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Wikipedia / Phase (waves)

In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function of some real (such as time) is an angle. It is expressed in such a scale that it varies by one full turn goes through each period goes through each complete cycle). It may be measured in any angular unit such as degrees or radians completes a full period.

This convention is especially appropriate for a sinusoidal, 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.)

) if the difference between them is a whole number of periods.

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.

spanning a whole turn, one gets the phase shift, phase offset, or phase difference is called the initial phase.

Mathematical definition

be its period (that is, the smallest positive real number). Then the at

is an arbitrary "origin" value of the argument, that one considers to be the beginning of a cycle.

This concept can be visualized by imagining a clock, measured clockwise.

where the function's value changes from zero to positive.

The phase expressed in degrees (from 0° to 360°, or from −180° to +180°) is defined the same way, except with "360°" in place of "2π".

Consequences

ranges over a single period.

with a specific waveform.)

Adding and comparing phases

respectively. Thus, for example, the sum of phase angles is 30° (, minus one full turn), and subtracting 50° from 30° gives a phase of 340° (, plus one full turn).

instead of 360.

Phase shift Phase shiftShiftPhase differenceDifference

is called the phase difference or phase shift. when the difference is zero, the two signals are said to be in phase; otherwise, they are out of phase with each other.

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.

when the phase difference is zero, the two signals will have the same sign and will be reinforcing each other. One says that constructive interference when the phases are different, the value of the sum depends on the waveform.

For sinusoids

radians), one says that the phases are opposite, and that the signals are in antiphase. Then the signals have opposite signs, and destructive interference occurs. Conversely, a phase reversal or phase inversion implies a 180-degree phase shift.

is a quarter of turn (a right angle, or ), sinusoidal signals are sometimes said to be in quadrature, e.g., in-phase and quadrature components of a composite signal or even different signals (e.g., voltage and current).

. The periodic changes from reinforcement and opposition cause a phenomenon called beating.

For shifted signals

. In the clock analogy, this situation corresponds to the two hands turning at the same speed, so that the angle between them is constant.

(in terms of the modulo operation

called simply the initial phase.

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.

is the length seen at the same time at a longitude 30° west of that point, then the phase difference between the two signals will be 30° (assuming that, in each signal, each period starts when the shadow is shortest).

For sinusoids with same frequency

is either identically zero, or is a sinusoidal signal with the same period and phase, whose amplitude is the difference of the original amplitudes.

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.

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.

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.

Formula for phase of an oscillation or a periodic signal

The phase of a simple harmonic oscillation or sinusoidal signal are constant parameters called the amplitude, frequency, and phase axis. The term phase can refer to several different things:

  • , in which case we would say the phase, and the phase.

  • have the same phase but are relative to their own specific references.

  • , or its principal value, is referred to as instantaneous phase, often just phase.

Absolute phase

See also

References