Method and apparatus for calibrating a multiport test system for measurement of a DUT

[0001] This application claims priority under 35 U.S.C. 120 to co-pending and commonly owned U.S. patent application Ser. No. 09/954,809 filed Sep. 18, 2001, which claims priority under 35 U.S.C. 119(e) to commonly-owned, U.S. Provisional Patent Application Serial No. 60/233,595 entitled, METHOD AND APPARATUS FOR LINEAR CHARACTERIZATION OF MULTI-TERMINAL SINGLE-ENDED OR BALANCED DEVICES, filed Sep. 18, 2000, now abandoned, which is hereby incorporated by reference in its entirety.

[0002] It is to be understood that according to this disclosure an unbalanced device has one signal carrying terminal for each single ended input and out put of the device and operates in the single ended mode. It is also to be understood that a balanced device has two signal carrying terminals for each balanced input and output pair of the device and operates in one of two modes, either a common mode (even mode) or a differential mode (odd mode).

[0003] Signal integrity and its characterization is an issue of growing importance as digital networks increase in speed and bandwidth. Traditionally, RF devices and digital devices have had little in common. However, as digital signals operate higher and higher in speed and approach a 1 gigabit per second (hereinafter Gb/S) threshold, the digital signal analysis tools and the RF signal analysis tools begin to overlap in function and requirements. In addition, with high speed digital signals, the harmonic contents of the digital signals are many times higher than a frequency of a fundamental tone, which results in a need for greater precision in connections between, for example, a device under test (hereinafter DUT) and testing devices that characterize these high speed digital DUTs.

[0004] A traditional differential time domain reflectometer (hereinafter TDR) system, which has been used to test digital DUTs, uses a step function as a driving signal to a DUT. The step function signal is used because there is no commercially available signal source that can generate an impulse function, which is a preferred driving signal for analog circuit characterization.

[0005] Due to harmonic frequencies content in fast-rise-time digital signals, one may think of high-speed digital circuits or interconnections as transmission lines, and consider the effects of, for example, reflected signals on the measurement of these circuit and interconnections. However, frequency domain instruments such as the Vector Network Analyzer (hereinafter VNA) that typically consider these effects have not historically been used to measure such digital circuits and interconnections. Instead, the differential TDR has been used to characterize high-speed digital circuits and components.

[0006] There are several reasons why the differential TDR has traditionally been used. For one, time-domain representations of digital signals, such as state-to-state transitions, need to be preserved and their signal characteristics as a function of time need to be characterized. In addition, most devices and systems transmitting high-speed digital data use differential signals, while most VNAs are designed for a single-ended or unbalanced, 2-terminal devices. However, with increasing data rates, the dynamic range of a very high-speed differential TD system is often inadequate for analyzing low-level signals such as crosstalk signals or the signal components responsible for generating electromagnetic interference (hereinafter EMI). In addition, the parasitic inductances and capacitances that exist in signal lines and interconnections are generally ignored at lower data rates, where traditional TDR systems have been used, but as the data rates become significantly high they can no longer be ignored. Further, the traditional differential TDR systems do not correct for the systematic sources of error in the measurement equipment, nor do they support de-embedding fixtures or interconnects use to characterize DUT.

[0007] As is known to those of skill in the art, a complete characterization of a single-ended linear 1-terminal or 2-terminal DUT can be achieved by measuring standard S-parameters with a Vector Network Analyzer (hereinafter VNA). A standard 2-port S-parameter matrix represents a frequency domain representation of all possible signal paths between any two terminals of a multi-terminal DUT, including forward, reverse and transmission and reflection characteristics of and between the two terminals of the DUT.

[0008] One known procedure for measurement of a multi-terminal DUT with a commercially available 2-port VNA, is to connect each port o the VNA to 2-terminals of the multi-terminal DUT, and to terminate the rest of the terminals of the multi-terminal DUT with high quality matching terminations. The 2-port VNA is used to measure or characterize the 2-terminals of the multi-terminal DUT with the remainder of the terminals terminated. This procedure is then repeated wherein two additional terminals of the multi-terminal DUT are measured and the remainder of the terminals are terminated with the high quality matching terminations, until all of the terminals of the multi-terminal device have been measured. Once all the terminals of the multi-terminal device haven been measured, the plurality of 2-port measurements made by the VNA are then processed to obtain the S-parameters of the multi-terminal DUT. This procedure is described, for example in the related art portion of U.S. Pat. No. 5,578,932 issued to the applicant of this application, which is herein incorporated by reference in its entirety.

