Alternating Current and Electric Power

Electrical energy powers nearly every aspect of modern life—from lighting and communication systems to medical devices such as ventilators, electrocardiographs (ECGs), and medical imaging equipment. While previous chapters focused on steady electric currents, most electrical systems encountered in everyday life operate using alternating current (AC), in which both voltage and current vary continuously with time.

Understanding AC is especially important in the health sciences. Many physiological signals—including those measured by electrocardiograms (ECGs) and electroencephalograms (EEGs)—are time-varying electrical signals. In addition, hospitals rely on AC power to operate medical equipment safely and efficiently. The concepts introduced in this chapter therefore connect directly to both biological systems and the technologies used to diagnose and treat patients.

Alternating Current

Up to this point, most examples—particularly those involving batteries—have used constant voltage sources. Once current is established in these circuits, it flows steadily in a single direction. This type of current is called direct current (DC).

Many real-world electrical systems, however—including electrical power grids, medical equipment, and laboratory instruments—operate using time-varying voltage sources. These sources produce alternating current (AC), in which the direction of charge flow reverses periodically.

When the voltage varies sinusoidally with time, the circuit is called an AC circuit. This is the form of electricity used in homes, hospitals, schools, and research facilities throughout most of the world. As shown in Figure 23.1, direct current produces constant voltage and current, whereas alternating current produces voltage and current that oscillate sinusoidally.

Household electricity typically operates at either 50 Hz or 60 Hz, depending on the country. A frequency of 60 Hz means the current reverses direction 60 times every second. These rapid oscillations make it possible to generate, transform, and transmit electrical energy efficiently over long distances.

AC Voltage and Current

The voltage produced by an alternating-current (AC) source changes continuously with time. For the electrical power supplied to homes and hospitals, this variation is very nearly sinusoidal. The instantaneous voltage is described by

[latex]V=V_0\sin(2\pi ft)[/latex]

where [latex]V[/latex] is the instantaneous voltage at time [latex]t[/latex], [latex]V_0[/latex] is the peak voltage, and [latex]f[/latex] is the frequency of the source measured in hertz (Hz). The voltage oscillates smoothly between [latex]+V_0[/latex] and [latex]-V_0[/latex], completing one full cycle every period [latex]T=1/f[/latex].

For a purely resistive circuit, Ohm's law applies at every instant. Since the resistance remains constant, the current also varies sinusoidally:

where the peak current is

Because both voltage and current reach their maximum, minimum, and zero values at exactly the same time, they are said to be in phase. This behavior is characteristic of purely resistive circuits and is illustrated in Figure 23.1. In later chapters, we will see that circuits containing capacitors or inductors can cause voltage and current to become out of phase.

The alternating nature of AC electricity has practical consequences. For example, lights powered directly from the electrical grid actually brighten and dim every half-cycle of the voltage. In a 60-Hz electrical system, this occurs 120 times each second. Although this flicker is normally too rapid for the human eye to perceive, it can create a stroboscopic effect when rapidly moving objects are illuminated. Similar considerations are important in medical imaging, high-speed photography, and motion-analysis systems.

Power in AC Circuits

In both DC and AC circuits, electrical power describes the rate at which electrical energy is transferred or converted into other forms of energy. For a resistor, this energy is usually converted into thermal energy, although in devices such as lamps, part of the energy is also emitted as light. The instantaneous power delivered to any electrical device is the product of the instantaneous voltage across it and the instantaneous current flowing through it:

[latex]P=IV[/latex]

For a purely resistive AC circuit, both the voltage and the current vary sinusoidally and remain in phase. Their instantaneous values are

[latex]V=V_0\sin(2\pi ft)[/latex]

[latex]I=I_0\sin(2\pi ft)[/latex]

Substituting these expressions into the power equation gives

[latex]P=I_0V_0\sin^2(2\pi ft)[/latex]

Unlike the voltage and current, which alternate between positive and negative values, the power delivered to a resistor is never negative. During the positive half-cycle, both the voltage and the current are positive, so their product is positive. During the negative half-cycle, both quantities become negative because the current reverses direction together with the voltage. Multiplying two negative values again produces a positive result. Consequently, a resistor continuously absorbs energy from the source regardless of the direction of current flow.

Notice that the power varies as the square of a sine function rather than as a sine function itself. The quantity [latex]\sin^2(2\pi ft)[/latex] oscillates between 0 and 1, and its average value over one complete cycle is exactly one-half. Therefore, the average power supplied by an AC source is

[latex]P_{\text{ave}}=\frac{1}{2}I_0V_0[/latex]

Average power is usually the quantity of greatest practical interest because it determines the total electrical energy consumed. A light bulb labeled "60 W," for example, dissipates an average power of 60 W even though its instantaneous power continually increases and decreases many times each second.

Root Mean Square (RMS) Values

One challenge with alternating current is that both the voltage and the current continually change sign. If we simply average either quantity over one complete cycle, the positive and negative halves cancel, giving an average value of zero. Clearly, however, a household electrical outlet is not equivalent to a 0-V source—it can still operate appliances and deliver significant amounts of energy.

