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It looks like it could be that Im in (a) is not the same as Im in (b), so Im_b<Im_a.
So they should have called Im in (a) Im1, and in (b) Im2. That would explain the reason for the lower values than calculated. They do in fact state that the Im is the maximum current, and in (b) it is less than in (a).
I took a real quick look though.
Also, it is usually not possible to discern when or when not they are using a sine wave based on a graphic drawing. Even if you drew a perfect sine wave over top of the graphic they gave you and it matched up perfectly, that still would not mean it was a true sine wave. It's only a sine wave when they state it is a sine wave or the context clearly suggests that it is. Here, they specifically state that it is an arbitrary waveform, otherwise we'd have no trouble calculating the RMS and average values.
You should have no trouble deriving the formulas yourself. You know the definition of RMS and average, which are just integrals. Hence, you can determine any of the values (for example f0=Irms/Iave) as exactly as you like and you don't need to use a square pulse approximation. Just use the definitions given, and apply them using your knowledge.
The problem is not about deriving the formulas. I don't even know what figure they have based their calculation about conduction angle on. The form factor I found, using the formulas given in the text, for 80 degrees conduction angle comes to be sqrt(1.4/3.142)/(1.4/3.142)=1.5 which is much different from the stated one, i.e. 2.3.
Anyway, I have tried it again but still the end result is quite different. I'm sorry if my method is wrong. Thank you.
Also, I don't think you are using the correct duty cycle in the book's formula. Because the signal is a rectified sine wave, the period is pi and not 2pi.
EDIT: I attached a calculation I did in mupad using the book formula and the proper integrals. My interpretation is that you need to you the angle 80 degrees compared to a full 360 degrees for the period of pi for the rectified sine wave. That's the best I can do to resolve the discrepancy.
Steve:
I have to agree with PG here that they dont want us to assume that we know the waveform, so it makes for a more interesting but also more inaccurate exercise. This kind of thing would occur on the floor of a shop where the tech is looking at the waveform on a scope and cant really tell what analytical waveform it has but still wants to be able to estimate the life of the diode.
Steve and PG:
I think you are both missing the point of my last post or you would not have any questions left, or maybe you just want to search for another possibility too. But if you do some calculations you'll find that none of the formulas match up with the results they show in the table, so that's why i came to the conclusion i did.
The main point is that the variable "Im" is a variable even within the context of the text. It is not a "constant variable", and certainly not a "constant", it's a total "variable".
Let me try to illustrate this...
We have a variable "r" written in a text somewhere. Sometimes it is a constant though even though they did not capitalize it as in:
v=i*r
but later in that same text they talk about the area of a circle:
A=pi*r^2
Now is 'r' in the first example the same as 'r' in the second example? Obviously it can not be. But it is also possible that in a more unknown text we might see:
x=i*r
and then later in that same text:
x=i*r
and the two 'r' are still different r's, even though they should not have been written in that somewhat confusing manner.
We see this from time to time, and this is usually referred to as "reusing" a variable. The reused variable is not exactly the same as the old one in that although it has the same general meaning it has a different numerical value.
A simple example would be for a circle:
D=2*r
where r is the radius, and the text tells us that we are calculating the diameter of a circle, and it might come out to 22 lets say, and that means that r=11. Then later in the text we see the same thing:
D=2*r
but what they want now is for us to calculate the diameter using a different r, so we come out with a different result to the same statement but they only show the same statement, not the values outright:
D=2*r
We are expected to know that in the second case r=7 not 11 this time.
I could be wrong of course, but it does seem that it is that simple and after all they do state that "Im" is the MAXIMUM value of the wave, and if you look at the two drawings one is clearly higher than the other.
Also, the values are lower than the otherwise 'expected' values by an amount that seems to agree with the drawings, if we use the max Im although i admit the drawings are not that great.
Again, i could be wrong because the context is limited and that's the way some of these text's are written unfortunately.
I also do not agree with this kind of writing. If there is the possibility of misinterpretation i like to make it more clear myself, but this is exactly why we have dialog, so that we can go back and forth to hash out what is really going on sometimes because it's not that clear to begin with. It's unfortunate that with a written book we dont have this option usually. Perhaps in the future with electronic texts we will have the option to ask questions directly to the author through some fast mechanism that isnt as bad as email.
