
741 F.Supp. 1426 (1990)
MICRO MOTION, INCORPORATED, Plaintiff,
v.
EXAC CORPORATION, Defendant.
No. C-89-1825 WHO.
United States District Court, N.D. California.
June 28, 1990.
*1427 John W. Keker, Paul V. Carroll, Keker & Brockett, San Francisco, Cal., John O. Tramontine, Jesse J. Jenner, David J. Lee, Fish & Neave, New York City, for plaintiff.
Sheldon W. Witcoff, Daniel A. Boehnen, Allegretti & Witcoff, Ltd., Chicago, Ill., Charles B. Cohler, Richard Haas, Lasky, Haas, Cohler & Munter, San Francisco, Cal., for defendant.

OPINION AND ORDER
ORRICK, District Judge.

I. BACKGROUND OF THE LITIGATION
This is an action for patent infringement involving devices known as nonintrusive Coriolis mass flowmeters, which measure the mass flow rate of fluids. Plaintiff, Micro Motion, Incorporated ("Micro Motion"), is a Colorado corporation with its principal place of business in Boulder, Colorado. Since 1977, Micro Motion has manufactured and sold Coriolis mass flowmeters and accessories. Micro Motion is a wholly-owned subsidiary of Emerson Electric Company. Defendant, Exac Corporation ("Exac"), is a California corporation, incorporated in 1983, with its principal place of business in San Jose, California. In 1983, Exac began the design of a Coriolis mass flowmeter, and, since 1984, Exac's entire business has been the manufacture and sale of Coriolis mass flowmeters and accessories. Exac is a wholly-owned subsidiary of Fisher Controls International, Inc., which in turn is a wholly-owned subsidiary of Monsanto Company.
Micro Motion filed suit against Exac alleging that Exac's 1100 and 2100 flowmeters[1] infringe Claims 8 and 57 of Micro *1428 Motion's U.S. Reissue Patent No. 31,450 ("'450 patent") and claim 1 of U.S. Patent No. 4,491,025 ("'025 patent") under the doctrine of equivalents.
Exac denies that its devices infringe Micro Motion's patents.[2] Exac's pleadings initially asserted that the patents in suit were invalid and unenforceable, but these issues have been resolved against Exac, and Exac no longer contests the enforceability or validity of Micro Motion's patents.[3]
This matter was initially tried to a jury between June 30 and July 27, 1987, with Judge Spencer M. Williams presiding. At that trial, Micro Motion alleged both literal infringement and infringement under the doctrine of equivalents. On July 29, 1987, the jury returned a verdict that none of the asserted patent claims had been infringed literally or under the doctrine of equivalents by the Exac meters.
Micro Motion moved for a new trial. On December 21, 1987, Judge Williams granted the motion and vacated the first jury verdict in its entirety. He also limited the issues on retrial to infringement under the doctrine of equivalents because Micro Motion had decided to waive the literal infringement issue. Judge Williams subsequently recused himself from the case, and it was reassigned to this Court. The parties filed a stipulated waiver of trial by jury, and the retrial before the Court commenced on March 12, 1990.
After carefully considering the sufficiency, weight, and credibility of the testimony of the witnesses, their demeanor on the stand, the documentary evidence admitted at trial, and the post-trial submissions of the parties, the Court finds that Micro Motion's patents in suit have been infringed under the doctrine of equivalents. This Opinion and Order comprises the findings of fact and conclusions of law required by Federal Rule of Civil Procedure 52(a).

