An automotive turbocharger functions only as a compressor where the driving force is the expansion of waste exhaust gasses. All four adiabatic and isobaric steps of the process are not present. Fuel is not burned in a turbocharger.
The thermodynamic functions of a turbocharger does not match the functions of a gas turbine, which is an example of a Brayton Cycle device.
The Wikipedia discussion has a very good discussion on the four steps of the ideal Brayton cycle
wat if we consider the engine also.. fuel burns in cylinder, exhaust goes to turbine side of turbo, which drives compressor, which pumps in air back to cylinder.