Angular Symmetrical Components

            Conventional Symmetrical components are divided into three categories such as Positive, negative and Zero Sequence Components. Now I proposed the different analysis of the symmetrical components using polar coordinate theory which we used in Vector fields namely Cylindrical coordinate system and Spherical Coordinate system.

            First we represent the three sequence voltages in three coordinate (Cartesian) system. I just name the space for representation of the magnitudes of symmetrical components as "Symmetical Component Space(SCS)". Alternatively, we may call the Space as "ZPN Space".

        The main purpose of this analysis is to simplify the voltage unbalance analysis, and factor estimation. Cartesian coordinate system of ZPN components are converted into two polar coordinate system.

A. Cylindrical Angular Symmetrical Components (CASC) 

        Line voltages are divided into positive and negative sequence components only. Zero sequence components are absent in line voltages. So the sequence voltages of line voltages will be represented in 2D plane (PN Plane). 

Fig 1. Cylindrical Coordinate representation/Polar coordinate of line voltages

        Effective magnitude of Positive and Negative Sequence Components is referred as "Delta Effective Line Voltage (DEL Voltage)". This voltage is equal to Quadratic mean of unbalanced line voltages. The power in unbalanced system of these line voltages will be equal to the same delivered by balanced three phase system of "DEL Voltage".

        The Geometric angle (as ahown in figure) between Positive sequence component and DEL Voltage (Not phase angle) is referred as "Tangent Angle of Unbalance (TAU)". It is determined as Tangent Inverse of Voltage Unbalance Factor which is the ratio between Negative Sequence and Positive Sequence. Negative Sequence in per unit is similar to the TAU in radians for smaller values. So the TAU is represented as Angular form of Negative Sequence Voltage.

B. Spherical Angular Symmetrical Components (SASC)

        Phase voltages are divided into three components namely, Positive, Negative and Zero sequence voltages. Spherical coordinate system is used for the representation of sequence components. Already the DEL voltage and TAU will represent the Positive and Negative sequence components. Zero sequence component will be represented in Polar form by converting it into spherical coordinate system as shown in figure below.

Fig 2: Spherical Angular Symmetrical Components (SASC)

        Effective magnitude of all Sequence Components is referred as "Star Effective Line Voltage (SEL Voltage)". This voltage is equal to the effective magnitude of all Phase Voltages. . The power in unbalanced star connected system will be equal to the same delivered by balanced three phase system of "SEL Voltage".

        The Geometric angle (as ahown in figure) between SEL Voltage and DEL Voltage is referred as "Zero Sequence Tangent Angle (ZESTA)". It is determined as Cosine Inverse of the ratio between DEL Voltage and SEL Voltage. It is represented as the Angular form of Zero Sequence Component.

        The four components from CASC and SASC is collectively referred as "Angular Symmetrical Components". 

1. Star Effective Line Voltage (SELV)

2. Delta Effective Line Voltage (DELV)

3. Zero Sequence Tangent Angle (ZeSTA)

4. Tangent Angle of Unbalance (TAU)

Conversion Table is given below:

        SELV, DELV, and ZeSTA will be easily determined by the magnitudes of phase voltages and line voltages. But the Tangent Angle of Unbalance is typical to estimate. TAU is depends on the sequence either positive or negative. Sequence dependent TAU will ranges from zero to 90 degree. Absolute TAU estimated from line voltage magnitudes irrespective of phase angles will ranges only from zero to 45 degrees.

The Estimation of TAU will be published Soon...

References:

1. Research Article: Click Here

2. Research Thesis: Click Here

http://hdl.handle.net/10603/564562