Circuits

Kirchhoff's Laws
Nodal Analysis
Mesh Analysis
Voltage-Current Divider
Thévenin-Norton Equivalent Circuits
Maximum Power Transfer
Operational Amplifier
Linear Amplifier
R-C-L Circuits

\usepackage{circuitikz}
\begin{document}
\begin{circuitikz}[american]
\draw (0, 0)
to[R=$R_{1}$] (2, 0);
\end{circuitikz}
\end{document}

\usepackage{circuitikz} 
\begin{document} 
\begin{circuitikz}[american, voltage shift=0.5] 
\draw (0,0) 
to[isource, l=$I_0$, v=$V_0$] (0,3) 
to[short, -*, i=$I_0$] (2,3) to[R=$R_1$, i>_=$i_1$] (2,0) -- (0,0); 
\draw (2,3) -- (4,3) to[R=$R_2$, i>_=$i_2$] (4,0) to[short, -*] (2,0); \end{circuitikz} 
\end{document}

\usepackage{circuitikz}
\begin{document}
\begin{circuitikz}[american]
\draw (0, 0)
to[battery] (0, 2)
to[ammeter] (0, 3)
to(4, 3);
\draw (2, 3)
to[switch] (2, 2)
to[bulb](2, 0);
\draw (3, 3)
to[switch] (3, 2)
to[bulb](3, 0);
\draw (4, 3)
to[switch] (4, 2)
to[bulb](4, 0);
\draw(4, 0) to(0, 0);
\draw(0, 0) 
to (-1, 0)
to [voltmeter] (-1, 2)
to (0, 2);
\end{circuitikz}
\end{document}

Passive circuit element: elements that absorb/dissapate Power
- Conventionally, we define this as when the current points into the positive terminal of voltage
- Capacitors
- Inductors
A planar circuit can be drawn on a 2d plane without any overlap
Natural response: how a circuit reacts if you remove all the independent sources

Parts of a Circuit

Ideal wire: ideal conductor with 0 resistance
Node: a point where 2 or more elements intersect
Path: from one node to another node, can't go through a node more than once
Branch: path with only 1 element
Loop: path that starts and ends in the same node
Series: when 2 elements share only 1 node exclusively
- chains count
Parallel: when 2 or more elements share the same pair of nodes

Resistance Formulas

Series >> $R_{T} = R_{1} + R_{2} + R_{3} + \dots$
Parallel >> $\frac{1}{R_{T}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + \dots$
- For 2 leads: $R_{T} = \frac{R_{1} R_{2}}{R_{1} + R_{2}}$
- Your total resistance will always be lower than your lowest resistance
- You can write $R_{1} ∣∣ R_{2} ∣∣ R_{n}$

derivation

$\begin{aligned} i_{s} & = i_{1} + i_{2} + \dots + i_{n} \\ = \frac{V}{R_{1}} + \frac{V}{R_{2}} + \dots + \frac{V}{R_{n}} \\ = V (\frac{1}{R_{1}} + \frac{1}{R_{2}} + \dots + \frac{1}{R_{n}}) \\ \frac{1}{R_{e q}} & = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \dots + \frac{1}{R_{n}} \end{aligned}$

How to Affect Resistance

$R \propto L$ length
$R \propto \frac{1}{A}$ cross-sectional area
$R \propto T$ temperature
$R \propto ρ$ resistivity of material
Resistance proportionality >> $R \propto \frac{ρ L T}{A}$