AFAIK. it makes no difference where you download either a game or an expansion pack, as long as you use the same player account on every machine. Once the purchase is made (you have not been clear about where you made the purchase), an ORIGIN digital copy is permanently tied onto your first account it was bought with.
From then on, using your player account, you can download it again, to another of your PCs.
I am going to pause here for a computer lesson. There are two separate events involved. A digital copy is transferred across a network from the Master Website to your local machine. That procedure, and ONLY that one, is a "Download". Following that transfer of the files, there is an event called an "Installation".
If you don't know the difference yet, and I haven't made it clear enough, look it up anywhere on the Internet using Yahoo, Google, any search engine.
There should NOT be any difficulty with the actual download operation.