[0009] As was described Combined Differential and Common-Mode Scattering Parameters: Theory and Simulation, David G. Bockelman and William R. Eisenstadt, IEEE Transactions on Microwave Theory and Techniques, Vol. 43, No.7, July 1995 (hereinafter the Bockleman IEEE Article). The standard S-parameters can be extended to mixed-mode S-parameters (hereinafter S_mm) that characterizes the linear performance of balanced devices. The mixed-mode S-parameter matrix S_mm are similar to conventional signal-ended S-parameter matrix S in that the S_mm characterize the stimuli and the response between any two terminal of a DUT. However, the S_mm also considers the stimulus and response mode of operation, in addition to the stimulus and response port.

[0010] Referring to FIG. 1(a), as is known to those of skill in the art, the single-ended 2-port S are defined in the format S_yz with yz representing the response and stimulus ports, respectively. Referring now to FIG. 1(b), the mixed-mode S-parameter matrix S_mm for balanced devices are defined as S_wxyz, with wx representing the additional response and stimulus modes of the DUT. The S_mm of FIG. 1(b) can characterize the linear performance of a balanced 2-terminal device. As with single-ended S-parameters S, each column represents a different stimulus condition and each row represents a different stimulus condition and each row represents a different response. In addition, the S_mm matrix can be divided into four quadrants, and the DUT can be considered a 2-port device operating in pure and conversion modes of operation, as illustrated in FIG. 1(b).

[0011] Referring to FIG. 1(b), the upper left-hand quadrant describes the behavior of the DUT when it is stimulate with a differential-mode signal and a differential-mode response is observed. This quadrant consists of four parameters: reflection parameters S_dd11, S_dd22 on both balanced terminals one and two of the 2-terminal DUT and transmission parameters S_dd21, S_dd12 in forward and reversed directions between terminals one and two of the 2-terminal DUT. Thus, this quadrant describes the performance characteristics of a 2-terminal DUT operating in the differential mode.

[0012] The lower right quadrant describes the behavior of the DUT when it is stimulated with a common-mode signal and a common-mode response is observed. This quadrant also consists of four parameters: reflection parameters S_cc11, S_cc22 on both balance terminals one and two of the DUT and transmission parameters S_cc21, S_cc12 in the forward and reverse directions between terminals one and two. This quadrant describes the fundamental performance characteristics of a 2-terminal DUT operating in the common-mode. It is to be appreciated by one of skill in the art that if one objective is to reduce the level of common-mode signals in a DUT, these parameters can be used to analyze the DUT behavior. In addition, a common-mode rejection of the DUT is often of interest, and it can be calculated by taking a ratio of a differential-mode gain, S_dd21, to the common-mode gain, S_cc21.

[0013] The lower left quadrant of FIG. 1(b) describes the behavior of the DUT when it is stimulated with a differential-mode signal and a common-mode response is observed. The S-parameters of this quadrant show how the DUT converts a differential-mode signal to a common-mode signal. Again, it is possible to examine reflections on each terminal and transmission of a signal in the forward and reverse directions in this mode of operation. It is to be appreciated that in a perfectly symmetrical balanced device, there should be no conversion between the differential mode and the common mode and all of these S-parameters should be zero.

[0014] This lower left quadrant is useful for describing the behavior of the DUT, for example, because any asymmetry in a DUT may result in an imbalance in an amplitude of signals on each side of a balanced pair of terminals of the DUT, or in a phase skew. A conversion of a stimulus signal from a differential-mode to a common-mode may, for example, cause signals to appear on ground returns in the DUT and can result in generation of electromagnetic interference (hereinafter EMI). In high-speed digital signal devices, any imbalance in the DUT can also result in degradation ob bit-error rates (hereinafter BERs).

[0015] The upper right quadrant of FIG. 1(b), describes the behavior of the DUT when it is stimulated with a common-mode signal and a differential-mode response is observed. These S-parameters characterize how the DUT converts a common-mode signal to a differential-mode signal. The mixed mode S-parameters in the upper right quadrant also characterize reflections on each terminal and transmission in a forward and reverse directions between terminals one and two of the DUT. Again, it is to be appreciated that if a balanced DUT is perfectly symmetrical, there will be no conversion between these modes, and all of these terms should be zero.

[0016] As discussed above, any conversion of a signal from a common-mode to a differential-mode may cause the DUT and any system that the DUT is to be used, to be susceptible to EMI. For example, noise that couples into the DUT and system from power supplies, ground connections, and the like may generally be introduced as a common-mode signal. If this common-mode noise signal is converted to the differential-mode, it may be superimposed on the actual DUT signal and degrade its signal-to-noise ratio. Therefore, the mode-conversion behavior of the DUT front common to differential mode may result in the DUT being susceptible to sources of noise and interference.

