---
title: "Characteristic impedance, SIL, propagation constants"
subject: "Power System Analysis I"
chapter: 5
key: "semester_4/power_system_analysis_i:characteristic-impedance-sil-constants"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/characteristic-impedance-sil-constants
site: BCT Engineering
---

# Characteristic impedance, SIL, propagation constants

Power System Analysis I · Chapter 5: Transmission Line Modeling

- Asked in: 2072 Magh, 4 marks
- Video: [Power Systems | Lecture - 20 | Characteristic or Surge Impedance and Surge Impedance Loading](https://www.youtube.com/watch?v=bT0dc2Cm2Xo) — Engineering Devotion
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/characteristic-impedance-sil-constants
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-5

## Question

Define characteristic impedance, SIL, phase shift constant and attenuation constant for an overhead transmission line.

## Short answer

Characteristic impedance Z_c = √(z/y) ≈ √(L/C) for lossless lines (~250–400 Ω). SIL = V_L²/Z_c, the load giving flat voltage. Attenuation constant α = Re(γ), phase shift constant β = Im(γ); α causes amplitude decay (neper/km), β causes phase shift (rad/km).

## Answer

Let the line have series impedance $z = r + j\omega L$ and shunt admittance $y = g + j\omega C$ per unit length.

#### Characteristic (surge) impedance, $Z_c$
The ratio of voltage to current of a travelling wave on an infinitely long line (or a line terminated in $Z_c$):

$$
Z_c = \sqrt{\frac{z}{y}} = \sqrt{\frac{r + j\omega L}{g + j\omega C}}\ \Omega
$$

For a lossless line, $Z_c = \sqrt{L/C}$, a pure resistance (about 400 Ω for single-conductor overhead lines, 250–300 Ω for bundled EHV lines, 40–60 Ω for cables). It does not depend on length.

#### Surge impedance loading (SIL)
The power delivered when the line feeds a resistive load equal to $Z_c$:

$$
\text{SIL} = \frac{V_L^2}{Z_c}\ \text{MW}\ (V_L \text{ in kV})
$$

At SIL the reactive power generated by the line capacitance equals that absorbed by its inductance, so the voltage is flat along the line.

#### Propagation constant
$\gamma = \sqrt{zy} = \alpha + j\beta$ per unit length.

#### Attenuation constant, $\alpha$
The real part of $\gamma$, in nepers per km. It shows how fast the **magnitude** of a voltage or current wave falls along the line ($e^{-\alpha x}$), because of losses in $r$ and $g$. For a lossless line, $\alpha = 0$.

#### Phase shift constant, $\beta$
The imaginary part of $\gamma$, in radians per km. It shows how much the **phase** of the wave changes per unit length. For a lossless line:

$$
\beta = \omega\sqrt{LC}, \quad \lambda = \frac{2\pi}{\beta} \approx 6000\ \text{km at 50 Hz}, \quad v = \frac{\omega}{\beta} \approx 3\times10^5\ \text{km/s}
$$

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For a lossless line, what is the characteristic impedance and how does it depend on line length?
   A. Zc = sqrt(LC); it grows with length
   B. Zc = sqrt(L/C), a pure resistance independent of length
   C. Zc = sqrt(L/C), and it is proportional to length
   D. Zc = sqrt(C/L), independent of length
   Answer: B. Zc = sqrt(z/y) reduces to sqrt(L/C) for a lossless line, a resistance that does not depend on length. (Key point: Characteristic impedance)

2. What happens along the line when it is loaded at SIL?
   A. Reactive power generated by line capacitance equals that absorbed by line inductance, so the voltage profile is flat
   B. The line absorbs the maximum reactive power
   C. The receiving voltage is double the sending voltage
   D. The line current is zero
   Answer: A. At SIL (load = Zc), Q generated by C balances Q absorbed by L, giving a flat voltage profile. (Key point: SIL meaning)

3. In gamma = alpha + j beta, what do alpha and beta describe?
   A. Alpha is the phase change per km, beta is the magnitude decay per km
   B. Alpha is the magnitude decay of the wave per km (nepers), beta is the phase shift per km (radians)
   C. Both describe magnitude decay
   D. Alpha is the wavelength and beta the velocity
   Answer: B. Alpha is the real part (amplitude falls as e^-alpha x due to r and g), beta the imaginary part (phase change per unit length). (Key point: Attenuation vs phase constants)

4. What is the SIL formula for a line with line voltage VL (kV) and surge impedance Zc?
   A. VL x Zc
   B. VL^2 / Zc in MW
   C. Zc / VL^2
   D. sqrt(3) VL / Zc
   Answer: B. SIL is the power delivered into a resistive load equal to Zc: VL^2/Zc. (Key point: SIL formula)

## Related in Power System Analysis I

- [200 mile line: Zc, propagation constant, SIL](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3A200-mile-zc-propagation-sil) — Power System Analysis I, chapter 5
- [400 km line at UPF: Zc, propagation, regulation](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3A400km-upf-zc-regulation) — Power System Analysis I, chapter 5

## Related in other subjects

- [Derive input impedance of transmission line](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aderive-input-impedance-line) — Electromagnetism, chapter 5
- [6° phase shift per 100 km of lossless line](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Aphase-shift-6-degree-100km) — High Voltage Engineering, chapter 3
- [Surge impedance and velocity of 220 kV line](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Asurge-impedance-220kv-200km) — High Voltage Engineering, chapter 3, asked 2×