Question

Find the capacitance between the conductors of a single phase 10 Km long line. The diameter of each conductor is 1.213cm. The spacing between the conductors is 1.25m

Answer 1

To find the capacitance per unit length of a single-phase transmission line, we can use the following formula for the capacitance between two conductors:

\[ C' = \frac{2 \pi \epsilon_0}{\ln\left(\frac{D}{r}\right)} \]

where:
- \( C' \) is the capacitance per unit length (F/m),
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.854 \times 10^{-12} \) F/m,
- \( D \) is the distance between the centers of the conductors (m),
- \( r \) is the radius of the conductors (m).

Given:
- Diameter of each conductor \( = 1.213 \) cm \( = 0.01213 \) m,
- Spacing between conductors \( = 1.25 \) m.

First, calculate the radius \( r \) of each conductor:

\[ r = \frac{\text{Diameter}}{2} = \frac{0.01213 \text{ m}}{2} = 0.006065 \text{ m} \]

Next, use the formula to find the capacitance per unit length:

\[ C' = \frac{2 \pi \times 8.854 \times 10^{-12}}{\ln\left(\frac{1.25}{0.006065}\right)} \]

Calculate \( \frac{1.25}{0.006065} \):

\[ \frac{1.25}{0.006065} \approx 206.39 \]

Now, calculate the natural logarithm:

\[ \ln(206.39) \approx 5.33 \]

Finally, substitute these values into the formula:

\[ C' = \frac{2 \pi \times 8.854 \times 10^{-12}}{5.33} \approx 5.3 \times 10^{-12} \text{ F/m} \]

So, the capacitance per unit length of the transmission line is approximately \( 5.3 \text{ pF/m} \).

Answer 2

To find the capacitance per unit length of a single-phase transmission line consisting of two conductors, we can use the formula for the capacitance of two parallel cylindrical conductors:

\[
C = \frac{2\pi \epsilon_0}{\ln\left(\frac{D}{r}\right)}
\]

where:
- \( C \) is the capacitance per unit length (in Farads per meter),
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.854 \times 10^{-12} \, \text{F/m} \),
- \( D \) is the distance between the centers of the conductors (in meters),
- \( r \) is the radius of the conductors (in meters).

### Step 1: Calculate the radius of the conductors
Given that the diameter of each conductor is \( 1.213 \, \text{cm} \):

\[
\text{Diameter} = 1.213 \, \text{cm} = 0.01213 \, \text{m}
\]
\[
r = \frac{\text{Diameter}}{2} = \frac{0.01213 \, \text{m}}{2} = 0.006065 \, \text{m}
\]

### Step 2: Determine the distance between the centers of the conductors
The spacing between the conductors is given as \( 1.25 \, \text{m} \). Therefore, the distance \( D \) between the centers of the conductors is:

\[
D = 1.25 \, \text{m}
\]

### Step 3: Calculate the capacitance per unit length
Now we can substitute the values into the capacitance formula:

\[
C = \frac{2\pi \epsilon_0}{\ln\left(\frac{D}{r}\right)}
\]

Calculating \( \frac{D}{r} \):

\[
\frac{D}{r} = \frac{1.25 \, \text{m}}{0.006065 \, \text{m}} \approx 206.5
\]

Now, calculate the natural logarithm:

\[
\ln\left(206.5\right) \approx 5.87
\]

Now substituting this back into the capacitance formula:

\[
C = \frac{2\pi (8.854 \times 10^{-12} \, \text{F/m})}{5.87}
\]
\[
C \approx \frac{2 \times 3.14159 \times 8.854 \times 10^{-12}}{5.87}
\]
\[
C \approx \frac{5.5707 \times 10^{-11}}{5.87} \approx 9.49 \times 10^{-12} \, \text{F/m}
\]

### Conclusion
The capacitance per unit length of the single-phase transmission line is approximately:

\[
C \approx 9.49 \, \text{pF/m}
\]

### Step 4: Total capacitance for the entire line
For a line length of \( 10 \, \text{km} = 10,000 \, \text{m} \), the total capacitance \( C_{\text{total}} \) can be calculated as:

\[
C_{\text{total}} = C \times \text{Length} = 9.49 \times 10^{-12} \, \text{F/m} \times 10,000 \, \text{m}
\]
\[
C_{\text{total}} \approx 9.49 \times 10^{-8} \, \text{F} = 94.9 \, \text{nF}
\]

Thus, the total capacitance between the conductors of the 10 km long single-phase line is approximately **94.9 nF**.