--- title: "Equivalent-π model of a long line" subject: "Power System Analysis I" chapter: 5 key: "semester_4/power_system_analysis_i:equivalent-pi-long-line" source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/equivalent-pi-long-line site: BCT Engineering --- # Equivalent-π model of a long line Power System Analysis I · Chapter 5: Transmission Line Modeling - Asked in: 2074 Bhadra, 6 marks - Video: [PS28 Equivalent Pi and T Circuits of Long Transmission Line](https://www.youtube.com/watch?v=54t-J2bwbAY) — Lectures in Electrical Engineering - 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/equivalent-pi-long-line - Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-5 ## Question What is an equivalent π-model of a long transmission line? Derive expressions for parameters of this circuit in terms of line parameters. ## Short answer An equivalent-π model is a lumped π circuit that exactly matches the long-line distributed-parameter ABCD constants. Series Z' = Z (sinh γl)/(γl) and each shunt Y'/2 = (Y/2)(tanh(γl/2))/(γl/2), where Z is total series impedance and Y total shunt admittance. ## Answer An **equivalent-Π model** of a long line is a lumped Π circuit (series $Z'$, shunt $Y'/2$ at each end) whose ABCD constants are **exactly equal** to those of the long line. It gives exact terminal results while being as simple as a nominal-Π circuit. ```text IS Z' IR o-->--+---[ Z' ]----+-->--o | | VS [Y'/2] [Y'/2] VR | | o-----+-------------+-----o ``` #### ABCD of the Π circuit and of the long line For any Π circuit: $$ A = D = 1 + \frac{Z'Y'}{2}, \quad B = Z', \quad C = Y'\left(1 + \frac{Z'Y'}{4}\right) $$ For the long line: $$ A = D = \cosh\gamma l, \quad B = Z_c\sinh\gamma l, \quad C = \frac{\sinh\gamma l}{Z_c} $$ #### Series arm $Z'$ Equating $B$: $$ Z' = Z_c\sinh\gamma l = \sqrt{\frac{z}{y}}\sinh\gamma l $$ Multiply and divide by $l\sqrt{zy}$ (note $Z = zl$, $\gamma l = l\sqrt{zy}$): $$ Z' = zl\,\frac{\sinh\gamma l}{l\sqrt{zy}} = Z\,\frac{\sinh\gamma l}{\gamma l} $$ #### Shunt arm $Y'/2$ Equating $A$: $$ \begin{aligned} 1 + \frac{Z'Y'}{2} &= \cosh\gamma l \\ \frac{Y'}{2} &= \frac{\cosh\gamma l - 1}{Z_c\sinh\gamma l} = \frac{1}{Z_c}\tanh\frac{\gamma l}{2} \end{aligned} $$ using $\cosh\gamma l - 1 = 2\sinh^2\frac{\gamma l}{2}$ and $\sinh\gamma l = 2\sinh\frac{\gamma l}{2}\cosh\frac{\gamma l}{2}$. Writing $\frac{1}{Z_c} = \sqrt{y/z}$ and multiplying and dividing by $\frac{\gamma l}{2}$: $$ \frac{Y'}{2} = \frac{Y}{2}\,\frac{\tanh(\gamma l/2)}{\gamma l/2} $$ (The $C$ equation is then automatically satisfied, because $AD - BC = 1$.) #### Summary | Element | Nominal-Π | Equivalent-Π | |---|---|---| | Series | $Z$ | $Z\frac{\sinh\gamma l}{\gamma l}$ | | Each shunt | $\frac{Y}{2}$ | $\frac{Y}{2}\frac{\tanh(\gamma l/2)}{\gamma l/2}$ | The correction factors $\frac{\sinh\gamma l}{\gamma l}$ and $\frac{\tanh(\gamma l/2)}{\gamma l/2}$ tend to 1 as $\gamma l \to 0$, so for shorter lines the equivalent Π reduces to the nominal Π. For long lines, $Z'$ is slightly less than $Z$ and $Y'$ slightly more than $Y$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. An equivalent-Π circuit of a long line is one whose: A. Series arm equals Z and shunt arms equal Y/2 B. ABCD constants exactly equal those of the long line C. Resistance is zero D. Capacitance is distributed Answer: B. It is a lumped Π giving exact terminal results. (Key point: Definition of equivalent-Π) 2. Equating B gives the series arm: A. Z' = Z cosh γl B. Z' = Z γl / sinh γl C. Z' = Z sinh(γl)/(γl) D. Z' = Z tanh(γl/2) Answer: C. Z' = Zc sinh γl = Z sinh(γl)/(γl). (Key point: Series arm Z') 3. The shunt arm of equivalent Π is: A. Y/2 x sinh(γl)/(γl) B. Y/2 x tanh(γl/2)/(γl/2) C. Y/2 x cosh(γl) D. Y x γl Answer: B. From A: Y'/2 = (1/Zc) tanh(γl/2) = (Y/2) tanh(γl/2)/(γl/2). (Key point: Shunt arm Y'/2) 4. As γl tends to 0, the equivalent Π: A. Becomes an open circuit B. Has zero series impedance C. Diverges D. Reduces to the nominal Π Answer: D. Both correction factors tend to 1. (Key point: Limit for short lines) ## Related in Power System Analysis I - [Equivalent-T of 300 km long line](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Along-line-equivalent-t-300km) — Power System Analysis I, chapter 5 - [Distributed parameters and current along a long line](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Adistributed-parameters-long-line-current) — Power System Analysis I, chapter 5 - [Long line by short model and vice versa](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Along-line-short-model-vice-versa) — Power System Analysis I, chapter 5, asked 2× - [Long line sending end voltage derivation](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Along-line-sending-voltage-derivation) — Power System Analysis I, chapter 5 - [Long line voltage and current at intermediate point](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Along-line-intermediate-point) — Power System Analysis I, chapter 5 - [Model validity and nominal-π ABCD with phasor](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Along-model-validity-nominal-pi) — Power System Analysis I, chapter 5 ## Related in other subjects - [Transmission parameters in terms of Y-parameters](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Atransmission-line-parameters-in-terms-of) — Electrical Circuit Theory, chapter 7 - [Determining single phase IM equivalent circuit parameters](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Aequivalent-circuit-parameter-tests) — Electric Machines II, chapter 3 - [Transformer equivalent circuit, referring parameters](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Aequivalent-circuit-primary-secondary-derivation) — Electrical Machines, chapter 2 - [Derive input impedance of transmission line](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aderive-input-impedance-line) — Electromagnetism, chapter 5