GATE EE

Electric Field and Distribution Systems GATE Exam Quick Notes

Lecture Notes

SEC 01

Electric Field and Insulators

Electric Field Distribution

Key Concepts for GATE

  • Electric Field Intensity (E):

    \[E = \frac{V}{d} \quad \text{(for uniform field)}\]

  • For Cylindrical Conductors:

    \[E_r = \frac{V}{r \ln(R_2/R_1)} \quad \text{(at radius r)}\]

  • Maxwell’s Stress Equation:

    \[\text{Mechanical Stress} = \frac{1}{2}\epsilon_0 E^2 \quad \text{N/m}^2\]

  • Electric Field at Conductor Surface:

    \[E_{max} = \frac{V}{r \ln(D/r)} \quad \text{(single conductor)}\]
1Corona and Breakdown

  • Critical Disruptive Voltage:

    \[V_c = m_0 g_0 \delta r \ln\left(\frac{D}{r}\right)\]
    where \(m_0 = 1\) (smooth conductor), \(g_0 = 21.21\) kV/cm

  • Visual Critical Voltage:

    \[V_v = m_v g_0 \delta r \left(1 + \frac{0.3}{\sqrt{\delta r}}\right)\ln\left(\frac{D}{r}\right)\]

  • Corona Loss:

    \[P_c = \frac{244}{\delta} (f + 25)\sqrt{\frac{r}{D}}(V - V_c)^2 \times 10^{-5} \text{ kW/km/phase}\]

  • Air Density Factor: \(\delta = \frac{3.92b}{273+t}\) where \(b\) = pressure (cm Hg), \(t\) = temperature (°C)

1Dielectric Strength and Breakdown

  • Breakdown Voltage:

    \[V_{breakdown} = E_{breakdown} \times d\]

  • Paschen’s Law:

    \[V_b = \frac{Bpd}{\ln(Apd) - \ln[\ln(1 + 1/\gamma)]}\]
    where \(p\) = pressure, \(d\) = gap distance

  • Townsend’s Avalanche Criterion:

    \[\gamma(\exp(\alpha d) - 1) = 1\]

  • Factors Affecting Breakdown:

    • Gas pressure and temperature

    • Electrode geometry

    • Frequency and waveform

    • Humidity and contamination

1Insulators - Types and Properties

  • Pin Type Insulators:

    • Up to 33 kV

    • Single piece construction

    • Good for moderate voltages

  • Suspension Insulators:

    • Above 33 kV

    • Multiple units in string

    • Flexible arrangement

  • Strain Insulators:

    • For dead ends and corners

    • Mechanical strength important

  • Shackle Insulators:

    • For LV lines

    • Vertical and horizontal mounting

  • Materials:

    • Porcelain

    • Glass

    • Polymer/Composite

  • Properties:

    • High dielectric strength

    • Weather resistance

    • Mechanical strength

    • Low power loss

1Insulator String Analysis

  • String Efficiency:

    \[\eta = \frac{\text{Voltage across string}}{n \times \text{Voltage across disc nearest to conductor}} \times 100\%\]

  • Voltage Distribution:

    \[V_1 : V_2 : V_3 = 1 : K(1+K) : K(1+2K+K^2)\]
    where \(K = \frac{C}{mC}\) (ratio of pin to earth capacitance to mutual capacitance)

  • Methods to Improve String Efficiency:

    • Longer cross-arms (reduce \(K\))

    • Grading of units (equal voltage distribution)

    • Using guard rings

    • Arcing horns

  • Grading Methods:

    • Capacitance grading

    • Resistance grading

1Insulator Testing and Failures

  • Tests on Insulators:

    • Routine tests (power frequency, impulse)

    • Type tests (mechanical, thermal)

    • Sample tests (radio interference)

  • Power Frequency Test:

    \[V_{test} = 2 \times V_{rated} \text{ (dry)}\]
    \[V_{test} = 1.5 \times V_{rated} \text{ (wet)}\]

  • Impulse Test:

    \[V_{impulse} = 2.5 \times V_{rated} \text{ (positive)}\]
    \[V_{impulse} = 2.2 \times V_{rated} \text{ (negative)}\]

  • Common Failures:

    • Flashover (external)

    • Puncture (internal breakdown)

    • Mechanical failure

    • Contamination effects

SEC 02

Distribution Systems

1Distribution System Configurations
1Radial System

  • Simplest and cheapest

  • Poor reliability

  • High voltage drop

  • Used in rural areas

  • Single source feeding

1Ring Main System

  • Closed loop configuration

  • Better reliability

  • Voltage drop reduced

  • Common in urban areas

  • Alternative supply paths

1Interconnected System
1Parallel Feeders
1Distribution System Components

  • Feeders:

    • No tappings along length

    • Designed for current carrying

    • Connect substation to distribution area

  • Distributors:

    • Multiple tappings for consumers

    • Designed for voltage drop

    • Supply power to service mains

  • Service Mains:

