AC Resistance vs. DC Resistance

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Frank Restly

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Application: We have a direct drive three phase 480V fan listed for 1170 RPM @ 60Hz on a variable frequency drive. We want to over speed the fan to operate at 1770 RPM @ 90 Hz.

Problem: We need to calculate the AC resistance of the conductors serving fan when operating at 90Hz.

The formula for AC resistance is as follows:

RAC= RDC * ( 1 + Ys + Yp )

YS = Skin Effect Factor
YP = Proximity Effect Factor (Three Conductors)

Ys = (2 * μ * f * ks / RDC )2 / (192 + (2 * μ * f * ks / RDC )2 )

z = (2 * μ * f * ks / RDC )2 / (192 + 0.8 * (2 * μ * f * ks / RDC )2 ) - Intermediate calculation

Yp = z * (d / s)2 * [ 0.312 * (d / s)2 + 1.18 / (z + 0.27) ]

μ (Magnetic Permiability of Free Air) = 1.2566 x 10-7
ks (Construction Factor) = 1
f = frequency (Hertz)
d = diameter of conductor
s = spacing between conductor centers

Referring to NEC Chapter 9, Table 9 - AC Resistance of copper conductors in PVC Conduit at 75 Deg C / 60 Hz, the following values are listed:

#6 - 0.4900 Ohms / 1000 Ft
#4 - 0.3100 Ohms / 1000 Ft
#3 - 0.2500 Ohms / 1000 Ft
#2 - 0.1900 Ohms / 1000 Ft
#1 - 0.1500 Ohms / 1000 Ft

Referring to NEC Chapter 9, Table 8 - DC Resistance of uncoated copper conductors at 75 Deg C, the following values are listed:

#6 - 0.4910 Ohms / 1000 Ft
#4 - 0.3080 Ohms / 1000 Ft
#3 - 0.2450 Ohms / 1000 Ft
#2 - 0.1940 Ohms / 1000 Ft
#1 - 0.1540 Ohms / 1000 Ft

The first problem that I have is that the AC resistance numbers listed in the NEC are in many cases smaller than the DC resistances listed. For instance, the NEC lists the AC resistance of #6 at 0.490 Ohms / 1000 Ft and the DC resistance at the larger value of 0.4910 Ohms / 1000 Ft. I don't see how that is possible given that the formulas I am using for the skin effect factor and proximity effect factor should generate positive values.

What is the formula that the NEC uses to calculate AC resistance, and how do I modify it to adjust for an operating frequency other than 60 Hz?
 
90Hz Won't Be a Problem

90Hz Won't Be a Problem

Probably not, but with direct drive fans and VFD's becoming more prevalent, I would like to know what my limitations are. Suppose instead of operating at 90Hz, I want to operate at 270Hz? As an FYI, the reactance formula for three cables in conduit is as follows:

XL = 2 * μ * f * ln ( 2a / d )

And so using NEC Chapter 9, Table 9, I can calculate the reactance of three copper conductors at 90Hz by multiplying the reactance at 60Hz by the ratio 90 / 60. For instance the reactance of three #6 copper conductors in PVC conduit is 0.051 Ohms / 1000 Feet at 60Hz and so would be 0.051 * 90 / 60 = 0.0765 Ohms / 1000 Feet at 90Hz.
 
Frank Restly said:
Probably not, but with direct drive fans and VFD's becoming more prevalent, I would like to know what my limitations are. Suppose instead of operating at 90Hz, I want to operate at 270Hz? As an FYI, the reactance formula for three cables in conduit is as follows:

XL = 2 * μ * f * ln ( 2a / d )

And so using NEC Chapter 9, Table 9, I can calculate the reactance of three copper conductors at 90Hz by multiplying the reactance at 60Hz by the ratio 90 / 60. For instance the reactance of three #6 copper conductors in PVC conduit is 0.051 Ohms / 1000 Feet at 60Hz and so would be 0.051 * 90 / 60 = 0.0765 Ohms / 1000 Feet at 90Hz.
Click to expand...
But how long is the run?
 
You can also look at this as a skin-effect problem- when does that start to affect the current flow. At 60 Hz, the skin effect is around 0.35", so for any single conductor smaller than 0.7" there won't be an effect. At 300Hz, the skin effect is 0.164", so you're limited to a wire smaller than 0.328", which would be around 0 AWG (and that's probably multi-stranded).

Take a look at http://daycounter.com/Calculators/SkinEffect/Skin-Effect-Calculator.phtml
 
How long is the run...

How long is the run...