I am attaching a drawing of this to illustrate how i think this should be interpreted. Note that you should ignore "Im" alone in the drawing of the two waves and just pay attention to "Im1" and "Im2", and also note that the red line shows the max amplitude of the second graph matches the max amplitude of the first graph, and that the factor between the two max amplitudes is roughly 90 percent, which matches the calculations vs the table pretty closely. So the table entries are correct if we just agree to believe that they did it right but they wont tell us the actual amplitude of the second graph numerically (unless of course we calculate it ourselves).
I think the actual factor between the table and the calculations assuming no change in amplitude would be roughly 1.3/1.4142 but that table value of "1.3" could have been rounded up or down so this (which comes out to 0.919) is very approximate. Try that with the other values and see if it works.
It is interesting though that for 180 degrees Im should be the max shown in the first graph. But if we use the max shown in the second graph, we get the right result
You may be right. I find book descriptions like this frustrating because they are not being clear on definitions. It may be as you say, or perhaps something else they defined is not right. However, we can bypass this type of thing just by assuming that the form factor definition is correct, and then using fundamental definitions of RMS and AVR with the correct limits as we define them for ourselves. I realize this does not help PG who may need to pass a test that uses the book as a reference, but so many books have mistakes that we are usually better off rechecking everything ourselves anyways. I could give so many examples of famous books riddled with mistakes.
I updated my post above to show the comparison at 80 degrees. I can run that program and make a table for comparison.
Assume angle is relative to 360 degrees for one period of a rectified sine wave. For example, their 80 degrees is 80 degrees out of 360.
Angle (deg)......Book-Table........Integrated-Value.......Book-Formula ....20......................5........................4.90......................4.24 ....40......................3.5.....................3.46......................3.0 ....60......................2.7.....................2.82......................2.45 ....80......................2.3.....................2.43......................2.12 ....100....................2.0.....................2.16......................1.90 ....120....................1.8.....................1.96......................1.73 ....140....................1.6.....................1.81......................1.60 ....160....................1.4.....................1.68......................1.50 ....180....................1.3.....................1.57......................1.41
These don't agree, but there is a correlation. So, I don't know for sure what the issue is, and it could be what you said. However, what I can say for sure (assuming I didn't make a dumb mistake) is that the Integrated-Values are correct because they are based on the right RMS and AVR values. This was the gist of my first post here. We can derive the formulas for themselves. Now, I did this numerically for the table, but it can be done exactly too. I let mupad do it out for me and I attached the formula as a picture. Note that angle is theta in degrees.
MrAl: Thank you for the detailed analysis. But the author had stated that the Im in both figures had equal magnitude.
Steve, your answer for 80° conduction angle is pretty close to the given one. Could you please give it a look to see where I went wrong with my approach? Thanks.
I think MrAL may be right about the Im value. The text is still ambiguous, but what he describes makes more physical sense.
As far as what you showed, I believe the limits in your integration are wrong and you seem to be using the wrong duty cycle. Here, D=to/T which would equate to 80/360, not 80/180.
See if this resolves your problem, or maybe I just misinterpreted what you did.
Edit: After posting it occurred to me that they are discussing SCR ratings, not TRIAC ratings. If a TRIAC were being used, it would carry both halves of the AC current, but if SCRs are in use, one would need two of them for the full AC waveform, and each SCR would be carrying the "half wave" waveform. This would mean that my table for the "half wave" waveform would apply.
End Edit.
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It's not clear to me whether table 4-2 in the image is intended to apply to a "full wave" phase controlled waveform:
or a "half wave" waveform:
I calculate the value for the "full wave" waveform like this, where the variable "ca" is conduction angle:
And I get this table of values for the "full wave" waveform:
But for the "half wave" waveform, I get these values:
Perhaps the values given in figure 4-2 were calculated before the days of easy access to computers. Maybe whoever did the calculations used a slide rule!
Yes, he may be. But I just thought that I should point out that the author had assumed Im to be the same in both figures. Thanks.
Electrician: Your reply was very helpful. Now I see my approach wasn't wrong. It's just that I was using "full wave" waveform, and yours and mine answers are the same for 80 degrees conduction angle, 1.6788. The text and Steve were using "half wave" waveform and that's why my answer wasn't matching with theirs. Thanks.
