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

# ABCD constants by nominal-T model

Power System Analysis I · Chapter 5: Transmission Line Modeling

- Asked in: 2071 Bhadra, 5 marks
- Video: [ABCD parameters of medium transmission line by Nominal T method explained by Irfan Mujawar](https://www.youtube.com/watch?v=HzHGlzKCBPI) — 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-t-abcd-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-5

## Question

Derive the expression of ABCD constants for a medium length transmission line using nominal-T model.

## Short answer

In the nominal-T model, A = D = 1 + ZY/2, B = Z(1 + ZY/4) Ω, C = Y S. Derivation: mid-point voltage V_C = V_R + (Z/2)I_R, then I_C = Y V_C, I_S = I_R + I_C, V_S = V_C + (Z/2)I_S.

## Answer

In the **nominal-T model** of a medium line, the whole shunt admittance $Y = j\omega C$ is placed at the mid-point of the line, and the series impedance $Z = R + jX$ is split into two halves, $Z/2$ on each side (per-phase values).

```text
 IS     Z/2            Z/2     IR
 o-->--[R/2 jX/2]--+--[R/2 jX/2]-->--o
                   |
 VS              [ Y ] Ic   (VC)     VR
                   |
 o-----------------+-----------------o
```

#### Derivation
**Step 1:** Voltage at the mid-point (across the capacitor):

$$
V_C = V_R + \frac{Z}{2}I_R
$$

**Step 2:** Current through the shunt branch:

$$
I_C = YV_C = YV_R + \frac{ZY}{2}I_R
$$

**Step 3:** Sending-end current:

$$
I_S = I_R + I_C = YV_R + \left(1 + \frac{ZY}{2}\right)I_R
$$

**Step 4:** Sending-end voltage:

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

**Step 5:** Comparing with $V_S = AV_R + BI_R$ and $I_S = CV_R + DI_R$:

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

#### Check

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

So the network is reciprocal, and since $A = D$ it is symmetrical. Compared with the nominal-Π model, $A$ and $D$ are the same, but the factor $(1 + ZY/4)$ moves from $C$ to $B$.

## Check yourself

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

1. In the nominal-T model, where are the series impedance and shunt admittance placed?
   A. Z at the middle and Y/2 at each end
   B. Z/2 on each side of a shunt Y at the mid-point
   C. Y split in two halves with Z at the sending end
   D. All of Y at the receiving end
   Answer: B. The impedance is split into Z/2 on both sides, with the whole shunt admittance Y at the mid-point. (Key point: Nominal-T model structure)

2. Which expression gives the sending-end current in the derivation?
   A. IS = IR
   B. IS = IR + Y VR
   C. IS = Y VR + (1 + ZY/2) IR
   D. IS = (1 + ZY/4) VR + Y IR
   Answer: C. VC = VR + (Z/2)IR, IC = Y VC, so IS = IR + IC = Y VR + (1 + ZY/2) IR. (Key point: Mid-point voltage and sending current)

3. What are the ABCD constants of the nominal-T model?
   A. A = D = 1 + ZY/2, B = Z(1 + ZY/4), C = Y
   B. A = D = 1 + ZY/2, B = Z, C = Y(1 + ZY/4)
   C. A = D = 1 + ZY/4, B = Z, C = Y
   D. A = D = 1, B = Z, C = Y
   Answer: A. Substituting into VS = VC + (Z/2)IS gives these; the factor (1 + ZY/4) is in B, whereas in nominal-Π it is in C. (Key point: Nominal-T ABCD result)

## Related in Power System Analysis I

- [Nominal-π model ABCD derivation](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Anominal-pi-abcd-derivation) — 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
- [Two lines in parallel: ABCD constants](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aparallel-lines-abcd-numerical) — Power System Analysis I, chapter 5
- [Line model classification and short-line ABCD](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aclassify-models-short-line-abcd) — 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×
- [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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