ElectricalPower System

On an infinitely long transmission line, the attenuation coefficient α is expressed in terms of ratio K of the two voltages on km apart on the line, in bels/km.The value of this coefficient is given by

α = logₑK

α = K²

α = 2logX₁₀K

None of above

Correct Answer

⚙️ TE • Technical Concept & PrincipleElectricalPower System

Option D

None of above

Quick Summary:

The attenuation coefficient $\alpha$ represents the rate at which signal amplitude decays per unit length along a transmission line. For a transmission line where $K = \frac{V_1}{V_2}$ represents the ratio of voltages at two points separated by a distance $d$ in kilometers, the attenuation in bels is given by $\log_{10}(K)$ , and in nepers by $\ln(K)$ .

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$\alpha = \frac{1}{d} \ln(\frac{V_1}{V_2})$ (in nepers/km)

$\alpha = \frac{1}{d} \log_{10}(\frac{V_1}{V_2})$ (in bels/km)

⚙️ Working Principle

As an electromagnetic wave propagates along a transmission line, energy is lost due to series resistance ( $R$ ) and shunt conductance ( $G$ ). The voltage at a distance $x$ is defined by $V(x) = V_0 e^{-\alpha x}$ , where $\alpha$ is the attenuation constant. Expressing this for a distance $d$ , we get $\frac{V_0}{V_d} = e^{\alpha d}$ , leading to $\alpha = \frac{1}{d} \ln(\frac{V_0}{V_d})$ nepers/km or $\alpha = \frac{1}{d} \log_{10}(\frac{V_0}{V_d})$ bels/km.

📌 Key Points

▸
The unit 'Bel' is defined based on the base-10 logarithm of a ratio.
▸
The unit 'Neper' is defined based on the natural logarithm ( $e$ ) of a ratio.
▸
Attenuation coefficient $\alpha$ is a property of the transmission line parameters $R, L, G,$ and $C$ .

✅ Advantages

▸
Provides a standardized way to quantify signal loss.
▸
Essential for calculating the maximum permissible length of communication and power lines.

❌ Disadvantages / Limitations

▸
Frequency dependence of parameters makes $\alpha$ non-constant across wide spectra.
▸
Requires precise measurement of voltage magnitudes at two distinct spatial points.

🛠️ Applications / Uses

▸
Telecommunication network design.
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Power transmission line efficiency analysis.
▸
Fiber optic signal propagation modeling.

📄 Additional Information

▸
Since $K$ represents the ratio over $d$ km, the coefficient is $\alpha = \frac{1}{d} \log_{10}(K)$ . If $d=1$ km, then $\alpha = \log_{10}(K)$ .
▸
Option A uses $\ln$ (natural log) which corresponds to nepers, not bels. Option C incorrectly places a 2 multiplier and uses $X_{10}$ notation.

📊 Diagram / Illustration

✅

D is correct — the attenuation coefficient in bels/km for a distance $d$ is given by $\alpha = \frac{1}{d} \log_{10}(K)$ , which does not match any of the provided formulas.

Core Concepts Used

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Transmission Line Theory Logarithmic Units (Bel/Neper) Signal Attenuation

💡 EXAM TIP

Always verify if the logarithm base matches the unit (Base 10 for Bels, Base e for Nepers) when solving attenuation problems in power systems.

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