---
title: "Bus admittance matrix and its elements"
subject: "Power System Analysis II"
chapter: 1
key: "semester_5/power_system_analysis_ii:ybus-definition-elements"
source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-definition-elements
site: BCT Engineering
---

# Bus admittance matrix and its elements

Power System Analysis II · Chapter 1: Interconnected Power System

- Asked in: 2072 Kartik, 5 marks
- Video: [Bus Admittance Matrix | Ybus Matrix | Derivation of Ybus Matrix | Load Flow](https://www.youtube.com/watch?v=KXYiU_3QcAY) — Dr. Hemant Mahala Academy 
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-definition-elements
- Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-1

## Question

What is a bus admittance matrix? Define its elements.

## Short answer

The bus admittance matrix Ybus relates injected currents I = Ybus V. Diagonal element Yii is the sum of all admittances connected to bus i; off-diagonal Yik equals negative of the series admittance between buses i and k, zero if no connection.

## Answer

The **bus admittance matrix** $Y_{bus}$ is an $n \times n$ matrix that relates the currents injected at the $n$ buses of a power network to the bus voltages measured from the reference (ground):

$$
\mathbf{I}_{bus} = \mathbf{Y}_{bus}\,\mathbf{V}_{bus},\qquad
I_i = \sum_{k=1}^{n} Y_{ik}V_k
$$

For a 3-bus system:

$$
\begin{bmatrix} I_1 \\ I_2 \\ I_3 \end{bmatrix} =
\begin{bmatrix} Y_{11} & Y_{12} & Y_{13} \\ Y_{21} & Y_{22} & Y_{23} \\ Y_{31} & Y_{32} & Y_{33} \end{bmatrix}
\begin{bmatrix} V_1 \\ V_2 \\ V_3 \end{bmatrix}
$$

#### Diagonal elements: self (driving-point) admittance
$$
Y_{ii} = y_{i0} + \sum_{k \ne i} y_{ik} = \left.\frac{I_i}{V_i}\right|_{V_k = 0,\ k \ne i}
$$

$Y_{ii}$ is the sum of all admittances connected to bus $i$, including series line admittances and shunt elements (half line charging, capacitors, reactors). It equals the current injected at bus $i$ per unit voltage at bus $i$ when all other buses are shorted to ground.

#### Off-diagonal elements: mutual (transfer) admittance
$$
Y_{ik} = -y_{ik} = \left.\frac{I_i}{V_k}\right|_{V_j = 0,\ j \ne k}
$$

$Y_{ik}$ is the negative of the series admittance between buses $i$ and $k$. It is the current injected at bus $i$ per unit voltage applied at bus $k$ with all other buses shorted. If buses $i$ and $k$ are not directly connected, $Y_{ik} = 0$.

#### Properties
- Symmetric: $Y_{ik} = Y_{ki}$ (no phase-shifting transformers).
- Sparse: most off-diagonal elements are zero in large networks.
- Usually diagonally dominant, since $Y_{ii}$ is the sum of the admittances that appear (with opposite sign) in the off-diagonal terms of its row; it is formed easily by inspection.
- Used directly in Gauss–Seidel and Newton–Raphson load flow.

## Check yourself

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

1. The bus admittance matrix relates:
   A. Bus voltages to line power flows
   B. Bus injected currents to bus voltages (I = Ybus V)
   C. Bus powers to bus angles
   D. Fault current to fault voltage only
   Answer: B. Ybus gives injected bus currents from bus voltages measured from reference. (Key point: Definition)

2. Y_ii equals the current injected at bus i per unit voltage at bus i when:
   A. All other buses are open-circuited
   B. All other buses are shorted to ground
   C. All other buses carry rated voltage
   D. Bus i is the slack bus
   Answer: B. Driving-point admittance is measured with the other buses shorted to ground (V_k = 0). (Key point: Self admittance meaning)

3. If buses i and k are not directly connected, Y_ik is:
   A. Equal to Y_ii
   B. Negative of Y_kk
   C. Zero
   D. Equal to the shunt admittance
   Answer: C. Y_ik = -y_ik, and there is no branch, so it is zero; this causes sparsity. (Key point: Mutual admittance)

4. Ybus is symmetric (Y_ik = Y_ki) provided that:
   A. The network has no phase-shifting transformers
   B. All buses are PQ buses
   C. There is no line charging
   D. The system has fewer than 4 buses
   Answer: A. A complex tap (phase shifter) breaks reciprocity and the symmetry. (Key point: Symmetry)

## 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
- [Bus classification and 4-bus Ybus](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Abus-classification-ybus-four-bus) — Power System Analysis II, chapter 1
- [Ybus and bus types (parallel lines)](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-bus-types-2074) — 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
- [Ybus for 3-bus lines with shunt susceptance](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-three-bus-shunt-2071) — Power System Analysis II, chapter 1

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