Phasor Representation Ac

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What is Phasor Representation?

A phasor representation is represented as a complex number that is functional in a sinusoidal manner. The time-invariant parameters are amplitude (A), initial phase (θ), and angular velocity (ω). This representation is based upon the concept known as analytic representation.

A constant which is postulated by covering the frequency and dependence of time is known as phasor and complex amplitude. The term ‘phaser’ is eliminated from the diagrammatic calculations. It is equivalent to the vectors, which are possible for phasers also.

The inventor of the phasor transform was Charles Proteus Steinmetz. He was working on General electric earlier in the 19th century.

Phasor Representation AC

Electrical circuits consist of a resistor, which is linked with an AC source. An inductor is also connected with this AC source along with a capacitor or the joint of any two or all of the enlisted components linked with an AC source.

In the case of a resistor, we know that the current is present in phase with the source of voltage. When it comes to the case of a capacitor, the current either increases or decreases the source of voltage to a certain amount.

We need to use the concept of phasors to co-relate the voltage and the current. Phasor representation shows the perfect inter-relationship between the current and the voltage in a diagrammatic form.

Phasor Representation of Alternating Quantities

Before going to this topic, assume that the alternating voltages and currents obey the sine law. The generators are constructed to generate EMFs, which are generally in sine waveform. The assumption mentioned above implies the easiest form of calculation. Alternative quantities are given in the waveform. These equations are implying spontaneous values are quite inconvenient.

It is an advantage to elaborate on a sinusoidal quantity (generally voltage or current) for the solution of AC problems. It is represented by a line that is rotating in the counterclockwise direction, having a definite length with the same angular velocity. It is just like the sinusoidal quantity. This kind of rotating line is called the phasor.

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Let’s consider the OA line (also called phasor) indicating the maximum scale value. It is the maximum value of an alternating quantity which can be termed emf i.e. OA = Emax.

It is rotating counterclockwise with an angular velocity ω (rad/sec) about the point O. 

It is shown in the figure above. 

At the outside end of the phasor, arrow mark is drawn, at least to pinpoint which end is considered to move and slightly to mention the accurate length of the phasor when more than one phasors appear to synchronize. 

In the figure; OA has made a rotation with an angular velocity θ, which is equal to ωt when the emf was passing at zero magnitudes from a certain occupied position.

OB = OA sin θ, OB is represented as the projection of OA on the Y-axis.

We know, OB= OA sin θ = Emaxsin ωt = e. 

This is the value of the emf at that position.

This proves that the projection of OA which acts on the vertical axis acts in the place of the instantaneous magnitude of emf.

It has been observed that the phasor OA rotating in the direction of counter-clockwise direction will express a sinusoidal quantity (generally current and voltage), If:

  1. The length of OA is equal to the maximum value obtained from the sinusoidal current or voltage to a relevant scale.

  2. OA is in a horizontal plane at the current stage, and the alternating quantity (which is voltage or current) is zero and increasing.

  3. The angular velocity of OA is in such a state that it accomplishes one revolution at the same time. The time is the same for completing one cycle, which is taken by alternating quantity.

Phasor Representation of AC Quantities

It is a vector that is passed down to state a sinusoidal function. Phasor rotates at an angular velocity ω about the origin. The values which are sinusoidally altering can be represented by the vertical component of phasors for a given problem, like v and i. At this circumstance, the highest value of the voltage and the current is represented as the magnitude of the phasors.

From the figure shown below, we can notice the relationship between the sinusoidal representation and a phasor concerning time. The value of the quantity is represented as the projection of the phasor on the vertical axis.

For illustration, the projection of the phasor on the vertical axis is given by vmsint and imsint respectively in the case of a current or a vector phasor. 

It concludes the magnitude of the current or the voltage at that position.

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It is simple to identify that one of the two quantities is in the equal phase with the help of the phasor diagram. For illustration, the phasors for the voltage and current are in the exact direction for all occurrences for a given circuit. The phase angle will be zero in between the voltage and the current.

Representation of AC and Voltage by Phasor Diagram

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It is sometimes quite difficult to represent mathematically, to visualize the angular or phasor difference among sinusoidal waveforms. To overcome these problems, we need to represent the sinusoids graphically within the phasor-domain form with the help of the phasor diagram. 

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FAQ (Frequently Asked Questions)

1. What is the Need to Draw a Phasor Diagram for Alternating Quantities?

Ans: The phasor diagram is drawn to state the relationship between two quantities. From the diagram, you can easily manipulate which quantity lags and the phase difference as well.

2. What are the Advantages of the Phasor Diagram?

Ans: The mathematical operations, such as addition, multiplications, etc., can be carried out conveniently using the phasor diagram. The calculation can include any common values such as RMS, peak values, phase angles, etc.

3. Illustrate the Characteristics Between Phasor and Vector?

Ans: A vector illustrates quantity and direction, whereas a phasor is a rotating vector. Phasor implies quantity and displacement calculated in the counterclockwise direction. Phasors are used to represent AC quantities.

4. A Series L-R-C Circuit is Connected to a Variable Frequency 300 V Source with L = 60 mH, C = 90 F and R= 50 Ω. Find Resonant Angular Frequency and the Source Frequency Which Drives the Circuit in Resonance?

Ans: Given, L = 60mH = 60 * 10⁻³⁶ H

C = 90μF = 90* 10⁻⁶ F

R = 500π, V = 300volts

In L-R-C, the resonant angular frequency is given by:

ω₀ = 1/√LC = 1/√60*10⁻³90*10⁻⁶ = 430 rad/s 

The actual/source frequency will be ω = 2πv

v = ω/2π = 430/2π = 68.43Hz.