Refer to the autotransformer circuit in Fig. 13.44. Calculate: (a) I1, I2, and Io if ZL = 8 + j6 Ω, and (b) the complex power supplied to the load.

Question

Refer to the autotransformer circuit in Fig. 13.44. Calculate: (a) [latex]\pmb{I_1, I_2}[/latex], and [latex]\pmb{I_o}[/latex] if [latex]\pmb{Z_L} = 8 + j6 Ω[/latex], and (b) the complex power supplied to the load.

Accepted Answer

(a) This is a step-up autotransformer with [latex]N_1  =  80, N_2  =  120, \pmb{V_1 } =120 \underline{/30^{\circ}}[/latex], so Eq. (13.67) can be used to find [latex]\pmb{V_2}[/latex] by

[latex] \pmb{\frac{ V _1}{ V _2}}=\frac{N_1}{N_1+N_2}=\frac{80}{200} [/latex]

or

[latex]\pmb{V _2}=\frac{200}{80} \pmb{V _1}=\frac{200}{80}\left(120 \underline{/ 30^{\circ}} ight)=300 \underline{/ 30^{\circ}} V \ \space \ \pmb{I _2}=\pmb{\frac{ V _2}{ Z _L}}=\frac{300 \underline{/ 30^{\circ}}}{8+j 6}=\frac{300 \underline{/ 30^{\circ}}}{10\underline{/ 36.87^{\circ}}}=30 \underline{/ -6.87^{\circ}} A [/latex]

But

[latex] \pmb{\frac{ I _1}{ I _2}}=\frac{N_1+N_2}{N_1}=\frac{200}{80} [/latex]

or

[latex] \pmb{I _1}=\frac{200}{80} \pmb{I _2}=\frac{200}{80}\left(30 \underline{/ -6.87^{\circ}} ight)=75 \underline{/ -6.87^{\circ}} A[/latex]

At the tap, KCL gives

[latex] \pmb{I _1+ I _o}= \pmb{I _2}[/latex]

or

[latex] \pmb{I _o}= \pmb{I _2- I _1}=30 \underline{/-6.87^{\circ}}-75 \underline{/-6.87^{\circ}}=45 \underline{/ 173.13^{\circ}} A [/latex]

(b) The complex power supplied to the load is

[latex]\pmb{S _2}= \pmb{V _2 I _2^*}=\left| \pmb{I _2} ight|^2 \pmb{Z _L}=(30)^2\left(10 \underline{/ 36.87^{\circ}} ight)=9 \underline{/ 36.87^{\circ}} kVA[/latex]

Question 13.11: Refer to the autotransformer circuit in Fig. 13.44. Calculat...

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