II. CORIOLIS FLOWMETERS
Micro Motion is the owner of the '450 reissue patent, entitled "Method and Structure for Flow Measurement." PX 1.[4] The Coriolis mass flowmeter of the '450 patent was invented by James Smith, founder and President of Micro Motion. The '450 reissue patent issued on November 29, 1983, on an application for reissue of original U.S. patent 4,187,721 (issued on February 12, 1980). PX 2.
Micro Motion is also the owner of the '025 patent, entitled "Parallel Path Coriolis Mass Flow Rate Meter." PX 3. The Coriolis mass flowmeters disclosed and claimed in the '025 patent were invented by Mr. Smith and Donald Cage, a Micro Motion engineer. The '025 patent issued on January 1, 1985.
*1429 These patents describe Coriolis mass flowmeters. Coriolis mass flowmeters are meters that, when placed in a pipeline carrying a fluid, directly and nonintrusively measure the rate of mass flow of that fluid through the pipe. RT-WHO 244-45.
Before direct mass flowmeters became commercially available, mass flow was measured indirectly with volumetric flowmeters. These meters were able to measure mass flow by utilizing two devices, one to measure fluid volume and one to measure fluid density. Mass flow could then be calculated mathematically from these two measurements. Volumetric flowmeters, however, tended to be costly and/or inaccurate, and the two devices required significantly more maintenance than a single meter. RT-WHO 41, 84-86, 126-27, 170, 180-81, 1131, 1138.
In order to accurately measure mass flow with a single device, Coriolis mass flowmeters were developed. These meters take advantage of a phenomenon known as the Coriolis force. Coriolis forces result when a mass moves radially from one point on a rotating surface to a second point. As the surface rotates, the velocity of the mass changes and the mass is accelerated in a direction perpendicular to its radial path. The acceleration of the mass generates a Coriolis force, which acts on the mass in the surface created by the rotation and perpendicular to the instantaneous radial movement. RT-WHO 249-57; PX 12.
Coriolis mass flowmeters, when inserted in a pipeline, direct fluid through a curved conduit that conducts the fluid away from the pipeline and then back to the pipeline. As the fluid is conducted through the conduit, the conduit is oscillated. RT-WHO 253-54. When the conduit oscillates in one direction, the Coriolis force acts in that direction on the fluid flowing away from the pipeline, and acts in the reverse direction on the fluid flowing back toward the pipeline. This causes the conduit to twist in response to these oppositely-directed forces. RT-WHO 255-57, 259, 271.
The amount of Coriolis-induced twist for a given angular velocity is proportional to the mass flowing through the oscillating conduit. Therefore, the greater the mass flow rate, the greater the Coriolis-induced twist. By measuring the amount of twist, it is possible to calculate the mass flow rate of the fluid. RT-WHO 258.[5]
The amount of twist caused by the Coriolis force can be measured by positioning sensors on opposite sides of the oscillating conduit, so that the two sensors differentially detect and measure the opposing movements of the conduit as the fluid is flowing. These sensors measure the difference between the time when one side of the conduit passes by a reference point and the time when the trailing side of the conduit passes that same reference point. The movement of each side of the conduit is essentially sinusoidal, and the output from each sensor can therefore be traced in the form of superimposed sine waves representing each side of the conduit. RT-WHO 260. The time difference between the superimposed waves can then be measured. This measurement is generally referred to as "delta t." RT-WHO 261-62. Delta t is proportional to the mass flow rate. RT-WHO 277-78.
The relationship between Coriolis-induced distortion and mass flow rate was known before any of the devices in suit were invented. For example, Sipin U.S. Patent 3,485,098 ("Sipin '098") predated Micro Motion's patents and includes the general principle of an oscillating mass flowmeter with curved conduits. PX 6; RT-WHO 272, 274-75. Pearson U.S. Patent 2,624,198 ("Pearson '198") and Roth U.S. Patent 2,865,201 ("Roth '201") also predated Micro Motion's patents and claimed to describe nonintrusive mass flowmeters. PX 4, 5; RT-WHO 273-74. The need for the conduit to direct fluid flow out from the pipeline and then back toward it was also known before Micro Motion's patents issued. RT-WHO 278.
*1430 While the necessary technology was accessible, however, no commercially successful nonintrusive mass flowmeter was manufactured before Micro Motion developed its meter. RT-WHO 43-44, 66, 135. Exac's own documents and witnesses have conceded this point. RT-WHO 1155-56; PX 54 at 16; PX 143 at 70. Early nonintrusive meters by Li and Bendix that measured mass flow directly were commercially unacceptable. RT-WHO 135-37. No evidence was presented at trial to demonstrate that flowmeters based on the Pearson '198 or Sipin '098 patents were ever made. The Decker Corporation built and tested prototypes of a flowmeter based on the Roth '201 patent, but there is no evidence that this meter was ever commercially produced. PX 5; Roth Dep. at 42-44. In fact, Micro Motion's first meters were initially greeted with skepticism by the industry. RT-WHO 137, 1155.
Micro Motion began producing Coriolis mass flowmeters in 1977. RT-WHO 46-48, 58. These early meters (Micro Motion Model B and Model C) were covered by the '450 patent, and contained a single, U-shaped flow tube. The commercial embodiment of the '450 patent used in the Model B meter utilized optical sensors to measure the Coriolis-induced twist. The commercial embodiment of the '450 patent found in the Model C meter retained the U-shaped flow tube but substituted magnetic velocity sensors for the optical sensors. RT-WHO 59-65, 302-03. Both of these commercial embodiments measured delta t at the point where the electrical signal passed through the zero-voltage point at the maximum rate, which occurred in these devices at the midplane of oscillation. RT-WHO 493, 587, 1900-01.
In 1983, Micro Motion introduced its Model D meter, covered by the '025 patent. This commercial embodiment of the '025 patent contained two parallel U-shaped flow tubes through which the fluid flowing through the pipe was directed. The Model D meters also used two magnet-and-coil velocity sensors, with the magnetic part of the sensor mounted on one conduit and the coil part mounted on the other conduit. PX 3. Like the Model B and C meters, this commercial embodiment measured delta t at the midplane of oscillation.
The Exac mass flowmeters in suit were designed by Exac's founders, Drs. Erik Dahlin and Alan Young. During the autumn of 1982, Drs. Young and Dahlin discussed the development of a mass flowmeter, and they decided to form a business in order to market such a meter. In March 1983, Dr. Dahlin met with venture capitalists to seek funding for the business. In April 1983, Dr. Dahlin submitted a business plan to these potential investors. At this time, Exac had not yet built a meter. RT-WHO 844-45; PX 20; Case Dep., Vol. I at 5-6, 9-10, 13-14, 59-60; Dahlin Dep., Vol. I at 49-51, 77, 356.
Prior to 1982, Drs. Young and Dahlin had experience with the Micro Motion C meter while employed by Zikonix. RT-SW 2447-53; PX 20 at 3. In 1982, Dr. Dahlin obtained and studied Micro Motion's '721 patent, later reissued as the '450 patent. In early 1983, Dr. Dahlin obtained and studied another patent, Cox U.S. Patent No. 4,127,028 ("Cox '028"), which depicts a flat loop and a crossover loop and whose text discloses the use of electromagnetic sensors in place of photo-electric sensors in a Coriolis mass flowmeter. RT-SW 2453-54; PX 7.
In June 1983, Dr. Dahlin asked Mr. Joseph Bottom of Microbio Resources to obtain for him information on the Micro Motion D meter. Mr. Bottom subsequently met with a Micro Motion salesman, who described and demonstrated the Model D meter and sent him information, literature, and drawings of the meter. Mr. Bottom in turn passed this information on to Dr. Dahlin in June 1983. RT-WHO 193-96, 199-200; PX 26. The first sketch by Dr. Dahlin of a Coriolis meter having two tubes with a split parallel flow path was made in mid-July 1983. RT-SW 2500.
Exac opened for business in July 1983. In mid-July 1983, Dr. Young asked Mr. William Tanner of Sensoray Company to order a Micro Motion C meter, without revealing that Exac was the intended recipient of the meter. RT-WHO 205-09; PX *1431 36, 38. Mr. Tanner placed this order and, in so doing, made several false statements to Micro Motion. RT-WHO 208-13; PX 38, 49. Mr. Tanner received the C meter on August 22, 1983, and delivered it to Exac. RT-WHO 210-11; PX 40, 41.
In September 1983, Dr. Young asked Mr. Tanner to order a D meter from Micro Motion, again without revealing Exac as the intended recipient of the meter. Young Dep., Vol. I at 392-94; RT-WHO 213-15; PX 43, 45. Mr. Tanner received a Model D meter on December 9, 1983, and delivered it to Exac. RT-WHO 215-16; PX 51, 52. Drs. Dahlin and Young testified that upon receiving the Micro Motion meters, they took the meters apart and studied them. Dahlin Dep., Vol. I at 84-85; Young Dep., Vol. I at 120-21.
Exac completed its first working model for a helical crossover loop mass flowmeter in late November or early December 1983. RT-WHO 852. Exac began commercially selling these meters approximately one year later. RT-WHO 852.
Exac contends that its devices differ from the Micro Motion patents in several respects, and that these differences make its meters noninfringing. These asserted differences are listed below, and are discussed in more detail in Section III, infra.
First, the Exac devices employ two parallel helical crossover loops[6] through which fluid is directed, rather than the "U"-shaped conduits used in the Micro Motion meters. RT-WHO 846, 849-50, 866; PX 95. The Exac devices also measure the Coriolis force in terms of "phase angle," which is a combination of frequency and delta t, rather than measuring delta t alone. RT-WHO 884-85, 897, 979-81, 1047; DX RT at 34. The phase angle is measured at the extreme of conduit oscillation, rather than at the midplane. DX RT at 9, 18, 35, 38-41. The Exac devices allegedly enlarge and magnify the Coriolis effect through mechanical amplification. RT-WHO 873-74. Finally, the Exac meters use a microprocessor to correct for the nonlinearity of operation of the meters, whereas Micro Motion's meters accurately measure mass flow without the need for correction by a microprocessor. RT-WHO 891; DX RT at 19, 29.[7]

III. INFRINGEMENT UNDER THE DOCTRINE OF EQUIVALENTS

A. General Legal Principles


1. The Legal Standard: Function, Way, Result
Micro Motion waived any claim that Exac's meters literally infringe the '450 or '025 patents. Rather, Micro Motion argues that the Exac meters infringe its patents under the doctrine of equivalents.
Infringement under the doctrine of equivalents is reached only when there is no literal infringement. Seattle Box Co. v. Industrial Crating & Packing, Inc., 731 F.2d 818, 828 (Fed.Cir.1984). Thus, the literal language of the patent does not define the "metes and bounds" of the claims when determining equivalence. Thomas & Betts *1432 Corp. v. Litton Systems, 720 F.2d 1572, 1579 (Fed.Cir.1983).
The doctrine of equivalents is a judicially-created equitable doctrine designed to prevent an infringer who does not literally infringe an invention from nonetheless "stealing the benefit of an invention." Graver Tank & Mfg. Co. v. Linde Air Products Co., 339 U.S. 605, 608, 70 S.Ct. 854, 856, 94 L.Ed. 1097 (1950) (quoting Royal Typewriter Co. v. Remington Rand, 168 F.2d 691, 692 (2d Cir.), cert. denied, 335 U.S. 825, 69 S.Ct. 50, 93 L.Ed. 379 (1948)). It represents an exception to the general rule that a patent's claims define the metes and bounds of patent protection. Texas Instruments, Inc. v. United States Int'l Trade Comm'n, 805 F.2d 1558, 1572 (Fed.Cir.1986). Because it is a relatively narrow exception, it should not be used to defeat the legitimate process of "inventing around" an adversely held patent. Id.
Under the doctrine of equivalents, infringement may be found if the Exac devices perform substantially the same function as the Micro Motion inventions, in substantially the same way, to obtain substantially the same result. Graver Tank, 339 U.S. at 608, 70 S.Ct. at 856 (citing Sanitary Refrigerator Co. v. Winters, 280 U.S. 30, 42, 50 S.Ct. 9, 13, 74 L.Ed. 147 (1929)). If any one of these three elements of equivalence are not present in the accused device, then there is no infringement of that claim. Lear Siegler, Inc. v. Sealy Mattress Co., 873 F.2d 1422, 1425-27 (Fed.Cir.1989). Micro Motion has the burden to prove infringement under this doctrine by a preponderance of the evidence. SRI Int'l v. Matsushita Elec. Corp., 775 F.2d 1107, 1123 (Fed.Cir.1985).[8]
Determination of infringement under this doctrine is a two-step process. First, the literal language of the claims must be construed to determine their meaning and scope. Any definitional questions are to be resolved by reference to the patent specifications, its prosecution history, the prior art, and expert testimony. PX 277, ¶ 4; RT-WHO 1573-75. Second, the construed claims must be compared with the accused device, to determine whether the accused device includes each element of the claims or its substantial equivalent. Snellman v. Ricoh Co., 862 F.2d 283, 286 (Fed.Cir.1988). There can be no infringement unless each and every claimed element of each asserted claim or its substantial equivalent is present in the accused device. Lemelson v. United States, 752 F.2d 1538, 1550-51 (Fed.Cir.1985). Where the construed claims are different from the accused device, only insubstantial changes between the construed claims and the accused device constitute infringement under this doctrine. Pennwalt Corp. v. Durand-Wayland, Inc., 833 F.2d 931, 934-35 (Fed. Cir.1987) (en banc), cert. denied, 485 U.S. 961, 108 S.Ct. 1226, 99 L.Ed.2d 426 (1988).
An equivalent of a claimed invention need not be described in the patent to constitute an infringement. D.M.I., Inc. v. Deere & Co., 755 F.2d 1570, 1574 (Fed.Cir. 1985). The trial court may, however, rely on a reference in the patent itself that discloses the accused element as an equivalent to the patented element in determining equivalence. See, e.g., Linde Air Products Co. v. Graver Tank & Mfg. Co., 86 F.Supp. 191, 199 (N.D.Ind.1947), aff'd in pertinent part, 167 F.2d 531 (7th Cir.1948), aff'd in pertinent part, 336 U.S. 271, 69 S.Ct. 535, 93 L.Ed. 672 (1949).
The law does not require proof of bad faith to establish infringement. Wilden Pump & Eng'g Co. v. Pressed & Welded Products Co., 655 F.2d 984, 989 (9th Cir.1981). Accordingly, even though the accused device may improve on a patented device through good faith design work, the *1433 accused device may nonetheless infringe the patented device. The fact that an accused device is an improvement over the claimed invention as a consequence of subsequent developments does not preclude a finding of infringement under the doctrine of equivalents. Ryco, Inc. v. Ag-Bag Corp., 857 F.2d 1418, 1426-27 (Fed.Cir. 1988).

