PE Exam Exam > PE Exam Notes > Electrical & Computer Engineering for PE > Cheatsheet: Transmission
Cheatsheet: Transmission
1. Transmission Line Fundamentals
1.1 Line Parameters
| Parameter | Description & Units |
|---|---|
| Resistance (R) | Ohms per unit length (Ω/km or Ω/mile); conductor resistance causing I²R losses |
| Inductance (L) | Henries per unit length (H/km or H/mile); magnetic field energy storage |
| Capacitance (C) | Farads per unit length (F/km or F/mile); electric field energy storage between conductors |
| Conductance (G) | Siemens per unit length (S/km or S/mile); leakage current through insulation |
1.2 Characteristic Impedance
| Formula | Variables |
|---|---|
| Z₀ = √[(R + jωL)/(G + jωC)] | Z₀: characteristic impedance (Ω); ω: angular frequency (rad/s) |
| Z₀ ≈ √(L/C) | Lossless line approximation |
1.3 Propagation Constant
| Formula | Variables |
|---|---|
| γ = α + jβ | γ: propagation constant; α: attenuation constant (Np/km); β: phase constant (rad/km) |
| γ = √[(R + jωL)(G + jωC)] | General form |
| β = ω√(LC) | Lossless line phase constant |
| α ≈ R/(2Z₀) + GZ₀/2 | Low-loss approximation |
1.4 Velocity and Wavelength
| Formula | Variables |
|---|---|
| v = 1/√(LC) | Velocity of propagation (m/s) for lossless line |
| v = c/√εᵣ | c: speed of light (3×10⁸ m/s); εᵣ: relative permittivity |
| λ = v/f = 2π/β | λ: wavelength (m); f: frequency (Hz) |
2. Transmission Line Models
2.1 Short Line Model (length < 80="" km="" or="" 50="">
- Shunt capacitance and conductance neglected
- Only series impedance Z = R + jωL considered
- ABCD parameters: A = D = 1; B = Z; C = 0
2.2 Medium Line Model (80-240 km or 50-150 miles)
| Configuration | Description |
|---|---|
| Nominal π | Series impedance Z with shunt admittance Y/2 at each end |
| Nominal T | Series impedance Z/2 at each end with shunt admittance Y in middle |
- Nominal π ABCD: A = D = 1 + YZ/2; B = Z; C = Y(1 + YZ/4)
- Nominal T ABCD: A = D = 1 + YZ/2; B = Z(1 + YZ/4); C = Y
2.3 Long Line Model (length > 240 km or 150 miles)
| Parameter | Formula |
|---|---|
| A = D | cosh(γl) |
| B | Z₀ sinh(γl) |
| C | sinh(γl)/Z₀ |
- l: line length; γ: propagation constant; Z₀: characteristic impedance
- For lossless line: A = D = cos(βl); B = jZ₀ sin(βl); C = j sin(βl)/Z₀
3. ABCD Parameters
3.1 General Equations
| Equation | Description |
|---|---|
| Vₛ = AVᵣ + BIᵣ | Sending end voltage in terms of receiving end |
| Iₛ = CVᵣ + DIᵣ | Sending end current in terms of receiving end |
| AD - BC = 1 | Reciprocity condition for passive networks |
3.2 Units and Symmetry
- A and D are dimensionless; for symmetric networks A = D
- B has units of impedance (Ω)
- C has units of admittance (S)
4. Power Flow and Voltage Regulation
4.1 Receiving End Relationships
| Formula | Variables |
|---|---|
| Pᵣ = VᵣIᵣ cos(θᵣ) | Pᵣ: receiving end real power; θᵣ: power factor angle at receiving end |
| Qᵣ = VᵣIᵣ sin(θᵣ) | Qᵣ: receiving end reactive power |
| Sᵣ = Pᵣ + jQᵣ = VᵣIᵣ* | Sᵣ: complex power; Iᵣ*: complex conjugate of receiving end current |
4.2 Voltage Regulation
| Formula | Description |
|---|---|
| VR = (|Vᵣ,NL| - |Vᵣ,FL|)/|Vᵣ,FL| × 100% | Voltage regulation percentage; NL: no-load; FL: full-load |
| VR = (|Vₛ|/|A| - |Vᵣ|)/|Vᵣ| × 100% | Using ABCD parameters with constant sending end voltage |
4.3 Transmission Efficiency
| Formula | Variables |
|---|---|
| η = Pᵣ/Pₛ × 100% | η: efficiency; Pₛ: sending end real power; Pᵣ: receiving end real power |
