Symmetrical Components Analyzer

Perform precise Fortescue transformations to analyze unbalanced 3-phase power systems. Visualize phasors, identify fault types (L-G, L-L), and calculate sequence components ($V_0, V_1, V_2$) for protective relay settings. Applicable standards include IEEE 141, IEEE C37, and IEC 60909.

Phase A
Phase B
Phase C

Engineering Insights: Mastering Symmetrical Components

The Need for Fortescue's Theorem

In 1918, Dr. Charles LeGeyt Fortescue presented a paper to the AIEE that revolutionized power system analysis. Before this, analyzing unbalanced 3-phase faults (like a Single-Line-to-Ground fault) required solving complex differential equations that were practically impossible for large grids.

The Core Concept: Fortescue proved that any unbalanced set of N phasors can be decomposed into N sets of balanced phasors. For a 3-phase system, an unbalanced set of vectors ($V_a, V_b, V_c$) is transformed into three balanced sets:

  • Positive Sequence ($V_1$): Three phasors of equal magnitude, spaced 120° apart, rotating in the standard phase sequence (A-B-C). This represents pure power generation.
  • Negative Sequence ($V_2$): Three phasors of equal magnitude, spaced 120° apart, rotating in the opposite sequence (A-C-B). This represents unbalance and produces opposing torque in motors.
  • Zero Sequence ($V_0$): Three phasors of equal magnitude and zero phase displacement (all in phase). These currents must flow through the ground or neutral conductor.

The Mathematics: The 'a' Operator

To perform these calculations manually or via software, we use the complex operator $a$. Just as $j$ rotates a vector by 90°, $a$ rotates a vector by 120°.

$$ a = 1\angle120^\circ = -0.5 + j0.866 $$ $$ a^2 = 1\angle240^\circ = -0.5 - j0.866 $$ $$ a^3 = 1\angle360^\circ = 1 + j0 $$

Using this operator, the transformation matrix is defined as:

$$ \begin{bmatrix} V_0 \\ V_1 \\ V_2 \end{bmatrix} = \frac{1}{3} \begin{bmatrix} 1 & 1 & 1 \\ 1 & a & a^2 \\ 1 & a^2 & a \end{bmatrix} \begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix} $$

Practical Applications in Fault Analysis

Understanding which sequence components are present allows engineers to identify the type of fault occurring in the system:

Fault Type Sequence Components Present Characteristics
3-Phase Balanced Positive ($V_1$) Only Ideal operation or symmetrical 3-phase bolted fault. No ground current.
Line-to-Ground (L-G) Pos + Neg + Zero ($V_1, V_2, V_0$) The most common fault. All three networks are connected in series. High $I_0$ indicates ground fault.
Line-to-Line (L-L) Pos + Neg ($V_1, V_2$) No Zero sequence (unless ground is involved). Networks are connected in parallel.
Double Line-to-Ground Pos + Neg + Zero ($V_1, V_2, V_0$) All three networks are connected in parallel.