---
title: "Nominal-π model ABCD derivation"
subject: "Power System Analysis I"
chapter: 5
key: "semester_4/power_system_analysis_i:nominal-pi-abcd-derivation"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/nominal-pi-abcd-derivation
site: BCT Engineering
---

# Nominal-π model ABCD derivation

Power System Analysis I · Chapter 5: Transmission Line Modeling

- Asked in: 2073 Bhadra, 5 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/nominal-pi-abcd-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-5

## Question

Draw a nominal π-model of a medium transmission line and derive the expressions to determine ABCD parameters of the model.

## Short answer

In the nominal-π model, half the total shunt admittance Y/2 is placed at each end and the total series impedance Z in the middle. Applying KCL and KVL yields A = D = 1 + ZY/2, B = Z, C = Y(1 + ZY/4). The network is symmetrical and reciprocal.

## Answer

In the **nominal-Π model** of a medium line (80–250 km), the total series impedance $Z = R + jX$ is lumped in the middle, and half of the total shunt admittance, $Y/2 = j\omega C/2$, is placed at each end.

```text
 IS       IL    Z = R + jX        IR
 o-->--+-----[ R ]--[ jX ]-----+-->--o
       |                       |
 VS  [Y/2] Ic1           Ic2 [Y/2]  VR
       |                       |
 o-----+-----------------------+-----o
```

#### Derivation (per phase)
**Step 1:** Current in the receiving-end capacitor:

$$
I_{C2} = \frac{Y}{2}V_R
$$

**Step 2:** Current in the series impedance:

$$
I_L = I_R + I_{C2} = I_R + \frac{Y}{2}V_R
$$

**Step 3:** Sending-end voltage:

$$
\begin{aligned}
V_S &= V_R + ZI_L = V_R + Z\left(I_R + \frac{Y}{2}V_R\right) \\
V_S &= \left(1 + \frac{ZY}{2}\right)V_R + ZI_R
\end{aligned}
$$

**Step 4:** Sending-end current:

$$
\begin{aligned}
I_S &= I_L + \frac{Y}{2}V_S \\
&= I_R + \frac{Y}{2}V_R + \frac{Y}{2}\left[\left(1 + \frac{ZY}{2}\right)V_R + ZI_R\right] \\
&= \left(Y + \frac{ZY^2}{4}\right)V_R + \left(1 + \frac{ZY}{2}\right)I_R \\
I_S &= Y\left(1 + \frac{ZY}{4}\right)V_R + \left(1 + \frac{ZY}{2}\right)I_R
\end{aligned}
$$

**Step 5:** Compare with $V_S = AV_R + BI_R$, $I_S = CV_R + DI_R$:

$$
A = D = 1 + \frac{ZY}{2}, \quad B = Z\ \Omega, \quad C = Y\left(1 + \frac{ZY}{4}\right)\ \text{S}
$$

#### Check

$$
AD - BC = \left(1 + \frac{ZY}{2}\right)^2 - ZY\left(1 + \frac{ZY}{4}\right) = 1 + ZY + \frac{Z^2Y^2}{4} - ZY - \frac{Z^2Y^2}{4} = 1
$$

$A = D$ shows the network is symmetrical, and $AD - BC = 1$ shows it is reciprocal (passive).

## Check yourself

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

1. In the nominal-Π derivation the first step is IC2 equal to:
   A. Y VR
   B. (Y/2) VR
   C. (Y/2) VS
   D. Z IR
   Answer: B. The receiving-end capacitor carries (Y/2)VR. (Key point: Receiving-end shunt current)

2. The current in the series impedance is:
   A. IR
   B. IR - IC2
   C. IR + (Y/2)VR
   D. IS + IC1
   Answer: C. IL = IR + IC2. (Key point: Series current)

3. Comparison gives:
   A. A = D = 1, B = Z, C = 0
   B. A = D = 1 + ZY/2, B = Z, C = Y(1 + ZY/4)
   C. A = 1 + ZY, B = Z/2, C = Y
   D. A = D = 1 + ZY/2, B = Z(1 + ZY/4), C = Y
   Answer: B. That is the Π result; option D belongs to the T model. (Key point: Result for A, B, C)

4. AD - BC = 1 shows the network is:
   A. Active
   B. Reciprocal (passive)
   C. Lossless only
   D. Unsymmetrical
   Answer: B. AD - BC = 1 is the reciprocity condition; A = D indicates symmetry. (Key point: Meaning of AD - BC = 1)

## Related in Power System Analysis I

- [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
- [ABCD constants by nominal-T model](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Anominal-t-abcd-derivation) — Power System Analysis I, chapter 5
- [Nominal-T phasor diagram and ABCD parameters](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Anominal-t-phasor-abcd) — Power System Analysis I, chapter 5, asked 2×
- [Nominal-T derivation, phasor, losses, regulation](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Anominal-t-derivation-losses) — Power System Analysis I, chapter 5
- [Nominal-π derivation, losses and UPF phasor diagram](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Anominal-pi-200km-derivation-losses) — Power System Analysis I, chapter 5
- [ABCD of nominal-π and nominal-T compared](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aabcd-nominal-pi-t-compare) — Power System Analysis I, chapter 5, asked 2×

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