Why does stator current increase with increasing load in a synchronous motor?

Isn't the current, coming from the 3-phase source of the stator, constant? We don't change the source, so how does the current change? Is a voltage induced because of the magnetic field of the rotor?

This leads me to another question which might help: Does the rotor's magnetic field of an induction motor affect the stator's current value?

Since the motor converts electrical energy to mechanical energy, the electrical input power must be equal to the mechanical power transmitted to the load plus power lost in the motor. Since the input voltage is constant, that would have to be reflected in the input current and power factor since power = voltage X current x power factor.

The mechanism by which electrical power is converted to mechanical power is explained using the equivalent circuit of the motor. Just as in a DC motor, a back EMF is generated in an AC synchronous motor. So the motor equivalent circuit is an AC back EMF generator in series with the internal impedance of the motor. The back EMF opposes the source voltage so that the stator current is proportional to the source voltage minus the back EMF divided by the internal impedance of the machine. With an AC machine, the voltage, current and impedance values are all complex numbers. The phase angle difference between the terminal voltage and the back EMF is determined by torque angle, the angle between the rotating stator and rotor magnetic fields. As the name implies, the torque angle is proportional to torque.

This leads me to another question which might help: Does the rotor's magnetic field of an induction motor affect the stator's current value?

I don't think that is related to the synchronous motor question. The induction motor rotor current is ultimately supplied by the stator, but the magnetic fields in both the stator and rotor are pretty much constant as long as the applied voltage and frequency are constant and the ratio of voltage to frequency is constant.

In an electric motor, almost all types, the current is proportional to the torque.

If the load increases, in other words, the torque increases and the current will increase.

The magnetic field in a motor are related to the voltages.

The rotor produces a field that turns at the same speed as the supply. When there is no phase difference between the fields, the current is almost zero.

The phase difference between the stator and the rotor is called the torque angle. This is the load on the motor.

Changes in the magnitudes of the magnetic fields, caused by changes in the voltage magnitudes, have little effect on the torque angle. These changes mostly alter the apparent power and are therefore related to the power factor.

(This is a simple explanation and I am making it a community answer so that terminology can be improved by those that understand more than I do. :-)