BACKGROUND OF THE INVENTION This invention relates to refrigeration devices for operating at cryogenic temperatures, and, more particularly, to orifice pulse tube cryocoolers. This invention was made with government support under Contract No. W-7405- ENG-36 awarded by the U.S. Department of Energy. The government has certain rights in the invention. Significant effort has been expended to develop efficient and reliable cryocoolers for many applications where cryogenic temperatures are needed. Initially, development was driven by defense needs for effective optical sensors in the IR spectrum. Commercial electronics companies have recently funded cryocooler development in order to access the capabilities of cryogenic CMOS circuitry and the potential capabilities of high temperature superconductors operating at liquid nitrogen (as opposed to liquid helium) temperatures. Many designs and products for both Stirling engine coolers and orifice pulse tube coolers have been developed and applied. In general, Stirling devices have been found to be more efficient (a factor of 2 is quoted in some literature) than orifice pulse tube cooler devices. However, the orifice pulse tube approach has better reliability due to fewer moving parts (in some designs, no moving parts). In many applications, the vibrations from a Stirling device are unacceptable and the orifice pulse tube is the preferred approach.

The cryogenic/liquefied industrial gases industry consists of the liquefaction/separation of air, the liquefaction of hydrogen, the liquefaction of helium, and the liquefaction of petroleum gases. The majority of liquefied gas product is formed in large-scale plants where energy consumption and power efficiency are important concerns. While the overall cycle from raw material to

Air liquefaction plants use an isentropic expansion step for the final cooling. In this approach, the pre-cooled, compressed gas is expanded through a turbine. By performing work in passing through the turbine, a high degree of cooling of the gas is ensured. The turbine drives a compressor that compresses the overhead gas (that part of the gas flow that did not condense during expansion) prior to re-injecting it into the liquefaction flow stream. Most research on improving gas liquefaction technology appears to focus on improving the design of the turbo-expanders to achieve better work extraction and improved condensation. In the liquefied natural gas market, a few establishments use refrigeration machines to cool and condense the product gas. These refrigeration-based systems use proprietary mixtures of light hydrocarbons (propane, ethylene, methane) whose refrigeration cycle is intricately integrated with the cooling of the natural gas (from which these refrigerator working fluids are originally obtained). It is possible that these refrigerants could be replaced by cryocoolers provided the overall process obtains adequate condensation efficiency.

Applications of cryocooling to superconductors fall into two groups: cooling of electronic components incorporating superconductors and cooling of large scale superconductor windings used as electromagnets in such devices as MRIs, NMR, particle accelerators, and power generators. Applications for these components include the power industry, the medical/diagnostic industry, the analytical instrument industry, and the high energy physics industry. Essentially all existing devices use a passive cryogen supply system in which the superconductor is supplied with cryogen from a reservoir. The reservoir must be periodically resupplied by a liquified gas supply company.

For purposes of comparison to a pulse tube refrigerator (PTR), a Stirling refrigerator may be regarded as consisting of several aligned components: hot compressor, piston, hot heat exchanger, regenerator, cold heat exchanger, and cold expander piston. A conventional PTR 10 shown in Figure 1A operates similarly, except that the cold expander piston is replaced with four stationary

components: pulse tube 24 with heat exchanger 26, orifice 12, and reservoir 28. Hot compressor t4, hot heat exchanger 16, regenerator 22, and cold heat exchanger 18 complete PTR 10.

Stiriing refrigerators are more efficient than PTR refrigerators for three reasons. First, work is absorbed and dissipated into waste heat in orifice 12 of

PTR 10, whereas work is efficiently recovered at the cold expander piston of the Stirling refrigerator and delivered back to the hot compressor piston. Second, the effective thermal conductance of pulse tube 24 often puts a greater thermal load on cold heat exchanger 18 than does the heat generated by friction and other losses at the cold expander piston in the Stirling refrigerator. Third, control of the time-phase relationship between mass flow and pressure is easily accomplished in the Stirling refrigerator, but is limited in the PTR. In the Stirling refrigerator, mass flow phase leads the pressure phase at the hot heat exchanger and lags pressure phase at the cold heat exchanger. In conventional PTRs the mass flow phase lags the pressure phase at both the hot heat exchanger 16 and cold heat exchanger 18, as shown in Figure 1 B.

