# Appendix 2: Ideal Circuit Elements

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- 89078

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

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.