2. Principles of Claim Construction
Claims are construed in light of the patent's specifications, the prosecution history, the prior art, and the testimony of expert witnesses. Smithkline Diagnostics, Inc. v. Helena Laboratories Corp., 859 F.2d 878, 882 (Fed.Cir.1988). Under the principle of claim differentiation, a broadly written claim cannot be limited by another, more narrowly written claim. Marsh-McBirney, Inc. v. Montedoro-Whitney Corp., 882 F.2d 498, 504 (Fed.Cir. 1989). Moreover, a limitation appearing in the specifications, the preferred embodiments, or the commercial embodiments, cannot be read into the patent's claims. Intervet America, Inc. v. Kee-Vet Laboratories, Inc., 887 F.2d 1050, 1053 (Fed.Cir. 1989); Laitram Corp. v. Cambridge Wire Cloth Co., 863 F.2d 855, 865 (Fed.Cir.1988), cert. denied, ___ U.S. ___, 109 S.Ct. 2069, 104 L.Ed.2d 634 (1989). The specifications can, however, be used to construe the literal meaning of words in the claims. McGill Inc. v. John Zink Co., 736 F.2d 666, 674 (Fed.Cir.), cert. denied, 469 U.S. 1037, 105 S.Ct. 514, 83 L.Ed.2d 404 (1984).

3. The Time at Which Equivalence is Determined
Prior to the commencement of trial, in response to in limine motions filed by the parties, the Court determined that, for purposes of trial, equivalence would be determined as of the date of alleged infringement by Exac. In electing to use the date of alleged infringement as the relevant time period, the Court rejected Exac's argument that equivalence should be determined at the time the patents in suit issued.
To support its position, Exac relied on two Supreme Court cases, old but never overruled, that state that there can be no infringement if the fact of equivalence of the two devices was not known at the date of the patent. Gould v. Rees, 82 U.S. (15 Wall.) 187, 194, 21 L.Ed. 39 (1872); Gill v. Wells, 89 U.S. (22 Wall.) 1, 28-29, 22 L.Ed. 699 (1874). Micro Motion, however, cited several recent Federal Circuit cases that hold that equivalence is determined at the time the alleged infringement occurs. See Atlas Powder Co. v. E.I. duPont de Nemours & Co., 750 F.2d 1569, 1581 (Fed.Cir. 1984); Texas Instruments, 805 F.2d at 1563; Hughes Aircraft Co. v. United States, 717 F.2d 1351, 1365 (Fed.Cir.1983).[9]
Both Exac and Micro Motion acknowledged the split in authority relevant to this issue in their pretrial briefs on the subject. Interestingly, the recent Federal Circuit cases, which clearly diverge from Gould and Gill, do not make reference to either case or explain why they adopt a different approach to the time at which equivalence is determined.
*1434 In resolving the conflict, the Court was persuaded by the special expertise of the Federal Circuit in patent cases, the relative recency of the Federal Circuit opinions on this issue, and the sound rationale asserted by the Federal Circuit for using the date of alleged infringement to determine equivalence. The Federal Circuit has noted that devices that change the patented invention due to technological advances that occur subsequent to the original patent may nonetheless infringe that patent under the doctrine of equivalents. Texas Instruments, 805 F.2d at 1563; Hughes Aircraft, 717 F.2d at 1365.
In this case, Exac's alleged infringement began in December 1983, when Exac first designed a working flowmeter, and continued throughout 1984, the year in which Exac began commercial sales of its meters. Accordingly, equivalence will be determined as of 1984 for the '450 patent. The '025 patent, however, did not issue until January 1, 1985, and infringement cannot occur absent an issued patent. The relevant date at which equivalence will be determined with respect to the '025 patent will, therefore, be January 1, 1985.[10]

4. The Range of Equivalents to Which Micro Motion's Patents are Entitled
Under the doctrine of equivalents, patents that represent a significant advance in the art are entitled to a broad range of equivalents. A patent need not be a "pioneer patent" to be entitled to a broad range of equivalents. Eibel Process Co. v. Minnesota & Ontario Paper Co., 261 U.S. 45, 63, 43 S.Ct. 322, 328, 67 L.Ed. 523 (1923).
Satisfaction of a long-felt but unfilled need for a device is evidence that that device is a significant advance in the art. Perkin-Elmer Corp. v. Computervision Corp., 732 F.2d 888, 895-96 (Fed.Cir.), cert. denied, 469 U.S. 857, 105 S.Ct. 187, 83 L.Ed.2d 120 (1984). Other factors that demonstrate the significance of an invention include unsuccessful attempts by others to accomplish the results of the patented invention, Studiengesellschaft Kohle mbH v. Dart Industries, Inc., 549 F.Supp. 716, 736 (D.Del.1982), aff'd, 726 F.2d 724 (Fed.Cir.1984), commercial success of the patented invention, Simmons Fastener Corp. v. Illinois Tool Works, 739 F.2d 1573, 1575-76 (Fed.Cir.1984), cert. denied, 471 U.S. 1065, 105 S.Ct. 2138, 85 L.Ed.2d 496 (1985), and praise or awards for the patented device. In re Piasecki, 745 F.2d 1468, 1474 (Fed.Cir.1984).
The Court finds that Micro Motion's patents in suit were a significant advance in the art, and are, therefore, entitled to a broad range of equivalents. Prior to the invention of the Micro Motion meters, there was a long-felt need in the industry for a flowmeter that would measure mass flow directly, nonintrusively, and accurately. RT-WHO 130-32, 160, 1129-31, 1138-39; PX 137 at 2; Maloney Dep., Vol. II at 5-6. Efforts to develop a nonintrusive mass flowmeter had begun decades before Micro Motion introduced its meters, but no commercially successful products emerged from these efforts. RT-WHO 47-48, 135-37, 275. The commercial success of Micro Motion's meters is demonstrated by the fact that sales of these meters doubled approximately every twelve months (RT-WHO 66-67), with cumulative sales of $40 million by 1984. RT-WHO 108-09; PX 68. Finally, Micro Motion's flowmeters received several industry awards. RT-WHO 51-52, 96-102; PX 12, 75-84.

