EE362-Electromechanical Energy Conversion-II

class: center, middle

# EE-362 ELECTROMECHANICAL ENERGY CONVERSION-II

# Torque in Induction Motors

## Ozan Keysan

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

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

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# Mechanical Power

### Linear Motion

## Power(W) = Force (N) x Speed (m/s)
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### Rotational Motion
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## Power(W) = Torque (Nm) x Rotational Speed (rad/s)

## \\(P = T \omega \\) \\( \quad \rightarrow T = \dfrac{P}{\omega} \\)

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### Can you guess a few applications that require high start-up torque?
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## Electric Cars: 
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[BMW i3](https://www.bmw.com.tr/tr/all-models/bmw-i/i3/2017/bmwi3-genel-bakis.html)

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## Start-up Torque

### For curious students: [BMW i3 Specs](http://hybridfordonscentrum.se/wp-content/uploads/2014/05/20140404_BMW.pdf),

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## Importance of Start-up Torque: [BMW i3 vs WV Golf GTI](https://www.youtube.com/watch?v=jhm3Iv3tAcA)

For curious students: [Tesla Induction Motor Info](https://www.technologyreview.com/s/410636/tesla-roadster/), [Reverse engineering a Tesla drivetrain](https://evannex.com/blogs/news/35045701-reverse-engineering-a-tesla-drivetrain), [BMW i3 Specs](http://hybridfordonscentrum.se/wp-content/uploads/2014/05/20140404_BMW.pdf),

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# Torque-Power Relation
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## \\( P\_{mech} = 3 I\_{2}'^2 \dfrac{(1-s)}{s}r'\_2 \\)
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## \\( T = \dfrac{P}{\omega} \\)
## What is \\(\omega\\) of the rotor?
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## \\(\omega_r\\)
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\\(= (1-s)\omega_s\\)

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# Torque-Power Relation

### \\( P\_{mech} = T (1-s) \omega\_s = 3 I\_{2}'^2 \dfrac{(1-s)}{s}r'\_2 \\)
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### \\( T \omega\_s = 3 I\_{2}'^2 \dfrac{r'\_2}{s} \\)
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# Generated Torque

## \\( T = 3 I\_{2}'^2 \dfrac{r'\_2}{s} \dfrac{1}{\omega_s} \\)
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, which is equal to:

## \\( T = \dfrac{P\_{ag}}{\omega\_s} \quad \\)
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 or \\( \quad T = \dfrac{P\_{mech}}{\omega\_r} \\)
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###\\(\omega_s\\) is the mechanical synchronous speed! \\(\quad \omega_s = \dfrac{2 \pi f_e}{(p/2)}\\)

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# Generated Torque
### \\( T = 3 I\_{2}'^2 \dfrac{r'\_2}{s} \dfrac{1}{\omega_s} \quad \\) We know:
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- ###  \\(\omega_s\\), if we know \\(f_e\\) and number of poles
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- ###  \\(s\\): if we know rotor speed
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- ###  \\(r'\_2\\) from locked-rotor test
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## How can we calculate \\(I'\_2\\)?

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## How can we calculate \\(I'\_2\\)?
<img src="http://myelectrical.com/Portals/0/SunBlogNuke/2/WindowsLiveWriter/InductionMotorEquivalentCircuit_D7EF/Induction%20Motor%20Equivalent%20Circuit%20-%20Simplified_thumb.png" alt="Drawing" style="width: 400px;"/>

- ## Inaccurate but easy: Move parallel branch to source side
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- ## More accurate: Calculate Thevenin equivalent as seen from the rotor side

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# Thevenin Equivalent Circuit
<img src="https://www.researchgate.net/publication/338011903/figure/fig2/AS:1151985076191266@1651666053906/Equivalent-Thevenin-circuit-of-AC-motor.png" alt="Drawing" style="width: 600px;"/>
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### \\(V\_{th} = \dfrac{(R\_c // jX\_m)}{R\_1 + jX\_1 + (R\_c // jX\_m)}V\_1\\)
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### \\(Z\_{th} = (R\_1 + jX\_1) // (R\_c // jX\_m)\\)

