Purpose
Memory controllers contain the logic necessary to read and write to DRAM, and to "refresh" the DRAM by sending current through the entire device. Without constant refreshes, DRAM will lose the data written to it as the capacitors leak their charge within a fraction of a second (not less than 64 milliseconds according to JEDEC standards).
Reading and writing to DRAM is performed by selecting the row and column data addresses of the DRAM as the inputs to the multiplexer circuit, where the demultiplexer on the DRAM uses the converted inputs to select the correct memory location and return the data, which is then passed back through a multiplexer to consolidate the data in order to reduce the required bus width for the operation.
Bus width is the number of parallel lines available to communicate with the memory cell. Memory controllers' bus widths range from 8-bit in earlier systems, to 512-bit in more complicated systems and video cards (typically implemented as four 64-bit simultaneous memory controllers operating in parallel, though some are designed to operate in "gang mode" where two 64-bit memory controllers can be used to access a 128-bit memory device).
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