Hi, and WELCOME TO TECHIMO - from the Folding@Home Team (#111)...!
I'm not entirely sure why you would need to change your IP address by disconnecting from the web - it should just carry on with little/no change in your browsing experience...
From what you've posted it sounds like it has done that anyway, which leads me to question why you are the one who wants to force the change of IP address...?? Off the top of my (slightly tipsy) head, the only reasons I can think of would be to avoid being tracked and/or discovered...
I would imagine that (at least) 99% of (LAN) networks don't need to hide their footsteps in this way, and if you're doing it so that someone can't access your network, then it is fairly likely that there is some form of software that can track you either by cookies and/or MAC address anyway......
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I've seen the light... It was green, flashy and attached to a Network Interface Card...
Whenever someone says "You can't miss it" I invariably do...