Roketsan Training

class: center, middle

# Roketsan Training - Session I

# Magnetic Circuits and Magnetic Materials

## Ozan Keysan

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

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

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# Electric Circuits vs. Magnetic Circuits

## Electric Materials
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## Conductors:
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 Copper, Aluminium, Iron...
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## Insulators: Plastic, Air...
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# Electric Circuits vs. Magnetic Circuits

## What is the best conductor?

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

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

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

### Gold Plated High Speed HDMI cable [>30.000 TL](https://www.hepsiburada.com/audioquest-cherry-cola-active-optical-hdmi-kablo-20m-p-HBV00000SX5L0): SCAM!

AudioQuest's sophisticated Noise-Dissipation System combines RF noise-absorbing carbon and directionality-controlled silver-plated conductors to minimize the distortion caused by RF interference, ensuring a fully immersive entertainment experience.

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# Electric vs. Magnetic Materials
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### In Electric Circuits we have Conductors 
### In Magnetic Circuits we have 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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# Electric vs. Magnetic Materials

## Insulators in electric circuits
## = Non-magnetic materials in magnetic circuits
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### (e.g. Plastic, Copper, Aluminium )

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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), [Diamagnetism of Water](https://www.youtube.com/watch?v=gJeqriqRYYE)

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# 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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# Conducting Materials: 
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# Electric Circuits:

## - Copper
## - Aluminium

# Magnetic Circuits:
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## - Iron
## - Electrical Steel

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# Insulating Materials: 
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## Electric Circuits:

## - Air, Plastics

## Magnetic Circuits:
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## - Air, Copper, Aluminium

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# Magnetic vs Electric Circuit
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## Conductivity of Copper vs 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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# Magnetic vs Electric Circuit
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## How about Magnetic Circuits?
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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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# Magnetic Circuits and Maxwell Equations:

#### $$\textrm{Gauss' Law}\quad \nabla \cdot \vec{E}  =  \frac{\rho}{\varepsilon_0} $$
#### $$\textrm{Gauss' Law ($\vec{B}$ Fields)} \quad \nabla \cdot \vec{B}  =  0 $$
#### $$\textrm{Faraday's Law} \quad \nabla \times\quad \vec{E}  = - \frac{\partial \vec{B}}{\partial t}$$
#### $$\textrm{Ampere's Law} \quad \nabla \times \vec{B}  =  \mu_0 \vec{J} + \mu_0\varepsilon_0\frac{\partial \vec{E}}{\partial t}$$

####For curious students:

#### - [Maxwell Equations](http://www.maxwells-equations.com/)
#### - [Who's afraid of Maxwell equations?](https://www.majr.com/wp-content/uploads/2018/04/Elya-Joffe-Whos-Afraid-of-Maxwells-Equations-Pittsburgh-EMC-Chapter.compressed.pdf)
#### - [An Intuitive Guide to Maxwell’s Equations](https://photonlines.substack.com/p/an-intuitive-guide-to-maxwells-equations)

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# Ampere's Law

### \$\oint_C {\vec{H}.d\vec{\ell}}\$=
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### \$ \iint_{closed S} \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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# Ampere's Law

## Magneto-Motive Force (\$\mathcal{F}\$)

## \$ \mathcal{F} = NI \$
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## Electric Circuits: \$\quad \quad V=IR\$

## Magnetic Circuits: \$ \quad \quad \mathcal{F} = \Phi \mathcal{R} \$
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# Maxwell Equations:

### $$\textrm{Gauss' Law ($\vec{E}$ Fields) }\quad \nabla \cdot \vec{E}  =  \frac{\rho}{\varepsilon_0} $$
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### $$\textrm{Gauss' Law ($\vec{B}$ Fields)} \quad \nabla \cdot \vec{B}  =  0 $$
--

## 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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## Electric Field can diverge

## (i.e. can get away from a charged particle)

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## Divergence of a Magnetic Field is Zero!

## (i.e. it can not run away, always come back to original point in loops)

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# Gauss Law (for Magnetism)

## $$\nabla \cdot {B} =0$$

## $$\oint_S {B dA} = 0$$

## Practical Meaning: 
### - There are no magnetic flux sources. 
### - No magnets with single pole!
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# Ohm's Law (for Magnetic Circuit)

## Electric Circuits: \$\quad \quad V=IR\$
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## Magnetic Circuits: \$ \quad \quad \mathcal{F} = \Phi \mathcal{R} \$
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## Electrical Resistance: \$\quad  R = \frac{\rho l}{A} = \frac{l}{\sigma A}\$
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## Magnetic Reluctance: \$\quad \quad \mathcal{R} = \frac{l}{\mu A}\$

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# Current Density vs. Magnetic Flux Density

## Current Density = Conductivity x Electric Field

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

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## Flux Density = Permeability x Magnetic Field

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

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#Permeability

## Permeability of free space:

## $$\mu_0 = 4 \pi 10^{-7} \quad H/m$$

## Common to define permeability relative to vacuum:

## $$\mu = \mu_o . \mu_r$$

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# Exercise

### Calculate the reluctance of the iron core in terms of \$\mu_0\$, assuming \$\mu_r=4000\$, \$l=500 mm\$, \$A_c=2500mm^2\$.

