EE361-Electromechanical Energy Conversion

class: center, middle

# EE-361
# Dynamic Equations
# Multiply-Excited Systems

## Ozan Keysan

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

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

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# Review:
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# Rotational Sytems:

## \\(F = - \dfrac{1}{2}\Phi^2\dfrac{d R(x)}{dx}\\)

### or  alternatively

## \\(F = \dfrac{1}{2}I^2\dfrac{d L(x)}{dx}\\)

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# Review:

# Rotational Sytems:

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

### or  alternatively

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

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# What is the torque in the following systems?

## a) If Coil#1 is excited only,

## b) If Coil#2 is excited only,

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

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

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

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

## Or it can be written as:
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## \\(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\\)
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# Stored Energy in Matrix Form
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### \\(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}\\)

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# 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)
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# Cylindrical Rotor, Salient Stator

<img src="./images/salient_stator.jpg" alt="Drawing" style="width: 300px;"/>
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### \\(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)
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# 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)
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# Combination with Mechanical Systems:
--

## Linear and Rotational Motion

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

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# Dynamic Equations: Damping
--

<img src="http://www.maplesoft.com/view.aspx?SI=3926/forced_oscillations125.gif" alt="Drawing" style="width: 400px;"/>
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### \\(F = B v = B \dfrac{dx}{dt}\\) : Dissipates energy (~Resistance)
### Overdamped, [underdamped](http://www.youtube.com/watch?v=j-zczJXSxnw) (similar to RLC circuits)

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# Dynamic Equations: Inertia

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

## or

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

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# 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} \\)

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# 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://www.youtube.com/watch?v=D5hSwaMxNIA), [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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# How do satellites align themselves?

<img src="https://www.hytera.com/iwov-resources/hytera/02_products/4_banner_image/catagory_satellite_banner.jpg" alt="Drawing" style="width: 900px;"/>
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# How do satellites align themselves?

## Jet Propulsion

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# How do satellites align themselves?

## Reaction Wheels

[Reaction wheel operation principle](https://www.youtube.com/watch?v=woCdjbsjbPg) 
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# How do satellites align themselves?

## Torquer
--

### A coil interacting with the earth's magnetic field.

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# Torquer
### A coil interacting with the earth's magnetic field.

<img src="https://www.isispace.nl/wp-content/uploads/2020/10/imtq-min.jpg" alt="Drawing" style="width: 500px;"/>
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# Torquer
### A coil interacting with the earth's magnetic field.

####[More information](http://www.slideshare.net/zuliana26/satellite-dynamic-and-control)
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## You can download this presentation from: [keysan.me/ee361](http://keysan.me/ee361)