Phase (waves)

This convention is especially appropriate for a sinusoidal function, since its value at any argument ${\displaystyle t}$ then can be expressed as ${\displaystyle \varphi (t)}$, the sine of the phase, multiplied by some factor (the amplitude of the sinusoid). (The cosine may be used instead of sine, depending on where one considers each period to start.)


Usually, whole turns are ignored when expressing the phase; so that ${\displaystyle \varphi (t)}$ is also a periodic function, with the same period as ${\displaystyle F}$, that repeatedly scans the same range of angles as ${\displaystyle t}$ goes through each period. Then, ${\displaystyle F}$ is said to be "at the same phase" at two argument values ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ (that is, ${\displaystyle \varphi (t_{1})=\varphi (t_{2})}$) if the difference between them is a whole number of periods.


The numeric value of the phase ${\displaystyle \varphi (t)}$ depends on the arbitrary choice of the start of each period, and on the interval of angles that each period is to be mapped to.


The term "phase" is also used when comparing a periodic function ${\displaystyle F}$ with a shifted version ${\displaystyle G}$ of it. If the shift in ${\displaystyle t}$ is expressed as a fraction of the period, and then scaled to an angle ${\displaystyle \varphi }$ spanning a whole turn, one gets the phase shift, phase offset, or phase difference of ${\displaystyle G}$ relative to ${\displaystyle F}$. If ${\displaystyle F}$ is a "canonical" function for a class of signals, like ${\displaystyle \sin(t)}$ is for all sinusoidal signals, then ${\displaystyle \varphi }$ is called the initial phase of ${\displaystyle G}$.

Phase comparison[edit]

Phase comparison is a comparison of the phase of two waveforms, usually of the same nominal frequency. In time and frequency, the purpose of a phase comparison is generally to determine the frequency offset (difference between signal cycles) with respect to a reference.^[3]


A phase comparison can be made by connecting two signals to a two-channel oscilloscope. The oscilloscope will display two sine signals, as shown in the graphic to the right. In the adjacent image, the top sine signal is the test frequency, and the bottom sine signal represents a signal from the reference.


If the two frequencies were exactly the same, their phase relationship would not change and both would appear to be stationary on the oscilloscope display. Since the two frequencies are not exactly the same, the reference appears to be stationary and the test signal moves. By measuring the rate of motion of the test signal the offset between frequencies can be determined.


Vertical lines have been drawn through the points where each sine signal passes through zero. The bottom of the figure shows bars whose width represents the phase difference between the signals. In this case the phase difference is increasing, indicating that the test signal is lower in frequency than the reference.^[3]

The phase of a simple harmonic oscillation or sinusoidal signal is the value of ${\textstyle \varphi }$ in the following functions: