Phase Mechanics

Phase, in mechanics of vibrations, is the fraction of a period (i.e., the time required to finish a full cycle) that to some extent completes after last passing through the reference or zero position. For example, the reference position for the hands of a clock is at the numeral 12, and therefore the big hand features a period of 1 hour. At \[\frac{1}{4}\] past the hour, the big hand features a phase of one-quarter period, having skilled a phase of 90⁰ or \[\frac{\pi}{2}\] radians. In this example, the motion of the big hand may be a uniform circular motion, but the concept of phase also applies to simple periodic motion like that experienced by waves and vibrating bodies.

Phase Mechanics

If the position yof an object, point or particle changes with respect to a simple harmonic motion, then the value of the position will change with respect to time t,according to the product of the amplitude, or maximum displacement, r, of the particle and a sine or cosine function composed of its angular velocity, denoted by the Greek letter omega (), the time t, and what is called the angle, symbolized by the Greek letter phi (Φ) : y = r sin (ωt +  Φ). The angle is known as the phase angle at any given time t, and the phase angle is equal towhen time is considered to be zero. 

According to phase mechanics, the phase itself is found to be a fractional value, i.e., the ratio of used time t to the total time period T, i.e., \[\frac{t}{T}\], this ratio is equal to the ratio of the phase angle to the angle of the one complete cyclei.e., 3600 or 2 Radians. Thus, from phase mechanics, it is understood that the phase for uniform circular (or harmonic motion) has the value \[\frac{(ωt+Φ)}{2\pi}\]. Using these expressions to the illustrations of the moving minute hand quoted above, Φ is zero (which says that zero phase angle at zero time), and corresponding angular velocity is 2π radians per hour, and time t is \[\frac{1}{4}\] hour, giving a phase of \[\frac{1}{4}\].

While relating or comparing the phases of multiple periodic motions, such as waves, these motions are said to be in phase when corresponding points reach maximum or minimum displacements together or simultaneously. If the crests of two waves executing periodic motion cross the same point or the line simultaneously or in other words at the same time, then they are in phase for the given position. At the same time, if the crest of one and the trough of the other cross simultaneously, then the phase angles differ by 180°, or π radians, and the waves are said to be out of phase180⁰.

The determination of phase difference is of fundamental importance in bringing the change in current technology. In the diagram added below, we see that two curves represent the voltage or the potential difference (V) and the electric current (I) in an alternating current (AC) electric circuit with pure inductance (consisting of the only inductor). The difference in phase angle between the voltage and the current is found to be 90⁰ i.e., the current and voltage curves are perpendicular to each other. From the analysis of the graph, the current is said to lag one-quarter cycle i.e., \[\frac{1}{4}\] in phase. This lag in current can be observed in the diagram. In alternating current power transmission, the terms multi-phase and polyphase are employed to currents that are out of phase with one another. For example, in a two-phase system, we know that two currents are flowing with a phase-angle difference of 90°,similarly, in a three-phase system which is most commonly used in home appliance, the currents differ in phase angle by120⁰.

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Phase Mechanics Equation

The phase shift or the phase difference of a Sinusoidal Waveform is denoted by the angle Φ (Greek letter Phi), and the phase shift or the phase difference is measured in terms of degrees or radians that the waveform has changed from a particular reference point along the horizontal zero axes. In other words, the phase difference is the lateral (parallel) difference between two or more waveforms along a common axis. One of the interesting facts is the sinusoidal waveforms of the identical frequency sometimes will have a phase difference.

The phase shift of an alternating waveform usually range from between 0 to its maximum time period T(say) of the waveform during one complete cycle and this can be anywhere along the horizontal axis between, Φ = 0 to 2π (radians) or Φ  = 0 to 3600depending upon the angular units utilised.

The phase shift will also be represented as a time shift of in seconds describing a fraction of the time period, T for example, +30 mS or – 40 s but commonly it is more popular to express phase shift as an angular measurement.

Then the phase mechanics equation for the instantaneous value of a sinusoidal voltage or current waveform will be modified to take into consideration the phase angle of the waveform and this new general expression will represent the phase mechanics equation.

A = A_max sin (ωt \[\pmΦ\])

Where,

A -The amplitude of the given sinusoidal wave

A_max-The maximum amplitude of the given sinusoidal wave

ω -The angular frequency

Φ -The phase angle measured in either degrees or radians explains whether the given sinusoidal waveform has shifted to the left or right from the reference point.

Now, from the above expression, we notice that, if the positive slope of the sinusoidal waveform crosses the horizontal axis prior to time t = 0, then the waveform is said to be shifted to the left so Φ >0, and the phase angle will be positive in nature i.e., +Φ giving a leading phase angle. In other words, it develops earlier in time than00producing an anticlockwise rotation of the sinusoidal wave vector.

Similarly, if the positive slope of the sinusoidal waveform crosses through the horizontal x-axis after some time (i.e., after t=0) then the waveform is said to be shifted to the right so Φ <0, and the phase angle will be noted as negative in nature (-Φ) producing a lag in the phase angle as it appears following in time than 00producing a clockwise rotation of the sinusoidal wave vector. Both are the special cases elaborating lag and lead phenomenon and both special cases are presented below.

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Now, if we compare the phases of multiple sinusoidal waves, the periodic motions of the sinusoidal waves are said to be in phase when corresponding points reach maximum or minimum displacements together or simultaneously. If the crests of the sinusoidal waves executing periodic motion pass through the same point or the line simultaneously or in other words at the same time, then they are in phase for the given position. At the same time, if the crest of one and the trough of the other cross simultaneously, then the phase angles differ by 180° or π radians, and the waves are said to be out of phase 180⁰

To determine the phase difference, we require the frequency and period of the waves considered. For example, an electronic oscillator is producing sine waves of a frequency 200 Hz. Dividing the frequency by 1 gives the period, or the time taken to complete each cycle, so \[\frac{1}{200}\]gives a period of 0.02 seconds. 

The Phase Difference Expression is Generally Given By:

PS = \[\frac{300\ast \Delta t}{T}\]

Did You Know?

• Estimating the phase difference between any two waves involves the comparison of two waves, and part of that comparison will be choosing which wave is considered as the first wave and which one is the second. 

• In electronics, generally, the second wave is ideally the output of an amplifier or other device, and the first wave will be the input signal. 

• In math, the calculation of the phase difference will be different than in the phase mechanics. In mathematics, the first wave may be an original function and the second a consequent or secondary function. For example, the first function may be y = cos x, and the second function may be y = sin xor vice versa. The order of the consideration of the waves will not affect the absolute value of the phase difference, but it will determine the polarity of the shift i.e, whether it is a positive phase shift or negative phase shift.