Fault Analysis and Symmetrical Components GATE EE Exam Quick Notes

Symmetrical Components

Symmetrical Components Theory

Fortescue’s Theorem

  • Any unbalanced 3-phase system can be decomposed into:

    • Positive sequence (ABC rotation)

    • Negative sequence (ACB rotation)

    • Zero sequence (in-phase components)

  • Transformation Matrix:

    \[\begin{bmatrix} V_a^0 \\ V_a^1 \\ V_a^2 \end{bmatrix} = \dfrac{1}{3} \begin{bmatrix} 1 & 1 & 1 \\ 1 & a & a^2 \\ 1 & a^2 & a \end{bmatrix} \begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix}\]
    where \(a = e^{j120^\circ} = -\dfrac{1}{2} + j\dfrac{\sqrt{3}}{2}\)

Key Relationships

\[V_a = V_a^0 + V_a^1 + V_a^2\]
\[V_b = V_a^0 + a^2V_a^1 + aV_a^2\]
\[V_c = V_a^0 + aV_a^1 + a^2V_a^2\]

Operator ’a’ Properties

Essential for GATE Problems

Definition

\[a = e^{j120^\circ} = -\dfrac{1}{2} + j\dfrac{\sqrt{3}}{2}\]

Important Properties

  • \(a^3 = 1\)

  • \(a^2 = e^{j240^\circ} = e^{-j120^\circ}\)

  • \(1 + a + a^2 = 0\)

  • \(a^* = a^2\) (conjugate)

  • \(|a| = 1\)

Common Calculations

  • \(a - a^2 = j\sqrt{3}\)

  • \(a^2 - a = -j\sqrt{3}\)

  • \(a^n = a^{n \bmod 3}\)

  • For balanced system: \(V_a^0 = V_a^2 = 0\)

GATE Tip

Memorize these properties - they appear frequently in numerical problems!

Inverse Transformation

From Sequence to Phase Components

  • Inverse Transformation Matrix:

    \[\begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & a^2 & a \\ 1 & a & a^2 \end{bmatrix} \begin{bmatrix} V_a^0 \\ V_a^1 \\ V_a^2 \end{bmatrix}\]

  • Phase Relationships:

    \[\begin{aligned} V_a &= V_a^0 + V_a^1 + V_a^2 \\ V_b &= V_a^0 + a^2V_a^1 + aV_a^2 \\ V_c &= V_a^0 + aV_a^1 + a^2V_a^2 \end{aligned}\]

For Current Components

Same transformation applies to currents:

\[I_a^0 = \dfrac{1}{3}(I_a + I_b + I_c)\]
\[I_a^1 = \dfrac{1}{3}(I_a + aI_b + a^2I_c)\]
\[I_a^2 = \dfrac{1}{3}(I_a + a^2I_b + aI_c)\]

Sequence Networks

  • Positive Sequence:

    • Normal balanced system

    • Generators present

    • Same as normal operation

  • Negative Sequence:

    • Opposite rotation

    • No voltage sources

    • Passive network only

  • Zero Sequence:

    • In-phase currents

    • Depends on ground connection

    • 3 times neutral current

Network Characteristics

  • Networks are independent

  • Connected only at fault point

  • Each has different impedance

  • Thevenin equivalent used

Important

Zero sequence current requires return path (ground or neutral)

\[I_0 = \dfrac{I_a + I_b + I_c}{3} = I_n\]

Sequence Impedances of Equipment

Key Points

  • Zero sequence can’t flow through \(\Delta\) connections

  • Zero sequence impedance depends on grounding

  • Negative sequence \(\approx\) Positive sequence for static equipment

Fault Analysis

Types of Faults

Classification and Statistics

Symmetrical Faults

  • Three-phase fault (LLL)

  • Balanced system maintained

  • Highest fault current

  • Easiest to analyze

  • Only positive sequence present

Unsymmetrical Faults

  • Single line-to-ground (LG) - 70-80%

  • Line-to-line (LL) - 15-20%

  • Double line-to-ground (LLG) - 5-10%

  • Unbalanced system

  • All three sequences present

Fault Statistics

  • LG: 70-80%

  • LL: 15-20%

  • LLG: 5-10%

  • LLL: 2-5%

GATE Focus

All fault types are equally important for GATE - don’t neglect any!