[0017] Accordingly, mode-conversion is one phenomenon that should be considered when examining the signal integrity of balanced DUTs. However, since these signals are ideally zero, the dynamic range provided by a conventional differential TDR systems is typically not sufficient.

[0018] It is to be appreciated that the tremendous growth of, for example, the Internet and the increasingly faster processing speed of computers and other digital devices is moving more products in to the domain of high-frequency data transmissions. Uncorrected sources of error in current test instruments may have been acceptable for lower data rate devices, but are no longer acceptable as the data rates of these devices are increasingly pushed higher. In addition, much of the test equipment used to characterize these digital devices provide scalar data. However, scalar data does not include enough information to adequately characterize and remove the sources of error. Further, these traditional test devices have issues of dynamic range and also may often provide and excessive amplitude incident signal to the DUT being tested.

[0019] The foregoing and other objects and advantages will be more fully appreciated from the following drawings in which:

[0020] FIG. 1(a) illustrates a single-ended, 2-port S-parameter matrix that is known to those of skill in the art;

[0021] FIG. 1(b) illustrates a mixed-mode, S-parameter matrix for balance devices that is known to those of skill in the art;

[0022] FIG. 2 illustrates on embodiment of a multiport test system according to this disclosure;

[0023] FIG. 3 illustrates an embodiment of a 4-terminal multiport test system according to this disclosure;

[0024] FIG. 4 illustrates an embodiment of a 9-port multiport test system comprising three test and one referenced channel receiver, according to this disclosure;

[0025] FIG. 5(a) illustrates an embodiment of a 6-port multiport test system comprising two test and one referenced channel receiver, according to this disclosure;

[0026] FIG. 5(b) illustrates another embodiment of a 6-port multiport test system comprising three test and one referenced channel receiver, according to this disclosure;

[0027] FIG. 6 illustrates one embodiment of an N-port calibration device as known to those of skill in the art;

[0028] FIG. 7 illustrates one embodiment of a single-pole, double throw switching device used in an embodiment of an N-port automatic calibration device, according to this disclosure;

[0029] FIG. 8(a) illustrates one condition of the single-pole, double-throw switch of FIG. 7;

[0030] FIG. 8(b) illustrates a second condition of the single-pole, double-throw switch of FIG. 7;

[0031] FIG. 8(c) illustrates a third condition of the single-pole, double-throw switch of FIG. 7;

[0032] FIG. 8(d) illustrates a fourth condition of the single-pole, double-throw switch of FIG. 7;

[0033] FIG. 9 illustrates the difference in phase presented by the single-pole, double-throw switch of FIG. 7 when operating in the third and fourth conditions;

[0034] FIG. 10 illustrates a reflection loss of the single-pole, double-throw switch of FIG. 7 when operating in the fourth condition;

[0035] FIG. 11 illustrates a schematic representation of a combined single-ended transmission line and a balanced transmission line, according to this disclosure;

[0036] FIG. 12 illustrates a schematic of the DUT of FIG. 1 with alternative port assignments of a 6-port MTS to the DUT of the FIG. 11;

[0037] FIG. 13 illustrates a schematic diagram of a balun DUT and one example of a port assignment, according to this disclosure;

[0038] FIG. 14 illustrates a schematic diagram of a two differential transmission line DUT and an exemplary port assignment, according to this disclosure;

[0039] FIG. 15 illustrates an embodiment of a 4-pair differential transmission line DUT and one exemplary port assignment, according to this disclosure;

[0040] FIG. 16 illustrates one example of assigning terminal numbers for input and output ports of a differential pair of transmission lines, according to this disclosure;

[0041] FIG. 17 illustrates one example of assigning terminal numbers for input and output ports of a differential pair of transmission lines, according to this disclosure;

[0042] FIG. 18 illustrates mixed-mode S-parameter measurements of a differential pair of transmission lines such as illustrated in FIG. 16, with an embodiment of a multiport test system of this disclosure;

[0043] FIG. 19 illustrates a time-domain impulse response representation of the mixed-mode S-parameter of FIG. 18;

[0044] FIG. 20 illustrates the pure differential forward transmission time-domain impulse response of the time-domain representation of FIG. 19;

[0045] FIG. 21 illustrates the transmission time-domain impulse response of a 3.35 Psec loss-less transmission line that is simulated according to an embodiment of an MT of this disclosure;

[0046] FIG. 22 illustrates a pure differential forward transmission time-domain step response of the FIG. 18;

[0047] FIG. 23 illustrates the transmission time-domain step response of a 3.35 psec loss-less transmission line that is simulated according to an embodiment of an MTS of this disclosure;