### Don't forget to convert to SI units (e.g. mm to m).
 
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# Exercise

### Now, calculate the reluctance again with the air-gap added as below. Relative permeability of air is 1, and assume the air-gap is 3 mm.

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#Draw Equivalent Circuits:
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<img src="./images/ee361/c_core3.png" alt="Drawing" style="width: 800px;"/>

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# Exercise

## With the same dimensions of the previous example:
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## Assume Nt=5, I=10 A

## Find B and H in the core.
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## [Solutions](./files/ee361-reluctance-solved.pdf)
---

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## What happens if you double the current?
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## \$ \mathcal{F} = NI \$
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## Current is doubled --> MMF is doubled
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## \$ \mathcal{F} = \Phi \mathcal{R} \$
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### [Breaking Bad - Magnets](https://youtu.be/J0IqVC8mTwk?t=47s)

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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/Vacuum (Linear)
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<img src="http://www.coolmagnetman.com/images/magfund071.gif" alt="Drawing" style="width: 400px;"/>
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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 limit => Saturation
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# Magnetic Saturation

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# Magnetic Saturation

## Residual Magnetism
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Compactcassette.jpg/1920px-Compactcassette.jpg" alt="Drawing" style="width: 320px;"/>
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<img src="./images/ee361/ego_karti.jpg" alt="Drawing" style="width: 350px;"/>

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# Hysteresis Loss
<img src="https://www.electronics-tutorials.ws/wp-content/uploads/2018/05/electromagnetism-mag19.gif" alt="Drawing" style="width: 500px;"/>

#### [How do magnets work?](https://youtu.be/hFAOXdXZ5TM?t=4m24s)
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# Hysteresis Loss
<img src="https://qph.fs.quoracdn.net/main-qimg-ac943b670cbcf5b5ced59b7e9bff9075" alt="Drawing" style="width: 750px;"/>

#### [How do magnets work?](https://youtu.be/hFAOXdXZ5TM?t=4m24s)
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## Hysteresis Loss \$\propto\$ Frequency
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# Hysteresis Loss

## Variable Hmax
<img src="./images/ee361/B-H_loop.png" alt="Drawing" style="width: 700px;"/>

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# Lenz's Law

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

### 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)

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# AC Resistance vs DC Resistance
## Which one is higher?

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- ## AC Resistance > DC Resistance
- ## Why?

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# Skin Depth
# Distribution of Current in a Wire
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/61/Skin_depth.svg/450px-Skin_depth.svg.png)
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# Skin Depth

### Due to circulating eddy currents
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Skineffect_reason.svg/293px-Skineffect_reason.svg.png)

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# Skin Depth

## Current Distribution: \$ J = J_S e ^{-d/\sigma}\$

## Skin depth: \$\sigma = \sqrt{\frac{2 \rho}{\omega \mu_r \mu_0}}\$

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# Skin Depth for Copper:
- ## 50 Hz --> 9.3 mm

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- ## 1 kHz --> 2.1 mm

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- ## 10 kHz --> 0.66 mm

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- ## 100 kHz --> 0.21 mm

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- ## 1 MHz --> 66 um

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## How to increase conductor area without increasing thickness?

## Bundled Conductors

### Also helps to reduce [Corona discharge](http://www.electricaleasy.com/2016/07/corona-discharge.html)
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## How to increase conductor area without increasing thickness?

## Foil Conductors
<img src="https://kllaminates.si/wp-content/uploads/2015/03/image002.jpg" alt="Drawing" style="width: 600px;"/>

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## Hollow Conductors
<img src="http://sub.allaboutcircuits.com/images/52013.jpg" alt="Drawing" style="width: 400px;"/>
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# AC Resistance

## AC Resistance [Calculator](http://www.clemson.edu/ces/cvel/emc/calculators/Resistance_Calculator/round.html)

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## Rule of Thumb: For 50 Hz power systems
## $$Rac \simeq 1.1 Rdc$$
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## Power Dissipated due to Eddy Currents

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

## Notice the \$f^2\$ element
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#Magnetic Losses

# Hysteresis Loss \$\propto\$ Frequency

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

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### Several Applications

- [Eddy Current Brake](https://www.youtube.com/watch?v=lywaL2tpLB0)

- [Eddy Current Science](https://www.youtube.com/watch?v=ezXVzc64qRE), [Eddy Current Seperator2](https://www.youtube.com/watch?v=zlwpEzcamcc)

- [Levitating Coil](http://www.youtube.com/watch?v=VydPQuLyEns)

- [Induction Heater](http://www.youtube.com/watch?v=aOsbIP3y2Wg), [Induction Cooking](https://www.youtube.com/watch?v=T3AI1eQ50iE), [Electroboom Eddy Curents](https://www.youtube.com/watch?v=u7Rg0TcHQ4Y)

- [Electromagnetic Retarder](https://www.youtube.com/watch?v=5FMoUEmSsdE), [Rötarder Sesi](https://www.youtube.com/shorts/kQygTyy1p08)

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