# Phaser Diagram

A Phaser Diagram is a graphical way to represent the relationship between size and direction between two or more variable quantities.

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Sinusoidal waveforms of the same frequency can have a Phase Difference between themselves, representing the angular difference of two sinusoidal waveforms.In addition, the terms "progress(leading)" and "lagging", and "intra-phase" and "out of phase", are commonly used with the generalized sinusoidal expression given as follows: A _{(t) )} = A m sin(ωt ± Φ), which represents sinusoid in the form _{of} a time zone.

But when presented mathematically in this way, it is sometimes difficult to visualize this angular or phaser difference between two or more sinusoidal waveforms.One way to overcome this problem is to graphically show sinusoids in **spatial** or phaser-field format using **Phaser Diagrams,** which is achieved by the rotating vector method.

Basically, a rotating vector simply called "**Phaser**" is a scaled line that represents a scaled line whose length represents both its size ("peak amplitude") and direction ("phase"), which at some point in time is "frozen".

A phaser is a vector with an arrowhead at one end that partially represents the maximum value of the vector amount (V or I) and partly the tip of the rotating vector.

In general, vectors are assumed to revolve around a fixed zero point at one end known as the "point of origin", while the angular velocity ( ω ) of the arrowhead representing the quantity is assumed to rotate freely **counterclockwise** at a full angular speed.This counterclockwise rotation of the vector is considered a positive rotation.Likewise, a clockwise rotation is considered a negative return.

Although the terms both vectors and phasers are used to describe a rotating line with both size and direction, the main difference between the two is that a vector size is the "peak value" of the sinusoid, while a phaser size is the "RMS value" of the sinusoid. In both cases, the phase angle and direction remain the same.

The phase of a quantity that changes at any time over time can be represented by a phaser diagram, so phaser diagrams can be considered "functions of time".A full sine wave can be generated by a single vector rotating at an angular speed of ε = 2πε, where ε is the frequency of the waveform.Then a **Phaser**is a quantity that has both "Size" and "Direction".

In general, when creating a phaser diagram, the angular velocity of the sine wave is always assumed to be ω in rad/sec.Consider the following phaser diagram.

### Phaser Diagram of a Sinusoidal WaveForm

Because the single vector rotates counterclockwise, the tip at point A will rotate to a full round of 360 ^{o} or 2π, representing a full loop.If the length of the moving tip is transferred to a chart as shown above at different angular intervals over time, a sinusoidal waveform will be drawn starting from the left over zero time.Each position along the horizontal axis indicates the time since zero time, t = 0.When the vector is horizontal, the tip of the vector represents angles of 0 ^{o} , 180 ^{o,} and 360 ^{o.}

Similarly, when the tip of the vector is vertical, it represents a positive peak value of 90 ^{o} or π/2 ( +Am ) and a negative peak value ( -Am ) at 270 ^{o} or 3π/2.The timeline of the waveform then represents the angle in degrees or radians at which the phaser moves.Therefore, we can say that a phaser represents a scaled voltage or current value of a rotating vector that is "frozen" at some point in time ( t ), and in our example above this is at an angle of 30 ^{o.}

Sometimes when analyzing alternative waveforms, especially when we want to compare two different waveforms on the same axis, we may need to know the position of the phaser representing the Varying Amount at a certain time over time.For example, voltage and current.In the waveform above, we assumed that the waveform started instantly at t = 0 with a corresponding phase angle in degrees or radian.

However, if a second waveform begins to the left or right of this zero point, or if we want to show the relationship between the two waveforms in phaser notation, then we will have to take into account this phase difference of the waveform, Φ.See the diagram below that we used in the previous Phase Difference tutorial.

### Phase Difference of a Sinusoidal WaveForm

To define these two sinusoidal quantities, the generalized mathematical expression will be written as follows:

The current lags behind the I voltage, the V angle is Φ and in our example above this is 30^{o.} Therefore, the difference between the two phasers representing the two sinusoidal amounts is the angle Φ, and the resulting phaser diagram will be as follows:

### Phaser Diagram of a Sinusoidal WaveForm

The phaser diagram is drawn on the horizontal axis to correspond to zero time ( t = 0 ).The lengths of the phasers are proportional to the voltage, ( V ) and current ( I ) values at the time the phaser diagram is drawn.Since the two phasers *rotate counterclockwise* as previously stated, the current phaser outperforms the voltage phaser by as much as the Φ angle, so the Φ angle is measured in reverse of the same clockwise direction.

