2. TRANSMISSION LINE PARAMETERS AND PERFORMANCE

 

2. TRANSMISSION LINE PARAMETERS AND PERFORMANCE

2.1 Line Parameters: Concepts of R, L, and C of Line Parameters and Types of Lines

Transmission lines are made up of conductors (usually copper or aluminum), and they have intrinsic electrical characteristics, called line parameters, which influence how the line behaves when transmitting electrical power.

The main parameters of a transmission line are:

1. Resistance (R):

   • The resistance represents the opposition to the flow of current in the conductors.
   • It depends on the material of the conductor, its length, and its cross-sectional area.
   • Resistance causes power losses in the form of heat due to the flow of current.
   • Formula: $R = \rho \frac{L}{A}$
     • $\rho$ = resistivity of the material
     • $L$ = length of the conductor
     • $A$ = cross-sectional area of the conductor

2. Inductance (L):

   • Inductance arises due to the magnetic field produced by the flow of current through the conductors.
   • The inductance depends on the spacing between the conductors, the number of conductors, and the physical properties of the transmission line.
   • The inductance also causes reactance, which opposes changes in current.
   • Formula: $L = \frac{\mu_0 \mu_r}{2\pi} \ln \left(\frac{D}{r}\right)$
     
     • $\mu_0$ = permeability of free space
     • $\mu_r$ = relative permeability of the conductor material
     • $D$ = distance between conductors
     • $r$ = radius of the conductor

3. Capacitance (C):

   • Capacitance in a transmission line occurs because of the electric fields between conductors. When the line is energized, the conductors store charge, creating capacitance.
   • The capacitance depends on the physical configuration of the line, the conductor's spacing, and the dielectric properties of the surrounding medium (air).
   • Formula: $C=\frac{2\pi ϵ}{\ln (\frac{D}{r})}$
     • $\epsilon$ = permittivity of the medium
     • $D$ = distance between conductors
     • $r$ = radius of the conductor

Types of Transmission Lines:

1. Short Line: Lines with a length of up to 50 km. For short lines, the effects of inductance and capacitance are minimal, so the line can be approximated as a simple resistor.
2. Medium Line: Lines ranging between 50 km and 200 km. Both inductance and capacitance have a significant effect.
3. Long Line: Lines longer than 200 km, where inductance, capacitance, and resistance need to be considered. Long lines experience significant voltage drops and require more complex models to predict performance.

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2.2 Performance of Short Line: Efficiency, Regulator, and Its Derivation, Effect of Power Factor, Vector Diagram for Different Power Factor

1. Performance of Short Line:

   • A short line is typically defined as a transmission line with a length of up to 50 km.
   • For short lines, the series impedance (R + jX) of the line is the primary concern, while the shunt admittance (capacitance) is neglected because of its minimal effect on short distances.
   • The performance of short lines is usually evaluated in terms of:
     • Voltage Regulation: The change in voltage at the load end when the load is disconnected.
     • Efficiency: The ratio of the power delivered to the load to the power supplied by the source.

2. Efficiency of Short Line:

   • The efficiency of the transmission line is given by the formula: $$\eta = \frac{P_{\text{load}}}{P_{\text{input}}} = \frac{V_{\text{load}}^2}{V_{\text{source}}^2}$$ where:
     • $P_{\text{load}}$ is the power delivered to the load,
     • $P_{\text{input}}$ is the total power supplied to the line,
     • $V_{\text{load}}$ and $V_{\text{source}}$ are the voltages at the load and source ends, respectively.

3. Regulator:

   • A voltage regulator is used to maintain a constant output voltage despite variations in the input voltage or load conditions. It is crucial for ensuring that the voltage at the load remains within acceptable limits.

4. Effect of Power Factor:

   • Power Factor (PF) is the ratio of real power (P) to apparent power (S), and it indicates the efficiency with which electrical power is used. It can be lagging, leading, or unity.
   • A lagging power factor (inductive load) causes higher current flow, leading to greater line losses.
   • A leading power factor (capacitive load) can cause over-voltage, potentially damaging equipment.
   • The vector diagram for different power factors illustrates the relationship between the voltage, current, and power in the system.

   Vector Diagram for Lagging Power Factor:

   • In this case, the current lags behind the voltage, meaning that the current waveform is delayed compared to the voltage waveform. This increases the total power loss due to the larger current.

   Vector Diagram for Leading Power Factor:

   • In this case, the current leads the voltage, meaning that the current waveform is ahead of the voltage waveform. This may cause over-voltage and unstable conditions in the system.

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2.3 Transposition of Conductors and Its Necessity

• Transposition refers to the process of switching the positions of conductors in a transmission line periodically, so that each conductor has an equal amount of time in the positions of the other conductors.

• Necessity of Transposition:

  • Balance the Line Parameters: In a multi-conductor transmission line, especially in three-phase systems, conductors are spaced at unequal distances, which results in unequal line parameters (inductance, capacitance). Transposition ensures that each conductor is subjected to equal inductance and capacitance, leading to a balanced system.
  • Reduce Radio Interference: Unbalanced transmission lines can cause electromagnetic interference. By transposing the conductors, the electromagnetic fields are more evenly distributed, reducing the possibility of unwanted radio frequency interference.
  • Minimize Voltage Unequalities: Without transposition, voltage imbalances can occur, which affects the overall stability and performance of the transmission line.

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2.4 Skin Effect, Ferranti Effect, and Proximity Effect

1. Skin Effect:

   • The skin effect refers to the phenomenon where alternating current (AC) tends to flow on the surface of a conductor rather than uniformly throughout its cross-section.
   • This occurs because the changing magnetic fields generated by the AC cause the current to concentrate near the surface of the conductor, leading to an increase in resistance and power losses at higher frequencies.
   • The skin effect is more pronounced in higher frequency currents (such as those in transmission lines) and at higher voltages.

2. Ferranti Effect:

   • The Ferranti effect is a phenomenon that occurs in long transmission lines, where the voltage at the receiving end of the line can be higher than the voltage at the sending end, especially under light load or no load conditions.
   • This effect is caused by the capacitance of the transmission line, which can cause the voltage to rise due to the reactive power stored in the line.
   • The Ferranti effect is more noticeable in long lines operating at high voltages, and it can lead to over-voltage conditions at the receiving end.

3. Proximity Effect:

   • The proximity effect occurs when two or more conductors carrying alternating current are placed close to each other. The alternating magnetic field of one conductor affects the current distribution in the nearby conductor.
   • As a result, the current is not evenly distributed across the cross-section of the conductor, leading to increased losses and reduced efficiency.
   • The proximity effect is more pronounced when the conductors are placed closer together or when the frequency of the current is high.

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Summary of Key Points:

1. Line Parameters: The key line parameters (R, L, and C) define the electrical characteristics of a transmission line and influence its performance.
2. Performance of Short Line: Short lines primarily experience power loss due to their resistance, and their efficiency and voltage regulation depend on their length and load conditions.
3. Transposition of Conductors: Transposing the conductors helps balance line parameters and reduce interference.
4. Skin, Ferranti, and Proximity Effects: These effects influence the current flow and voltage profile in transmission lines, affecting their efficiency and stability.

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Practice Questions:

1. What are the primary parameters that affect the performance of a transmission line? Explain the concepts of resistance, inductance, and capacitance.
2. Describe the performance of a short transmission line. What factors influence its efficiency?
3. What is the Ferranti effect and how does it impact long transmission lines?
4. Explain the necessity of conductor transposition in transmission lines.
5. What is the skin effect and how does it impact high-frequency currents in transmission lines?