EE568-Selected Topics in Electrical Machines

class: center, middle

# EE-568 Selected Topics in 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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# Basics of Energy Conversion
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# Lorenz Force

## \\(\vec{F} = q (\vec{E} + \vec{v} \times \vec{B})\\)
--

### In a purely magnetic system (no electric field) :
## \\(\vec{F} = q \vec{v} \times \vec{B}\\)
--

### or alternatively force density (\\(N/m^3\\))):
## \\(\vec{f} = \rho \vec{v} \times \vec{B} = \vec{J} \times \vec{B}\\)

#### \\( \rho \\): Charge density (C/m3) , \\( \vec{v} \\): velocity

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#Lorenz Force

## \\(\vec{F} = \vec{J} \times \vec{B}\\)

![](http://sharkphysics.weebly.com/uploads/8/5/9/5/8595301/265734720.png?452)
---
# Lorenz Force Applications

- ### [Force Demo](http://www.youtube.com/watch?v=K9ks_DNPAFQ)
- ### [Homopolar Motor](http://www.youtube.com/watch?v=kJKuNcgbW-o)
- ### [Wolrd's Simplest Electric Train](https://www.youtube.com/watch?v=J9b0J29OzAU)

- ### [Navy Railgun](https://www.youtube.com/watch?v=NmFeRYPNP88)
- ### [Aselsan Tufan](https://www.youtube.com/watch?v=O5GtuQk3t44)
- ### [Aselsan Tufan-2](https://www.youtube.com/watch?v=MxloiA5mSSk)

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# Determine the direction of rotation

---
#What would happen in the device below?

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# Basics

- ## Forces and Torque
--

- ## Energy Balance
--

- ## Stored Energy (Magnetic and Mechanical)
--

- ## Losses
---
# Losses

- ## Electric Losses (Ohmic Losses)
--

- ## Magnetic Losses (Hysteresis Loss)
--

- ## Mechanical Losses (Friction etc)

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# Lossless System

### Electric Energy Input = Stored Magnetic Energy + Mechanical Work

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# Energy Balance

### Ignore resistive and friction losses
--

### \\(\Delta W\_{electrical} = \Delta W\_{magnetic} + \Delta W\_{mechanical}\\)
--

### \\( Vi\, \mathrm{d}t = i \, \mathrm{d}\lambda = \Delta W\_{magnetic} +  f_{mech} \, \mathrm{d}x\\)

### or
--

## \\( \Delta W\_{magnetic} =  i \, \mathrm{d}\lambda -  f_{mech} \, \mathrm{d}x\\)

---
# Energy Balance
## Same can be obtained from the power relations

## \\( \dfrac{\partial W\_{magnetic}}{\partial t}  =  V.i - f_{mech} \dfrac{\partial x}{\partial t} \\)
--

## \\( \dfrac{\partial W\_{magnetic}}{\partial t}  =  i \dfrac{\partial \lambda}{\partial t} - f_{mech} \dfrac{\partial x}{\partial t} \\)

---

# Review: Magnetic Energy
![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/12/Coenergy.svg/640px-Coenergy.svg.png)
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# Review: Magnetic Energy

## \\(W\_{stored} = \int_0^\lambda i(\lambda) d\lambda \\)
--

## or from B-H curve

## \\(W\_{stored} = \int \_{volume } (\int_0^B H dB) \\)
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# Magnetic Energy

### In Linear Systems:
--

## Magnetic Energy = Magnetic Co-Energy
--

## Magnetic Energy + Magnetic Co-Energy = \\(\lambda i\\)
--

### Thus (only in linear systems)
## W(magnetic) = \\(\dfrac{1}{2} \lambda i = \dfrac{1}{2} L i^2 =\dfrac{1}{2L} \lambda^2 \\)

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# Singly-Excited Electromechanical System

---
# Force from the Stored Energy

## Derivative of Energy w.r.t. position gives the force!
---
# Force from Stored Energy
## \\(W\_{magnetic} = i d\lambda - f dx\\)
### Take derivative
--

### Some useful reading:

- #### [MIT From Lasers to Motors](http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/readings/MIT6_007S11_forces.pdf)

- #### [Fitzgerald-Electromechanical Energy Conversion](http://cnx.org/contents/XKs0ES7Y@1/Chapter-3Electromechanical-Ene)

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# Force from Stored Energy
### \\(Force = - \dfrac{\partial W\_{mag}(\lambda, x)}{\partial x} |\_{\lambda = constant}\\)

