Electric Field and Distribution Systems GATE Exam Quick Notes

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:

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

Corona 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)

Dielectric 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

Insulators - 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

Insulator 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

Insulator 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

Distribution Systems

Distribution System Configurations

Radial System

  • Simplest and cheapest

  • Poor reliability

  • High voltage drop

  • Used in rural areas

  • Single source feeding

Ring Main System

  • Closed loop configuration

  • Better reliability

  • Voltage drop reduced

  • Common in urban areas

  • Alternative supply paths

Interconnected System

  • Highest reliability

  • Multiple feeders

  • Complex protection

  • Used in cities

  • Redundant supply

Parallel Feeders

  • Load sharing

  • Improved reliability

  • Reduced losses

  • Better voltage regulation

Distribution 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

Voltage Drop Calculations

  • Single-phase 2-wire System:

    \[V_d = I(R \cos\phi + X \sin\phi)\]

  • Three-phase 3-wire System:

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

  • DC Distribution:

    \[V_d = 2IR \quad \text{(for 2-wire)}\]
    \[V_d = IR \quad \text{(for 1-wire earth return)}\]

  • Percentage Voltage Regulation:

    \[\% \text{VR} = \frac{V_{no-load} - V_{full-load}}{V_{full-load}} \times 100\]

  • Voltage Drop with Uniformly Distributed Load:

    \[V_d = \frac{I \times L \times Z}{2}\]

Distribution System Design Considerations

  • Voltage Levels:

    • Primary: 11kV, 22kV, 33kV

    • Secondary: 415V, 230V

    • LV: 110V, 220V, 440V

  • Load Characteristics:

    • Residential: lighting, appliances

    • Commercial: motors, HVAC

    • Industrial: large motors, furnaces

  • Design Factors:

    • Load density and growth

    • Reliability requirements

    • Voltage regulation limits

    • Short circuit levels

    • Economic considerations

Distribution 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

  • Commercial Losses:

    • Energy theft

    • Metering errors

    • Billing inefficiencies

Distribution 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

Per-Unit System

Per-Unit System Basics

  • Definition:

    \[\text{Per-unit value} = \frac{\text{Actual value}}{\text{Base value}}\]

  • Base Quantities Selection:

    • Choose 2 base quantities (usually \(S_B\) and \(V_B\))

    • Others are derived from fundamental relationships

  • Base Quantities:

    • \(S_B\) = Base VA (typically 100 MVA)

    • \(V_B\) = Base voltage (typically rated voltage)

    • \(I_B = \frac{S_B}{\sqrt{3}V_B}\) (3-phase), \(I_B = \frac{S_B}{V_B}\) (1-phase)

    • \(Z_B = \frac{V_B^2}{S_B}\) (3-phase), \(Z_B = \frac{V_B^2}{S_B}\) (1-phase)

Per-Unit Calculations

Single-Phase System

\[Z_{pu} = Z_{actual} \times \frac{S_B}{V_B^2}\]
\[I_{pu} = I_{actual} \times \frac{V_B}{S_B}\]
\[P_{pu} = P_{actual} \times \frac{1}{S_B}\]

Three-Phase System

\[Z_{pu} = Z_{actual} \times \frac{S_B}{V_{LL}^2}\]
\[I_{pu} = I_{actual} \times \frac{\sqrt{3}V_{LL}}{S_B}\]

Example

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

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

Base Conversion

Base Conversion Formula

When changing from old base to new base:

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

Example

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

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

Transformer Base Conversion

For transformers, voltage bases change with turns ratio:

\[V_{B,secondary} = V_{B,primary} \times \frac{N_2}{N_1}\]

Per-Unit System Applications

  • Power Flow Analysis:

    • Simplifies calculations

    • Voltage levels normalized

    • Easy comparison of system parameters

  • Short Circuit Analysis:

    • Direct addition of impedances

    • No need for voltage transformation

    • Simplified fault calculations

  • Transformer Modeling:

    • Eliminates ideal transformer

    • Impedance referred to common base

    • Simplified equivalent circuits

  • System Studies:

    • Stability analysis

    • Load flow studies

    • Protection coordination

Per-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

Typical 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: \(X = 0.15-0.25\)

GATE Tip

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

Key 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:

    \[V_c = m_0 g_0 \delta r \ln\left(\frac{D}{r}\right)\]

Important Points for GATE

Electric Field & Insulators

  • Corona occurs when field exceeds critical value

  • String efficiency always less than 100%

  • Grading improves voltage distribution

Distribution Systems

  • Radial: simplest, Ring: better reliability, Interconnected: highest reliability

  • Voltage drop depends on R, X, load current, and power factor

  • Primary distribution: 11kV, Secondary: 415V typically

Per-Unit System

  • Choose 2 base quantities, derive others

  • Base conversion essential for multi-voltage systems

  • Simplifies power system analysis significantly