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Modeling the Behavior of PM DC Brushed Motor

Modeling the Behavior of PM DC Brushed Motor

Modeling the Behavior of PM DC Brushed Motor

(OP)
I have looked long and hard about clear and concise information on the web, so please forgive me if some of this seems trivial to you, but hope that people in the future will be able to use this as a reference as well.

I am in the process of working to model the acceleration profile for a small single person direct drive electrical vehicle and running into some issues fully understanding DC PM brushed motors torque vs. speed properties.  Essentially the model is going to use the max torque/speed curve of the motor as a input to calculate a maximum theoretical velocity as a function of time equation. To make things simpler I will list the current specs of our given motor (http://www.electricmotordepot.com/products/ME0708-Electric-PM-Motor-%28Etek-Replacement%29.html and http://www.electricmotorsport.com/store/ems_ev_parts_motors_me0708.php):

k= speed const = 72 [RPM/V] = 0.13 [Nm/A]
v= voltage = 48 [V]
R= armature resistance = 0.010 [Ohms]
imax= max peak current = 300 [A] for 1 [min]
icont= max continuous current = 150 [A]
stall torque = 38 [Nm]

Based on my current understanding, the amount of torque is driven strictly by the current applied to the motor while the speed is controlled by applied voltage and governed by the following formula:

n= motor speed at the shaft [RPM]
t= torque at the shaft [Nm]

E=k*w
t=k*I
v=E+I*R

sub-together:
n=(v/k-(t*R/k^2)*(60(2*pi))

solve for t:
t=-(k(k*n-120*pi*v))/(120+pi*R)

According to this equation, a torque at a speed of 0[RPM] under full v=48[v] would be 624 [Nm].  Clearly this is greatly above the 38[Nm] peak stall torque. Why does this equation seem to incorrectly model this?  Is it simply because the current needed would be too great for the motor as shown below?

t/k=I = 624[Nm]/0.13[Nm/A] = 4800 [A]

Is this the reason the peak stall torque is 38[Nm] due to imax (300[A]) limiting this? Is it correct to say that in order to keep under the 38[Nm] limit while still operating at imax of 300[A] that we would have to greatly drop our voltage supplied to a quite small value?

39[Nm] = 0.13[Nm/A] * 300 [A]

Or is this all bunk and this isn't working out for some other reason?  Is the 38[Nm] stall torque the limiting factor in terms of a physical constraint, because according the the above equation, it sounds like in theory the motor should be able to spit out scary amounts of torque?
 
Also, I have seen diagrams of both 1) linear (as described by the equations above) and 2) constant, then exponential decay speed/torque curves for DC PM brushed motors, whats with this?  In one of the books I'm reading they started talking about field weakening and constant torque in what I thought to be the PM DC brushed section, does field weakening apply to this type of motor and what is the explanation behind the two types of speed/torque curves?

Is there a better way to model a speed/torque curve for the application of developing a theoretical acceleration profile, or is this the proper way to do it? Do the steady-state equations used above apply to acceleration, or is this a poor assumption?

After this is a bit cleared up, I'll ask the second part of my question about correctly modeling this motor.

Thank you for your time.
 

RE: Modeling the Behavior of PM DC Brushed Motor

Step back a bit and review DC motor characteristics. When the motor is turning it will generate a back EMF almost directly proportional to the speed. The unloaded motor will accelerate until the applied voltage minus the back EMF causes enough current flow to power the losses. There may be only a few volts difference between the applied voltage and the back EMF. At this point it also only takes a few RPM from an overhauling load and the motor will become a generator and it's terminal voltage will rise as it delivers power back to the source.
Read this over a few times. Get comfortable with the back EMF being very close to the applied voltage. The current will be the [applied voltage minus the back EMF] divided by the effective resistance of the armature circuit. Both the effective voltage and the armature circuit resistance are quite small and the I2R supplying the losses will be small.
For a motor driving a load at near rated speed, the load slows the motor and the back EMF decreases. There is a greater difference between the back EMF and the applied voltage and the this drives a proportionately greater current through the armature circuit resistance. The motor slows until equilibrium is reached when the I2R of the armature balances the load and the losses.
When the motor is energized at standstill, the torque will be limited by the field strength and the current will be limited by the armature circuit resistance and any external current limiting devices. The current will be very high.
As the motor accelerates the armature current will drop as the back EMF builds. Initially accelerating torque will be constant until the torque developed by the I2 equals the torque limit of the field strength. As the motor continues to accelerate the torque will drop off linearly with speed until equilibrium is reached at the motor rated current.
This is for acceleration. For steady state operation you may encounter thermal limits and external controls may be needed.
Interesting problem. Is this a motor project or a modeling exercise?

