In Fig. 31-38, a three-phase generator G produces electrical...

In Fig. $31-38,$ a three-phase generator G produces electrical power that is transmitted by means of three wires. The electric potentials (each relative to a common reference level) are $V_{1}=$ $A \sin \omega_{d} t$ for wire $1, V_{2}=A \sin \left(\omega_{d} t-120^{\circ}\right)$ for wire $2,$ and $V_{3}=$ $A \sin \left(\omega_{d} t-240^{\circ}\right)$ for wire $3 .$ Some types of industrial equipment (for example, motors) have three terminals and are designed to be connected directly to these three wires. To use a more conventional two-terminal device (for example, a lightbulb), one connects it to any two of the three wires. Show that the potential difference between any two of the wires (a) oscillates sinusoidally with angular frequency $\omega_{d}$ and $(\mathrm{b})$ has an amplitude of $A \sqrt{3}$

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Key Concepts

Three-Phase AC Systems

A three-phase AC system consists of three sinusoidally varying voltages that are offset in phase by 120° relative to each other. This arrangement is common in power generation and distribution because it provides a constant power transfer over each cycle and allows efficient operation of motors and other industrial equipment.

Line-to-Line Voltage in Three-Phase Systems

In three-phase systems, devices are often connected across two phases instead of one phase and neutral. The voltage measured between any two phases, known as the line-to-line voltage, is obtained by subtracting one phase voltage from another. This results in a new sinusoidal voltage whose amplitude and phase depend on the difference between the original phase angles.

Phasor Representation and Sinusoidal Analysis

Phasor representation is a powerful technique for analyzing AC circuits where sinusoidal functions are expressed as rotating vectors in the complex plane. Through the application of this method and trigonometric identities, the potential difference between any two phases is shown to retain the same angular frequency as the original sources while its amplitude scales as a function of the phase difference, in this case leading to a ?3 multiplication factor.

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