Question: Suppose we have two transformers with same turns ratio, \$N\_2/N\_1\$. First transformer \$A\$ is of \$100:10\$ and the second transformer \$B\$ is \$1000:100\$. Both have same turns ratio so both can step down the voltage by 10 times theoretically. If we assume size of wires are identical at primary and secondary, current and power ratings are same (I am not sure). But eventhough turns ratio is the same for both transformers, the number of turns is different in primary and secondary.

So my question is does the number of turns in turns ratio matter? I have a feeling that \$ B \$ has some advantages. But I am unable to figure out.

Answer: 
Obviously, B has some advantages indeed. Otherwise all transformers would use a tiny number of turns. More turns generate more magnetizing inductance \$L\_M \propto N^2\$.

Imagine a transformer with the secondary open, which is essentially a large inductor. If you connect an AC voltage with frequency \$f\$ to the primary, it will see an impedance of \$2 \pi f L\_M\$. If this impedance is too low, a lot of current will flow through the primary and be wasted with no effect. This excess current will be also present in addition to the regular load current when the secondary is connected. Therefore the excess current should be minimized and \$L\_M\$ should be maximized. In practise, there is a compromise between a sufficiently large \$L\_M\$ and a small/cheap transformer, so \$N\$ will be neither <10 nor extremely high.

Examples:

* For an SMPS transformer, where the lowest frequency is perhaps 100 kHz you don't need a lot of \$L\_M\$ to prevent excessive primary current, so a rather low number of turns will be fine.
* For a microphone transformer, you have a low end frequency of perhaps 20 Hz and in addition you don't want to load the signal source too much, so you need a lot of \$L\_M\$, which can mean 1000s of turns on primary.
* 50 Hz mains transformers also need large \$L\_M\$, because the primary voltage is high, frequency is low and you want the idle current to be low.
Transformer design has to optimize a lot of different variables. And I am not an expert. But the transformer primary needs to have enough turns so that it acts like a big inductor and prevents excessive current from flowing. If you take a particular transformer core, you can either wind the primary with many turns of fine wire (high voltage primary) or wind it with fewer turns of thicker wire (lower voltage primary).

But in both cases you MUST fill the whole winding area with copper to make sure the transformer will work well and have low resistance and handle its rated power.

You cannot just put one turn of thin wire. That is basically a short circuit, not a transformer. So, if you choose a transformer core based on power requirements, then choose your operating voltage, then you can calculate how many turns are needed to prevent oversaturation of the core. THEN you choose a wire diameter which will fill the available space efficiently.

That is more or less how the design process works. Once the primary is settled, you choose the number of secondary turns to get the desired secondary voltage. Once you know the number of secondary turns, you choose a wire diameter that fills the space effectively.

Generally, you end up with a lot of turns on your primary for 50 or 60 Hz power transformers.