EE281-Electric Circuits

class: center, middle

# EE-281 ELECTRIC CIRCUITS

## Ozan Keysan

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

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

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# Review

## Ohm's Law (Resistivity, Conductivity)

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## Kirchhoff's Current Law

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## Kirchhoff's Voltage Law

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## Node, Branch, Loop, Mesh

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# Analysis Methods

## Scalability?

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# Analysis Methods

## 1-Nodal Analysis
### Unknown: Node Voltages
### Uses Kirchhoff's Current Law

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## 2-Mesh Analysis
### Unknown: Mesh Currents
### Uses Kirchhoff's Voltage Law

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# Steps for Nodal Analysis

## 1-Choose a reference node (ground).
## 2-Label node voltages.
## 3-Write KCL equations for each node.
## 4-Solve the equations.
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# Choosing Reference Node

- ### Node that interconnects the most branches.
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- ### Usually at the bottom of circuit.
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- ### Label ground with one of the symbols below:

![](http://www.clipartbest.com/cliparts/9iR/xn5/9iRxn56ie.png)
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# Label each node voltage:
## Except the ground node!

![](https://www.rose-hulman.edu/cleo/video/streamer.php?id=240&action=attr&attr=statement_diagram
)

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# Write KCL for each node

## Incoming Currents + Outgoing Currents = 0
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## The most practical way:
### Use \$-\$ sign for incoming currents.
### Use \$+\$ sign for outgoing currents.

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# Example-1:
## Calculate node voltages:

![](https://raw.githubusercontent.com/ozank/ee281/master/images/node_example1.png)
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# Example-1: Label nodes

![](https://raw.githubusercontent.com/ozank/ee281/master/images/node_example1b.png)

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# Example-1:
### Answer: \$V_1 = 40/3 V \quad V_2 = 20 V \$

![](https://raw.githubusercontent.com/ozank/ee281/master/images/node_example1.png)
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# Example-2:
![](https://raw.githubusercontent.com/ozank/ee281/master/images/node_example2.png)
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# Example-2:
### Answer: \$V_1 = -6 V \quad V_2 = -42 V \$
![](https://raw.githubusercontent.com/ozank/ee281/master/images/node_example2.png)
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# Dependent Sources
## Dependent Voltage Source
![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Voltage_Source_%28Controlled%29.svg/200px-Voltage_Source_%28Controlled%29.svg.png)

## Shown with a square sign
## Its value is dependent on another variable
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# Dependent Sources
## Dependent Current Source
![](https://www.digilentinc.com/classroom/Electronics101/assets/KT-DepSrcs-Fig1b.png)

## Shown with a square sign
## Its value is dependent on another variable
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# Dependent Sources
## Dependent Current Source
![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Voltage_controlled_current_source.svg/500px-Voltage_controlled_current_source.svg.png)

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# Example-3:
![](https://raw.githubusercontent.com/ozank/ee281/master/images/node_example3.png)
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# Example-3:

### Answer: \$V_1 = 4.8 V \quad V_2 = 2.4 V \quad V_2 = -2.4 V \$

![](https://raw.githubusercontent.com/ozank/ee281/master/images/node_example3.png)

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#Example:
## Obtain resistance matrix for:
![](https://raw.githubusercontent.com/ozank/ee281/master/images/node1.png)

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#Example:
## Obtain resistance matrix for:
![](https://raw.githubusercontent.com/ozank/ee281/master/images/node2.png)

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# Example:
## Resistance Matrix:
$$
\begin{bmatrix}
\frac{1}{R_1}+\frac{1}{R_2} & -\frac{1}{R_2} & 0 \\\
-\frac{1}{R_2} & \frac{1}{R_2}+ \frac{1}{R_3}+\frac{1}{R_4} & -\frac{1}{R_4} \\\
0 & -\frac{1}{R_4}  & \frac{1}{R_4}\\\
\end{bmatrix}
\begin{bmatrix}
V_a \\\
V_b \\\
V_c \\\
\end{bmatrix}
=
\begin{bmatrix}
I_1 \\\
0 \\\
-I_2 \\\
\end{bmatrix}
$$
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# Example:
## Remember!

## Conductance = 1/ Resistance

## $$G = \frac{1}{R}$$
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#Conductance Matrix:
## (Sometimes it is more convenient)

$$\begin{bmatrix} G_1+G_2 & -G_2 & 0 \\\ -G_2 & G_2+ G_3+G_4 & -G_4 \\\
0 & -G_4  & G_4 \end{bmatrix}
\begin{bmatrix}
V_a \\\
V_b \\\
V_c
\end{bmatrix}
=
\begin{bmatrix}
I_1 \\\
0 \\\
-I_2
\end{bmatrix}
$$

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# No Escape from Linear Algebra!

# $$Ax=b $$

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# $$x=A^{-1}b $$
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# Any questions?

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