Equivalent DC circuit of 3 phase induction motor

I’m currently trying to model a simplified circuit of an inverter + 3 phase induction motor. For now, I want to represent the DC link as an ideal independent voltage source. To keep this example simple, I also want to keep frequency and input voltage the same, though I expect to be changing these to simulate a VFD

I’m using https://myelectrical.com/notes/entryid/251/induction-motor-equivalent-circuit as reference

The output I want is motor torque and resulting current on the DC link.

So, given $V_{DC}$, what would $I_{DC}$ be in terms of inverter equations and induction motor equations?

I think I’m most of the way there, but I can’t find a full set of equations for this, only separate equations for motors and inverters.

^{simulate this circuit – Schematic created using CircuitLab}

For input voltage to the motor, I think this is correct
$$V_{L} = frac{3}{sqrt{2}pi}V_{DC}\
V_{P} = frac{V_{L}}{sqrt{3}}$$
It’s my understanding that torque is per phase
$$T = frac{1}{2pi n_s}frac{R_{2}}{s}frac{E_{2}^2}{frac{R_{2}}{s}^2+X_{2}^2}$$
So total motor torque
$$T_{3phi} = 3T = frac{3}{2pi n_s}frac{R_{2}}{s}frac{E_{2}^2}{frac{R_{2}}{s}^2+X_{2}^2}$$
To calculate back emf $E_2$:
$$E_{2} = V_{P}-frac{I_{1}}{Z_{1}}$$
Impedance per phase:
$$Z_1=R_{1} + jX_{1}\
Z_m=jX_{m}\
Z_2=frac{R_{2}}{s} + jX_{2}\
Z=Z_{1}+frac{Z_{m}Z_{2}}{Z_{m}+Z_{2}}$$
Would the calculation back to $I_{DC}$ be as follows?
$$I_{1} = frac{V_{P}}{Z}$$
$$I_{L} = frac{I_{1}}{sqrt{3}}$$
$$I_{DC} = frac{3}{sqrt{2}pi}I_{L}$$

EDIT:
Specifically, I want to know if the equations I have above are correct for determining the average current on the DC link (neglecting most losses for now)