MrAl: Thank you for the detailed analysis. But the author had stated that the Im in both figures had equal magnitude.
Steve, your answer for 80° conduction angle is pretty close to the given one. Could you please give it a look to see where I went wrong with my approach? Thanks.
PG:
Well at this point i was looking for what could cause the values in the Table to be so far off, so i was also looking for a method that might have led to the error in the Table. This means i looked at many different things including the possibility that they used a function like sin(wt)^2 instead of sin(wt). But what i found never made any sense, so i am starting to think that this was just one of those big mess ups in the text, as i'll explain a bit more in a second.
Steve:
Thanks, but now i am suspecting that there was a big error in the Table simply because of that 180 degree entry. I'll explain more.
Steve & PG:
Because of the 180 degree entry i have to suspect the entire Table, and that could be the whole reason for the error. The logic behind this suspicion isnt too complicated either.
If we take their text formula as is without question and simplify it, we end up with the same formula that can be found in other places:
fo=RMS/AVG=sqrt(T/t)
where fo is the "form factor" as described in the text. Note Im disappeared, and we end up relying on the average value for the scaling later.
Now if we take a rectangular wave (which is what they are talking about all along) and calculate the form factor for a 180 degree conduction angle and peak amplitude 1, we can for example set T=2 and t=1 and we get:
fo=sqrt(2)=1.4142 approximately.
Now we know that the RMS value of a 180 degree conduction angle rectangular wave is 1/sqrt(2)=0.7071. What we need is to get that value from knowing the average AVG and the form factor fo. This would simply be:
Estimated_RMS=AVG*fo
The rectangular wave with 180 degree conduction angle has AVG=0.5, so we get:
Estimated_RMS=0.5*sqrt(2)
and that equals 1/sqrt(2) also.
So we got the right result for the 180 degree square wave using the form factor and average value of the 180 degree square wave. So there should be no doubt whatsoever here, at least for this one example.
So knowing we got the right result for 180 degrees (and they did want us to use a rectangular wave for the estimation, not the true value of the RMS of a given wave) we compare fo computed with f0 from the Table and we find that they are simply NOT the same values. Thus, i can see no way that the Table can be relied upon to be used when calculating the estimated RMS values using the procedure they outlined. It only takes one wrong value to screw up a table, and so i suggest that we try to find another table and ignore this one, or simply make our own table. We can vary the conduction angle from 20 degrees to 180 degrees as they did, and maybe Steve did that already.
Another point is that in today's world of programmable calculators and computers this method is probably deprecated anyway simply because it's not that hard to measure two or more points on the wave and then estimate based on the RMS values of the resulting triangular areas. We can then decide how much safety factor we want to include in the design rather than settle for some arbitrarily introduced value from a method that sometimes will come out overrated and sometimes come out underrated. You can make a table of both methods using say a sine wave and see the difference, then decide for yourself after comparing values that come from wide pulses and values that come from short pulses and see what the difference is.
o knowing we got the right result for 180 degrees (and they did want us to use a rectangular wave for the estimation, not the true value of the RMS of a given wave) we compare fo computed with f0 from the Table and we find that they are simply NOT the same values.
It's not clear that the table is derived from the rectangular approximation. The very last thing they say at the bottom is "Table 4-2 gives the form factor as a factor of conduction angle (theta). The conduction angle is the duration for which the SCR is on. It is measured as shown in Figure 4-7"
Figure 4-7 shows a phase controlled half sine wave. One might infer from this that the table values were calculated from a half sine, not a rectangular pulse. So as in so much of the rest of the note, it's not clear what they have done.
Here is a table showing form factors derived from both the half sine and from a rectangular approximating pulse. The second column is for the half sine and the third column is for the rectangular approximation. I was surprised to see that the fo values for the rectangular pulse are less than for the half sine, because near the beginning they say that using the rectangular pulse gives some safety factor since the RMS value of the rectangular pulse is greater than that of the half sine.
Both columns are less for some conduction angles, and greater for some, than the values in table 4-2. I think they just didn't do their arithmetic very well. They do say near the beginning, "Determining the RMS value for a nonsinusoidal waveform like the one shown in Figure 4.6 is quite difficult." Fifty years ago it was definitely more difficult than now.
