The Beta configuration is the classic Stirling engine configuration and has enjoyed popularity from its inception until today. Stirling's original engine from his patent drawing of 1816 shows a Beta arrangement. A photograph of Robert Stirling, the original patent drawing, as well as an animated model of Stirling's engine is clearly shown in an interesting website by Bob Sier. From the figure below we see that unlike the Alpha machine, the Beta engine has a single power piston and a displacer, whose ideal purpose is to "displace" the working gas at constant volume, and shuttle it between the expansion and the compression spaces through the series arrangement cooler, regenerator, and heater. In actual engines the linkage driving the piston and displacer will move them such that the gas will compress while it is mainly in the cool compression space and expand while in the hot expansion space. This is clearly illustrated in the adjacent animation which was produced by Richard Wheeler (Zephyris) of Wikipedia.
A detailed description of the ideal Beta machine cycle is presented in our Engineering Thermodynamics - Chapter 3b web resource. Refer also to the animation of the Beta machine by Matt Keveney - Single Cylinder Stirling Engine, showing clearly the principle of operation.
Apart from Stirling's original engine, an important early Beta engine is Lehmann's machine on which Gusav Schmidt did the first reasonable analysis of Stirling engines in 1871. Andy Ross has recently built a small working replica of the Lehmann machine, as well as a model air engine, both based on single cylinder Beta configuration.
Rolf Meijer of Philips, Holland, developed his famous vibrationless rhombic drive Beta engines in the early 1960s. A detailed description of this engine can be found in the Beta model Stirling engine website by Pierre Gras.
Probably the most ingenious Stirling engines yet devised are the free-piston engines invented and developed by William Beale at Ohio University in the late 1960s. Legend has it that while teaching about the rhombic drive engine he suddenly realised that "this engine will still run if we simply throw away this complex drive mechanism - Eureka!". He then formed the company Sunpower, Inc., which has been the leader in the development of free-piston Stirling engines and cryocoolers to this day. Most of Sunpower's engines are Beta arrangements and employ no mechanical linkage system. The main aspect of the free piston machine is that the output power can be obtained through a linear alternator, allowing the entire system to be hermatically sealed. Sunpower is also involved in the manufacture Stirling cycle croygenic coolers for liquifying oxygen. Over the years Sunpower has transformed Athens, Ohio into a hotbed of Stirling cycle machine activity, which now includes three R&D/manufacturing companies.
Stirling Technology, Inc. is a spinoff of Sunpower, and was formed in order to continue the development and manufacture of the 5 kW ST-5 Air engine. This large Beta type engine burns biomass fuel (such as sawdust pellets or rice husks) and can function as a cogeneration unit in rural areas. It is not a free-piston engine, and uses a bell crank mechanism to obtain the correct displacer phasing.
Global Cooling, Inc is a spinoff of Sunpower, and was formed mainly in order to develop free-piston Stirling cycle coolers for home refrigerator applications. These systems, apart from being significantly more efficient than regular vapor-compression refrigerators, have the added advantage of being compact, portable units using helium as the working fluid (and not the HFC refrigerants such as R134a, having a Global Warming Potential of 1,300). A schematic diagram followed by an animated schematic of a typical cooler (both courtesy of Global Cooling) are shown below:
More recently Global Cooling decided to concentrate their development efforts on systems in which there are virtually no competitive systems - cooling between -40°C and -80°C, and they established a new company name: Stirling Ultracold.