K. Kanao et al., "A Miniature Pulse Tube Refrigerator for Temperatures below 100 K," 34 Cryogenics, ICEC Supplement, pp. 167-169 (1994), reports that a PTR orifice can be replaced with a small tube connecting the pulse tube with the reservoir, where the flow impedance between the reservoir and the pulse tube is adjusted by selecting tubes of differing diameter and length to optimize PTR performance. Zhu et al., "Phase Shift Effect of the Long Neck Tube for the Pulse Tube Refrigerator," Proceedings of the 9 ^th International Cyrocoolers Conference held June 1996 (Preprint - to be published), further discusses the effect of a long neck tube inserted between the pulse tube hot end and the reservoir. Replacing the orifice with a long neck tube is taught to produce a phase shift that can be changed by changing the diameter and length of the long neck tube.

It will be appreciated that these references discuss only the effect of replacing conventional orifice 12 with a long neck tube connecting pulse tube 24

with reservoir 28. While PTR performance optimization is discussed, there is no discussion or analysis relating to the optimization. In accordance with the present invention, the effect of acoustic impedance on PTR performance is analyzed and a variable acoustic impedance is introduced to optimize PTR performance.

Accordingly, it is an object of the present invention to control the phase relationship between mass flow and pressure to improve the operating efficiency of a PTR.

Yet another object of the present invention is to recover power from the orifice end of the PTR.

Additional objects, advantages and novel features of the invention will be set forth in part in the description which follows, and in part will become apparent to those skilled in the art upon examination of the following or may be learned by practice of the invention. The objects and advantages of the invention may be realized and attained by means of the instrumentalities and combinations particularly pointed out in the appended claims.

SUMMARY OF THE INVENTION To achieve the foregoing and other objects, and in accordance with the purposes of the present invention, as embodied and broadly described herein, the apparatus of this invention may comprise a PTR having a pulse tube and a reservoir with a compliance value C. A variable acoustic impedance connects the pulse tube and the reservoir. The variable acoustic impedance includes a tube member that forms an inertance having a value L and a first variable acoustic resistance having a value R _s , wherein the acoustic impedance formed by the values C, L, and R _s has a phase angle that is variable to achieve optimum cooling efficiency.

In another aspect of the present invention, an acoustic transmission line connects the pulse tube and the driver for returning power from the pulse tube to the driver to further increase the PTR operating efficiency.

BRIEF DESCRIPTION OF THE DRAWINGS The accompanying drawings, which are incorporated in and form a part of the specification, illustrate the embodiments of the present invention and, together with the description, serve to explain the principles of the invention. In the drawings:

FIGURES 1A and 1B depict a prior art PTR and corresponding pressure- mass flow phase relationship.

FIGURES 2A and 2B depict a PTR according to the present invention and pressure-mass flow phase relationship adjusted for efficient operation.

FIGURES 3A and 3B depict exemplary embodiments of variable acoustic inertances.

FIGURE 4 depicts a PTR with an acoustic transmission line for energy return according to one aspect of the present invention. FIGURES 5A and 5B schematically depict a variable impedance network and phase diagram according to one embodiment of the present invention. FIGURE 6 is a complex phase diagram showing experimental results. FIGURE 7 is a complex phase diagram showing experimental results from an inertance with parallel variable valve resistances. FIGURES 8A and 8B are complex phase diagrams showing experimental results.

DETAILED DESCRIPTION OF THE INVENTION In efficient Stirling-cycle cryocoolers, the phase angle for the oscillating mass flow leads the phase angle for the oscillating pressure at the hot end of the regenerator, and lags behind it at the cold end. In orifice pulse tube refrigerators (PTRs), the mass flow phase angle leads the pressure phase angle at both ends of the regenerator, resulting in lower efficiency. The phase shift between oscillating pressure and oscillating mass flow at the cold end of the regenerator is determined in part by the purely resistive nature of the "orifice" of the orifice

pulse tube refrigerator, so that the pressure difference across the orifice is in phase with the mass flow through it. In accordance with our invention, the phase shift between mass flow and pressure at the cold end is shifted to the more efficient Stiriing values by adding an inertance in series with the orifice. The word "inertance" is an acoustics term connoting both inertia and inductance, because it is due to inertial effects of moving gas and is the acoustic analog of electrical inductance. In a further aspect of our invention, power previously dissipated in the orifice can be recovered by the system compressor through inertial effects in an acoustic transmission line. As illustrated in Figure 1A, the conventional orifice pulse tube refrigerator

10 (PTR) may be regarded as a conventional Stirling refrigerator in which the cold moving parts have been replaced by stationary components. The cold-end piston of the Stirling refrigerator is replaced with pulse tube 24, hot heat exchanger 26, orifice 12, and reservoir 28. Energy is supplied by compressor 14, which may be a conventional piston engine or a thermoacoustic engine. The basic operation of a PTR is reviewed in R. Radebaugh, "A Review of Pulse Tube Refrigeration," 35 Adv Cyrogenic Eng., pg 1191 (1990).

Referring now to Figure 2A, our invention provides inertance 32 in series with a resistive element 13 to shift gas mass flow phase at cold heat exchanger 18, as shown in Figure 2B. Resistive element 13 may be a valve, variable orifice, baffles, or any other device that provides a resistance to fluid movement. Further, as shown in Figure 4, the use of inertia in acoustic transmission line 34 can be used to feed some of the power that would otherwise be dissipated in orifice 12 or resistive element 13 (Figures 1A and 2A, respectively) back to compressor 14. In other words, our invention reduces the effects of two of the three causes of reduced efficiency generally discussed above for PTRs. For purposes of this description, like numbered parts in Figures 2A, 3A, 3B, and 4 perform like functions and may not be separately discussed for each figure.

In the harmonic approximation, where the oscillatory pressure and mass flow are considered to be essentially sinusoidal in time, a lumped-impedance

model closely analogous to a simple ac electrical circuit illustrates the principle. See, e.g. L. E. Kinsler et al., Fundamentals of Acoustics. Chapter 10, "Resonators, Ducts, and Filters," pp. 225-243, John Wiley and Sons (1982). In the conventional PTR, an orifice or valve forms a purely resistive impedance and the reservoir is a compliance, analogous to an electrical capacitor. Oscillating pressure is analogous to oscillating voltage, and oscillatory volumetric mass flow is analogous to oscillating current.

As is often done in analysis of ac electric circuits, complex variables represent amplitudes and phases of oscillatory quantities. The amplitude of the oscillating pressure is the amplitude of the oscillating volumetric mass flow is , and the phases of the complex numbers p. and U. reflect the time phases of the oscillations. The compliance of the reservoir is

C= Vlγp _m (1) where V is the volume of the reservoir, p _m is the mean pressure (i.e., the average pressure), and γ is the ratio of isobaric to isochoric specific heats. The adiabatic compressibility of an ideal gas is 1 / γ p _m . Just as in an electrical circuit, the complex impedance of a compliance is Z _c = 1 /jωC, where j = V-T and ω = 2πf, with / denoting the frequency of the oscillations. Hence, the impedance of a compliance is a negative imaginary number, so the impedance Z of the RC "circuit" formed by orifice 12 and compliance reservoir 28 (Figure 1 A) must lie in the fourth quadrant in a plot of lm(Z) vs. Re(Z), i.e., a negative phase shift in the complex impedance plane. In practice, before the present invention, the highest efficiency PTRs have required that Z be as real as possible, so the compliance reservoir volume was typically rather large to provide a large C and a concomitant small Z _c .

