The phase connection between voltages and currents in a circuit is shown graphically in phasor diagrams.
They make use of a reference phasor and a coordinate system. An angular difference, or phase difference, can exist between sinusoidal waveforms of the same frequency.
This connection may be described using terms like “lead,” “lag,” “in-phase,” and “out-of-phase”. The waveform in the time domain is represented by the generalized sinusoidal formula A(t) = Am sin(ωt ± Φ).
Mathematically visualizing the angular difference can be difficult, though. This is overcome by phasor diagrams, which use rotating vectors to graphically depict the waveforms in the phasor domain.
These vectors, which are often referred to as phase vectors, show the amplitude and direction of AC values. They are stuck at a certain moment of time.
An arrow is used to symbolize a vector, the arrowhead indicates the vector quantity’s highest value, and the arrow end revolves around a fixed point called the “point of origin.” For positive values, vectors spin counter-clockwise, for negative values, they rotate clockwise.
While vectors and phasors both depict spinning lines, with magnitude and direction, in reactance-based AC circuits, phasors are used to express the “rms value” of sinusoids. Their angular velocity, direction, and phase angle are all the same.
The phase of an alternating quantity with a single vector spinning counter-clockwise at an angular velocity of ω = 2π̒f to create a full sine wave is depicted in a phasor diagram.
While phasors may be expressed mathematically in shapes like rectangular, polar, or exponential waveform, vectors obey the parallelogram law of addition and subtraction.
For example (a + jb). As a result, phasor terminology specifies the voltage and current’s useful (rms) value.
In phasor diagram construction the angular velocity of a sine wave is commonly considered to be expressed as ω in rad/sec. Take a look at the phasor diagram below.
Sinusoidal Waveform Analysis using Phasor Diagrams
As the vector rotates counterclockwise, we see that point A completes one full revolution of 360 degrees or 2π radians, it represents one complete cycle.
When you plot the moving tip’s length at different angles generates a sinusoidal waveform starting at zero time.
The horizontal axis shows time elapsed since t = 0, with the tip at 0, 180, and 360 degrees when horizontal, and at +Am at 90 degrees or π/2, and -Am at 270 degrees or 3π/2 when vertical.
We find that waveform’s time axis reflects the phasor’s angle in degrees or radians. Phasors can be used to represent scaled voltage or current values frozen at a specific angle, like 30 degrees.
It is important that you know the phasor’s position in analyzing alternating waveforms, especially when comparing voltage and current. Consider the phase difference (Φ) between waveforms if they start at different points or to represent their relationship in phasor notation.
Sinusoidal Waveform and its Phase Difference
These two sinusoidal values can be represented mathematically as follows:
- V(t) = Vm sin(ωt)
- i(t) = Im sin(ωt – Φ)
In the illustration above, the angle Φ indicates the 30° lag between the current (i) and voltage (v). The phasor diagram that results will be equal to angle Φ, which is the difference between the two phasors that signify the two sinusoidal variables.
Sinusoidal Waveform Phasor Diagram
As can be seen, the phasor diagram is based on the horizontal axis at time zero (t = 0), with phasor lengths proportional to voltage (V) and current (I) values.
Now, the current phasor lags behind the voltage phasor by an angle Φ, since both rotate counterclockwise.
Because of the shared frequency, the current phasor lags behind the voltage phasor at t = 30°.
The current phasor intersecting the horizontal zero axis serves as the new reference, with the voltage phasor “leading” by angle Φ. One phasor acts as the reference while others either lead or lag after it.
Phasor Diagrams: The Process of Adding Phasors
As we learned above, phasors are useful for summing sinusoids of the same frequency. You will see, when waveforms are in-phase, they can be added together like DC values.
However if they are not in-phase, the phase angle must be considered. We can use Phasor diagrams to add them together using the parallelogram law.
For example suppose, if V1 has a peak voltage of 20 volts and V2 has a peak voltage of 30 volts, with V1 leading V2 by 60 degrees, we can find the total voltage, VT, by constructing a phasor diagram and parallelogram.
The Process of Combining Two Phasors through Phasor Addition
If you want to find the phasor sum V1 + V2 accurately, the graphical method involves drawing the phasors to scale on graph paper and then you can measure the length of the resultant r-vector.
But, you may find this method time-consuming leading to inaccuracies if not drawn precisely. Alternatively you may also try an analytical method called as the Rectangular Form.
In this method you can determine the vertical and horizontal directions of the voltages and calculate the vertical and horizontal components of the resultant r-vector, VT.
In the Rectangular Form it utilizes the cosine, and sine rules to determine the resultant value. In the Rectangular Form we express the phasor as Z = x ± jy, where x represents the real part and y represents the imaginary part.
