We present a homogeneous set of accurate atmospheric parameters for a complete sample of very and extremely metal-poor stars in the dwarf spheroidal galaxies ( dSphs ) Sculptor , Ursa Minor , Sextans , Fornax , Boötes I , Ursa Major II , and Leo IV .
We also deliver a Milky Way ( MW ) comparison sample of giant stars covering the -4 < [ Fe/H ] < -1.7 metallicity range .

We show that , in the [ Fe/H ] \succsim - 3.7 regime , the non-local thermodynamic equilibrium ( NLTE ) calculations with non-spectroscopic effective temperature ( T _ { eff } ) and surface gravity ( log g ) based on the photometric methods and known distance provide consistent abundances of the Fe i and Fe ii lines .
This justifies the Fe i /Fe ii ionisation equilibrium method to determine log g for the MW halo giants with unknown distance .

The atmospheric parameters of the dSphs and MW stars were checked with independent methods .
In the [ Fe/H ] > -3.5 regime , the Ti i /Ti ii ionisation equilibrium is fulfilled in the NLTE calculations .
In the log g - T _ { eff } plane , all the stars sit on the giant branch of the evolutionary tracks corresponding to [ Fe/H ] = -2 to -4 , in line with their metallicities .
For some of the most metal-poor stars of our sample , we hardly achieve consistent NLTE abundances from the two ionisation stages for both iron and titanium .
We suggest that this is a consequence of the uncertainty in the T _ { eff } -colour relation at those metallicities .
The results of these work provide the base for a detailed abundance analysis presented in a companion paper .