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#Torque

### \\(T\_e = \dfrac{3 V\_{th}^2}{(R\_{th}+\dfrac{r'\_2}{s})^2 + (X\_{th}+X'\_2)^2}\dfrac{r'\_2}{s \omega\_s}\\)
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## If you're in a hurry, move the parallel branch to motor terminals and replace:
### \\(V\_{th} \rightarrow V\_1 \quad R\_{th} \rightarrow R\_1 \quad X\_{th} \rightarrow X\_1\\)
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#Torque

### \\(T\_e = \dfrac{3 V\_{1}^2}{(R\_{1}+\dfrac{r'\_2}{s})^2 + (X\_{1}+X'\_2)^2}\dfrac{r'\_2}{s \omega\_s}\\)

### \\(V\_{th} \rightarrow V\_1 \quad R\_{th} \rightarrow R\_1 \quad X\_{th} \rightarrow X\_1\\)
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# Torque Characteristics
## Can you guess the waveform wrt rotor speed?

### \\(T\_e = \dfrac{3 V\_{th}^2}{(R\_{th}+\dfrac{r'\_2}{s})^2 + (X\_{th}+X'\_2)^2}\dfrac{r'\_2}{s \omega\_s}\\)

## [Torque Graphs](https://docs.google.com/spreadsheets/d/1YVq94hV64z6VSiN8q-v7XydcfLR3xLdcp-5GhdYZg6Y/edit?usp=sharing)

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#Typical Torque Curve of an Induction Motor

<!--
#Typical Current Curve of an Induction Motor

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# Torque characteristics

## For small values of slip:
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 Torque is proportional to slip
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## For large values of slip: 
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 Torque is inversely proportional to slip
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## Rated slip is usually smaller than 0.05

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# Start-up Torque

## Substitute s=1 in the torque equation
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### \\(T\_{start} = \dfrac{3 V\_{th}^2}{(R\_{th}+r'\_2)^2 + (X\_{th}+X'\_2)^2}\dfrac{r'\_2}{\omega\_s}\\)

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## Round #2
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### [Tesla Model S P85D  vs Lamborghini LP570-4 Super Trofeo](https://youtu.be/r4CnSS4OG4A?t=34s)
<img src="http://www.dsf.my/wp-content/uploads/2014/03/Lamborghini-BlancpainSuper-Trofeo-Race-Series8.jpg" alt="Drawing" style="width: 500px;"/>

#### [Tesla P100D vs Lamborghini Hurcan](https://youtu.be/5Rj4Av2ddsY?t=1m3s)

#### More info: [Tesla ModelS P100D 0-100 km in 2.4 seconds](http://www.roadandtrack.com/new-cars/videos/a32342/the-tesla-model-s-p100d-ludicrous-plus-can-do-0-60-in-239/), [Tesla Induction Motor Info](https://chargedevs.com/newswire/elon-musk-cooling-not-power-to-weight-ratio-is-the-challenge-with-ac-induction-motors/)

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# Maximum Torque Point

### \\(T\_e = \dfrac{3 V\_{th}^2}{(R\_{th}+\dfrac{r'\_2}{s})^2 + (X\_{th}+X'\_2)^2}\dfrac{r'\_2}{s \omega\_s}\\)

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# Maximum Torque Point

## \\( T\_e = \dfrac{P\_{ag}}{\omega\_s} \quad \\)

### Maximum torque point = Maximum airgap power point
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# Maximum Torque Point

## What is the condition for maximum airgap Power?
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### Maximum Power Transfer Theorem: \\(\dfrac{r'\_2}{s}= \sqrt{R\_{th}^2+ (X\_{th}+X'\_2)^2}\\)
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## Slip for maximum torque

### \\(s\_{maxT}= \dfrac{r'\_2}{\sqrt{R\_{th}^2+ (X\_{th}+X'\_2)^2}}\\)
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## Maximum Torque (substitute s)

### \\(T\_{max} = 3 \dfrac{ 0.5 V\_{th}^2}{\omega\_s}\dfrac{1}{(R\_{th}+\sqrt{R\_{th}^2 + (X\_{th}+X'\_2)^2}}\\)

### Notice that \\(s\_{maxT}\\) depends on \\(r'\_2\\) but \\(T\_{max}\\) doesn't.

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## You can download this presentation from: [keysan.me/ee362](http://keysan.me/ee362)