Symmetrical (3-Phase) Fault

LLL Fault Analysis

  • Characteristics:

    • Balanced fault (ABC phases remain symmetrical)

    • Most severe in terms of current magnitude

    • Simplest to analyze

    • Only positive sequence network active

  • Boundary Conditions:

    • \(V_a = V_b = V_c = 0\) (at fault point)

    • \(I_a + I_b + I_c = 0\) (balanced)

    • \(I_a^0 = I_a^2 = 0\)

    • \(V_a^0 = V_a^2 = 0\)

Analysis

  • Only positive sequence active

  • \(I_a^1 = \dfrac{V_{prefault}}{Z_1}\)

  • \(I_f = |I_a^1| = \dfrac{V_{prefault}}{Z_1}\)

Fault Current Calculation

\[I_f = \dfrac{V_{prefault}}{Z_1}\]
Short Circuit MVA: \(\text{SC MVA} = \sqrt{3} \times V_{LL} \times I_f\)

Example:

For 11 kV system with \(Z_1 = j0.2\) pu: \(I_f = \dfrac{1}{j0.2} = -j5\) pu

Single Line-to-Ground (LG) Fault

Most Common Fault (70-80%)

Boundary Conditions

Assume fault on phase ’a’:

  • \(V_a = 0\) (fault point grounded)

  • \(I_b = I_c = 0\) (other phases isolated)

  • \(I_a = I_f\) (fault current)

Sequence Analysis

From boundary conditions:

  • \(I_a^0 = I_a^1 = I_a^2 = \dfrac{I_f}{3}\)

  • \(V_a^0 + V_a^1 + V_a^2 = 0\)

  • Networks connected in series

Network Connection

All three sequence networks in series:

\[I_a^1 = \dfrac{V_{prefault}}{Z_0 + Z_1 + Z_2}\]

Fault Current

\[I_f = 3I_a^1 = \dfrac{3V_{prefault}}{Z_0 + Z_1 + Z_2}\]

Important

LG fault current depends on all three sequence impedances. Zero sequence impedance significantly affects fault current.

Line-to-Line (LL) Fault

No Ground Connection

Boundary Conditions

Assume fault between phases ’b’ and ’c’:

  • \(V_b = V_c\) (phases shorted)

  • \(I_a = 0\) (phase ’a’ isolated)

  • \(I_b = -I_c\) (equal and opposite)

  • \(I_b + I_c = 0\)

Sequence Analysis

From boundary conditions:

  • \(I_a^0 = 0\) (no zero sequence)

  • \(I_a^1 = -I_a^2\)

  • \(V_a^1 = V_a^2\)

  • Positive and negative networks in parallel

Network Connection

Positive and negative networks in parallel:

\[I_a^1 = \dfrac{V_{prefault}}{Z_1 + Z_2}\]

Fault Current

\[I_f = |I_b| = |I_c| = \dfrac{\sqrt{3}V_{prefault}}{Z_1 + Z_2}\]

Key Point

LL fault current is independent of zero sequence impedance. No ground path required.

Double Line-to-Ground (LLG) Fault

Complex Network Connection

Boundary Conditions

Assume fault on phases ’b’ and ’c’ to ground:

  • \(V_b = V_c = 0\) (both phases grounded)

  • \(I_a = 0\) (phase ’a’ isolated)

  • \(I_b \neq I_c\) (unequal currents)

  • \(I_b + I_c = I_f\) (total fault current)

Sequence Analysis

From boundary conditions:

  • \(I_a^0 + I_a^1 + I_a^2 = 0\)

  • \(V_a^1 = V_a^2\)

  • Positive in series with (Negative || Zero)

Network Connection

Complex parallel-series combination:

\[I_a^1 = \dfrac{V_{prefault}}{Z_1 + \dfrac{Z_2 Z_0}{Z_2 + Z_0}}\]

Fault Current

\[I_f = I_b + I_c = 3I_a^0\]
where:
\[I_a^0 = I_a^1 \times \dfrac{Z_2}{Z_2 + Z_0}\]

Complexity

LLG fault analysis is most complex - requires careful network manipulation.

