=KlJsVqc0ywM)

---
# Electric Circuits

## Analogy to Hydraulic System

### - Voltage --> Pressure
### - Electric Current --> Water Current
### - Hydraulic Resistance --> Resistance

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# Magnetic Circuits

## Magnetic circuits are analogous to electric circuits:

<img src="magnetic_electric_circuit.png" alt="Drawing" style="width: 800px;"/>
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# Magnetic Circuits

## Materials:

| Electric Circuits | Magnetic Circuits |
| -- | -- | -- |
| **Conductors:** Copper, Aluminum | **Ferromagnets**: Iron, Electrical Steel |
| **Insulators:** Air, Plastic | **Non-magnetics:** |
||Copper, Water (Diamagnetic)|
|| Air, Aluminum (Paramagnetic) |

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# Magnetic vs Electric Circuit
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## Copper is \$10^{22}\$ times more conductive than air.
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## What about permeability of Iron vs Air?
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## Iron is JUST 3000 times more "permeable" than air

## Therefore, air is a circuit element in magnetic circuits!

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# Magnetic Circuits

## Flux (Wb)
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## Flux Density (Wb/m\$^2\$ or T)
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## Magneto-motive Force (MMF), (A.turns)

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# Ohm's Law (for Magnetic Circuit)

### Electric Circuits:

### $$V=IR$$
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### Magnetic Circuits

### $$\mathcal{F} = \Phi \mathcal{R}$$
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### Reluctance

### $$\mathcal{R} = \frac{l}{\mu A}$$
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#  A note on symbols.

## $$\mathcal{F} = \Phi \mathcal{R}$$

## In the [textbook](https://books.google.com.tr/books?id=7XkxDQAAQBAJ&pg=PA60&lpg=PA60&dq=current+linkage+MMF&source=bl&ots=sEturQtDGU&sig=bm5WgK8_CeKLG82Hq526OWLDn7E&hl=tr&sa=X&ved=0ahUKEwj4lezSoLDZAhVKCcAKHR8RBKsQ6AEIJzAA#v=onepage&q=current%20linkage%20MMF&f=false):
--

- ## Magneto-motive force = NI = [Current Linkage](http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=121-11-46)
--

## $$ NI = MMF =\mathcal{F} = \Theta$$

### For latest offical, symbols refer to [IEC Standars](http://www.electropedia.org/iev/iev.nsf/index?openform&part=121)

### [Conventional Magnetic Symbols](https://www.d.umn.edu/~snorr/ece4501s4/MAGCKTS.PDF)

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# Current Density vs. Magnetic Flux Density

## Current Density = Conductivity x Electric Field

## $$\vec{J}=\sigma \vec{E}$$

--
## Flux Density = Permeability x Magnetic Field

## $$\vec{B} = \mu \vec{H}$$

---

# Magnetic Circuits

### Magnetic circuits are analogous to electric circuits:

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# Linearity vs. Non-linearity
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## $$\vec{B} = \mu \vec{H} $$
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# Linear B-H Characteristics

- ## Air

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# B-H Curve of Air (Linear)
--

<img src="http://www.itacanet.org/wp-content/uploads/2011/06/mag-600x521.jpg" alt="Drawing" style="width: 300px;"/>
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- ## Slope gives permeability (\$\mu = \frac{B}{H}\$)
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- ## Constant Slope = Linear Material

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## Non-Linear B-H Characteristics

- ### Iron, Cast-Steel ...

## B is limited by saturation

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# Magnetic Saturation

## Residual Magnetism
<img src="http://www.edmsauce.com/wp-content/uploads/2014/05/Sonys-New-Cassette-Tape-Holds-64750000-Songs.jpg" alt="Drawing" style="width: 320px;"/>
--
<img src="http://www.rayhaber.com/wp-content/uploads/ego_karti.jpg" alt="Drawing" style="width: 350px;"/>
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# Hysteresis Loss
<img src="http://www.electronics-tutorials.ws/electromagnetism/mag19.gif" alt="Drawing" style="width: 400px;"/>
--
<img src="http://www.stanelcorftechnologies.com/img/hysteresis-loss.jpg" alt="Drawing" style="width: 320px;"/>

--
## Hysteresis Loss \$\propto\$ Frequency
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# Hysteresis Loss

## Variable Hmax
<img src="http://www.insulcoresolutions.com/images/Hysteresis-L-web.jpg" alt="Drawing" style="width: 600px;"/>

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# Lenz's Law

## Why there is a negative sign?
## \$e = -\frac{d \Phi}{dt}\$

### An induced electromotive force (emf) always gives rise to a current whose magnetic field opposes the original change in magnetic flux.

