---
title: "Zbus vs Ybus; off-nominal transformer model"
subject: "Power System Analysis II"
chapter: 1
key: "semester_5/power_system_analysis_ii:ybus-off-nominal-transformer"
source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-off-nominal-transformer
site: BCT Engineering
---

# Zbus vs Ybus; off-nominal transformer model

Power System Analysis II · Chapter 1: Interconnected Power System

- Asked in: 2079 Bhadra, 2+4 marks
- Video: [BUS ADMITTANCE MATRIX - PART 15 - ADDING OFF-NOMINAL TAP CHANGING TRANSFORMER IN A LINE](https://www.youtube.com/watch?v=tdZWwWjinW8) — zenmurali Sachithanandam
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-off-nominal-transformer
- Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-1

## Question

What are difference between bus impedance matrix (Zbus) and bus admittance matrix (Ybus)? Develop Ybus for regulating transformers whose off-nominal turn ratio is 1:t and draw the equivalent π-model for this transformer.

## Short answer

Ybus is sparse, formed by inspection, used for load flow; Zbus is full, built algorithmically, used for fault analysis. For an off-nominal transformer 1:t with leakage admittance y, Ybus = [[t²y, –ty],[–ty, y]]. The π-model has series ty, shunt t(t‑1)y on the tap side, and (1–t)y on the other side.

## Answer

#### Zbus versus Ybus
| Point | $Y_{bus}$ | $Z_{bus}$ |
|---|---|---|
| Relation | $\mathbf{I} = \mathbf{Y}_{bus}\mathbf{V}$ | $\mathbf{V} = \mathbf{Z}_{bus}\mathbf{I}$ |
| Structure | Sparse | Full (dense) |
| Formation | By inspection, simple | Building algorithm or inversion |
| Elements | Driving-point and transfer admittances | Driving-point and transfer impedances |
| Main use | Load flow (G-S, N-R) | Short-circuit (fault) studies |
| Modification | Easy | Needs building steps |

#### Ybus for a regulating (off-nominal tap) transformer
Model the transformer as an ideal transformer of ratio $1 : t$ in series with its per-unit leakage admittance $y = 1/z$ placed on the tap (bus q) side. Bus p is on the "1" side, bus q on the "t" side.

```text
  p   Ip      1 : t     x    y     Iq   q
  o----->----)|(------o--[===]----<-----o
  Vp              Vx = t Vp
```

Ideal transformer: $V_x = tV_p$. Let $I_{xq} = y(V_x - V_q)$ be the current leaving node x towards q, so $I_q = -I_{xq}$. Being lossless, $V_pI_p^* = V_xI_{xq}^*$, which with real $t$ gives $I_p = tI_{xq}$:

$$
\begin{aligned}
I_q &= y(V_q - tV_p) = -tyV_p + yV_q \\
I_p &= t\,y(tV_p - V_q) = t^2yV_p - tyV_q
\end{aligned}
$$

In matrix form:

$$
\begin{bmatrix} I_p \\ I_q \end{bmatrix} =
\begin{bmatrix} t^2y & -ty \\ -ty & y \end{bmatrix}
\begin{bmatrix} V_p \\ V_q \end{bmatrix}
$$

So the transformer adds $t^2y$ to $Y_{pp}$, $y$ to $Y_{qq}$ and $-ty$ to $Y_{pq}$, $Y_{qp}$.

#### Equivalent π-model
Match a π circuit (series $A$, shunts $B$ at p and $C$ at q) to the matrix: $Y_{pq} = -A$, $Y_{pp} = A + B$, $Y_{qq} = A + C$:

$$
A = ty,\qquad B = t^2y - ty = t(t - 1)y,\qquad C = y - ty = (1 - t)y
$$

```text
  p          t*y            q
  o--------[=====]----------o
  |                         |
 [ t(t-1)y ]          [ (1-t)y ]
  |                         |
 ===                       ===
```

If $t = 1$, the shunts vanish and the model becomes the simple series admittance $y$. For $t \ne 1$ the two shunt branches have opposite signs (one acts inductive, the other capacitive), which is how the tap changes reactive power flow and bus voltage.

## Check yourself

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

1. Which statement correctly contrasts Ybus and Zbus?
   A. Ybus is full and Zbus is sparse
   B. Ybus is sparse and formed by inspection; Zbus is full
   C. Both are sparse
   D. Ybus is used mainly for fault studies
   Answer: B. Ybus is sparse and easy to form, used in load flow; Zbus is full, used for faults. (Key point: Zbus vs Ybus)

2. For off-nominal ratio 1:t with leakage admittance y on the tap side, Y_pp is:
   A. y
   B. t y
   C. t^2 y
   D. y / t
   Answer: C. Ip = t^2 y Vp - t y Vq, so Ypp = t^2 y. (Key point: Transformer Y_pp)

3. The mutual terms Y_pq = Y_qp are:
   A. -t y
   B. t^2 y
   C. -y/t
   D. y
   Answer: A. From the current equations, the coefficient of Vq in Ip is -t y. (Key point: Transformer mutual term)

4. In the equivalent pi model, the shunt at bus q is:
   A. t(t-1)y
   B. (1-t)y
   C. t y
   D. zero for all t
   Answer: B. C = Y_qq - A = y - t y = (1-t)y; the p-side shunt is t(t-1)y. (Key point: Pi model shunts)

## Related in Power System Analysis II

- [Ybus derivation for 3-bus π-model network](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-three-bus-derivation) — Power System Analysis II, chapter 1, asked 2×
- [Ybus, Zbus and 4-bus Ybus formulation](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-zbus-four-bus) — Power System Analysis II, chapter 1
- [Ybus with nominal π model, line lengths](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-km-lines-pi-model) — Power System Analysis II, chapter 1, asked 2×
- [Bus admittance matrix and its elements](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-definition-elements) — Power System Analysis II, chapter 1
- [Ybus modified by phase-shifting transformer](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-phase-shifter-midpoint) — Power System Analysis II, chapter 1
- [Node equations and Ybus, six-bus network](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-six-bus-node-equations) — Power System Analysis II, chapter 1

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