Symmetrical components are an approach to solve problems on unbalanced polyphase systems. CL Fortesque purposed this method in 1918. According to the Fortesque theorem, any three-phase unbalanced system quantities (such as current, voltage, or other sinusoidal quantities) can be resolved into three balanced systems of phasors known as symmetrical components. This resolved balanced system contains three sequence networks that can be solved separately on a single-phase basis. Thus solved problems in terms of symmetrical components can be brought back to the actual condition by phasor addition of these quantities or by superposition.

3-Phase System Symmetrical Components

The three symmetrical components are positive phase sequence, negative phase sequence, and zero phase sequence. These sequence components differ from each other in terms of phase sequence. It means that each of these quantities will go through a maximum value in a different order.

Positive Phase Sequence

In the positive phase sequence system, the phase of line current or voltage goes through the same phase sequence as that of normal supply. Assuming a conventional counter-clockwise direction of rotation, the positive phase sequence phasors are represented in the figure below.

Positive phase sequence symmetrical components

Figure: Positive Phase Sequence Components

A balanced three-phase system under normal conditions of operation will only contain a positive phase sequence. Also for a system under a 3-phase fault, the system will constitute positive phase sequence components only.

Positive phase sequence components are represented by subscript 1. All the three phasors are of equal magnitude with 120-degree separation. These can be represented in terms of operator ‘a’ as:

Where,

Negative Phase Sequence

In the negative phase sequence system, the phase of line current or voltage rotates counter-clockwise direction but attains a maximum in reverse order i.e A-C-B as shown in the figure below.

Negative phase sequence symmetrical components

Figure: Negative Phase Sequence Components

This type of situation arises when the system is unbalanced during an unsymmetrical fault condition. The unsymmetrical fault condition will contain the negative phase sequence component along with the positive phase sequence component. A zero-phase sequence component will also be present in the case of fault to earth.

Negative phase sequence components are represented by subscript 2. All the three phasors are of equal magnitude with 120-degree separation. These can be represented in terms of operator ‘a’ as:

Zero Phase Sequence

In the zero phase sequence system, the phase of line currents or voltages is represented by three equal phasors in phase as shown below.

Zero phase sequence symmetrical components

Figure: Zero Phase Sequence Components

This condition arises in the case of faults where earth return is present. Generally a 3-phase system with a fourth wire or earth return from which the earth currents can return to the system under fault condition, this sequence component is present. During earth fault, along with zero phase sequence components, positive and negative phase sequence components are also present.

Zero phase sequence components are represented by subscript 0.

Ao = Bo = Co

Representation of Symmetrical Components

The phasors of an unbalanced three-phase system can be represented in terms of symmetrical components as:

In terms of ‘a’ operator these can be represented as:

In matrix form, these can be represented in terms of operator ‘a’ as:


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