--- title: "Model validity and nominal-π ABCD with phasor" subject: "Power System Analysis I" chapter: 5 key: "semester_4/power_system_analysis_i:long-model-validity-nominal-pi" source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/long-model-validity-nominal-pi site: BCT Engineering --- # Model validity and nominal-π ABCD with phasor Power System Analysis I · Chapter 5: Transmission Line Modeling - Asked in: 2070 Bhadra, 2+4+2 marks - Video: [ABCD parameters of medium transmission line by nominal Pi method explained by Irfan Mujawar.](https://www.youtube.com/watch?v=K62VWxxG8d8) — Dr. Irfan Mujawar - 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-model-validity-nominal-pi - Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-5 ## Question Explain why all lines can be represented by the long transmission line model whereas all the lines cannot be represented by the short transmission line model. Derive expressions of ABCD parameters of a nominal π-model of a medium transmission line. Also draw the phasor diagram for lagging power factor load. ## Short answer The long-line model is exact for any length because it solves distributed-parameter equations; the short-line model neglects shunt capacitance, so it fails for lines >80 km. Nominal-π ABCD: A=D=1+ZY/2, B=Z, C=Y(1+ZY/4). For lagging load, phasor diagram shows V_S built from V_R plus voltage drop across Z, and I_S = I_L + Ic1. ## Answer #### Why the long-line model fits all lines but the short-line model does not - The **long-line model** uses distributed parameters and solves the line equations exactly: $A = \cosh\gamma l$, $B = Z_c\sinh\gamma l$, $C = \sinh\gamma l/Z_c$. No approximation is made, so it is valid for any length. For a short line $\gamma l$ is small, so $A \to 1$, $B \to Z$, $C \to 0$, which is exactly the short-line result. - The **short-line model** neglects shunt capacitance ($C = 0$, $I_S = I_R$). For medium and long lines the charging current is large, so this model gives wrong $V_S$, $I_S$, power factor and regulation, and cannot show the Ferranti effect. So it is valid only for short lines (up to about 80 km). #### ABCD parameters of the nominal-Π model Total series impedance $Z$ in the middle; $Y/2$ at each end. ```text IS IL Z = R + jX IR o-->--+-----[ R ]--[ jX ]-----+-->--o | | VS [Y/2] Ic1 Ic2 [Y/2] VR | | o-----+-----------------------+-----o ``` $$ \begin{aligned} I_L &= I_R + \frac{Y}{2}V_R \\ V_S &= V_R + ZI_L = \left(1 + \frac{ZY}{2}\right)V_R + ZI_R \\ I_S &= I_L + \frac{Y}{2}V_S = Y\left(1 + \frac{ZY}{4}\right)V_R + \left(1 + \frac{ZY}{2}\right)I_R \end{aligned} $$ $$ A = D = 1 + \frac{ZY}{2}, \quad B = Z, \quad C = Y\left(1 + \frac{ZY}{4}\right) $$ #### Phasor diagram (lagging p.f. load, $V_R$ reference) 1. $I_R$ lags $V_R$ by $\phi_R$; $I_{C2} = j\frac{\omega C}{2}V_R$ leads $V_R$ by 90°. 2. $I_L = I_R + I_{C2}$. 3. $V_S = V_R + I_LR + jI_LX$ ($I_LR$ parallel to $I_L$, $I_LX$ 90° ahead of $I_L$). 4. $I_{C1}$ leads $V_S$ by 90°; $I_S = I_L + I_{C1}$. ```text IC2 ^ + tip of VS | VS . | | . | IL.X | . d | O---------------------------> VR \ \ phiR \ IL.R \ v IL + v IR IS = IL + IC1 (IC1 is 90 deg ahead of VS) ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why can the long-line model represent lines of any length? A. It ignores the shunt capacitance B. It uses distributed parameters and solves the line equations exactly, so no approximation is made C. It assumes a lossless line D. It is only a lumped pi circuit Answer: B. The exact solution with cosh/sinh of γl is valid for any length and reduces to the short-line result (A = 1, B = Z, C = 0) as γl becomes small. (Key point: Why long model fits all lengths) 2. Why does the short-line model fail for medium and long lines? A. It neglects shunt capacitance, so it ignores the large charging current and cannot show the Ferranti effect B. It ignores the series resistance C. It requires the load to be zero D. It uses hyperbolic functions Answer: A. The short-line model sets C = 0 and IS = IR, which is wrong when charging current is significant. (Key point: Limitation of short-line model) 3. What are the ABCD constants of the nominal-Π model? A. A = D = 1 + ZY/2, B = Z, C = Y(1 + ZY/4) B. A = D = 1 + ZY/2, B = Z(1 + ZY/4), C = Y C. A = D = 1, B = Z, C = Y D. A = 1 + ZY, B = Z, C = Y/2 Answer: A. With Y/2 at each end, VS = (1 + ZY/2)VR + Z IR and IS = Y(1 + ZY/4)VR + (1 + ZY/2)IR. (Key point: Nominal-Π ABCD) 4. In the lagging-p.f. phasor diagram for the nominal-Π model, how is VS built? A. VS = VR + IL(R + jX), where IL = IR + IC2 and IC2 leads VR by 90° B. VS = VR + IR(R + jX), with IR as the line current C. VS = VR - IL(R + jX) D. VS = VR + IC1 (R + jX) Answer: A. The series current IL is IR plus the capacitor current at the receiving end; the drop IL R (parallel to IL) and IL X (90° ahead) is added to VR. 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