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I have two different transformers. The output of the first one is 15 volts and 1500 mA. The output of the second one is 5 volts and 500 mA.

If I connected them in series (to get 20 volts) what will happen to the current?

If I connected them in parallel (to get 2000 mA) What will happen to the voltage?

Is there any danger in the previous connections?

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1) If you connect them in series, properly, you'll get 20 volts out but the current out of the pair must be externally limited to 500 mA.

2) If you connect them in parallel, the transformer with the 500 mA secondary will be destroyed first.

3) See 2).

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    \$\begingroup\$ If you connect them in series you'll get either 10V or 20V. The latter is the proper connection. Either way, the current will be 500mA. \$\endgroup\$ – Brian Drummond Oct 9 '14 at 11:48
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    \$\begingroup\$ IMO in this case 'must be limited to' is a better wording than 'will be limited to' or 'will be'. It is the designer who must do the limiting, the transformers won't do it for you (as is implied by 2) ). \$\endgroup\$ – Wouter van Ooijen Oct 9 '14 at 14:36
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    \$\begingroup\$ @WoutervanOoijen: Good catch, thanks. It's fixed. :-) \$\endgroup\$ – EM Fields Oct 9 '14 at 14:51

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