5. Level of Ordinary Skill in the Art
An important factor in determining infringement under the doctrine of equivalents is whether a person reasonably skilled in the art would have known that the changes in the accused device were substantially equivalent to the patented claims. Graver Tank, 339 U.S. at 609, 70 S.Ct. at 856. This hypothetical person must have recognized the equivalence of the substituted element or elements at the time of the alleged infringement. Atlas Powder, 750 F.2d at 1581.
Factors to consider in determining the level of ordinary skill in the art include: *1435 (1) educational level of the inventor; (2) the type of problems encountered in the art; (3) prior art solutions to those problems; (4) the rapidity with which innovations are made in the art; (5) the sophistication of the technology; and (6) the educational level of workers active in the field. Environmental Designs, Ltd. v. Union Oil Co., 713 F.2d 693, 696 (Fed.Cir.1983), cert. denied, 464 U.S. 1043, 104 S.Ct. 709, 79 L.Ed.2d 173 (1984).
The hypothetical person of ordinary skill in the art is one "who is attempting to solve the problems the inventor addressed," General Battery Corp. v. Gould, Inc., 545 F.Supp. 731, 750 (D.Del.1982), although he is not required to match the level of expertise of the inventor himself. Kimberly-Clark Corp. v. Johnson & Johnson, 745 F.2d 1437, 1454 (Fed.Cir. 1984). Accordingly, the Court concludes, for purposes of this suit, that this hypothetical person should possess experience in the design and development of flowmeters, and not merely experience in the use of these meters.
Applying these considerations to the facts of this case, the Court finds that the person of ordinary skill in the art in 1984 was one who was a design engineer with a college degree in mechanical engineering or the equivalent, and who had several years of experience in the design and development of flow measurement and control instruments.

B. Application of the Doctrine of Equivalents to the Patents in Suit


1. Claim 8 of the '450 Patent

a. Construction of Claim 8

Claim 8 of the '450 patent is a dependent claim that incorporates by reference all of independent Claim 1. RT-WHO 316-22; PX 98. Claim 8 contains three elements, and describes the sensors ("means to measure") to be used on "[a] flow meter as set forth in claim 1." Therefore, construction of this claim also necessarily involves construction of Claim 1 of the '450 patent.

(1) Elements of Claim 1
Claim 1 of the '450 patent describes Micro Motion's flowmeter in broad terms, and contains four elements: a support, a "U"-shaped, continuous conduit, a means for oscillating the conduit, and a means to measure the Coriolis forces. PX 1. In addition to these four elements, the parties agreed, for purposes of this litigation, that Claim 8 requires that the conduit described in Claim 1 be planar.
The first element of Claim 1 is "a support." The accused Exac devices literally contain this element. RT-WHO 356.
The second element is "a `U'-shaped, continuous conduit" that is (1) free of pressure sensitive joints, (2) solidly mounted, (3) cantilevered, and (4) has an oscillation axis and a perpendicular deflection axis, each of which has a different resonant frequency. By stipulation of the parties, this second element also requires that the conduit be planar. As described more fully below, with the exception of planarity, the Exac devices literally satisfy this second element of Claim 1.
Exac's conduits are not "U" shaped in the literal sense. Although the term "U shaped" could be interpreted to refer to a literal "U," however, the references to this term in the patent's specifications make clear that this is not what the patent requires. The specifications state that, "[t]hough described as `U shaped,' [the conduit] may have legs which converge, diverge, or are skewed substantially. A continuous curve is contemplated.... [T]he physical configuration of `U' shaped conduit 14 is not critical." RT-WHO 358; PX 1. Moreover, one of the preferred embodiments of the '450 patent depicts a conduit that is not a literal "U" shape. PX 1, fig. 9. Interpreting this claim in light of the patent specifications, the Court finds that Claim 1 requires a functional, rather than a literal, "U"-shaped conduit. RT-WHO 453.
A functional "U"-shaped conduit is one that directs fluid away from the support, then parallel to the support, then back toward the support. At trial, this was referred to as an "out-across-back flow pattern". *1436 The conduits in the Exac devices, although in the shape of helical crossover loops, do direct the fluid away from the support, parallel to the support, and then back to the support, thus meeting the requirements of a functional "U" shape. RT-WHO 357-58, 454.
In addition to being functionally "U" shaped, the conduit described in the second element of Claim 1 must also be continuous, free of pressure sensitive joints, solidly mounted, cantilevered, planar, and must have an oscillation and a deflection axis with different resonant frequencies. Exac's helical conduit literally satisfies the requirements of a continuous curve that is free of pressure sensitive joints and is solidly mounted to the support. RT-WHO 354. Exac's conduits are also cantilevered within the literal scope of Claim 8. RT-WHO 354, 375-77. The term "cantilevered" refers to a structure that has its center of mass at a distance from its support. RT-WHO 1624. Exac's founders, Drs. Young and Dahlin, have previously referred to their crossover loop as a cantilevered structure. RT-WHO 1084-85; PX 8 at 16; RT-SW 920-21. Finally, Exac's conduits have an oscillation axis and a deflection axis with different resonant frequencies. RT-WHO 354-55, 1649-50.
Micro Motion concedes, however, that the helical crossover conduits used in the Exac devices are not planar. Given the construction of Claim 8 that was accepted for purposes of this trial, the Exac conduits are, therefore, not within the literal scope of the conduits called for by Claim 8, despite the fact that all the requirements of Claim 1, element 2, are literally met in the Exac devices. The Court must, therefore, determine whether the nonplanar Exac conduits are the substantial equivalent of the Micro Motion planar conduits.
The third element of Claim 1 is a means for oscillating the conduit relative to the support. This element is literally present in the Exac devices. RT-WHO 364.
The fourth element of Claim 1 is a means to measure the Coriolis forces tending to distort the conduit. This element is defined in more detail in Claim 8, which is dependent on Claim 1.
The "means to measure the mass flow rate" described in Claim 8 has three elements. First, there must be two symmetrically mounted sensors. Second, each sensor must "output a signal" when the adjacent sideleg of the conduit "passes through the midplane of oscillation." Third, the sensor must measure "the time lag between signal outputs by the sensor," or delta t. RT-WHO 261-63, 321; PX 98; Smith Dep. at 194-96. Each of these three elements is discussed separately below.

(2) The Symmetry Element
Claim 8 calls for two sensors, mounted "at symmetrical positions relative to the deflection axis" of the conduit. Exac's meters, like Micro Motion's, have two sensors. Therefore, the Court must determine what is meant by the term "symmetrical" as used in the patent claims.
At trial there was much dispute regarding the meaning of "symmetrical" in this context. Micro Motion introduced evidence that "symmetrical" referred to symmetry about an axis. RT-WHO 368-73; PX 98. Exac presented testimony that the term "symmetry" referred to planar (mirror image) symmetry. RT-WHO 1204-16, 1381-82, 1395, 1399-1400. This factor is significant from the standpoint of equivalence because the Exac sensors do not possess planar symmetry, though they do have symmetry about an axis.
The Court finds that the term "symmetrical" in Claim 8 refers to symmetry about an axis or line. Exac's attempt to define this term by reference to Micro Motion's Model L Instruction Manual (RT-WHO 555-60) is improper, because claims are only to be construed with reference to specifications, the prosecution history, prior art, and expert testimony. See McGill, 736 F.2d at 671-75. The expert testimony conflicted regarding how common it is in the flowmeter industry to use the term "symmetry" to refer to axial symmetry. RT-WHO 368-73, 1401, 1407-08. The Court resolves this conflict in favor of Micro Motion, and finds that "symmetry" in Claim 8 *1437 refers to symmetry about a line, not planar symmetry. Accordingly, the Exac devices are symmetrical within the literal scope of Claim 8, because they have symmetry about a line. RT-WHO 1381.

(3) Measurement at the Midplane
The second element of Claim 8 requires that each of the two sensors must be "adapted to output a signal as the adjacent portion of the `U' shaped conduit passes through the mid-plane of oscillation." Exac construes this language to mean that the sensors must make a measurement when the conduit passes through the midplane of oscillation. Micro Motion responds that the claim requires only that the sensors output a signal when the conduit passes through the midplane, and does not require that the sensors actually make a measurement at the midplane.
While Micro Motion's interpretation of the claim language is a reasonable one, ambiguous language in the claim is to be construed in light of the patent specifications. After reviewing the specifications, the Court finds that Claim 8 literally requires that the sensors make a measurement at the midplane of oscillation.
In fact, measurement at the midplane was considered by inventor James Smith to be quite important. The specifications of the '450 patent state that measurement as the conduit legs pass through the midplane of oscillation is "a most significant aspect of the present invention," because this "avoids the necessity of maintaining constant frequency and amplitude." PX 1, col. 6, 11. 27-42. Moreover, the reissue oath and the prior art statement filed by Mr. Smith in connection with the '450 patent distinguished the prior art because it failed to "recognize the procedure for, or advantages of, making time difference measurements of the distorted sections of the sensing tube passing through the mid-plane of oscillation." DX G at 23 (emphasis added).
Accordingly, the Court construes Claim 8 to literally require that the sensors on the meter make a measurement at the midplane of oscillation. The Exac sensors make a measurement at the extreme of oscillation, and are, therefore, not within the literal scope of Claim 8. Because the Court finds that this second element of Claim 8 is not literally satisfied by the accused Exac devices, it must determine whether Exac's method of measurement at the extreme is substantially equivalent to the midplane measurement described in the '450 patent.

