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Sinusoidal and Phasor Representation Consider the function
f ( t ) = rejω^ t 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. t = 4^ πω
c. t = 2^ πω
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. re j^ ωt^ = r cos ωt + jr sin ω t
Figure 2 If the current through a circuit lags the voltage by θ , such that: V ( t )= Vm sin ω t i = Im sin( ω t − θ )
Then the values of V and i can be plotted against ωt , 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
Figure 4(b) shows the phasor plot of voltage and current in the frequency domain. Hence V and I are fixed in space if the frequency ω is constant but takes up different positions as ω is varied.
Figure 4
