Phase Relationship of a Sinusoidal Waveform

Thus far, we have considered only sine waves that have maxima at $\pi/2$ and $3\pi/2$, with a zero value at 0, $\pi$, and $2\pi$, as shown in [Fig. 1]. If the waveform is shifted to the right or left of $0^\circ$, the expression becomes where v is the angle in degrees or radians that the waveform has been shifted. If the waveform passes through the horizontal axis with a positive-going (increasing with time) slope before $0^\circ$, as shown in [Fig. 1], the expression is At $wt = \alpha = 0^\circ$, the magnitude is determined by $A_m \sin \theta$. If the waveform passes through the horizontal axis with a positive-going slope after $0^\circ$, as shown in [Fig. 2], the expression is And at $wt = \alpha = 0^\circ$, the magnitude is $A_m \sin \theta$, which, by a trigonometric identity, is $A_m \sin \theta$. If the waveform crosses the horizontal axis with a positive-going slope $90^\circ (\pi/2)$ sooner, as shown in [Fig. 3], it is called a cosine wave; that is, or The terms lead and lag are used to indicate the relationship between two sinusoidal waveforms of the same frequency plotted on the same set of axes. In [Fig. 3], the cosine curve is said to lead the sine curve by $90^\circ$, and the sine curve is said to lag the cosine curve by $90^\circ$. The $90^\circ$ is referred to as the phase angle between the two waveforms. In language commonly applied, the waveforms are out of phase by $90^\circ$. Note that the phase angle between the two waveforms is measured between those two points on the horizontal axis through which each passes with the same slope. If both waveforms cross the axis at the same point with the same slope, they are in phase. The geometric relationship between various forms of the sine and cosine functions can be derived from [Fig. 4]. For instance, starting at the$\ sin \alpha$ position, we find that $\cos \alpha $is an additional $90^\circ$ in the counterclockwise direction. Therefore, $\cos \alpha = \sin(\alpha + 90^\circ)$. For $\sin \alpha$ we must travel $180^\circ$ in the counterclockwise (or clockwise) direction so that $-\sin \alpha = \sin( \alpha \pm 180^\circ)$, and so on, as listed below: In addition, one should be aware that If a sinusoidal expression should appear as the negative sign is associated with the sine portion of the expression, not the peak value $E_m$. In other words, the expression, if not for convenience, would be written Since the expression can also be written A plot of each will clearly show their equivalence. There are, therefore, two correct mathematical representations for the functions. The phase relationship between two waveforms indicates which one leads or lags, and by how many degrees or radians.