Filename: HDMATH.TXT
WWW URL: http://www.firmware.com/support/bios/hdmath.htm
FTP URL: ftp://ftp.firmware.com/text/hdmath.txt
Author: Terry Slade, Micro Firmware Technical Support
Revision: 07/27/97
Summary: Some notes on exact sizes of BIOS limitations on hard drive
sizes.
Just some rough notes to provide a reference for some confusing numbers. See also:
OVER4GB.TXT http://www.firmware.com/pb4ts/over4gb.htm
METRIC.TXT http://www.firmware.com/pb4ts/metric.htm
A sector is 512 bytes - this is true on all hard drives in the PC world and also applies to floppy disks.
This refers to formatted data capacity - a sector actually uses more than 512 bytes. 512 bytes are available for data storage. Throughout this document and in most discussions of hard drive capacity, all figures assume formatted capacity (512 bytes/sector).
A megabyte (MB) can be properly defined in two different ways:
"decimal megabyte" = 1,000,000 bytes (10 to the 6th power)"binary megabyte" = 1,048,576 bytes (2 to the 20th power)
(also easily calculated as 1024 x 1024)
(Some of the values for MB and GB in this document are shown in
parenthesis to indicate binary MB and GB rather than decimal).
To convert decimal MB to binary MB:
decimal MB x 1,000,000
---------------------- = binary MB
1,048,576To convert binary MB to decimal MB:
binary MB x 1.048576 = decimal MB
DOS FDISK shows drive capacity in binary megabytes.
Windows 3.x File Manager shows drive capacity in binary megabytes.
CMOS setup in older ROM BIOSes will show drive capacity in binary megabytes.
CMOS setup in newer ROM BIOSes (including all MFI BIOS upgrades) will show drive capacity in decimal megabytes.
CHKDSK shows capacity in decimal megabytes (actually total bytes).
Drive manufacturers report drive capacities in decimal megabytes.
"decimal gigabyte" = 1,000,000,000 bytes (10 to the 9th power)"binary gigabyte" = 1,073,741,824 bytes (2 to the 30th power)
(1024 x 1024 x1024)
Cylinders limit is 1024 cylinders numbered 0-1023 (10 bits) Heads limit is 256 heads numbered 0-255 (8 bits) Sectors limit is 63 sectors numbered 1-63 (6 bits)1024 x 256 x 63 = 16,515,072 sectors
16,515,072 x 512 = 8,455,716,864 bytes 8.4GB (7.9GB)
Cylinders limit is 65536 cylinders numbered 0-65535 (16 bits) Heads limit is 16 heads numbered 0-15 (4 bits) Sectors limit is 255 sectors numbered 1-255 (8 bits)65536 x 16 x 255 = 267,386,880 sectors
267,386,880 x 512 = 136,902,082,560 bytes 136.9GB (127GB)
Cylinders limit is 1024 cylinders numbered 0-1023 (10 bits) Heads limit is 16 heads numbered 0-15 (4 bits) Sectors limit is 63 sectors numbered 1-63 (6 bits)1024 x 16 x 63 = 1,032,192 sectors
1,032,192 x 512 = 528,482,304 bytes 528MB (504MB)
(cylinders are halved and heads doubled until cylinders are 1024 or less)
0-1024 cyls 16 heads
1025-2048 cyls 32 heads
2047-4096 cyls 64 heads
4097-8192 cyls 128 heads 8192-16384 cyls 256 heads 16385-32768 cyls 512 heads
Many older BIOSes erroneously allocate only 12 bits for the cylinder field in CMOS RAM.
Cylinders limit is 4096 cylinders numbered 0-4095 (12 bits) Heads limit is 16 heads numbered 0-15 (4 bits) Sectors limit is 63 sectors numbered 1-63 (6 bits)4096 x 16 x 63 = 4,128,768 sectors
4,128,768 x 512 = 2,113,929,216 bytes 2.1GB (1.97GB)
All versions of DOS including WIN95 are unable to handle the value 256 (or any value over 255) for the number of heads. Although this is a problem with the operating system, a newer BIOS should compensate for it, normally by using Assisted LBA translation, which limits the total heads value to 255 or by using a revised form of ECHS transalation, which adjusts 16 heads to 15 and increases the cylinder values by 16/15 before performing the normal ECHS translation.
8192 cylinders causes 16 heads to translate to 256 8191 cylinders causes 16 heads to translate to 128Cylinders limit is 8191 cylinders numbered 0-8190 (13 bits) Heads limit is 16 heads numbered 0-15 (4 bits) Sectors limit is 63 sectors numbered 1-63 (6 bits)
8191 x 16 x 63 = 8,256,528 sectors
8,256,528 x 512 = 4,227,342,336 bytes 4.2GB 4227MB (3937MB)
16384 causes 16 heads to translate as 256 heads. 16385 would cause 16 heads to translate as 512 heads which is beyond the limits that a translating BIOS can handle.Cylinders limit is 16384 cylinders numbered 0-16383 (14 bits) Heads limit is 16 heads numbered 0-15 (4 bits) Sectors limit is 63 sectors numbered 1-63 (6 bits)
16384 x 16 x 63 = 16,515,072 sectors
16,515,072 x 512 = 8,455,716,864 bytes = 8456MB = 8.5GB (8064MB, 7.875GB)
The ATA-3 spec says that a drive with over 8 heads should use no more than 16383 cylinders. One of the first 8.4GB drives, the Quantum Fireball SE uses these values:
16383 x 16 x 63 = 16,514,064 sectors
16,514,064 x 512 = 8,455,200,768 bytes 8.5GB 8455MB (8064MB, 7.875GB)
The maximum translated values would be:1024 x 255 x 63 = 16,450,560 sectors
16,450,560 x 512 = 8,422,686,720 bytes 8.4GB 8423MB
Maximum untranslated CHS values:
16320 x 16 x 63
The maximum translated values would be:1024 x 240 x 63 = 15,482,880 sectors
15,482,880 x 512 = 7,927,234,560 bytes = 7927MB = 7.9GB
Maximum untranslated CHS values:
15360 x 16 x 63
In translation algorithms, any non-whole numbers are rounded down to the nearest whole number. Depending on the factory CHS values used on a particular drive, there may or may not be a small loss in capacity due to rounding of non-whole numbers when the parameters are translated by the BIOS. This loss may vary with different types of translation.
Factory CHS 9924 x 16 x 63 = 10,003,392 total sectors
5,121,736,704 bytes 5.1GB
(4884MB, 4.77GB)Assisted LBA 622 x 255 x 63 = 9,992,430 sectors
5,116,124,160 bytes 5.1GB
(4879MB, 4.76GB)
loss = 10962 sectors 5,612,544 bytes 5.6MB (5.35MB)Standard ECHS 620 x 256 x 63 = 9,999,360
5,119,672,320 bytes 5.1GB
(4882.5MB 4.77GB)
loss = 4032 sectors 2,064,384 bytesRevised ECHS 661 x 240 x 63 = 9,994,320
10585 x 15 x 63
loss = 9072 sectors 4,644,864 bytes
Factory CHS 4092 x 16 x 63 = 4,124,736 sectorsECHS 1023 x 64 x 63 = 4,124,736 sectors - no loss
4092 is divided by 4 to get cylinders to 1024 or below. 4092 is divisible by 4 with no remainder, so there is no rounding.