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<!DOCTYPE html>
<html>
<head>
<title>EE281-Electric Circuits</title>
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class: center, middle
# EE-281
# Circuit Analysis with Phasors
## Ozan Keysan
[ozan.keysan.me](http://ozan.keysan.me)
Office: C-113 <span class="meta">•</span> Tel: 210 7586
---
#Phasors
## In polar form
##\\(z = x + j y = r \angle \phi\\)
###or
##\\(z = r ( cos (\phi) + j sin (\phi)) \\)
---
# Phasors
## Time domain:
## \\(v(t)=V_m cos (wt + \phi)\\)
--
## Phasor domain:
## \\(\mathrm{V} =V_m \angle \phi\\)
---
#Phasor Domain
# Derivation: \\(j \omega\\)
# Integration: \\(\dfrac{1}{jw}\\)
---
#Phasor Circuit Analysis: Resistor
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/phasor_resistor.png" alt="Drawing" style="width: 400px;"/>
### Time Domain: \\(v = i R\\)
--
### Phasor Domain: \\(\mathrm{V} = \mathrm{I} R\\)
---
#Phasor Circuit Analysis: Resistor
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/phasor_resistance_VI.png" alt="Drawing" style="width: 400px;"/>
## Same phase for voltage and current
---
#Phasor Circuit Analysis: Inductor
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/phasor_inductor.png" alt="Drawing" style="width: 300px;"/>
### Time Domain: \\(v = L \dfrac{di}{dt}\\)
--
### Phasor Domain: \\(\mathrm{V} = j \omega L \mathrm{I}\\)
---
#Phasor Circuit Analysis: Inductor
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/phasor_inductor_VI.png" alt="Drawing" style="width: 400px;"/>
##Current lags voltage 90 degrees
---
#Phasor Circuit Analysis: Capacitor
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/phasor_capacitor.png" alt="Drawing" style="width: 300px;"/>
### Time Domain: \\(i = C \dfrac{dv}{dt}\\)
--
### Phasor Domain: \\(\mathrm{I} = j \omega C \mathrm{V}\quad\\) or \\( \quad \mathrm{V} = \dfrac{1}{j\omega C} \mathrm{I}\\)
---
#Phasor Circuit Analysis: Capacitor
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/phasor_capacitor_VI.png" alt="Drawing" style="width: 400px;"/>
## Current leads voltage 90 degrees
---
#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}\\)
---
# Impedance
## How do you combine a resistor and inductor?
<img src="http://www.radio-electronics.com/images/inductor-with-resistance-01.gif" alt="Drawing" style="width: 400px;"/>
--
## \\(Z\_{eq}=Z\_1 + Z\_2 \\)
--
## \\(Z\_{eq}= R + j \omega L \\)
--
## \\(Z\_{eq}= R + j X = |Z| \angle \theta \\)
---
#Example:
## Determine impedance, \\(v(t)\\) and \\(i(t)\\)
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/impedance_ex1.png" alt="Drawing" style="width: 800px;"/>
---
#Example:
## Determine impedance, \\(v(t)\\) and \\(i(t)\\)
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/impadence_ex2.png" alt="Drawing" style="width: 800px;"/>
---
#Example:
## Find the input impedance, (\\(\omega = 50 rad/s\\))
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/impedance_ex3.png" alt="Drawing" style="width: 800px;"/>
---
#Example:
## Find the input impedance, (\\(\omega = 10 rad/s\\))
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/impedance_ex4.png" alt="Drawing" style="width: 800px;"/>
---
#Example:
## Determine voltage \\(v_o (t)\\)
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/impedance_ex5.png" alt="Drawing" style="width: 800px;"/>
---
#Example:
## Determine voltage \\(v_o (t)\\)
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/impedance_ex6.png" alt="Drawing" style="width: 800px;"/>
---
# Any questions?
## You can download this presentation from: [keysan.me/ee281](http://keysan.me/ee281)
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