Mindyard | Two buses, i and j, are connected with a transmission line of admittance Y, at the two ends of which there are ideal transformers with turns ratio as shown below. Bus admittance matrix for the system is [ − t i t j Y t j 2 Y t i 2 Y − t i t j Y ] , [ t i t j Y − t j 2 Y − t i 2 Y t i t j Y ] , [ t i 2 Y − t i t j Y − t i t j Y t j 2 Y ] , [ t i t j Y − ( t i − t j ) 2 Y − ( t i − t j ) 2 Y t i t j Y ]

Two buses, i and j, are connected with a transmission line of admittance Y, at the two ends of which there are ideal transformers with turns ratio as shown below. Bus admittance matrix for the system is

Options

$[\begin{matrix} - t_{i} t_{j} Y & t_{j}^{2} Y \\ t_{i}^{2} Y & - t_{i} t_{j} Y \end{matrix}]$

$[\begin{matrix} t_{i} t_{j} Y & - t_{j}^{2} Y \\ - t_{i}^{2} Y & t_{i} t_{j} Y \end{matrix}]$

$[\begin{matrix} t_{i}^{2} Y & - t_{i} t_{j} Y \\ - t_{i} t_{j} Y & t_{j}^{2} Y \end{matrix}]$

$[\begin{matrix} t_{i} t_{j} Y & - {(t_{i} - t_{j})}^{2} Y \\ - {(t_{i} - t_{j})}^{2} Y & t_{i} t_{j} Y \end{matrix}]$

Correct Answer :

Option C

$[\begin{matrix} t_{i}^{2} Y & - t_{i} t_{j} Y \\ - t_{i} t_{j} Y & t_{j}^{2} Y \end{matrix}]$

Solution :

Correct Answer:

The correct bus admittance matrix (Y_bus) for the given system is:

$[\begin{matrix} t_{i}^{2} Y & - t_{i} t_{j} Y \\ - t_{i} t_{j} Y & t_{j}^{2} Y \end{matrix}]$

Step-by-Step Explanation:

1. Understanding the Given System from the Image:

From the given diagram:
- Bus i has a voltage V_i and is connected to an ideal transformer with a turns ratio of 1 : t_i.
- Bus j has a voltage V_j and is connected to an ideal transformer with a turns ratio of t_j : 1.
- The transmission line connected between the secondaries of the two transformers has admittance Y.

2. Voltage and Current Relationships across the Ideal Transformers:

Let the voltages on the inner side (the transmission line side) of transformer i and transformer j be V'_i and V'_j respectively.

For the transformer at Bus i with turns ratio 1 : t_i:
$V_{i}^{'} = t_{i} V_{i}$

For the transformer at Bus j with turns ratio t_j : 1:
$V_{j}^{'} = t_{j} V_{j}$

3. Transmission Line Currents:

The current flowing through the line admittance Y from the inner terminal i to inner terminal j is given by Ohm's Law:
$I_{i}^{'} = Y (V_{i}^{'} - V_{j}^{'})$

Substituting V'_i = t_iV_i and V'_j = t_jV_j:
$I_{i}^{'} = Y (t_{i} V_{i} - t_{j} V_{j})$

4. Relating Line Currents to Bus Injected Currents:

By power conservation or ideal transformer equations (I₁ N₁ = I₂ N₂):

At Bus i:
$I_{i} = t_{i} I_{i}^{'} = t_{i} [Y (t_{i} V_{i} - t_{j} V_{j})] = t_{i}^{2} Y V_{i} - t_{i} t_{j} Y V_{j}$

At Bus j:
$I_{j} = t_{j} I_{j}^{'} = t_{j} [Y (t_{j} V_{j} - t_{i} V_{i})] = - t_{i} t_{j} Y V_{i} + t_{j}^{2} Y V_{j}$

5. Matrix Formulation:

Expressing the nodal current equations in matrix form:
$[\begin{matrix} I_{i} \\ I_{j} \end{matrix}] = [\begin{matrix} t_{i}^{2} Y & - t_{i} t_{j} Y \\ - t_{i} t_{j} Y & t_{j}^{2} Y \end{matrix}] [\begin{matrix} V_{i} \\ V_{j} \end{matrix}]$

Thus, the bus admittance matrix Y_bus is:
$Y_{b u s} = [\begin{matrix} t_{i}^{2} Y & - t_{i} t_{j} Y \\ - t_{i} t_{j} Y & t_{j}^{2} Y \end{matrix}]$

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Two buses, i and j, are connected with a transmission line of admittance Y, at the two ends of which there are ideal transformers with turns ratio as shown below. Bus admittance matrix for the system is

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