Symmetrical Components and Sequence impedances

Symmetrical Components:

A system of three unbalanced phasors can be resolved in the following three symmetrical components:

Positive Sequence: A balanced three-phase system with the same phase sequence as the original sequence.They are displaced from each other by 120° & they are of equal magnitudes.

Negative sequence: A balanced three-phase system with the opposite phase sequence as the original sequence.They are displaced from each other by 120° & they are of equal magnitudes.

Zero Sequence: Three phasors that are equal in magnitude and are in the same phase.

Physical Significance of Sequence Components:

• If a set of positive sequence voltages is applied to the stator winding of the alternator,the direction of rotation of stator field is same as the rotor.
• If the direction of stator field is opposite to that of the rotor, the set of voltages are negative sequence voltages.
• The zero sequence voltages are single phase voltage and therefore they give rise to an alternating field in space.

Sequence Components:

First of all,we must clear about the operator used in sequence components:

The operator relation holds

So,

Unsymmetrical components can be written in the matrix form as;

Then,symmetrical components can be calculated as;

Defining the vectors V _a012 and V _abc as

By this way,symmetrical components can be calculated.

Sequence Impedances:

The voltage drop caused by a current of the certain sequence depends on the impedance of that part of circuit and current of that sequence only.The impedance of a circuit when positive sequence current alone is flowing is called impedance to positive sequence current or simply positive sequence impedance.It is normally represented by Z₁.

Similarly, Negative sequence impedance and zero sequences impedance can be defined.They are normally represented by Z₂ and Z₀ respectively.

For the static devices such as transformers and transmission lines, positive sequence impedance Z₁ is equaled to negative sequence impedance Z₂ .i.e.Z₁=Z₂,for the static devices.

But zero sequence impedance for static devices depends on the return path.Such as grounded neutral wire isolated neutral or delta connected.For the rotating device,positive sequence impedance is generally not equal to negative sequence impedance.But zero sequence impedance is generally less than positive and negative sequence impedances.

For the symmetrical components of impedance such that Ohm's law is satisfied:

We have,

From the original unbalanced phasors,

So we can write,

the impedance is a symmetric 3 x 3 matrix that consists of self-impedances on the diagonal and mutual impedances (between phases) on the off-diagonals.

Hence,By the Ohm's law;