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Forum Index : Electronics : Another Saturation Tester Design
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wiseguy Guru Joined: 21/06/2018 Location: AustraliaPosts: 1156 |
Instead of a heap of words followed by a schematic I will do it in reverse, so here is the schematic. I have been prompted to provide some details so here it is. Or for a clearer image 2019-02-17_102257_ST_Schematic.pdf Why do I want a saturation tester? I consider the output choke to be a very important component in the OZ inverter topology if maximum efficiency is to be achieved. I still make reference to the OZ inverter as the power elements are essentially unchanged for the toroid design, bridged Mosfet power stage and a 50Hz modulated HF PWM driving section, regardless of whether it is driven by a nano or an EG8010/EGSS002. I will be choosing the nano for my modulated pwm driver. I am going to have to provide the rest of this text in installments due to other conflicting time restraints sorry.... TBC If at first you dont succeed, I suggest you avoid sky diving.... Cheers Mike |
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Warpspeed Guru Joined: 09/08/2007 Location: AustraliaPosts: 4406 |
That should make a really nice tester Mike. Cheers, Tony. |
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wiseguy Guru Joined: 21/06/2018 Location: AustraliaPosts: 1156 |
Hi Tony, thanks - it works better than I expected. Instead of driving the inductor with a 50% square wave and variable frequency I decided to try a different approach. I have excess SG3524's around so chose that for the PWM generator. Using a pot I can create a PWM pulse as narrow as ~ 300nS, at the max pot setting it becomes a 50% square wave of the frequency determined by pot 2 which has 2 ranges. The max pulse width at the minimum range & minimum pot is ~1.6mSecs the maximum on the highest range max setting is ~ 15uSecs. So in effect I have a variable pulse width generator at a variable repetition rate. I also wired the SG3524's current limit input with an adjustable current limit after proving that a saturated inductor indeed becomes a dead short..... & killed a FET. The current sense is a 0.005 ohm shunt and is the best place to view the current ramp. I purchased one of the 4/40/400A active current probes with a BNC for the CRO - but am quite unimpressed with the performance compared to just monitoring the 0.005 ohm shunt. I chose an arbitrary inductor of 30A which yields 150mV for 30A across the shunt as a starting point. I had to provide some voltage offsets for the shunt current/voltage feedback and the current target setting trim-pot. This allows the current limit to fully disable output drive at minimum current setting. Note the current is not reduced at all until the current applied to the current +/- pins has increased beyond the internal 200mV current sense threshold. I empirically set the current limit by setting the trim pot near minimum and driving an inductor into saturation and then slowly adjusting the current limit until the max amplitude was 250mv on the CRO which corresponds to 50A. The Mosfets I used are IRFB4710's (~70A @ 100V) but the HY4008's would be excellent also. The U30D20 power diodes I user were also a TO247 mount. I provided a small DC/DC converter to supply the isolated upper FET gate driver - these are only a few dollars, I also provided for the usual diode fed bootstrapped upper FET supply voltage but havent proved that it works. As I expected that the voltage required for testing would be at least 30V, I used a cheap ali express step-down converter to provide the 12V power for the circuitry. So inductors can be tested with voltages from 15 to at least 40V. Whilst viewing a repetitive current Ramp of 20A the circuit draws around 60mA from the 30V supply. The PCB is 50mm x 100mm and the main capacitor over hangs one end. The Mosfets are soldered to the PCB, the 2 diodes are connected to the FETs and PCB with 1.5mm copper wires. TBC If at first you dont succeed, I suggest you avoid sky diving.... Cheers Mike |
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Warpspeed Guru Joined: 09/08/2007 Location: AustraliaPosts: 4406 |
I must get around to rebuilding my own inductor tester again properly. Its still an ugly quickly cobbled together rats nest. Currently putting together an old 12Hp single cylinder Wisconsin 240v generator. The field winding has gone down to ground and needs rewinding which should be a simple job. Plan to make a proper electronic voltage regulator for it. Another odd job for my inductor tester. I need to know how much dc field current is really needed to saturate the magnetic circuit, so I know the maximum volts and amps the field winding really needs. The inductor tester here hardly ever gets used, but it can be a really handy piece of test equipment to have around sometimes. Cheers, Tony. |
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wiseguy Guru Joined: 21/06/2018 Location: AustraliaPosts: 1156 |
