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Advanced/Random Walks/parts/1 circuits.tex
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@ -1,10 +1,12 @@
\section{Circuits}
An \textit{electrical circuit} is a graph with a few extra properties,
called \textit{current}, \textit{voltage}, and \textit{resistance}.
called \textit{current}, \textit{voltage}, and \textit{resistance}. \par
In the definitions below, let $X$ be the set of nodes in a circuit.
\begin{itemize}[itemsep=3mm]
\item \textbf{Voltage} is a function $V(n): N \to \mathbb{R}$ that assigns a number to each node of our graph. \par
\item \textbf{Voltage} is a function $V: X \to \mathbb{R}$ that assigns a number to each node of our graph. \par
In any circuit, we pick a \say{ground} node, and define the voltage\footnotemark{} there as 0. \par
We also select a \say{source} node, and define its voltage as 1. \par
@ -15,15 +17,18 @@ called \textit{current}, \textit{voltage}, and \textit{resistance}.
\footnotetext{
In the real world, voltage is always measured \textit{between two points} on a circuit.
Voltage is defined as the \textit{difference} in electrical charge between two points.
Here, all voltages are measured with respect to our \say{ground} node.
Hence, voltage is a function of two nodes.
This detail isn't directly relevant to the problems in this handout, so you mustn't worry about it today. \par
Just remember that the electrical definitions here are a significant oversimplification of reality.
\vspace{2mm}
Note that this is different than current and resistance, which aren't functions
of two arbitrary nodes --- rather, they are functions of \textit{edges}
(i.e, two adjecent nodes).
}
\item \textbf{Current} is a function $I(e^\rightarrow): N \times N \to \mathbb{R}$ that assigns a number to each
\textit{oriented edge} $e^\rightarrow$ in our graph. An \say{oriented edge} is just an ordered pair of nodes $(n_1, n_2)$. \par
\item \textbf{Current} is a function $I: X^2 \to \mathbb{R}$ that assigns a number to each
\textit{oriented edge} in our graph. An \say{oriented edge} is just an ordered pair of nodes $(n_1, n_2)$. \par
\vspace{1mm}
@ -31,7 +36,7 @@ called \textit{current}, \textit{voltage}, and \textit{resistance}.
Naturally, $I(a, b) = -I(b, a)$.
\item \textbf{Resistance} is a function $R(e): N \times N \to \mathbb{R}^+_0$ that represents a certain edge's
\item \textbf{Resistance} is a function $R: X^2 \to \mathbb{R}^+_0$ that represents a certain edge's
resistance to the flow of current through it. \par
Resistance is a property of each \textit{link} between nodes, so order doesn't matter: $R(a, b) = R(b, a)$.
\end{itemize}
@ -50,10 +55,14 @@ the case! Any circuit obeys \textit{Ohm's law}, stated below:
$$
V(a, b) = I(a,b) \times R(a,b)
$$
\note{
$V(a, b)$ is the voltage between nodes $a$ and $b$. If this doesn't make sense, read the footnote below. \\
In this handout, it will be convenient to write $V(a, b)$ as $V(a) - V(b)$.
}
This handout uses two notations for voltage: two-variable $V(a, b)$ and one-variable $V(a)$. \par
The first represents the voltage between points $a$ and $b$, better reflecting reality (see the footnote below).
The second measures the voltage between $a$ and ground, and is more convenient to use in equations.
\textbf{Try to use the single-variable notation in your equations.}
Convince yourself that $V(a, b) = V(a) - V(b)$.
\vfill
\definition{Kirchoff's law}
@ -64,25 +73,46 @@ Formally, we can state this as follows:
\vspace{2mm}
Let $x$ be a node in our circuit and $B_x$ the set of its neighbors. We than have
Let $x$ be a node in our circuit and $N_x$ the set of its neighbors. We than have
$$
\sum_{b \in B_x} I(x, b) = 0
\sum_{b \in N_x} I(x, b) = 0
$$
which must hold at every node \textbf{except the source and ground vertices.} \par
\hint{Keep this exception in mind, it is used in a few problems later on.}
\vfill
\pagebreak
\begin{instructornote}
Be aware that some students may not be comfortable with these concepts from physics,
nor with the circuit notation on the next page.
\vspace{2mm}
It may be a good idea to give the class a quick lecture on this topic,
explaining the basics of electonic circuits and circuit diagrams.
\vspace{2mm}
Things to cover:
\begin{itemize}
\item All the definitions on the previous page, in detail.
\item What's an Ohm, an Amp, a Volt?
\item Measuring voltage. Why is $V(a, b) = V(a) - V(b)$?
\item What does the $\Omega$ in the picture below mean?
\item Circuit symbols in the diagram below.
\end{itemize}
\vspace{2mm}
You could also draw connections to the graph flow handout,
if the class covered it before.
\end{instructornote}
Consider the circuit below. This the graph from \ref{firstgraph}, turned into a circuit by:
Consider the circuit below. \textbf{This the graph from \ref{firstgraph}}, turned into a circuit by:
\begin{itemize}
\item Replacing all edges with $1\Omega$ resistors
\item Attaching a 1 volt battery between $A$ and $B$
@ -107,7 +137,8 @@ It exists only to create a potential difference between the two nodes.
\problem{}<onecurrents>
From the circuit diagram above, we immediatly know that $V(A) = 1$ and $V(B) = 0$. \par
What equations related to the currents out of $x$ and $y$ does Kirchoff's law give us?
What equations related to the currents out of $x$ and $y$ does Kirchoff's law give us? \par
\hint{Current into $x$ = current out of $x$}
\vfill