    • Final connection to consumers

    • Short length

    • Include energy meters

  • Distribution Transformers:

    • Step down voltage

    • 11kV/415V typical

    • Located near load centers

  • Switchgear:

    • Circuit breakers

    • Isolators

    • Protective relays

  • Metering Equipment:

    • Energy meters

    • CT/PT combinations

    • Communication systems

1Voltage Drop Calculations
1Distribution System Design Considerations
1Distribution System Losses

  • Technical Losses:

    \[P_{loss} = I^2 R = \frac{P^2 R}{V^2 \cos^2\phi}\]

  • Transformer Losses:

    • Core losses (constant)

    • Copper losses (variable with load)

  • Line Losses:

    \[P_{line} = 3I^2R \quad \text{(3-phase)}\]

  • Loss Reduction Methods:

    • Higher voltage levels

    • Improved power factor

    • Load balancing

    • Proper conductor sizing

    • Distribution transformer location optimization

1Distribution System Protection

  • Protection Devices:

    • Fuses (HRC, rewirable)

    • Circuit breakers (SF6, vacuum)

    • Reclosers and sectionalizers

    • Lightning arresters

  • Protection Schemes:

    • Overcurrent protection

    • Earth fault protection

    • Distance protection

    • Differential protection

  • Coordination:

    • Time grading

    • Current grading

    • Discrimination

  • Fault Types:

    • Line to ground

    • Line to line

    • Three phase faults

    • Double line to ground

SEC 03

Per-Unit System

1Per-Unit System Basics

  • Base Quantities Selection:

  • Base Quantities:

    • 1Per-Unit Calculations
    1Single-Phase System
    \[Z_{pu} = Z_{actual} \times \frac{S_B}{V_B^2}\]
    \[I_{pu} = I_{actual} \times \frac{V_B}{S_B}\]
    \[P_{pu} = P_{actual} \times \frac{1}{S_B}\]
    1Three-Phase System
    \[Z_{pu} = Z_{actual} \times \frac{S_B}{V_{LL}^2}\]
    \[I_{pu} = I_{actual} \times \frac{\sqrt{3}V_{LL}}{S_B}\]
    1Example

    A 100 MVA, 11 kV generator has \(X_d'' = 0.2\) pu. Find actual reactance:

    \[X_{actual} = X_{pu} \times \frac{V_B^2}{S_B} = 0.2 \times \frac{11^2}{100} = 0.242 \Omega\]
    1Base Conversion
    1Base Conversion Formula

    When changing from old base to new base:

    \[Z_{pu}^{new} = Z_{pu}^{old} \times \left(\frac{V_B^{old}}{V_B^{new}}\right)^2 \times \frac{S_B^{new}}{S_B^{old}}\]
    1Example

    Given: \(Z = 0.1\) pu on 50 MVA, 11 kV base Find: \(Z\) on 100 MVA, 11 kV base

    \[Z_{new} = 0.1 \times \left(\frac{11}{11}\right)^2 \times \frac{100}{50} = 0.2 \text{ pu}\]
    1Transformer Base Conversion

    For transformers, voltage bases change with turns ratio:

    \[V_{B,secondary} = V_{B,primary} \times \frac{N_2}{N_1}\]
    1Per-Unit System Applications
    1Per-Unit System Advantages

    • Normalizes values to common base

    • Eliminates need for phase conversion

    • Simplifies analysis of multi-voltage systems

    • Makes transformer analysis easier

    • Equipment parameters fall in narrow ranges

    • Facilitates computer analysis

    • Reduces computational errors

    1Typical Per-Unit Values

    • Generators: \(X_d = 1.0-2.0\)

    • Transformers: \(X = 0.05-0.15\)

    • Transmission lines: \(X = 0.3-0.5\) per 100 km

    • Motors: \(X = 0.15-0.25\)

    1GATE Tip

    Always verify base values when given per-unit quantities. Check if single-phase or three-phase bases are used!

    1Key Formulas Summary for GATE

    1. Electric Field at Conductor Surface:

      \[E_{max} = \frac{V}{r \ln(D/r)}\]

    2. String Efficiency:

      \[\eta = \frac{V_{\text{string}}}{n \times V_{\text{disc nearest to conductor}}} \times 100\%\]

    3. Voltage Drop (3-phase):

      \[V_d = \sqrt{3} I(R \cos\phi + X \sin\phi)\]

    4. Per-unit Impedance:

      \[Z_{pu} = Z_{actual} \times \frac{S_B}{V_B^2}\]

    5. Base Conversion:

      \[Z_{pu}^{new} = Z_{pu}^{old} \times \left(\frac{V_B^{old}}{V_B^{new}}\right)^2 \times \frac{S_B^{new}}{S_B^{old}}\]

    6. Corona Critical Voltage:

      \[V_c = m_0 g_0 \delta r \ln\left(\frac{D}{r}\right)\]
    1Important Points for GATE
    1Electric Field & Insulators
    1Distribution Systems
    1Per-Unit System