Given circuit shows an ideal autotransformer (k=1 and no power losses).
(a) Apply Kirchhoffs laws to write down two equations to express the voltages V1 and V2 in terms of the self and mutual voltages induced across the two coils.
(b) Hence obtain an expression for the voltage ratio \( \frac{V_1}{V_2} \) for the condition
L1 = 4L2.
(c) Determine the supply voltage and current with a 10 Ω load carrying
100 A and L1 = 4L2.
Question (a):
KVL loop 1:
\( V_1=j\omega L_1I_1+j\omega L_2I_1j\omega L_2I_2+j\omega MI_2j\omega MI_1 \)
KVL loop 2:
\( V_2=j\omega L_2I_2+j\omega L_2I_1+j\omega MI_1 \)
Question (b):
if \( L_1=4L_2 \)
then \( M=k\sqrt{L_1L_1}=1\sqrt{4L_2L_2}=2L_2\)
so \( V_1=j\omega 3L_2I_1+j\omega L_2I_2 \)
and \( V_2=j\omega 3L_2I_1j\omega L_2I_2 \)
\( \frac{V_1}{V_2}=\frac{j\omega 3L_2I_1+j\omega L_2I_2}{j\omega 3L_2I_1j\omega L_2I_2}\)
Is that correct? And what's next from here?
Question (c):
\( V_2=1000V\)
That's all I've got on this.
Your help is greatly appreciated.
(a) Apply Kirchhoffs laws to write down two equations to express the voltages V1 and V2 in terms of the self and mutual voltages induced across the two coils.
(b) Hence obtain an expression for the voltage ratio \( \frac{V_1}{V_2} \) for the condition
L1 = 4L2.
(c) Determine the supply voltage and current with a 10 Ω load carrying
100 A and L1 = 4L2.
Question (a):
KVL loop 1:
\( V_1=j\omega L_1I_1+j\omega L_2I_1j\omega L_2I_2+j\omega MI_2j\omega MI_1 \)
KVL loop 2:
\( V_2=j\omega L_2I_2+j\omega L_2I_1+j\omega MI_1 \)
Question (b):
if \( L_1=4L_2 \)
then \( M=k\sqrt{L_1L_1}=1\sqrt{4L_2L_2}=2L_2\)
so \( V_1=j\omega 3L_2I_1+j\omega L_2I_2 \)
and \( V_2=j\omega 3L_2I_1j\omega L_2I_2 \)
\( \frac{V_1}{V_2}=\frac{j\omega 3L_2I_1+j\omega L_2I_2}{j\omega 3L_2I_1j\omega L_2I_2}\)
Is that correct? And what's next from here?
Question (c):
\( V_2=1000V\)
That's all I've got on this.
Your help is greatly appreciated.
Attachments

28.2 KB Views: 46