However, if the waveforms t = 30 are frozen at ^{that} time, the corresponding phaser diagram will resemble the one shown on the left.Since the two waveforms are at the same frequency, the current phaser once again lags behind the voltage phaser.

However, since the current waveform crosses the horizontal zero axis line at this moment, we can use the current phaser as our new reference and accurately say that the voltage phaser "directs" the current phaser at an angle. In both cases, a phaser is designated as a reference phaser, and all other phasers will be front or behind according to this reference.

## Adding a Phaser

Sometimes when studying sinusoids, for example, in an AC series circuit, it is necessary to combine two alternative waveforms that are not in-phase with each other. If it is in-phase, that is, if there is no phase shift, they can be combined in the same way as DC values to find the algebraic sum of the two vectors. For example, if the two voltages of 50 volts and 25 volts, respectively, are "in-phase" together, they are combined to create a voltage of 75 volts (50 + 25).

However, if they are not in-phase, that is, they do not have the same directions or starting point, then the phase angle between them must be taken into account, so that they are put together using phaser diagrams to determine the result phaser or vector sum using the parallelogram law.

Consider the two AC voltages in which the V1 has a peak voltage of 20 volts and the V2, which has a peak voltage of 30 volts, in which v1 directs the V2 with 60o. The total voltage of the two voltages, VT,can be found first by drawing a phaser diagram representing two vectors, and then by creating a parallelogram in which the two sides have V1 and V2 voltages, as shown below.

### Adding Phasers of Two Phasers

By drawing the two phasers to be scaled on graphics paper, the Phaser total V1 + V2 can be easily found by measuring the length of the diagonal line known as the "result r-vector" from the zero point to the intersection of the 0-a structure lines. The disadvantage of this graphics method is that it is time consuming when drawing phasers to scale.

In addition, while this graphical method gives an accurate enough answer for most purposes, it can generate an error if it is not scaled correctly.Then one way to ensure that the right answer is always obtained is an analytical method.

Mathematically we can collect two voltages first by finding their "vertical" and "horizontal" directions, and after that we can calculate both "vertical" and "horizontal" components for the resulting "r vector" V _{T} .This analytical method, which uses the cosine and sinus rule to find this result value, is often called **a Rectangular Form.**

In rectangular form, the phaser is divided into a real part, x, and a virtual part, creating the expression y generalized Z = x ± jy.(We will discuss this in more detail in the next tutorial).This gives us a mathematical expression that represents both the magnitude and phase of sinusoidal voltage as follows:

### Description of Complex Sinusoid

Therefore, adding two vectors, A and B, using the previous generalized expression is as follows:

## Phaser Collection Using Rectangular Form

The voltage, 30 volt V _{2} , indicates the reference direction along the horizontal zero axis, then has a horizontal component as follows, but there is no vertical component.

- • Horizontal Component = 30 cos 0
^{o}= 30 volts - • Vertical Component = 30 sin 0
^{o}= 0 volts - This then gives us the voltage rectangular expression
_{V2}: 30 ± j0 of the region

Voltage, V _{1} /20 volt voltage, V _{2} x 60 ^{o} , then it has both horizontal and vertical components as follows.

- • Horizontal Component = 20 cos 60
^{o}= 20 x 0.5 = 10 volts - • Vertical Component = 20 sin 60
^{o}= 20 x 0.866 = 17.32 volts - This then gives us the voltage rectangle expression V
_{1}: 10 ± j17.32

The resulting voltage, V _{T} , is found by collecting horizontal and vertical components as follows.

