EE281-Electric Circuits

class: center, middle

# EE-281
# Sinusoids & Phasors

## Ozan Keysan

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

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

---
# Where's the Energy Coming From?
--

## Generation --> Transmission --> Distribution

<img src="https://raw.githubusercontent.com/ozank/ee361/master/images/electric_grid.png" alt="Drawing" style="width: 800px;"/>
---
<img src="https://raw.githubusercontent.com/ozank/ee361/master/images/turkiye_elektrik_iletim.png" alt="Drawing" style="width: 800px;"/>
---
# Electric System of Turkey
--

## Capacity: 61.2 GW
--

## Max. Consumption: 40 GW (~20 million kettles)
--

## Per Capita Consumption: 2850 kWh/year
--
(1/5th of USA)

---
# Why do we need AC?
--

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

---
# AC Transmission Lines
<img src="http://i.huffpost.com/gen/1682046/thumbs/o-HIGH-VOLTAGE-TRANSMISSION-LINE-facebook.jpg" alt="Drawing" style="width: 800px;"/>

---
#Transformers

A three-phase 225 MVA, 275 kV power transformer, which weights 241 tonnes (Courtesy of Scottish Hydro)
---
# Direct-Current Grid
### - The voltage cannot increased (without power electronics).
### - Low-voltage transmission is inefficient
### - Transformers don't work with DC
--

# Alternating-Current Grid
### - Transformers can be used to step-up voltage
### - Thinner/Cheaper transmission cables
### - Energy can be transferred to long distances
---
# Not only in Power Engineering

## Sinusoids 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/))

[homer](https://www.youtube.com/watch?v=QVuU2YCwHjw&feature=youtu.be&t=1m)

---
# Sinusoid

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

## \\( V(t) = V\_m sin (\omega t) \\)

#### or

## \\( V(t) = V\_m cos (\omega t) \\)
---
# \\( V(t) = V\_m sin (\omega t) \\)

- ## \\( V\_m \\) : Amplitude
- ## \\( \omega\\) : Angular velocity (radians/second)
- ## \\( t \\) : Time (seconds)
---
# \\( V(t) = V\_m sin (\omega t) \\)

## \\(f\\): Frequency in Hertz (Hz)
--

## \\(\omega = 2 \pi f\\): Frequency radians/s
--

## \\(T\\): Period (seconds)
## \\(T = \dfrac{1}{f} = \dfrac{2\pi}{\omega}\\)
---
# Phase Difference: \\( V(t) = V\_m sin (\omega t + \phi) \\)
--

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

<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/phase_difference.png" alt="Drawing" style="width: 500px;"/>
---
# Phase Difference

### \\(sin(\omega t \pm \pi) = \\)
--
\\( - sin(\omega t)\\)
--

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

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

### \\(cos(\omega t \pm \pi/2) = \\)
--
\\( \mp sin(\omega t)\\)
---
# 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;"/>
---
#Sinusoid Operations
## \\(3 cos (\omega t) - 4 sin (\omega t) = ?\\)
--

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

### [Graphical representation](https://docs.google.com/spreadsheets/d/1Kdm0kixL6wKhGP5Zfe-r_ukD5cu7suq9UgxDRV96atM/edit?usp=sharing)
---
# Vector Representation
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/sinusoid_phasor.png" alt="Drawing" style="width: 600px;"/>

### [Graphical representation](https://docs.google.com/spreadsheets/d/1Kdm0kixL6wKhGP5Zfe-r_ukD5cu7suq9UgxDRV96atM/edit?usp=sharing)

---
#Sinusoid Operations
## \\(3 cos (\omega t) - 4 sin (\omega t) =\\)
##\\( 5 cos(\omega t + 53.1^o)\\)

### [Graphical representation](https://docs.google.com/spreadsheets/d/1Kdm0kixL6wKhGP5Zfe-r_ukD5cu7suq9UgxDRV96atM/edit?usp=sharing)
---
#Examples
## Find amplitude, phase, period an frequency:

