WRIM & LRS start time calculation 1

[sberbece]

Hello !
Please help me with the equations to calculate the starting time of a WRIM motor, started with an LRS.
Data known:
-Motor voltage rating;
-Shaft power output;
-Motor rotor inertia;
-Load inertia;
-In = I1 at full speed @ full load;
-Locked rotor torque: 260% Tn
-I2 and V2;
-synchronous and full load speed;
-motor’s equivalent circuit, R1, X1, Rm, Xm, R’2, X’2;
-motor efficiency @: 50%, 75% and 100 % load;
-motor power factor @: 50%, 75% and 100 % load;

Maintaining the starting current <= 1.6In, need to find out:
-Tm = Motor’s maximum torque, corresponding to Ist = 1.6In;
-LRS starting resistance to obtain the Tm at start;
-LRS resistance band, i.e. Rmax and Rmin;
-Motor’s acceleration time;

It is true that these days any power software analysis programs do these computations easily. However, I’d like to be able to do it manually, before using the computer.

Thank you in advance for your support.

Regards,
Stefan

 

[7anoter4]

There are some approximations for calculation of torque as a function of slip but there are
far to be accurate as the rotor winding resistance and reactance strongly depends on skin effect.
In traditional design of induction machines, skin effect is represented by correction coefficients Kr and Kx depend on shape of the rotor slot, the conductor material and the rotor current frequency. This dependence is not linear.
Kr=rotor ac resistance/rotor dc resistance [typically 1 to 5]
Kx=rotor ac slot leakage inductance/rotor dc slot leakage inductance. [usually 0.2 to 1]
The Manufacturer usually provides torque slip and current slip curves for different values of exterior resistance. See, for instance, thread237-250935 .If you have not these curves you can measure the torque using suitable equipment.
The approximate formula for torque slip appreciation is based on Steinmetz Model considering R2 and X2 [L2 !] constant or linearly variable with the slip.
You need to know the transformation coefficient from rotor elements to stator as R'2 is referred to stator. So, in order to calculate R'ext you need the stator to rotor ratio kw= kw1/kw2 .The Manufacturer gives the rotor voltage.
In this case the rotor to stator ratio kw =E2/E1 it is, approximately, kw=Rotor rated voltage/Stator rated voltage.
The current coefficient then is 1/kw and the resistance and reactance coefficient is kw^2.
The slip corresponding to the maximum [break-down] torque will be:
sk = R2'/SQRT(R1^2+Xst^2) where:
R'2=R'2rotor+R'2exterior R'2rotor=rotor winding resistance at start; R'2exterior= exterior connected resistance required at start.
Xst =X1+X2'start
I1start=U1/sqrt(3)/(SQRT((R1+R'2start)^2+Xst^2) I1rated=HP1rated*.746/sqrt(3)/pf/eff*1000
If you intend to maintain I1start<=1.6*Irated you have to solve inequality:
U1/sqrt(3)/(SQRT((R1+R'2start)^2+Xst^2)<1.6*Irated .From here :
R'2start>R1-sqrt(R1^2-c) where U1/sqrt(3)/1.6/Irated=kist U1=motor rated voltage [stator]
c=kist^2-Xst^2-R1^2

 

[7anoter4]

Correction:
I1 rated formula has to be:
I1rated=HP1rated*.746/sqrt(3)/pf/eff*1000/U1 U1=motor rated voltage [stator]
kw =E'2/E2 E'2=E1 then kw= stator voltage/rotor voltage I'2=I2*kw R'2 =R2/kw^2. X'2=X2/kw^2.
For other values of slip in order to simplify the calculation you may neglect Xm and Rfe and the stator current formulae is as follows:
I1(s) =U1/SQRT [(R1+R'2) ^2+(X1+s*X'2start) ^2] =I'2(s)
The power transferred from stator to rotor Pi=3*R'2*I'2^2/s and the torque will be Pi/rpmsynchron.
So, you have to know synchrony rpm[s=0], or stator poles number and the stator current frequency [60 or 50 Hz].
The Tbreakdown can be calculated if you know sk [from above].
Knowing Tbreakdown and Tstart you may appreciate the average Tmotor=0.45*(Tbreakdown+Tstart) for A or B NEMA Type.
You have to know the load average torque TL=KL*Tloadrated. KL=1/3 for centrifugal pump or fan, for instance.
Tacc=Tmotor-TL [the acceleration torque].
The acceleration time will be then:
tstart=wk^2*(rpmsynchron -0)/Tacc/308
where wk^2 it is the sum of motor wk^2 and load wk^2.

 

[sberbece]

Thank you 7another4!