5.5.3Phasor expressions of impedance

The ultimate purpose of phasors is to simplify AC circuit analysis, so this is what we will explore now. Consider the problem of defining electrical opposition to current in an AC circuit. In DC (direct-current) circuits, resistance (R) is defined by Ohm’s Law as being the ratio between voltage (V ) and current (I):

R = VI

There are some electrical components, though, which do not obey Ohm’s Law. Capacitors and inductors are two outstanding examples. The fundamental reason why these two components do not follow Ohm’s Law is because they do not dissipate energy like resistances do. Rather than dissipate energy (in the form of heat and/or light), capacitors and inductors store and release energy from and to the circuit in which they are connected. The contrast between resistors and these components is remarkably similar to the contrast between friction and inertia in mechanical systems. Whether pushing a flat-bottom box across a floor or pushing a heavy wheeled cart across a floor, work is required to get the object moving. However, the flat-bottom box will immediately stop when you stop pushing it due to the energy loss inherent to friction, while the wheeled cart will continue to coast because it has kinetic energy stored in it. When a resistor disconnected from an electrical source, both voltage and current immediately cease. Capacitors and inductors, however, may store a “charge” of energy when disconnected from a source (capacitors retaining voltage and inductors retaining current).

The relationships between voltage and current for capacitors (C) and inductors (L) are as follows:

Expressed verbally, capacitors pass electric current proportional to how quickly the voltage across them changes over time. Conversely, inductors produce a voltage drop proportional to how quickly current through them changes over time. The symmetry here is beautiful: capacitors, which store energy in an electric field that is proportional to the applied voltage, oppose changes in voltage. Inductors, which store energy in a magnetic field that is proportional to applied current, oppose changes in current. The manner in which a capacitor or an inductor reacts to changes imposed upon it is a direct consequence of the Law of Energy Conservation: since energy can neither appear from nothing nor simply vanish, an exchange of energy must take place in order to alter the amount of energy stored within a capacitor or an inductor. The rate at which a capacitor’s voltage may change is directly related to the rate at which electric charge (current) enters or exits the capacitor. The rate at which an inductor’s current may change is directly related to the amount of electromotive force (voltage) impressed across the inductor.

When either type of component is placed in an AC circuit and subjected to oscillating signals, it will pass a finite amount of alternating current. Even though the mechanism of a capacitor’s or inductor’s opposition to current (called reactance) is fundamentally di erent from that of a resistor (called resistance), just like inertia di ers in its fundamental nature from friction, it is still convenient to express the amount of electrical opposition in a common unit of measurement: the ohm (Ω). To do this, we will have to figure out a way to take the above equations and manipulate them to express each component’s behavior as a ratio of VI .

Furthermore, recall that reactance (X) is a scalar quantity, having magnitude but no direction. Impedance (Z), on the other hand, possesses both magnitude and direction (phase), which is why the imaginary operator j must appear in the impedance expressions to make them complete. The impedance o ered by pure inductors and capacitors alike are nothing more than their reactance values (X) scaled along the imaginary (j) axis (phase-shifted 90o).

These di erent representations of opposition to electrical current are shown here for components exhibiting 50 ohms, resistances (R) and reactances (X) shown as scalar quantities near the component symbols, and impedances (Z) as phasor quantities on the complex plane: