# What is sine wave torque constant and voltage constant for electric motors

I am wondering if anyone can explain to me what the sine wave torque constants and the voltage constants are and how to calculate them for a specific electric motor? Thanks!

The torque constant and the voltage constant are characteristics that apply to permanent magnet DC (PMDC) motors (with brushes and commutator) and brushless motors. When you talk about a "sine wave torque constant" that only applies to brushless motors (I'll come back to this later).

Starting with the PMDC motor, we can model it using 3 equations:

$T=k_T*I$

$E=k_E*\omega_m$

$V_S=E+R_a*I+V_b$

$k_T$ and $k_E$ are the torque constant and voltage constant (sometimes called the back-emf constant), respectively. Assuming SI units, they have units of $Nm/A$ and $V/(rad/sec)$, respectively. $I$ is current DC current, $T$ is electromagnetic torque in $Nm$, $E$ is the back-emf, $\omega_m$ is the angular velocity in $rads/sec$, $V_S$ is the supply voltage, $R_a$ is the armature resistance and $V_b$ is the voltage drop of the brushes.

If you hook the PMDC motor up to another motor and back-drive your motor at a known speed ($\omega_m$) and measure the voltage produced across the terminals ($E$), then you can determine $k_E$ from the second equation. You should get the same value of $k_E$ at various speeds.

If you assume an ideal PMDC motor, $k_T = k_E$, assuming SI units. So if you find $k_E$ using the above method, you will get a pretty good estimate of $k_T$ as well. However, you can measure $k_T$ by mounting the motor to a dynamometer, applying various torques to the motor from 0 to stall torque, and measuring current draw for those different torques. You can then plot torque vs. current and find the best-fit slope of that line. That slope is your $k_T$.

For 3 phase brushless motors, these 2 constants are similar in principle, but the specifics differ. Because 3 phase brushless motors can have different shapes for their back-emf waveform and because they can be driven with different shaped currents (sinusoidal or trapezoidal) and a number of other issues, the relationship between $k_T$ and $k_E$ is complicated and changes depending on the specifics of the motor/drive combination. The key to keeping it all straight is to remember that $k_T$ and $k_E$ in a PMDC motor assumed that the back-emf ($E$) was a mean, rectified DC value and $I$ was the DC input current. If you keep that in mind, then the equations still hold for brushless motors per phase. So, for example, one issue that comes up is that if you want to measure $k_E$, you usually can only measure the line-to-line voltage, not the phase voltage. So brushless motor manufacturers often give you the line-to-line values for motor constants, not the phase values. Sometimes motor manufacturers will give you the peak values for these constants, other times they will give you the RMS values for these constants. They won't always tell you which one they give you.

Sometimes you will see motor manufacturers distinguish between sine wave torque constant and square wave torque constants. If you are using a trapezoidal drive (6 step) drive, then you should use the square wave torque constant. If you are using a sinusoidal drive, you should use the sine wave torque constant. The ratio between these 2 is (ideally) $\frac{2*\sqrt3}{\pi} = 1.103$.

You can measure $k_E$ and $k_T$ for a brushless motor in basically the same way as with a PMDC motor. However, with a brushless motor, keep in mind you are probably measuring line-to-line voltages and make sure you keep track of whether you are measuring peak or RMS values of voltage and current.