Estimating Power from slip -> temperature correction 5

[electricpete]

I have always thought that motor power could be estimated based on linear relationship with slip: 0 power at zero slip; nameplate power at nameplate slip.

I just saw IEEE 939-1995 (Energy Efficiency) section 6.15 which suggests that slip will increase by approx 0.35% per degrees C above 25C (winding temperature).


They go on to state "Most motor manufacturers, when specifying the full-load RPM, take their readings at 25C or on a cold motor".

That's the first I've ever heard of it. What use is a ficticous number based on 25C winding temp? Winding temp is sure to increase above that during starting and running.

Am I reading this right?

 

[rhatcher]

Hey electricpete,

That is a new one on me. However, I may be able to shed some light on part of it. I would guess that the that "slip will increase by approx 0.35% per degrees C above 25C (winding temperature)" to be related to increasing winding resistance with temperature. Keep in mind that the amount of resistance in the stator of an AC motor affects both torque and slip, similar to the effect of resistance in the rotor circuit (actually the same effect as rotor current is reflected to the stator). Low resistance equates to low slip and high efficiency and high resistance equates to higher starting torque but higher slip and lower efficiency. Again, whether this is what they are talking about is a guess based on the source and my interpretation of what the connection is....

I hate to say it based on the source, but my instinct is to disagree with the idea that most motor nameplates are based on a cold motor or a 25C winding temperature motor. That just does not sound right as the measured result would be so transient as to be meaningless. My understanding is that nameplate ratings are (or should be) based on testing at full load, rated ambient temperature, and after achieving steady state temperature.

With respect to using 25C as a standard temperature for anything to do with motors, I am at a loss.... Most motors ratings are based on an ambient of 40C, and will have a winding temperature (obviously) greater than that based on design features and load. I would guess that the windings in most motors are at a temperature of 25C for only a few moments during or right after starting regardless of how low the ambient temperature is (within reason), so I cannot explain the logic behind that one.

 

[rhatcher]

Hey electricpete,

That is a new one on me. However, I may be able to shed some light on part of it. I would guess that the statement "slip will increase by approx 0.35% per degrees C above 25C (winding temperature)" is related to increasing winding resistance with temperature. Keep in mind that the amount of resistance in the stator of an AC motor affects both torque and slip, similar to the effect of resistance in the rotor circuit (actually the same effect as rotor current is reflected to the stator). Low resistance equates to low slip and high efficiency and high resistance equates to higher starting torque but higher slip and lower efficiency. Again, whether this is what they are talking about is a guess based on the source and my interpretation of what the connection is....

I hate to say it based on the source, but my instinct is to disagree with the idea that most motor nameplates are based on a cold motor or a 25C winding temperature motor. That just does not sound right as the measured result would be so transient as to be meaningless. My understanding is that nameplate ratings are (or should be) based on testing at full load, rated ambient temperature, and after achieving steady state temperature.

With respect to using 25C as a standard temperature for anything to do with motors, I am at a loss.... Most motors ratings are based on an ambient of 40C, and will have a winding temperature (obviously) greater than that based on design features and load. I would guess that the windings in most motors are at a temperature of 25C for only a few moments during or right after starting regardless of how low the ambient temperature is (within reason), so I cannot explain the logic behind that one.

 

[electricpete]

Thanks rhatcher - your reaction was the same as mine.

I don't have the document in front of me but it seems there were some conflicting cues as to whether the temperature in question was ambient temperature or winding temperature. Ambient temperature might make a little more sense. I'll look at the document later and report back.

Based on the equivalent circuit, I think that rotor temperature (not stator winding temp) would have the most direct influence on slip vs power. But rotor temperature is of course not an accessible measurement. But worthwhile to keep in mind that slip vs speed characteristics would change dramatically as the rotor cools after starting.... and therefore wait a few hours after start to take slip measurements.

 

[electricpete]

One possible explanation as to how we could determine nameplate speed "at 25C winding temperature" would be to perform equivalent circuit parameter measurements with the machine at 25C. Then compute torque speed curve from the equivalent circuit.

Here are some excerpts from IEEE 739-1995 section 6.16 (with items highlighted in attempt to discern whether we are correcting for ambient or winding temp deviation from 25C):

"[A]Measurement of slip to determine motor load is a useful technique when dealing with the typical energy audit involving a number of motors because the variation in 1- motor nameplate data, 2 - voltage at motor 3 - TEMPERATURE OF THE MOTOR STATOR [emphasis added] may tend to average out on a statistical basis. The slip technique is not nearly as accurate or useful when the load of a single motor is being evaluated.

