Transmission Lines and Cables Part 2
Calculating Transmission Line Inductances and Capacitances
Now, let’s delve into the calculation of transmission line inductances and capacitances, focusing on a balanced geometry as shown. In this configuration, the distance between any two conductors is ( D ), and ( R ) represents the radius of any conductor. Under balanced conditions, the currents sum to zero: $I_a + I_b + I_c = 0$.
Self-Inductance Calculation
To compute the self-inductance ( L_a ), we apply the basic definition of inductance: flux linkage divided by current. The total flux linkage for phase ( a ) includes contributions from its own current as well as currents in phases ( b ) and ( c ), applied via superposition.
Flux Linkage Due to Self-Current
The flux linkage due to the current in phase ( a ) itself is calculated using Ampere’s law, integrating the magnetic field intensity over the conductor’s length. This results in an expression that accounts for the magnetic field surrounding the conductor.
Flux Linkage Due to Other Currents
Similarly, we calculate the flux linkage in phase ( a ) due to currents in phases ( b ) and ( c ). Each contribution is integrated over the appropriate distance, accounting for the magnetic fields generated by these currents.
Total Flux Linkage and Self-Inductance
By summing up the contributions from all phases and applying the principle of superposition, we obtain the total flux linkage for phase ( a ). Dividing this by ( I_a ) yields the self-inductance per phase.
Geometric Mean Distance
If the geometry is not balanced, we can calculate the geometric mean distance to approximate the inductance. This accounts for variations in conductor separation.
Effect of Bundling
Bundling conductors affects inductance, reducing it compared to the single-conductor case. The reduction factor depends on the bundle configuration and can be calculated accordingly.
Transmission Line Capacitances
Capacitances between conductors must also be considered. Electric field concepts are applied to compute the capacitance per unit length between a conductor and a hypothetical neutral point.
Effect of Bundling on Capacitance
Bundling increases capacitance, with specific factors depending on the bundle configuration. These factors adjust the calculated capacitance accordingly.
Practical Considerations and Tools
Values for resistance and capacitance vary based on factors like voltage and conductor spacing. Sophisticated programs like EMT DC can accurately calculate these parameters, including frequency-dependent effects for transient studies.
Equations
Resistance formula: $R = \rho \frac{L}{A}$
Skin depth formula: $\delta = \sqrt{\frac{2}{\omega \mu \sigma}}$
Self-inductance formula: $L_a = \frac{\lambda_a}{I_a} = \frac{2 \times 10^{-7}}{\pi} \ln{\frac{2D}{R}}$
Flux linkage due to other currents formula: $\lambda_a = \lambda_a^{(1)} + \lambda_a^{(2)} + \lambda_a^{(3)}$
Flux linkage due to current in phase B formula: $\lambda_a^{(1)} = \frac{2 \times 10^{-7}}{\pi} \left(\frac{D}{2} + \sqrt{\left(\frac{D}{2}\right)^2 + R^2}\right)$
Flux linkage due to current in phase C formula: $\lambda_a^{(3)} = \frac{2 \times 10^{-7}}{\pi} \left(\frac{D}{2} + \sqrt{\left(\frac{D}{2}\right)^2 + R^2}\right)$
Geometric mean distance formula: $D = \sqrt{D_1 D_2}$
Capacitance between conductors formula: $C = \frac{\pi \epsilon}{\ln{\frac{D}{R}}}$
Typical values: $\begin{align*}
R & : 10-50 \text{ m}\Omega/\text{km} \\
L & : 0.5-2 \text{ mH}/\text{km} \\
C & : 5-10 \text{ }\mu\text{F}/\text{km}
\end{align*}$
Conclusion
Understanding the intricacies of transmission line inductances and capacitances is crucial for designing efficient and reliable power transmission systems. By accurately modeling these parameters, engineers can optimize system performance and ensure grid stability.