Still I don't understand how the sense antenna works and how the combined effects of loop and sense antenna works. Can you please elaborate on that?

Sorry, not clear to me

@mins, sorry not clear to me.

@mins "Thus if we subtract the two currents, the instantaneous change (the sin2πf factor) is cancelled from the result, and the result is sensitive to the signal bearing. The resulting pattern has a heart shape (cardioid)" could you pls explain this part in greater detail?

@mins "Now the polarity of the signals: In the loop the sign is positive when the bearing is in [270..(0)..90] and negative in [90..(180)..270]." I just don't understand the polarity part. Its a sine wave right? How can you have polarity of a sine wave? Pls explain. Sorry for asking again and again. Thanks.

@Toughquestions: Yes the sign of the instantaneous amplitude changes every half cycle, but also note $-\sin x$ is opposite to $\sin x$ regardless of $x$, I call that "opposite polarities" (accurately: the phase difference is 180°). When the loop plane is aligned with the transmitter bearing, the signal reaches one side of the wire turn first and then progresses towards the other side. Which side is first reached determines the signal phase (polarity): $\sin x$ or $-\sin x$. This is the key point. On the contrary the sense antenna signal is not dependent on the bearing.

@mins let's take the 90 degree and 270 degree examples. For 90 degree, no current in the loop, only current flows is in the sense and that's a sine wave. The same is true for 270 degree. How can we differentiate if the NDB is forward or behind? Thanks.

@Toughquestions: For 270/90° bearings, the transmitter is left or right of the loop plane, no current is induced in the loop, this is a null used to determine a precise (but ambiguous) bearing without the sense antenna, which is indeed useless. Then, to remove the ambiguity, the loop is turned by 90° to have the maximums. The sense antenna is connected to remove the bearing ambiguity. The 2 steps could be done on the maximums, but as explained, the maximums are less precise. Modern ADF antennas are not loops, and don't rotate, the principle is slightly different (phase-comparison).

@Toughquestions: If you want to continue asking for complements, you should delete all your previous comments which are now unnecessary (and I'll delete the corresponding answers).

@mins pls explain this part in greater details. "On one side Asin2πft+Asin2πft=2Asin2πft On the other side −Asin2πft+Asin2πft=0 The 180° ambiguity has been removed". I'll delete everything once I understand. Sorry and thanks again.

Does one side mean one side of the loop or 0 degree with the transmitter? If one side means one side of the loop, then shouldn't we add 3 signals? I mean 2 signals for the 2 sides of the loop and 1 signal for the sense?

@Toughquestions: Sorry for the previous message being sort of twice the same content (forgot to remove my initial version) and latex formula between $ not rendered.

@Toughquestions: As for 3 or 2 signals added --> there are only 2 currents, the current in the loop and the current in the sense antenna. Do get confused because the current in the loop can have a value when the transmitter bearing is 0° and another value when it's bearing is 180°. Bearing means the relative direction of the transmitter relative to the loop plane....

@Toughquestions: ... The bearing changes when we rotate the loop antenna (or we change the aircraft heading) or when the transmitter location on the ground changes. This is a relative direction, the definition of a bearing.