Roketsan Training

class: center, middle

# Roketsan Training - Session III

# Electromechanical Energy Conversion

## Ozan Keysan

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

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

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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}\\)

###  \\( \vec{J}\\): Current Density (A/m2), \\( \vec{B}\\): Magnetic Field Density (T)

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### Let's integrate force density  \\(\vec{F}\\) over the current carrying wire volume
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## \\( F = B I L \\)

### B: Magnetic Field Density, I: Current, L: Length of the wire
--

#### but don't forget the cross operator between B and I

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### Let's integrate force density  \\(\vec{F}\\) over the current carrying wire volume

## \\( F = B \times I L \\)

### Force is maximum when B and I are perpendicular to each other

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# Lorenz Force Applications

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

- ### [Electromagnetic Aircraft Launcher](https://www.youtube.com/watch?v=T1Icd3MFmWc)
- ### [Navy Railgun](https://www.youtube.com/watch?v=NmFeRYPNP88), [Railgun-2](https://www.youtube.com/watch?v=_VvXXtT3HoU)
- ### [Aselsan Tufan](https://www.youtube.com/watch?v=O5GtuQk3t44)
- ### [Aselsan Tufan-2](https://www.youtube.com/watch?v=MxloiA5mSSk)

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# Electromagnetic Launcher (Railgun)

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## ASELSAN TUFAN

## 8 MJ, 2000 m/s exit velocity, 2 Million Amps peak current

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# Electromagnetic Launcher (Railgun)

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#Lorenz Force

## \\( F = B \times I L \\)

### Exercise:

- ### How can we increase the force in a electromagnetic system?
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- ### Increase Magnetic Field (more magnet, more field current)
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- ### Increase Current
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- ### Increase the volume of conductor (longer, larger area, more number of turns)
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## Limitations in a Practical System

## Increasing B (Magnetic Field Density)

## Limits of the permanent magnets ( Brem less than 1.4 T, B airgap around 0.7 T)
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## Magnetic Saturation (steel laminations: 1.5 T, cobalt SMC: 1.8-2 T)
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## Efficiency and volume issues with field windings (electromagnets)

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## Limitations in a Practical System

## Increasing Current (Electrical Loading)
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- ## Heating (Copper losses proportional with \\( I^2\\)
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- ## Resistance also increases with temperature (feed forward mechanism)
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- ## Short overload is possible due to thermal mass

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## Limitations in a Practical System

## Increasing Copper Length:
--

- ## Increases resistance, and losses
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## Increasing cross section area of the wire
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- ## Increases eddy current losses, increases mass
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## Using more number of turns

- ## Increases volume, but usually more preferable to increasing cross section area
																							 
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### How can we calculate non-current carrying electromagnetic systems?

## A Solenoid
<img src="https://www.intercyprus.com/cdn/shop/files/61AmVqjLRZL._SL1447_55e31f4b-c0d1-43c8-b4e2-a6bc8801dc31.jpg" alt="Drawing" style="width:400px;"/>

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# Simplified Representation of a Solenoid

## How can you calculate the generated force?

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# A very Important Rule:

# There is no direct connection between electrical systems and mechanical force!

## Electrical energy is converted to magnetic energy, then magnetic energy is converted to mechanical energy

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## Electric Energy Input =
--

## Losses (Heat) + Stored Magnetic Energy + Mechanical Work

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# Equivalent Circuit of a Solenoid

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### Let's ignore the losses for now
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### Electric Energy Input =  Stored Magnetic Energy + Mechanical Work

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# Review: Stored 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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# Review: Stored Magnetic Energy

### In linear magnetic systems (no saturation)

## 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

<img src="./images/emec_relay.png" alt="Drawing" style="width: 800px;"/>
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# Example:

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# Example:

### Find magnetic stored energy as a function of x. (I=10 A)

<img src="./images/emec_ex2.png" alt="Drawing" style="width: 800px;"/>
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# Example Cont:

### Find magnetic stored energy if N=1000, g=2mm, d=15cm, l=10cm

<img src="./images/emec_ex2.png" alt="Drawing" style="width: 800px;"/>
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# Force from the Stored Energy

## Derivative of Energy w.r.t. position gives the force!
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# 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](https://feevuted.edu.vn/wp-content/uploads/2018/02/Fitzgerald-2002-Electric-Machinery_Cambrige.pdf)

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

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## 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} \\)
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# Example: Previous Example

### Derive the force as a function of position

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# Rail Gun Example:

###[Mini Coil Gun](http://www.youtube.com/watch?v=lIEgg7d8ZsU), [Large Rail Gun](http://www.youtube.com/watch?v=wa_vuX5_oAk), [Part-2](http://www.youtube.com/watch?v=rAwvnXYH488), [How it works](http://www.youtube.com/watch?v=PQNHsS-BNs4)

### [Reading assignment](http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/lecture-notes/MIT6_007S11_lec14.pdf)

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# Summary

## Magnetic Circuit Tries
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- ## To reduce \\(W\_{magnetic}\\) if \\(\Phi\\) is constant
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- ## To maximize the inductance
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- ## To minimize the reluctance (\\(L=N^2/R\\))

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### How about rotational systems?

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# What would happen in this circuit?

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## Mechanical Power & Energy:
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## Linear Motion: 
--
\\(P = F v = F \dfrac{dx}{dt}\\) Watt
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## Rotational:
--
\\(P=T \omega = T \dfrac{d\theta}{dt} \\) Watt
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## Linear Motion: \\(W = \int P dt = F x \\) Joule
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## Rotational: \\(W=  \int P dt = T \theta \\) Joule
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## Linear Acceleration: 
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## \\(F = m a = m \dfrac{dv}{dt}\\)
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## Rotational Acceleration:
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## \\(T=J \dfrac{d\omega}{dt} \\) Watt

## J: Rotational Inertia (\\(kgm^2\\))
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# Rotational Sytems:
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## Remember in linear systems:

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

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# Rotational Sytems:

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

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

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

## 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}\\)

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# 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
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### [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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# Summary

## Magnetic Circuit Tries
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- ### 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\\).
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# Multiply-Excited Systems

## What happens if there are more than one coils?

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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 Systems
## 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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# Combination with Mechanical Systems:
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## 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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# Summary

- ## Multiply excited systems still tries to minimize total stored magnetic energy

- ## Derivative of self inductances and mutual inductance can work together or oppose each other.

- ## Magnetic forces interact with the mechanical systems and generate a system response

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## You can download this presentation from: [keysan.me/ee361](http://keysan.me/ee361)