## \\(v(t) = 12 cos (50t + \pi/3)\\)
--

## Represent AC grid voltage as sinusoid
---
#Examples
## Represent following sinusoids in vector diagram:

### \\(-10 cos (\omega t +50^o)\\)
### \\(12 sin (\omega t -10^o)\\)
---
#Examples

### Calculate:
### \\(Vcos(wt)-Vcos(wt-2\pi/3)\\)

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

## \\(z = x + j y\\)
###where \\(j = \sqrt{-1}\\)
---
# Complex Numbers
## \\(j = \sqrt{-1}\\) or better as: \\(j^2 = -1\\)

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

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

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

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

###or

##\\(z = r ( cos (\phi) + j sin (\phi)) \\)
---
# Phasor Operations

## Addition
## \\(z\_1 + z\_2 = (x_1 +x_2) + j(y_1+y_2)\\)
--

## Substraction
## \\(z\_1 - z\_2 = (x_1 - x_2) + j(y_1 - y_2)\\)

---
#Phasor Operations

## Multiplication
### \\(z\_1 z\_2 = x_1 x_2 + j x_1 y_2 + j y_1 x_2 - y_1 y_2\\)
--

## or simply
## \\(z_1z_2 = r_1 r_2 \angle (\phi_1 + \phi_2)\\)
---
# Phasors
## In exponential form:
## \\(z = x + j y = r e^ {j\phi}\\)

--
## Euler's formula:
## \\(e^{jx}=cos (x) + j sin(x)\\)
---
## Euler's identity: \\(e^{j \pi} +1 =0 \\)

#### "The most remarkable equation in mathematics", Richard Feynman

#### [more info](http://en.wikipedia.org/wiki/Euler%27s_formula), [video explanation](http://www.youtube.com/watch?v=mgNtPOgFje0), [proof](https://docs.google.com/presentation/d/1oMNjkDp-LieSGnZEwNpceG8KTIvbnus9olu3KqnM5bg/present#slide=id.i0), [proof2](http://www.integralworld.net/collins30.html), [Euler's formula](http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/), [better explained](http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/)

---
# Phasors

##\\(e^{j\phi}=cos (\phi) + j sin(\phi)\\)

##Thus:
--

## \\(cos(\phi)=Re(e^{j\phi})\\)
## \\(sin(\phi)=Im(e^{j\phi})\\)
---
# Phasors

## Time domain:
## \\(v(t)=V_m cos (wt + \phi)\\)
--

## Phasor domain:
## \\(\mathrm{V} =V_m \angle \phi\\)
---
# Phasors

## Time domain:
## \\(v(t)=V_m sin (wt + \phi)\\)
--

## Phasor domain:
## \\(\mathrm{V} =V_m \angle \phi - 90^o\\)
---
#Phasors
![](https://c.mql5.com/2/3/phasor.gif)

#### [Animation](http://www.animations.physics.unsw.edu.au/jw/phasor-addition.html),  [Animation2](http://en.wikipedia.org/wiki/Phasor)
---
#Phasor Domain
## Derivation
## \\(v(t)=V_m cos (wt + \phi)\\)
--

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

## In Phasor domain: \\(j \omega \mathrm{V}\\)
---
#Phasor Domain
## Derivation: \\(j \omega \mathrm{V}\\)
## Integration:  \\(\dfrac{\mathrm{V}}{jw}\\)
---
#Examples
## Evaluate the following:

## \\((5+j2)(-1+j4)-5\angle 60^o\\)
---
#Examples
## Transform sinusoid to phasor

## \\(i = 6 cos (50t - 40^o)\\)
## \\(v = -4 sin(30t + 50^o)\\)
---
#Examples
## Find the sinusoid

## \\(\mathrm{I} = -3 + j4\quad A\\)
## \\(\mathrm{V_1} = 8e^{-j20^o}\quad V\\)
## \\(\mathrm{V_2} = j 8e^{-j20^o}\quad V\\)
---
#Examples
#Calculate \\(i_1 + i_2\\)

## \\(i_1 = 4 cos (wt + 30^o)\\)
## \\(i_2 = 5 sin (wt - 30^o)\\)
---
#Examples
#Calculate the following equation using phasors

###\\(4i + 8 \int i dt - 3 \dfrac{di}{dt} = 50 cos (2t + 75^o)\\)
---
# Any questions?

## You can download this presentation from: [keysan.me/ee281](http://keysan.me/ee281)