waross said:
I may have been wrong to ignore armature resistance.
You tell me.
I agree with your approach - we would get the right qualitative answer by ignoring the armature resistance, except at extremely heavy loads which would probably consistute severe overload.
To quantify it for this situation, I'd plot the two lines (T vs N) and look for the intersection. Neglecting armature resistance is fine for qualitative purposes as long as we are operating to the right (higher N) of the intersection of the two curves.
Find that intersection by solving two equations (one per curve) in two unknowns (T and N).
The general form of the torque speed line is
T = Tz - N (Tz/Nn)
The original line (ref temperature subscript 0) is:
T = Tz0 - N (Tz0/Nn0)
The cold line at -30F is
T = [Tz0/0.78^2] - N ([Tz0/0.78^2] / [0.78*Nn0])
To find the intersction, set them equal:
Tz0 - N (Tz0/Nn0) = [Tz0/0.78^2] - N ([Tz0/0.78^2] / [0.78*Nn0])
…divide through by Tz0…
1 - (N/Nn0) = 1/0.78^2 - (N/Nn0)/0.78^3
…move N/Nn0 terms to the right and constant terms to left…
(1 - 1/0.78^2) = N/nN0 (1 - 1/0.78^3)
… solve N/Nn0…
N/N0 = (1 - 1/0.78^2) / (1 - 1/0.78^3) = 0.58
T/Tz0 = 1 - N N/Nn0 = 1 - 0.58 = 0.42
So for these particular numbers, the intersection of the original line and the cold line is (N,T) = (0.58*Nn0, 0.42*Tz0). As long as speed remains above 58% of original no-load speed (equivalently Torque remains below 42% of original zero-speed torque) then we reach the correct qualitative conclusion by neglecting armature resistance. It's a very safe bet.
Let me circle back to associate a more generic solution starting with the specific solution:
N/N0 = (1 - 1/0.78^2) / (1 - 1/0.78^3)
Let's subsitute 0.78 = 1-x where x represents fractional decrease in resistance (in our specific case x=0.22, we will limit our general consideration to x<< 1)
N/N0 = (1 - 1/[1-x]^2) / (1 - 1/[1-x]^3)
N/N0 = (1 - [1-x]^<-2>) / (1 - [1-x]^<-3>)
Substitute [1-x]^p ~ 1-px for x<<1
N/N0 ~ (1 - [1--2x]) / (1 - [1--3x])
N/N0 ~ -2x / -3x = 2/3
So for any small decrease in temperature (resistance), we can neglect armature resistance in qualitative prediction as long as no-load speed doesn't decrease to 2/3 of nominal. It's still a very safe bet.
Also neglected was additional external resistance in series with the armature…if it does not change temperature as much (perhaps portion of the supply wiring is in a temperature controlled area) then the effects of motor armature resistance variation are even smaller (we could neglect armature resistance variation over an even wider range of speeds and torques).
TLDR - Bill you were right as usual. It's probably intuitive to you after many years of working with dc motors. For me, dc motors are something I'm still learning about, so I took the opportunity to explore it in my own way.
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(2B)+(2B)' ?
