Symmetrical Components and Symmetrical & Unsymmetrical Fault Analysis

 

1. SYMMETRICAL COMPONENTS – BASICS

Definition

Symmetrical components is a method of resolving unbalanced 3-phase quantities into three balanced sets:

Used to analyze unsymmetrical faults easily.

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Why needed?

Unbalanced system → difficult to analyze directly

Symmetrical components → convert into balanced systems → easy analysis

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2. TYPES OF SEQUENCE COMPONENTS

1. Positive Sequence (Normal system)

Balanced system

Phase sequence:

$$a \rightarrow b \rightarrow c$$

Equal magnitude, 120° apart

Example:

$$V_a, V_b, V_c$$

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2. Negative Sequence

Reverse phase sequence:

$$a \rightarrow c \rightarrow b$$

Occurs during faults.

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3. Zero Sequence

All three phasors equal in magnitude and phase:

$$V_a = V_b = V_c$$

Occurs when neutral is involved.

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3. SYMMETRICAL COMPONENT TRANSFORMATION

Let:

Phase voltages:

$$V_a, V_b, V_c$$

Sequence voltages:

$$V_0, V_1, V_2$$

Where:

• $V_0$ = zero sequence
• $V_1$ = positive sequence
• $V_2$ = negative sequence

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Operator "a"

VERY IMPORTANT FOR GATE

$$a = 1 \angle 120°$$

Properties:

$$a^2 = 1 \angle 240°$$ $$1 + a + a^2 = 0$$ $$a^3 = 1$$

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Voltage Transformation Equation

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

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4. SEQUENCE IMPEDANCES

Three sequence impedances:

• Positive sequence impedance → $Z_1$
• Negative sequence impedance → $Z_2$
• Zero sequence impedance → $Z_0$

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Important GATE point

For transmission line:

$$Z_1 = Z_2$$

But

$$Z_0 \ne Z_1$$

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5. FAULT ANALYSIS – BASICS

Fault → abnormal condition in power system

Examples:

• Short circuit
• Line touching ground

Fault current becomes very high.

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6. TYPES OF FAULTS

Symmetrical Fault

Balanced fault

Example:

3-phase fault (LLL)

Rare but most severe.

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Unsymmetrical Fault

Unbalanced fault

Types:

1. Single line to ground (LG)

2. Line to line (LL)

3. Double line to ground (LLG)

Most common: LG fault (70%)

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7. SYMMETRICAL FAULT ANALYSIS (3-PHASE FAULT)

Only positive sequence network used.

Fault current:

$$I_f = \frac{V}{Z_1}$$

Where:

V = prefault voltage

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Key point:

Negative and zero sequence currents = 0

$$I_2 = 0$$ $$I_0 = 0$$

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8. UNSYMMETRICAL FAULT ANALYSIS

Use symmetrical components.

Use sequence networks.

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9. SEQUENCE NETWORK CONNECTION FOR DIFFERENT FAULTS

VERY IMPORTANT FOR GATE

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1. Single Line to Ground Fault (LG)

Most common fault.

Sequence networks connected in series.

Diagram concept:

$$Z_1 + Z_2 + Z_0$$

Fault current:

$$I_f = \frac{3V}{Z_1 + Z_2 + Z_0}$$

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Key point:

$$I_0 = I_1 = I_2$$

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2. Line to Line Fault (LL)

Sequence networks:

Positive and negative only.

Zero sequence not present.

Connection:

$$Z_1 + Z_2$$

Fault current:

$$I_f = \frac{\sqrt{3}V}{Z_1 + Z_2}$$

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Key point:

$$I_0 = 0$$

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3. Double Line to Ground Fault (LLG)

Sequence networks connected in parallel combination.

Fault current:

$$I_f = \frac{3V}{Z_1 + \frac{Z_2 Z_0}{Z_2 + Z_0}}$$

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10. SUMMARY TABLE (VERY IMPORTANT FOR GATE)

Fault Type	Sequence Networks Used
3-phase fault	Positive only
LG fault	Z₁, Z₂, Z₀ in series
LL fault	Z₁, Z₂ only
LLG fault	Z₁, Z₂, Z₀ parallel

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11. FAULT SEVERITY ORDER (IMPORTANT)

Highest current:

3-phase fault

Then:

LLG fault

Then:

LL fault

Then:

LG fault

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12. SEQUENCE CURRENT RELATION

LG fault:

$$I_0 = I_1 = I_2$$

LL fault:

$$I_0 = 0$$

3-phase fault:

$$I_2 = I_0 = 0$$

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13. ADVANTAGES OF SYMMETRICAL COMPONENT METHOD

Simplifies analysis

Converts unbalanced system → balanced system

Easy to calculate fault current

Standard method used in power system analysis

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14. GATE EXAM MOST IMPORTANT FORMULAS

LG fault:

$$I_f = \frac{3V}{Z_1 + Z_2 + Z_0}$$

LL fault:

$$I_f = \frac{\sqrt{3}V}{Z_1 + Z_2}$$

3-phase fault:

$$I_f = \frac{V}{Z_1}$$

Operator:

$$1 + a + a^2 = 0$$

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15. ONE-PAGE REVISION SHEET

Positive sequence → normal system

Negative sequence → reverse sequence

Zero sequence → equal phase

3-phase fault → positive sequence only

LG fault → all sequences in series

LL fault → positive and negative only

LG fault most common

3-phase fault most severe