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Ideal Circuit Elements

Module by: Don Johnson

Summary: This module provides examples of the elementary circuit elements; the resistor, the capacitor,and the inductor, which provide linear relationships between voltage and current.

The elementary circuit elements—the resistor, capacitor, and inductor— impose linear relationships between voltage and current.

Resistor

Resistor
resistor.png
Figure 1: Resistor. v=Ri v R i
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 R R known as the resistance. Resistance has units of ohms, denoted by ΩΩ, named for the German electrical scientist Georg Ohm. Sometimes, the v-i relation for the resistor is written i=Gv i G v , with G G, the conductance, equal to 1R 1 R . Conductance has units of Siemens (S), and is named for the German electronics industrialist Werner von Siemens.
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

Capacitor
capacitor.png
Figure 2: Capacitor. i=Cddtvt i C t v t
The capacitor stores charge; as current is the rate of change of charge, integrating the capacitor's v-i relation yields q=Cv q C v . The charge stored in a capacitor is proportional to the voltage. The constant of proportionality, the capacitance, has units of farads (F), and is named for the English experimental physicist Michael Faraday. 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 v-i relation can be expressed in integral form.
vt=1C∫-∞tiαdα v t 1 C α t i α (1)

Inductor

Inductor
inductor.png
Figure 3: Inductor. v=Lddtit v L t i t
The inductor stores magnetic flux, with larger valued inductors capable of storing more flux. Inductance has units of henries (H), and is named for the American physicist Joseph Henry. The integral form of the inductor's v-i relation is
it=1L∫-∞tvαdα i t 1 L α t v α (2)

Sources

Sources
vsource.pngisource.png
Subfigure 4.1
Subfigure 4.2
Figure 4: The voltage source on the left and current source on the right are like all circuit elements in that they have a particular relationship between the voltage and current defined for them. For the voltage source, v= v s v v s for any current ii; for the current source, i=- i s i i s for any voltage vv.
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-i relation is v= v s v v s regardless of what the current might be. As for the current source, i=- i s i i s regardless of the voltage. Another name for a constant-valued voltage source is a battery, and can be purchased in any supermarket. Current sources, on the other hand, are much harder to find; we'll learn why later.

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