Phasors in AC Circuit Analysis


Phasor is a vary important mathematical concept used frequently for Alternating Current(AC) circuit analysis. I will try to explain some basic properties of phasors that you studied(or did not study) in school. My main purpose is to help you build up sufficient background for the next article to be published which is about transformer vector group. But in general understanding the phasor will help you in analyzing the AC circuits more efficiently. Just little middle school maths.


Phasors

The  voltage so the current  produced from the generator are alternating in nature. Hence the whole transmission and distribution system voltage and  current are alternating in nature. (see figure below). Here the value of voltage increases from 0 to Vmax then decreases to become zero then its polarity changes and again  it becomes -Vmax and then returns back to zero again. Now one cycle or 360 degree is completed. As the shape of this wave is the trigonometric sine wave so it is called sinusoidal wave. Also see Fig-A in which two sinusoidal waves are drawn



In a 60 Hz system, in one second the voltage or current wave completes 60 such cycles.

Any sinusoidal wave whether it represent current or voltage cycles or something else are actually represented in the form of trigonometric equation  V= Vsin (x), where V is the magnitude and x is the angle covered. If we will start solving AC circuits using this type of equations then it will be extremely difficult to solve the large AC circuit problems.

The concept of phasor is used to simplify any AC circuit problem.

The beauty is that any sinusoidal wave can be represented by a phasor. The phasor is like vector. It has magnitude and arrow direction as shown in Fig-A. Remember that current or voltage are not vectors.

Phasor is a mathematical tool which has made calculations in electrical engineering simple. In mathematics the term complex number or complexor is used in place of phasor.

The sinusoidal voltage wave can be equivalently represented by a phasor rotating anticlockwise, centered at origin. If there are two voltage waves  then they can be represented by two phasors, the length of each phasor proportional to the magnitude of respective voltage. see Fig-A. In the figure red and green phasors correspond to red and green sinusoidal waves respectively. Here in this case we have taken the magnitude of green phasor about half that of red phasor and the green phasor is 60 degrees behind the red phasor. As the phasors always rotate in anticlockwise direction, from the phasor diagram it is clear that the green phasor is behind the red phasor. In the sinusoidal waves diagram you may feel it confusing. Just think that red wave started its positive cycle at '0' degree (origin), but the green wave still have to go 60 degrees to start its positive journey. Also imagine that green wave will attain its maximum (Vmax/2) after  rotating 60 degrees after red wave has attained its maximum (Vmax).   Carefully compare both the sinusoidal and phasor representation. In simple AC circuits the phase difference between voltage and current waves arises due to reactive circuit elements like inductors and capacitors. The angle of phase difference depends on the numerical values of the reactive elements and active elements (resistance).


On paper, if  you draw  1 inch phasor for 11 kV voltage then 3 inch should be the length of 33 kV phasor.  Both these phasors rotating about the origin in anticlockwise direction. For 60 Hz system  all the phasors will complete 60 rotations (or 60 cycles) in one second. The two phasors will never cross each other. The angle between the phasors the phase difference will never change as the phasors rotate.

A phasor can be written in the form,  A + j B. It is called Rectangular form (other is Polar form which we will not analyze here ).

In rectangular form A is the active part and B is reactive part and j is a symbol or operator


                                         examples V1=3 +j 2, V2=5+j 7  etc.



Addition of Phasors

Two phasors can be added and gives another phasor.

        
         If     V1=3 +j 2 and  V2=5+j 7

         Then   V1+V2 = (3+5) + j (2 +7) = 8 + j 9

The two phasors can be added geometrically to get the same above result. One phasor's head is joined to other phasor's tail. Final or resultant phasor is the arrow from first phasor's tail to other phasor's head. see the Fig-B and analyze carefully.



In this manner you can add several phasors to get the resultant phasor. I have illustrated the addition of five numbers of phasors in Fig-C below, you can add thousands (if you have time and patience). The phasors are just rearranged to get the resultant phasor R. I have taken the advantage of colours to better visualize. Remember you can add five phasors any order (without changing the direction and angle of each ), that means any one can be first and any one can be last, but still you will get the same phasor R (same length and angle). try it.




Subtraction of Phasors

Similarly one phasor can be subtracted from the other,

         V2-V1 = (5-3) + j (7-2) = 2+j 5.

Remember that V2-V1 can be written as V2 + (- V1). Geometrically to find -V1, just reverse the direction of V1. Now add V2 to this reversed V1 to get V2-V1. Try it.

Phasors can also be multiplied and divided. We will discuss whenever we require.

I am concluding this session about phasor hoping that it will help you better understand the future articles in electrical systems.

Magnitude of Phasor

A phasor has a magnitude. Simply speaking it is the length of the phasor. It is a value. Symbolically the magnitude of the phasor V1 is  written as |V1|.

