How do HDMI refresh rates and bandwidth capabilities work?

Dealing with refresh rates and cable bandwidth can be tricky with HDMI because the cable isn't all that matters, the actual display also impacts performance. As such, let's look closely at what HDMI cables are able to accomplish in these areas and then talk about how those specs apply to actual usage when it comes to connecting to a display.
 
Basic HDMI cables
Category 1 HDMI cables are the standard HDMI wire that you would use to connect various devices, and they are capable of supporting refresh rates of either 24hz or 60hz. In terms of bandwidth, they are able to move data at 75Mhz, which equates to moving data at a rate of 2.25 Gbps.
 
High-Speed HDMI cables
Category 2 HDMI cables are specifically designed to move more data to displays and to do it faster. The refresh rates remain standard at 24hz or 60hz, but the bandwidth climbs to 340Mhz, which leads to a data rate of 10.2 Gbps.
 
What to make of refresh rates
There are two key issues with refresh rates - what the cable can handle and what the television can handle. The 60hz cap on HDMI cables means that an HDMI wire is moving data at a rate of 60 frames-per-second. However, television refresh rates can go much higher. In this situation, the television would play each frame multiple times, leading to a much smoother image. 
 
The importance of bandwidth
Bandwidth becomes a major issue when dealing with 4K displays, which high-speed HDMI cables can support. Category 1 HDMI wires are limited to a 720p/1080i display configuration. When moving data at 60hz to a 4K display, the cable needs to carry a large amount of data. However, that data escalates when the television runs at higher than 60hz, as the cable needs to transport all of the data associated with those duplicate frames.
In isolation, the refresh rates and bandwidth of HDMI cables aren't particularly complex, but the way those capabilities intersect with display capabilities creates complexity.

 
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HDMI Backwards Compatibility