First I will say there are two different versions of the induction motor equivalent circuit.
1 - The version we are all familiar with is the "steady state" equivalent circuit.
2 - There is also a "transient" equivalent circuit, which is much better for predicting transients (although it still typically employs assumptions such as neglecting saturation and neglecting slotting effects). Attached is the induction motor transient equivalent circuit from Paul Krause's "Analysis of Electric Machinery".
Use of the attached transient equivalent circuit requires choice of a radian speed "w" of the reference frame. The typical choice is w = we (the frequency of the electrical supply), in which case our power supply voltage which is sinusoidal in the stationary reference frame becomes dc in the simulation reference frame.
The model on the slide has everything needed to do a state space simulation. We can choose either the four currents or the four flux linkages as state variables (if we know one, we know the other per 4 equations in 4 unknowns at the bottom).
The magnetic effects are split into two types of circuit elements – the inductances and the dependent voltage sources.. In a synchronous reference frame under steady state conditions, the voltage across the inductances is 0. So the entire steady state behavior is predicted by those dependent voltage sources, with the inductances in the circuit playing no role in steady state (other than as coefficients determine flux linkage as input to dependent voltage sources).
So we know for a fact the dependent sources control the steady stage behavior with inductance elements playing no role. I
believe that for the first cycle of a DOL starting transient, the reverse is true... the inductance elements control the transient and the dependent voltage sources have little role. Perhaps the fact that the voltage sources are initially zero helps us believe this assumption (It is subject for validation and comment whether that is true and whether it can be intuitively deduced from the equivalent circuit).
If we accept the questionable premise that the dependent voltage sources play no role in the first cycle of DOL start, then we can cross the dependent voltage sources out of the circuit. What is left are two d-q circuits with only the elements from the equivalent circuit including R2 which would be the value of the variable R2/s resistance when s=1. If we reverse the dq transformation back into a/b/c variables and assume balanced (*) conditions to support per-phase equivalent circuit, we recover the normal ("steady state") equivalent circuit. (* if you are uneasy about applying the concept of "balanced" view to transient analysis, view it in the Laplace domain with your inductance elements as s*L rather than j*w*L, and your three voltage sources differeing by factor exp[-s*2*pi/3] and you can see it applies within the assumptions of linear system.).
So how valid is that approach especially given the questionable assumption? It leads to analysis of the DOL starting transient using the steady state equivalent circuit which leads to the equation that you gave. It is my perception that it is pretty close. I know that NEMA MG-1 states the max theoretical transient inrush current is 2*sqrt(2) * I_lrc_rms, which is consistent with this approach.
What I would like to do is to do a few simulations to compare the two approaches. That is the only way to answer the question that I know of.
Do I have the right equation above for modeling the transient inrush current with time over the first couple of cycles.
I am going to investigate how closely that matches the more exact transient approach.
If not do you know where I can find the correct equation.
The more exact transient analysis approach requires numerical simulation as described in Krause or other references. I can provide spreadsheet or perhaps some on-line free references if you are interested in pursuing this approach.
I want to use this equation to see how the current magnitude changes with resistance values in the motors and/or cables. What would you recommend as the best approach to this?
You could certainly lump external resistances into R1 in the transient numerical simulation and (if it's valid to begin with) in the steady state equivalent circuit approach.
Note that if you had larger cables with significant inductance and wanted to model that external inductance, the same simple approach would NOT work.. you couldn't lump them in with L1 because L1 is involved in the transformations between flux and current.
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(2B)+(2B)' ?