In one aspect of the present invention, an inertance 32 is placed in series with resistive element 13 (Figure 2A) to allow access to the first quadrant (positive phase shift) in the complex impedance plane. The simplest inertance is

a tube of length / and cross-sectional area A, in which the inertia of the moving gas contributes an inertance

L = pll A (2)

In K. M. Godshalk et al., "Characterization of 350 Hz Thermoacoustic Driven Orifice Pulse Tube Refrigerator with Measurements of the Mass Flow and Pressure," 4_1 Adv in Cryogenic Eng. (1996), pp. 1411-1418, the effect of the inertance of the pulse tube itself on the phase angle between the mass flow and pressure at the cold end of the pulse tube is discussed. It was found that a pulse tube could be designed to obtain phasing of the mass flow similar to the phasing found in a Stirling cycle refrigerator; i.e., the mass flow leads the pressure at the warm end of the pulse tube and lags the pressure at the cold end of the pulse tube. This was found to be feasible at 350 Hz due to the shorter wavelength of sound in helium at this high frequency.

The complex impedance of an inertance Z _L = jωL is proportional to frequency so that the relatively high operating frequency of the Godshalk device made the effect of the pulse tube inertance readily apparent. But conventional

PTRs operate at frequencies well below 350 Hz and typically below 120 Hz. Our invention permits inertance effects to be realized and phase control obtained at frequencies much lower than 350 Hz.

Figure 2A illustrates a PTR 30_having a separate inertance 32, in accordance with this invention. PTR 30 is generally a conventional PTR with hot end piston 14, a regenerator 22, hot 16 and cold 18 heat exchangers at opposite ends of regenerator 22, pulse tube 24 and hot heat exchanger 26. Inertance is introduced by the addition of separate acoustic inertance 32 in series with resistance 13 and compliance reservoir 28. As shown in Figure 2B, the mass flow of the operating fluid is advantageously phase shifted relative to the fluid pressure.

Inertance 32 and orifice 13 may be formed as a variable complex impedance network, e.g., with a fixed inertance and variable acoustic resistance elements R _s and R _p as shown and discussed below for Figure 5A. However, it

will be appreciated from the discussion below that substantial areas of the first quadrant of the complex acoustic impedance plane can be accessed by a network having two variable values: a fixed inertance and variable resistors R _s and R _p , a variable inertance and variable ft., or a variable inertance and variable R _p .

A variable inertance may be formed as shown in Figures 3A and 3B. Figure 3A illustrates a variable length tube, which may be conveniently formed by a trombone slide 42, interacting with fixed tube segments 44A and 44B. Figure 3B illustrates a variable area tube, where rod 48 slides within tube 46 to obtain a variable average area for fluid flow in tube 46 and a concomitant variable acoustic inductance.

In one exemplary design of a large PTR with 4 kW of gross cooling power, at the hot end of the regenerator and I p _m ~ 0.09 at the entrance to the orifice, it is estimated that the power required to drive the refrigerator would be reduced by up to 20% when the phase of Z, at the orifice, is shifted from 0° to +25°.

To verify that the realization of such a phase shift is possible, without investment in expensive hardware, an impedance network was constructed having globe orifice valves for resistance R _s and R _p , a right circular cylinder reservoir for compliance C, and tubing for inertance L . The apparatus was designed to be similar, in the strictest technical sense of the word, to the design required for the 4 kW helium design, but at greatly reduced power. A half-scale model was built, filled with 2.5 MPa argon and operated at 23 Hz with p _m ~ 0.09. This is accurately similar to the full-scale, 40 Hz, 3 MPa helium design with 0.09. The primary advantage of the half-scale model is that all powers are reduced by a factor of 16, so that only 250 W must be supplied to test the behavior of the 4 kW application. All dimensionless variables, such as Reynolds' number, Mach number, and length ratios, are identical so that the physics is the same as for full scale.

For the test apparatus, a thermoacoustic driver was used (see, e.g., G. Swift, "Analysis and Performance of a Large Thermoacoustic Engine," 92 J.Acoust. Soc. Am., pp 1405 (1992)) rather than a piston drive. But the nature of the source of oscillating pressure is irrelevant for this invention. In a first 23 Hz test apparatus, the inertance was a 2 meter length of copper tubing with 1.1 cm inside diameter, so that L=8.6x10 ⁵ kg/m ⁴ and α?L=1.2x10 ⁸ Pa-sec m ³ . With / approximately equal to λ l2π (where λ = a l f is the wavelength of sound and a is the speed of sound), this inertance actually has some transmission line characteristics (a 4% effect). The compliance reservoir was a right circular cylinder with internal volume V=2.3x10 ^"3 m ³ , so that

C=5.5x10- ¹⁰ m ⁴ sec ² /kg and -/ΛuC=1.2x10 ⁷ Pa-sec/m ³ .

