Simulating motor w/ non-simult transition of breaker contacts
Simulating motor w/ non-simult transition of breaker contacts
(OP)
2 questions:
Q1 – Does anyone have suggestions or references for simulation of motor current response to non-simultaneous opening or closing of motor supply breaker contacts.
Q2 – Does anyone have example of solved (simulated) problem so I can validate my simulation when I do it?
Discussion: Krause's "Analysis of Electric Machinery" 2nd Ed doesn't discuss any simulations involving non-simultaneous contact opening or closing (seems like a big gap to me).
His model seems to assume that the voltage at the neutral of the motor wye point is the same as the voltage at the neutral of the power supply as was briefly discussed here
thread237-251952: Motor transient simulation – basis for assumption that Vn is 0?
I have been thinking about it some more, and I believe that I have a little better understanding (my thoughts, not confirmed by any references...couldn't find any that discussed this aspect much).
First I will define some notation. Each voltage is defined by as a difference between 2 points, the 1st point is the 1st post script, and the 2nd point is the 2nd postscript. m indicates quantity measured at the motor, n indicates quantity measured at the power supply. So the variables w'll need are:
Vam_mn = Voltage at motor terminal A, referenced to motor neutral
Vam_pn = Voltage at motor terminal A, referenced to Powersupply neutral
Vap_pn = Voltage at power supply terminal A, referenced to Powersupply neutral. This is the same as Vam_pn unless we have inserted an impedance between the source voltage and the motor (which is what I intend to do).
If we look at a motor in the stationary ref frame where the motor-parameters are the same for each phase, then my posulate is that its voltages must satisfy:
Vam_mn + Vbm_mn + Vcm_mn = 0. (my posulate)
The reason is that ia + ib + ic = 0 and in absence of homopolar flux path: Phi_A + Phi_B + Phi_C = Phi_total = 0 and therefore d/dt (Phi_total) = 0 where Phi_A is total flux linked by phase A stator winding. The voltages Va_mn etc are made up of sums of resistive terms proportional to currents and inductive terms proportional to rate of change of fluxes. I have a tough time working through all the terms to prove this rigorously in terms of the full model, but it has a lot of intuitive appeal to me (*).
If I'm right about that, it explains why Krause's model works if we meet the conditions:
1 - the power source satisfies Vap_pn + Vbp_pn + Vcp_pn = 0
AND
2 – the source is directly connected to the motor.
If we meet those conditions, there is no voltage drop between motor and source based on 2 and we'll have
Vap_pn = Vam_pn, Vbp_pn = Vbp_pn, Vcp_pn = Vcm_pn
Also we know that Vam_pn = Vam_mn + Vmn_pn, and similar for phases B and C
Substituting into 1 Vam_pn = Vam_mn + Vmn_pn, and similar for the other 2 phases we have
(Vam_mn + Vmn_pn)+ (Vbm_mn + Vmn_pn) +(Vcm_mn + Vmn_pn) = 0
Vam_mn + Vbm_mn + Vcn_mn +3*Vmn_pn = 0
My postulate above is that Vam_mn + Vbm_mn + Vcn_mn = 0. If mypostulate above is correct, then we have proven that Vmn_pn = 0 as long as we meet 1 and 2. It explains why Krause's approach of assuming the motor neutral voltage is the same as the power supply neutral voltage under the limited class of problems defined by conditions 1 and 2.
Now I am wanting to put a resistor in between the Powersource and the motor to simulate non-simultaneous contact closing or opening. No longer is conditions 1 and 2 met and I believe no longer can I assume motor neutral voltage same as power supply neutral voltage.
There is a little bit of chicken-or-the-egg problem in trying to do a simulation of this scenario. Lets use currents as state variables. You could calculate motor neutral voltage if you knew the currents and their derivatives. But you can't calculate the current derivatives until you know the motor neutral voltage. I think there may be a very simple solution: If we know the currents we know: Vam_pn, Vbm_pn, Vcm_pn (these are simply the source voltage corrected for voltage drop in the inserted resistance between source and motor).