| Losses = Pₛ - Pᵣ | Total line losses (W or MW) |
5. Surge Impedance Loading (SIL)
5.1 Definitions
| Parameter | Formula |
|---|---|
| Surge Impedance | Zₛ = √(L/C) |
| SIL | SIL = Vᵣ²/Zₛ |
| SIL (3-phase) | SIL = (Vₗₗ²)/Zₛ (MW); Vₗₗ: line-to-line voltage (kV) |
- At SIL loading, receiving end voltage equals sending end voltage for lossless line
- No reactive power exchange; unity power factor at characteristic impedance
5.2 Loading Conditions
| Loading | Effect |
|---|---|
| Below SIL | Capacitance dominates; receiving end voltage rises (Ferranti effect) |
| At SIL | Flat voltage profile; no net reactive power |
| Above SIL | Inductance dominates; voltage drops along line |
6. Ferranti Effect
6.1 Description
- Receiving end voltage exceeds sending end voltage at light load or no-load
- Caused by line capacitance charging current
- More pronounced on long, lightly loaded lines
- Leading current produces voltage rise due to inductive reactance
6.2 Voltage Rise Formula
| Formula | Condition |
|---|---|
| Vᵣ/Vₛ = 1/cos(βl) | Lossless line at no-load; βl: electrical length |
| Vᵣ ≈ Vₛ[1 + (ωCVₛl)²Z/2] | Approximate formula for short to medium lines |
7. Conductor Configurations
7.1 Geometric Mean Distance (GMD)
| Configuration | Formula |
|---|---|
| Single-phase (2 conductors) | GMD = D (spacing between conductors) |
| Three-phase equilateral | GMD = D (side length of triangle) |
| Three-phase asymmetric | GMD = ∛(D₁₂ × D₂₃ × D₃₁) |
7.2 Geometric Mean Radius (GMR)
| Configuration | Formula |
|---|---|
| Solid cylindrical conductor | GMR = r × e^(-1/4) = 0.7788r; r: conductor radius |
| Stranded conductor | GMR provided by manufacturer or GMR ≈ 0.7788 × r_equivalent |
| Bundled conductors (n per phase) | GMRᵦ = ⁿ√(GMR × d₁ × d₂ × ... × dₙ₋₁); d: spacing between sub-conductors |
7.3 Inductance Formulas
| Formula | Application |
|---|---|
| L = 2 × 10⁻⁷ ln(GMD/GMR) | Inductance per phase (H/m) for three-phase line |
| L = 0.2 ln(GMD/GMR) | Inductance per phase (mH/km) |
| Xₗ = ωL = 2πfL | Inductive reactance (Ω/km or Ω/mile) |
7.4 Capacitance Formulas
| Formula | Application |
|---|---|
| C = 2πε₀εᵣ/ln(GMD/r) | Capacitance per phase (F/m); ε₀ = 8.854 × 10⁻¹² F/m |
| C = 0.0556εᵣ/log₁₀(GMD/r) | Capacitance per phase (μF/km); εᵣ ≈ 1 for air |
| Xc = 1/(ωC) | Capacitive reactance (Ω·km or Ω·mile) |
8. Bundled Conductors
8.1 Purpose
- Reduce corona and radio interference
- Decrease inductive reactance
- Increase capacitance
- Improve power transfer capability
8.2 Bundle GMR
| Number of Sub-conductors | GMR Formula |
|---|---|
| 2 | GMRᵦ = √(GMR × d) |
| 3 | GMRᵦ = ∛(GMR × d²) |
| 4 | GMRᵦ = 1.09 × ⁴√(GMR × d³) |
- d: spacing between sub-conductors in bundle
- GMR: geometric mean radius of individual conductor
9. Transposition
9.1 Purpose and Effect
- Balance impedances among three phases
- Each conductor occupies each position for one-third of line length
- Reduces voltage unbalance and inductive interference
- GMD becomes geometric mean of all phase-to-phase distances
9.2 Average GMD for Transposed Line
| Formula | Variables |
|---|---|
| GMD = ∛(D₁₂ × D₂₃ × D₃₁) | D₁₂, D₂₃, D₃₁: distances between phases |
10. Corona
10.1 Description
- Ionization of air surrounding conductors at high voltage gradients
- Causes power loss, audible noise, radio interference, ozone production
- Occurs when electric field strength exceeds dielectric strength of air (~30 kV/cm at sea level)
10.2 Critical Disruptive Voltage
| Formula | Variables |