Ok back to the discussion of why I built a saturation tester. I would like to preface the rest of this discussion by first stating that this is somewhat of a mix of part conjecture, part calculated, part experience and at this stage somewhat unproven and is essentially my own view & gut feel. I have presented below a worst case for low battery voltage in (43V). The inductor between the FET bridge stage and the Toroid primary winding has to be capable of providing the whole average sinusoidal current for the 50Hz sinusoidal toroidal drive required at full power - without saturating at the HF & current involved. So if we use some arbitrary figures to try to make sense of this lets assume a nominal 48V battery system. Lets assume 43V is considered end of discharge (battery empty) and 54V is battery fully charged. Lets also assume the inverter is capable of 4kW @ 230VAC. For simplicity we will assume 100% efficiency to make the maths easier, we really know that everything over 90% is a bonus so in real life there may be 10% losses to also consider. At 43VDC in & 230VAC output, requires us to generate ~ 325VAC peak, using the nominal 8 to 1 ratio represents a primary peak voltage of ~ 40.6V. 4kW equates to 98.5A RMS at 40.6V (93A RMS from the battery at 43V). The peak current would then be around 140A. The choke can be thought of as the series inductor in a switching regulator, but in this case we have a modulated half sinusoid reference. The current in the choke will have a saw-tooth current that ramps up and down & that has an average of 140A at the peak of the sinewave. If we plug this into a formula, a correctly gapped 7.5uH choke will have 4 turns and will have a current capability of 140A average, ramping between 112 A to 168A at the upper and lower peaks of the sawtooth. A core similar to a PM 114/93 pot core which is nearly a 2kG lump of ferrite is required and gapped with an Al value of ~630nH, the centre leg area is ~1,380sqmm and the volume is ~ 344,000cumm. It will handle the whole range of battery voltage without saturating. If the core were to enter saturation the FETs can be killed or and the choke core can run hot and efficiency takes a dive, if the inductor is too small, more HF switching energy is presented to the Toroid which can increase eddy current losses and the transformer can heat up more than expected. Between these two extremes is the "sweet point" we would hope to attain to keep the core and choke losses under control. I also believe that a correctly sized capacitor used directly across the primary after the choke will or may help to reduce unwanted HF from entering the toroidal transformer that was designed for highest efficiency at 50Hz/60Hz not 20+kHz. Whilst a smaller capacitor across the toroidal output has the advantage of being increased in value by the turns ratio when reflected back to the primary, some of the HF switching is already creating higher eddy current losses in the toroid lamination tape which I would hope to minimise further with a primary capacitor. I obtained some large ferrite Ecores 70-54-32 from TSC normally~ $20US per pair but I got them considerably cheaper. If I stack 2 of them as 1 core, the combined area is around 1400sqmm and the volume ~ 326,000cumm which is comparable to the PM 114/93 pot core. Then I need to do some saturation tests to ensure the new inductor wont saturate for the above conditions. The aerosharp C core some are using for the primary choke is a cost effective answer albeit a bit lossy compared to a well designed ferrite choke. The iron core also has a much better "soft" saturation response to an overload capability compared to ferrite. Of course running at 50kHz would halve the choke size and make the ferrite a more attractive proposition with better headroom. The above will need to be proved by running efficiency checks for the various configurations which will eventually tell us what yields the best results. If at first you dont succeed, I suggest you avoid sky diving.... Cheers Mike |
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Warpspeed Guru Joined: 09/08/2007 Location: AustraliaPosts: 4406 |
Have to agree with all of that Mike. Its a really excellent explanation. I am surprised that such large pieces of ferrite can now be had for only twenty dollars. This is something quite new, and I am sure only the Chinese could manage to do it. Anyhow, you have certainly nailed the requirements for a 140 Amps plus as a minimum saturation requirement for the example 48v inverter given. The choice of material comes down to gapped ferrite or gapped grain oriented silicon C cores. The differences in characteristics are rather large. Both can work, and there are valid arguments both ways. Personally I prefer steel cores and I will give my reasons why further down the page. I think we can all agree that the purpose of the choke is to reduce the peak switching current through the mosfets by introducing some series impedance between the bridge and the highly capacitive transformer primary load. The primary of the transformer will be inductive only up to its parallel self resonant frequency which might be in the very low Khz region. Above that, it will appear capacitive, and very strongly capacitive at the PWM switching frequency and its harmonics. The amount of series inductance determines the current switching peaks through the mosfets which comprise average load current plus these switching current spikes. The mosfet losses during the fully Rdson part of the cycle will be relatively low, but while the mosfets are switching through the linear region, there