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I have a baby swing that eats D batteries. It uses 4 of them. I measured the current being used by the swing with everything turned on and at full (swing at max speed with baby in it, spinning things turned on, sound on at full volume, etc.). When I set my meter to measure up to 10A, and set the dial to 10A, it gets up to 0.560 or so, which I am assuming is about 600mA max. Does that sound correct? I just want to double check my work. I am planning on soldering a female DC power jack to the battery terminals to allow a power adapter. I should be able to get any power adapter that supplies 6V DC at over 600mA, correct? I am going to get one rated at 6V and 1A. For safety, I am planning on attaching an in-line fuse holder with a 1A fuse. Does this all sound safe enough? I did some research, and this should be perfectly safe. Anybody have any issues with this that I am overlooking? I appreciate it!
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You seem to have all bases coverd on this. I can't see anything you have missed. Good luck.
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That all looks quite good. 

I would check the current draw immediately when the swing is turned on and has to work harder to start the motion.  Similarly, check the current if you hold the swing in place while it is running.  If the motor stalls the current could increase enough to blow the fuse, but I think most baby swings have some mechanism that keeps this from happening, so it may not increase much at all.  If those tests don't raise any red flags I would consider you good to go.
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The instantaneous draw from dead stop might be too quick for your meter to measure.

Do the motor stall test that jroorda suggests to get an idea of what it would be.
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Sounds good. I will give it a try. I am fairly certain the swing works with a ratcheting type of operation that does not care if the swing is actually in motion or not. But I will try to test anyway. The 1A inline fuse should protect me anyway. Is there a good rule of thumb for where to place the fuse? I figured on placing it on the positive lead, just behind the female DC jack.
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Fuse placement sounds reasonable.
If you use a switching regulator, they are usually self protecting against shorted outputs anyway.
Example:
http://www.dipmicro.com/store/DCA-0610

I use these 5V, 7.5V, 12V supplies all the time.  Might have a 9V too.
5V for no-regulator designs, 7.5V for logic level only Arduino's, 12V to power LED strips with TPIC6B595s to sink current.
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