--- title: "Typical ABCD values of long lines" subject: "Power System Analysis I" chapter: 5 key: "semester_4/power_system_analysis_i:long-line-abcd-typical-values" source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/long-line-abcd-typical-values site: BCT Engineering --- # Typical ABCD values of long lines Power System Analysis I · Chapter 5: Transmission Line Modeling - Asked in: 2072 Asoj, 5 marks - 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/long-line-abcd-typical-values - Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-5 ## Question What are the typical values of ABCD constants in long transmission lines? Express these constants in terms of line series impedance and shunt admittance where applicable. ## Short answer For a long EHV line (300–500 km), A = D ≈ 0.85–0.95 ∠ 0.5°–2°, B ≈ 100–200 ∠ 75°–87° Ω (a little less than Z), C ≈ (0.9–2) × 10⁻³ ∠ ~90° S (a little less than Y). ## Answer For a long line the ABCD constants are given exactly by hyperbolic functions of $\gamma l$. Expanding them in series puts them in terms of the total series impedance $Z = zl$ and total shunt admittance $Y = yl$ (note $\gamma^2l^2 = ZY$ and $Z_c = \sqrt{Z/Y}$). #### Exact expressions $$ A = D = \cosh\gamma l, \quad B = Z_c\sinh\gamma l, \quad C = \frac{\sinh\gamma l}{Z_c} $$ #### Series (in terms of Z and Y) $$ \begin{aligned} A = D &= 1 + \frac{ZY}{2} + \frac{Z^2Y^2}{24} + \frac{Z^3Y^3}{720} + \cdots \\ B &= Z\left(1 + \frac{ZY}{6} + \frac{Z^2Y^2}{120} + \cdots\right) \\ C &= Y\left(1 + \frac{ZY}{6} + \frac{Z^2Y^2}{120} + \cdots\right) \end{aligned} $$ Also, $B = Z\frac{\sinh\gamma l}{\gamma l}$ and $C = Y\frac{\sinh\gamma l}{\gamma l}$. In all cases $AD - BC = 1$. #### Typical values For a well-designed 50 Hz overhead EHV line of 300–500 km ($\gamma l \approx 0.3$–0.5 rad, nearly all imaginary): | Constant | Typical value | Remarks | |---|---|---| | A = D | 0.85–0.95 ∠ 0.5°–2° | Less than 1, small positive angle | | B | 100–200 ∠ 75°–87° Ω | A little less than $Z$ | | C | (0.9–2) × 10⁻³ ∠ about 90° S | A little less than $Y$ | Example: a 450 km, 400 kV line with $r = 0.033$ Ω/km, $L = 1.067$ mH/km, $C = 0.0109$ μF/km has $A = 0.886\angle 0.71^\circ$, $B = 145.8\angle 84.6^\circ\ \Omega$ (with $|Z| = 151.6\ \Omega$), $C = 1.48\times10^{-3}\angle 90.2^\circ$ S (with $|Y| = 1.541\times10^{-3}$ S). Points to note: - $|A| < 1$ means no-load $V_R = V_S/|A| > V_S$ (Ferranti effect). - $B$ and $C$ are both slightly smaller than $Z$ and $Y$, because $|\sinh\gamma l/\gamma l| < 1$ when $\gamma l$ is nearly imaginary. - A short line has $A = 1$, $C = 0$; as length increases $|A|$ falls and $|C|$ rises. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which set gives the exact ABCD constants of a long line? A. A = D = cosh(γl), B = Zc sinh(γl), C = sinh(γl)/Zc B. A = D = sinh(γl), B = Zc cosh(γl), C = cosh(γl)/Zc C. A = D = cosh(γl), B = sinh(γl)/Zc, C = Zc sinh(γl) D. A = 1 + ZY/2, B = Z, C = Y Answer: A. Solving the distributed line equations gives A = D = cosh(γl), B = Zc sinh(γl) and C = sinh(γl)/Zc. The last option is the nominal-Π approximation. (Key point: Exact ABCD of long line) 2. For a typical 300-500 km EHV line, what is |A| = |D| like? A. Greater than 1, around 1.1 B. Exactly 1 C. Less than 1, about 0.85-0.95, with a small positive angle D. Zero, because the line is mostly inductive Answer: C. Because cosh(γl) is about 1 + ZY/2 and ZY is negative real-ish, |A| falls below 1 (0.85-0.95). (Key point: Typical magnitude of A) 3. What does |A| < 1 imply for a long line at no load? A. The receiving-end voltage is lower than the sending-end voltage B. The receiving-end voltage VR = VS/|A| is higher than VS (Ferranti effect) C. The line current is zero at the sending end D. The line has no losses Answer: B. At no load IR = 0, so VS = A VR and VR = VS/|A|, which exceeds VS when |A| < 1. (Key point: Ferranti effect from |A|<1) 4. As the line gets longer, how do |A| and |C| change compared with a short line (A = 1, C = 0)? A. |A| rises above 1 and |C| stays zero B. |A| falls and |C| rises C. Both |A| and |C| rise D. Both |A| and |C| fall Answer: B. Longer length increases shunt charging effects, so |A| falls below 1 and |C| grows from zero. 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