We have so far considered direct current (dc) sources and circuits with dc sources. These currents do not change direction with time. But voltages and currents that vary with time are very common. The electric mains supply in our homes and offices is a voltage that varies like a sine function with time. Such a voltage is called alternating voltage (ac voltage) and the current driven by it in a circuit is called the alternating current (ac current). Today, most of the electrical devices we use require ac voltage. This is mainly because most of the electrical energy sold by power companies is transmitted and distributed as alternating current. The main reason for preferring use of ac voltage over dc voltage is that ac voltages can be easily and efficiently converted from one voltage to the other by means of transformers. Further, electrical energy can also be transmitted economically over long distances. AC circuits exhibit characteristics which are exploited in many devices of daily use. For example, whenever we tune our radio to a favourite station, we are taking advantage of a special property of ac circuits – one of many that you will study in this chapter.

Figure 7.1 shows a resistor connected to a source ε of ac voltage. The symbol for an ac source in a circuit diagram is . We consider a source which produces sinusoidally varying potential difference across its terminals. Let this potential difference, also called ac voltage, be given by v = v sin(wt) where v is the amplitude of the oscillating potential difference and ω is its angular frequency. To find the value of current through the resistor, we apply Kirchhoff’s loop rule (∑ ε(t) = 0), to the circuit to get v sin(wt) = iR, or i = v sin(wt)/R. Since R is a constant, we can write this equation as i = i sin(wt) where the current amplitude i is given by i = v/R. The voltage and current are in phase with each other, meaning they reach their maximum and minimum values at the same time.

The instantaneous power dissipated in the resistor is p = i²R. Following the sinusoidal nature, we represent the average power over a cycle as P = R, where is the average of i² over one cycle. Since the oscillating current alternates between positive and negative, the average value becomes positive (meaning energy is indeed consumed). The average power can be expressed as P = (1/2)i²R, utilizing the effective or rms current I for simplicity.

To express the power in a similar framework as in DC circuits, we need to redefine the current and voltage in an AC circuit into effective values called root mean square (rms) values. The rms current I is derived from the maximum current as I = im/√2 (approximately 0.707 im), and similarly, the rms voltage V can be expressed as V = vm/√2. These rms values allow the equations for power in AC circuits to resemble those in DC circuits, which is greatly beneficial for analyzing energy consumption.

The average power dissipated in AC circuits varies based on the relationship of the current and voltage introduced by the phase angle. The power factor is defined as cosφ, where φ represents the phase angle between the current and voltage. It plays a critical role in determining how effectively the electrical energy is converted into useful work. In scenarios involving inductive or capacitive circuits, cosφ can be less than 1, leading to a lower effective power usage.

Transformers are used to adjust the level of alternating voltage for efficient power transmission. They consist of two coils of wire wound around a magnetic core. When an AC voltage is applied to the primary coil, it creates an alternating magnetic field that induces voltage in the secondary coil. The ratio of turns between the primary (Np) and secondary (Ns) coils determines if the transformer steps up or down the voltage; for instance, if Np < Ns, the output voltage (Vs) is greater than the input (Vp). Transformers are vital for long-distance transmission, allowing high voltage to minimize energy loss.