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There's a whole bunch of methods for designing a power transformer, but I like to use this method because it gives a little overestimate of the results, giving you a fairly good safe margin when using it, to not overheat or blow out the transformer. Okay so the design steps are as it follow:
1. You gotta know the primary voltage (220/110V). Also you got to know the nominal secondary voltage and power. Let's assume we want 30V@5A from 220V . So we can assume the secondary power is 150W.
2. Once you know the secondary nominal power rating, you gotta figure out the primary's power. It's given by the formula Ppr=Psec/k. k is the conversion efficiency constant and is around 0.9 for transformers over 50W. It can be as low as 0.77 for low power transformers (1-5W) and around 0.85 for midpower transformers (10-20W). So the primary needs to have a power rating of 166W
3. Now we got to determine the cross sectional area of the core. You can find how to measure it on the web given your core. It's usually a multiplication of one side of the core by the other. So S=sqrtPpr. In our example it's S=sqrt166=12.8cm^2. Always round up a little when building it, for instance for our example an appropriate core will have a cross section of 13.5cm^2
4. Next you gotta find the wire turns. For the primary the formula is Npr=40xVpr/S. So we get Npr=40.220/12.8= 687 turns for the primary. The secondary is given as Nsec=44xVsec/S. Nsec=44.30/12.2=103 turns for the secondary.
5. The last thing we need to determine is the wire diameter based on the current through it. It's given by the formula d=0.02xsqrtI. The current is in mA and the given result for the diameter in mm. I'm not exactly sure for what current density is the formula given, but it hasn't let me down so far. So for the primary we have I=P/V. I=166/220=754mA. d=0.02xsqrt754=0.55mm. Same applies for the secondary. d=0.02xsqrt5000=1.4mm
6. With this the calculations finish and you can start building. A few tips. C type cores are around 20% more efficient than E-I cores and will heat up less if properly secured. Use double coated wire for the primary, it's always beneficial to secure your transformer good. If you want your output characteristic less soft and kneeley, you can use 3:2 interleaving if your winding window allows it. Just wind 1/3 of the primary, then 1/2 of your secondary, then again 1/3 of your primary, again the other half of the secondary and finish it up with the last 1/3 of your primary. This is beneficial, especially if you've got a constructively large core. Every interleavement should be carefully insulated with high voltage dielectric, like waxed paper, or some other good dielectric, whatever suits you best. Always insulate the transformer as good as your budget allows it. From double enamel insulation of the wires to good HV insulation between the windings.
The results here give a 10% theoretical margin of error, but in reality due to losses and whatnot, everything should add up.
A few notes on the methodics of the calculations. Most results, coefficients and calculations are slightly increased, from a practical point of view (according to the author) for less transformer heating. Also the secondary voltage given, is for the nominal load of 150W. Working on an open circuit or with a lighter load, will increase the secondary voltage by a few volts.
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