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# Eddy Current Losses
### Created by Induced Voltage in a Conductor

#### [Magnet in a Copper Tube](http://www.youtube.com/watch?v=keMpUaoA3Tg)
--

### Power Dissipated: \$P= \frac{ V \pi^2 B_{peak}^2 d^2 f^2}{k \rho}\$

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# Hysteresis Loss \$\propto\$ Frequency

# Eddy Loss \$\propto\$ Frequency\$^2\$

---
# Applications

### Eddy Current Brake
### Induction Heater

[Levitating Coil](http://www.youtube.com/watch?v=VydPQuLyEns), [Induction Heater](http://www.youtube.com/watch?v=aOsbIP3y2Wg)

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## Assumptions we used to make in EE361, but not in this course

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## - All flux contained in magnetic circuit (no leakage)
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## - Uniform flux distribution in the core
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## - Flux travels straight in airgaps (no fringing)
--

## - In some cases infinite permeability is assumed

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# Faraday's Law of Induction
--

### \$\nabla \times \vec{E}  = - \dfrac{\partial \vec{B}}{\partial t}\$

### which equals to:

--
### $$\oint_C \vec{E} dl  = - \dfrac{d\Phi}{dt} = -\dfrac{d}{dt}\iint \vec{B} dA$$

---
# Faraday's Law of Induction

### Induced Voltage in a Coil

## $$ e= \oint \vec{E} dl = -\dfrac{d \Phi}{dt}$$

## $$ e = -\dfrac{d \Phi}{dt} = -\dfrac{d}{dt}\int \vec{B} dA$$
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# Ways to generate voltage in a coil
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- ## Time variation of flux (e.g. AC excitation)
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- ## Due to conductor movement in a stationary field
--

- ## Rotation (e.g. motors)
--

- ## Deformation of coil (Area changes)

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# Faraday's Law of Induction

## Total voltage in a coil with N turns
## \$V_{coil} = N\dfrac{d \Phi}{dt} = \dfrac{d \lambda}{dt}\$
---

# Flux Linkage

### \$\lambda\$ : Flux Linkage (Warning, in the textbook shown by [\$\Psi\$)](http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=121-11-24)
### \$\Phi\$ : Flux per turn

### \$\lambda = N_{turns} \Phi \$ (turn. Weber)

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# Relation between Magnetic and Electric Circuits

--
<img src="http://2.bp.blogspot.com/-Lm15GER7L20/VieZ0DjOoRI/AAAAAAAAAXA/RQHcpv7aBfk/s1600/inductor%2Bsymbol.jpg" alt="Drawing" style="width: 300px;"/>
--

# \$ L = \dfrac{d \lambda}{d I}\$

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# Relation between Magnetic and Electric Circuits

## What happens if you double the number of turns?
--

## Inductance increases with \$N^2\$
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# Inductance

## Inductance is the flux linkage created per ampere

## \$ L =  N \dfrac{d \Phi}{dI}= \dfrac{d \lambda}{d I}\$

## \$ L =  \dfrac{N^2}{R}\$

---
# Mutual Inductance:

## Magnetic Interaction between two coils.

## Exists if current in one coil induces voltage in another coil.

[Example](https://www.miniphysics.com/uy1-mutual-inductance.html)

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# Mutual Inductance:

## \$ V_{2} = M \frac{dI_1}{dt} \$
<img src="https://raw.githubusercontent.com/ozank/ee361/master/images/mutual_flux.png
" alt="Drawing" style="width: 380px;"/>

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# Magnetic Stored Energy

## What is the energy stored in an inductor?
--

## \$W = \dfrac{1}{2} L i^2 \$

## But why? And what if L is variable?

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# Magnetic Energy
## \$dw = \mathcal{F}(t) d \Phi(t) \$

--
## \$W = \int_{\Phi_1}^{\Phi_2} \mathcal{F} d \Phi \$

--
## Magnetic Energy Density
## \$W = \int_{B_1}^{B_2} H d B \$

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# Total Energy Stored (Linear Systems)

## Initial B=0

## \$W = \dfrac{1}{2}R \Phi^2 =\dfrac{1}{2}\dfrac{\mathcal{F}^2}{R}\$
or
## \$W = \dfrac{1}{2}\dfrac{Volume}{\mu} B^2 \$

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## You can download this presentation from: [keysan.me/ee564](http://keysan.me/ee564)