(4) Measurement of Delta T
The third element of Claim 8 defines the "means to measure the mass flow rate" of Claim 1, by calling for the measurement of the "time lag between signal outputs by the sensors." This measurement was referred to at trial as delta t. The Court finds that the reference in Claim 8 to "means to measure the mass flow rate" is the same as the reference in Claim 1 to the "means to measure the magnitude of Coriolis forces," and that both phrases refer to a means to measure delta t.
Although Claims 1 and 8, construed together, require that the device measure mass flow rate by measuring delta t, Claim 1 does not call for any specific means of measuring delta t.[11] Therefore, the Court finds that various types of sensors could be used for this purpose. Different types of motion sensors, such as position sensors, velocity sensors, and acceleration sensors, were known in 1984 to be interchangeable. RT-WHO 290, 719-20, 744; RT-SW 2479-80; Smith Dep. at 69-70.
Despite the interchangeable nature of these sensors, however, Exac's meters literally measure phase angle, and not delta t. RT-WHO 1420-21. Thus, the third element of Claim 8 is not literally satisfied by the Exac devices, and the Court must scrutinize this element under the doctrine of equivalents.

b. Function, Way, Result of Claim 8

Based on the foregoing, the Court concludes that most, but not all, of the elements *1438 of Claim 8 are literally present in the accused Exac devices. As described above, three elements of Claim 8 are not literally satisfied by the Exac devices. These are: the planarity requirement of Claim 1, the requirement that the meters measure delta t, and the requirement that this measurement be made at the midplane of oscillation.
Under the doctrine of equivalents, however, infringement may nonetheless be found if the Exac devices perform substantially the same function as the Micro Motion inventions, in substantially the same way, to obtain substantially the same result. Graver Tank, 339 U.S. at 608, 70 S.Ct. at 856. The relevant inquiry is whether, at the time of the alleged infringement (here 1984), a person of ordinary skill in the art would have understood that what was substituted in the accused device for the literal elements in the patent was equivalent. The function/way/result analysis must be applied to each element of the asserted claims and to the device as a whole. Hughes Aircraft, 717 F.2d at 1364. Having applied this analysis to the three elements of Claim 8 that are not literally present in the Exac meters, the Court finds that each of these elements is present by equivalence in the Exac devices.

(1) The Planarity Element
The absence of planarity in the Exac meters establishes that this element of Claim 8 is not literally present in the accused devices. Accordingly, the function/way/result test must be applied to compare the patented conduits with the Exac conduits under the doctrine of equivalents.
The "function" of the patented conduits is to oscillate while fluid is traveling through them, thereby producing a Coriolis force that will twist the conduits by an amount that can be measured in order to determine the mass flow rate of the fluid. RT-WHO 377, 411, 419. The patented conduit performs this function in the way described in the patent: it is solidly mounted and cantilevered, free of pressure sensitive joints, and it has different resonant frequencies about its oscillation and deflection axes. RT-WHO 377-78, 411, 419. The result achieved is deformation or twisting of the conduit that can be accurately measured to provide the mass flow rate. RT-WHO 378, 411, 419.
The accused Exac conduits clearly perform the same function and achieve the same result. RT-WHO 411, 419. Exac contends, however, that its conduits measure mass flow rate in a different way because they are nonplanar.
Exac contended at trial that its nonplanar conduit was subject to hydrostatic, inertial, and centrifugal forces that tend to twist its helical conduits independently of the Coriolis force caused by the flowing fluid. RT-WHO 495-504, 1364. By contrast, these forces are minimized by Micro Motion's planar conduit design. RT-WHO 505-07. Because the Exac nonplanar conduit is sensitive to these extraneous forces, Exac argues that its conduit cannot be equivalent to the Micro Motion planar conduit in the way it measures mass flow rate.
Micro Motion, however, presented convincing testimony that such extraneous forces are significant only if they occur at the drive frequency and in phase with the Coriolis force (RT-WHO 379-85, 590-93), and that such an occurrence would be unlikely. RT-WHO 385. Moreover, while Exac contends that its nonplanar conduits are sensitive to these extraneous forces, its meters do not monitor changes in the hydrostatic pressure or centrifugal forces, nor do the meters do anything to compensate for changes in these forces during measurement of mass flow rate. RT-WHO 389, 699, 981-83, 988-89, 1014-15, 1018. Many of the effects of these forces can be removed from the mass flow measurement by the process of initially "zeroing" the meter so that they are not a factor in the mass flow calculation. RT-WHO 986-90, 1015, 1052-55. Other effects of these extraneous forces are compensated for by minimal "fine tuning" adjustments to scale factors stored in Exac's microprocessor. RT-WHO 1016-19, 1056-62.
Despite Exac's contention that the sensitivity of its conduits to these extraneous *1439 forces means it is not equivalent to Micro Motion's conduit, it is apparent that ultimately Exac's meters do not compensate for or correct the effect of these forces (beyond the initial zeroing of the meter), and any resulting forces that are not zeroed out simply result in an error in mass flow measurement. RT-WHO 1011-13, 1048-50, 1055. Accordingly, this difference between the Exac and Micro Motion conduits is not found to be substantial. Based on the foregoing, the Court finds that the Exac nonplanar conduit and the Micro Motion planar conduit are equivalent.
Micro Motion also presented evidence that the equivalence of nonplanar and planar conduits would have been known to a person of ordinary skill in the art in 1984. The Cox '028 patent, issued in 1978, disclosed a planar, substantially "U"-shaped conduit with a narrowed neck, as well as a circular crossover conduit. PX 7, figs. 1 & 5. The Cox '028 specifications disclose that these conduit shapes are interchangeable and equivalent embodiments. PX 7 at col. 1, 1. 55  col. 2, 1. 8; col. 6, 11. 16-20. These disclosures in the Cox '028 patent demonstrate that a person of ordinary skill in the art in 1984 would have been aware that a crossover conduit in the form of a continuous curve was equivalent to a planar, "U"-shaped conduit. RT-WHO 365-67.

(2) Measurement of Delta T
The second element of the Micro Motion patents not literally present in the Exac devices is the requirement that the sensors on the meter measure delta t. This difference, therefore, must also be analyzed for infringement by equivalence, using the function/way/result test.
The "function" of the "means to measure" element of Claim 8, of which delta t measurement is a part, is to measure the amount of twist in the oscillating conduit or conduits in order to accurately measure mass flow rate. RT-WHO 398. The "way" this result is achieved is to utilize motion sensors to determine the time lag between the passage of the sidelegs of the oscillating conduit past a reference point. RT-WHO 398-99. The "result" achieved is a measurement of delta t, which can be electronically converted to an accurate measurement of mass flow rate.[12] RT-WHO 399.
Exac presented evidence to show that, because its meters measure phase angle rather than delta t, the "way" its meters measure mass flow is not substantially similar to Micro Motion's measurement of delta t. This contention, however, is without merit. The Exac devices measure phase angle by first measuring delta t and then multiplying by the frequency of oscillation. Exac's microprocessor then determines the mass flow rate by using a mathematical equation that is based on phase difference and frequency. RT-WHO 979, 997-98; Schindler Dep. at 102-03; Swanson Dep. at 220-22; RT-SW 1797-1801; PX 63.
The evidence at trial, however, demonstrated that measurement of the phase angle of a sinusoidal signal is equivalent to measurement of delta t, because phase angle divided by frequency is equal to delta t. RT-WHO 399-400, 1432-33. Using mathematical principles familiar to a college engineering student, Micro Motion demonstrated at trial that the equation used by Exac's microprocessor to determine mass flow rate is easily translated into a form in which the mass flow rate is equal to delta t, multiplied by a "nonlinearity correction factor" based on phase difference, and a temperature compensation factor. RT-WHO 983-1048.
The testimony at trial established that the "nonlinearity correction factor" is a small number which in essence "fine tunes" the delta t measurement. RT-WHO 612, 1802. The need for temperature compensation, though not addressed by the Micro Motion patent, is one that was well understood by persons of ordinary skill in the art in 1984. RT-WHO 584, 1790, 1804; PX 276 at ¶¶ 30-31. Based on the foregoing, the Court finds that Exac's measurement of phase angle is substantially *1440 the same as Micro Motion's direct measurement of delta t, and that these two measurements were known to be equivalent in 1984. PX 275 at ¶ 16; PX 276 at ¶ 41.

(3) Measurement at the Midplane of Oscillation
The third element of Claim 8 that is not literally present in the Exac devices is the requirement that delta t be measured at the midplane of conduit oscillation.[13] This element is not literally satisfied by the accused devices because they measure delta t at the extreme of conduit oscillation. Accordingly, the Court must determine whether measurement at the extreme of oscillation is substantially the same as measurement at the midplane. This element of Claim 8 has the same function, way, and result as the second element of Claim 8 described above.
Micro Motion presented testimony at trial that delta t is the same at every point throughout the oscillatory path of the conduits. RT-WHO 262, 403-06. Some of this testimony came in the form of admissions by Exac personnel. Dahlin Dep. at 188-89, 201; RT-SW 2490-91; PX 111. Because delta t is the same throughout the sinusoidal cycle, Micro Motion contends that measuring delta t at any location in the cycle, including the extremes of oscillation, will provide the same value of delta t as measuring at the midplane. From this Micro Motion concludes that measuring delta t at the extreme of oscillation is the equivalent of measuring delta t at the midplane. PX 275; PX 276 at ¶¶ 39-41.
Exac presented some evidence at trial to refute the conclusion that delta t is the same throughout the oscillatory cycle. RT-WHO 631-37. Dr. Graeme Fowler, a witness called by Micro Motion, testified at the first trial that delta t is the same everywhere, but he recanted that testimony at this trial because he claimed it had been based on the erroneous assumption that the magnitude of oscillation of each side leg is the same, when in fact, the magnitudes are different. RT-WHO 626-37, 644-46. Further, Exac's counsel suggested that the video demonstrations by Dr. Durgin, Micro Motion's technical expert, in fact revealed a "taper" between delta t signals that would not have been present if delta t were, in fact, the same throughout the oscillatory cycle. RT-WHO 790-92.
The evidence most credible to the Court, however, demonstrated that delta t is the same at all points in the oscillatory cycle. RT-WHO 262, 313-14, 403-06; Dahlin Dep. at 188-89, 201; PX 111. Exac's own founder, Dr. Dahlin, testified at the first trial that "if you have two sine waves here, the time differential or phase angle is the same no matter. You look on those everywhere in these when we have two identical sine waves, it's the same time difference and phase angle everywhere." RT-SW 2490-91.
Moreover, Micro Motion persuasively demonstrated that the reason the Exac devices measure at the extreme of oscillation is a function of the type of sensors used, rather than any substantive difference in delta t at various points in the oscillatory cycle. In 1984, it was well-known that the best place to process time-varying electrical voltage signals is the point at which the signals cross through zero voltage. RT-WHO 286-91, 404-05, 746; PX 275 at ¶ 7; PX 276 at ¶ 38. For position and acceleration sensors, that point occurs when the conduit is at the midplane of oscillation. RT-WHO 730, 740-41. For velocity sensors, that point is when the conduit is at the top or bottom (the extremes) of oscillation. RT-WHO 397-98, 737.
The Exac sensors generate velocity voltage signals, which measure zero voltage crossings when the conduit is at the extremes of oscillation. The Micro Motion commercial model meters also use velocity sensors, but the velocity voltage signal is integrated before measuring delta t. This integration converts the voltage signals to position signals. Consequently, the Micro Motion meters measure zero voltage crossing when the conduit is at the midplane. RT-WHO 745-46. From the standpoint of equivalence, however, both types of sensors *1441 are measuring at the point of their respective zero voltage crossings. The Court, therefore, finds the measurements to be substantially the same, despite the different positions of the respective conduits at the moment of measurement.
The next issue the Court must resolve in determining equivalence is whether a person of ordinary skill in the art in 1984 would have understood that delta t is the same at every point throughout the oscillatory cycle. Exac suggested at trial that, even assuming that delta t is, in fact, the same throughout the oscillatory cycle, persons of ordinary skill in the art in 1984 would not have known this principle. DX RT at 34-42. Exac argues that such persons would have relied on the repeated statements made by Micro Motion in its patents and in its commercial literature that it was not possible to measure delta t away from the midplane because delta t went to zero at the extremes of oscillation. RT-WHO 1530-34. Therefore, Exac argues that the fact that its devices make measurements at the extremes of oscillation cannot possibly be an equivalent to measuring at the midplane because the fact that delta t is the same throughout the cycle of oscillation was only learned subsequent to 1984.
Exac's contention that Micro Motion's patents teach that delta t goes to zero at the extremes of oscillation was refuted by Micro Motion's experts at trial. RT-WHO 403-04; PX 275 at ¶¶ 3, 4, 8, 9; PX 276 at ¶ 37. These experts also testified that a college freshman or sophomore in an introductory physics course would have understood that delta t remains the same throughout the oscillatory cycle (RT-WHO 481-83), as would a person of ordinary skill in the art in 1984. RT-WHO 589. The fact that the patents themselves do not say that delta t is the same everywhere is not significant, as a patent need not describe in the specifications that which is well known in the art. In re Wands, 858 F.2d 731, 735 (Fed.Cir.1988).
Moreover, the fact that Micro Motion may have made statements about the necessity of measuring delta t at the midplane of oscillation that subsequently proved to be erroneous does not prevent a finding of infringement under the doctrine of equivalents. All these statements indicate is that the inventor and marketers of the Micro Motion meters were unaware at first of the complete theory of the inventions. An inventor is not required to understand the theory of how his invention works, so long as the patent adequately discloses to a person of ordinary skill in the art how to make and use the invention. Diamond Rubber Co. v. Consolidated Rubber Tire Co., 220 U.S. 428, 435-36, 31 S.Ct. 444, 447-48, 55 L.Ed. 527 (1911). "[A] patentee is not responsible for the correctness of [his] theories and explanations when their correctness is not related to validity of the claims under consideration." Raytheon Co. v. Roper Corp., 724 F.2d 951, 959 (Fed.Cir.1983), cert. denied, 469 U.S. 835, 105 S.Ct. 127, 83 L.Ed.2d 69 (1984). In other words, an inventor is entitled to the merits of his invention, even if they surpass his expectations or go beyond what was known or commercially available at the time of the invention. Johnson & Johnson v. W. L. Gore & Assoc., 377 F.Supp. 1353, 1355 (D.Del.1974).
The evidence at trial demonstrated that Micro Motion at one time believed, and promulgated the belief, that it was important to measure delta t at the midplane of oscillation. This belief is not relevant to infringement under the doctrine of equivalents, however, because a patentee's invention is not necessarily limited by his own misconception of its theory of operation, and because persons of ordinary skill in the art in 1984 understood that delta t was the same everywhere despite the erroneous statements of Micro Motion. Accordingly, the Court finds that Exac's measurement at the extremes of oscillation is equivalent to the measurement of the patented devices at the midplane of oscillation.
Based on the foregoing, the Court finds that Claim 8 of the '450 patent is infringed by the Exac devices under the doctrine of equivalents.