will be both high current and high voltage simultaneously across the mosfet, which produces a very short but explosive releases of heat. The larger the choke inductance we can come up with, the lower these mini explosive bursts of heat within the mosfets will be. There is no simple answer to what is a "correct" and precise choke inductance requirement. More is definitely better, but practical and cost reasons are going to place some upper limit on final physical choke size. Now the whole purpose of the choke is to reduce high frequency ripple current through both the mosfet bridge and the transformer primary. High frequency current is the cause of higher mosfet switching loss, skin effect in the wire, and eddy current and hysteresis loss in both choke and transformer. Losses at 50Hz will be negligible from all these causes, but very significant at 23 Khz (which is around 460 times higher). In other words, everything is going to run significantly hotter without a choke, idling power will be up, and stresses on the mosfets will be higher at full power. Now the choke itself is going to introduce some extra losses of its own, both in the wire and in the core. That is unavoidable. But there will be a very worthwhile overall improvement in inverter efficiency. The reduction of overall losses will be much greater than any additional losses added by the choke. That is the key. Now we come down to choosing a suitable core material for our choke. We are after two things, maximum achievable inductance, and the highest possible final saturation current. Choke losses will be mentioned later. Permeability of the core material is not relevant in a choke, because we can adjust the air gap get the best final trade off between saturation and inductance with either material. The biggest difference between steel and ferrite is the saturation level, and high frequency core loss. Ferrite varies a fair bit depending on grade and operating temperature, but most fall over above about 0.35 Teslas. Silicon steel can be run up to 1.7 teslas in a transformer, and the peak flux is going to be x1.4 times that or probably around 2.3 Teslas at actual saturation. Its difficult to give an exact saturation figure because the onset of saturation is very gradual. But the silicon steel manufacturers suggest 1.7T for transformer design so I will use that figure. So immediately we can see that given identical core cross sectional area, a steel core is going to take about 2.3/0.35 roughly about six times the ampere turns to saturation. We can either use that to get six times the amps at saturation in our choke, or fit six times as many turns onto a steel choke and get 36 times the inductance. Given that steel is usually cheaper than ferrite, it makes for a much smaller and lower cost choke if we choose to use a steel core. Although those $20 ferrite monsters of Mike's do look pretty good from the cost perspective, a ferrite core would still need to be made six times larger in cross sectional area to equal the ampere turns capability of a steel core. Ah, but what about the increased high frequency core loss of a steel core !! This is where the rubber meets the road in the ferrite versus steel argument. High frequency core loss only arises from the high frequency ripple current in the winding. If the choke were fed with pure dc current, there would be zero core losses. I don't know what the relative losses are, but ferrite would have negligible core loss, which can be assumed to be absolutely zero in an application like this. Iron laminations will be very lossy at 23 Khz, but what the actual losses are will be totally dependent on the choke high frequency ripple current. If we can get the choke ripple current right down very low, the high frequency core losses will also be very much reduced. As the whole purpose of fitting the choke in the first place is to reduce high frequency ripple current, if we can build in enough inductance, we not only have a very effective choke, core loss will shrink to insignificance. As we can theoretically fit on six times the turns and get thirty six times the inductance onto a similar sized steel core as onto an identical cross sectioned ferrite core, the steel losses will not be high enough to worry about in practice. If using a ferrite core for ultra low choke losses means we probably have to give up a lot of inductance. That will increase the high frequency ripple current, and that increases the eddy current core losses in the transformer. So there is little achieved in having an ultra low loss choke of we are heating up the transformer core and mosfets more than necessary. As one final thought, all of the high powered commercial grid tie inverters use steel cored chokes, and they definitely work. If there were worthwhile gains to be had by using ferrite, it would have become the standard choke material. Cheers, Tony. |
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renewableMark Guru Joined: 09/12/2017 Location: AustraliaPosts: 1678 |
Why not use Iron and ferrite in series? Half way down the page HERE (part 15, where there is a pic of a power brd.) Poida talks about differences. That big Iron one we did at your place Tony got put in series with two sets of THESE($64) It would be interesting to see that same test Poida did with both in series. My caveman approach was to use both and get the best of both worlds, (prob doesn't work that way though) Cheers Caveman Mark Off grid eastern Melb |
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Warpspeed Guru Joined: 09/08/2007 Location: AustraliaPosts: 4406 |
No reason why both a ferrite core choke and a steel cored choke cannot be run in series as you yourself are doing. No problem with that, if the iron choke is working properly, the ferrite choke can be removed completely without any great change in the idling current. Try it and see. The problem with testing is that frequent blow ups can be the result of many different causes that may be working in combination. Fixing one issue completely may offer no obvious immediate improvement. Everything has to be working just right, to have a robust trouble free inverter, and the choke is just one part of the whole "system". Transients on the observed CRO voltage waveform can be a result of monitoring, and may not really be there. A much better indication might be to monitor the primary ripple current of the transformer through a Hall current sensor. That has only magnetic coupling and hopefully will eliminate grounding problems between the oscilloscope ground and inverter ground. Any difference in ground will likely cause circulating currents to flow in the screen of the oscilloscope probe which will produce a voltage drop, that will be then added directly to the signal seen at the oscilloscope. A 47uH steel choke and a 47uH ferrite choke will have no difference in operation except for two effects. As the magnetic cores go into saturation, the inductance of both will fall away, and that effect is not frequency dependent. Its only the permeability of the magnetic circuit that falls due to excessive ampere turns. Grain oriented silicon steel has about a 6:1 advantage over ferrite for ampere/turns versus core cross sectional area. The losses in the core will definitely be much higher with laminated steel, and those losses will increase with frequency, meaning the core just runs hotter. But that has zero effect on the inductance, or how well the choke operates at higher frequencies. Its still going to be the same 47uH and work just as well, even if the core runs smoking hot to do it ! In practice its the copper losses and temperature rise in the wire that will determine how hot our choke runs at full power. We can get away with thinner wire in our choke than required for the transformer primary, because the transformer primary requires a much longer length of wire. The choke will be smaller and have few turns, so we can use thinner gauge of wire, and because there is not a lot of wire length in the choke, the losses can be kept as a low proportion of total conduction loss. (mosfets + choke + primary) Its helpful to work out the respective wire lengths, wire cross sections, and respective conduction losses in both transformer primary and choke. If we can usefully reduce the wire cross section in the choke we may possibly be able to fit on several more turns without significantly increasing the total copper loss. And inductance is proportional to turns squared, so even one extra turn is a significant gain if it will fit onto the core. Always fit as many turns as will fit onto your choke without excessive heating, then adjust the air gap to suit. That is the golden rule. Tinker (Klaus) is the recognized master of filling chokes with maximum possible amount of copper. Doing that, you will get the most possible out of the ampere turns that a particular core is capable of giving, without resorting to a larger core size. Cheers, Tony. |
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wiseguy Guru Joined: 21/06/2018 Location: AustraliaPosts: 1156 |
Thanks for the detailed responses Tony - I essentially concur with all your comments. The main reasons for considering using only Ferrite at the moment is that I have access to quite a few of the large Ferrite E-cores I mentioned, I paid ~$120 for 24 of them (24 halves). So if I stack 2 or 3 pairs together it is a simple cost effective solution - I am considering just flat copper sheet (with typical copper foil winding type construction) for the few turns I will need, I haven't done the sums yet re thickness. Depending on performance my next choice is the sendust type materials especially KoolMu & HiFlux cores which can operate up to as high as 1.5 Tesla and have a more gentle saturation curve. But these are not generally cheap & if cheap from China, it will still be expensive with the freight. Before Klaus chimes in to remind me I still dont have a working inverter yet I KNOW ! but it is getting closer... Will send out my stand-alone version of the Nano1 control board this week and order the remaining bits for it - I can almost see & smell the smoke coming..... If at first you dont succeed, I suggest you avoid sky diving.... Cheers Mike |
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Warpspeed Guru Joined: 09/08/2007 Location: AustraliaPosts: 4406 |