|---|---|
| V₀ = g₀ × m × r × ln(D/r) | V₀: disruptive voltage (kV); g₀: breakdown gradient (21.1 kV/cm at 25°C, 76 cmHg) |
| m = 0.93-0.98 | Surface condition factor; m = 1 for smooth, polished conductor |
- r: conductor radius; D: spacing between conductors
- Air density factor δ = 3.92b/(273 + t); b: pressure (cmHg); t: temperature (°C)
10.3 Peek's Formula for Corona Loss
| Formula | Condition |
|---|---|
| Pᶜ = (241/δ) × (f + 25) × √(r/D) × (V - V₀)² × 10⁻⁵ | Corona power loss (kW/km/phase); f: frequency (Hz) |
11. Skin Effect and Proximity Effect
11.1 Skin Effect
- AC current concentrates near conductor surface
- Effective resistance increases with frequency
- Skin depth δₛ = √(2ρ/(ωμ)); ρ: resistivity; μ: permeability
- For copper at 60 Hz, skin depth ≈ 8.5 mm
11.2 Proximity Effect
- Magnetic field from adjacent conductors distorts current distribution
- Further increases AC resistance beyond skin effect
- Significant in bundled conductors and cables
11.3 AC Resistance
| Parameter | Description |
|---|---|
| Rₐc = kₛ × Rdc | kₛ: skin effect factor (1.0-1.1 for overhead lines at 60 Hz) |
| Rₐc = (kₛ + kₚ) × Rdc | kₚ: proximity effect factor; included for cables and bundles |
12. Sag and Tension
12.1 Catenary Equations
| Formula | Variables |
|---|---|
| S = wL²/(8T) | S: sag (m); w: weight per unit length (N/m); L: span length (m); T: tension (N) |
| S = wL²/(8H) | H: horizontal component of tension; for small sag H ≈ T |
| L_conductor = L + 8S²/(3L) | Actual conductor length accounting for sag; parabolic approximation |
12.2 Effects of Temperature and Ice/Wind Loading
| Formula | Description |
|---|---|
| w_total = √(w_c² + w_w²) + w_i | w_c: conductor weight; w_w: wind load; w_i: ice weight |
| ΔL/L = αΔT + (T₂ - T₁)/(AE) | α: thermal expansion coefficient; A: cross-sectional area; E: Young's modulus |
12.3 Ruling Span
- Equivalent span that represents multiple unequal spans between dead-end towers
- L_ruling = √[(L₁³ + L₂³ + ... + Lₙ³)/(L₁ + L₂ + ... + Lₙ)]
- Used for sag-tension calculations in actual line design
13. Insulator Considerations
13.1 Types
| Type | Application |
|---|---|
| Pin | Distribution lines up to 33 kV |
| Suspension | Transmission lines above 33 kV; multiple discs |
| Strain | Dead-end towers and sharp angle turns |
| Post | Substations and low-profile applications |
13.2 Voltage Distribution
- Voltage distributes non-uniformly across suspension insulator string due to capacitance to ground
- Unit nearest conductor has highest voltage stress
- String efficiency = (Voltage across string)/(n × Voltage across unit nearest ground); n: number of units
- Grading rings or guard rings improve voltage distribution
14. Thermal Limits and Ampacity
14.1 Heat Balance Equation
| Parameter | Description |
|---|---|
| I²R = q_c + q_r - q_s | Heat generated by current = convection + radiation - solar heating |
| q_c | Convective heat loss; function of wind speed and temperature difference |
| q_r | Radiative heat loss; function of surface temperature and emissivity |
| q_s | Solar heat gain; function of solar intensity and conductor surface area |
14.2 Ampacity Factors
- Maximum allowable conductor temperature (varies by conductor type: 75°C-100°C ACSR)
- Ambient temperature
- Wind speed (perpendicular wind increases cooling)
- Solar radiation intensity
- Conductor emissivity and absorptivity
- Elevation (affects air density and convection)
14.3 Emergency Ratings
- Short-term ratings exceed normal thermal limits
