EE361-Electromechanical Energy Conversion

class: center, middle

# EE-361 ELECTROMECHANICAL ENERGY CONVERSION

## Ozan Keysan

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

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

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# Why do we need AC?
<img src="https://www.thetechoutlook.com/wp-content/uploads/2021/03/Elon-Musk-tweets-about-it-would-be-better-if-Nikola-tesla-and-Thomas-Edison-were-together.jpg" alt="Drawing" style="width: 700px;"/>

## [War of Currents](http://en.wikipedia.org/wiki/War_of_Currents)

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# Alternating-Current Grid

## - Transformers can be used to step-up voltage

## - Thinner/Cheaper transmission cables

## - Energy can be transferred to long distances

## - More Efficient (Including the extra power loss of the transformer)

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# Why do we need AC?

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# Not only in Power Engineering

## Sinusoids exists in:

### - Nature: Waves, Motion...

### - Mechanical Engineering: Vibration, [Natural frequency](http://www.youtube.com/watch?v=j-zczJXSxnw)

### - Computer Science: File Compression, Image Processing, [Fourier Transform](http://jackschaedler.github.io/circles-sines-signals/dft_introduction.html) ([mp3](http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-face), [jpeg](http://david.li/filtering/))

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# Sinusoid

## Sinusoid: Signal that has the form of the sine or cosine function.

## \\( V(t) = V\_{max} sin (\omega t) \\)

#### or

## \\( V(t) = V\_{max} cos (\omega t) \\)
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# \\( V(t) = V\_{max} sin (\omega t) \\)

## \\( V\_{max}\\) : Amplitude
 ## \\( \omega\\) : Angular velocity (radians/second)
 ## \\( t \\) : Time (seconds)
--

## \\(f\\): Frequency in Hertz (Hz)
## \\(\omega = 2 \pi f\\): Frequency radians/s

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# Phase Difference: \\( V(t) = V\_{max} sin (\omega t + \phi) \\)
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## \\(\phi\\): Phase (radians)

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# Can you answer the following?
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## \\( cos (\omega t - \pi)\\) = 
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\\( - cos (\omega t) \\)
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## \\( cos (\omega t - \pi /2)\\) = 
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\\( sin (\omega t) \\)
--

## \\( sin (\omega t + \pi /2)\\) = 
--
\\( cos (\omega t) \\)

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# Phase Difference

<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/phase_rotate_a.png" alt="Drawing" style="width: 350px;"/><img src="https://raw.githubusercontent.com/ozank/ee281/master/images/phase_rotate_b.png" alt="Drawing" style="width: 350px;"/>
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# What about sinusoid operations?

## \\(3 cos (\omega t) + 4 sin (\omega t) = ?\\)
--

## \\(A cos (\omega t) + B sin (\omega t + \phi) = ?\\)

### [Graphical representation](https://www.desmos.com/calculator/yjdj9jyqpc)

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#Phasors
## Phasor: a complex number that represents the amplitude and phase of a sinusoid
--

## \\(z = x + j y\\)
###where \\(j = \sqrt{-1}\\)
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# Complex Numbers
## \\(j = \sqrt{-1}\\) or better expressed as:  \\(j \times j = -1\\)

[Better explained: complex numbers](http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/)
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#Phasors
## In polar form

##\\(z = x + j y = r \angle \phi\\)
--

##where
##\\(r = \sqrt{x^2 + y^2}\\)
## \\(\phi = tan^{-1}(\dfrac{y}{x})\\)
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#Phasors
## In polar form

##\\(z = x + j y = r \angle \phi\\)

###or

##\\(z = r ( cos (\phi) + j sin (\phi)) \\)
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# Phasors

## Remember phasors are there to make your life EASIER!
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## Derivation in phasor domain?
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## \\(i(t)= cos (wt + \phi)\\)
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## \\(\dfrac{d i(t)}{dt}=?\\)
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## In Phasor domain: \\(j \omega \\) ( i.e rotation by 90 degrees!)
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#Phasor Circuit Analysis: Inductor

