Your laptop runs on batteries, so why shouldn't your router? Finding an AC outlet isn't a problem if you're working in a hotel room, but cords are troublesome when you're in a coffee shop and nearly impossible when you're working out of your car or you're outdoors. Every travel router requires electrical power, but some can operate on either AC or battery power. Some routers plug straight into the wall, while others rely on an adapter and a cable (this is arguably a better option, because you'll get better range if the router is higher off the floor).
Most travel routers will have at least one ethernet port, so that you can plug into a hard-wired ethernet connection and create a wireless network. Routers with two ports enable you to create a secure wireless network with the WAN (wide-area network) port, and an even more secure wired network with the other. Most travelers will use the wired connection to support one PC. If you carry a wired switch, you could support as many wired clients as there are ports on the switch.
Most travel routers support only the 2.4GHz frequency band. The upside to this band is range; the downside is it's very crowded and suffers from interference with microwave ovens and other electronic devices. Dual-band routers support the less-crowded 5GHz frequency band, but you should expect to get less range there.
Most travel routers can be configured to operate in one of five modes, although not every router supports every mode. In router mode, you connect the travel router to a DSL or cable modem and it becomes a fully functional wireless router. In WAP (Wireless Access Point) mode, the travel router shares a wired broadband connection wirelessly with one or more clients. In WISP (Wireless Internet Service Provider) mode, the device becomes a client to a wireless Internet service (such as a Wi-Fi hotspot) and shares that connection with its wireless clients. In mobile router mode, you plug in a 3G, 4G, or LTE USB adapter and share the wireless data plan tied to that device. In repeater mode, the device will extend the range of a wireless router or wireless access point. It doesn't create a network of its own in this mode. And in bridge mode, the travel router makes a wireless connection to a network and is hardwired to a client that doesn't have a wireless adapter of its own.
Some wireless routers deliver more throughput than others. A router that supports one spatial stream using the 802.11n standard can deliver throughput of 150Mbps. A router that supports two spatial streams using that standard can deliver 300Mbps of throughput, and so on. A router that supports the 802.11ac standard can deliver throughput of 433Mbps per spatial stream.
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