Bill
--------------------
"Why not the best?"
Jimmy Carter

RE: Modeling the Behavior of PM DC Brushed Motor

The torque-speed curve is a steady state characteristic of the motor. You are looking for the transient behavior of the motor-load system. The only accurate way to do this is to solve the differential equation/s with the voltage as the input and the differential of the current with reference to time.
 

RE: Modeling the Behavior of PM DC Brushed Motor

(OP)
Thanks you both for your responses!  This is neither a motor design project nor a modeling exercise, rather we are building a small electric vehicle and I'm trying to learn and model as much as possible along the way.  With that said I understand this model won't be spot on, but would like to see if I can at least get it close to being able to create a informed estimate which we can then base the optimization of our direct drive gear ratio on. Not to sound like a broken record, but first off let me check that my understanding now is correct and state this for people coming across this in the future. Please forgive the verbosity of the post, but I didn't want to miss anything.

The current being drawn is directly proportional to the torque of the system:

I*K=Torque

In transient conditions the voltage applied is a function of speed, current and the differential of inductance with respect to time, thus:

Vapp = I*R+E+L*(dI/dt)  --  (Current times resistance)+(back EMF proportional to armature speed)+(inductance*differential in current with reference to time)

I = (Vapp-E-L*(dI/dt)/R)

Thus to solve this differential equation you need to have 'E' (the back EMF) in terms of time, or in other words the armature speed as a function of time.

In terms of my modeling situation, the approach I am taking for the grander scope is to solve the differential equation for velocity at a given time using the following equations.

Frr = force of rolling resistance
Fdrag = force of drag as a function of velocity
Fw = traction force at the wheels based on the torque the motor can put out at as a function of velocity
m = mass

(dV/dt) = (Fw(V) - Fdrag(V^2) - Frr) / m

Fw is based off of the current velocity of the vehicle and is the reason I am trying to establish this baseline torque curve for the motor as the torque from the motor is convertible to a force at the wheel based on gearing and wheel size.

The differential equation for the current drawn (I) relies on the back EMF being able to be wrote as a function of velocity, but the velocity differential equation can't be solved till the current is known.  Thus I seem to be left with the option to define a velocity over time function and solve the differential equation for the current, which would make the efforts of solving all of this pointless if I am defining it initially.  Is this all correct, or have I made a mistake?

After reviewing the post by waross, a few books and playing with the equations, I think I have a solid understanding of the dynamics in a transient situation. While the difference in applied voltage (max 48v) and back EMF will draw current far higher than within the ratings of the motor when at low speeds, the controller will limit the applied voltage (a bit misleading since it's doing it through PWM) so that we don't surpass our max rated torque. The primary limiting factor in terms of our motor is the max rated continuous current of 150A and thus the primary limiting factor in terms of max torque output.  The 150A threshold covers nearly the entire operating speed range of the motor.  Thus, I assume the controller varies the voltage applied to the motor so that when the pedal is floored, max rated current (150A) is drawn by the motor over a range of speeds in a way that is proportional to the back EMF at the current operating speed.  This only occurs when you are trying to accelerate as fast as possible from a standstill.  Thus this eliminates the aforementioned issues of defining the back EMF as a function of velocity and the differential of current with respect to time (becomes 0) if the current is going to be limited in most situations by the 150A threshold.  While there is technically some differential of time in the feedback loop of the controller varying the voltage and everything that is changing and thus the current in reality is varying slightly, from my understanding these variations would be nearly incalculable for a real lie situation with out knowing the exact way the proprietary controller does this.  Is this a correct assumption or a over simplification?

