# what is the DC motor RPM formula?

#### Maverick Ng

Joined Jul 12, 2017
5
what is the DC motor RPM formula?

#### WBahn

Joined Mar 31, 2012
26,398
What kind of DC motor? What kind of load? In terms of what variables?

#### WBahn

Joined Mar 31, 2012
26,398
Not much of a data sheet.

If you find a formula, how are you going to then "prove that this is the formula to find out rpm answer."

Have you ever used a cordless screwdriver?

Does it turn at the same speed when you are screwing a long screw into a hard piece of wood? Or does the speed depend on how much resistance there is to the screw?

So if someone asked you for a formula for the speed of a cordless screwdriver, what would you tell them?

#### WBahn

Joined Mar 31, 2012
26,398
Another way to think about it is what you are asking is very much akin to asking someone for a formula for the speed of a car.

#### MrAl

Joined Jun 17, 2014
8,984
https://www.alibaba.com/product-detail/12v-dc-motor-specifications-FF-130RH_781291487.html

i just want to know what is the formula of rpm, i want used to prove that this is the formula to find out rpm answer.
Hi,

First, that does not seem to show data for your particular motor. It's also too hard to see the curves. How specific do you need to be?

Second, the data sheet for the motor should show the speed/torque/power curves. You can glean information from that for your application. You can get some points from those curves and either tabularize them and use interpolation or do a curve fit on them, or just use the graphical data as is.

Third, if you want to estimate the speed/torque curve you can find the no load point and stall point and draw a straight line between then and use that as an approximate speed/torque curve. It's better to use the data sheet curves though if you can find them.

If you want more detail then we'd have to turn to some equations for a motor but i dont know how deep you want to get into this.

#### Papabravo

Joined Feb 24, 2006
18,409
There is a basic equation for torque. It is the product of current and a torque constant Kt. If the motor has no load and you know the moment of inertia for the rotor you can calculate the angular acceleration. From the angular acceleration you can compute the velocity. Here is where things get tricky. As the motor begins to move there is a back EMF which reduces the torque. The faster it moves the greater the reduction. At a certain speed the torque actually drops to zero. That is why an accurate speed torque curve is what you need

#### Maverick Ng

Joined Jul 12, 2017
5
then in the datasheet of dc motor there's a rpm value
so how they calculate the rpm is that value?
how to calculate out?
what is the formula or equation?

#### Papabravo

Joined Feb 24, 2006
18,409
then in the datasheet of dc motor there's a rpm value
so how they calculate the rpm is that value?
how to calculate out?
what is the formula or equation?
What makes you think it is calculated rather than empirically determined?

#### Maverick Ng

Joined Jul 12, 2017
5
What makes you think it is calculated rather than empirically determined?
i have 12v dc motor with 6100 rpm, and now if i apply only 5v. so how many was the rpm?

#### Papabravo

Joined Feb 24, 2006
18,409
i have 12v dc motor with 6100 rpm, and now if i apply only 5v. so how many was the rpm?
I have no earthly idea; motors respond to current -- not voltage.

• cmartinez

#### WBahn

Joined Mar 31, 2012
26,398
i have 12v dc motor with 6100 rpm, and now if i apply only 5v. so how many was the rpm?
Depends on the motor and, more so, on the conditions. What is the load on the motor? Does that load change with speed? Probably. But how?

If the motor is unloaded, then MOST permanent magnet motors have a speed that is roughly linear with respect to applied voltage. This is because the back-emf is pretty linear with motor speed and the torque needed to run the motor with no load is generally small and fairly linear with speed. But many loads are highly nonlinear with motor speed, so all bets are off.

Joined Jul 18, 2013
25,207
i have 12v dc motor with 6100 rpm, and now if i apply only 5v. so how many was the rpm?
RPM is related to voltage, that motor is rated for 12vdc then it should rotate at 6100 rpm with 12v applied if it has been rated correctly.
Conversely if you rotated the motor from an external source and measure the generated voltage at a particular rpm, it can be obtained empirically.
With 5v applied at no load the rpm will be in direct proportion to the voltage .
Max.

#### MrAl

Joined Jun 17, 2014
8,984
Hello again,

A simplified static solution would look like:
w=V/k1-T/k2*R

where k1 and k2 are constants for the motor and R is the equivalent armature resistance, V is the applied voltage and w is the angular speed.

