EE281-Electric Circuits

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

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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

## Time domain:
## \\(v(t)=V_m cos (wt + \phi)\\)
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## Phasor domain:
## \\(\mathrm{V} =V_m \angle \phi\\)
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#Phasor Domain
# Derivation: \\(j \omega\\)
# Integration:  \\(\dfrac{1}{jw}\\)
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#Phasor Circuit Analysis: Resistor

### Time Domain: \\(v = i R\\)
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### Phasor Domain: \\(\mathrm{V} = \mathrm{I} R\\)
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#Phasor Circuit Analysis: Resistor

## Same phase for voltage and current
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#Phasor Circuit Analysis: Inductor

### Time Domain: \\(v = L \dfrac{di}{dt}\\)
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### Phasor Domain: \\(\mathrm{V} = j \omega L \mathrm{I}\\)
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#Phasor Circuit Analysis: Inductor

##Current lags voltage 90 degrees
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#Phasor Circuit Analysis: Capacitor

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

## Current leads voltage 90 degrees
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#Impedance
#### Impedance is the equivalent of resistance in phasor domain
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### \\(Z = \dfrac{V}{I} \quad\quad V = ZI\\)
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## Resistor: \\(R\\)
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## Inductor: \\(j \omega L\\)
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## Capacitor: \\(\dfrac{1}{j \omega C}\\)
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# 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;"/>
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## \\(Z\_{eq}=Z\_1 + Z\_2 \\)
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## \\(Z\_{eq}= R + j \omega L \\)
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## \\(Z\_{eq}= R + j X = |Z| \angle \theta \\)
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#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;"/>
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#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;"/>
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#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;"/>
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#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;"/>
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#Example:
## Determine voltage \\(v_o (t)\\)
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/impedance_ex5.png" alt="Drawing" style="width: 800px;"/>
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#Example:
## Determine voltage \\(v_o (t)\\)
<img src="https://raw.githubusercontent.com/ozank/ee281/master/images/impedance_ex6.png" alt="Drawing" style="width: 800px;"/>
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# Any questions?

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