Thus, i can see no way that the Table can be relied upon to be used when calculating the estimated RMS values using the procedure they outlined. It only takes one wrong value to screw up a table, and so i suggest that we try to find another table and ignore this one, or simply make our own table. We can vary the conduction angle from 20 degrees to 180 degrees as they did, and maybe Steve did that already.
Another point is that in today's world of programmable calculators and computers this method is probably deprecated anyway simply because it's not that hard to measure two or more points on the wave and then estimate based on the RMS values of the resulting triangular areas. We can then decide how much safety factor we want to include in the design rather than settle for some arbitrarily introduced value from a method that sometimes will come out overrated and sometimes come out underrated. You can make a table of both methods using say a sine wave and see the difference, then decide for yourself after comparing values that come from wide pulses and values that come from short pulses and see what the difference is.
I should point out that there is a difference between what I did and what TheElectrician did. I considered the rectified sine wave to be a full wave sine wave with period of pi. Hence, I considered the conduction angel to be relative to pi being 360 degrees, not 2pi being 360 degrees. You can see that our tables are different, even though the numbers are similar, and our formulas are different, even though they look similar.
Don't ask me which way is correct because the text is ambiguous in too many ways for my liking. However, if I were to define it myself, I would consider the repetition period to be 360 degrees.
I think that whoever created the table just didn't have the numerical calculating power available that we have today. The calculations are, as they said, quite difficult, and whatever method they used, slide rule or log tables, substantial error accumulated.
Hi PG,
I looked at your problem and I thought the text was rather confusing.
The subject seems to me, to be about determining the internal power dissipation in a Silicon Controlled Rectifier. The text introduces the concept of form factor in order to show that conduction angle and average current are central to calculating the power dissipation in and SCR.
This kind of data has been made available in data books for many years; like since SCR's were available commercially. I suggest you have a look at data for say a BTW23 and a BT151.
The authors of the text would have made more sense of the subject had they referred to some real life information. To me, the introduction of a square wave-form was completely nonsensical. Most SCR applications use sine wave sourced power supplies because, in the control of serious amounts of power, the power source is the AC power mains. There are some exceptions of course, where SCR's are used to switch a DC supply, and in this case, the SCR has to be force commutated to switch it off. In modern usage, IGBT's have replaced SCR's, because of the ease of turning them off.
Coming back to the BTW23, the calculation of dissipation is given for a range of conduction angles and average current. There is a tabulation which shows that the tabulation in your text example is for a sine wave form and is based on a 'per half cycle' (see BT151). The same page of the datasheet shows the size of heat sink required, and this is the fundamental need of the subject of your text. The BTW23 datasheet shows also, the device dissipation for DC operation (form factor of 1 ).
I think you will gain more good knowledge of your subject if you look at some real device data and then try your maths on dissipation calculations (see your post #4)
Yes, i agree it is not clear what they used to develop the table, but from all of our investigations none of us can determine exactly how they made the table or what waveform they used. That's the main reason why i was suggesting that the table be ignored except in the case where the instructor wants to follow the table for some reason and then i can only suggest that he be queried directly about how to construct the table or why it should be used at all. The table is in error by an amount which i consider too large to be a simple rounding or similar error for the day and age the book seems to have been written, but you may still be right about your conclusion about the reason for the miscalculation being an untimely one because the author could have copied that table from a much older reference. Only a question to the author could clear this up i believe.
I also agree that we might infer that the table values were calculated from a half sine rather than a rectangular, but that doesnt change my conclusion that the table values are incorrect by an unacceptable amount.
Why bother with tables...
I had mentioned in a previous post that i would not do it this way. I suggested at least a two point method that would measure the wave at at least two points (per slope) and use that to estimate the RMS values, then decide separately (and more reasonably) what the safety margin should be.
But the purpose of still talking about the table was not to suggest that we always use this kind of table, but one of the questions was referrenced to the table so myself and others thought it would be a good idea to try to prove or disprove the table, then go from there. A table for sine waves (or half sines or whatever) would certainly be a good idea for techs working on thyristor circuits in line operated equipment where they would like to quickly estimate the RMS values to double check the device rating, perhaps just to get a feel for how the device will behave temperature wise.