*1442 2. Claim 57 of the '450 Patent

a. Construction of Claim 57

Claim 57 of the '450 patent is a dependent claim that incorporates by reference all of independent Claim 38.[14] PX 97. Claim 38 describes "a flowmeter" that contains the same elements previously discussed in Section III B 1, supra. The asserted elements of the conduit described in Claim 38 and the resulting infringement analysis under the doctrine of equivalents need not be repeated here because they are identical to that presented in the preceding discussion of Claim 8. RT-WHO 411-12. Accordingly, for the reasons stated above, the Court finds that independent Claim 38 of the '450 patent has been infringed by the Exac devices under the doctrine of equivalents.
Infringement of an independent claim, however, such as Claim 38, does not necessarily mean that claims dependent on that claim are also infringed. Wahpeton Canvas Co. v. Frontier, Inc. 870 F.2d 1546, 1552 n. 9 (Fed.Cir.1989). The Court must, therefore, rule separately on the issue sue of infringement of Claim 57.
Claim 57 of the '450 patent contains one additional element not present in Claim 8. It requires that the resonant frequency of the patented conduit about its oscillation axis must be lower than the resonant frequency of the conduit about its axis of symmetry.
There was little testimony on this point at trial, and the issue appears to have been essentially conceded by Exac. The evidence that was offered at trial demonstrated that Exac's conduits have an oscillation axis with a lower resonant frequency than that of the deflection axis. RT-WHO 354-55, 1649-50. Thus, the Court finds that this additional element is literally present in the Exac devices.
Given that the one additional element presented by dependent Claim 57 is literally present in the Exac devices, and the remaining elements of Claim 57 (which are expressed in Claim 38 and incorporated by reference) have already been found to be infringed under the doctrine of equivalents, the Court finds that Claim 57 as a whole has been infringed by Exac's devices under the doctrine of equivalents.

3. Claim 1 of the '025 Patent
The '025 patent, issued on January 1, 1985, marked a significant change from the concept of the '450 patent. The '025 patent introduced the design of two parallel flow tubes in place of a single oscillating conduit. Parallel flow tubes were introduced by Micro Motion to correct certain problems that had arisen with the commercial embodiments of the '450 patent. The meters disclosed in the '450 patent, when installed, had been susceptible to external vibrations that affected the accuracy of their mass flow measurement. RT-WHO 343-44.
In the devices covered by the '025 patent, fluid flowing toward the Micro Motion meter is directed through two parallel conduits. PX 89. These conduits are oscillated in opposite directions, so that any effect from external vibration is equal and opposite in the two tubes, thus cancelling out the effects of vibration. Cage Dep. at 79, 81-83, 113-114; RT-WHO 342-43. As a result, the '025 commercial meters could be mounted directly in the fluid line, instead of having to be mounted to a support. RT-WHO 344. Although the '025 patent did not issue until January 1, 1985, Micro Motion began marketing the preferred embodiment of the '025 patent as its Model D meter in December 1982. RT-WHO 90-91. Micro Motion alleges that the Exac parallel tube flowmeters infringe Claim 1 of the '025 patent under the doctrine of equivalents.

a. Construction of Claim 1

Claim 1 of the '025 patent is an independent claim. PX 99. The parties agreed for purposes of this trial that the claim language *1443 requires a planar conduit[15] (RT-WHO 336-37, 418; PX 99), an element that is clearly not literally satisfied by the non-planar Exac helical crossover conduits.
Claim 1 of the '025 patent is in many respects almost identical to Claims 8 and 57 of the '450 patent, and the claim constructions adopted in sections 1 and 2, above, will be applied where they are relevant to this claim.
Claim 1 contains six elements. First, the claim requires an inlet manifold and an outlet manifold. The Court finds that this element is literally present in the accused Exac devices. RT-WHO 416.
The second element of this claim describes "two U-shaped flow tubes" with certain specified characteristics. With the exception of the requirement that the conduits be planar, the Court finds that all of the characteristics of this second element are literally present in the Exac devices.
In reaching this finding, the Court is aware that the second element of Claim 1 refers to "U-shaped flow tubes" without enclosing the "U" in quotation marks, as was done in the '450 patent. The lack of quotation marks in the '025 patent, however, does not indicate that the '025 patent requires conduits in the shape of a literal "U." This is because the '025 patent itself specifically incorporates the teachings of the '450 patent, which the Court has construed previously to require only a functional "U" shaped conduit. PX 3 at col. 1, 1. 29  col. 2, 1. 42. Therefore, because the '025 patent incorporates the '450 patent by reference, Claim 1 of the '025 patent is similarly construed to require that the conduits be in the shape of a functional, not a literal, "U." RT-WHO 619-21. The Exac helical crossover conduits literally meet the requirement of a functional U because they direct the flow of fluid away from the support, then parallel to the support, and then back toward the support. RT-WHO 357-58, 454, 1773-74.
Exac's conduits also literally contain the remaining features described by the second element of Claim 1. Exac's conduits are free of pressure sensitive joints (RT-WHO 417), mounted in parallel fashion and fixedly attached (RT-WHO 417), and cantilevered. RT-WHO 417. The Exac conduits have essentially equal moments of inertia and essentially equal spring constants (RT-WHO 418), and the resonant frequency of the torsion axes is different from the resonant frequency of the bending axes and from the odd harmonic values of the resonant frequency of the bending axes. RT-WHO 418-19. Therefore, the only characteristic of the second element of Claim 1 that is not literally present in the Exac devices is the requirement that the conduits be planar.
The third element of Claim 1 is a "means for inputting to each flow tube[] ... essentially equal amounts of fluid from an inlet plenum." The fourth element of Claim 1 requires a similar exit plenum to drain fluid from the flow tubes.
The precise definition of "plenum" was disputed at trial. Exac's experts contended its devices contained a flow-splitter, but did not contain a plenum. RT-WHO 1252.
Exac introduced testimony at trial that the plenums called for by the third and fourth elements of Claim 1 must have symmetrical boundaries and must be large enough to produce essentially equal, uniform fluid pressure. Exac also presented evidence that its flowsplitter has none of these qualities. RT-WHO 1252-55.
The Court, however, rejects Exac's interpretation of Claim 1, because Claims 6 and 7 of the '025 patent, which are dependent from Claim 1, recite these very same limitations. The rule of claim differentiation prohibits the Court from reading limitations from dependent (narrow) claims into an independent (broader) claim because to do so would render the dependent claims redundant. Marsh-McBirney, 882 F.2d at *1444 504. The specifications of the '025 patent, which can properly be considered in claim construction, also contain these same limitations. PX 3, col. 6, 11. 29-58. This limiting language, however, refers only to a preferred embodiment of the patent and, moreover, limitations from a patent's specifications cannot be read into its claims. Intervet America, 887 F.2d at 1053; Laitram Corp., 863 F.2d at 865.
The Court may also consider expert testimony when construing claims. Smithkline Diagnostics, 859 F.2d at 882. Micro Motion introduced expert testimony that, according to Webster's Ninth New Collegiate Dictionary, a plenum is a space that is full of matter. PX 275 at ¶¶ 22-23. Under this definition, however, the Exac flowsplitter is not a literal plenum. The flowsplitter is full of matter, in this case fluid, but it is not clearly a "space," because it resembles a continuation of the pipeline more than a discreet chamber. Because the Court concludes that the Exac flowsplitter is not a literal plenum, the third and fourth elements of Claim 1 must accordingly be analyzed under the doctrine of equivalents.
The fifth element of Claim 1 calls for a means of sinusoidally driving the conduits at their resonant frequency, in a manner similar to the tines of a tuning fork. Exac's meters contain a coil and magnet device that excites the conduits in precisely this fashion. RT-WHO 421. This element, then, is literally present in the Exac devices.
The sixth element of Claim 1 is the "means to measure" element. Micro Motion argued that the claim language does not specify any particular point at which delta t is to be measured. The Court, however, disagrees. The language of Claim 1 requires that delta t be measured at the midplane of oscillation. The claim expressly states that the patented device has a "means to measure the time interval between the passages of the first side legs ... through the respective midplanes of oscillation, and the passage of the second side legs, caused by Coriolis forced deflection about said torsion axes, to pass through the respective midplanes of oscillation at a later time." PX 3, 99.
Nonetheless, having previously found in its discussion of the '450 patent that the Exac devices measure the equivalent of delta t, and that measurement of delta t at the extreme of oscillation is the equivalent of measuring it at the midplane of oscillation, the Court finds that this sixth element of Claim 1 is present by equivalence in the accused Exac devices.