- The accurancy of nameplate data is adressed in MG-1-1993 section 12.46 The standard states that the variation from the nameplate speed shall not exceed 20% of the difference between sync speed and rated speed, when measured at rated voltage, frequency, AND WITH AN AMBIENT TEMPERATURE OF 25C [emphasis added]. In practice, most manufacturers can hold a 5% tolerance.

[C] - Most motor manufactueres, when specifying full load rpm, take their readings at 25C or on a cold stator. A common error in reading full load rpm with a tachometer WITH THE MOTOR HOT [emphasis added] is that the reading gives a misleading overload appearance.

[D] - The increase in percent slip per degree C RISE [emphasis added] is in the range of 0.342 to 0.38 for motors 10hp and above.

[E]For example, a 10hp 1725rpm motor is running at 2C ABOVE AN AMBIENT OF 25C [emphasis added]. The true full load slip is then calculated as
0.342% per degree C RISE [emphasis added]*20C = 6.84%
1.0684*(1800-1725)=1.0684*75=80.13 full load slip.
The corrected full load speed is 1800-80.13=1719.87rpm"

Paragraph A, C, and D all seem to imply that the temperature in question is winding temperature. Paragraph B, implies that the ambient temperature is the relevant temperature. Paragraph E talks about a rise which again seems to imply winding temperature is the temperature in question. Note that 2C is used in the problem statement and 20C in the calcs... I assume the 2C was a typo since the calculation result can't be a typo..... although 2C would be a credible rise only if we're talkiong about a rise in ambient above 25C.

Given the apparent enormous 20% error allowed by the standard, perhaps any calculation of this sort is doomed?
(Note that voltage affects the calculation but can also be easily corrected for.)

 

[nbucska]

Hi Rhatcher: Have you heard about specmanship ? <nbucska@pcperipherals.com>

 

[electricpete]

In my above message, the place where the bold started was supposed to be marked as paragraph letter B. I guess that when I enclose letter B in square brackets it turns into a font-control character.

 

[electricpete]

According to the equivalent circuit model of an induction motor, the slip at a given power (in the linear range of slip vs power) should be proportional to
Rrotor/(V^2) and relatively independent of the other circuit parameters (doesn't depend on Rstator).

Since the temperature coefficient for copper is 0.4% per deg C, then it follows that rotor resistance of a copper rotor bar circuit should vary in that manner.

I believe that aluminum rotor bars are more common for smaller motors 100HP and under. I'm not sure what the temperature coefficient for aluminum is, but I wouldn't be surprised if it is in the neighborhood of 0.35% per degrees C.

If anyone is interested - here is how I had my computer analyse the circuit:
T := V^2/((R1+R2/s)^2+(X1+X2)^2)*R2/s/wsync

dT_ds := 2*V^2/((R1+R2/s)^2+(X1+X2)^2)^2*R2^2/s^3/wsync*(R1+R2/s)-V^2/((R1+R2/s)^2+(X1+X2)^2)*R2/s^2/wsync

substituting s=0 into the above
RunningTSlope := V^2/R2/wsync

This is the slope of T vs S. Since Speed is constant to within a couple of percent this is also the slope of Power vs S. The slip at a given power is inversely proportional to the slope.

slip ~ Rrotor/(V^2)

 

[electricpete]

The above form for dT_ds is a little unruly for hand calcs using s=0 since s appears in the denominator of some terms. My computer came up with this equivalent expression

dT_ds := -V^2*R2*(-R2^2+R1^2*s^2+s^2*X1^2+2*s^2*X1*X2+s^2*X2^2)/(R1^2*s^2+2*R1*s*R2+R2^2+s^2*X1^2+2*s^2*X1*X2+s^2*X2^2)^2/wsync

 

[jbartos]

