There is two types of transformers in the market- linear transformer and switch mode power transformer. The transformers design is different. Sometimes it is just referred as ac transformer. Linear transformer mostly used in radio, project kits, VCD and etc while switch mode transformer can be found in Monitor, Tv, Fax and so on.
A transformer consists of two or more coils wound on the same core. For power transformers, the core material is usually iron. For a radio-frequency transformer, the core material is usually ferrite or air. The basic property of a transformer is to change AC voltage. A transformer cannot change direct current voltage. A step down transformer has a lower AC output voltage at its secondary winding than the AC input voltage to its primary winding. Conversely, a step up transformer has a higher secondary than primary voltage. The transformer laminations or coating of shellac, enamel or varnish is to insulate adjacent turns from shorts between winding. I will not go too details about how transformer work and transformer calculation as you can read more details from most electrical and electronic transformers books.
How to test transformers: Transformer failures are relatively rare in most electronic equipment, compared with other components. If too much current goes through a coil or winding, the winding heats up and can either open up completely, or the insulation between turns of wire can break down, causing the coil to be partly or completely shorted.
Just like a coil, you can check continuity of any primary or secondary transformer winding. The actual value of DC resistance you read is rather meaningless, but at least you will know whether a winding is open. Checking linear transformer is quite straight forward. Apply the ac voltage to the primary winding and expect output voltage at the secondary side. No or low output voltage means the transformer winding has open or shorted winding. Look for any evidence of overheating, such as darkened or blackened areas or a burnt smell.
About how to test a power transformer-it is a little bit complicated. You cannot apply ac voltage to the equipment and expect the desire output at the secondary side without connect to load otherwise over voltage will happen (because you have removed all the secondary load) and will blow the power section which consist of power fet, pwm ic and etc. Switch mode power supply are designed to run with load while linear transformer can be test without load.
The best way to test a switch mode power transformer is first remove it from the board. Check only the primary winding as secondary winding failures are exceedingly rare. How do we know which one is the primary winding? Trace from the big filter cap the positive pin to see where the circuit goes. It will go to one of the pin of the transformer. Then look for the power fet middle pin which is the Drain and it will lead to another pin of the transformer. This two pins is the primary winding of the transformer. Checking the resistance of the primary winding only can reveal whether it is open but it cannot check if it has developed a shorted winding. Only by using a ringer test you will know if it shorted or not. A shorted turn in the primary winding can cause the power section to blow.
The cheapest ringer tester that i came across is Dick Smith LOPT METER or flyback tester. Just connect the probes to the primary winding and the result are instantly seen from the Led bar graph. 4-8 bar represent a good winding, 1-2 bar represent partial short and if the led goes off means there is a heavy short in the winding. Besides checking the transformer it can be use to check B+ COIL, horizontal yoke coil and flyback primary winding and ballast (choke).
The expensive ringer tester is from sencore equipment. Sencore LC102 and LC103C besides checking capacitor it also have the capability to check all kind of winding. Whether it is a low voltage transformer, high voltage transformer, current transformer, pulse transformer, electric transformer, custom transformer this equipment will be able to check and test it.
If you do not wish to invest on this tester you may use a light bulb to check it. Remove one of the primary winding pin and connect a 60 watt light bulb. One end to the circuit while the other to the transformer pin. If the bulb light and didn't goes off suspect a shorted winding and if the light goes off, the winding is ok.
Transformer repair is not easy and it require a good skill to do it. Most of the transformer repairer preferred to rewind bigger winding such as fan, car alternator and starter. Switch mode power transformer is a little bit difficult to rewind because of the small size and low price.
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