### Introduction

When we are listening to a song, we perceive the sinusoidal sound waveforms as music. Their amplitude gives us how loud the signal is and the frequency tells us if the sound is low or high pitched. However, the third important parameter, which is the phase, is harder to experience by ears.

This tutorial will clarify and give more details about the phase parameter that we already explored in one of our previous tutorials about Phase Splitters. The first section will, therefore, give a presentation of the concepts of phase and phase difference as a reminder.

In the second section, we detail more aspects of the concept of phase shift and we focus on a particular case when the signals are not synchronized.

The third and last section will finally present the important role of the phase difference in the interference phenomenon.

### Presentation

The phase of a sine signal is often noted with Φ and measured in radians (rad) or degrees (°) and can vary between **-π and +π rad** or **-180° and +180°**.

On a graph, the phase of an AC signal represents the initial state of its related sine function at the origin of time :

The phase Φ of a signal can be of three different nature and dictates the position of the waveform around the vertical axis :

- Equal to 0 (° or rad) such as for the signal y
_{1}(t) which act as a reference signal - Be positive such as for the signal y
_{2}(t) - Be negative such as for the signal y
_{3}(t)

The phase of a single signal is not very relevant because whether the AC waveform is of electrical or mechanical nature, the perception will remain unchanged if the signal has a phase or not. What is more important and can clearly be perceived is a **phase difference**, also called a **phase shift** between two signals of the same frequency.

### Phase difference

#### Between signals of the same frequency

What is important to keep in mind during this section is that we only talk about a phase shift between two signals **with identical frequency**. Consider therefore two signals of the same frequency with different phases and possibly different amplitudes: y_{1}(t)=Asin(ωt+Φ_{1}) and y_{2}(t)=Bsin(ωt+Φ_{2}). We define the phase difference as the quantity **ΔΦ**_{21}**=Φ _{2}**

**-Φ**

_{1.}

In **Figure 1**, we have ΔΦ_{21}=+Φ_{2}, ΔΦ_{31}=-Φ_{3}, and ΔΦ_{32}=-Φ_{3-}Φ_{2}. A positive phase difference, such as ΔΦ_{21}, indicates that the signal y_{2}(t) temporarily precedes the reference signal y_{1}(t), we also say that **y _{2}(t) leads y_{1}(t)**. A negative phase difference, such as ΔΦ

_{31 }and ΔΦ

_{32}, indicates that the signal y

_{3}(t) follows the signals y

_{1}(t) and y

_{2}(t), we also say that

**y**.

_{3}(t) lags y_{1}(t) and y_{2}(t)Among all of the values between -180° and +180° or -π and +π rad that a phase difference can take, a few ones can be highlighted and illustrated in the following **Figure 2** :

An opposite phase is characterized by a phase shift of **+180° or +π rad**, which is strictly identical to -180° or -π rad. If the reference signal is V_{ref}=v_{ref}sin(ωt), then the opposite signal is V_{opp}=v_{ref}sin(ωt+π)=-v_{ref}sin(ωt), therefore, V_{ref}+V_{opp}=0.

Quadrature signals are characterized by a phase shift of +90° or +π /2 rad for the “advance” and -90° or -π /2 rad for the “delay”.

#### Between the current and voltage signals

In this subsection, we focus specifically on the phase shift of the **current (I)** and **voltage (V)** signals across an electrical dipole and investigate its consequences regarding power.

In the DC regime, the dissipated power (P) across a dipole is given by the product of the voltage and current :

In the AC regime, this representation is no longer true since both the voltage and current are alternating. Consider the voltage across the dipole to be **V=V _{rms}√2.sin(ωt)** and the current of the same frequency presenting a phase difference of +ΔΦ:

**I=I**. With V

_{rms}√2.sin(ωt+Φ)_{rms}and I

_{rms }being the root mean square values.