To describe the effective voltage and current of an AC source, we use root mean square (rms) values. The rms value is defined so that it produces the same average power in a resistor as an equivalent DC source. In other words, if a resistor connected to an AC source and the same resistor connected to a DC battery produce the same amount of heating, then the DC voltage equals the rms voltage of the AC source.

For sinusoidal voltages and currents, the rms values are

[latex]V_{\text{rms}}=\frac{V_0}{\sqrt{2}}[/latex]

[latex]I_{\text{rms}}=\frac{I_0}{\sqrt{2}}[/latex]

Using these definitions, the average power can be written in the familiar form

[latex]P_{\text{ave}}=I_{\text{rms}}V_{\text{rms}}[/latex]

This equation has exactly the same appearance as the power equation for DC circuits. The difference is simply that the voltage and current are rms values rather than constant values.

Nearly all electrical equipment is rated using rms values unless otherwise stated. For example, the "120 V" supplied by a typical household outlet is actually an rms voltage. The voltage does not remain fixed at 120 V. Instead, it oscillates smoothly between approximately +170 V and −170 V because

[latex]V_0=\sqrt{2}V_{\text{rms}}\approx170~\text{V}[/latex]

Similarly, current ratings for circuit breakers and electrical wiring are rms values, and the power ratings printed on household appliances correspond to average power consumption.

For students in the health sciences, rms quantities are particularly important because they are the standard used for the design of hospital electrical systems, medical equipment, and laboratory instrumentation. Engineers use rms values to determine wire sizes, safety limits, and the power consumed by devices ranging from infusion pumps to imaging equipment.

AC Ohm's Law and Power

For a purely resistive AC circuit, Ohm's law can be written using rms values exactly as it is for a DC circuit:

[latex]I_{\text{rms}}=\frac{V_{\text{rms}}}{R}[/latex]

Likewise, the familiar power relationships become

[latex]P_{\text{ave}}=I_{\text{rms}}V_{\text{rms}}[/latex]

[latex]P_{\text{ave}}=\frac{V_{\text{rms}}^2}{R}[/latex]

[latex]P_{\text{ave}}=I_{\text{rms}}^2R[/latex]

These equations are identical in form to those used for DC circuits. This similarity is one of the main reasons rms quantities are used so extensively: they allow AC circuits to be analyzed with the same mathematical framework developed for steady currents. Whether calculating the power dissipated by a heating element, determining the current drawn by a medical device, or estimating the load on a hospital electrical circuit, engineers typically work directly with rms values.

It is important to remember that these equations apply only to purely resistive circuits, where the voltage and current remain in phase. Many practical circuits also contain capacitors and inductors, which cause the current and voltage to become out of phase. In those situations, additional concepts such as reactance and power factor are needed to fully describe the circuit's behavior.

Electrical Safety: AC and DC Hazards

Electricity is an essential part of modern medicine and everyday life, but it can also present serious hazards. Although people often associate electrical danger with high voltage, voltage alone does not determine the severity of an electric shock. The physiological effects of electricity depend on several factors, including the amount of current flowing through the body, how long the current flows, the path it takes, and whether the current is alternating (AC) or direct (DC).

For example, static electricity can involve voltages of several thousand volts, yet the resulting shocks are usually harmless because they involve extremely small amounts of charge delivered over a very short time. In contrast, the much lower voltages found in household electrical outlets can be dangerous because they are capable of driving substantial currents through the body.

The severity of an electric shock depends primarily on:

• the magnitude of the current passing through the body,
• the duration of exposure,
• the path the current follows (especially if it passes through the heart or brain), and
• the frequency of the current.

For currents of similar magnitude, alternating current at the frequencies used in electrical power systems (50–60 Hz) is generally more hazardous than direct current. One reason is that alternating current repeatedly stimulates nerves and muscles, often producing sustained muscle contractions that make it difficult for a person to release a live conductor. In addition, currents in this frequency range are particularly effective at disrupting the heart's normal electrical activity, increasing the risk of ventricular fibrillation, a life-threatening condition in which the heart can no longer pump blood effectively.

Direct current can also be dangerous, especially at high currents or voltages, but its physiological effects differ. A large DC current typically causes a single strong muscle contraction when contact is first made, whereas AC continually stimulates muscles as it alternates direction. Regardless of whether the source is AC or DC, sufficiently large currents passing through the body can cause severe burns, respiratory paralysis, cardiac arrest, or death.

During the late nineteenth century, the relative merits of AC and DC became the focus of a famous technological competition known as the War of the Currents. Thomas Edison promoted direct-current power systems, while George Westinghouse and Nikola Tesla advocated alternating current. Although Edison argued that DC was safer, AC ultimately became the standard for electrical power distribution because it could be transformed efficiently to very high voltages for transmission and then reduced again to safer voltages for consumer use.

Today, nearly all electrical power systems use AC transmission because high-voltage transmission greatly reduces energy losses. Modern electrical safety relies on protective technologies such as insulation, grounding, circuit breakers, ground-fault circuit interrupters (GFCIs), and careful engineering standards rather than on the choice of AC or DC alone.