Appendix 2: Ideal Circuit Elements
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The elementary circuit elements—the resistor, capacitor, and inductor— impose linear relationships between voltage and current.
Resistor
The resistor is far and away the simplest circuit element. In a resistor, the voltage is proportional to the current, with the constant of proportionality RR, known as the resistance. \[v(t) = R i(t)\]
Resistance has units of ohms, denoted by \(\mathrm{\Omega}\), named for the German electrical scientist Georg Ohm. Sometimes, the \(v\text{-} i\) relation for the resistor is written \(i = Gv\), with \(G\), the conductance, equal to \(\frac{1}{R}\). Conductance has units of Siemens \((\mathrm{S})\), and is named for the German electronics industrialist Werner von Siemens.
When resistance is positive, as it is in most cases, a resistor consumes power. A resistor's instantaneous power consumption can be written one of two ways. \[p(t) = Ri^{2} (t) = \frac{1}{R} v^{2} (t)\]
As the resistance approaches infinity, we have what is known as an open circuit: no current flows, but a non-zero voltage can appear across the open circuit. As the resistance becomes zero, the voltage goes to zero for a non-zero current flow. This situation corresponds to a short circuit. A superconductor physically realizes a short circuit.
Capacitor
The capacitor stores charge and the relationship between the charge stored and the resultant voltage is \(q = Cv\). The constant of proportionality, the capacitance, has units of farads \((\mathrm{F})\), and is named for the English experimental physicist Michael Faraday. As current is the rate of change of charge, the \(v \text{-} i\) relation can be expressed in differential or integral form. \[i(t) = C \frac{\text{d} v(t)}{\text{d}t}\] \[v(t) = \frac{1}{C} \int\limits_{-\infty}^{t} i (\alpha) \ \text{d} \alpha\] If the voltage across a capacitor is constant, then the current flowing into it equals zero. In this situation, the capacitor is equivalent to an open circuit. The power consumed/produced by a voltage applied to a capacitor depends on the product of the voltage and its derivative: \[p(t) = C v(t) \frac{\text{d} v(t)}{\text{d}t}\]
This result means that a capacitor's total energy expenditure up to time \(t\) is concisely given by \[E(t) = \frac{1}{2} C v^{2} (t)\]
This expression presumes the fundamental assumption of circuit theory: all voltages and currents in any circuit were zero in the far distant past \((t = -\infty)\).
Inductor
The inductor stores magnetic flux, with larger valued inductors capable of storing more flux. Inductance has units of henries \((\mathrm{H})\), and is named for the American physicist Joseph Henry. The differential and integral forms of the inductor's \(v \text{-} i\) relation are \[v(t) = L \frac{\text{d} i(t)}{\text{d}t}\] \[i(t) = \frac{1}{L} \int\limits_{-\infty}^{t} v (\alpha) \ \text{d} \alpha\] The power consumed/produced by an inductor depends on the product of the inductor current and its derivative: \[p(t) = L i(t) \frac{\text{d} i(t)}{\text{d}t}\]
Its total energy expenditure up to time \(t\) is given by \[E(t) = \frac{1}{2} L i^{2} (t)\]
Sources
a)
b)Sources of voltage and current are also circuit elements, but they are not linear in the strict sense of linear systems. For example, the voltage source's \(v \text{-} i\) relation is \(v = v_{s}\) regardless of what the current might be. As for the current source, \(i = -i_{s}\) regardless of the voltage. Another name for a constant-valued voltage source is a battery, and it can be purchased in any supermarket. Current sources, on the other hand, are much harder to acquire; we'll learn why later.