It can be shown that the active power dissipated in the dipole in AC regime is given by **Equation 1**:

The term cos(Φ) is known as the **power factor **and gives the efficiency of a receptor to absorb the power of a source. This factor is a real number between 0 and 1 and these two extrema reflect very different behavior :

- If cos(Φ)=1 the dipole is considered
**purely resistive**, the phase shift between the voltage and current is zero. The dipole does not present any inductive or capacitive behavior. - If cos(Φ)=0 the dipole is
**purely reactive**, the phase shift between the voltage and current is maximum, equal to ±90° or ±π/2 rad. In this case, the dipole does not consume any power but instead returns it to the circuit.

The power given in **Equation 1 **is called the **active power (P)**, the product V_{rms}×I_{rms }is known as the **apparent power** and noted **S**. It is the power that would be dissipated if the component was purely resistive. The quantity V_{rms}×I_{rms}×sin(Φ) is the **reactive power **and noted **Q**. These quantities can be linked thanks to the phase shift ΔΦ in the same complex power diagram :

#### Between signals of similar frequency

In this section, we consider two signals y_{1}(t) which is the reference and y_{2}(t) phase-shifted of Φ of similar frequencies but not strictly identical: ω_{1}≠ω_{2}. Usually, a phase-shift can only be defined for two signals of the same frequency, but in this particular case, it still makes sense to define a phase-difference as the frequencies are similar. If the frequencies are too different, usually when ω_{1}>2ω_{2 }for example, it does not make sense to define it since the phase difference varies as much as the signal itself.

In the case where the signals are of similar frequency, the phase difference is not constant anymore but slowly varies with time: **ΔΦ(t)=(ω _{2}-ω_{1})t+Φ**.

The superposition of these two signals is interesting because of the creation of a **beating phenomenon** such as illustrated in **Figure 4**:

The beating takes his name from the acoustic domain where this phenomenon is particularly audible and easy to experience, however, it also appears in optics, electronics, mechanics etc… The beating is actually a particular case of **interference**, which we focus on in the next section.

### Interference

We can see in **Figure 4 **that a superposition of sine waveforms results sometimes in addition to the amplitudes when the signals are in phase or in a subtraction when the signals are in opposition of phase. This phenomenon is known as **interference **and takes place when the signals are of the same frequency.

Consider again two sinusoidal waveforms of identical frequency : y1(t)=A_{1}sin(ωt+Φ1) and y_{2}(t)=A_{2}sin(ωt+Φ_{2}). Let’s call y_{3}(t) the superposition y_{1}(t)+y_{2}(t) and A_{3} its amplitude. It can be shown that the amplitude of y_{3}(t) satisfies the following equation :

We can notice that the phase difference between y_{1}(t) and y_{2}(t) plays an important role in the final amplitude of the resulting signal. Two cases are interesting to highlight :

- ΔΦ
_{12}=0, the signals are in phase and the amplitude A_{3}is maximal as satisfies A_{3}^{2}=(A_{1}+A_{2})^{2}. In this case, we say that the interference between y_{1}and y_{2}is**constructive**. - ΔΦ
_{12}=±π rad, the signals are in opposition of phase and the amplitude A_{3}is minimal and satisfies A_{3}^{2}=(A_{1}-A_{2})^{2}. In this case, the interference between y_{1}and y_{2}is**destructive**.

When the phase difference is between these two extrema, we can plot a graph showing the evolution of A_{3} as a function of ΔΦ_{12} :

In this figure, we choose for the sake of simplicity A_{1}=A_{2}. We can see again that when ΔΦ_{12}=0, A_{3}=A_{1}+A_{2}=2 and when ΔΦ_{12}=±180°, A_{3}=A_{1}-A_{2}=0.

### Conclusion

This tutorial has given a detailed presentation about the concept of phase and phase difference, moreover, it has pinpointed its importance through some examples.

First of all, we present what is the phase of a signal and in which unit it is measured. The concept of phase alone, however, is not very relevant and this is why we focus on the following sections about the phase difference or phase shift.

