Figure shows the extended view of a $$2$$ pole $$dc$$ machine with $$10$$ armature conductors. Normal brush positions are shown by $$A$$ and $$B,$$ placed at the inter polar axis. If the brushes are now shifted, in the direction of rotation to $$A'$$ and $$B'$$ as shown, the voltage waveform $${V_{A'B'}}$$ will resemble

The figure above shows coil $$1$$ and $$2,$$ with dot markings as shown, having $$4000$$ and $$6000$$ turns respectively. Both the coils have a rated current of $$25$$ $$A.$$ Coil $$1$$ is excited with single phase, $$400$$ $$V,$$ $$50$$ $$Hz$$ supply.

The coils are to be connected to obtain a single-phase, $${{400} \over {1000}}\,\,V,$$ auto-transformer to drive a load of $$10$$ $$kVA.$$ Which of the options given should be exercised to realize the required auto-transformer?

The star-delta transformer shown above is excited on the star side with balanced, $$4$$-wire, $$3$$-phase, sinusoidal voltage supply of rated magnitude. The transformer is under no load condition.

With both $$S1$$ and $$S2$$ open, the core flux waveform will be

$$$F\left(x,y\right)=\left(x^2\;+\;xy\right)\;{\widehat a}_x\;+\;\left(y^2\;+\;xy\right)\;{\widehat a}_y$$$. Its line integral over the straight line from (x, y)=(0,2) to (2,0) evaluates to