Sinusoidal and Phasor Representation: Complex Numbers, Waveforms, and Phasor Transform, Lecture notes of Engineering

Download Sinusoidal and Phasor Representation: Complex Numbers, Waveforms, and Phasor Transform and more Lecture notes Engineering in PDF only on Docsity! Sinusoidal and Phasor Representation Consider the function tjretf ω=)( This is a complex number of absolute value r but varies with time t. A few sketches of the function are made for: a. t = 0 b. ω π 4 =t c. ω π 2 =t These sketches are shown in Figure 1. Figure 1 The line segment of length r will rotate in a counterclockwise direction at a constant angular velocity ω. The projection of this rotating vector along the real axis produces a cosine waveform and along the imaginary axis a sine waveform source as shown in Figure 2. tjrtrre tj ωωω sincos += Figure 2 If the current through a circuit lags the voltage by θ, such that: tVtV m ωsin)( = )sin( θω −= tIi m Then the values of V and i can be plotted againstωt, the waveforms in Figure 3(b) is produced. Figure 3(a) shows a pair of line segments which rotate in the counterclockwise direction at a constant velocity ω in the complex plane. Since both rotate at the same velocity, the phase angle θ between them is maintained and the current lags the voltage by this angle θ. It is clear that the projection of the rotating line segments on the imaginary axis produces the waveforms in Figure 3(b). Figure 3