EE361-Electromechanical Energy Conversion

class: center, middle

# EE-361 ELECTROMECHANICAL ENERGY CONVERSION

## Ozan Keysan

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

## \\(\phi\\): Phase (radians)

## \\(\phi > 0\\): the signal leads (reaches its peak earlier)

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# Can you answer the following?
--

## \\( cos (\omega t - \pi)\\) = 
--
\\( - cos (\omega t) \\)
--

## \\( cos (\omega t - \pi /2)\\) = 
--
\\( 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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# Where does a phasor come from?
## Euler: \\(e^{j\theta} = cos\theta + j\,sin\theta\\)
--

## \\(V\_m cos(\omega t + \phi) = Re\\{ V\_m e^{j\phi} \, e^{j\omega t}\\}\\)
--

## \\(e^{j\omega t}\\): the same rotation for every signal in the circuit, so drop it
## \\(\mathbf{V} = V\_m \angle \phi\\): only the amplitude and phase remain

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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}\\)
--

### Amplitude: peak value in circuit theory, RMS value in power engineering (this course)

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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!
--

## Differentiation in phasor domain?
--

## \\(i(t)= sin (\omega t + \phi)\\)
--

## \\(\dfrac{d i(t)}{dt}=?\\)
--

## 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\\)
--

## 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 (325 V peak)
--

## \\( 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}\\)

--
## In Plain English:

## The RMS value of an AC signal is equal to the value of direct current (DC) which would have the same power dissipation as the AC.
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# RMS of a Sinusoid
## \\(V\_{rms} = \sqrt{\dfrac{1}{T}\int\_0^T V\_m^2 cos^2(\omega t)\, dt}\\)
--

## \\(\quad\quad = \sqrt{\dfrac{V\_m^2}{2}} = \dfrac{V\_m}{\sqrt{2}}\\)
--

## 230 V\\(\_{rms}\\) → 325 V peak

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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?
--

[Edit Graph](https://www.desmos.com/calculator/in0czoq74w)
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# In Power Engineering RMS values are almost always used instead of peak values !

# We also use RMS values in phasor diagrams !

## \\(P = V I cos\theta\\) with RMS values: no ½ factor

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# Peak or RMS phasors?
<table style="width:100%; font-size:25px; border-collapse:collapse; margin-top:10px; line-height:1.35">
<tr style="border-bottom:2px solid #333"><th style="width:24%"></th><th>Circuit theory</th><th>Power engineering (EE361)</th></tr>
<tr style="border-bottom:1px solid #ccc"><td><b>Phasor magnitude</b></td><td>Peak: V\\(\_{m} \\) </td> <td>RMS:  V\\(\_{rms}\\) = V\\(\_{m} / \sqrt{2}\\)</td></tr>
<tr style="border-bottom:1px solid #ccc"><td><b>Average power</b></td><td>\\(P = \dfrac{1}{2} V\_{m} I\_{m} cos\theta \quad \\)</td><td>\\(\quad P = V I cos\theta\\)</td></tr>
</table>
--

###

### Same circuit, same angles: only the scale changes by \\(\sqrt{2}\\)
### In this course every V, I and phasor is RMS unless stated otherwise!

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## Peak vs RMS
<iframe src="interactive/rms_peak_explorer.html?embed" width="800" height="600" style="border:0"></iframe>

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# Common Peak/RMS mistakes
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### \\(P = V\_{peak}^2 / R\\) → 2× too high
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### Mixing a peak voltage with an RMS current → \\(\sqrt{2}\\) error
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### Using \\(\sqrt{2}\\) for non-sinusoidal waveforms (square, triangle, PWM)
--

### Forgetting that multimeters, nameplates and the "230 V" grid are all RMS

#### Practice at home (with a short quiz): [keysan.me/presentations/interactive/rms_peak_explorer.html](https://keysan.me/presentations/interactive/rms_peak_explorer.html)

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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?

[Edit Graph](https://www.desmos.com/calculator/wr3iw7kdiv)

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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?

## Real power only dissipates in resistive components.
## Ideal L and C store energy for a quarter cycle and return it in the next: the average power is zero. Reactive power describes this back-and-forth energy exchange.

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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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## Explore: Phasors, Waveforms & Power
<iframe src="interactive/ac_power_explorer.html?embed" width="790" height="530" style="border:0"></iframe>
#### Try it yourself: [keysan.me/presentations/interactive/ac_power_explorer.html](https://keysan.me/presentations/interactive/ac_power_explorer.html)

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

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

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# Complex Power
## \\(S = V\_{rms} I\_{rms}^\* = P + jQ\\)
--

## \\(P = V I cos\theta \quad\\) (W)
## \\(Q = V I sin\theta \quad\\) (var), positive for inductive loads
## \\(|S| = V I = \sqrt{P^2 + Q^2} \quad\\) (VA)

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# Example
### A 230 V (rms), 50 Hz supply feeds a series load: R = 8 Ω, L = 19.1 mH. Find: \\(Z, I, \theta\\), PF, P, Q, S
--

### \\(X\_L = 2\pi 50 \times 0.0191 = 6\,\Omega\\)
### \\(Z = 8 + j6 = 10 \angle 36.9° \,\Omega\\)
--

### \\(I = 230/10 = 23\\) A (rms), lagging by 36.9°
--

### PF = cos(36.9°) = 0.8 lagging
--

### P = \\(I^2R\\) = 4.23 kW, Q = \\(I^2X\\) = 3.17 kvar, S = 5.29 kVA
#### Check it on the explorer slide: the RL preset is this load

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# Why do we care about power factor?
## A 10 kW load on a 230 V supply:
## PF = 1.0 → I = 43.5 A, PF = 0.8 → I = 54.3 A
--

### 25% more current → \\(1.25^2\\) = 56% more cable loss (\\(I^2R\\)), plus larger cables and transformers
--

### That's why industrial consumers pay penalties for reactive energy
--

### Solution: a parallel capacitor supplies Q locally. For the previous example, about 191 µF brings PF to 1.

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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)