Starting Current of Motor Through VFD

Of all types of ways to start an AC induction motor, ONLY a VFD can possibly do so without exceeding the motor full load amps, as long as you have all the time in the world to accelerate it, and your load does not require 100% of the motor torque as load torque.

To accelerate a load from a stop, you need enough torque to move the mass (inertia) of the connected load + the motor inertia, then to overcome the friction and any other minor influencs such as windage, plus just a little more. When someone SIZES a motor, those same issues go into the full load torque requirements as well, except the extra. That "little more" is referred to as "Average Accelerating Torque" (AAT), which factors into the RATE of acceleration. The rate of acceleration is determined by the maximum acceleration time you can live with. That then is a factor of what you require at the LOAD side, as well as what the MOTOR can withstand thermally. The motor thermal limitation is a factor of the current applied to it and the thermal time constant inherent it it's design, meaning the rate at which it can cool itself. When running at full voltage and full speed, the current is equal to the torque output (and efficiency), so it is primarily load dependent. But when started with ANY method other than a VFD, the current is very high compared to the work being performed, so there is excessive heating taking place, which is limiting the time you can take to accelerate it. So tarting is always a game played between the load acceleration time and the thermal damage curve of the motor.

But when you have a VFD, the drive is capable of making the motor create its rated Full Load Torque at ANY speed, even theoretically zero speed (requires a special version). So that means that since the motor can deliver its rated torque continuously during acceleration WITHOUT exceeding the Full Load Amp rating of the motor, which it can technically do so forever. So then since you do NOT need to worry about the motor thermal constant during acceleration, you are left with ONLY the requirements that the LOAD may have for acceleration rate. If there are none, you can take a minute, an hour, a day, a week, a month or a year to fully accelerate that load, it's all the same to the drive and motor.

Caveats: 1) the motor must be designed such that it's thermal constant does not rely upon a shaft connected cooling fan, because when the motor is turning slow, so is the fan, and generally the fans are designed to cool the motor when it is at full speed, so the convection cooling is reduced at slow speeds. So you must use at least a TENV motor design, or better, a motor with a blower that is powered separately from the motor shaft so that it delivers full cooling all the time. 2) simplistic V/Hz drives are not going to be able to accurately control torque below about 25% speed, so acceleration at the low end like that can often require upward of 150% current. Vector drives mitigate that.

Bottom line, there is no formula that you can use, it is totally dependent upon external issues affecting your maximum acceleration time. So once you determine THAT, you can use standard load acceleration formulae to calculate the AAT you will require, then from that you can determine the effect that torque will have on the motor thermal model, which will become the limiting factor on the acceleration time. But if the motor thermal limit is below the external load acceleration time limits, then you need not be concerned.

All true, accurate and explains exactly why a VFD works so well on an induction motor. However, this is not exactly what the OP asked for. The OP asked for a formula for calculating the starting current using a VFD on an induction motor. To do this I prefer using the Steinmetz model of an induction motor. A careful analysis of this motor will reveal that this model is valid for any frequency with but a few simple caveats. Some of the loss parameters will change in a second order relationship if eddy current loops in the stator laminations become to severe and the slip between stator and rotor does not induce a stalling effect. For those willing to plow through a more nuanced model that includes these effects they should look here.

Here is a nice paper using the Steinmetz model that includes the mechanical loads of a compressor.

You will find many equations in these papers that should allow someone with the necessary mathematical skills to calculate any current from start to locked rotor condition.