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In simple terms, a synchronous motor operates on the fundamental idea that the rotor should rotate at the same speed as the rotating magnetic field produced by the stator. For better understanding, imagine a track with synchronized swimmers who match their movements with the music beat. Here, the stator's rotating magnetic field is like the rhythm of the music, and the rotor in the synchronous motor 'dances' to this beat, maintaining the same pace irrespective of changes in its 'dance moves' or load.

A key takeaway is that synchronous motors need a separate DC power source for the rotor, quite distinct from the three-phase AC power supplied to the stator. This DC source magnetizes the rotor, enabling it to lock in step with the rotating magnetic field. It's analogous to giving the swimmers magnetic shoes that can only move in time with the magnetic pulses matching the rhythm of the song.

The rotating magnetic field is the motor's heartbeat. Created by alternating currents in the three-phase stator windings, this field whirls around the stator at a steady pace – the synchronous speed. Take a carousel for instance - as it turns, the horses go round and round at a set speed; similarly, the rotating magnetic field spins inside the motor. Through a clever design of the winding layout, the phase difference in the electricity supply generates a magnetic field that rotates in one direction, harnessing the beauty of physics to get the job done without any physical movement.

Magnetic locking, also known as 'magnetic coupling', is what makes a synchronous motor 'synchronous'. When you place two magnets near each other, they will snap into place, lining up exactly; this is the essence of magnetic locking. The rotor's DC-energized electromagnet becomes a synchronized dance partner to the rotating magnetic field. No matter the load on the motor, this locking keeps the rotor spinning in perfect harmony with the stator field.

Think of the rotor as a compass needle, always finding and aligning with the stator's magnetic 'north'. Under varying load, the rotor may wobble in its alignment, but never loses the rhythm, maintaining a constant dance, or in technical terms, a constant speed.

The torque angle is the measure of how much the rotor 'lags' or 'leads' behind the stator's magnetic field. Let's compare this to a game of tug of war; when the game starts, the rope is straight. As soon as one team pulls, the rope tilts – that tilt represents the torque angle. In our synchronous motor, when load increases, our metaphorical 'team' on the rotor side pulls harder, increasing this angle without letting go of the rope. The angle can change, but the synchronous speed remains unchanged, like the game does not speed up or slow down because one team pulls harder.

Synchronous speed is the speed at which the rotor must turn to remain in sync with the stator's magnetic field. It's a calculated pace, determined by the frequency of the AC supply and the number of poles in the motor, like the exact tempo needed for an orchestra to stay on beat with the conductor's baton. Even when more instruments (load) join in, necessitating more effort (torque) from the musicians, the beat (synchronous speed) does not change.

Dating back to its fundamentals, the synchronous speed (\(n_s\)) is given by the equation \(n_s = \frac{120f}{P}\), where \(f\) is the supply frequency, and \(P\) is the number of poles. This makes it clear that the synchronous motor's speed is locked to the electrical power grid's beat and is unfazed by the crescendos or diminuendos of the workload.