You will find that, this expression effectively represents both the magnitude and phase of the sinusoidal voltage.
Understanding Complex Sinusoid
v = Vm cos(Φ) + jVm (sinΦ)
Thus, when you add together vectors A and B, using the general expression mentioned earlier, the result is:
A = x + jy
B = w + jz
A + B = (x + w) + j(y + z)
Adding Phasors in Rectangular Form
As can be seen, voltage V2 with a magnitude of 30 volts is directed along the horizontal zero axis, resulting in a horizontal component of 30 volts and no vertical component.
Horizontal Component = 30 cos 0o = 30 volts Vertical Component = 30 sin 0o = 0 volts From the above we get the rectangular expression for voltage V2 as: 30 + j0.
Next, we find, voltage, V1 with a magnitude of 20 volts leads voltage V2 by 60o, resulting in both horizontal and vertical components.
Horizontal Component = 20 cos 60o = 20 x 0.5 = 10 volts Vertical Component = 20 sin 60o = 20 x 0.866 = 17.32 volts This gives us the rectangular expression for voltage V1 as: 10 + j17.32.
This gives us the resultant voltage, VT, by adding the horizontal and vertical components with each other. VHorizontal = sum of real parts of V1 and V2 = 30 + 10 = 40 volts VVertical = sum of imaginary parts of V1 and V2 = 0 + 17.32 = 17.32 volts.
Now, if you want to find the magnitude of voltage VT, you can simply use Pythagoras’s Theorem for a 90o triangle.
VT = √[(Real or Horizontal Component)2 + (Imaginary or Vertical Component)2 ]
VT = 402 + 17.322
VT = 43.6V
The phasor diagram that follows will be:
How to do Phasor Subtraction of Phasor Diagrams
Here we can see, phasor subtraction closely resembles the previous rectangular addition method, but now, the vector difference is represented by the opposite diagonal of the parallelogram, formed by V1 and V2 voltages.
How to do Phasor Subtraction of Two Phasors
Here, we are subtracting the horizontal and vertical components instead of adding them together.
- A = x + jy
- B = w + jz
- A – B = (x – w) + j(y – z)
Understanding 3-Phase Phasor Diagrams
In the past we studied single-phase AC waveforms with a rotating coil in a magnetic field.
However if you are placing three identical coils at an electrical angle of 120o to each other on the same rotor shaft, you can generate a three-phase voltage supply.
A balanced three-phase voltage supply consists of three sinusoidal voltages with equal magnitudes and frequencies, but here, they are out-of-phase with each other by 120o electrical degrees.
The three phases are commonly color-coded as Red, Yellow, and Blue, with Red being the reference phase. The rotation sequence for a three-phase supply is Red, Yellow, Blue (R, Y, B).
Similar to single-phase phasors, the phasors for a three-phase system also rotate in an anti-clockwise direction around a central point, indicated by the arrow labeled ω in rad/s.
Please find the phasors for a three-phase balanced star or delta connected system as shown below.
Three-phase Phasor Diagrams
Here we find that the magnitudes of the phase voltages are equal, but their phase angles are different.
The three coil windings are connected at points a1, b1, and c1 to establish a shared neutral connection for all three phases.
If you consider the red phase as the point of reference, the voltage of each phase can be determined in relation to the common neutral as indicated below:
- Phase (Red): VRN = Vm sinθ
- Phase (Yellow): VYN = Vm sin(θ – 120o)
- Phase (Blue): VBN = Vm sin(θ – 240o) or VBN = Vm sin(θ + 120o)
Thus, if the reference voltage is VRN, the phase sequence becomes R – Y – B. Here, yellow phase lags VRN by 120 degrees, and the blue phase lags VYN by 120 degrees.
Alternatively you can say that the blue phase voltage, VBN, leads the red phase voltage, VRN, by 120 degrees. In a balanced three-phase system, with a fixed 120-degree relationship between sinusoidal voltages, their phasor sum is always zero: Va + Vb + Vc = 0.
Conclusion
As we learned above, phasor diagrams project rotating vectors onto a horizontal axis, representing instantaneous values. They apply solely to sinusoidal AC quantities and can depict two or more stationary sinusoids at any given moment.
The reference phasor is aligned with the positive x-axis, with other phasors drawn relative to it. Phasors rotate anticlockwise leading or lagging behind the reference.
The length of a phasor signifies the rms value. Note that, different-frequency sinusoids can’t coexist on the same diagram. Yes, vectors of the same frequency can be added or subtracted to form a resultant vector. In a balanced three-phase system, each phasor is displaced by 120 degrees.
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