Normalized and Relative Symmetrical Components

Introduction

Classical symmetrical components express an unbalanced three-phase system in terms of positive, negative and zero sequence components. While these quantities accurately describe the system, they do not directly indicate the relative contribution of each sequence.

NARSC - Normalized and Relative Symmetrical Components 


The Advanced Symmetrical Components (ASC) framework introduces two new dimensionless representations:

  • Normalized Symmetrical Components (NSC) – referenced to the Symmetrical Effective Phase (SEP) Voltage.

  • Relative Symmetrical Components (RSC) – referenced to the Symmetrical Effective Line (SEL) Voltage.

These representations simplify the interpretation of voltage unbalance and provide a geometric understanding of three-phase systems.

1. Relative Symmetrical Components (RSC)

Relative Symmetrical Components are obtained by dividing each sequence component by the Symmetrical Effective Line (SEL) Voltage.


The three parameters satisfy


where

  • τ (TAU) = Tangent Angle of Unbalance

  • η (ETA) = Earth Tangent Angle

Thus, the complete state of voltage unbalance can be represented using only two angles, TAU and ETA.

2. Normalized Symmetrical Components (NSC)

Normalized Symmetrical Components are obtained by dividing each sequence component by the Symmetrical Effective Phase (SEP) Voltage.

The normalized components satisfy the unit-sphere relationship

This equation shows that every three-phase system corresponds to a point on the surface of a unit sphere.

Relationship Between RSC and NSC

The two representations are related by



Therefore,

  • RSC describes the relationship between positive and negative sequence components using the SEL reference.

  • NSC provides a fully normalized three-dimensional representation referenced to SEP.

Physical Interpretation

Relative Symmetrical Components (RSC)

  • Independent of overall voltage magnitude.

  • Describes line-voltage asymmetry.

  • Represented on a unit circle.

  • Controlled by the unbalance angle TAU.

Normalized Symmetrical Components (NSC)

  • Fully normalized representation.

  • Includes zero-sequence contribution.

  • Represented on a unit sphere.

  • Controlled by both TAU and ETA.

Example: V+=400V, V-=10V, V0=3V

Relative Symmetrical Components
RPS=0.99988=99.988%,
RNS=0.025=2.5%
RZS=0.0075=0.75%
Normalized Symmetrical Components
NPS=0.99966=99.966%,
NNS=0.024992=2.4992%
NZS=0.0074997=0.74997%
For Lower value of Neutral voltage, RSC is equal to NSC.

Summary

The proposed Advanced Symmetrical Components framework extends the classical theory by introducing two normalized coordinate systems.

Relative Symmetrical Components (RSC) use the Symmetrical Effective Line Voltage as the reference and describe the relative distribution of positive and negative sequence components.

Normalized Symmetrical Components (NSC) use the Symmetrical Effective Phase Voltage as the reference and provide a complete three-dimensional description of the positive, negative and zero sequence components.

Together, RSC and NSC offer a compact, geometrically meaningful, and physically intuitive framework for analysing balanced and unbalanced three-phase systems using only two characteristic angles, TAU and ETA.

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