Before revision: The quarter wave transformer is an alternative to a stub; but, whereas a stub is terminated in a short (or open) circuit and the length is chosen so as to produce the required impedance transformation, the λ/4 transformer is in series with the load and its length and characteristic impedance are designed to produce the required impedance transformation.
After revision: As with the physical transmission line, the relationship between the characteristic impedance, Z0, input impedance, Zin and load impedance, ZL is:The quarter wave transformer is an alternative to a stub; but, whereas a stub is terminated in a short (or open) circuit and the length is chosen so as to produce the required impedance transformation, the λ/4 transformer is in series with the load and its length and characteristic impedance are designed to produce the required impedance transformation.