Phasor: Streamlining The AC Analysis (#1)
This transition to phasor analysis is one of the most effective methods, which helps to make the analysis of AC circuits clearer and less time-consuming. It transforms the point of view from the intricate and variable time domain to a more easily comprehensible frequency domain through which circuit behavior can be controlled with more efficiency.
Understanding the Time Domain
In electrical engineering, the time domain is where one sees the voltage and currents as they vary with time. It can also be noted that for most of the AC systems, the waveforms observed are sinusoidal, which means that they move between a positive and negative value in a very periodic manner. These waveforms are characterized by three main properties:
Amplitude
This is the highest value of the waveform which gives the maximum voltage or current at any instant of time.
Frequency
This is the frequency with which the waveform fluctuates; expressed in cycles per second, or Hertz (Hz). It describes the number of times that one waveform repeats itself in one second.
Phase
This shows the displacement of the waveform with respect to a given timeline of reference. In other words, it indicates the instant in the waveform’s cycle at which the waveform can be observed.
Whenever engineers solve AC circuits in the time domain, they have to look at how these sinusoidal waveforms interact with each other and with other elements such as resistors, inductors, and capacitors. Each of these components reacts differently to AC signals:
Resistors
These components oppose the flow of current and the voltage across a resistor always leads or lags the current through it in phase.
Inductors
These components resist alterations in the present, thus making the voltage to lead current by a phase of up to 90 degrees.
Capacitors
These store and release the energy in the form of electric fields and thus the voltage lags behind the current by up to 90 degrees.
However, when there are more than one sinusoidal sources or components in a circuit, the voltages and or currents may not all phase with each other but they may be out of phase. This phase difference poses a problem in the analytical process, because the two waveforms cannot be added together in a straightforward manner. However, you have to take account of their amplitudes and their phase angles, and to do that the use of trigonometric identities is necessary.
For instance, if you have two sinusoidal voltages in quadrature that are out of phase, the net result is not the algebraic sum of their respective amplitudes. You have to take into consideration how one waveform is ahead or behind the other, which makes it slightly complicated. This type of analysis can get very complicated mathematically as soon as one has to consider several interacting components in a circuit. All mathematical operations, including addition and subtraction or multiplication of waveforms, require that particular attention be paid to the amplitude as well as the phase of the signals.
Phasor Transformation
To overcome the difficulties that arise from the time-domain analysis, engineers bring in to use the phasors. A phasor is a complex number that preserves a sinusoidal waveform’s features of amplitude and phase but does not take into account the waveform’s time dependence.
If you have a sinusoidal function, when you convert it to a phasor, what you really do is lock it at a certain point of the waveform, and what you are only interested in is the steady state condition. This change transforms the analysis from the time domain to the frequency domain, where time-dependent trigonometric functions are not required. However, here you deal with the complex numbers which are the amplitude and phase of the waveform.
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In the frequency domain:
It makes various operations that would otherwise require the application of trigonometry to become much easier. For example:
One good thing about phasors is that they do not require the use of trigonometrical identities since they deal directly with the important attributes of the waveform. This makes solving circuit equations much easier and less prone to mistakes as compared to the conventional approach.
Application
However, the real strength of the phasor analysis comes to light when dealing with AC circuits in a steady-state regime. In these cases, the engineers are more interested in the global behavior of the circuit rather than the value of voltage and current at every instance of time.
Steady-state conditions imply that all the transient phenomena in a circuit—those that occur at the moment the circuit is switched on, for example, have died down, and the voltages and currents in the circuit are periodic.
In this steady-state, the use of phasors allows engineers to:
Simplify the Analysis
As the phasors eliminate the time dependency in the equations, the engineers can easily concentrate on the various relationships between the real and imaginary parts of the voltage and current. This makes it easier to assess how several parts will connect and how the circuit will work in general.
Enhance Efficiency
Phasor analysis also makes calculations easy and is very useful in gaining a better understanding of the circuit’s behavior. Whenever engineers require a fast overview of voltage drops, current flows, and power distribution, they do not need to analyze the time-domain waveforms in detail.
Improve Clarity
This is because when working in the frequency domain, the analysis is made easier and the amount of mental effort required from engineers is minimized. They are better placed to draw the circuit’s behavior in their mind and decide on the design or the problem-solving approach.
For instance, if an engineer is required to find the total current passing through a circuit loaded with different voltage sources, he or she can convert each of the voltage sources into the phasor form, perform the desired arithmetic computation, and then convert the answer back to the time domain if necessary. This process is much simpler than the attempt to add the time domain sinusoidal waveforms directly, with the help of which extensive trigonometric calculations are to be performed.
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Yes, phasor is easier to understand for a pure sinusoidal AC system. However, now the load is more non-linear, resulting in harmonics.