EE361-Electromechanical Energy Conversion

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# EE-361 ELECTROMECHANICAL ENERGY CONVERSION

## Ozan Keysan

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

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

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# Review

## Magnetic circuits are analogous to electric circuits:

- ### EMF __> MMF (NI)

- ### Current __> Flux (\$\Phi\$)
  ### Current Density __> Flux Density (B)

- ### Resistance __> Reluctance

# Review

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

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

## Faraday's Law
## \$V_{coil} = -N\frac{d \Phi}{dt} = -\frac{d \lambda}{dt}\$

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# What is an Inductor?
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![](http://www.coilws.com/images/Inductors%20Group.jpg)

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# Relation between Magnetic and Electric Circuits
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## \$ V\_L = N \dfrac{d \Phi}{dt}= N \dfrac{d \Phi}{di} \dfrac{di}{dt}\$
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## \$ V\_L =L \dfrac{di}{dt}\$

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# Inductance

## Inductance is the flux linkage created per ampere

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

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# Inductance

## If the inductance of an inductor with N turns is L. 
## What happens to inductance if number of turns is doubled?
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### In linear systems (e.g. air-cored inductor):

## \$ L =   \dfrac{d \lambda}{d I} =N \dfrac{\Phi}{I}\$
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## \$ \Phi =   \dfrac{ N I}{\mathcal{R}} \$
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## \$ L =  N \dfrac{\Phi}{I} = N \dfrac{N I }{I \mathcal{R}} =  \dfrac{N^2 }{\mathcal{R}}\$
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# Inductance

## \$ L =   \dfrac{N^2}{\mathcal{R}} = \dfrac{N^2 \mu A }{l}\$

## Inductance increases with the square of number of turns \$N^2\$!

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## Try it: Double the turns

#### Try it yourself: [keysan.me/presentations/interactive/magnetic_circuit_explorer.html](https://keysan.me/presentations/interactive/magnetic_circuit_explorer.html)
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#Example

## Toroidal Inductor

<img src="https://5.imimg.com/data5/WK/MH/MY-36512415/toroidal-core-inductor-500x500.jpg" alt="Drawing" style="width: 500px;"/>
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# 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://ozank.gitbooks.io/ee361/content/images/mutual_flux.png" alt="Drawing" style="width: 380px;"/>
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# Mutual Inductance:

## \$ V_{2} = M \frac{dI_1}{dt} \$

### Coupling Coefficient (k): ratio between self inductances and mutual inductance

### \$ k = \dfrac{M}{\sqrt{L\_1 L\_2} } \$

#### k ≈ 1: closely coupled coils on a shared iron core (e.g. transformers)
#### k ≈ 0: air-cored coils far apart that don't link each other

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# Example

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### What is the equivalent inductance if two inductors are connected in series?
<img src="https://image.shutterstock.com/shutterstock/photos/2050849352/display_1500/stock-photo-electronic-components-to-receiving-radio-frequency-signal-on-green-printed-circuit-board-planar-2050849352.jpg" alt="Drawing" style="width: 380px;"/>
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### \$L_{eq} \ne L_1 + L_2\$ if there is mutual inductance between coils
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### Equivalent inductance can be larger or smaller depending on the mutual inductance 
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## Try it: Positive or Negative Mutual Coupling

#### Try it yourself: [keysan.me/presentations/interactive/coupled_coils_explorer.html](https://keysan.me/presentations/interactive/coupled_coils_explorer.html)

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## Why we use twisted pair cables?
<img src="https://stl.tech/wp-content/uploads/2022/05/Picture5-3.jpg" alt="Drawing" style="width: 800px;"/>

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## Why we use twisted pair cables?

### Twisting minimizes the mutual inductance to nearby noise sources: the voltages induced in neighbouring loops cancel, so the net flux linkage is close to zero.
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# Mutual Inductance:

## Stored Energy in two inductances
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## \$= \dfrac{1}{2} L\_{1}i\_1^2 + \dfrac{1}{2} L\_{2}i\_2^2 +  M i\_1 i\_2\$
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### In the matrix form

### \$W\_{mag}=\dfrac{1}{2}  \begin{bmatrix}  i\_1 & i\_2  \end{bmatrix} \begin{bmatrix}  L\_{1} & M \\\ M & L\_{2} \end{bmatrix} \begin{bmatrix}  i\_1 \\\ i\_2  \end{bmatrix}\$
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# AC Excitation
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## \$\Phi(t) = \Phi_{max} sin (\omega t)\$

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### Linear magnetic material

<img src="http://www.ibiblio.org/kuphaldt/electricCircuits/AC/02307.png" alt="Drawing" style="width: 800px;"/>
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#AC Excitation Non-linear B-H material

<img src="http://upload.wikimedia.org/wikipedia/commons/8/81/Power_Transformer_Over-Excitation.gif" alt="Drawing" style="width: 500px;"/>
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## Try it: Linear vs. Saturating Core

#### Try it yourself: [keysan.me/presentations/interactive/ac_excitation_explorer.html](https://keysan.me/presentations/interactive/ac_excitation_explorer.html)

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## Variation of inductance with current

### If the core saturates with current, inductance value (proportional with the slope of BH curve) get smaller!

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# AC Excitation
### Saturation of non-linear magnetics with AC current

<img src="http://www.ibiblio.org/kuphaldt/electricCircuits/AC/02336.png" alt="Drawing" style="width: 800px;"/>
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# Excitation Current

## \$ i\_e(t) = i\_c(t) + i\_m(t) \$

### Core-loss component \$i\_c\$: in phase with v, supplies hysteresis and eddy-current losses
### Magnetizing component \$i\_m\$: 90° behind v, creates the flux
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## Try it: Core-loss and Magnetizing Current

#### Try it yourself: [keysan.me/presentations/interactive/ac_excitation_explorer.html](https://keysan.me/presentations/interactive/ac_excitation_explorer.html)
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# Non-linear B-H with Hysteresis

#Phasors in Inductors

### In ideal inductor current lags voltage by 90 degrees

[Need a Reminder?](http://keysan.me/presentations/ee281_phasor_circuit_analysis.html)

#Excitation Current

## $$I_e(t) = I_c(t) + I_m(t)$$
### Excitation Current = Core Loss Component + Magnetizing Component

## What about the resistance of copper and leakage flux?
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## You can download this presentation from: [keysan.me/ee361](http://keysan.me/ee361)