We have 4 unknowns (Vam_mn, Vbm_mn, Vcn_mn, Vmn_pn) and 4 equations (Vam_pn = Vam_mn + Vmn_pn, Vbm_pn = Vbm_mn + Vmn_pn, Vcm_pn = Vcm_mn + Vmn_pn and by postulate Vam_mn+Vbm_mn + Vcn_mn = 0). The solution is
Vam_mn = Vam_pn – Vaverage
Vbm_mn = Vbm_pn – Vaverage
Vcm_mn = Vbm_pn – Vaverage
where Vaverage = (Vam_pn + Vbm_pn + Vcm+pn)/3
(* - as we make the system in between the motor and the power supply more complex by adding resistances etc between source and motor, it would be very non-intuitive to assume that the motor would somehow "knew" what the power supply neutral voltage was and set its own neutral accordingly, since all the motor sees at it's terminals is phase-to-phase voltages. In contrast it is very intuitive to assume as per my posulate that the motor looks at the 3 phase-to-phase voltages and establishes a neutral voltage which depends only on those... the one and only neutral voltage required to maintain the sum of the phase-to-motor-neutral voltage at zero).
Sorry I am not trying to sound like a researcher. I know this problem has been solved by many people before and is probably very trivial in the scheme of things, but I haven't been successful in finding anyone else's written solution to this problem/question. If my discussion or postulate sounds incorrect or misguided please let me know.
Q1 – Does anyone have suggestions or references for simulation of motor current response to non-simultaneous opening or closing of motor supply breaker contacts.
Q2 – Does anyone have example of solved (simulated) problem so I can validate my simulation when I do it?
Discussion: Krause's "Analysis of Electric Machinery" 2nd Ed doesn't discuss any simulations involving non-simultaneous contact opening or closing (seems like a big gap to me).
His model seems to assume that the voltage at the neutral of the motor wye point is the same as the voltage at the neutral of the power supply as was briefly discussed here
thread237-251952: Motor transient simulation – basis for assumption that Vn is 0?
I have been thinking about it some more, and I believe that I have a little better understanding (my thoughts, not confirmed by any references...couldn't find any that discussed this aspect much).
First I will define some notation. Each voltage is defined by as a difference between 2 points, the 1st point is the 1st post script, and the 2nd point is the 2nd postscript. m indicates quantity measured at the motor, n indicates quantity measured at the power supply. So the variables w'll need are:
Vam_mn = Voltage at motor terminal A, referenced to motor neutral
Vam_pn = Voltage at motor terminal A, referenced to Powersupply neutral
Vap_pn = Voltage at power supply terminal A, referenced to Powersupply neutral. This is the same as Vam_pn unless we have inserted an impedance between the source voltage and the motor (which is what I intend to do).
If we look at a motor in the stationary ref frame where the motor-parameters are the same for each phase, then my posulate is that its voltages must satisfy:
Vam_mn + Vbm_mn + Vcm_mn = 0. (my posulate)
The reason is that ia + ib + ic = 0 and in absence of homopolar flux path: Phi_A + Phi_B + Phi_C = Phi_total = 0 and therefore d/dt (Phi_total) = 0 where Phi_A is total flux linked by phase A stator winding. The voltages Va_mn etc are made up of sums of resistive terms proportional to currents and inductive terms proportional to rate of change of fluxes. I have a tough time working through all the terms to prove this rigorously in terms of the full model, but it has a lot of intuitive appeal to me (*).
If I'm right about that, it explains why Krause's model works if we meet the conditions:
1 - the power source satisfies Vap_pn + Vbp_pn + Vcp_pn = 0
AND
2 – the source is directly connected to the motor.
