Because the inductive component of the voltage across a superconducting coil is 10(2) - 10(4) times larger than its loss component, in the most of electric methods for AC loss measurements compensating voltage proportional to partial derivative I/partial derivative t is needed, I being the current in the coil. This paper presents a description of an air-core transformer whose mutual inductance M can be smoothly changed in a very large interval and the secondary voltage of which has a negligible phase shift with regard to partial derivative I/partial derivative t in the primary as well as strongly suppressed noise voltage. It also describes a very sensitive method of measuring AC losses in superconducting coils based on the use of such mutual as well as of a selective nanovoltmeter. This method eliminates the need off an integrator with small drift and phase shift error, of a high resolution voltage divider with sliding contact and of a shunt resistor with small phase shift which are required by other analog electric methods. Experimental compensating coil and their parameters as well as AC Toss measurements in different superconducting coils are also described.