This article by Stefan Vorkoetter originally appeared in the April 2002 issue of QuietFlyer magazine and is reproduced here with permission.

Motor Math

Don't let the word "math" in the title scare you. There won't be much of it, but I thought it would be worthwhile explaining just how one can calculate the performance of a motor.

Last month, we looked at propeller basics (I cleverly left the word "math" out of the title that time), and one of the things we came up with was the following formula:

power = k rpm³ diameter⁴ pitch

This formula expresses the relationship between a propeller's pitch, diameter, rpm, and other characteristics (represented by fudge factor k). For an average propeller, k is about 5.3x10^-15 if pitch and diameter are in inches, and power is in Watts.

The power term in the formula is the amount of power needed to turn the propeller at the specified rpm. In other words, it is the required propeller input power. Since the propeller is attached to the motor shaft (we'll only consider direct-drive for now), the propeller's input power must equal the motor's output power.

The Ideal Motor

If there were such a thing as a perfect 100% efficient motor, then the motor's output power would be equal to the motor's input power. In other words, all the Watts flowing into the motor would make it to the propeller, and hence become the propeller's input power.

In reality, there are many losses built into a motor. Some of these are electrical losses, such as the resistance of the windings and brushes. Some are magnetic losses, such as the armature's resistance to being magnetized by current flowing through the windings. There are also mechanical losses, such as brush and bearing friction, and drag as the armature rotates through the air inside the motor.

The Real Motor

For the purpose of predicting motor performance, a real motor can be treated as if it consisted of three components: an ideal motor, a resistor, and a current sink. The sketch below illustrates this. Note that this is not really what's inside a motor, but rather is just a circuit that approximates the characteristics of a motor.

Mathematically, we model a real motor as containing an ideal motor, a resistor, and a current sink. Click to enlarge.

The ideal motor has only one characteristic, and that is the rpm it produces per Volt of input. This is universally known by the symbol K_v, and is expressed in rpm/Volt. (Actually, there is a second characteristic, K_t, which is the amount of torque the motor produces per Amp of input, but this is always inversely related to K_v, so if you know K_v, you can compute K_t and vice versa. The equation relating these two is K_v K_t = 1355 when K_t is expressed in inch-ounces per Amp).

The resistor also has one characteristic, its resistance, measured in Ohms. The resistor represents the winding resistance of the real motor, which is known by the symbol R_a (the "a" stands for armature). The resistance serves no useful purpose, but is an unavoidable physical attribute of the motor.

Finally, the current sink draws a fixed amount of current, and throws it away. The amount of current it draws is the no-load current of the real motor, which is known by the symbol I₀ (I is the universal symbol for current, and the "0" refers to zero load). This is measured in Amps.

Now lets see how these all fit together.

First we need to know the voltage reaching the ideal motor, which we'll call V_i ("i" for ideal). This equals the voltage at the real motor terminals (we'll call it V), minus the voltage lost in the resistor. This in turn is equal to the total current (which we'll call I) times the value of the resistor. So,

V_i = V - I R_a.

Next we'll need the current reaching the ideal motor, I_i. The total current flowing into the real motor is divided between the current sink and the ideal motor. Since the current sink draws a fixed amount of current, the current reaching the ideal motor is just,

I_i = I - I₀.

Now that we have the voltage and current flowing into the ideal motor, we can multiply them together to get the power. Furthermore, since the ideal motor is 100% efficient, the power flowing into it equals the power used by the propeller. So,

(V - I R_a) (I - I₀) = k rpm³ diameter⁴ pitch.

Finally, we need to make use of the motor's K_v value somewhere in our formulae. The ideal motor spins at an rpm directly proportional to the voltage it receives (V_i). Therefore,

rpm = (V - I R_a) K_v.

These last two equations form the basis of any calculations we may wish to perform for a motor. We have nine variables and two equations, so if we know the values for any seven of these variables, we can solve for the other two. Fortunately, we generally know everything except rpm and I.

An Example

Let's consider a common example, an Astroflight Cobalt 05 motor, 8x4 propeller, and seven SCR type cells. For this motor, we can look up the following: K_v=2125, I₀=2.5, and R_a=0.045. We'll assume that under load, seven SCR cells will deliver about 7V. For k, we'll use the value for an average propeller, which is 5.3x10^-15. Our diameter and pitch are 8 and 4 respectively. What we want to find out is the current and the rpm. So if we plug all the numbers we know into our two equations, we get:

(7 - 0.045 I) (I - 2.5) = 5.3x10^-15 rpm³ x 8⁴ x 4

rpm = (7 - 0.045 I) x 2125

Unfortunately, solving these is non-trivial. The easiest way is with a mathematical software package such as Maple, but most of us don't have such a program. If you're mathematically inclined and very patient, you can solve them by hand, either symbolically or numerically.

Trial and error is a viable method too. Simply guess a value for I, and use the second equation to compute rpm. Then substitute your guessed I and computed rpm into the first equation and see if it holds true. For example, if we guess that I is 22 Amps, then the second equation gives us that rpm is 12771. If we put these values into the first equation, we get 117.2 = 180.9, which is clearly not true. So, we'll need to try a different guess for I. If that makes the first equation worse, then we'll need to go the other way in guessing I. After repeating this a few times, we'll find that I is 29.4, and rpm is 12067.

Complications

These equations are adequate for getting an approximate prediction of motor performance with a given propeller, but there are a few shortcomings. In the real world, the voltage V is not known exactly, because it actually depends on the current I. The higher the current, the lower the voltage, due to the internal resistance of the cells, and also the speed control. The principals are still the same, but solving the equations becomes much more complicated.

Specialized electric flight software packages, such as MotoCalc or ElectriCalc take all these factors into account and eliminate the need for the electric modeler to solve these equations directly.

Conclusion

You certainly don't need to know all this to successfully fly electric models, but if you like to experiment with power systems, or are designing your own models, a basic understanding of the underlying mathematics can make life a lot easier.
 
 

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Last updated Wednesday May 28, 2008.	E-mail Stefan
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