The performance equations of any branch ‘i’ in impedance form will be'

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Option A: $V_i + e_i = Z_i 	imes I_i$

Explanation & Concepts:
In power system network analysis, any branch 'i' of a network consists of a series impedance $Z_i$ and an internal source $e_i$. The branch performance equation describes the relationship between the terminal voltage $V_i$, internal emf $e_i$, and the current $I_i$ flowing through the branch.

Working Principle: According to Kirchhoff's Voltage Law applied to a branch, the terminal voltage $V_i$ plus the internal source emf $e_i$ must balance the voltage drop across the branch impedance $Z_i I_i$. The branch impedance $Z_i$ accounts for the resistive and reactive properties of the line or component, while $e_i$ represents any active voltage source within that branch.

Key Formulas: $V_i + e_i = Z_i 	imes I_i$ — Branch performance equation in impedance form; $I_i = Y_i(V_i + e_i)$ — Admittance form of the branch equation

Key Points: The equation $V_i + e_i = Z_i I_i$ assumes the convention where $e_i$ is a series voltage source.; Branch performance equations are fundamental for building the Z-bus (impedance matrix) for power flow studies.; In the absence of an internal source, $e_i$ becomes zero, simplifying the expression to $V_i = Z_i I_i$.

Advantages: Allows systematic formation of bus impedance matrices.; Easily accounts for series compensation and internal line voltage sources.

Disadvantages: Does not account for shunt elements directly without modification.; Requires complex arithmetic as Z is a complex quantity (R + jX).

Applications: Power system load flow analysis.; Short circuit calculations and fault analysis.

Additional Info: Option B ($V_i = Z_i 	imes I_i$) is only true for passive branches without internal EMF sources.. Options C and D are mathematically incorrect representations of the KVL loop equation for the specified branch model.

Answer: A is correct — The performance equation $V_i + e_i = Z_i 	imes I_i$ correctly satisfies Kirchhoff's Voltage Law for a branch containing a series impedance and a voltage source.

Exam Tip: Remember that in admittance form ($Y = Z^{-1}$), the equation becomes $I_i = Y_i(V_i + e_i)$, which is frequently tested in nodal analysis.

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Option B: $V_i = Z_i 	imes I_i$

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Option C: $V_i - e_i = Z_i$

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Option D: $V_i - e_i = Z_i 	imes I_i$

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