# Time Domain: \\(v = L \dfrac{di}{dt}\\)
--

# Phasor Domain: \\(\mathrm{V} = j \omega L \mathrm{I}\\)
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#Phasor Circuit Analysis: Inductor
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/phasor_inductor.png" alt="Drawing" style="width: 600px;"/>

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#Phasor Circuit Analysis: Inductor

## Inductor current lags inductor voltage 90 degrees
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## Phasor Circuit Analysis: Capacitor

### Time Domain: \\(i = C \dfrac{dv}{dt}\\)
--

### Phasor Domain:  \\( \quad \mathrm{V} = \dfrac{1}{j\omega C} \mathrm{I}\\)
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## Phasor Circuit Analysis: Capacitor

## Capacitor current leads capacitor voltage 90 degrees

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#Impedance
#### Impedance is the equivalent of resistance in phasor domain
--

### \\(Z = \dfrac{V}{I} \quad\quad V = ZI\\)
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## Resistor: \\(R\\)
--

## Inductor: \\(j \omega L\\)
--

## Capacitor: \\(\dfrac{1}{j \omega C}\\)
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# Impedance
## How do you combine a resistor and inductor?
--

## \\(Z\_{eq}=Z\_1 + Z\_2 \\)
--

## \\(Z\_{eq}= R + j \omega L \\)
--

## \\(Z\_{eq}= R + j X = |Z| \angle \theta \\)

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# What is the magnitude of grid voltage?

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# Low-Voltage Grid = 230 Vrms
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## \\( V(t) = V\_{rms} \,\sqrt{2} \,cos ( 2 \pi \,f \,t) \\)
## \\( V(t) = 230 \,\sqrt{2} \,cos ( 2 \pi \,50 \,t) \\)

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# What is RMS?

# RMS = \\( \sqrt{\dfrac{1}{T}\int_0^T (f(t))^2 dt}\\)

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## In Plain English:

## The RMS value of an AC signal is equal to the value of direct current (DC) which would have the same effect or energy change as the AC.
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# What is RMS?

## What is the average power (in Watts) dissipated in a resistor of 1 \\(\Omega\\) while carrying a 1 A(rms) current?
--

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# What is RMS?

# In Power Engineering RMS values are almost always used instead of peak values !

# We also use RMS values in phasor diagrams !

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# Active and Reactive Power
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### What is the average power (in Watts) dissipated in an inductor with an impedance of   \\(j 1 \, \Omega\\) while carrying a 1 A(rms) current?

[Edit Graph](https://www.desmos.com/calculator/in0czoq74w)

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# Active and Reactive Power

### What is the average power (in Watts) dissipated in an inductor with an impedance of   \\(j 1 \, \Omega\\) while carrying a 1 A(rms) current?

## Remember: Real power only dissipates in resistive components. Ideal inductors or capacitors don't dissipate real power, but uses reactive power.

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# Active and Reactive Power

## What is \\(V\_{rms} \times I\_{rms} \,\\)?
--

## \\(V\_{rms} \times I\_{rms} \neq P \\)
--

## \\(S = V\_{rms} \times I\_{rms} \\) (in VA), Apparent Power
--

## Active Power (P) in Watts equal to

## \\(P= V\_{rms}  I\_{rms} cos(\theta) \\) (in W)
--

## \\(cos(\theta)\\): Power factor

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# Unity Power Factor

## In resistive circuits (or in compensated circuits)
## Usually the desired case for electric grid.

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# Lagging Power Factor

## In inductive (RL) circuits

<img src="https://cdn1.byjus.com/wp-content/uploads/2021/02/lagging-power-factor.png" alt="Drawing" style="width: 800px;"/>
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# Leading Power Factor

## In Capacitive (RC) circuits

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# \\(S = \sqrt{P^2 + Q^2}\\)

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# \\(S = \sqrt{P^2 + Q^2}\\)

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## Important AC Power Equations

#### [EE Portal](https://electrical-engineering-portal.com/complex-power-analysis)

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# Any questions?

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