# Sinusoidal Waveforms Phase Relations

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 $$\bbox[10px,border:1px solid grey]{A_m \sin(wt\pm\theta)} \tag{1}$$ where v is the angle in degrees or radians that the waveform has been shifted.
Fig. 1: Defining the phase shift for a sinusoidal function that crosses the horizontal axis with a positive slope before $0^\circ$.
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 $$\bbox[10px,border:1px solid grey]{A_m \sin(wt+\theta)} \tag{2}$$ 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 $$\bbox[10px,border:1px solid grey]{A_m \sin(wt-\theta)} \tag{3}$$ And at $wt = \alpha = 0^\circ$, the magnitude is $A_m \sin \theta$, which, by a trigonometric identity, is $A_m \sin \theta$.
Fig. 2: Defining the phase shift for a sinusoidal function that crosses the horizontal axis with a positive slope after $0^\circ$.
Fig. 3: Phase relationship between a sine wave and a cosine wave.
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, $$\bbox[10px,border:1px solid grey]{\sin(wt+90) =\sin(wt+{\pi \over 2}) = \cos(wt) } \tag{4}$$ or $$\bbox[10px,border:1px solid grey]{\sin(wt) =\cos(wt-90) =\cos(wt-{\pi \over 2})}\tag{5}$$ 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.
Fig. 4: Graphic tool for finding the relationship between specific sine and cosine functions
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: $$\bbox[10px,border:1px solid grey]{ \begin{split} \cos \alpha &= \sin( \alpha + 90)\\ -\cos \alpha &= \sin( \alpha - 90)=\sin( \alpha + 270)\\ \sin \alpha &= \cos( \alpha - 90)\\ -\sin \alpha &= \sin( \alpha \pm 180)\\ \end{split} }$$
In addition, one should be aware that $$\bbox[10px,border:1px solid grey]{ \begin{split} \sin (-\alpha) &= -\sin( \alpha)\\ \cos (-\alpha) &= \cos(\alpha)\\ \end{split} }$$ If a sinusoidal expression should appear as $$e = -E_m \sin(wt)$$ 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 $$e = E_m(- \sin(wt))$$ Since $$- \sin(wt) = \sin(wt \pm 180)$$ the expression can also be written $$e = E_m \sin(wt \pm 180)$$ 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.
Example 1: What is the phase relationship between the sinusoidal waveforms of each of the following sets?
a. $v = 10\sin(wt + 30^\circ)$, $i = 5\sin(wt + 70^\circ)$
b. $v = 10\sin(wt + 60^\circ)$, $i = 5\sin(wt - 20^\circ)$
Solution:
a. See Fig. 5.
$i$ leads $v$ by $40^\circ$, or $v$ lags $i$ by $40^\circ$.
Fig. 5:
b. See Fig. 6.
$i$ leads $v$ by $80^\circ$, or $v$ lags $i$ by $80^\circ$.
Fig. 6: