Originally Posted by

**AndresKiani**
Is there a rigorous proof that explains why **physical dimensions behave as algebraic quantities**. This concept allows Physicists to incorporate dimensional analysis in their calculations and theories. I'm interested in a proof or rigorous argumentation to why this works. Any help would be great.

I thought maybe this relates to the Original Poster's question...

The following is a geometric proof demonstrating BLDC motor parameters behaving as algebraic quantities which can be represented geometrically.

Geometric Depiction:

Suppose I have a 100kv (100 rpm per applied volt) BLDC motor with a fixed volume of copper winding and I measure the winding resistance lead to lead as 1 ohm, but I am free to change the length and cross section of the copper winding, while retaining the same copper volume. The geometric proof demonstrates that in this scenario, changes in KV (rpm per volt) are proportional to the square root of changes in conductance. For example suppose the original winding is 1 unit length and 1 unit cross section area. Now I divide the winding length by sqrt(3) [1.73205], new length = 0.5773 and I multiply the cross section by sqrt(3) [1.73205], new cross section area = 1.73205 to keep the same copper volume. The resistance drops to 1/3 of the original value while the conductance increases by a factor of 3. When this happens, as depicted in the geometric proof, the KV increases by a factor of sqrt(3) [1.73205], new kv value = 173.205 (square root of the conductance change factor with fixed copper volume). Another case of an increase in conductance x 3 leading to a change factor of KV of 1.73205 is keeping the same number of turns (same conductor length and volume) but changing the "termination" of the windings from Wye to Delta.

This has been previously posted elsewhere at:

code to change the motor amp limit - Page 14 - vedder.se forums
The "traditional" formula for describing changes in KV is as follows (no change in termination):

KN=C

K=C/N

K = kv = new kv (max rpm per volt) no load

N = turns = new # of wire turns per tooth

C = constant = original kv x original # turns

Changing termination wye to delta increases kv by a factor of 1x1.73205 [sqrt(3)]

Changing termination delta to wye decreases kv by a factor of 1/1.73205 [1/sqrt(3)]

^but notice "turns," "constant," and "termination" are not SI base unit derived variables.

I have written a new formula that gives identical results using only SI derived variables (no turns, constant or termination variables).

My formula for describing changes in KV is as follows:

D=sqrt(E/(V*N))

E=N*V*D^2

V=E/(N*D^2)

N=E/(V*D^2)

D = Change Factor of KV (rpm/v)

E = Change Factor of Conductor Resistivity (ohm-meters)

V = Change Factor of Conductor Volume (meters^3)

N = Change Factor of Conductor Resistance (ohm)

For example, changing termination wye to delta, change factor of Resistance is 1/3 (conductance change factor 1x3) while the change factor of KV is 1.73205.

Another example is shortening the winding length by a factor of 1/sqrt(3) and increasing the cross section by a factor of 1x1.73205... also leading to a change factor of Resistance is 1/3 (conductance change factor 1x3) while the change factor of KV is equal to 1.73205.