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Hello, I'm new here and wish to know what is the relationship of speed, frequency and voltage of an induction motor. I plan to program a microcontroller for controlling the speed of three-phase induction motor. Anyway, I face a problem here, if I vary the amplitude, what's going on with the frequency?? I need some formula of calculation in order to help me to decide what freqeuncy is suitable for certain amplitude.

Hello Syee
Welcom to the group.
An induction motor is a pseudosychronous device and essentially operates at the frequency of the supply connected to it. You vary the speed by varying the frequency applied to it.
To operate, the motor requires flux in the iron, or more particularly in the gap between the rotor and stator. The rotor would typically operate at a speed slightly lower than the rotational speed of the stator field (dependant on the frequency). This difference in speed is known as the slip, and is load dependant. The slip causes the flux to cut through the rotor windings, inducing a current to flow in the rotor. The frequency of the current flow is equal to the slip frequency and can never be zero.
The induction motor is designed as a compromise between iron loss and copper loss. If you reduce the turns on the stator, the copper loss would reduce due to reduced resistance, but the flux in the iron would increase causing an increase in the iron loss. If the flux in the iron is too high, the losses increase dramtaically and the iron is said to be saturated. The object is to operate close to saturation to minmise the copper loss but not in saturation to minimise the iron loss. The rated voltage reflects the optimum flux at the rated frequency. As you reduce the frequency applied to the motor, you need to reduce the voltage in order to retain the same flux density. If you do not, the flux density will increase and the iron will saturate.
The inverter would normally exhibit a constant v/f ratio from the rated frequency down to a low frequency. In simplistic terms, you can draw a straigh line for zero volts at zero Hz up to rated volts at rated frequency.
At low frequencies, with a constant v/f ratio, the flux will reduce due to the ratio between the reactive component and the resistive component of the stator, so it is normal to provide some voltage boost at very low frequencies to increase the low frequency torque. This is only necessary for sticky loads.

Essentially, if you lok at the equivalent circuit of an induction motor, you want to keep the current flowing through the magnetising coil constant at all frequencies. This will provide constant flux.

Does that answer your question??
Best regards,
Mark.