If we meet those conditions, there is no voltage drop between motor and source based on 2 and we'll have
Vap_pn = Vam_pn, Vbp_pn = Vbp_pn, Vcp_pn = Vcm_pn
Also we know that Vam_pn = Vam_mn + Vmn_pn, and similar for phases B and C
Substituting into 1 Vam_pn = Vam_mn + Vmn_pn, and similar for the other 2 phases we have
(Vam_mn + Vmn_pn)+ (Vbm_mn + Vmn_pn) +(Vcm_mn + Vmn_pn) = 0
Vam_mn + Vbm_mn + Vcn_mn +3*Vmn_pn = 0
My postulate above is that Vam_mn + Vbm_mn + Vcn_mn = 0. If mypostulate above is correct, then we have proven that Vmn_pn = 0 as long as we meet 1 and 2. It explains why Krause's approach of assuming the motor neutral voltage is the same as the power supply neutral voltage under the limited class of problems defined by conditions 1 and 2.
Now I am wanting to put a resistor in between the Powersource and the motor to simulate non-simultaneous contact closing or opening. No longer is conditions 1 and 2 met and I believe no longer can I assume motor neutral voltage same as power supply neutral voltage.
There is a little bit of chicken-or-the-egg problem in trying to do a simulation of this scenario. Lets use currents as state variables. You could calculate motor neutral voltage if you knew the currents and their derivatives. But you can't calculate the current derivatives until you know the motor neutral voltage. I think there may be a very simple solution: If we know the currents we know: Vam_pn, Vbm_pn, Vcm_pn (these are simply the source voltage corrected for voltage drop in the inserted resistance between source and motor).
We have 4 unknowns (Vam_mn, Vbm_mn, Vcn_mn, Vmn_pn) and 4 equations (Vam_pn = Vam_mn + Vmn_pn, Vbm_pn = Vbm_mn + Vmn_pn, Vcm_pn = Vcm_mn + Vmn_pn and by postulate Vam_mn+Vbm_mn + Vcn_mn = 0). The solution is
Vam_mn = Vam_pn – Vaverage
Vbm_mn = Vbm_pn – Vaverage
Vcm_mn = Vbm_pn – Vaverage
where Vaverage = (Vam_pn + Vbm_pn + Vcm+pn)/3
(* - as we make the system in between the motor and the power supply more complex by adding resistances etc between source and motor, it would be very non-intuitive to assume that the motor would somehow "knew" what the power supply neutral voltage was and set its own neutral accordingly, since all the motor sees at it's terminals is phase-to-phase voltages. In contrast it is very intuitive to assume as per my posulate that the motor looks at the 3 phase-to-phase voltages and establishes a neutral voltage which depends only on those... the one and only neutral voltage required to maintain the sum of the phase-to-motor-neutral voltage at zero).
Sorry I am not trying to sound like a researcher. I know this problem has been solved by many people before and is probably very trivial in the scheme of things, but I haven't been successful in finding anyone else's written solution to this problem/question. If my discussion or postulate sounds incorrect or misguided please let me know.
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(2B)+(2B)' ?





RE: Simulating motor w/ non-simult transition of breaker contacts
m indicates quantity measured at the motor, n indicates quantity measured at the power supply
should have been
m indicates quantity measured at the motor, p indicates quantity measured at the power supply
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(2B)+(2B)' ?
RE: Simulating motor w/ non-simult transition of breaker contacts
But what is the point of this study? Why does the non-simultaneous opening or closing matter?
RE: Simulating motor w/ non-simult transition of breaker contacts
RE: Simulating motor w/ non-simult transition of breaker contacts
Apparently the programmers of ATP built some relationships or assumptions into their program which yield results equivalent to Vam_mn + Vbm_mn + Vcm_mn = 0. That's a good data point for me. Thanks!
