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: 
Number of turns on a transformer matters greatly. The turns ratio is one of many considerations in designing a transformer. The following are high-level considerations when designing a transformer.

* As Tobalt stated, *magnetizing inductance* is important. Too few turns will consume excess current, even without a load.
* You need a minimum of turns to prevent *saturation* of the core. Number of turns controls the *flux density*. If the core saturates, your primary starts to look like a short circuit to the driver. You can also control when a transformer saturates by changing the effective core area.
* Increasing number of turns will increase *leakage inductance* (can be modeled as a series inductance either in the primary or secondary leads) which may be undesirable. There are cases where leakage inductance is desirable. Leakage inductance can also be controlled by controlling the physical positioning of the windings and can increase the cost of increased build complexity.
* Increasing the number of turns will lower the *resonant frequency* of the transformer (larger inductance, larger self-capacitance). You want to keep the resonant frequency above the highest frequency of operation by at least 5x (my rule of thumb).
* Wire diameter and type (single, bunched, Litz) is important and affects the *winding (copper) loss*. For a transformer operated at a single frequency, there is an optimum wire diameter which balances AC copper losses (eddy losses: *proximity effect* which is prominent at frequencies below apx 1 MHz, and *skin effect*) and DC copper losses. Proximity effect can be reduced by using bunched or Litz wire, skin effect can be reduce by using Litz wire - both are expensive options. Copper loss is an important consideration in power transformers.
* *Core losses* are affected by flux density. Less turns have a higher flux density which mean higher core loss. Core loss is an important consideration in power transformers
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.