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Inside Story - February 2003 (Derivations)

Maximum power - A formula for the current at the point of maximum power for a given motor can be derived from formula in this article. This information is an addendum to the column, "Inside Story - February 2003".

Splash How have all the formulas been derived?
We can measure the idle current directly. In theory also the internal resistance could be measured directly, but all our formulas are quite sensitive to , so it has to be as exact as possible, which is difficult to measure because the values are so low. We choose another way and calculate the values for and by solving a set of simultaneous equations. (for the math guys, this is the fun part!) We start with two copies of formula (8), one for each measurement we have made for one motor:
(8-1)
(8-2)
We get two equations, each containing the same two unknown values. to eliminate from (8-1) we convert (8-2) by dividing both sides by the terms in the parentheses into:
(8-3)
...and replace in (8-1) by the right side of (8-3):
(9-1)
Now we sort the values until we have an equation for :
step 1: multiply both sides by
(9-2)
step 2: eliminating the parentheses:
(9-3)
step 3: sorting terms with and without :
(9-4)
step 4: putting outside the parentheses:
(9-5)
step 5: dividing both sides by :
(9-6)
...and we finally have calculated (not measured) ! we put this into formula (8-1) to get :
(8-4)
Now we have measured and calculated all the significant motor constants: , , and , which we can use in turn to calculate all the values of equations (2) - (8). To do this we just have to measure and in each operation point, means, with each combination of motor, battery, gearbox and prop that we are interested in!
The value of , which is calculated through the "back door" will, when used in our formulas, exactly deliver the voltage, current, and rpm that we put into our formulas in the beginning. So a lot of possible measuring errors will be compensated.
Formula (10) did also not fall from heaven. I will show you how it is developed. At first we had formula (4):
(4)
...that we multiply out and get:
Still (4)
to find the current at which maximum power is achieved we build the first derivation of to :
(14)
will be zero at the maximum power because at lower current the power rises, so the first derivation is positive, and at higher current the power goes down, so the first derivation is negative. Since the power will change continuously when variating the current, we know that the first derivation has to cross the zero line exactly at the point of maximum power. We use this:
(14)
We add on both sides:
(14-1)
We divide by on both sides and get:
(14-2)
That is identical to formula (10), which was to prove.
Now that we start to like mathematics :-) we will go on and show how to develop formula (11). We start with formula (7):
(7)
That is (by multiplying out and reducing) the same as:
(7)
We build the first derivation of to and set it to zero, as above:
(15)
We add on both sides:
(15-1)
Multiply by :
(15-2)
Divide by :
(15-3)
Building the square root:
(15-4)
That is identical to formula (11), which is also proven now.
The interested reader who was willing to follow me that far will be able to prove formula (12) also! Tip: eliminate in formula (7) by using formula (11), multiply out and reduce! Have fun!

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