Disk Geometry

Hard disk drives are called by that name because they are not floppy (as in floppy disk drives). They are organized as a concentric stack of disks or "platters":

Each platter has two surfaces (although in practice the outer surfaces on the top and bottom of the stack are often unused because of physical space considerations), and each has its own read/write head (which reads and writes data magnetically on the surface). The data is stored on concentric circles on the surfaces known as tracks:
Corresponding tracks on all surfaces on a drive, when taken together, make up a cylinder:
Since an individual data block is one sector of a track:
blocks can be addressed by specifying the cylinder, head and sector numbers of the block ("CHS").
It is interesting to note that most IDE (Intelligent Drive Electronics) disk drives pretend to have a far different geometry than they actually have physically. For instance, a typical hard drive might tell the operating system that it has 16,384 cylinders, 80 heads and 63 sectors per track; since 80 heads implies at least 40 platters, each with 2 surfaces, we see immediately that this cannot represent the physical contents of a hard drive which is just 3/4 of an inch thick! In fact, most hard drives have only one or two platters, and the geometry they pretend to have is there for the software's sake, as we shall see below.
IDE disk drives can address blocks logically, treating the disk drive as a series of blocks numbered from 0 to n. It might seem, then, that organizing the drive in this geometrical fashion is unnecessary. There are still two reasons why disk geometry is relevant:
  1. as we noted previously, antiquated programming on BIOS chips prevents many computers from loading an operating system from a cylinder number larger than 1023;
  2. on multitasking computers, outstanding read and write data requests for a hard drive are scheduled by track number to minimize head movement.
    If the requests were serviced on a first come first served basis (FCFS, or first in first out - FIFO), the head would in general have to move randomly back and forth over the surfaces. This makes access times unnecessarily large, since it takes extra time for the head to change direction, and moreover, there will simply be more head movement than if the head were to move smoothly across the surface, reading and writing data as the appropriate tracks appeared under the heads.
Consider our disk drive which has 16,384 cylinders, 80 heads and 63 sectors per track. For historical reasons, a sector has 512 bytes (which suggests that early operating systems allocated just 9 bits for the sector part of a disk address; 512 = 2 9). What is the capacity of such a disk? We can use units to calculate how much storage is available. Our basic equation is
? bytes = 1 disk
The conversion factors appropriate to this hard disk are

Notice that the traditional nomenclature is "heads per cylinder" and "sectors per track". Since there is only one head able to access any given track (on any given surface), we have an implicit conversion factor available to us:

1 track / head
Note too that two of these conversion factors correspond to powers of 2:
16,384 = 2 14
63 = 2 6 - 1
This suggests that these values are intimately related to the number of bits which are used to store the cylinder and sector numbers, respectively.

The computation of the hard drive capacity then proceeds as follows:

? bytes = 1 disk * (16,384 cylinders / disk) * (80 heads / cylinder) * (1 track / head) *
(63 sectors / track) * (512 bytes / sector)

= 42,278,584,320 bytes

This is a large and rather inconvenient number. We adopt the following "metric" prefixes when discussing storage space:

Using these definitions, our result would be expressed in GB as :

42,278,584,320 bytes / (1,073,741,824 bytes / GB) = 39.375 GB
These definitions represent common usage among memory manufacturers and operating systems programmers, in contradiction to the strict powers of ten used by scientists throughout the world, as well as by hard drive manufacturers:

It is important to realize that the manufacturers of most disk drives quote capacities which are considerably larger than the actual usable amount of space available on the drive. This is because some space is unavoidably lost when the "raw" tracks are separated into sectors: sectors have gaps between them which are needed for timing reasons, and they also have addresses and checksums for error checking. So our 39.375 GB would be described by a manufacturer as something noticably larger than the 42.279 GB which strict powers of ten would imply.

We turn next to the computation of speed, throughput and utilization.

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©2002, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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