In the first paragraph of the second section, we define the phase shift ΔΦ and give some vocabulary related to particular cases of phase difference: in-phase (ΔΦ=0°), the opposition of phase (ΔΦ=±180°) and the quadratures (ΔΦ=±90°).

In a second subsection, we highlight the importance of the phase-shift between the current and voltage in a circuit. The power dissipated in any electronic component is directly proportional to the cosine of the phase shift, which is called the power factor.

In the last section, we link and explain the interference phenomenon to the phase shift parameter. The beating phenomenon which is explained previously in the article is a particular case of interference.

## FAQs

### What is the difference between phase shift and phase difference? ›

**The phase difference between two waveforms is often called a phase shift**. A phase shift of 360 degrees is a time delay of one cycle, or one period of the wave, which actually amounts to no phase shift at all. A phase shift of 90 degrees is a shift of 1/4 of the period of the wave, etc.

**What is phase shift in electronics? ›**

A 'phase shift' is **the minute variation amongst the two waveforms**. In the areas of both electronics and mathematics, a phase shift is the delay present between these two waveforms that share the same frequency or period.

**What is phase difference in digital electronics? ›**

Phase difference, also called phase angle, is **a measure in degrees of how much one wave leads another wave or lags behind that wave**. For example, Figure 2 shows two waves with the same amplitude and frequency but with different phase patterns.

**What is the difference between phase shift and time shift? ›**

**Time delay shifts all frequencies by the same amount of time, whereas phase shift delays some frequencies longer than others**. In fact, an all-pass filters center frequency is defined at the frequency at which the phase shift is 90 degrees.

**What is an example of a phase shift? ›**

Example: **3 sin(100t + 1)**

period is 2π/100 = 0.02 π phase shift is C = 0.01 (to the left) vertical shift is D = 0.

**What is the phase difference? ›**

The phase difference is **the difference in the phase angle of the two waves**. Path difference is the difference in the path traversed by the two waves. The relation between phase difference and path difference is direct. They are directly proportional to each other.

**What is the concept of phase and phase difference? ›**

Phase: The position of the moving particle of a waveform is called “Phase” and is measured in “Radians or degrees”. Phase difference: The time interval by which a wave leads by or lags by another wave is called “Phase difference” or “Phase angle”.

**What causes a phase shift in a circuit? ›**

This phase shift occurs because **the inductive reactance changes with changing current**. Recall that it is the changing magnetic field caused by a changing current that produces inductive reactance.

**What is the purpose of phase difference? ›**

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.

**How do you measure phase difference? ›**

Phase difference can be measured on an oscilloscope by **determining the time delay between two waveforms along with their period**. All periodic signals can be described in terms of amplitude and phase.

### Why 120 degree phase shift? ›

The phase angle shift of 120° is **a function of the actual rotational angle shift of the three pairs of windings**. If the magnet is rotating clockwise, winding 3 will generate its peak instantaneous voltage exactly 120° (of alternator shaft rotation) after winding 2, which will hit its peak 120° after winding 1.

**What is the relation between phase difference and time difference? ›**

The frequency of a signal is given by f, and the time the (in secs) regarding one degree of phase is the **= 1 / (360f) = T / 360**. Therefore, a one-degree phase shift on a 5 MHz signal shows a time shift of 555 picoseconds.

**What is the relation between time shift and phase shift? ›**

**The time interval for 1° of phase is inversely proportional to the frequency**. If the frequency of a signal is given by f, then the time t_{deg} (in seconds) corresponding to 1° of phase is t_{deg} = 1 / (360f) = T / 360. Therefore, a 1° phase shift on a 5 MHz signal corresponds to a time shift of 555 picoseconds.

**What is another name for phase shift? ›**

noun. ['trænˈzɪʃən'] a change from one place or state or subject or stage to another.

**What is a real life example of phase change? ›**

Examples of Phase Changes

For example, you have probably witnessed **freezing, melting, and vaporization** just by making ice, melting ice, and boiling water. Condensation often occurs on the outside of cold beverage containers. This is when the humid air changes directly to a liquid on the surface of the container.

