Upvote:7
No, just register your Swedish passport online for free, and you're good to go.
Go to this site https://online.immi.gov.au/lusc/login
Create an account
Click on "New application"
Under "Application group", select "Visitor" and then "eVisitor (651)"
Fill out the registration form and then click "sumbit applications"
You'll get a confirmation email within 15-20 minutes that the registration has been successful. Then simply take your passport and go to Australia.
Pro Tip: you can now use the Smartgates as a Swedish passport holder. It's an automated border crossing system which is very easy