--
## For Linear Systems
### \\(Force = - \dfrac{\partial}{\partial x} (\dfrac{\lambda^2}{2 L(x)}) =\dfrac{ \lambda^2}{2 L(x)^2} (\dfrac{d L(x)}{dx}) \\)
--

### \\(Force = \dfrac{i^2}{2} \dfrac{d L(x)}{dx} \\)
---
# Example:

### Derive the force as a function of position

---

# Example: [Junkyard Magnet](https://www.youtube.com/watch?v=XBWy9gzGGd4)
--

### Derive the force:

---
# Virtual Work Method

--
## When \\(\Phi\\) is constant:

### \\(Force = - \dfrac{\partial W\_{mag}(\Phi, x)}{\partial x} |\_{\Phi = constant}\\)

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# Virtual Work Method
--

## Use Co-energy (\\(W\_{mag}'\\)) instead of Magnetic Energy

--
## When MMF (\\(NI\\)) is constant:

### \\(Force = \dfrac{\partial W'\_{mag}(\mathrm{F}, x)}{\partial x} |\_{\mathrm{F} = constant}\\)

---

# Summary

## Magnetic Circuit Tries
--

- ## To reduce \\(W\_{magnetic}\\) if \\(\Phi\\) is constant
--

- ## To maximize the inductance
--

- ## To minimize the reluctance (\\(L=N^2/R\\))

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## Mechanical Power & Energy:
--

## Linear Motion: 
--
\\(P = F v = F \dfrac{dx}{dt}\\) Watt
--

## Rotational:
--
\\(P=T \omega = T \dfrac{d\theta}{dt} \\) Watt
--

## Linear Motion: \\(W = \int P dt = F x \\) Joule
--

## Rotational: \\(W=  \int P dt = T \theta \\) Joule
---

## Linear Acceleration: 
--

## \\(F = m a = m \dfrac{dv}{dt}\\)
--

## Rotational Acceleration:
--

## \\(T=J \dfrac{d\omega}{dt} \\) Watt

## J: Rotational Inertia (\\(kgm^2\\))
---

# What would happen in this circuit?

<img src="./images/single_phase_reluctance_motor.png" alt="Drawing" style="width: 700px;"/>
---
# Can you guess the torque expression?

<img src="./images/single_phase_reluctance_motor.png" alt="Drawing" style="width: 700px;"/>
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# Rotational Systems:
--

## Remember in linear systems:

### \\(f = - \dfrac{\partial W\_{mag}(\Phi, x)}{\partial x} |\_{\Phi = constant}\\)

### or  alternatively use Co-energy:

### \\(f = \dfrac{\partial W'\_{mag}(\mathcal{F}, x)}{\partial x} |\_{\mathcal{F} = constant}\\)

---
# Rotational Systems:

## Take the derivative wrt \\( \theta \\) not \\( x \\):
--

### \\(T = - \dfrac{\partial W\_{mag}(\Phi, \theta)}{\partial \theta} |\_{\Phi = constant}\\)

### or  alternatively

### \\(f = \dfrac{\partial W'\_{mag}(\mathcal{F}, \theta)}{\partial \theta} |\_{\mathcal{F} = constant}\\)

#### [More information](http://engineering.nyu.edu/mechatronics/Control_Lab/Criag/Craig_RPI/SenActinMecha/EM_Motion_Fundamentals_2.pdf)
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# Rotational Systems:

## Take the derivative wrt \\( \theta \\) not \\( x \\):
--

## \\(T = - \dfrac{1}{2}\Phi^2\dfrac{d R(\theta)}{d \theta} |\_{\Phi = constant}\\)

### or  alternatively

## \\(T = \dfrac{1}{2}I^2\dfrac{d L(\theta)}{d \theta} |\_{i = constant}\\)

---
# How can we achieve a constant rotation?
--

## Single Phase Reluctance Motor
--

[Magnetorquer](https://blog.satsearch.co/2019-08-21-magnetorquers-an-overview-of-magnetic-torquer-products-available-on-the-global-marketplace-for-space.html)
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# Single Phase Reluctance Motor
--

### [Magnetic Flux](https://www.youtube.com/watch?v=hDJnLt7cBTY),  [Micro-stepping for higher accuracy](https://www.youtube.com/watch?v=eyqwLiowZiU)

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# Reluctance Motors

### [More info](http://electrical-engineering-portal.com/characteristics-and-work-principles-of-switched-reluctance-sr-motor)
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# Reluctance Motors

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# Who is this guy?