Suggestions to electricpete (Electrical) Jul 6, 2001 marked ///\\\:
I have always thought that motor power could be estimated based on linear relationship with slip: 0 power at zero slip; nameplate power at nameplate slip.
///Often presented in texbooks.\\I just saw IEEE 939-1995 (Energy Efficiency) section 6.15 which suggests that slip will increase by approx 0.35% per degrees C above 25C (winding temperature).
///Assumptions are missing since more variables are involved, e.g. shaft power or torque, etc.\\They go on to state &quot;Most motor manufacturers, when specifying the full-load RPM, take their readings at 25C or on a cold motor&quot;.
///This is considered a room temperature since the test engineers are not supposed to be shivering and meters are supposed to be calibrated at about this temperature. Normal motor service conditions are from 0°C to 40°C. &quot;Golden middle&quot; is 20°C or perhaps 25°C to make it more compliant with the room temperature. Incidentally, water cooled motors service conditions are for the ambient temperature range from 10°C to 40°. Then, the 25°C fits nicely in the middle. See Smeaton R.W., &quot;Switchgear and Control Handbook,&quot; Second Edition, McGraw-Hill Book Company, 1987, page 9-9\\That's the first I've ever heard of it. What use is a ficticous number based on 25C winding temp?
///It is not so fictitious. Just imagine various exaggerations, e.g. 0°C\\Winding temp is sure to increase above that during starting and running.
///Yes, it means the conductor temperature.\\Am I reading this right?
///Almost.\\

 

[electricpete]

jbartos - thanks for the info. I may not have explained it well, but as shown in the above numerical example, the IEEE 739 method does take into account the load in computing slip, and then applies a temperature-based correction factor. Also they suggest an additional correction inversely proportional to square of the voltage (as fraction of rated voltage).

Some more possibly relevant info in IEEE 112-98 section 4.3.2.2. &quot;Slip measurements should be corrected to the specified stator temperature as follows:
S_s = S_t(t_s+k)/(t_t+k)
where S_s =is slip corrected to specified stator temp t_s
S_t = slip at measured stator winding temp t_t
t_s = specified stator temp (section 5.1.1)
k=234.5 for 100% IACS copper, 225 for aluminum, based on a volume conductivity of 62% (based on ROTOR conductor material).&quot;

Section 5.1.1 addresses computation of stator losses... near rated stator winding temp. The above does not specifically address temperature associated with Nameplate slip, but apparently NEMA MG-1 does (25C). But this IEEE112 calculation does better illustrate the same method and the underlying logic (that rotor temperature is the temperature of importance but is assumed to be equal to stator temperature for lack of better info).

 

[jbartos]

Suggestions to the previous posting marked by ///\\\:
jbartos - thanks for the info. ///You are welcome.\\\ I may not have explained it well, but as shown in the above numerical example, the IEEE 739 method does take into account the load in computing slip, and then applies a temperature-based correction factor. Also they suggest an additional correction inversely proportional to square of the voltage (as fraction of rated voltage). ///It is appropriate to consider all relevant variables when it comes to analysis of relationships. If one considers motor torque- speed approximate equation for the speed with low slip, then there will be several potential parameters/variables that could be considered beside temperature (implied in some parameters/variables).\\Some more possibly relevant info in IEEE 112-98 section 4.3.2.2. &quot;Slip measurements should be corrected to the specified stator temperature as follows:
///IEEE Std 112-1984 par 5.4.2
Ss = St(Ts+k)/(Tt+k)
where Ss =is slip corrected to standardized stator temp Ts
St = slip at measured stator winding temp Tt
Ts = specified temperature for resistance correction (section 5.3.1)
Tt = observed stator winding temperature during load test
k=234.5 for 100% IACS copper, 225 for aluminum, based on a volume conductivity of 62% (based on ROTOR conductor material).&quot;
Par. 5.3.1 The specified temperature used in making resistance corrections (see 5.2.2.5.4) shall be determined by one of the following, listed in order of preference.
1) Measured temperature rise by resistance from a rated load temperature test plus 25°C
2) Specified temperature given in and applicable product standard, such as ANSI/NEMA MG1-1978. This is related to the stator loss RI**2 which is modified to:
a) 1.5 x RI**2 for 3-phase
b) 2.0 x RI**2 for 2-phase
Par. 5.2.2.5.4 Total losses in efficiency formula equal the sum of the stator and rotor RI**2 corrected to the specified temperature for resistance correction, Ts, core loss, friction and windage losses, and stray load loss. When the rated load temperature rise has not been measured, the resistance of the windings shall be corrected to the following temperature:
Class of Insulation System Temperature °C
A 75
B 95
F 115
H 130
This reference temperature shall be used for determining RI**2 losses at all loads. If the rated temperature rise is specified at that of a lower class of insulation system, the temperature for resistance correction, Ts, shall be that of the lower insulation class, e.g. Ts = 25°C + 95°C for room temperature 25°C and the motor with insulation class B temperature rise 95°C.\\Section 5.1.1 addresses computation of stator losses... near rated stator winding temp. The above does not specifically address temperature associated
with Nameplate slip, but apparently NEMA MG-1 does (25C). But this IEEE112 calculation does better illustrate the same method and the underlying logic
(that rotor temperature is the temperature of importance but is assumed to be equal to stator temperature for lack of better info).
///See the previous comments.\\

