Direct current (DC) refers to an electric current that flows in a single direction, maintaining a constant polarity. This is in contrast to alternating current (AC), which periodically changes its direction and magnitude. AC forms the backbone of modern electricity transmission and distribution systems due to its efficient long-distance transmission capabilities.

In homes, the power supplies use sinusoidal sources to provide electricity. These sources generate a voltage that varies sinusoidally over time. In other words, the voltage oscillates between a maximum and minimum value in a smooth, wave-like pattern.

Mathematically, this variation is represented by a harmonic function known as a sinusoid. The sinusoid is defined by three primary parameters: its amplitude (the peak value), angular frequency (the rate at which it oscillates), and phase (its position relative to a reference point).

The sinusoidal waveform repeats its pattern every T seconds, where T is referred to as the period of the function. The period is inversely related to the angular frequency, representing the duration of one complete cycle of the waveform.

On the other hand, the sinusoid frequency is defined as the number of cycles that occur per second. It is the reciprocal of the period and is typically measured in Hertz (Hz).

To understand the concept of phase, consider two sinusoids with the same frequency. If one sinusoid reaches its maxima and minima earlier than the other, then the first sinusoid is said to "lead" the second. In this case, the two sinusoids are out of phase, and the shift between them is known as the phase difference.

On the contrary, if the two sinusoids reach their maxima and minima simultaneously, they are considered to be in phase. This means they have a phase difference of zero.

Understanding these fundamental concepts related to AC is crucial for various applications, from designing electronic circuits to analyzing power systems. The principles of AC underpin many technologies in our daily lives, including household appliances, telecommunications systems, and power generation facilities.