Mike, I think we are on the same page with this. Ferrite will certainly work, but it needs to be rather large, and the cost then becomes prohibitive. But if you can snagged a bunch of really huge ferrite cores somehow, its certainly a workable solution. My own low frequency inverters do not require chokes, so I am not heavily into the choke testing and developing thing. But I definitely like the copper sheet idea. Something like 1.2mm thick annealed copper, 75mm wide with very thin insulation would be a very simple way to fit maximum possible copper into a rectangular window. Any kind of round wire just has too much wasted space. If the sheet is hammered over a very rigid rectangular mandrel, the core halves could then slip in. Those KoolMu and HiFlux Molypermaloy toroids are wonderful things and extremely low loss, but the lack of an adjustable air gap means you must order the cores in the exact permeability grade you think you need. I have a few odd sizes in various permeabilities here, have had them for years, but they always seem to be unsuitable for any project of the moment. Cheers, Tony. |
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wiseguy Guru Joined: 21/06/2018 Location: AustraliaPosts: 1156 |
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renewableMark Guru Joined: 09/12/2017 Location: AustraliaPosts: 1678 |
What's your opinion if the choke is too oversized, say 50% bigger than it needs to be? Cheers Caveman Mark Off grid eastern Melb |
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Warpspeed Guru Joined: 09/08/2007 Location: AustraliaPosts: 4406 |
Mike, a lively technical discussion is always welcome ! [quote]Can we reduce the choke ripple current to almost zero - no way as then it no longer stores and releases energy and one of the main properties that increased our efficiency is lost.[/quote] We can reduce ripple current as low as you wish to go, but it can never be absolutely zero. Is it possible to add a resistor into a dc circuit to reduce current to zero ? No because even with an infinitely large resistor, there would still be an infinitely small current. But never zero. [quote]I keep on harping about an inverter as being a series switching regulator with an active flywheel diode because that is exactly what its modelled equivalent and operation is doing.[/quote] That is an excellent explanation of how this all works. The only difference is that our "regulator" output voltage is smoothly modulated and going up and down like a yo-yo at 50Hz instead of pure dc. [quote]I think there is a way to obtain a fairly precise choke inductance value and energy storage capability and it is as simple as calculating for a switching regulator choke at the two worst case extremes of lowest voltage and highest current and vice versa, ensuring we are avoiding getting too close to saturation at either extreme ie 54V fully charged to 43V discharged. [/quote] Yes indeed, we can calculate exactly what any value of choke inductance is going to do at the 50% duty cycle point. At full 100% duty cycle we have full dc rail voltage and no switching at all. At full 0% duty cycle we have dc ground and also no switching going on at all. At 50% duty cycle is violently pulling the choke alternately first one way, then the other, and we then have the highest peak to peak ac ripple current through the choke. I stole this picture off the internet, it shows how ripple falls to a minimum at the exact 100% duty cycle, and at the extreme left hand edge of the screen the duty cycle looks to be about 50%. What I am suggesting, is we set some minimum acceptable target inductance for worst case ripple current, and try to do better than that if we possibly can. Mark, We are fortunate that the PWM switching frequency (around 23Khz) is so very far above our 50Hz power frequency, around x460 times higher. The choke impedance will have 460 times as much of an effect at the PWM frequency as at 50Hz. If the choke inductance was made fifty times larger or a hundred times larger it would certainly start to block some of our 50Hz power and cause us some problems. But we could easily go twice or even five times higher in inductance without any problems at all. If we set a minimum of say 50uH, at 50% duty cycle and with 50v dc supply and 23Khz, that would give us a peak to peak triangular ripple current of just under 22 amps. That works out to around 12.7 amps rms of ripple. That might be around 10% ripple at the sort of flat out power levels we are talking about here, and would be considered fairly low for a switching power supply. So 50uH may arguably a realistic minimum target that would work fine. But if you can do 100uH or 150uH it would work even better. It does start to become impractically large though, and you would be well into the point of diminishing returns. I cannot remember how your own choke ended up, but a vague recollection of about 70uH or slightly above that, was a very satisfactory result for us at the time. Cheers, Tony. |
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wiseguy Guru Joined: 21/06/2018 Location: AustraliaPosts: 1156 |