I played with this a bit in mathematica, which can be seen in the attached image and wrote up the following description http://entplex.org/forums/MotorCurrentTorque.jpg

When starting at a stand still, the controller applies a low voltage to the motor so that the current doesn't excede the rated 150 amps.  Under a situation where maximum acceleration is desired, the voltage is increased slowly as the velocity of the vehicle increases, resulting in a continuos draw of 150A over a range of speeds.  Since the torque that a motor is applying is directly proportional to the current I, a continuos current of 150A mean that the torque in this range is continuos, no?  Eventually a point is reached where the current drawn is no longer constrained by the max rated current of the motor and the full applied voltage is neared, thus the current decreases due to the difference between the applied voltage and back EMF being very small.  beyond this point, the modeling situation becomes far more complex and the assumption of being able to draw the max rated current goes out the door.

So, does this all sound about right, or am I still missing a critical point and making stupid assumptions?

RE: Modeling the Behavior of PM DC Brushed Motor

Hi entplex

I want to put a slightly different view on what your trying to achieve and that is why don't you specify the mechanical output requirements first ie:- mass of passenger, max acceleration req to reach a maximum speed and maximum speed with a given load both on the flat and incline, with these parameters set you can size your motor power and torque accordingly.

Regards

desertfox

RE: Modeling the Behavior of PM DC Brushed Motor

"Fw is based off of the current velocity of the vehicle and is the reason I am trying to establish"

No, you are overcomplicating it

Fw= k*I

where k includes Kt and gear ratios and wheel sizing.

 you are missing coulomb friction and maybe some viscous friction from force equation

You have 2  equations which must be solved.

first solve
1)M*sv+k2*v^2+K3*v+k4=k5*I I=150
for v


while testing
2) I=(V-k1*v)/(R) for I>150 I
as soon as
I<150
you solve the coupled pair

1)M*sv+k2*v^2+K3*v+k4=k5*I
2)V=(sL+R)*I+k1*v)

L-winding inductance
s=d/dt
V=voltage
v= vehicle velocity

 

RE: Modeling the Behavior of PM DC Brushed Motor

corredtion
the force equations should include additional load that is not known apriori, so the criterion for coupling the equations should come from from a variety of postulated load profiles.

RE: Modeling the Behavior of PM DC Brushed Motor

(OP)
@desertfox:
The approach you have described and linked to is the way we initially decided upon this motor as a rough way of establishing a baseline.  The problem I am now trying to tackle is modeling the maximum theoretical acceleration profile of the vehicle, which requires the approach that zekeman outlined.

@ zekeman:
Your first post is in essence the approach I was describing.  Since over the majority of our speed range, the 150 A current is the limiting factor (see the graphs of current over RPM and torque over RPM in the previously linked image http://entplex.org/forums/MotorCurrentTorque.jpg) I assume a constant torque of 18 [N*m] based on 150A * Kt. This is only a safe assumption since the majority of our operating range is with-in the continuous current limited speed range and based on the assumption that our controller will vary the net voltage applied to the motor so that the current is maintained at 150A. Both of these assumption can't be made in all situations, but in our case I don't see any reason they wouldn't apply.  Once the 150A limit no longer applies, then yes the approach of solving the two equations i.e I < 150 then you do have to use the coupled pair as the aforementioned assumption no longer applies.  

You are completely correct that I still need to factor in coulomb friction (i.e. will the drive wheel spin out), but with regard to viscous friction, is that not accounted for by the force of drag in the model, or are you referring to something else?  Here is a image with the calculations, including the equations and graphs.  While working out my understanding of the motor system I have chose to simplify the physical model and exclude incline, drive train efficiency, compression of the suspension, etc.  http://entplex.org/forums/TheoreticalMaxAcceleration1.jpg

@zekeman (second post): Approaching this from the other end of selecting a loading profile and then calculating current draw would indeed be a valid approach, but it would be more of a brute force method and requires solving the coupled equations.

With regard to solving the coupled pair, I don't fully understand how to go about this.  In the linked image, I solve M*sv+k2*v^2+K3*v+k4=k5*I  to get velocity as a function of time based on the fixed current.  Due to my limited understanding of E and M, how would you solve V=(sL+R)*I+k1*v)?  Are you suggesting I solve for current as both a function of applied voltage and velocity?  This might just simply be a matter of my understanding of multi-variable and partial differential equations needing to be developed, is this so?

Once again I'd like to thank every body for helping with this and being patient.

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