As in our example V1 = 3+j2

Using Pythagoras theorem we get the magnitude of V1 as
                  
                                                                |V1| = √(32+22)

For more see j and a operator







Three Phase Transformer Basics



The last post was about  single phase transformer. The theory is quite easy to understand. It is time for the three phase transformer. The basic theory remain the same. The three phase transformer can be realized by properly connecting three numbers of single phase transformer or designed as a single unit. The three nos of single phase type requires more materials and costlier where as the single unit three phase transformer requires less materials and so cheaper. When a winding fault occurs in one unit of the three single phase type then only that particular unit is replaced by a similar unit, but a winding fault in three phase single unit type requires replacement of the complete transformer.  The three single phase type requires more space.


The three single phase type transformer are mainly used for extra high tension bulk power transmission and at generating stations. In this case four single phase transformers are used. Three single phase units are connected to grid and the fourth one is kept ready. Many times this type of arrangement is also done in Hydro power stations or other hilly areas where transportation of large single three phase unit  is not convenient or road permit is not available. Where space availability at the switch yard is less, the single unit 3-phase type should be preferred. 

We have already discussed about single phase type. In the diagram it is shown how three numbers of single phase units can be connected for D-Y arrangement (primary in delta and secondary in star or Y). Three single phase units also called bank of transformers.




In the diagram the windings of the same phase are colored same for easy understanding. The Vector(phasor) diagram shown below is also colored. The arrows denote the voltage in a particular winding. The magnitude of  red, green and blue arrows are same, denoting the same magnitude of voltage in all the three windings. The directions are 120 degrees apart, means the voltage waves are 120 degrees phase displaced from each other . The direction of arrow for example is BA, which is due to the polarity (dot mark) of winding shown ( you can think that 'A' is positive with respect to 'B'). similarly AC and BC. A balanced three phase system will always form the sides of the equilateral triangle. It is simple to remember that as the winding connections form the delta or star so also their respective voltage phasors.






The delta side has a three-phase balanced supply, so also the voltage induced in the windings of the star(Y) side.


For example in the delta (primary) side the voltage across the green color winding is CB(from C to B). The basic theory(recall the single phase case) says that the voltage induced in the star (secondary) green color winding is Nb (from N to b).
So in the phasor diagram we have shown both green and pointing in the same direction. Similarly it is easy to think about the other windings (color wise). BA has the same direction as Na and AC has the same direction as Nc.


The important thing is that CB is the line voltage but Nb is the phase voltage Also BA and and AC are line voltages where as Na and Nc are phase voltages.



Using geometry, the line voltage and phase voltage magnitude and phase angle can be calculated. The line voltage magnitude is calculated by multiplying phase voltage with square root of 3.


                           Vline = 1.73 x Vph



From our phasor diagram it is clear that there is 30 degree phase difference between Na and NA. More accurately voltage phasor Na is 30 degrees ahead of phasor NA. That is why voltage phasor ba is also 30 degrees ahead of phasor BA. 


In a three phase transformer, the voltage is not only stepped up or stepped down, but also a phase displacement occurs between the  primary and secondary sides. There are several other ways of connecting the primary and secondary side windings which is done as per the requirements of customer.



The above theory is true for both types of transformers. As both primary and secondary side windings and connection are visible in above type, so we have chosen the above for our analysis. Next we describe in brief the single unit three phase type transformer and also some common transformer considerations.



For more detail about vector group click this link


Three Phase Transformer (single Unit)

In the diagram below is shown a three phase transformer basic design.The transformer has three limbs. Each phase winding is wound around one of the limbs. Each limb has two windings the primary and secondary windings. On each limb, the low voltage winding is placed nearer to the steel core and the high voltage winding is placed over the low voltage winding. Insulation is placed between the low voltage and high voltage windings. Also the low voltage winding is insulated from the core. The reason of placing the low voltage winding nearer to the core is the requirement of less insulation.  See the diagram below. In the diagram 'Top View' it is clearly shown how the primary and secondary windings are wound on the same leg of the core for each phase.





 To reduce eddy current losses the core of a transformer is made of thin sheets of silicon steel stacked together. The sheets of silicon steels are insulated from each other. The core assembly is put inside the steel tank filled with oil. The oil of the transformer acts as insulator between the winding coils and steel tank. The oil of the transformer also helps in cooling the transformer(more to discuss later). The transformer oil also dampen the noise originating from the core assembly.

A balanced electrical system analysis is done in per phase basis. The transformer per phase impedance is very important for this purpose. The manufacturers of transformer provide this per phase impedance along with other transformer parameters and test values. The transformer is designed for achieving a specified impedance value. For distribution transformer the impedance is around 4 ohm and for large power transformer it may be 15 ohm or more as specified by the customer.

We will discuss more about transformer in future. In the next post I hope for a little change