# VFD Power Savings physical explanation - cubic vs square relationship

I am trying to either explain or derive where the power savings come from with a variable frequency drive (VFD). I am well aware of all the literature and generally accepted assumption that energy savings resulting from a VFD are proportional to the cube of the ratio of the reduced speed to the original speed of the motor. This is from the affinity laws.

However, if I consider the rotational power equation where power is equal to torque multiplied by rotational velocity OR the fluid power equation where power is equal to pressure multiplied by flow rate, it would appear to me that power should be directly proportional to speed.

Or, if I consider that VFDs maintain a constant volts to hertz ratio and assume that motor current is proportional to load and then use the electrical power equation where power equals current multiplied by voltage multiplied by power factor - I would assume that power is proportional to the square of the ratio of the reduced speed to the original speed.

Can anyone clarify things? Where is the discrepancy or what am I missing here? And are the affinity laws just stated as fact? Are they fundamental principles or do they have a derivation relating to fluid flow.

I will update this question with equations to make it easier to read.

Any thoughts or explanations are greatly appreciated!

Thanks!

• This is way outside my area. But when you run a pump faster, the back pressure and flow rate both go up, so that is not linear. Maybe square law? Concerning the rotational power equation, it may be a mistake to assume that torque remains constant. That is up to the mechanical load. With an agricultural irrigation pump, the torque required to spin the pump will be higher at higher rotational speeds, not constant. So as you speed up your VFD, the current will rise along with voltage. So all of your assumptions seem to be invalid. – mkeith Jun 1 '16 at 4:48