The proof would look something like this:
Vam_mn = Ia*R1 + d/dt (Lambda_a)
Vbm_mn = Ib*R1 + d/dt (Lambda_b)
Vcm_mn = Ib*R1 + d/dt (Lambda_c)
Vam_mn + Vbm_mn + Vcm_mn = (Ia + Ib + Ic ) + d/dt (Lambda_a + Lambda_b + Lambda_c)
We already know Ia + Ib + Ic = 0. If we can show Lambda_a + Lambda_b + Lambda_c=0 then we have proved that the sum of the voltages is 0. We know all the fluxes crossing the airgap sums to 0 (no homopolar flux path, everything that crosses goes back across somewhere else). There is just a little more work required to show the flux linkages Lambda sum to 0... will try that when I get a chance (unless anyone else wants to). In the meantime, based on my own thoughts and confirmation from your ATP results I am pretty confident it is the correct result.
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(2B)+(2B)' ?
RE: Simulating motor w/ non-simult transition of breaker contacts
There are other more obscure scenario's I'm looking at.... consider it curiosity.
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(2B)+(2B)' ?
RE: Simulating motor w/ non-simult transition of breaker contacts
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(2B)+(2B)' ?
RE: Simulating motor w/ non-simult transition of breaker contacts
RE: Simulating motor w/ non-simult transition of breaker contacts
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(2B)+(2B)' ?
RE: Simulating motor w/ non-simult transition of breaker contacts
RE: Simulating motor w/ non-simult transition of breaker contacts
Here is the missing part of the 15 Nov 10 8:10 proof. Specifically we prove here that the stator flux linkages sum to 0:
Break stator flux linkage Lambda in to 2 terms:
LambdaLeakA,B,C = associated with leakage reactance and leakage flux (flux that does not cross the airgap)
LambdaMagA,B,C = associated with magnetizing reactance and magnetizing flux (flux that does cross the airgap)
The stator leakage flux linkage for A phase is LambdaLeakA = Ia*L1 where L1 is leakage reactance. The sum of stator leakage flux linkage for three phases is Ia*L1 + Ib*L1 + Ic*L1. It is easy to see that this portion of the flux linkage sums to zero since the three phase currents sum to 0 in wye.
Now look at the A phase magnetizing flux linkage LambdaMA. Typically it is associated with magnetizing reactance. But instead of using the circuit model which gets messy, we will focus on the physical picture which captures the same considerations:
ASSUMPTIONS of Krause's model
At any point in time, the magnetizing flux is assumed sinusoidally distributed in space:
Phi(theta) = Phimax*cos(theta + thetaPhi).
The windings are assumed sinusoidally distributed. For phase A, the function describing turns linking a given position theta is given by
Na(theta) = Namax * cos(theta + thetaNa).
Nb and Nc are similar but shifted by 120 degrees.
The flux linkage for phase A would be a
LambdaA = Integral {Phi(theta) * N(theta)} dtheta from theta = 0 to 2*pi
The above integral of 2 sinusoids from 0 to 2*pi is maximum when they are in phase and is 0 when when the sinusoids are 90 apart... it is the same formulation we use when considering power factor angle between voltage and current... which is also it is conveniently described in vector / dot-product notation:
LambdaMA ~ [Phimax Angle ThetaPhi] dot [Nmax Angle ThetaNa]
LambdaMB ~ [Phimax Angle ThetaPhi] dot [Nmax Angle ThetaNb]
LambdaMC ~ [Phimax Angle ThetaPhi] dot [Nmax Angle ThetaNC]
LambdaMA + LambaMB + LambdaMC = [Phimax Angle ThetaPhi] dot {[Nmax Angle ThetaNa] + [Nmax Angle ThetaNb] + [Nmax Angle ThetaNc ] }
The quantity in curly brackets { } is sum of three vectors which have same magnitude but are 120 degrees apart and so they sum to 0 and the dot product is 0. So Magnetizing portion of flux linkage is 0. And combining with 0 leakage reactance we know the total stator flux linkages sums to 0. Therefore the derivative of sum of stator flux linkages must be 0. This was the missing piece of my proof 15 Nov 10 8:10 which supports the conclusion above that sum of voltages to neutral are 0 under this model.
That completes the proof imo. But I still feel better knowing ATP gave the same results.
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(2B)+(2B)' ?