- Allowable for limited duration (15 min, 4 hr ratings common)
- Risk of increased sag and annealing of conductor
15. Power Transfer Capability
15.1 Maximum Power Transfer
| Formula | Condition |
|---|---|
| P_max = (VₛVᵣ)/(|B|) sin(δ) | Lossless line; δ: angle between Vₛ and Vᵣ; B from ABCD parameters |
| P_max = (VₛVᵣ)/Xₗ | Short line; Xₗ: line reactance; maximum at δ = 90° |
| P = (VₛVᵣ)/Xₗ sin(δ) | Power-angle equation for lossless line |
15.2 Stability Considerations
- Steady-state stability limit at δ = 90° for lossless line
- Practical operating limit δ = 30°-45° for margin
- Compensation (series capacitors, shunt reactors) extends transfer capability
15.3 Voltage Stability
- Occurs when system cannot maintain voltage under increased load
- Critical when P approaches maximum deliverable power
- Reactive power support essential at receiving end
16. Compensation Techniques
16.1 Series Compensation
| Method | Effect |
|---|---|
| Series capacitors | Reduces effective line reactance; increases power transfer; improves voltage regulation |
| Degree of compensation | k = Xc/Xₗ; k = 0.25-0.75 typical; Xc: capacitive reactance; Xₗ: line reactance |
16.2 Shunt Compensation
| Device | Application |
|---|---|
| Shunt reactors | Absorb reactive power; control overvoltage on lightly loaded long lines |
| Shunt capacitors | Supply reactive power; improve power factor; boost voltage under load |
| SVC (Static Var Compensator) | Dynamic reactive power control; combination of reactors and capacitors |
| STATCOM | Advanced voltage-source converter; fast dynamic response |
16.3 Benefits of Compensation
- Increased power transfer capability
- Improved voltage profile along line
- Reduced losses
- Enhanced system stability
- Mitigation of Ferranti effect (shunt reactors)
17. Overhead vs. Underground
17.1 Comparison
| Aspect | Overhead |
|---|---|
| Cost | Lower installation cost; easier maintenance access |
| Capacitance | Lower (GMD larger); less charging current |
| Inductance | Higher (larger spacing) |
| Ampacity | Higher (better cooling from air convection) |
| Reliability | More susceptible to weather and lightning |
| Aspect | Underground Cable |
|---|---|
| Cost | 5-15 times higher installation cost; difficult fault location |
| Capacitance | Higher (smaller spacing, higher permittivity); significant charging current |
| Inductance | Lower (close conductor spacing) |
| Ampacity | Lower (limited heat dissipation from soil) |
| Reliability | Protected from weather; lower fault rate but longer repair time |
17.2 Cable Ampacity Factors
- Soil thermal resistivity (ρ_thermal: 90-120°C·cm/W for dry soil, 50-70 for moist)
- Burial depth and spacing between cables
- Load factor and daily/seasonal variations
- Cable insulation temperature rating (XLPE: 90°C, EPR: 105°C)
18. Conductor Types
18.1 Common Conductors
| Type | Description |
|---|---|
| AAC | All Aluminum Conductor; lightweight; low strength; short spans |
| AAAC | All Aluminum Alloy Conductor; better strength-to-weight than AAC |
| ACSR | Aluminum Conductor Steel Reinforced; most common; high strength; aluminum-to-steel ratios vary |
| ACAR | Aluminum Conductor Alloy Reinforced; aluminum alloy core for reinforcement |
| ACCC | Aluminum Conductor Composite Core; carbon fiber core; high temperature operation (up to 200°C) |
18.2 Conductor Selection Factors
- Current-carrying capacity (ampacity)
- Mechanical strength (tension and sag requirements)