The motor is rated at 25HP. The distance between the drive and fan motor is about 150 feet depending on how the conduit is routed. This is a roof top fan and motor with variable speed drive mounted remotely in a conditioned space.
 
Zbang...

Zbang...

Ultimately what I would like to know is what formula did the NEC writers use to calculate the AC resistance in Chapter 9 of the NEC. Why are these AC resistances in some cases smaller than the listed DC resistances in Chapter 9?
 
Frank Restly said:
Ultimately what I would like to know is what formula did the NEC writers use to calculate the AC resistance in Chapter 9 of the NEC. Why are these AC resistances in some cases smaller than the listed DC resistances in Chapter 9?
Click to expand...
Yes, that seems a bit odd. At the frequencies concerned I wouldn't have thought there would be a significant difference in resistance.
 
Frank Restly said:
The motor is rated at 25HP. The distance between the drive and fan motor is about 150 feet depending on how the conduit is routed. This is a roof top fan and motor with variable speed drive mounted remotely in a conditioned space.
Click to expand...
In terms of power rating and distance both are relatively small. It's about 18.5kW and 50 metres. We've done many pumping station installations where the motors, particularly on borehole pumps, can be several hundred metres away from the VSD. That has other issues to be dealt with but I don't recall skin effect ever being factored into cable ratings.
 
Frank Restly said:
Ultimately what I would like to know is what formula did the NEC writers use to calculate the AC resistance in Chapter 9 of the NEC. Why are these AC resistances in some cases smaller than the listed DC resistances in Chapter 9?
Click to expand...
That does seem weird. Reactance should always add to total impedance, shouldn't it? That's how it seems to me but I have been out of school for a long time.
 
ggunn said:
That does seem weird. Reactance should always add to total impedance, shouldn't it? That's how it seems to me but I have been out of school for a long time.
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Yes, but it is resistance, not impedance, that was mentioned.
 
ggunn said:
But aren't the numbers in Table 9 actually impedance numbers?
Click to expand...

My understanding is that they are actually resistance, taking only skin effect into consideration. I found this on a site about ac/dc resistance ratios.


Note: Columns 1 and 2 include skin effect only. For close spacing such as
multi-conductor cables or several cables in the same conduit, there will be an
additional apparent resistance due to proximity loss. This varies with spacing
(insulation thickness) but for most purposes can be neglected without serious error.
Use Column 3 for:
(a) Multiple-conductor metallic-sheathed cable.
(b) Multiple-conductor non-metallic sheathed cables in metal conduit.
(c) Two or more single-conductor non-metallic sheathed cables in same metallic
conduit.
 
ggunn said:
But aren't the numbers in Table 9 actually impedance numbers?
Click to expand...


I would expect that the Chapter 9 values for "AC resistance" would really be the magnitude of impedance.

To explain what I mean, impedance is a complex number consisting of real resistance and imaginary reactance. I would expect that "AC resistance" is the Pythagorean theorem combination of the two components of impedance.
 
Carultch said:
I would expect that the Chapter 9 values for "AC resistance" would really be the magnitude of impedance.

To explain what I mean, impedance is a complex number consisting of real resistance and imaginary reactance. I would expect that "AC resistance" is the Pythagorean theorem combination of the two components of impedance.
Click to expand...

If that were the case, then how much reactance is figured in? The table says resistance, not impedance. I am comfortable with thinking that AC resistance does not factor in reactance, just skin and prox effects.
 
The title of table 9 is

Table 9 Alternating-Current Resistance and Reactance for 600-Volt Cables, 3-Phase, 60 Hz, 75°C (167°F) — Three Single
Conductors in Conduit

There are columns for AC resistance and reactance.
 
Carultch said:
I would expect that the Chapter 9 values for "AC resistance" would really be the magnitude of impedance.

To explain what I mean, impedance is a complex number consisting of real resistance and imaginary reactance. I would expect that "AC resistance" is the Pythagorean theorem combination of the two components of impedance.
Click to expand...

I agree, so how could AC resistance/impedance ever be less than DC resistance? The hypotenuse can never be shorter than the base.
 
K8MHZ said:
If that were the case, then how much reactance is figured in? The table says resistance, not impedance. I am comfortable with thinking that AC resistance does not factor in reactance, just skin and prox effects.
Click to expand...
At 60Hz the skin effect is negligible for small wires. What is a prox effect?

For figuring voltage drop and conductor size adjustment, resistance and impedance are the same thing, aren't they? I think that AC "resistance" in Table 9 is actually impedance; why else would there be a difference between AC resistance in steel and AC resistance in PVC?
 
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