Final answer:
The construction of phasors involves converting time harmonic signals into a complex exponential form and representing them as vectors in the complex plane. Examples are given for various sinusoidal functions demonstrating the conversion to phasors.
Step-by-step explanation:
The task is to construct the phasors of given time harmonic signals. A phasor is a complex number representing the amplitude and phase of a harmonic oscillator, like a sinusoidal wave. When representing sinusoids like cosine and sine functions, we typically use Euler's formula to convert them into complex exponential form, which can then be represented as a phasor in the complex plane.
For example, the phasor for the signal 2cos(ωt + π/2) is 2∠π/2, which can be converted to 2i in the complex plane since cos(θ) + isin(θ) = eiθ and cos(π/2) = 0 and sin(π/2) = 1. Similarly, the phasor for signal 6sin(ωt + π/2) is 6∠(π/2 + π/2) = 6∠π. For signal 0.5sin(ωt - π/3 - α), the phasor is 0.5∠(-π/3 - α). Lastly, the signal 0.5sin(ωt + π - βx) + 7cos(ωt + π - βx) can be split into the sum of two phasors, one for the sine part and one for the cosine part, and added vectorially.