Fault Current Comparison

Relative Magnitudes

General Relationships

For typical power systems where \(Z_0 > Z_1 \approx Z_2\):

  1. Three-phase fault: \(I_{3\phi} = \dfrac{V_{prefault}}{Z_1}\) (Maximum)

  2. Line-to-line fault: \(I_{LL} = \dfrac{\sqrt{3}V_{prefault}}{Z_1 + Z_2} \approx 0.866 \times I_{3\phi}\)

  3. Double line-to-ground: \(I_{LLG}\) depends on \(Z_0\) (Variable)

  4. Single line-to-ground: \(I_{LG} = \dfrac{3V_{prefault}}{Z_0 + Z_1 + Z_2}\) (Usually minimum)

GATE Insight

  • If \(Z_0 >> Z_1, Z_2\): \(I_{3\phi} > I_{LL} > I_{LLG} > I_{LG}\)

  • If \(Z_0 << Z_1, Z_2\): \(I_{LG}\) can be maximum!

  • Always calculate - don’t assume relative magnitudes

Fault Analysis Procedure

Step-by-Step Approach

  1. Pre-fault Analysis:

    • Draw single-line diagram

    • Calculate pre-fault voltage at fault point

    • Convert all impedances to per-unit on common base

  2. Sequence Network Formation:

    • Draw positive sequence network (with generators)

    • Draw negative sequence network (passive, no sources)

    • Draw zero sequence network (depends on grounding)

  3. Network Connection:

    • Connect networks according to fault type

    • Calculate Thevenin equivalent impedances

  4. Sequence Current Calculation:

    • Calculate sequence currents using appropriate formulas

    • Transform to phase quantities using inverse transformation

  5. Final Results:

    • Calculate fault current magnitude and phase

    • Determine voltage at various buses

    • Check results for reasonableness

Key Formulas for GATE

Quick Reference

Symmetrical Components

\[\begin{aligned} V_a^1 &= \dfrac{1}{3}(V_a + aV_b + a^2V_c) \\ V_a^2 &= \dfrac{1}{3}(V_a + a^2V_b + aV_c) \\ V_a^0 &= \dfrac{1}{3}(V_a + V_b + V_c) \end{aligned}\]

Fault Currents

\[\begin{aligned} I_{f(3\phi)} &= \dfrac{V_{prefault}}{Z_1} \\ I_{f(LG)} &= \dfrac{3V_{prefault}}{Z_0 + Z_1 + Z_2} \\ I_{f(LL)} &= \dfrac{\sqrt{3}V_{prefault}}{Z_1 + Z_2} \\ I_{f(LLG)} &= 3I_a^0 = \dfrac{3V_{prefault}}{Z_1 + \dfrac{Z_2Z_0}{Z_2 + Z_0}} \times \dfrac{Z_2}{Z_2 + Z_0} \end{aligned}\]

Operator ’a’ Properties

\(a = e^{j120°}\), \(a^3 = 1\), \(1 + a + a^2 = 0\), \(a - a^2 = j\sqrt{3}\)

Special Cases and Practical Notes

GATE Problem Solving Tips

Solidly Grounded System

  • \(Z_0\) is small

  • LG fault current can be very high

  • Zero sequence impedance mainly from equipment

Ungrounded System

  • \(Z_0 = \infty\) (open circuit)

  • No LG fault current flows

  • Line-to-ground "fault" becomes line-to-line fault through capacitance

Impedance Grounded System

  • \(Z_0\) includes grounding impedance

  • LG fault current is limited

  • Common in distribution systems

GATE Strategy

  • Always identify grounding type first

  • Check units - per-unit or actual values

  • Verify sequence impedance values are reasonable

  • Use symmetry to check answers

Summary and Quick Review

Essential Points for GATE

Must Remember

  • Operator ’a’ properties

  • Sequence network connections for each fault

  • Boundary conditions for each fault type

  • Transformation matrices

  • Typical sequence impedance values

Common Mistakes

  • Confusing sequence network connections

  • Wrong boundary conditions

  • Forgetting factor of 3 in LG fault

  • Mixing up \(a\) and \(a^2\)

  • Incorrect per-unit conversions

Problem Solving Strategy

  • Read problem carefully

  • Identify fault type

  • Draw sequence networks

  • Apply boundary conditions

  • Connect networks correctly

  • Calculate step by step

  • Verify final answer

Time Management

  • Memorize standard formulas

  • Practice network connections

  • Use shortcuts for calculations

  • Check answers quickly