Calculation of Symmetrical Components


We are now aware of the concept of Sequence components of current / voltage. If you have miss this concept, please read Concept of Symmetrical Components.

Now we are at a stage to calculate the zero, positive and negative sequence components of current / voltage. As already discussed any three phase unbalanced voltage / current can be resolved into three set of balanced vectors. Thus we will use this concept to calculate the positive, negative and zero sequence components of voltages. Mind that the same philosophy is applicable for current also.



Before going into the calculation part, let us introduce ourselves with an operator λ. λ is an operator which when multiplied to any vector quantity, rotates the vector by an angle of 120° in anticlock wise direction without changing the magnitude of the vector. This means that λ must have a magnitude unity. From this definition we can write λ as below.

λ = ei2π/3

   = Cos(2π/3) + jSin(2π/3)

   = -0.5 + j0.866

Why not to explore more properties of λ? Sure, we must…

λ2 = ei4π/3

   = Cos(4π/3) + jSin(4π/3)

   = Cos(2π - 2π/3) + jSin(2π - 2π/3)

   = Cos(2π/3) - jSin(2π/3)

   = -0.5 - j0.866

and

λ3 = ei6π/3= ei2π

    = Cos(2π) + jSin(2π)

    = 1

⇒ λ3– 1 = 0

⇒ (λ + 1)(1 + λ2 + λ) = 0

As (λ + 1) cannot be zero, therefore

1 + λ2 + λ = 0

Thus to summarize the properties of operator λ,

λ3= 1

λ4= λ3. λ = λ

1 + λ2 + λ = 0

Consider the figure below where a three phase unbalanced voltages Va, Vband Vc are resolved into three set of balanced voltages.



According to the Concept of Symmetrical components,

Va = Va1+ Va2 + Va0  …………………(1)

Vb = Vb1+ Vb2 + Vb0 ………………….(2)

Vc = Vc1+ Vc2 + Vc0 …………………..(3)

But taking Va1 reference and applying the concept of operator λ,

Vb1 = λ2Va1

Vc1 = λVa1

Similarly for Negative Sequence we can write as

Vb2 = λVa2

Vc2 = λ2Va2

Fortunately for Zero Sequence,

Va0 = Vb0= Vc0

Thus from equation (2) and (3),

Vb = λ2Va1+ λVa2 + Vb0  ………………(4)

Vc = λVa1+ λ2Va2 + Vc0  ……………….(5)

Now, multiplying equation (4) by λ and (5) by λ2 and adding them to equation (1), we get

Va + λVb+ λ2Vc

= Va1(1+ λ3+ λ3) + Va2(1+ λ2+ λ4) + Va0(1+ λ + λ2)

= 3Va1 + Va2(1+ λ + λ2)

= 3Va1

⇒ Va1 = (Va+ λVb + λ2Vc ) / 3  …………………(6)

For getting negative sequence component, multiply equation (4) by λ2 and (5) by λ & add them to equation (1),

Va + λ2Vb+ λVc

= Va1(1+ λ4+ λ2) + Va2(1+ λ3+ λ3) + Va0(1+ λ + λ2)

= 3Va2

⇒ Va2 = (Va+ λ2Vb + λVc) / 3  ……………………(7)

For Zero Sequence component, add equation (1), (4) and (5),

Va + Vb+ Vc

= Va1(1+ λ+ λ2) + Va2(1+ λ+ λ2) + 3Va0

= 3Va0

⇒ Va0 = (Va+ Vb + Vc) / 3  ……………………(8)

Therefore from equation (6), (7) and (8), we have completely calculated the positive, negative and zero sequence voltages.

In the same way, we can calculate the three components of currents. For currents we can write as below.

Ia1= (Ia + λIb + λ2Ic ) / 3 

Ia2= (Ia + λ2Ib + λIc) / 3 


Ia0= (Ia + Ib + Ic) / 3  

Comments

Post a Comment

Popular posts from this blog

Concept of Subtransient, Transient & Steady State

Image
The concept of Subtransient, Transient and Steady State arises in case of fault in an Alternator. Let us assume a sudden short circuit in three phase of alternator. The fault current will flow in all the three phases of alternator and its waveform will be as shown in figure below. When the alternator is short-circuited, the currents in all the three-phases rise rapidly to a high value of about 10 to 18 times of full load current, during the first quarter cycle. The flux crossing the air gap is large during a first couple of cycles. The reactance during these first two or three cycle is least and the short circuit current is high. This reactance is called subtransient reactance and is denoted by X". The first few cycles come under sub-transient state. After a first few cycles, the decrement in the r.m.s. value of short circuit current is less rapid than the decrements during the first few cycles. This state is called the Transient  State and the reactance in th...
Read more

Open Circuit and Short Circuit Characteristics of Synchronous Machine

Image
Open Circuit Test and Short Circuit Test are performed on a Synchronous Machine to find out the parameters of Synchronous Machine and hence to have an idea of their performance.  Open Circuit Test of Synchronous Machine is also called No Load, Saturation or Magnetizing Characteristics for the reason which will be clear after going through the post. For getting the Open Circuit Characteristics of Synchronous Machine, the alternator is first driven at its rated speed and the open terminal voltage i.e. voltage across the armature terminal is noted by varying the field current. Thus Open Circuit Characteristic or OCC is basically the plot between the armature terminal voltage E f versus field current I f while keeping the speed of rotor at rated value. It shall be noted that for OCC, the final value of E f shall be 125% of the rated voltage. Figure below shows the connection diagram for performing the Open Circuit Test of Alternator. As clear from the figure a...
Read more

Transformer Cooling Classes

Image
Losses in the transformer are of the order of 1% of its full load kW rating. There are mainly two types of losses in a Transformer i.e. Core Loss and Ohmic Loss. Because of these losses the winding of transformer as well as Core gets heat up and we need to provide sufficient cooling to remove heat and maintain a constant winding and Core temperature. Various types of cooling methodologies are used for Transformer but before going to each methodology, we will have a look at the General Method of designating a particular cooling method i.e. class. In the nameplate of Transformer, four letter represents the cooling method of the Transformer. The above table is self explanatory, therefore if you see ONAN on the Name Plate of Transformer then it means that Oil with flash point less than 300° is used as cooling media and Natural convection is used for removing heat. In secondary side of cooling mechanism, Air with Natural convection is used by employing Radiator. ...
Read more