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
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.