To permit variation of both magnitude and phase of the complex impedance, an acoustic impedance network with inertance L and compliance C was formed in the configuration shown in Figure 5A, with two variable resistances (e.g., valves) R _p and R _s . This configuration allowed the complex impedance Z of the network to be set at desired points within the shaded area in Figure 5B. When the value of R _p is infinite (i.e., valve R _p is closed), only the series combination of R _s , L, and C contributes to the impedance. Since L and C are fixed by physical dimensions and gas characteristics, this case provided an upper bound for the imaginary part of Z in Figure 5B:

Im(Z) < coL - 1 / ω C ~1.1 x10 ^β Pa-sec/m ³ . Similarly, when R _s is infinite, only R _p and C contribute to the impedance. This provided the lower bound in Figure 5B:

Im(Z) > -1 lωC~ -1.2x10 ⁷ Pa-sec/m ³ . Dynamic pressure transducers measuring p _{E 1} and p _{c 1} were located at the entrance to this impedance network and in the compliance, respectively, in order to determine the network impedance Z = p _E _ I U _l and the power

E = - e(/?£ ₎ ιf ι) absorbed by the impedance, where the tilde denotes the

complex conjugate and Re denotes the real part. The volumetric mass flow U was determined from the pressure oscillations in the compliance: The volumetric mass flow £/, into the compliance delivers an extra mass m- = pU- 1 jω to the compliance, so the gas density in the compliance oscillates as p ι= m ^→ I V = ρU _\ I jωV . These oscillations in the compliance are nearly adiabatic, so the compressibility of the gas in the compliance is \ lγ p _m . Hence the oscillating density p causes an oscillating pressure

Pc,\ - P _\ ϊ Pm ¹ P ⁼ U _\ Y Pm I ) ^{ω V} ■ Solving for volumetric mass flow yields

The complex impedance and the power are therefore given by

and

One-inch globe valves were used as the variable resistances R _p and R _s in the first test apparatus. Copper tubing through which cooling water flowed was wrapped around the valves to remove the heat generated by the oscillating gas passing through the valves. The network end of the thermoacoustic engine was similarly wrapped for cooling purposes. The valves were set at a variety of openings, the thermoacoustic driver was adjusted to maintain p _m , the complex pressures p _£ , and p _{c 1} were measured, and values of impedance were calculated using Eq. (4). The results are shown in Figure 6. The circles tangent to the imaginary axis in Figure 6, calculated using Eq. (5), represent contours of equal power delivered to the network at an acoustic pressure amplitude p _{E 1} =0.08p _m .

In Figure 6, the experimental data are separated into two categories. With

R _s exercised and R _p closed, the data are denoted as "Inertance Only" (open circles). These data are close to the expected upper bound of lm(Z) discussed above. When R _p and R _s are both exercised, the data are denoted as "Combined" (open squares). These data cover a wide area of the complex impedance plane; intermediate points in the plane can be reached by suitable choices of openings of the two valves.

The leftmost "Inertance only" point represents the impedance with valve ft _p closed and valve R _s fully open. In this case, Re(Z) represents the loss in the inertance itself due to turbulence and viscous dissipation in the tube. For this large tube, the internal loss is small enough that the phase of Z can be greater than 85°.

As shown in Figure 6, it is possible to adjust both the phase and magnitude of the network impedance over broad ranges, which is identical to adjusting the complex ratio of pressure amplitude to volumetric mass flow in a

PTR load. Because of the scale factor (factor of 16 here), this invention clearly will work for a 4 kW application using helium.