[0048] FIG. 24(a) illustrates a pure differential reflection time-domain step response of the differential transmission line according to an embodiment of the multiport test set o this disclosure;

[0049] FIG. 24(b) illustrates a pure differential impedance profile of the differential transmission line according to an embodiment of the multiport test set of the disclosure;

[0050] FIG. 25 illustrates a simulated input waveform than can be provided according to an embodiment of the multiport test set of this disclosure;

[0051] FIG. 26 illustrates a simulated output response of a differential transmission line in response to the simulated input signal of FIG. 25, according to a multiport test set of this disclosure;

[0052] FIG. 27 illustrates another example of a simulated input form according to a multiport test system of this disclosure;

[0053] FIG. 28 illustrates a simulated output response of the differential transmission line in response to the simulated input signal of FIG. 27, according to an embodiment of a multiport test set of this disclosure;

[0054] FIG. 29 illustrates another embodiment of a simulated input signal that can be provided with an embodiment of a multiport test set according to this disclosure;

[0055] FIG. 30 illustrates a simulated output response of a differential transmission line in response to the simulated input response of FIG. 29, according to an embodiment of the test system of this disclosure;

[0056] FIG. 31 illustrates an eye diagram which can be provided according to an embodiment of a multiport test system of this disclosure;

[0057] FIG. 32 illustrates another embodiment of an eye diagram of a differential transmission line DUT at an increased data rate, that can be simulated with an embodiment of a multiport test system of this disclosure;

[0058] FIG. 33 illustrates mixed-mode S-parameters of a differential transmission line DUT cascaded with a 25 Psec loss-less transmission line embedded at an output of the DUT, that can be simulated with an embodiment of a multiport test system of this disclosure; and

[0059] FIG. 34 illustrates an eye diagram of the differential transmission line DUT embedded with the loss-less transmission line of FIG. 33, that can be provided with an embodiment of the multiport test system of this disclosure.

[0060] Numerous advantages, novel features, and objects of the invention will become apparent from the following detailed description of the invention when considered in conjunction with the accompanying drawings. In the FIG.s, each identical, or substantially similar component that is illustrated in various FIG.s is represented by a single numeral or notation. For purposes of clarity, not every component is labeled in every FIG., nor is every component of each embodiment of the invention shown where illustration is not necessary to allow those ordinary skill in art to understand the invention.

[0061] It is to be understood that according to this disclosure an unbalanced device has one signal-carrying terminal for each single ended input and output of the device and operates in the single ended mode. It is also to be understood that a balanced device has two signal carrying terminals for each balanced input and output pair of the device and operates in one of two modes, either a common mode (even mode) or a differential mode (odd mode). It is further to be appreciated that a multi-terminal device according to this disclosure is considered to be a device having more than two terminals. For example, one balanced pair of a DUT may comprise two signal terminals that may be coupled to two corresponding ports of a vector network analyzer (hereinafter VNA). Accordingly, a multi-terminal device can have more than two connections to a VNA. Therefore, additional hardware is needed to connect a multi-terminal device to a 2-port VNA. It is to be understood that according to this disclosure, a multiport test set (hereinafter MTS) is any test set that is used to measure a multi-terminal device having more than two terminals and may comprise a 2-port VNA and additional hardware that couples a multi-terminal DUT to the 2-port VNA. However, it is to be appreciated that a 2-port VNA need not be used. Instead a multiport VNA can be used and an MTS according disclosure may comprise any multiport VNA having a same number of ports as a multi-terminal DUT, and may also comprise a multiport VNA and any hardware that may be used to couple the multi-terminal device to the multiport VNA, where the multi-terminal device has more terminals than the multiport VNA.

[0062] One aspect of some embodiments of the method and apparatus for characterizing multi-terminal devices of this disclosure is that there is provided an ability to measure mixed-mode scattering parameters S_mm of balanced devices. In addition, another aspect of some embodiments of the method and apparatus of this disclosure is that the precision and accuracy provided by a VNA can be used to measure and characterize digital devices under test (hereinafter DUT). Still another aspect of some embodiments of the method and apparatus of this disclosure is that a VNA maybe used to provide characteristics and measurements that previously were unavailable to engineers concerned with signal integrity in high-speed digital systems. Still another aspect of some embodiments of the method and apparatus of this disclosure is that the results of a testing and characterization of, for example, a balanced DUT can be presented to the tester in either of the time domain or the frequency domain.

[0063] As was discussed in the description of the related art, the traditional differential time domain reflectometer (hereinafter TDR) systems use a step function as a driving signal to a DUT instead of an impulse function, which is desired. This is due to a lack of a source that can generate an impulse function, which is rich in harmonic content. In other words, traditional TDR's do not provide an impulse function. A step response or an impulse response or any singularity response can be modeled in a traditional VNA with a modeled step function driving signal each have several disadvantages, some of which described in the description of the related art.

