How precise is the internal clock of a modern PC?

It really is a feature of the history of the PC.
The original IBM-PC used a chip called the Real Time Clock which was battery backed up (Do you remember needing to change the batteries on these ?) These operated when the machine was powered off and kept the time. The frequency of these was 32.768 kHz (2^15 cycles/second) which made it easy to calculate time on a 16 bit system.
This real time clock was then written to CMOS which was available via an interrupt system in older operating systems.

A newer standard is out from Microsoft and Intel called High Precision Event Timer which specifies a clock speed of 10MHz
Even newer PC architectures take this and put it on the Northbridge controller and the HPET can tun at 100MHz or even greater.
At 10Mhz we should be able to get a resolution of 100 nano-seconds and at 100MHZ we should be able to get 10 nano-second resolution.

The following operating systems are known not to be able to use HPET: Windows XP, Windows Server 2003, and earlier Windows versions, older Linux versions

The following operating systems are known to be able to use HPET: Windows Vista, Windows 2008, Windows 7, x86 based versions of Mac OS X, Linux operating systems using the 2.6 kernel and FreeBSD.

With a Linux kernel, you need the newer “rtc-cmos” hardware clock device driver rather than the original “rtc” driver

All that said how do we access this extra resolution?
I could cut and paste from previous stackoverflow articles, but not – Just search for HPET and you will find the answers on how to get finer timers working

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