- Cost (material and installation)
- Conductivity (aluminum ≈ 61% IACS; copper ≈ 100% IACS)
- Weight and span length capability
19. Line Performance Equations
19.1 Sending End Calculations
| Parameter | Formula Using ABCD |
|---|---|
| Sending end voltage | Vₛ = AVᵣ + BIᵣ |
| Sending end current | Iₛ = CVᵣ + DIᵣ |
| Sending end power | Sₛ = VₛIₛ* = Pₛ + jQₛ |
19.2 Circle Diagrams
- Receiving end power circle: locus of Sᵣ for constant Vₛ, Vᵣ as load varies
- Sending end power circle: locus of Sₛ for constant Vₛ, Vᵣ as load varies
- Radius = VₛVᵣ/|B|; center offset by reactive power component
- Useful for visualizing operating limits and stability margins
20. Typical Parameter Values
20.1 Resistance
- Overhead lines: 0.01-0.15 Ω/km depending on conductor size
- Underground cables: 0.02-0.20 Ω/km depending on conductor size and material
20.2 Reactance
| Parameter | Typical Range |
|---|---|
| Overhead Xₗ | 0.3-0.5 Ω/km (60 Hz); increases with spacing |
| Overhead Xc | 200-400 kΩ·km; decreases with voltage level |
| Cable Xₗ | 0.08-0.15 Ω/km (close spacing reduces inductance) |
| Cable Xc | 10-50 kΩ·km (much lower than overhead due to high capacitance) |
20.3 Charging Current
- Overhead at 230 kV: approximately 0.5-0.7 A/km
- Overhead at 500 kV: approximately 1.5-2.0 A/km
- Underground cable: 10-40 A/km depending on voltage and cable type
20.4 Surge Impedance
- Overhead single circuit: 300-400 Ω
- Overhead bundled conductors: 230-300 Ω
- Underground cables: 30-80 Ω
The document Cheatsheet: Transmission is a part of the PE Exam Course Electrical & Computer Engineering for PE.
All you need of PE Exam at this link: PE Exam
About this Document
Apr 20, 2026
Last updated
Related Exams
Document Description: Cheatsheet: Transmission for PE Exam 2026 is part of Electrical & Computer Engineering for PE preparation.
The notes and questions for Cheatsheet: Transmission have been prepared according to the PE Exam exam syllabus. Information about Cheatsheet: Transmission covers topics
like and Cheatsheet: Transmission Example, for PE Exam 2026 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Cheatsheet: Transmission.
Introduction of Cheatsheet: Transmission in English is available as part of our Electrical & Computer Engineering for PE
for PE Exam & Cheatsheet: Transmission in Hindi for Electrical & Computer Engineering for PE course.
Download more important topics related with notes, lectures and mock test series for PE Exam
Exam by signing up for free. PE Exam: Cheatsheet: Transmission
Description
Cheatsheet: Transmission of Electrical & Computer Engineering to help you remember important concepts with short tricks. Start learning for PE Exam exam & improve retention with EduRev.
Information about Cheatsheet: Transmission
In this doc you can find the meaning of Cheatsheet: Transmission defined & explained in the simplest way possible. Besides explaining types of
Cheatsheet: Transmission theory, EduRev gives you an ample number of questions to practice Cheatsheet: Transmission tests, examples and also practice PE Exam
tests
Related Searches
ppt, Objective type Questions, Free, Extra Questions, mock tests for examination, pdf , Cheatsheet: Transmission, MCQs, Previous Year Questions with Solutions, Semester Notes, past year papers, Summary, study material, practice quizzes, Cheatsheet: Transmission, Important questions, Exam, shortcuts and tricks, Cheatsheet: Transmission, video lectures, Viva Questions, Sample Paper;