 

[electricpete]

jbartos - I'm glad you're interested in discussing this item. I was looking for feedback on my understanding. But I don't understand your latest message. Your comments in bold are followed by my response.
It is appropriate to consider all relevant variables when it comes to analysis of relationships. If one considers motor torque- speed approximate equation for the speed with low slip, then there will be several potential parameters/variables that could be considered beside temperature (implied in some parameters/variables).
Are you saying that important variables have been overlooked? The approach outlined above is that
S(P,Twdg,V)=Snp*(P/Pnp)*0.0035(T?-25C)*(Vnp/V)^2
where:
P = actual load (SHP)
Pnp = Nameplate load
Twdg = winding temp
T? = winding temp per IEEE 739 (although I would use ambient temperature here).
V = actual terminal voltage
Vnp = nameplate voltage
S(P,Twdg,V) = actual slip under actual conditions measured for this calc.
Snp = nameplate slip = syncronous speed minus nameplate speed.
(In my application, measured S is known and the above equation would be rearranged to solve for an estimate of the unknown actual shaft load P).
Are you suggesting any other variables that significantly impact operating slip? I can see that perhaps voltage imbalance and harmonics might come into play on very small scale... but much lower than the order of magnitude of the other errors (particularly the potentially huge correction for temperature if we follow IEEE739... and the huge 20% error allowed by NEMAMG1).

Par. 5.3.1 The specified temperature used in making resistance corrections (see 5.2.2.5.4) shall be determined by one of the following, listed in order of preference.
1) Measured temperature rise by resistance from a rated load temperature test plus 25°C
2) Specified temperature given in and applicable product standard, such as ANSI/NEMA MG1-1978. This is related to the stator loss RI**2 which is modified to:
a) 1.5 x RI**2 for 3-phase
b) 2.0 x RI**2 for 2-phase
Par. 5.2.2.5.4 Total losses in efficiency formula equal the sum of the stator and rotor RI**2 corrected to the specified temperature for resistance correction, Ts, core loss, friction and windage losses, and stray load loss. When the rated load temperature rise has not been measured, the resistance of the windings shall be corrected to the following temperature:
Class of Insulation System Temperature °C
A 75
B 95
F 115
H 130
This reference temperature shall be used for determining RI**2 losses at all loads. If the rated temperature rise is specified at that of a lower class of insulation system, the temperature for resistance correction, Ts, shall be that of the lower insulation class, e.g. Ts = 25°C + 95°C for room temperature 25°C and the motor with insulation class B temperature rise 95°C.
#1-Your paragraph numbers don't correspond to IEEE 112-1998.
#2-As I stated above, all of the temperatures related to insulation class discussed in IEEE112 above are used for calculating losses and efficiency. To my knowledge, none of the reference temperatures in IEEE112 have anything to do with the temperature associated with the speed listed on the motor nameplate (if I'm wrong then please let me know). The only source of info I know of regarding that temperature is NEMA MG-1-1993 section 12.46; or from more recent MG-1-1998 Revision 1 section 12.47, which states: &quot;The variation from nameplate or published data speed of alternating current single-phase and polyphase medium motors shall not exceed 20% of the difference between syncronous speed and rated speed when measured at rated voltage, frequency and load and with an AMBIENT [emphasis added] temperature of 25C).&quot;

That statement from NEMA seems to clearly indicated that rated speed is based on normal full-loaded conditions (with ambient of 25C). Therefore, &quot;T?&quot; for correcting slip data (see equation above) should be ambient temperature (assuming motor has reached stable temperature near full load). This provides a correction for any deviation in ambient temperature beyond the assumed 25C. We might get a little more sophisticated in estimating actual winding or rotor temp, but most practical approaches I can think of will provide a relatively small correction (compared with the huge IEEE739 discussed below).

However it seems clear to me that the author if IEEE739 intended for us to use our best estimate of Twinding in place of &quot;T?&quot; in the above equation (as was done in their example). This gives a very large correction to nameplate slip... on the order of 80C*0.35%/C ~ 28% of nameplate slip value... which seems unjustified.

