1.00 0°
1.00 240°
1.00 120°
Phase A
Phase B
Phase C
Zero Seq
Positive Seq
Negative Seq
A Phase:
Zero
Pos
Neg
Sum
Phase A head-to-tail
B Phase:
Zero
Pos
Neg
Sum
Phase B head-to-tail
C Phase:
Zero
Pos
Neg
Sum
Phase C head-to-tail
Phase A with IA0, IA1, IA2
Phase B with IB0, IB1, IB2
Phase C with IC0, IC1, IC2
Phasors:
Ia
Ib
Ic
Iab (Ia-Ib)
Ibc (Ib-Ic)
Ica (Ic-Ia)
Line & Phase Current Phasors

Mathematical Calculations and Theory

1. Sequence Component Transformations

Phase Domain to Sequence Domain (Forward Transform)

The sequence components are calculated from phase currents using the alpha operator method:

Alpha Operators:

α = ej2π/3 = cos(120°) + j·sin(120°) = -0.5 + j0.866

α² = ej4π/3 = cos(240°) + j·sin(240°) = -0.5 - j0.866

α³ = 1

Sequence Component Equations:

I₀ = (1/3) × (IA + IB + IC)

I₁ = (1/3) × (IA + α·IB + α²·IC)

I₂ = (1/3) × (IA + α²·IB + α·IC)

Note: I₀ = Zero sequence, I₁ = Positive sequence, I₂ = Negative sequence

Sequence Domain to Phase Domain (Inverse Transform)

Phase currents are reconstructed from sequence components:

Phase Current Equations:

IA = I₀ + I₁ + I₂

IB = I₀ + α²·I₁ + α·I₂

IC = I₀ + α·I₁ + α²·I₂

2. Phase Rotation (ABC vs CBA)

ABC Rotation (Standard)

In ABC rotation, phases follow the sequence A → B → C with 120° separation:

Phase A: 0°

Phase B: -120° (or 240°)

Phase C: +120°

CBA Rotation (Reverse)

In CBA rotation, phases follow the sequence C → B → A with 120° separation:

Phase A: 0°

Phase B: +120°

Phase C: -120° (or 240°)

Important: In CBA rotation, positive and negative sequence components are swapped in the user interface while maintaining ABC calculation standards internally.

3. Angle Unit Conversions

Degrees to Radians

radians = degrees × (π/180)

Example: 180° = 180 × (π/180) = π radians

Radians to Degrees

degrees = radians × (180/π)

Example: π radians = π × (180/π) = 180°

π Radians Display Format

This toolkit displays radians as multiples of π for easier interpretation:

π_multiplier = degrees ÷ 180

Display: π_multiplier + "π"

Examples: 90° = 0.5π, 180° = 1.0π, 270° = 1.5π, 360° = 2.0π

4. Fault Condition Calculations

Single Line-to-Ground (SLG) Fault

Only one phase carries current, typically phase A:

IA = Ifault ∠ 0°

IB = 0 ∠ 0°

IC = 0 ∠ 0°

Resulting Sequence Components:

I₀ = I₁ = I₂ = Ifault/3

All sequence components are equal in magnitude and phase

Phase-to-Phase Fault

Two phases carry equal and opposite currents (B and C phases):

IA = 0 ∠ 0°

IB = Ifault ∠ 0°

IC = Ifault ∠ 180°

Resulting Sequence Components:

I₀ = 0 (no zero sequence)

I₁ = -I₂ (positive and negative sequences are equal and opposite)

Balanced 3-Phase Condition

All phases carry equal magnitude currents with 120° phase separation:

ABC Rotation:

IA = I ∠ 0°

IB = I ∠ 240° (or -120°)

IC = I ∠ 120°

CBA Rotation:

IA = I ∠ 0°

IB = I ∠ 120°

IC = I ∠ 240° (or -120°)

Resulting Sequence Components:

I₀ = 0, I₂ = 0 (only positive sequence exists)

I₁ = I (positive sequence equals phase magnitude)

5. Complex Number Operations

Polar to Rectangular Conversion

Given: Magnitude |I| and Angle θ

Real part: Re = |I| × cos(θ)

Imaginary part: Im = |I| × sin(θ)

Complex form: I = Re + j·Im

Rectangular to Polar Conversion

Given: Real part Re and Imaginary part Im

Magnitude: |I| = √(Re² + Im²)

Angle: θ = arctan(Im/Re)

Polar form: I = |I| ∠ θ

6. Line Current Calculations

Line-to-Line Currents

Line currents are the differences between phase currents:

IAB = IA - IB

IBC = IB - IC

ICA = IC - IA

Note: Line currents form a closed triangle: IAB + IBC + ICA = 0