GATE Electrical Engineering (EE) 2020 Solved Paper

Question : 47 of 65

Marks: +1, -0

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 ratios as shown. Bus admittance matrix for the system is

$[\table {t}_{ {i}}^{2} Y , - {t}_{ {i}} {t}_{ {j}} Y ; - {t}_{ {i}} {t}_{ {j}} {Y} , {t}_{ {j}}^{2} {Y}]$
$[\table {t}_{ {i}} {t}_{ {j}} {Y} , - {t}_{ {j}}^{2} {Y} ; - {t}_{ {i}}^{2} {Y} , {t}_{ {i}} {t}_{ {j}} {Y}]$
$[\table -t_{i} t_{j} Y , t_{j}^{2} Y ; t_{i}^{2} Y , -t_{i} t_{j} Y]$
$[\table t_{i} t_{j} Y , -(t_{i}-t_{j})^{2} Y ; -(t_{i}-t_{j})^{2} Y , t_{i} t_{j} Y]$

Solution:

Let current

{I}_{ {i}}

{V}_{1}= {V}_{ {i}} {t}_{ {i}} \;\text " and "\; {V}_{2}= {V}_{ {j}} {t}_{ {j}}

{I}_{1}=\;{ {I}_{ {i}}}/{ {t}_{ {i}}} \& {I}_{2}=\;{ {I}_{ {j}}}/{ {t}_{ {j}}}

\;{I}/{t_{i}}=(V_{i} t_{i}-V_{j} t_{j}) y ⇒ I_{i}=V_{i} t_{i}^{2} ⋅ y V_{j} t_{j} ⋅ y ⋅ t_{j}

Similarly

\;{ {I}_{ {j}}}/{ {t}_{ {j}}}=( {V}_{ {j}} {t}_{ {j}}- {V}_{ {i}} {t}_{ {i}}) {y}

{I}_{ {j}}=- {V}_{ {i}} {t}_{ {i}} {t}_{ {j}} {y}+ {V}_{ {j}} {t}_{ {j}}^{2} ⋅ {y}

∴ [ {Y}]=[\table {yt}_{ {i}}^{2} , - {yt}_{ {i}} {t}_{ {j}} ; - {yt} {t} {t}_{ {j}} , {yt}_{ {j}}^{2}]

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