I think that rhatcher, jbartos and I are all in agreement on the basic approach (nameplate slip can't possibly be based on 25C winding temp). Personally, I think that the approach suggested by section 6.16 of IEEE739 is a lot of hot air.

 

[rhatcher]

Hi electricpete,

I have been away for a while and have obviously missed much. I must say that the consensus is correct that rotor resistance is usually thought of as the more influential factor in slip than stator resistance... In fact, it is presented in one reference that stator resistance may be disregarded for some generalized calculations. I am still reviewing the details of my references to get a feel for the magnitude of the effect of stator resistance as it does come into play to some degree. How much (and why) is the question. I do not have my references in front of me now and if I did, I do not think I could answer directly at this time as I am still in thought on this.

At this time it is clear to me in a common sense way that stator resistance is important in determining the amount of voltage available for generating flux, which in turn is proportional to torque and slip. It is also clear to me in a mathematical way that for equivalent circuits, stator resistance is usually associated with core loss and rotor resistance is usually associated directly with torque. I am trying to resolve my common sense (gut) feel with the equivalent circuit to see if the IEEE quoted difference makes sense. ( I will warn you that based on time, available references, and mathematical ability it may be difficult for me to do so without more info...)

In looking at the data as presented, I am keeping in mind that the referenced difference is 0.342% per degree celcius, meaning for a +/- 1C variation a per unit multiplier of + 1.00342/-0.99658 would be applied respectively. This is not much at all. My feeling is that the increased resistance losses (I²R) of the stator with increased temperature will not only affect the efficiency to an small degree based on heat loss but will also affect the slip based on the reduced available flux generating voltage in the air gap.

To the next point though, I am still at a loss to resolve the question of the 25C standard ambient temperature. The quoted material from IEEE seems contradictory on a few points which don't need to be brought up again. In particular though, I would agree with the idea that for a good design that speed should not vary more than 20% from nameplate speed when measured at 25C ambient, but I still believe that nameplate speed is based on ambient of 40C (unless otherwise specified) and full load with the expected temperature rise of the winding based on insulation class. To me I take this clause to mean that if you operate the motor at a lower ambient that the efficiency will be somewhat improved and slip somewhat reduced from nameplate value.

I will warn all reading this (after the fact) that I am not an expert and that I have been on vacation for a while so my mind is even more scattered than normal....

 

[electricpete]

rhatcher- I'm glad to have someone to bounce these ideas off of. Here's some more discussion about the role of rotor resistance (vs stator resistance).

There was a result above which was derived from the equivalent circuit model of motor (neglecting the magnetizing branch):
RunningTSlope = V^2/R2/wsync
Meaning the change in Torque per change in slip fraction is equal to V^2/R2/wsync. The units work out because V^2/R is watts and power/rotational speed=Torque. (slip is unitless).

One factor which makes this reasonable is that for low values of s approaching zero, the equivalent circuit resistance R2/s will become very large and will dwarf any of the series resistances. The range of s where R2/s >>> Z1+sL2 defines the range where we can consider Torque to be a linear function of s.... which is generally assumed to encompass the operating range.

As s increases beyond this range, the series impedances Z1+sL2 will become significant compared to R2/s, creating a non-linear portion of the curve.

Here's another intersting thought experiment.

You're probably aware the rotor field travels at syncronous speed. (the currents in rotor are at slip frequency... rotor field rotates at slip speed relative the the rotor and at slip speed + rotor speed = synconous speed relative to stationary reference).

If the frequency of the rotor field as seen by stator is constant, then the only way the stator &quot;knows&quot; what is going on in the rotor (speed and load) is by the magnitude (not frequency) of the rotor current. While the EQUIVALENT circuit has a relatively constant voltage applied to rotor, a constant X2 and a slip-varying resistance R2/s, the ACTUAL physical rotor has a slip-varying voltage sV0 applied, a slip-varying X2 (sL2) and a constant resistance R2.

At small slip, X2<<R2, and rotor current is approx
I=(V0*s)/R2.

Now compare this case to another case.... Take the same stator and replace the rotor with another rotor of higher resitance R2'=2*R2 AND operate it as a lower speed quivalent to 2*s, the current will remain the same
I=(V*2s)/(2R2). Since rotor current magnitude is the same and freqquency is the same, the stator will react the same and draw the same power Pin these two situations. Assume for purposes of power calculation that the speed is roughly constant (Since speed hasn't changed by more than a few percent). That means that the T=P/(2PiSpeed) is roughly the same in these two cases. Draw the TvsS curve... these two cases have the same T at differing speeds of s and 2s. The &quot;slope&quot; of the T vs s curve in these two cases differs by a factor of 2 which corresponds to the difference in rotor resistance.