For a large PTR with the configuration shown in Figure 5A, currently being tested with 3 MPa helium at 40 Hz, values of Z determined from Eq. (4) and using measured values of p _E , and p _c , are shown in Figure 7. These values of

Z were obtained by adjustment of R _s and ft _p over selected, narrow ranges of all possible adjustments. Positive phase shifts were clearly obtained. One exemplary operating point (highlighted at lower right in Figure 7) provided 1275 W of cooling power as measured by the liquefaction rate of a natural gas stream, while requiring 13 kW of input power to the PTR from a thermoacoustic driver. Another exemplary operating point (highlighted at upper left in Figure 7) provided 1515 W of cooling power while requiring 12.6 kW of input power. This is the most powerful PTR ever reported.

Most applications for PTRs require orders of magnitude less gross cooling power than 4 kW, and hence orders of magnitude larger impedances. Difficulties

arise at large inertance L = pll A, for two reasons: / cannot be increased beyond roughly λllπ, and A cannot be decreased to the point that dissipative effects overwhelm inertial effects.

Larger values for L, R _p and ft, were tried in the above test apparatus for a small PTR. Tubing inner diameters of 3.3 mm and 1.7 mm were used, each 2 m long, resulting in L=9.6x10 ⁶ kg/m ⁴ , and L=3.6x10 ⁷ kg/m ⁴ , respectively. Data for each are presented in Figures 8A and 8B.

In each case, the small section of tubing required when connecting valve R _p to the compliance has sufficient nuisance inertance to overwhelm the large compliance (which was sized for the earlier experiments shown in Figure 6); accordingly, impedance data denoted as "R _s Closed" have positive imaginary Z. As evident in Figures 8A and 8B, a variety of phase angles and powers delivered to the load are available using appropriate adjustments of the valves R _p and R _s . Indeed, it appears that all impedances between the triangles and the circles are accessible by suitable adjustment of R _p and R _s .

Figure 8B shows that the phase of Z can be near 70° even for scale powers as low as 2 W (equivalent to an operating power of 32 W in a full-scale helium PTR), further showing the usefulness of inertance for improving the efficiency of PTRs. Further use of inertial effects to improve PTR efficiency is illustrated in

Figure 4, where an acoustic transmission line 34 feeds power that would otherwise be dissipated in an orifice in previous PTRs back to volume 36 for input to compressor 14. Acoustic transmission line 34 is analogous to an electric transmission line; its nature is most simply appreciated by a lumped LC impedance approximation. An acoustic transmission line can transmit acoustic power while changing the relative magnitude and phase of p. and U..

The boundary between the "inertance" picture and the "transmission line" picture is not exact, but roughly occurs near I * λllπ: shorter lengths for acoustic transmission line 34 behave essentially as lumped inertances with U

independent of position in the tube, and longer lengths behave essentially as transmission lines, with significant dependence of U _λ on position. However, lumped inertances such as the 2-m long tubes discussed above have slight transmission-line character, and transmission lines with length not much greater than λllπ and located not too near a pressure anti-node have much in common with lumped inertances.

Figure 4 illustrates feedback of power to the back side 36 of compressor piston 14- Most piston compressors in use for PTRs have such a "back side" already, filled with the working gas, so that leakage past compressor piston 14 does not result in loss of working gas from the system and so that the piston does not have to support a large average pressure difference, such as from 3 MPa to atmospheric pressure.

Typically, back side 36 of compressor 14 contains a volume of working gas roughly comparable to the volume of the PTR 40 assembly. In one computer simulation, it was assumed that back volume 36 was three times larger than PTR 40 volume, so that the amplitude of the pressure oscillations in back volume 36 was about 1/3 the amplitude of the oscillations in PTR 40, and about 180° out of phase. This calculation was done for a 40 Hz, 3.1 MPa helium PTR having almost 1 kW of cooling power. The pressure oscillations at hot end 16 of regenerator 22 were assumed to be 310 kPa in amplitude, with a phase of -90°

