autotransformers currents
Consider Kirchhoff's current law:


  At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node.


Consider the central node. We see that it has \$255A\$ leaving to the right, and \$250A\$ entering from the top. There must be another \$5A\$ coming from somewhere, so that:

$$ 250A + 5A = 255A $$

This can be intuitively seen if one considers that charge can be neither created nor destroyed. If there is \$255C\$ of charge leaving each second, there must be exactly \$255C\$ of charge entering each second as well.

Lenz's law says something different. It simply says that the direction of the back-EMF (\$200V\$ and \$10000V\$ in this case) is such that it opposes the direction something else is trying to shove the current. In this case, say at some instant in the AC cycle the voltage source on the left is trying to shove current in the down direction through the autotransformer, so the autotransformer responds with a back-EMF that opposes that, and tries to shove the current back up. If it went the other way, we'd be violating the law of conservation of energy.

How the currents split between the halves of the autotransformer will then be determined by the windings ratio of the autotransformer, and the impedance of the load.



Here's another direction to approach it. For an ideal (auto)transformer, \$P_{in} = P_{out}\$. If we have a transformer that steps voltage down, then to keep input and output power equal, it must also step current up. Thus:

$$ P_{in} = P_{out}\\
V_{in} I_{in} = V_{out} I_{out}\\
V_{in} &gt; V_{out}\\
I_{in} &lt; I_{out}$$

If the output current is greater than the input current, where's the extra current come from? The answer is that \$5A\$, which is made possible by the reactance of the transformer. By forcing the output voltage to be lower than the input voltage (because the voltage ratios are constrained by the windings ratio of the transformer), current is the only variable free to change to avoid violating the law of conservation of energy. So, it does.