[0064] With the known TDR test systems the digital DUTs are typically measured by injecting a known digital signal into the DUT and measuring an output from the DUT. However, there are limitations to TDR test systems, such as, the test instrument may not be able to provide an input signal having all desired characteristics of the input signal. In addition, such TDR systems typically have limits on stability and accuracy when generating and measuring step signals. Further, lack of dynamic range in the TDR instrument can exacerbate or mask a presence of similar fluctuations induced by the DUT. Still further, DUT amplitude limits maybe far different than operating levels of the TDR instrumentation used. For example, the driving signal may have to be attenuated to safe levels for the DUT, but this may degrade a dynamic range of the TDR instrument and introduce an additional component in to the DUT test path that cannot be removed from a measurement with a TDR system. Moreover, scalar receivers are used in such TDR instrumentation, but scalar signals do not have enough information to correct for systematic error coefficients introduced, for example, by interfaced equipment such as test sets and test fixtures.

[0065] As is known to those of skill in the art, a characterization of a single-ended linear 1-terminal or 2-terminal DUT can be achieved by measuring standard S-parameters with a VNA. It is to be appreciated that a multi-terminal DUT that is to be measured at each terminal can be thought of as a series of 2-terminal measurements that can be measured with a 2-port VNA. There are N (N-1)/2 2-terminal signal paths for a DUT with N-terminals. For, example, a 4-terminal DUT has a six possible signal paths between the four terminals of the DUT. It can be appreciated that as a number of terminals of the DUT increase, the number of signal paths increases significantly. For example, an eight-terminal DUT has 28 2-terminal signal paths and a sixteen-terminal DUT has 120 2-terminal signal paths. Accordingly, it can be appreciated that characterizing a multi-terminal DUT is becoming increasingly more difficult, especially where multi-terminal devices are becoming more common due to the use of balanced topologies in digital systems.

[0066] One common approach for measuring multi-terminal devices with a 2-port VNA is known as the round-robin technique. Since only two terminals can be measured at a time, the terminals that are not measured are terminated preferably in a characteristic impedance that matches the terminal so as to prevent reflections of the measurement signal being used to characterize the DUT, back into the DUT. Thus, the VNA is coupled to two terminals of the multi-terminal DUT and the rest of the terminals are terminated with preferably high quality terminations. The terminated terminals and the terminals coupled to the VNA are then rotated by a systematical procedure until all of the terminals of the DUT have been measured. The multiple 2-terminal measurements are then processed to obtain the final S-parameters S of the multi-terminal DUT.

[0067] However, one problem with this approach is that signals reflected by the non-perfect terminations may still show up at the measured terminals, especially at frequencies above 500 MHz. For example, because the terminal impedances of the DUT rarely match the available terminations used to terminate the terminals of the DUT, there typically results reflected signals from these terminations. In addition, multiple connections and disconnections of the terminals of the DUT to the VNA and to the terminations are made. Thus, connector repeatability can contribute measurement error. Also, connector wear can be accelerated. Further, a potential for operator error exists and increases with the number of terminals of the DUT. Moreover, redundant measurements are made with this procedure. An additional disadvantage of measuring a multi-terminal DUT with a 2-port VNA comprises that any internal reflections from terminals with not perfectly matched terminations can add significant error to the measurement. This can be readily apparent in a DUT with low terminal-to-terminal isolation. In addition, multiplexing a VNA to a DUT with a plurality of terminals can result in an insertion of additional multiplexing equipment in the measurement pathways. Accordingly, the repeatability of the multiplexing equipment can contribute to measurement errors of the DUT. Further, different switch configurations used for routing the measurement signals to the DUT can present different impedances to the terminals of the DUT, thus contributing to measurement error.

[0068] Thus, with the above-described round robin technique any VNA error correction cannot include effects of unused terminals in the correction and the VNA relies on the quality of the terminations on the unused terminals of the DUT to suppress any reflected signals introduced by the unused terminals. This problem can be accentuated at higher frequencies. Also, many multi-terminal DUTs may not have an impedance value that is a standard impedance value such as 50 or 75. In addition, terminating an unused terminal with a non-standard termination may not be practical. Further, it may be difficult to get accurate measurements of the DUT due to the many connections and disconnections to the multi-terminal DUT. Still further, the measurement accuracy can be degraded by multiple redundant reflection measurements. For example, for a 4-terminal device each reflection coefficient can be measured three times. However, it is not readily apparent which measurement is the most accurate without knowing the reflection coefficient for each termination and using a correction algorithm that incorporates all the VNA ports. Still further, the calibration of the VNA and the measurement of the DUT can be time consuming and thus can be inefficient and impractical, for example, in a production environment. Moreover, the procedure is prone to errors, and therefore there is a need for a higher level or operator skill and experience.