**What are 7 examples of phase changes? ›**

Solid → Liquid | Melting or fusion |
---|---|

Liquid → Gas | Vaporization |

Liquid → Solid | Freezing |

Gas → Liquid | Condensation |

Solid → Gas | Sublimation |

**What is phase shifting and how does it work? ›**

A phase shifting transformer (PST) is **a specialised type of transformer, typically used to control the flow of active power on three-phase electric transmission networks**. It does so by regulating the voltage phase angle difference between two nodes of the system.

**Is phase shift always positive? ›**

**Phase shift is positive (for a shift to the right) or negative (for a shift to the left)**. The easiest way to find phase shift is to determine the new 'starting point' for the curve.

**What is the phase difference in three phase system? ›**

A three-phase AC power supply is a power supply composed of three AC potentials with the same frequency, the same amplitude, and a phase difference of **120°** from each other. Three-phase alternating current has many uses, and most of the alternating current electrical equipment in the industry.

**What is phase difference diagram? ›**

Phase Difference is **used to describe the difference in degrees or radians when two or more alternating quantities reach their maximum or zero values**. Advertisement. Phasors are an effective way of analysing the behavour of elements within an AC circuit when the circuit frequencies are the same.

### What is the symbol for phase difference? ›

(symbol: ϕ) The difference in phase between two sinusoidally varying quantities of the same frequency.

**What is the phase difference between two phase system? ›**

Two-phase electrical power was an early 20th-century polyphase alternating current electric power distribution system. Two circuits were used, with **voltage phases differing by one-quarter of a cycle, 90°**. Usually circuits used four wires, two for each phase.

**What do you mean by phase and phase difference of alternating current? ›**

When capacitors or inductors are involved in an AC circuit, the current and voltage do not peak at the same time. **The fraction of a period difference between the peaks expressed in degrees is said to be the phase difference**. The phase difference is <= 90 degrees.

**What is the effect of phase shift? ›**

Let's start with the bad news first: phase shifts **can cause acoustic problems**. Sound waves that are phase-shifted at the same frequency can falsify the sound through so-called interference. The signals can cancel each other out or amplify each other. This is referred to as room modes.

**How do you control phase shift? ›**

Single Phase Shift Control (SPS): SPS is conventional control method and it is the most widely used control method in researches. Because, its theoretical and experimental analysis are simple. **Triggering signals are defined by creating δ between Stage A (Q switches) and Stage B (S switches)**.

**What is the principle of phase difference? ›**

In principle, **the phase difference between two sinusoidal signals can be determined by measuring the time that elapses between the two signals crossing the time axis**. However, in practice, this is inaccurate because the zero crossings are susceptible to noise contamination.

**What happens when phase difference is 90? ›**

Phase Quadrature: **Two waves are said to be in phase quadrature** if their phase difference is 90 degrees (positive or negative).

**How do you find the phase difference between two voltages? ›**

The most preferred way to find out the value of the phase angle is using the formula, **tanϕ=χL−χCR⇒ϕ=tan−1(χL−χCR)**. We are given that V=V0sin(1000t+ϕ). Comparing this value of the alternating voltage against the general formula, V=V0sin(ωt+ϕ), we find that the value of the angular frequency(ω)is 1000.

**What units can phase difference be measured in? ›**

Phase difference is measured in **fractions of a wavelength, degrees or radians**. In the diagram (above), the phase difference is ¼ λ. This translates to 90^{o} ( ¼ of 360^{o} ) or π/2 ( ¼ of 2π ). We can understand how this equation is constructed by introducing ω (omega) , the angular velocity (units rad s^{-}^{1} ).

**What is the range of phase difference? ›**

The phase difference of an alternating waveform can range from **0 to T throughout one entire cycle**, and this can be anywhere along the horizontal axis between ∅=0 to 2π or ∅=0 to 360 degree depending on the angular units employed.