(with respect to an arbitrary reference phase). PTR design calculations indicated that the pressure oscillations at end 26 of pulse tube 24 should be 276.5 kPa in amplitude at a phase of -92.4°, with a volumetric mass flow of 8.32x10 ³ m ³ /s at -122.0° for best thermodynamic performance of PTR 40. Under these conditions, 1 kW of acoustic power must be removed from end 26 of pulse tube 24; without the use of transmission line 34 (i.e., in a configuration such as Figure 2A) this power would be absorbed in orifice 13 and converted into waste heat. Compressor back volume 36 ensured that the pressure oscillations in back volume 36 would have an amplitude of 100 kPa at a phase of +90°. Hence, the design of transmission line 34, assuming that it had

to connect between a first location with 276.5 kPa in pressure amplitude at a phase of -92.4°, with volumetric mass flow 8.32x10 ³ m ³ /s at a phase of -122°, and a second location with 100 kPa at +90°, causes line 34 to transmit as much of the 1 kW power as possible from the first location to the second location. Calculations show that a transmission line comprised of two tubes in series (one tube with 1.65 cm diameter and 7.89 m long connected in series with a downstream tube with 3.71 cm diameter and 3.44 m long) will deliver 814.9 W of acoustic power back to the compressor. The remaining

185.1 W is absorbed by turbulent losses in transmission line 34 and converted to waste heat.

Power can also be fed back to a thermoacoustic compressor, such as would be used in the thermoacoustic cryocooler described in U.S. Patent 4,953,366. In this case, transmission line 34 can be attached to the thermoacoustic resonator at a location where the pressure amplitude in the standing wave has a suitable amplitude and phase.

It is important to realize that this feedback-of-power idea achieves two efficiency advantages for PTRs: It retains the advantage of favorable phase shift between p, and U ₁ at the cold end of the PTR, described in the inertance discussions above, and it provides additional efficiency improvement by returning power to the compressor which would otherwise be dissipated in a prior-art PTR or in a PTR with simple inertance.

The embodiment of Figure 5A with both R _s and R _p variable (such as by using valves), is preferred for cases where the PTR must be operated at various cooling powers and cold temperatures, because this embodiment provides real- time control of both magnitude and phase of Z. An ft-L series embodiment with

R variable and L variable (see Figures 3A and 3B) may also be used for the same situations.

For power feedback to the driver, an acoustic transmission line in the form of one tube or two tubes in series is preferred for simplicity and high efficiency. More complicated systems, including lumped LC systems, would also work.

Adjustability can be provided with one or more valves and/or tubing have variable length and/or area, as discussed above for variable inertance.

Accordingly, a primary advantage of our invention is increased efficiency in PTRs by providing optimal phasing between _f and U. at the cold end of the regenerator, without in any way compromising the simplicity and low cost of

PTRs. This advantage is especially important for PTRs with very low cold temperature such as 100 K.

A secondary advantage is the ability to reduce the size of the compliance reservoir without decreasing the efficiency. This is possible because a positive imaginary impedance jωL can cancel a negative imaginary impedance 1 / jωC.

Another secondary advantage is the freedom to enlarge the pulse tube without sacrificing phase shift between pressure and mass flow at the cold heat exchanger. An enlarged pulse tube may permit an increase of the ratio of net to gross cooling power, with even higher resultant PTR efficiency. An advantage of the two-valve embodiment or the valve-plus-slide- trombone embodiment is the ability to adjust both magnitude and phase of the total impedance over wide ranges while the PTR is operating.

A primary advantage of the acoustic transmission line feedback is a further increase in efficiency. For large PTRs, this is as important as the primary advantage, above; for small PTRs, this is less than the primary advantage. For large PTRs at relatively high cold temperature, such as near 200 K, for food deep freezers or for precooling of gas in preparation for liquefaction in a further stage of refrigeration, feedback (with due attention to the phase between p, and U ₁ at the cold end) can improve the efficiency of PTRs by a factor of 2. A further advantage of the acoustic transmission line feedback is the elimination of the compliance reservoir.

The foregoing description of the invention has been presented for purposes of illustration and description and is not intended to be exhaustive or to limit the invention to the precise form disclosed, and obviously many modifications and variations are possible in light of the above teaching. The

embodiments were chosen and described in order to best explain the principles of the invention and its practical application to thereby enable others skilled in the art to best utilize the invention in various embodiments and with various modifications as are suited to the particular use contemplated. It is intended that the scope of the invention be defined by the claims appended hereto.