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# James Dyson

### [Digital Motor](https://www.youtube.com/watch?v=6kcGltYWz3k), [Operating Principle](https://www.youtube.com/watch?v=yajwmhe96pg), [Manufacturing](https://www.youtube.com/watch?v=L5IROOrHg5w)

---
# Dyson uses Reluctance Motors

### [Digital Motor](https://www.youtube.com/watch?v=6kcGltYWz3k), [Operating Principle](https://www.youtube.com/watch?v=yajwmhe96pg), [Manufacturing](https://www.youtube.com/watch?v=L5IROOrHg5w)

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# Reluctance Motor Example:

## Calculate the torque expression
--

###  with constant current.
--

### with sinusoidal current
--

### [Torque Graphs](https://docs.google.com/spreadsheets/d/1MDdJiaDq6uIzYzGMhTugNqbFBcJJClO-m_LqdGe1ZQ4/edit?usp=sharing)

### [Solutions](https://github.com/ozank/ozank.github.io/raw/master/files/ee361_emec_reluctance_motor_example.pdf)

---
# Summary

## Magnetic Circuit Tries
--

- ### To maximize the inductance, to minimize the reluctance (\\(L=N^2/R\\))

- ### To decrease the magnetic energy (increase co-energy)

## Rotational systems are similar to linear systems, but take the derivative of magnetic energy in terms of \\(\theta\\) instead of \\(x\\).

---
# What is the torque?
## Cylindrical Rotor, Cylindrical Stator

<img src="./images/round_rotor_stator.jpg" alt="Drawing" style="width: 300px;"/>
---
# What is the torque?
## Cylindrical Rotor, Salient Stator

<img src="./images/salient_stator.jpg" alt="Drawing" style="width: 300px;"/>
---
# What is the torque?
## Salient Rotor,  Cylindrical Stator

<img src="./images/salient_rotor.jpg" alt="Drawing" style="width: 300px;"/>
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# Multiply-Excited Systems

## What happens if both of the coils are excited?

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# Multiply-Excited Systems
<img src="./images/multiply_excited_block.jpg" alt="Drawing" style="width: 700px;"/>
## Electrical Energy = Magnetic Energy + Mechanical Energy
--

### \\(dW\_{mag} (\lambda\_1,\lambda\_2,\theta) = i\_1 d\lambda\_1 + i\_2d\lambda\_2 - T d\theta\\)
---
# Mutual Inductance

### Write down the voltage equation of Inductor 2.

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# What is the stored energy in coil1?

### \\(W\_{mag1}= \dfrac{1}{2}i^2 L\\) 
--
 \\(\quad \rightarrow\\) Not Correct!

### \\(dW\_{mag1}= i\_1 d\lambda\_1 \\)
--

### \\(dW\_{mag1}= i\_1 (L\_{11}di\_1  + L\_{12}di\_2) \\)

---
# Total stored energy (coil1+coil2)?
## \\(dW\_{mag}= i\_1 d\lambda\_1 + i\_2d\lambda\_2 \\)
--

## Or it can be written as:
--

## \\(W\_{mag}=\int dW\_{mag} \\)
## \\(= \dfrac{1}{2} L\_{11}i\_1^2 + \dfrac{1}{2} L\_{22}i\_2^2 +  M i\_1 i\_2\\)
---
# Stored Energy in Matrix Form
--

### \\(W\_{mag}=\dfrac{1}{2}  \begin{bmatrix}  i\_1 & i\_2  \end{bmatrix} \begin{bmatrix}  L\_{11} & M \\\ M & L\_{22} \end{bmatrix} \begin{bmatrix}  i\_1 \\\ i\_2  \end{bmatrix}\\)
--

### Generalized case
### \\(W\_{mag} = \dfrac{1}{2} \mathbf{I}_t \mathbf{L} \mathbf{I}\\)

### An application of multiply excited systems: [Contactless Surgery](https://www.youtube.com/watch?v=ocE3MjF77Wk) [More information](http://robohub.org/minimally-invasive-eye-surgery-on-the-horizon-as-magnetically-guided-microbots-move-toward-clinical-trials/)

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# Torque in Multiply Excited Sytems
## still depends on the derivative of \\(W\_{mag}\\)
--

### \\(T\_{mech}= \dfrac{1}{2} \dfrac{dL\_{11}}{d\theta}i\_1^2 + \dfrac{1}{2} \dfrac{dL\_{22}}{d\theta}i\_2^2\\)
### \\(\quad\quad\quad + \dfrac{dM}{d\theta} i\_1 i\_2\\)
--

