Park/Clarke Transform and Power

I'm looking to determine the electrical power input to a 3-phase BLDC motor, driven by TI's DRV8305 motor controller. I can't use the DC power flowing into the system because I'm trying to capture the efficiency loss of the uC, gate driver, and switching MOSFETS. There are sensors on the phase voltage and phase current. For the controller, it uses a power VARIANT Clarke transform, as shown here, for BOTH the voltage and current:

Then it uses a standard Park transform to get to the DQ0 reference frame:

From my understanding, Theta is chosen such that my current and voltage will align with my D axis only, and presto, I have a DC Current and DC Voltage. However, because my Clarke Transform was power variant, I must scale my P=I*V by some factor in order to get the correct answer?

From my understanding, a solution would look like:

$$P_{ABC} = I_{DQO-INVAR}(t)*V_{DQO-INVAR}(t)$$ $$P_{ABC} = P_{\theta}I_{\alpha\beta\gamma-INVAR}(t)*P_{\theta}V_{\alpha\beta\gamma-INVAR}(t)$$ $$P_{ABC} = P_{\theta}T_{INVAR}I_{ABC}(t)*P_{\theta}T_{INVAR}V_{ABC}(t)$$ $$P_{ABC} = P_{\theta}*sqrt(3/2)*T_{VAR}I_{ABC}(t)*P_{\theta}*sqrt(3/2)*T_{VAR}V_{ABC}(t)$$ $$P_{ABC} = 3/2*P_{\theta}T_{VAR}I_{ABC}(t)*P_{\theta}*T_{VAR}V_{ABC}(t)$$

Where $P_{\theta}$ is the Park Matrix, $T_{INVAR}$ is the power invariant Clarke Matrix, and $T_{VAR}$ is the power variant Clarke Matrix.

This would lead me to believe that I need to scale my power variant DQ0 output by 3/2 in order to get the actual power draw? Please confirm or deny.

I've looked for a solution, but it does not seem to be a common question. I'm forced to use the power variant Clarke transformation because TI's Instaspin Library is built on it. I'm rather unfamiliar with three-phase power. If there's any other information relevant to the problem, I'll be able to provide it. A good reference would be much appreciated.

• I don't understand your last equations, about INVAR, TVAR,...What matters is the way the voltage space vector is formed, the inverse Clarke and type of output switching (SVPWM,..), then $P=I_qV_q$. The power is product of q-axis components, like in PMDC motor where $P=I_aV_a$ – Marko Buršič Jul 7 '16 at 6:24

Always, the product of $I_{DC}$ and $V_{DC}$ gives you the power. The FOC needs to measure stator currents, not the DC current, which isn't clear from your description if you really measure DC current, voltage. First, from stator currents ia,ib,ic (or ia,ib for symetric load as AC motor is) you transform into $\alpha\beta$ coordinate system and then into dq coordinate system. The $i_q$ is proportional to the output torque, hence the elecrical power can be computed with the formula $P=M\Omega = k_ii_q\Omega$, where $\Omega$ is the rotor speed $[\dfrac{rad}{s}]$
In FOC the voltage is set with PI controller, so that inverter outputs the setpoint $i_q,i_d$ and there is no voltage feedback (unless you have BEMF sensing for sensoroless control). Since you mention about FOC, then you have DQ currents and speed from encoder and this is enough to calculate the electromagnetic output power.