Can a transformer handle more current than its specifications?

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Marcis Rancans
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How large current does go through resistor (how to calculate it?) and can it draw more current than written on specs of transformer (in this case burning it down)?
Do they mean 0,5 amps are max current safe to draw without burning down?

Transformer is connected to wall power (220 V, 60 hz). On transformer is written specs which say that output is 6V and 0,5 amps.

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Your calculation is correct.

Transformers are dumb. They don't know what they are rated for. There is no control circuit that maintains or limits it (unless one is added separate). All they do is provide a voltage ratio. As you have found, the resistance of the load determines the amperage, so if the transformer isn't rated for what gets pulled, it may burn-up.
 
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+1 to that.

Your calculation shows that a 5 ohms load would draw too much current for that transformer.

Try calculating the minimum resistor allowed.