### \\(T\_{mech} = \dfrac{1}{2} \mathbf{I}_t \dfrac{d\mathbf{L}}{d\theta} \mathbf{I}\\)

---
# Cylindrical Rotor, Cylindrical Stator
<img src="./images/round_rotor_stator.jpg" alt="Drawing" style="width: 300px;"/>
--

### \\(T=i\_1 i\_2 \dfrac{\partial M}{\partial \theta}\\) : (\\(L\_{11}\\),\\(L\_{22}\\) constant)
---
# Cylindrical Rotor, Salient Stator

<img src="./images/salient_stator.jpg" alt="Drawing" style="width: 300px;"/>
--

### \\(T=\dfrac{1}{2}i\_2^2 \dfrac{\partial L\_{22}}{\partial \theta} + i\_1 i\_2 \dfrac{\partial M}{\partial \theta}\\) : (\\(L\_{11}\\) constant)
---
# Salient Rotor, Cylindrical Stator

<img src="./images/salient_rotor.jpg" alt="Drawing" style="width: 300px;"/>
--

### \\(T=\dfrac{1}{2}i\_1^2 \dfrac{\partial L\_{11}}{\partial \theta} + i\_1 i\_2 \dfrac{\partial M}{\partial \theta}\\) : (\\(L\_{22}\\) constant)

---
# Salient Rotor, Cylindrical Stator

<img src="./images/salient_rotor.jpg" alt="Drawing" style="width: 300px;"/>
--

### \\(T=\dfrac{1}{2}i\_1^2 \dfrac{\partial L\_{11}}{\partial \theta} + i\_1 i\_2 \dfrac{\partial M}{\partial \theta}\\) : (\\(L\_{22}\\) constant)

---
# Cylindrical Machine vs. Salient Pole
--

### Salient Pole: Position dependent air-gap reluctance(and hence inductance; \\(L=N^2/R\\))

---

# Salient Pole Rotor Synchronous Machines

### Airgap is not uniform.

### There are both reluctance and synchronous torque components

---
## Power in Salient Pole Machines

### \\(T=\dfrac{1}{2}i\_1^2 \dfrac{\partial L\_{11}}{\partial \theta} + i\_1 i\_2 \dfrac{\partial M}{\partial \theta}\\) : (\\(L\_{22}\\) constant)
---
# Combination with Mechanical Systems:
--

## Linear and Rotational Motion

---
# Dynamic Equations: Ideal Spring
![](http://upload.wikimedia.org/wikipedia/commons/9/9d/Simple_harmonic_oscillator.gif)
---
# Dynamic Equations: Ideal Spring

![](http://upload.wikimedia.org/wikipedia/commons/9/9d/Simple_harmonic_oscillator.gif)
## \\(F = k (x - x_0)\\): No energy dissipation (~Ideal Inductor)

---
# Dynamic Equations: Damping
--

<img src="http://www.maplesoft.com/view.aspx?SI=3926/forced_oscillations125.gif" alt="Drawing" style="width: 400px;"/>
--

### \\(F = B v = B \dfrac{dx}{dt}\\) : Dissipates energy (~Resistance)
### Overdamped, [underdamped](http://www.youtube.com/watch?v=j-zczJXSxnw) (similar to RLC circuits)

---
# Dynamic Equations: Inertia

## \\(F=ma\\)
--

## or

## \\(F = m \dfrac{dv}{dt} = m \dfrac{d^2 x}{d^2  t} \\)

---
# Dynamic Equations: Mechanical Side

## \\(f\_{mech} = M \dfrac{d^2 x}{d^2  t} + B \dfrac{dx}{dt} \\)
## \\(\quad \quad \quad + k (x-x\_0) + f\_{external} \\)

---
# Bose's Active Suspension System

### [Bose suspension system](https://www.youtube.com/watch?v=3KPYIaks1UY), [Bose suspension will be mass produced](https://www.digitaltrends.com/cars/boses-revolutionary-adaptive-suspension-gets-a-reboot-for-2019/)

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# Bose Ride

### [Bose Ride](https://youtu.be/fFiBoIlHLKA?t=95), [Bose ride presentation](https://www.youtube.com/watch?v=wQSNlJEdOZE), [Truck Driver comments-1](https://youtu.be/L1RyhBzS3PU?t=70), [Truck Driver comments-2](https://youtu.be/iFzdHUJOv1g?t=40)

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## You can download this presentation from: [keysan.me/ee568](http://keysan.me/ee568)