[0069] It is to be appreciated that a characterization of a DUT operating in any of a pure single ended mode, a differential mode or a common mode can also be achieved by measuring standards S-parameters S with a VNA, and converting the standard S-parameters to mixed-mode S-parameters S_mm. However, a multi-terminal DUT may also have a combination of single-ended and balanced terminals. Thus, in addition to pure single ended and pure balanced modes, there are mode conversions between types of terminals of a multi-terminal DUT.

[0070] However, additional problems with measuring a multi-terminal DUT with a VNA are that there has been no general solution for transforming the S-parameters S of a DUT measured with the VNA to multiple single-ended and multiple balanced parameters that can exist for a DUT. In addition, there is a lack of testing tools for characterizing and measuring a signal integrity in high-speed DUTs, since neither the know differential TDR systems not the known VNAs are adequate in view of the above-discussed problems.

[0071] As known to those of skill in the art, a VNA is a frequency domain apparatus that provides standard S-parameters S of a DUT measured with a VNA. According to one aspect of some embodiments of the method and apparatus of this disclosure, a VNA may be used to measure and characterize balanced DUTs and to provide characteristics of these devices that have not previously been available to engineers concerned with signal integrity in high-speed digital systems. Nevertheless, while an S-parameter representation of a DUTs performance can comprehensively characterize the performance of a balanced or mixed-mode DUT, it is also to be appreciated that a frequency domain S-parameters representation does not directly indicate how the DUT is performing to one of skill in the art in the signal integrity field. This is because S-parameters provide a frequency domain representation, while high-speed digital data signals for digital DUTs are typically analyzed in the time domain. However, according to one aspect of some embodiments of the method and apparatus of this disclosure, a VNA can also be configured to transform the frequency domain S-parameters to the time domain with a transform, such as an Inverse Fourier Transform.

[0072] Another aspect of some embodiments of the method and apparatus of this disclosure takes advantage of the principle that the product of two frequency-domain signals is equal to a convolution in the time-domain of the Inverse Fourier Transform of these two frequency signals. Referring to Equation (1):

[0073] An input-driving signal X (f) in the frequency domain can be applied to a linear circuit that has a known through response S₂₁(f) in the frequency domain, which has previously been characterized by a VNA, to determine an output Y(f) of the DUT in the frequency domain. It is to be understood throughout this disclosure that the frequency domain is represented by (f) and the time domain is represented by (t). Where the input driving signal X(f) has a unity amplitude as it is swept in frequency over a frequency range of interest for testing a DUT, the output Y(f)S₂₁(f), where X(f)1. It is known to those of skill in the art that an input-driving signal having a unity amplitude in the frequency domain corresponds to a unit impulse function (t). Thus, taking an inverse for a transform of Equation (1) where the input signal is the unit impulse function (t), results in the output signal in the time domain y(t)s₂₁ (t), which is a complete characterization of the DUTs forward transmission coefficient in the time domain when the input signal x(t) is the unity impulse function (t). Thus, the time domain response of a DUT can be determined from the convolution of the measured S-parameter data of the DUT with the impulse function. In other words, the forward transmission coefficient s₂₁ (t) is the time-domain impulse response of the DUT. In a similar manner, it is to be appreciated that the time domain impulse response of the reflection coefficients s₁₁, s₂₂ and the reverse transmission coefficients s₁₂ of the measured S-parameters of a 2-terminal DUT can be determined by a similar convolution procedure.

[0074] According to one aspect of some embodiments of the method and apparatus of this disclosure, a VNA can be used to measure the S-parameters of a DUT, and the time-domain reflection (hereinafter TDR) response and the time-domain transmission (hereinafter TDT) response of the DUT can be determined by taking the Inverse Fourier Transform of the S-parameters of the DUT measured with a VNA. It is to be appreciated that the TDR response responds to the reflection coefficient, e.g. S₁₁, S₂₂, for a 2-terminal DUT, and the TDT response corresponds to the transmission coefficients, e.g., S₁₂, S₁₂, of the 2-terminal DUT.