Not a whole lot of practical value, but an interesting excercize.

 

[jbartos]

Suggestions to electricpete posting on Jul 12, 2001:
1. The temperature T at S(P,Twdg,V) in (T?-25C) is winding temperature per IEEE7939 since the ambient temperature is supposed to be in 0 to 40C for aircooled motor and 10 to 40C for water cooled motor.
The temperature rise is linked to NEMA insulation Class and it is uniquely set by the manufacturer per each motor type.
Then, Twinding = Trise + Tambient which must be smaller or equal to the peak, which is for Tambient = 40C. Since the motors may have different Trise that the Twinding will be different. Also, the T ambient is used different from Tambient = 40C. Usually, Tambient = 25C
Then, if you are investigating the motor parameters, Tambient is somewhat arbitrary in 0 to 40C for aircooled motor, however, it is used Tambient = 25C to meet standards.
The slip relationship with respect to the mentioned parameters or variables appears to be reasonable.
2. There is a different approximate slip relationship for the slip near the slip equal to zero and different relationship for the slip from 0 to 1. Now, it depends, which relationship you are interested most. I think, it is the relationship for slip near 0.
3. I did not have IEEE 112-1998 available so I used IEEE 112-1978, which should not deviate much from the more recent one.
4. The motor temperature associated with other motor nameplate parameters is the NEMA Insulation Class Temperature which is linked to Motor Temperature Rise, Trise, and it is unigue for each motor type having the same NEMA insulation Class.

 

[electricpete]

In previous discussion I made the assumption that R2/S was much larger than all the series impedances in the equivalent circuit for s small... which I assumed to be the linear range of T vs S.

But this leads to a contradiction.
It predicts that I = V/(R2/S) = sV/R2
And Power=R2I^2 = (S^2)(V^2)/R2.

But this gives power proportional to slip squared (for small s), which does not make sense. Likewise, since speed is approx constant for small s, torque will be proportional to power which is proportional to s^2. Where did I make a mistake?

 

[rhatcher]

Hi electricpete,

There are a few things I want to touch on, starting with the previous post and going back.

I₂ = E₁/(R2/S) = SE₁/R2

From the equivalent circuit, power across the air gap is

P = qI₂²(R₂/S)
where &quot;q&quot; is the # of phases.

Your mistake was in the substitution of the equivalent circuit variables into the basic Ohm's law equation &quot;P=I²R&quot;. In this case R is equal to &quot;R₂/S&quot;, P is the power transferred across the air gap, and V is the counter-emf &quot;E₁&quot;, and I is the reflected rotor current &quot;I₂&quot;. Doing the substitution as you intended:

P=I²R = {SE₁/R₂)²*{R₂/S) = SE₁²/R2

Sorry, but I just realized it is quitting time here at work (aka &quot;happy hour&quot. I will get back to you later with some more thoughts. I hope this helps so far...

PS: it is quite tedious to post equations here so that they make sense...

 

[electricpete]

rhatcher - you are right on the money. I owe you a star for setting me straight.

The equivalent circuit does preserve the proper magnitude of I2. I could use R2*I^2 for calculating I^2*R losses in the rotor (I have been playing around with rotor heating calcs during startup lately... which is probably where I got my wires crossed). Total airgap power requires us to use resistance R2/S as you said. Power output would be related to the difference I^2*(R2/S - R2), which is pretty darned close to I^2*R2/S for small S. Power is proportional to s, and all is well.

By the way.. what font control code do you use for subscripts or superscripts

 

[rhatcher]

I am not in a position to continue the technical aspects of this thread since I have been to happy hour and am going to continue the party very soon. (X-)) However, I can tell you that there is a variety of options available for manipulating text and, if you are interested, adding smiley faces to posts in this forum. Simply select 'preview post' below the window where you type your post. You will see a window showing your post as it will appear on the forum. (By the way, I find this useful for proofreading.) Scroll down and at the bottom of your screen you will find a section of the page titled 'Editing Tips'. In this section is a link (called 'here') for both 'TGML Tags' and 'Emoticons/Smileys'. TGML Tags shows how to show text in bold, italics, superscript, subscript, etc. Smiley faces are what they are.

Cya later