I like the idea of the digital scope too, as long as their company can afford that kind of thing. Lucky today they are cheaper than they were when i worked in the industry.
I am including a table here also. This table contains all four methods of how we might have viewed the table, including the original table values. Here we can quickly glance at any of the four possible calculation methods and see that the table does not agree with any of them. Thus i believe this concludes that the table is just simply in error so should be discarded.
I will post the table here and right after that explain what the entries are...
(see attachment for a better picture of this)
The major headings SINE, RECTANGULAR, are for interpreting the waveform as either sine or rectangular, and the TABLE is the original table values.
The left most column is the conduction angle, which applies to either of the other colums.
Just under SINE and RECTANGULAR appear 180 and 360 for the sine and rectanglur waves. "180" is the column that is for data that is calculated considering the sine wave to be full wave rectified but the period is also 180 degrees, so essentially we have a sine that is a full wave rectified sine and we conduct once every 180 degrees, not once every 360 degrees, where sub headings 180 degrees and 360 degrees refer to the original line frequency sine wave. So for a conduction angle of 20 degrees, the first column 3.4571 refers to the form factor for a full wave rectified sine where the conduction occurs at both 0 to 20 degrees and also at 180 to 200 degrees. The second sine column for 20 degrees we see 4.8891, and that refers to the form factor for a full wave rectified sine but here the conduction only occurs between 0 and 20 degrees and nowhere else over the full 360 degree original sine wave line waveform.
The values are all rounded to 5 signifcant figures.
As you can see the table had to be made using 'dots' to provide spacing because the editor isnt working right again. I've included a picture which looks better
The table is in error by an amount which i consider too large to be a simple rounding or similar error for the day and age the book seems to have been written
The image we're discussing is obviously much older than the 1998 publication date of the book that PG1995 got it from; the book no doubt reprinted it from old SCR application information. It looks like application information from a General Electric data sheet from the early days of SCRs. I suppose we could ask the book's author, Ashfaq Ahmed, but I'm pretty sure he didn't produce it in the late 90s.
When I said "...they didn't do their arithmetic very well.", I had in mind much more than a single rounding or other error. To calculate the table values using a half sine requires integration. Imagine doing a numerical integration with a slide rule, using perhaps the trapezoid rule for the numerical integration. Many calculations are required, with many rounding, or other, errors along the way. The table values only have 2 significant figures, which suggests they are the result of accumulating many slide rule calculations.
Column 3 of your table, which is identical to column 2 of my table in post #13, comes closest to matching the values in table 4-2 of the image. The discrepancies are greatest for large conduction angles. Consider that for the large conduction angles, more data points would have to be calculated for the numerical integration, so we might expect more accumulated errors, and a worse final result.
I like the idea of the digital scope too, as long as their company can afford that kind of thing. Lucky today they are cheaper than they were when i worked in the industry.
A digital scope suitable for work on grid frequency circuits can be had for well under $1000; google "Rigol". Any company can afford that. Analog scopes are almost unobtainable as new instruments; one has to get used ones on eBay.
Yes the image might have come from an older reference, and i mentioned that in my previous post. And i mentioned asking the author so that we could find out if he did in fact copy it from an older reference. But since he most likely has experience in the subject area we could also see if he would want to correct the entries too.
But i can not agree that with a difference of 1.3 versus 1.4142 is a simple accumulated error of some type. The integration is easy enough to do on a half sine wave waveform, without the need for numerical integration. I would expect the original author (old or new) to be able to do that without too much trouble.
To get a value of 1.3 the calculation would had to have resulted in a more accurate result range from 1.25 to 1.34999..., so it cant be a rounding error either.
Even if it was in fact a slide rule error of some type that still suggests that we dont use the table for anything.
But i think the main point is that we see an error but we just dont know where that error came from, so ignore the table for now, that's all. And as Steve suggested we can calculate our own table and make it as accurate as we want.
If the scope is cheap enough for the company to buy, then the chances are they already have one. Whether or not they have that feature though is another question so they might have to buy another one which they may or may not want to do. There's too many variables though to say for sure who wants to buy one and who doesnt.
But the subject of this thread was to figure out what how the form factor related to the conduction angle and the waveshape and i think if the table was more accurate this would have been solved a long time ago.
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