I have reconciled the differences in the equations and now I am sure my equation ep1 above is correct if the alpha is based on 20C (call it alpha20). Also I derived an alternate form below which matches the Cable Handbook if alpha is based on 0C (call it alpha0).
Attached is an excerpt from above-referenced ABB handbook. On the right side of the page you see the same equation as in the cable handbook. There parameter B (similar to Beta in cable handbook) is listed as "the reciprocal of temperature coefficient of resistivity at 0C", while the resistivity itself is given at 20C. The only reason I provided this attachment is to establish that the result is based on use of resistance standard at 20C and temperature coefficient at 0C.
Start with the expression I derived as posted 21 Feb 09 12:41
tfinal = ln{<Tfinal*R1+R0>/<R0+Tinit*R1>} * C/[I1^2*R1] [eq A]
As previously stated this result was derived using the following relationship:
R(T) = R0 + R1 * T [equation B]
where we have not yet defined R0, R1, T, but the result is valid for any collection of definitions of R0, R1, T that give the correct value R(T).
Let us start by defining T and R0 in a very natural way:
T is temperature in degrees C
R0 is resistance at 0 degrees C
Now we wish to determine the proper values of R1 assuming we are using 20C as the temperature for the reference resistance (call it R20) and 0C as the reference temperature for temperature coefficient (call it alpha0.... not much different than alpha20).
Using R0 and alpha0 as a reference, the general relationship for resistance would be:
R(T) = R0 * (1 + alpha0 * T) = R0 + R0 * alpha0 * T [equation C]
We can use this equation C to find R0 in terms of R20 and alpha0:
R20 = R0 * (1 + alpha0 * 20)
Solve for R0
R0 = R20 / ( 1 + alpha0 * 20)
We can also compare equation B to equation C to solve for R1
R(T) = R0 + R1 * T [equation A]
R(T) = R0 + R0 * alpha0 * T [equation B]
By inspection of the above equations A and B, we can see that
R1 = alpha0 * R0
And plugging in the equation for R0 into the result for R1 we have:
R1 = alpha0 * R20 / ( 1 + alpha0 * 20)
Divide numerator and denominator by alpha0
R1 = R20 / ( 1/alpha0 + 20)
So now we have expressions for R0 and R1 in terms of R20 and alpha0. Let us plug them back into equation A.
Start with original equation A
tfinal = ln{<Tfinal*R1+R0>/<Tinit*R1+R0>} * C/[I1^2*R1]
Divide numerator and denominator inside parantheses by R1
tfinal = ln{<Tfinal+R0/R1>/<Tinit + R0/R1>} * C/[I1^2*R1]
Plug our new definitions of R0 and R1 into the above equation:
R1 => R20 / ( 1/alpha0 + 20)
R1/R0 => alpha0
tfinal = ln{<Tfinal+1/alpha0>/<Tinit + 1/alpha0>} * C * ( 1/alpha0 + 20) /[ R20 * I1^2] [equation D]
Now if we we revisit the previous equation from the Electric Cables Handbook Chapter 9, pages 152-153 without making any "corrections" this time, we see they it is the same as equation D above.
Equation 9.1 was
I^2 = [K^2 * S^2 / T ] * ln {(Theta1 + beta) / (Theta0 + beta)}
Let's make some substitutions from their notation to my notation:
T -> tfinal
Theta1 -> Tfinal
Theta0 -> Tinit
beta -> 1/alpha0
I^2 = [K^2 * S^2 / tfinal ] * ln {(Tfinal + 1/alpha0) / (Tinit + 1/alpha0)} (EQ 9.1A)
Swap the location of tfinal and I^2
tfinal = [K^2 * S^2 / I^2 ] * ln {(Tfinal + 1/alpha0) / (Tinit + 1/alpha0)} (EQ 9.1B)
Now look at K^2 * S^2
{K^2} {S^2} = {Qc * (1/alpha + 20) / Rho20} * {Area^2}
where Qc is (heat capacity) in Joules * degC /Volume
Break this up into three factors: [Qc*Area], [1/Rho20*Area], and [1/alpha + 20]:
[Qc * Area] = C expressed on a per-length basis
[1 / Rho20 *Area] = 1/R20 where R20 is expressed on a per-length basis
[1/alpha + 20] = no clarification needed.
That gives us
S^2 * K^2 = C * (1/R20) * [1/alpha + 20]
where again C and R20 are per-length
Susbsituting K^2 * S^2 = C * 1/R20 * [1/alpha + 20] back into (EQ 9.1B) gives us:
tfinal = [C * (1/alpha + 20)]/ [ I^2*R20 ] * ln {(Tfinal + 1/alpha0) / (Tinit + 1/alpha0)} (EQ 9.1C)
After all this we see that my equation D which was fully derived above matches equation 9.1C which came from the Cable Handbook and so
the two solutions agree. ![[bigsmile]](/data/assets/smilies/bigsmile.gif "[bigsmile] [bigsmile]")
Assuming there were no difference between alpha0 and alpha20 (i.e. assuming the same alpha could be used in the two different equations T = R0*[1+ alpha * T] and T = R20[1+ alpha * <T-20>].... not a 100% correct assumption even for a linear relationship) then both of these equations should give the same solution as the equation ep1 that I previously posted 22 Feb 09 12:57. Strictly speaking, if the alpha is associated with 20C (as is most common), then equation ep1 is more accurate. If the alpha is associated with 0C, then EQA and EQ 9.1C are more accurate. I see various definitions of alpha used in different references. My impression is that 0C is used as a base point when we are using a higher-order approximation (either T = a + b T + c T^2..etc or the Calendar /Van-Dusen equation). If we want to use a single parameter for a linear approximation, then it is most often based on 20C or else it is based on connecting the endpoints of the interval 0C-100C and exact at one point in the center of that interval... which might happen to be close to 20C).
I am certainly splitting hairs to worry about the difference between alpha0 and alpha20. Either way, there is some error from using a linear approximation on a curve that deviates from linear if we go far above 100C, and likely other larger errors in assumptions for a calculation of this type.
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