From this simple expression we see that if the torque is zero then we have;
w=V/k1

which is completely linear so a 12v motor at no load run with 6v will run at 1/2 the speed.

However, there is always some torque T because of the imperfections in the motor, and these may be significant so we dont see a perfectly linear relationship with most regular motors we use. For example, a PC fan motor speed is not exactly proportional to voltage and this is very evident by the fact that a 12v motor does not run with only 1v applied, whereas if it was perfectly linear it would run at 1/12 the top end speed.
The 12v DC fans i tested start turning at 3v or 4v if i remember right. After that they may be fairly linear where we have pseudo drive voltage v such that: v=V-3 or v=V-4 possibly.

We can look at a better model, but we'll end up with something similar in the static case where we apply a DC voltage and wait for the motor speed to become perfectly constant.

#### Maverick Ng

Joined Jul 12, 2017
5
Hello again,

A simplified static solution would look like:
w=V/k1-T/k2*R

where k1 and k2 are constants for the motor and R is the equivalent armature resistance, V is the applied voltage and w is the angular speed.

From this simple expression we see that if the torque is zero then we have;
w=V/k1

which is completely linear so a 12v motor at no load run with 6v will run at 1/2 the speed.

However, there is always some torque T because of the imperfections in the motor, and these may be significant so we dont see a perfectly linear relationship with most regular motors we use. For example, a PC fan motor speed is not exactly proportional to voltage and this is very evident by the fact that a 12v motor does not run with only 1v applied, whereas if it was perfectly linear it would run at 1/12 the top end speed.
The 12v DC fans i tested start turning at 3v or 4v if i remember right. After that they may be fairly linear where we have pseudo drive voltage v such that: v=V-3 or v=V-4 possibly.

We can look at a better model, but we'll end up with something similar in the static case where we apply a DC voltage and wait for the motor speed to become perfectly constant.
how to find the value of K1 & K2?

Also how to find the value of R?

Joined Jul 18, 2013
25,207
This is some notes I put together when I was looking for similar information and found them in various sources:
A simplified model of a DC motor can be derived assuming the armature inductance to be zero and ignoring the resonance effect.
With these stipulations the equations are:

1. V=Ia R + Ke omega (Ia=armature current, R=armature resistance,
Ke=electr. constant, omega=speed)

2. Tg=Kt Ia (Tg=constant, Kt=torque constant)

3. Tg=J d(omega)/dt (J=inertia, d(omega)/dt=accel.)

The DC motor transfer function is:
Gm(s)=(1/Ke)/(1+s(Rj/KtKe)), which can be written Gm(s)=(1/Ke)/(1+sTm)
where Tm=mechanical time constant.
To measure the parameters you are asking for, use the following:

A. Measure the armature resistance as below, then apply voltage to the motor without load and measure the current and speed. From equation 1. you can easily derive Ke.

B. Apply nominal current to the motor (with the shaft locked) by means
of a variable voltage source. Measure the torque on the shaft. From this you can derive the torque constant Kt=Torque/Amp.

C. You will find that Kt is approx. equal to Ke

D. For the inertia you can obtain it by calculation from the size and
material of the rotor.

Note 1: inductance can be ignored- the electrical time constant is
very short compared to the mech time constant so that it can usually be
ignored.
You can measure the mech time constant by running the motor up to
speed at no load, disconnecting the supply and letting it coast down- plot speed vs time and fit to exponential N=No(e^-t/Tm) time to drop to 36.8% of original speed is the time constant.

Note2: If it is a permanent magnet motor, you can determine the internal emf by spinning it at rated speed and measuring the open circuit voltage. The voltage at any other speed will be directly proportional to speed.
To measure the winding resistance, lock the rotor so it doesn't turn and measure the current with a small voltage applied (so as not to exceed rated current) Don't not use a multimeter's ohm range.
If you want to find the inductance, you should use a scope- apply a voltage, rotor locked and look at the current trace vs time.
This will be of the form i=K[1-e^Rt/L] where i is the current at time t.
In most cases the inductance can be ignored as its effects are generally swamped by the mechanical inertia in transient cases and is of little importance for steady state.
Max.