We can reduce ripple current as low as you wish to go, but it can never be absolutely zero. I see where you are going but doesn't increasing inductance to reduce the ripple have 2 counter productive effects of increasing ampere turns and getting closer to saturation unless core volume is also increased ? (must admit I am only considering & referring to ferrite chokes) I stole this picture off the internet, it shows how ripple falls to a minimum at the exact 100% duty cycle, and at the extreme left hand edge of the screen the duty cycle looks to be about 50%. I think you should give the picture back.... That picture tells a good tale - it tells me that at the peak of the sinewave there is no headroom left from the supply voltage - to meet the peak, the PWM went to 100% and there was no more. If there had been a few volts of headroom the ripple would not have gone to zero. Except for the zero crossing where no switching or current briefly flows, there should be a variable height sawtooth ramp over almost the whole sinewave. Of course the P-P ripple and the rising and falling ramps will have different slope rise and fall ramp angles according to load and Vin. What I am suggesting, is we set some minimum acceptable target inductance for worst case ripple current, and try to do better than that if we possibly can. Agreed ! Mark, Sorry for the late reply - busy day. If the choke is double the minimum required, my view is that the peak to peak ripple is reduced, there is higher copper losses than there needs to be, the FETs probably enjoy a small peak current reduction of ~ 10% and maybe you are a bit closer to onset saturation losses at higher loads than you need to be. Not sure if Tony concurs - I deleted his quote before I answered this. If at first you dont succeed, I suggest you avoid sky diving.... Cheers Mike |
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Warpspeed Guru Joined: 09/08/2007 Location: AustraliaPosts: 4406 |
Yes indeed, increasing the turns definitely increases the dc flux level (ampere turns) and carries us closer to saturation, but our secret weapon is the air gap. The air gap only has an effect on the dc component of the magnetic flux. The ac component is controlled only by core cross sectional area, as per Faraday's law, exactly as it applies in a transformer. If we use a large number of turns, and also a very large air gap together, we can reduce the ac flux swing in the core, while keeping below saturation from the dc component by spreading the gap. We deal with each problem separately as we design our choke. Its the reason why we always try to completely fill our choke with the maximum possible number of turns that will fit, and then tweak the air gap for the best compromise between inductance and saturation level. Its all fascinating stuff. The dc component (actually 50Hz) through our choke reaches maximum at the peak of our mains cycle. But the high frequency ac ripple component is always maximum at the 50% PWM duty cycle point, and actually decreases up near the peak of the mains cycle. It really depends if we are using unipolar or bipolar PWM modulation how the PWM ripple amplitude changes at various points throughout the 50Hz sine wave. Cheers, Tony. |
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Warpspeed Guru Joined: 09/08/2007 Location: AustraliaPosts: 4406 |
There is an excellent book "switchmode power supply handbook" by Keith Billings. I just looked this up on Amazon and was SHOCKED by the price. I guess its now out of print and has become very rare and valuable. My copy was obtained brand new in 1994 and was a bargain at $170.00 back then. https://www.amazon.com/Switchmode-Power-Supply-Designers-Handbook/dp/0029478200/ref=sr_1_3?ie=UTF8&qid=1550705944&sr=8-3 &keywords=switchmode+power+supply+handbook Anyhow, here are a couple of pages of pure gold by the good Mr Billings. Cheers, Tony. |
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wiseguy Guru Joined: 21/06/2018 Location: AustraliaPosts: 1156 |
Any chance you could post images from the rest of the book Amazon listed it as no 16,884,004 in its best seller list For something so popular I guess $703.97 its a steal..... Excellent information which really illustrates & supports/confirms our understanding. If at first you dont succeed, I suggest you avoid sky diving.... Cheers Mike |
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Warpspeed Guru Joined: 09/08/2007 Location: AustraliaPosts: 4406 |
A few years ago I stumbled on an electronic copy of the entire book. Someone had scanned every page. It was of little interest to me at the time because I had the real thing. Don't know if its still out there on the intenet somewhere. Its possible, but it may have been shut down. There must be several hundred pages, and its especially good for magnetics design. Cheers, Tony. |
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nickskethisniks Guru Joined: 17/10/2017 Location: BelgiumPosts: 458 |
Hi, if I want to test an inductor with a pulse of let's say 80% duty. Could I replace 1 mosfet with a diode (or body diode) in the circuit in the first page? I want to test an inductor for a buck converter. I made the saturation tester of the first page and it worked verry well, the interpretation of the scope is another thing... |
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poida Guru Joined: 02/02/2017 Location: AustraliaPosts: 1419 |
a google search with the text switch mode power supply handbook by keith billings yields results. The 3rd one down is a direct link to download the pdf. Which I did. This is the 3rd edition and so it has a co-author. look at pages 474 or so for those that Madness shows above. Easier to read. wronger than a phone book full of wrong phone numbers |
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