Phase (waves)
Phase (waves)
Mathematical definition
![{\displaystyle \phi (t)=2\pi \left[\!\!\left[{\frac {t-t_{0}}{T}}\right]\!\!\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8667c57b10a88168acb01cf9516d856f5c03426)
![{\displaystyle [\![\,\cdot \,]\!]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c32b253126c14108caf99c7ffbeccb71dc7f7a26)
![{\displaystyle [\![x]\!]=x-\left\lfloor x\right\rfloor }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d36be2de0ba1ee61c54030e6a363ac841ecb99a)
![{\displaystyle \phi (t)=2\pi \left(\left[\!\!\left[{\frac {t-t_{0}}{T}}+{\tfrac {1}{2}}\right]\!\!\right]-{\tfrac {1}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/770ba0b8cfac7a5fd702b8f6a5063b27130320f6)
Consequences
- for all.
The phase is zero at the start of each period; that is
- for any integer.
Adding and comparing phases
Since phases are angles, any whole full turns should usually be ignored when performing arithmetic operations on them. That is, the sum and difference of two phases (in degrees) should be computed by the formulas
- and
![{\displaystyle 360\,\left[\!\!\left[{\frac {\alpha +\beta }{360}}\right]\!\!\right]\quad \quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/92b4181dd4ce3b785781bda9f93a9a1015a7df51)
![{\displaystyle \quad \quad 360\,\left[\!\!\left[{\frac {\alpha -\beta }{360}}\right]\!\!\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2605a5deb796fc44f1fb93adfe219897ecd3ad4b)
respectively. Thus, for example, the sum of phase angles 190° + 200° is 30° (190 + 200 = 390, minus one full turn), and subtracting 50° from 30° gives a phase of 340° (30 - 50 = -20, plus one full turn).
Phase shift
General definition
In the clock analogy, each signal is represented by a hand (or pointer) of the same clock, both turning at constant but possibly different speeds. The phase difference is then the angle between the two hands, measured clockwise.
The phase difference is particularly important when two signals are added together by a physical process, such as two periodic sound waves emitted by two sources and recorded together by a microphone. This is usually the case in linear systems, when the superposition principle holds.
For sinusoids
For shifted signals
- .
![{\displaystyle \varphi =2\pi \left[\!\!\left[{\frac {\tau }{T}}\right]\!\!\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26e80a24c454efd1885272666aac02d604428d8f)
Therefore, when two periodic signals have the same frequency, they are always in phase, or always out of phase. Physically, this situation commonly occurs, for many reasons. For example, the two signals may be a periodic soundwave recorded by two microphones at separate locations. Or, conversely, they may be periodic soundwaves created by two separate speakers from the same electrical signal, and recorded by a single microphone. They may be a radio signal that reaches the receiving antenna in a straight line, and a copy of it that was reflected off a large building nearby.
For sinusoids with same frequency
- and.
.
A real-world example of a sonic phase difference occurs in the warble of a Native American flute. The amplitude of different harmonic components of same long-held note on the flute come into dominance at different points in the phase cycle. The phase difference between the different harmonics can be observed on a spectrogram of the sound of a warbling flute.[3]
Phase comparison
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.[2]
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.[2]
Formula for phase of an oscillation or a periodic signal
The phase of an oscillation or signal refers to a sinusoidal function such as the following:
It can refer to a specified reference, such as , in which case we would say the phase of is , and the phase of is .
It can refer to , in which case we would say and have the same phase but are relative to their own specific references.
In the context of communication waveforms, the time-variant angle , or its principal value, is referred to as instantaneous phase, often just phase.
See also
In-phase and quadrature components
Instantaneous phase
Lissajous curve
Phase angle
Phase cancellation
Phase problem
Phase velocity
Phasor
Polarization
Coherence, the quality of a wave to display a well defined phase relationship in different regions of its domain of definition
Absolute phase