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
Any given core material, cross section and frequency has a maximum volts per turn. If you want to use the winding at a certain voltage, then you need enough turns to support that voltage.

For instance, low frequency mains transformer steel will only run up to a peak field of 1.7 T or so before it saturates. If you had a core that was 10 mm x 20 mm, and wanted to run it at 50 Hz, then the fastest the field swings, for a sine wave voltage, is 2pi.B.f = 6.28x1.7x50 = 534 T/s. In the 200 mm2 of the core, that's a peak flux change rate of 0.1 Weber/s, meaning that core at that frequency will only support 0.1 V/turn.

A 240 Vrms mains winding on that core would need to support 340 Vpeak, so would need 3400 turns minimum.