[0075] As will be apparent from the detailed disclosure infra, some embodiments of the method and apparatus of this disclosure provide the following advantages over conventional VNA and TDR measuring devices for the characterization of multi-terminal DUTs and, in particular, multi-terminal mixed-mode, high-speed, digital DUT. One advantage of some of the embodiments of the method and apparatus of this disclosure is that a comprehensive correction routine for a multiport VNA can be provided to enhance an accuracy of S-parameter measurements of a DUT made with the multiport VNA. Another advantage of some embodiments of the method and apparatus of this disclosure are that any of the following are reduced to a number of connections between the DUT and a VNA, a time needed to calibrate a VNA or measure the DUT, operator experience and skill, and effort used for calibration of a multiport VNA. Thus, there is provided an increased efficiency of calibration of a multiport VNA with some embodiments of multiport VNA and multiport calibration device of this disclosure. Still another advantage of some embodiments of the method and apparatus of this disclosure is that there is a provided a general solution for determining single-ended, balanced, and mixed-mode S-parameters for any terminal configuration of a multi-terminal DUT. Still another advantage of some embodiments of the method and apparatus of this disclosure is that signal-integrity parameters and, in particular, parameters, such as, the time domain response to a singularity function of the DUT are provided to analyze the performance of the high-speed, digital, DUT.

[0076] According to another aspect of some embodiments of the method and apparatus of this disclosure, to mixed-mode S-parameters S_mm of a DUT as determined by a multiport VNA of this disclosure, can be converted from the frequency domain to the time domain, for example, with an Inverse Fourier Transform. One aspect of some embodiments of the method and apparatus of this disclosure is that once the time-domain impulse response of the digital DUT is known, an output response of the DUT to any digital waveform can be determined by a convolution of the input digital waveform with the time-domain impulse response of the DUT, according to a Discrete Time Convolution Transformation. Still another aspect of some embodiments of the method and apparatus of this disclosure is that any digital input waveform can be synthesized. For example, the impulse function, as discussed above can be provided as the input waveform or, for example, known imbalanced conditions can be provided as the input waveform to the digital DUT

[0077] Another advantage of some embodiments of the method and apparatus of this disclosure is that a very high dynamic range can be attained with the multiport VNA of this disclosure. Still another advantage of some embodiments of the method and apparatus of this disclosure is that the S-parameters of the digital DUT can only be measured once to fully characterize the digital DUT. Still another advantage of some embodiments of the method and apparatus of this disclosure is that the DUTs response under various input signal conditions can be evaluated rapidly, and a comparison of the DUTs response for different input waveforms can be performed easily.

[0078] Referring now to FIG. 2, there is illustrated one embodiment of a multiport test system (hereinafter MTS) according to this disclosure. The MTS comprises N-ports. Each port has a respective signal separation device 107, 108 . . . 109, 118 . . . 117, and 116 and is coupled to a VNA receiver 104, 105 . . . 106, 115 . . . 114, and 113. The MTS also comprises a signal generator 100 that generates the test signals used to test a DUT 128, and a coupler 101 that couples the test signal as a reference signal to a reference channel receiver 102. The signal generator also provide the test signal to a single pole, multi-throw switch 103, that routes the test signal to any of the N-terminals of the MTS and the N-terminals of the DUT. Each output of the single-pole, multi-throw switch 103 is coupled to a corresponding single-pole, double-throw switch 110, 111 . . . 112, 121 . . . 120, and 119. According to one alternative embodiment of the MTS, N single-pole, double-throw switches can be integrated into switch 103 such that switch 103 is an absorptive type switch. Thus, the test signal can be fed to any terminal of the DUT 128 with an appropriate switch configuration.

[0079] FIG. 2 illustrates, by way of example, that test signal can be fed to terminal 1 of the DUT while all other terminals of the DUT are terminated into corresponding single-pole, double throw switch terminations 123, 124, 125, 126, and 127 of the single-pole, double-throw switches 111, 112, 119, and 121.

[0080] According to one aspect of this embodiment of the MTS of this disclosure, each pole 129, 130, 131, 132, 133, and 134 of each single-pole double-throw switch 110-112 and 119-121 is connected to a corresponding signal separation device 107-109 and 116-18. According to this embodiment, there are a same number of signal separating devices as there are ports of the MTS. It is to be appreciated that his embodiment of the MTS enables both reflected and transmitted signals from the DUT coupled to the MTS to be measured simultaneously.

[0081] FIG. 2 illustrates, by way of example, one mode of operation of the MTS in which all terminals except terminal 1 of the DUT which is coupled to the signal 100 by single-pole, double-throw switched 110 and single-pole, multi-throw switch 103, are terminated by respective internal loads 123, 124, 125, 126, and 127 which are coupled to respective poles 130, 131, 132, 133, and 134 of respective single-pole, double-throw switches 111-112 and 119-121. Thus, FIG. 1 illustrates by way of example that the test signal has been routed to terminal 1 of the DUT 128 and that all other terminals of the DUT have been terminated. It is to be appreciated that according to this embodiment of the MTS, the load match presented by the internal loads of the MTS to the terminals of the DUT will remain the same throughout an entire measurement sequence of the DUT, except for the condition where the incident signal is routed to a particular terminal of the DUT (in the illustrated case, terminal 1 of the DUT).