- V
_{Horizontal}= Sum of the actual parts of V_{1}and V_{2}= 30 + 10 = 40 volts - V
_{Vertical}= Sum of virtual parts of V_{1}and V_{2}= 0 + 17.32 = 17.32 volts

Now that both real and virtual values are found as the magnitude of the voltage, V _{T} is determined using **pythagorean theorem** for the triangle of 90 ^{o} as follows.^{}

The resulting phaser diagram will be as follows:

### V _{T Result}Value

## Phaser Extraction

Phaser extraction is very similar to the above rectangular collection method, but this time the vector difference is the other diagonal of the paralleloken between the two voltages of V1 and V2, as shown.

### Vector Extraction of Two Phasers

This time we remove both horizontal and vertical components instead of "adding" them together.

## 3-Phase Phaser Diagram

Previously, we looked at single-phase AC waveforms in which a single multi-turn coil rotates within a magnetic field.However, if three identical coils, each with the same number of coil rotations, are placed together at an electric angle of 120 o on^{the}same rotor shaft, a three-phase voltage source will be produced.

A balanced three-phase voltage source consists of three separate sinusoidal voltages, all equal in size and frequency, but exactly 120^{degrees} out of phase with each other.

Standard practice is to color-encode three phases in Red, Yellow, and Blue to define each phase with a red phase as a reference phase.The normal rotation order for a three-phase feed is Red, followed by Yellow, followed by Blue, ( R , Y , B ).

As with the above single phase phasers, phasers representing a three-phase system rotate counterclockwise around a central point, as shown by the arrow marked with ω in rad/s.Phasers for a three-phase balanced star or triangular connected system are shown below.

### Three Phase Phaser Diagram

All phase voltages are equal in size, but differ only at phase angles.The three windings of the coils are connected at points a _{1} , b _{1} and c _{1} to form a common neutral connection for three separate phases.Then, if the red phase is taken as a reference phase, each separate phase voltage can be defined in relation to the common neutral.

### Three Phase Voltage Equations

If the red phase voltage is taken as the reference voltage as previously stated,_{the} phase sequence will be R – Y – B, so that_{}the voltage in the yellow phase outperforms the V_{RN}by 120^{o}and the voltage in the blue phase is 120^{o.} but we can also say the blue phase voltage, V_{BN} red phase voltage, V_{RN}120^{o.}

One last point about a three-phase system. Three separate sinusoidal voltages are said to be "balanced" as they have a constant relationship of 120^{o} between each other, so in a balanced set of three-phase voltages the phaser total will always be zero: V_{a} + V_{b} + V_{c} = 0

## Phaser Schema Summary

In simplest terms, phaser diagrams are projections of a rotating vector to a horizontal axis that represents an instantaneous value.Since a phaser diagram can be drawn to represent any moment of time and therefore any angle, the reference phaser of an alternative quantity is always drawn along the direction of the positive x-axis.

- Vectors, Phasers and
**Phaser Diagrams**apply only to sinusoidal AC alternative sizes. - A Phaser Diagram can be used to represent two or more stable sinusoidal amounts at any time.
- Usually the reference phaser is drawn along the horizontal axis and other phasers are drawn at that moment.All phasers are drawn according to the horizontal zero axis.
- Phaser diagrams can be drawn to represent more than two sinusoids.They can be voltage, current or any other variable quantity, but they must all
**have the same**frequency . - All phasers are drawn counterclockwise.It is said that all phasers in front of the reference phaser are "pioneers", while all phasers behind the reference phaser are said to be "delayed".
- In general, the length of a phaser represents the rms value of the sinusoidal amount rather than its maximum value.
- Sinusoids of different frequencies cannot be shown in the same phaser diagram due to the different speeds of the vectors.At any time, the phase angle between them will be different.
- Two or more vectors can be aggregated or removed together, and
**the result**can become a single vector called a Vector. - The horizontal side of a vector is equal to the actual or "x" vector.The vertical side of a vector is equal to a virtual or "y" vector.The hypotension of the resulting right-angled triangle is equivalent to the vector "r".
- In a balanced three-phase system, each phaser is
^{replaced}by 120.

In the next course on AC Theory, we will look at representing sinusoidal waveforms in Rectangular form, Polar form and Exponential form as Complex Numbers.