163 lines
5.7 KiB
TeX
163 lines
5.7 KiB
TeX
\section{Circuits}
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An \textit{electrical circuit} is a graph with a few extra properties,
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called \textit{current}, \textit{voltage}, and \textit{resistance}. \par
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In the definitions below, let $X$ be the set of nodes in a circuit.
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\begin{itemize}[itemsep=3mm]
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\item \textbf{Voltage} is a function $V: X \to \mathbb{R}$ that assigns a number to each node of our graph. \par
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In any circuit, we pick a \say{ground} node, and define the voltage\footnotemark{} there as 0. \par
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We also select a \say{source} node, and define its voltage as 1. \par
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\vspace{1mm}
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Intuitively, you could say we're connecting the ends of a 1-volt battery to our source and ground nodes.
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\footnotetext{
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In the real world, voltage is always measured \textit{between two points} on a circuit.
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Voltage is defined as the \textit{difference} in electrical charge between two points.
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Hence, voltage is a function of two nodes.
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\vspace{2mm}
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Note that this is different than current and resistance, which aren't functions
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of two arbitrary nodes --- rather, they are functions of \textit{edges}
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(i.e, two adjacent nodes).
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}
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\item \textbf{Current} is a function $I: X^2 \to \mathbb{R}$ that assigns a number to each
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\textit{oriented edge} in our graph. An \say{oriented edge} is just an ordered pair of nodes $(n_1, n_2)$. \par
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\vspace{1mm}
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Current through an edge $(a, b)$ is a measure of the flow of charge from $a$ to $b$. \par
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Naturally, $I(a, b) = -I(b, a)$.
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\item \textbf{Resistance} is a function $R: X^2 \to \mathbb{R}^+_0$ that represents a certain edge's
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resistance to the flow of current through it. \par
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Resistance is a property of each \textit{link} between nodes, so order doesn't matter: $R(a, b) = R(b, a)$.
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\end{itemize}
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\vspace{2mm}
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It is often convenient to compare electrical circuits to systems of pipes. Say we have a pipe from point $A$ to point $B$.
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The size of this pipe represents resistance (smaller pipe $\implies$ more resistance), the pressure between $A$ and $B$
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is voltage, and the speed water flows through it is to current.
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\definition{Ohm's law}
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With this \say{pipe} analogy in mind, you may expect that voltage, current, and resistance are related:
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if we make our pipe bigger (and change no other parameters), we'd expect to see more current. This is indeed
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the case! Any circuit obeys \textit{Ohm's law}, stated below:
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$$
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V(a, b) = I(a,b) \times R(a,b)
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$$
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This handout uses two notations for voltage: two-variable $V(a, b)$ and one-variable $V(a)$. \par
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The first represents the voltage between points $a$ and $b$, better reflecting reality (see the footnote below).
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The second measures the voltage between $a$ and ground, and is more convenient to use in equations.
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\textbf{Try to use the single-variable notation in your equations.}
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Convince yourself that $V(a, b) = V(a) - V(b)$.
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\vfill
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\definition{Kirchoff's law}
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The second axiom of electrical circuits is also fairly simple. \textit{Kirchoff's law} states that the sum of all currents connected to
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a given edge is zero. You can think of this as \say{conservation of mass}: nodes in our circuit do not create or
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destroy electrons, they simply pass them around to other nodes.\par
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Formally, we can state this as follows:
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\vspace{2mm}
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Let $x$ be a node in our circuit and $N_x$ the set of its neighbors. We than have
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$$
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\sum_{b \in N_x} I(x, b) = 0
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$$
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which must hold at every node \textbf{except the source and ground vertices.} \par
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\hint{Keep this exception in mind, it is used in a few problems later on.}
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\vfill
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\pagebreak
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\begin{instructornote}
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Be aware that some students may not be comfortable with these concepts from physics,
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nor with the circuit notation on the next page.
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\vspace{2mm}
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It may be a good idea to give the class a quick lecture on this topic,
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explaining the basics of electronic circuits and circuit diagrams.
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\vspace{2mm}
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Things to cover:
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\begin{itemize}
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\item All the definitions on the previous page, in detail.
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\item What's an Ohm, an Amp, a Volt?
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\item Measuring voltage. Why is $V(a, b) = V(a) - V(b)$?
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\item What does the $\Omega$ in the picture below mean?
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\item Circuit symbols in the diagram below.
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\end{itemize}
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\vspace{2mm}
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You could also draw connections to the graph flow handout,
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if the class covered it before.
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\end{instructornote}
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Consider the circuit below. \textbf{This the graph from \ref{firstgraph}}, turned into a circuit by:
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\begin{itemize}
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\item Replacing all edges with $1\Omega$ resistors
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\item Attaching a 1 volt battery between $A$ and $B$
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\end{itemize}
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\vspace{2mm}
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Note that the battery between $A$ and $B$ isn't really an edge.
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It exists only to create a potential difference between the two nodes.
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\begin{center}
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\begin{circuitikz}[american voltages]
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\draw
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(0,0) node[above left] {$A$ (source)}
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to[R, l=$1\Omega$, *-*] (2,0) node[above] {$x$}
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to[R, l=$1\Omega$, *-*] (4,0) node[above] {$y$}
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to[R, l=$1\Omega$, *-*] (6,0) node[above right] {$B$ (ground)}
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to[short] (6, -1) node[below] {$-$}
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to[battery,invert,l={1 Volt}] (0, -1) node[below] {$+$}
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to[short] (0, 0)
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;
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\end{circuitikz}
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\end{center}
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\problem{}<onecurrents>
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From the circuit diagram above, we immediately know that $V(A) = 1$ and $V(B) = 0$. \par
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What equations related to the currents out of $x$ and $y$ does Kirchoff's law give us? \par
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\hint{Current into $x$ = current out of $x$}
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\vfill
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\problem{}
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Use Ohm's law to turn the equations from \ref{onecurrents} into equations about voltage and resistance. \par
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Find an expression for $V(x)$ and $V(y)$ in terms of other voltages, then solve the resulting system of equations.
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Does your result look familiar?
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\begin{solution}
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\setlength{\abovedisplayskip}{0pt} % Fix spacing on top
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\begin{flalign*}
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V(x) &= \frac{V(A) - V(y)}{2} &&\\
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V(y) &= \frac{V(x) - V(B)}{2} &&
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\end{flalign*}
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\end{solution}
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\vfill
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\pagebreak
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