GATE Electrical Engineering (EE) 2021 Solved Papers

Question : 61 of 65

Marks: +1, -0

A 3-Bus network is shown. Consider generators as ideal voltage sources. If row 1,2 and 3 of the

{Y}_{ {BUS}}

matrix correspond to Bus 1,2 and 3 , respectively, then

Y_{\;\text "BUS "\;}

of the network is:

$[\table -\;{1}/{2} j , \;{1}/{4} j , \;{1}/{4} j ; \;{1}/{4} j , -\;{1}/{2} j , \;{1}/{4} j ; \;{1}/{4} j , \;{1}/{4} j , -\;{1}/{2} j]$
$[\table -4 j , 2 j , 2 j ; 2 j , -4 j , 2 j ; 2 j , 2 j , -4 j]$
$[\table -\;{3}/{4} j , \;{1}/{4} j , \;{1}/{4} j ; \;{1}/{4} j , -\;{3}/{4} j , \;{1}/{4} j ; \;{1}/{4} j , \;{1}/{4} j , -\;{3}/{4} j]$
$[\table -4 j , j , j ; j , -4 j , j ; j , j , -4 j]$

Solution:

{I}_{1}+ {I}_{3}+ {I}_{4}= {I}_{2}

..............(1)

{I}_{1}=( {V}_{1}- {V}_{2})(- {j} 1)

{I}_{1}=(- {j} 1) {V}_{1}+( {j} 1) {V}_{2}

.....(2)

{I}_{2}= {V}_{2}(- {j} 1)

.......(3)

{I}_{3}=( {V}_{3}- {V}_{2})(- {j} 1)

I_{3}=(-j 1) V_{2}+(j 1) V_{3}

....(4)

{I}_{4}=( {V}_{4}- {V}_{2})(- {j} 1)

{I}_{4}=( {j} 1) {V}_{2}-( {j} 1) {V}_{4}

......(5)
From equation (1) & (3)

{I}_{1}+ {I}_{3}+ {I}_{4}= {V}_{2}(- {j} 1)

{V}_{2}=( {j} 1)[(- {j} 1) {V}_{1}+( {j} 1) {V}_{2}+( {j} 1) {V}_{2}+(- {j} 1) {V}_{3}+( {j} 1) {V}_{2}-( {j} 1) {V}_{4}]

{V}_{2}= {V}_{1}-3 {V}_{2}+ {V}_{3}+ {V}_{4}

4 {V}_{2}= {V}_{1}+ {V}_{3}+ {V}_{4}

V_{2}=\;{1}/{4} V_{1}+\;{1}/{4} V_{3}+\;{1}/{4} V_{4}

.......(6)
Now,

I_{1}=(1 j 1) V_{1}+(j 1)[\;{1}/{4} V_{1}+\;{1}/{4} V_{3}+\;{1}/{4} V_{4}]

I_{1}=-j \;{3}/{4} V_{1}+j \;{1}/{3} V_{3}+j \;{1}/{4} V_{4}

......(7)

I_{3}=(-j 1)[V_{3}-\;{1}/{4} V_{1}-\;{1}/{4} V_{3}-\;{1}/{4} V_{4}]

I_{3}=j \;{1}/{4} V_{1}-j \;{3}/{4} V_{3}+j \;{1}/{4} V_{4}

....(8)

I_{4}=(j 1)[\;{1}/{4} V_{1}+\;{1}/{4} V_{3}+\;{1}/{4} V_{4}]-(j 1) V_{4}

I_{4}=j \;{1}/{4} V_{1}+j \;{1}/{4} V_{3}-J \;{3}/{4} V_{4}

......(9)

[\table I_{1} ; I_{2} ; I_{3}]=[\table -j \;{3}/{4} , j \;{1}/{4} , j \;{1}/{4} ; j \;{1}/{4} , -j \;{3}/{4} , j \;{1}/{4} ; j \;{1}/{4} , j \;{1}/{4} , -j \;{3}/{4}][\table V_{1} ; V_{3} ; V_{4}]

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