### Why is 3 phase voltage 440 volts? ›

In Short, either Phase 1 and 2, Phase 2 and 3 or Phase 3 and 1 are positive while at the same time, either Phase 3, Phase 2 or Phase 1 are negative respectively. That's why two phases in a three phase system have 400V instead of 600, 660 or 690V.

**What happens when phase difference is 180? ›**

**Destructive interference** occurs when the maxima of two waves are 180 degrees out of phase: a positive displacement of one wave is cancelled exactly by a negative displacement of the other wave. The amplitude of the resulting wave is zero.

**Can phase shift be greater than 180? ›**

By definition, phase margin is (180 + phase angle) when the open loop gain is unity (= 0 dB). Positive phase margin thus indicates a stable closed loop. I just wanted to post an edit about this :) **A phase margin bigger than +180 degrees can be considered a phase margin less than -180 degrees**.

**What happens if phase difference is not constant? ›**

You can have a non-constant phase difference between the two sources. The result is that **the nulls will appear in different places** because the correct path lengths to cause destructive interference are different. As you adjust the phase difference, the nulls will appear to "move."

**What is phase shift between current and voltage? ›**

Capacitance in AC Circuits

Therefore a phase shift is occurring in the capacitor, the amount of phase shift between voltage and current is **+90° for a purely capacitive circuit, with the current LEADING the voltage**. The opposite phase shift to an inductive circuit.

**What is the phase difference between voltage and current? ›**

The phase difference between current and voltage in an AC circuit is **4Π radian**.

**What is the difference between phase difference and phase angle? ›**

**A phase difference is referred to as the difference between the frequencies of two waves within the same point of time** (Electrical4u, 2022). The angular component of the periodic wave is referred to as the phase angle also. "A∠θ, where A is referred to as the magnitude and θ symbolises the wave of phase angle".

**What is relation between phase difference and time difference? ›**

The time period in which waves travel between a node and antinode is the time difference. **When two or more alternating quantities approach their limit or zero values, then phase difference is used to define the difference in degrees or radians**.

**What is a phase shift in circadian rhythm? ›**

A phase shift in your circadian rhythms means that **your bedtime and wake-up time will move earlier in the day (phase advance) or later in the day (phase delay)**. The PRC is important because it can determine when to time light and melatonin correctly in order to advance (or delay) your circadian phase.

**Is phase difference always 90 degrees? ›**

**Not necessarily**. As others said, it is load dependent. For resistive load => both voltage and current remain in phase. For R-L ot L load (inductive) => current lags voltage by 90 degree (pure inductive) or less than it (R-L or R-L-C for ).

### What is phase difference in RLC circuit? ›

The value of the phase difference ϕ depends on the values of R, C, and L. **i(t)=I0sin(ωt−ϕ)**, where I0 is the current amplitude and ϕ is the phase angle between the current and the applied voltage. The phase angle is thus the amount by which the voltage and current are out of phase with each other in a circuit.

**What is phase difference in phasor diagram? ›**

A complex number that is used to represent a sinusoidal voltage or current is called a phasor. The magnitude of the phasor is the same as the maximum value of the sinusoidal waveform, and **the phase of the phasor is equal to the phase difference between the sinusoidal waveform and a cosine waveform**.

**Does phase shift depend on frequency? ›**

**The time interval for 1° of phase is inversely proportional to the frequency**. If the frequency of a signal is given by f, then the time t_{deg} (in seconds) corresponding to 1° of phase is t_{deg} = 1 / (360f) = T / 360. Therefore, a 1° phase shift on a 5 MHz signal corresponds to a time shift of 555 picoseconds.

**Does phase shift affect frequency? ›**

**The frequency is inversely proportional to the time interval for 1 degree of phase**. The frequency of a signal is given by f, and the time the (in secs) regarding one degree of phase is the = 1 / (360f) = T / 360. Therefore, a one-degree phase shift on a 5 MHz signal shows a time shift of 555 picoseconds.

**What causes phase shift? ›**

This phase shift occurs because **the inductive reactance changes with changing current**. Recall that it is the changing magnetic field caused by a changing current that produces inductive reactance.