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The above problem was asked in my test. I marked option D. But the solution says it should be option C.

Can any one please explain me what is y12' in the solution and from where it is coming ?

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The common systematic nodal voltage analysis Y-matrix hasn't such term as y12'. Where it comes from - that's possible only to guess. I guess an error. The solution has been written for different network. There a distributed capacitance of a long line is approximately divided equally to nodes 1 and 2.

Actually that can be also a drawing convention, the distributed capacitance is drawn at one end, but there's a rule how it should be taken into the account in calculations. Unfortunately I haven't your book.

If we assume the drawing covention case to be true, then capacitive impedance -10j which has been drawn between 2 and gnd, is actually distributed along the cable and calculated approximately as divided equally to nodes 1 and 2.

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Yes,
in modelling the transmission line into simple network system there are three cases:
1. short line where the capacitance of the line is neglected.
2. medium line where capacitance is not neglected but modelling is done in two ways:

a) total capacitance (or shunt impedance) of line is divided into two parts and lumped at two ends, this is called pi model
b) total reactance (or series impedance) of line is divided into two parts and at the center the lumped total capacitance of line is placed, this is called T-model.

For power system analysis the use pi model is mainly used. In above question total capacitance is given and its division. The modelling of transmission line is left to solve.

Long lines don't use lumped parameters.

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    \$\begingroup\$ Make sure you format your answers, use correct punctuation \$\endgroup\$
    – Voltage Spike
    Apr 1 '18 at 6:19

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