\chapter{Circuit theory}



Circuits do not demonstrate electrical phenomena on its most fundamental level; the most fundamental level to study (classical) electromagnetism on would be with Maxwell's equations.

That said, circuits prove extremely useful in the application of electromagnetism in devices.

There is not even physical necessity to work with circuits as opposed to a general graph, however since batteries are designed to have positive and negative charges on opposite ends (rather than a 'positive' piece and 'negative' piece; wouldn't that be inconventient), most of the time our graphs will be circuits.

\section{Circuit}

Closed circuit
Open circuit

Node (Electrically common point)
Component

Node-component, vertex-edge; it is applied graph theory!


\section{Properties}
So that circuits model electromagnetic phenomena, we give the circuit the following properties

Voltage of a path
Current on a component
Resistance of a path
Conductance of a path
Power on a component
Energy along a path

In electromagnetism, voltage is electric potential between 2 points in space by an electric field and is path independent, so we actually only really require 2 nodes for this measurement rather than a path (this will be Kirchoff's voltage law).

Resistance however is more geometric and is indeed dependent on the path; so resistance does not have the same luxury as voltage.

In circuit theory we always consider the direction of current being between adjacent nodes, so we choose the component connecting said nodes.

Though in reality electrons (negatively charged particles) are the ones making the motion to the protons (positively charged particles), this boils down to convention in circuit theory; either conventional current or electron flow. Due to this being a simplified graph analysis, it makes no difference which convention is chosen, however one must take care to be consistent. 


\section{Basic components}
We consider components as 'black boxes'; components that have a mathematical behaviour with no regard for how they are created.

Wire
Voltage source
Current source
Resistor
Antenna
Diode

\section{Circuit topologies}
Topology in this sense ma. The mathematical field called topology actually arose from graph theory; the same theory that circuit theory descends from.

Series
parallel

\subsection{Equivalent resistance}
\[R_S = \sum R_n\]
\[R_P = \frac{1}{\sum \frac{1}{R_n}}\]
\subsection{Equivalent impedance}


\section{Ohm's law}
\[ V=IR \]
\[ P=I^2 R \]
\section{Kirchoff's law}

Let $V_n$ be the voltages of a path of $n$ components forming a loop, then the following holds by the conservation of energy.
\[ \sum V_n = 0 \]

Let $I_n$ be the currents of all components of some node, then the following holds by the conservation of charge.
\[ \sum I_n = 0 \]
\subsection{Division principles}
\[ I_k = \]

\[\]

\section{Thevenin-Norton theorem}

\section{Maximum power transfer}


\section{Capacitance}
Capacitor
\[C=\frac{Q}{V}\]
\[I=C\frac{dV}{dt}\]
\subsection{Equivalent capacitance}
\subsection{Transient analysis of capacitors}

\section{Inductance}
Inductor
\[L=\frac{\Phi}{I}\]
\[V=L\frac{dI}{dt}\]
\subsection{Equivalent inductance}
\subsection{Transient analysis of inductors}



\section{Diodes}


\section{Rectifiers}


\section{Phasor}
Complex numbers arise from solving impedance by taking the Fourier transform of differential equations.
\subsection{Transfer function}
\subsection{Impedance}
















\chapter{Electronics}
Gain
Negative resistance
Propagation delay