b. Function, Way, Result of Claim 1

All of the elements of Claim 1 of the '025 patent are literally present in the accused Exac meters, with the exception of the requirements that the conduits be planar, that the devices contain an inlet and an outlet plenum, and that the devices measure delta t at the midplane of oscillation. With respect to the planarity and "means to measure" elements of this claim, however, the Court's previous discussion of the '450 patent established the equivalence of planar and nonplanar conduits, delta t and phase angle, and midplane measurement versus measurement at the extremes of oscillation. Accordingly, the only remaining issue for the Court to resolve as to Claim 1 is whether the Micro Motion plenum is equivalent to the Exac flowsplitter under the doctrine of equivalents.
The Court finds that the Micro Motion plenum and the Exac flowsplitter are equivalents. While the Exac flowsplitter is not a literal plenum, it performs substantially the same function, in substantially the same way, to achieve substantially the same result. RT-WHO 420-21.
The "function" of the plenum in the patented devices is to divide the fluid flowing into the conduits into two essentially equal amounts and to direct that fluid into the parallel flow tubes. The "way" the plenum accomplishes its function is to provide a manifold with one inlet opening and two exit openings. The "result" is an essentially equal flow of fluid through the two conduits, which permits an accurate measurement of mass flow. RT-WHO 420.
The Exac flowsplitter performs the same function in substantially the same way. *1445 The evidence at trial indicated that the Exac flowsplitter divides the flow of fluid in the pipeline so that fifty-six percent of the fluid goes through one flowtube and forty-four percent of the fluid goes through the parallel tube. RT-WHO 1251-52; DX RT at 46. The Court finds that this division of flow is "essentially equal," as required by Claim 1 of the '025 patent. PX 275 at ¶ 24. Exac's founders have admitted that its flowmeters split the fluid flow "in half" (RT-WHO 1030; PX 13 at 7) or "approximately in half." RT-SW 2483. Finally, the evidence at trial established that a person of ordinary skill in the art would have been aware, in January 1985 (the date the '025 patent issued), that the Exac flowsplitter was equivalent to the plenum in the patent. PX 276 at ¶ 44.

CONCLUSION
Based upon the above findings of fact and conclusions of law, the Court finds that Micro Motion has established by a preponderance of the evidence that the accused Exac devices infringe Claims 8 and 57 of the '450 patent and Claim 1 of the '025 patent under the doctrine of equivalents. Accordingly,
IT IS HEREBY ORDERED that:
1. The trial in this matter is to reconvene at a date set by the Court in order to determine the amount of damages to be awarded to Micro Motion.
2. A Status Conference will be held on Wednesday, July 11, at 2:00 p.m., at which time the parties will present to the Court a stipulation concerning the trial date, a damages study, and the issues that remain to be tried.
NOTES
[1]  These flowmeters combine an EX model sensor (i.e., EX 12, EX 120, EX 1200, EX 6000, or EX 9000) with a Series 7100 or Series 8100 transmitter. The parties stipulated for purposes of this trial that the designs of the Exac 1100 and 2100 model flowmeters are the same insofar as the issues of infringement are concerned. See Joint Pretrial Statement at 8 (filed Jan. 16, 1990).
[2]  Exac has also asserted affirmative defenses and antitrust counterclaims against Micro Motion alleging wrongful acquisition, procurement, maintenance, and enforcement by Micro Motion of patents that are invalid and not infringed, and other wrongful conduct in the marketplace. The proceedings with respect to the antitrust claims were severed and stayed by orders filed October 15, 1985, and April 8, 1986, pending final determination of the patent issues.
[3]  Exac ultimately withdrew its defense that Claims 8 and 57 of the '450 patent are invalid. With respect to the validity of Claim 1 of the '025 patent, the parties agreed to be bound by the final decision of the then-pending reexamination by the United States Patent and Trademark Office ("PTO") of the patentability of the '025 patent. The PTO subsequently confirmed the patentability of this claim. Exac's patent unenforceability defense was tried before Judge Spencer M. Williams in June of 1987. At the conclusion of this trial, Judge Williams held that Exac had failed to prove that the '450 and '025 patents were unenforceable. Micro Motion, Inc. v. Exac Corp., 686 F.Supp. 789, 797 (N.D.Cal.1987).
[4]  "PX" refers to plaintiff's trial exhibits; "DX" refers to defendant's trial exhibits; and "RT-WHO" refers to the trial transcript. Excerpts from designated deposition and former trial testimony are indicated by the witness' name followed by "Dep." (for deposition testimony) or "RT-SW" (for former trial testimony before Judge Williams). Where more than one deposition was designated, the volume number of the cited deposition is indicated.
[5]  The amount of twist is too small to be detected by the unaided human eye. At trial, the parties provided video animations, high-speed photography and other demonstrations to portray this phenomenon.
[6]  In a helical crossover loop, fluid enters the conduit from one side, makes a 360-degree loop and crosses over itself, exiting from the opposite side.
[7]  Exac also contends that the patent claims in suit must be construed to require a conduit that oscillates without torsion when there is no fluid flowing through the conduit. Exac believes that the presence of torsion in the no-flow condition in the Exac meters makes its meters noninfringing. However, none of the asserted patent claims require that the conduit oscillate without torsion when there is no flow. RT-WHO 457-60, 578-79; PX 1.

Similarly, Exac contends that the asserted patent claims must be construed to require flowmeters that are insensitive to variations in pressure. Yet, none of the asserted claims require such pressure independence. PX 97, 98, 99. Finally, Exac argues that the patents should be construed to require flowmeters that are insensitive to conduit temperature variations. Exac's conduits require a compensation for temperature. Again, however, the asserted claims do not require temperature insensitivity. RT-WHO 414-16. Moreover, there was evidence at trial that a person of ordinary skill in the art in 1984 would have known that the spring constant of resilient metal is dependent on temperature, so that it would have been unnecessary to disclose the fact of temperature sensitivity or means to compensate for it in the patents in suit. RT-WHO 582-85.
[8]  The doctrine of prosecution history estoppel may limit the reach of the doctrine of equivalents under certain circumstances. However, prosecution history estoppel applies only where a patent claim was rejected by the PTO based on prior art, and the patentee amends or limits the claim in order to overcome the examiner's prior art rejection. Bayer Aktiengesellschaft v. Duphar Int'l Research, B. V., 738 F.2d 1237, 1242-43 (Fed.Cir.1984). In this case, claims 8 and 57 of the '450 patent and claim 1 of the '025 patent were never rejected by the examiner. PX 126 at ¶¶ 18-20; RT-WHO 668-74, 680-81. Accordingly, there can be no prosecution history estoppel that would limit the application of the doctrine of equivalents to the claims in suit.
[9]  Micro Motion also suggests that the Supreme Court adopted the Federal Circuit's approach in two cases. Sanitary Refrigerator, 280 U.S. 30, 50 S.Ct. 9; Temco Co. v. Apco Co., 275 U.S. 319, 48 S.Ct. 170, 72 L.Ed. 298 (1928). In these cases, the Court found infringement even though the infringing devices were made after the patents in suit issued. Micro Motion suggests that the Temco and Sanitary Refrigerator Courts, to have found infringement, must, therefore, have determined equivalence at the time of infringement because in each case the accused device did not exist at the time the original patent issued.

Unfortunately, neither the Temco or Sanitary Refrigerator opinions state the test applied in determining equivalence. It is possible that these decisions looked at the state of the art at the time of the original patent in determining equivalence, even though the infringing devices themselves were not manufactured until years later.
Moreover, Exac notes that a Supreme Court case more recent than Temco and Sanitary Refrigerator used the date of the patent as the relevant date for determining equivalence. Halliburton Oil Well Cementing Co. v. Walker, 329 U.S. 1, 13, 67 S.Ct. 6, 12, 91 L.Ed. 3 (1946). Micro Motion, however, correctly points out that this statement in Halliburton is dicta. Therefore, the Court finds that these three cases are of no assistance in resolving this conflict.
[10]  Functionally, this is the same as the 1984 date to be applied to the '450 patent because the date of issue of the '025 patent is the first day of 1985.
[11]  Claim 1 simply describes a "means to measure the magnitude of Coriolis forces tending to elastically distort the `U' shaped conduit about the deflection axis...." PX 1.
[12]  The way delta t is used to compute mass flow rate is a matter of simple mathematics, since mass flow is a function of pipe geometry constants and delta t. PX 1 at col. 11, 11. 41-42.
[13]  This requirement is also a part of the "means to measure" element of Claim 8.
[14]  Again, as with Claim 8, the parties have agreed that this claim literally requires a planar conduit.
[15]  Subpart iv of the second element of Claim 1 requires that each of the parallel conduits must have "a bending axis ... located in the same plane as and perpendicular to the side legs of said U-shaped flow tube, and a torsion axis located in the same plane as the side legs of said U-shaped flow tube...."
