What is Phasor Representation?
AC phasor representation is an extremely important part of physics; it is taught in chapter 7 alternating current in class 12 physics, NCERT books that follow the CBSE curriculum. There are two types of currents that are studied in depth in class 12 physics that are alternating-current AC and direct current DC. This article mainly talks about alternating current AC.
The concept of AC phasor representation can only be cleared after studying direct current DC. DC current is a direct current that doesn’t change direction with time. However, voltages and currents that change with time are common. It is common in the sense that it is used in our homes and offices as the electric mains supply is a voltage which varies like a sign function with time. Such voltages are called alternating voltage or AC voltage and the current which is generated by it in a circuit is called alternating current or AC current. Nowadays the most common source of distributing electricity in homes and offices is an AC voltage. This is mainly because electrical energy which is sold by power companies is distributed as alternating current. AC voltage is extremely efficient in the sense that it can easily and efficiently be converted from one voltage to the other with the help of transformers. Another reason for its efficiency is that it can be helpful in distributing electricity over long distances.
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 for 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 imply 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.
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 OF. It is shown in the figure above. At the outside end of the phasor, an 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 counterclockwise direction will express a sinusoidal quantity (generally current and voltage). If the length of OA is equal to the maximum value obtained from the sinusoidal p current or voltage to a relevant scale. OA is in a horizontal plane at the current stage, and the alternating quantity (which is voltage or current) is zero and increasing. 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 quantities.
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. In 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 Vm sin t and i(m)sin t respectively in the case of a current or a vector phasor. It concludes the magnitude of the current or the voltage at that position.
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
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.
FAQs on Phasor Representation Ac
1. What is a phasor in the context of AC circuits?
A phasor is a rotating vector used to represent a sinusoidally varying quantity like an alternating current (AC) or voltage. Its length corresponds to the peak value (amplitude) of the quantity, and it rotates in a counter-clockwise direction at the same angular frequency (ω) as the AC quantity. The instantaneous value of the current or voltage at any time 't' is given by the projection of this rotating phasor onto the vertical axis.
2. How are alternating voltage and current expressed using phasors?
An alternating voltage (v) and current (i) are represented by their respective phasors. The standard equations they represent are:
- Voltage (v): v = Vm sin(ωt), where Vm is the peak voltage.
- Current (i): i = Im sin(ωt + φ), where Im is the peak current and φ is the phase angle.
A phasor diagram visually shows these two rotating vectors. The angle 'φ' between them indicates the phase difference, showing whether the current leads or lags the voltage in the circuit.
3. What do the terms 'leading' and 'lagging' signify on a phasor diagram?
In a phasor diagram, these terms describe the relative phase relationship between two alternating quantities, typically voltage and current:
- Leading: A quantity is said to 'lead' another if its phasor is positioned further in the counter-clockwise direction of rotation. For instance, in a purely capacitive circuit, the current leads the voltage by 90° (π/2).
- Lagging: A quantity 'lags' another if its phasor is behind in the counter-clockwise rotation. In a purely inductive circuit, the current lags behind the voltage by 90° (π/2).
4. How do the phasor diagrams for a pure resistor, inductor, and capacitor differ in an AC circuit?
The key difference is the phase angle (φ) between the voltage (V) and current (I) phasors:
- For a Resistor: The voltage and current are in phase (φ = 0°). Their phasors are parallel and rotate together.
- For an Inductor: The voltage leads the current by 90° (φ = +90° or +π/2). The voltage phasor is 90° ahead of the current phasor.
- For a Capacitor: The voltage lags the current by 90° (φ = -90° or -π/2), which means the current leads the voltage. The current phasor is 90° ahead of the voltage phasor.
5. Why is a phasor representation used for AC circuits instead of just sinusoidal equations?
While sinusoidal equations (like v = Vm sin(ωt)) are accurate, they become very complex to add or subtract, especially in circuits with multiple components (like a series LCR circuit). Phasor diagrams simplify this analysis by transforming a difficult trigonometric problem into a much simpler geometric one. Adding alternating quantities is reduced to performing vector addition on their phasors, making it easier to find the resulting amplitude and phase angle.
6. How is a phasor different from a standard vector?
While both have magnitude and direction, they represent fundamentally different concepts. A vector typically represents a physical quantity in space, such as force or velocity. In contrast, a phasor is a complex number, often visualised as a 'rotating vector' in a complex plane. Its magnitude is the amplitude of a sinusoidal wave, and its angle represents the wave's phase. A phasor is not a spatial quantity; it's a mathematical tool for analysing time-varying quantities like AC.
7. How does a phasor diagram represent the EMF produced in an AC generator?
In an AC generator, a coil rotating in a magnetic field induces a sinusoidal EMF (electromotive force), expressed as e = e₀ sin(ωt). This is represented by a single phasor whose length equals the peak EMF (e₀). This phasor rotates counter-clockwise with an angular velocity ω. At any given moment 't', the instantaneous EMF (e) is equal to the vertical projection of this rotating phasor.
8. Why is the counter-clockwise direction chosen as the standard for phasor rotation?
The counter-clockwise rotation is a mathematical convention that aligns with the standard representation of angles in trigonometry and the complex plane. In mathematics, positive angles are measured counter-clockwise from the positive horizontal axis. Since phasors are a graphical application of complex numbers (e.g., Vm * e^(jωt)) to solve AC circuit problems, adopting this universal convention ensures consistency and avoids confusion in calculations and diagrams.
