Once in a while I get asked why a motor can’t be designed that has a back emf constant, Ke, that is independent of the the torque constant, Kt. If only I could. I would solve the worlds energy needs. Of course, it cannot be done. Not only can it not be done but Ke and Kt are always proportional.
The reason they are proportional is because the same flux and turns that produces back emf voltage also produce torque. Showing in detail why this is so would be too complicated for a short article. Instead an example will serve as proof, and also show you how you can calculate the coversion factor.
Let us consider a motor that happens to have an efficiency of 100%, just to simplify our discussion. With this motor:
Pin = Pout
Arbitrarily we can assume a power of 1 watt, and equally arbitrarily we can assume the supply is a one volt one ampere supply. So Pin is 1 V x 1 A = 1 W. Then:
Pin = 1 W = Pout = Speed * Torque = 1V/Ke * 1A*Kt * C
Where C is a conversion factor that depends on the units of Kt and Ke. For example if Kt has units of N-m/A and Ke has units of V/s (radians is unitless) then the torque will have units of N-m and the speed will have units of s^-1, and C is 1. Since 1 N-m/s = 1 W, a little algebra shows the with these units the value of Kt must equal Ke.
The metric system is so easy….
If Kt has units of ozf-in/A and Ke has units of V/kRPM then the torque will have units of ozf-in and the speed will have units of kRPM, and C is:
2*pi radians/rev*1000/k*60s/minute / 141.612 ozf-in/N-m
The first part converts kRPM to s^-1 and equals 104.72. The second part converts ozf-in to N-m.
Keeping in mind that 1 N-m/s = 1 W, C is then 1/1.352,or the value of Kt, when Kt has units ozf-in/A, is 1.352 times the value of Ke, when Ke has units of V/kRPM. Any motor data you see where this is not true is due to errors in actual measurements.
And the laws of physics are safe for another day.