[0082] It is understood by those of skill in the are that the DUT 128 can be measured by the MTS of FIG. 2 over a desired signal frequency range of characterization, by providing the test signal by signal generator 100 and sweeping the test signal over the desired frequency range of test of the DUT. In response, a ratio of the measured signals with the MTS to the reference channel signals can be measured at each terminal of the DUT. With this embodiment of the MTS of this disclosure, it is to be appreciated that an efficiency of measurement of the DUT is maximized by presenting a single sweep of the test signal to each terminal of the DUT. Each receiver channel 104-10 and 113-115 can be compared to the reference channel 102 during the test signal sweep. In addition, reflection of the test signal from the incident terminal and the transmission of the test signal from the incident terminal to all of the terminals can be obtained simultaneously.

[0083] Thus, according to some embodiments of the method and apparatus of this disclosure, an MTS may have a same number of test channels as there are DUT terminals. For example, a 4-terminal DUT can be coupled to a 4-port MTS. As known to those of skill in the art, a 4-terminal DUT has sixteen standard S-parameters that characterize the DUT in a single-ended or unbalanced mode. With a 4-port MTS having the same number of ports as a 4-terminal DUT, the test signal need only be swept across the frequency range of interest and provided as an incident signal to each of the four ports, one sweep for each port for a total of four sweeps. At each sweep of the test signal, as set of four test to reference channel measurements can be made by the four port MTS. Thus, with four frequency sweeps of the test signal generator over the desired frequency range of measurement of the DUT, all sixteen S-parameters can be measured. It is to be appreciated that according to some aspects of this disclosure, the configuration of the MTS of FIG. 2 is preferred.

[0084] However, it is to be appreciated that as the number of terminals of a DUT increase, it becomes economically unfeasible to have the same number of MTS test channels as DUT terminals. For example, it becomes economically unfeasible to continue adding receiver test channels. Moreover, commercially available VNAs typically have a fixed number of channels, which may not be configured to accommodate increasing number of terminals on a DUT. To overcome this limitation, according to some embodiments of the method and apparatus of this disclosure the test channels of the MTS can be shared during measurement of the DUT.

[0085] Referring now to FIG. 3, there is illustrated a 4-terminal MTS according to another embodiment of the method and aparatus of this disclosure. The 4-port MTS of FIG. 3 comprises shared test channels 202, 203. With this arrangement, port 1, 2, 3, and 4 of the DUT and ports I and 2, and ports 3 and 4 of the MTS share a respective receiver channel 202, 203. According to this embodiment, a double-pole, double-throw switch 200 multiplexes the test channel 202 between ports and 1 and 2 of the MTS. In addition, a double-pole, double-throw switch 201 multiplexes the test channel 203 between ports 3 and 4 of the MTS. Each port of the MTS has its own signal separation device 205, 206, 207 and 208 that is multiplexed to the corresponding test channel via the respective double-pole, double-throw switch 200, 201 .This embodiment of the MTS also comprises as signal generator 209 that provides the test signal used to test the DUT, and couple 210 couples the test signal as a reference signal to a reference channel 211. The signal generator 209 also provides the test signal to a single-pole, multi-throw switch 212 that routes the test signal to any of the ports of the MTS.

[0086] With the embodiment of the MTS of FIG. 3, at each sweep of the test signal, a set of two test-to-reference-channel measurements can be made by the MTS. Thus, with a total of eight sweeps over the frequency range of interest of the test signal, all sixteen S-parameters of the DUT can be measured. However, it is to be appreciated that this embodiment has a reduced measurement speed of the DUT as compared to the embodiment of FIG. 2 if the same source and receiver assemblies are used in both embodiments.

[0087] It is also to be appreciated that according to one aspect of this embodiment of the MTS, an attenuation of the test signal from each DUT terminal to a respective test channel via respective multiplexing switch 200, 201, should be high enough to mitigate any affects of impedance changes when a switch position is changed. According to this embodiment, this is accomplished by providing the switches 2001, 201 after the signal separating devices 205-208. However, the attenuation provided by this arrangement can also affect a transmission dynamic range of the MTS system. The reduction in transmission dynamic range is dependent on the swept frequency range. For example, a typical reduction in transmission dynamic range is approximately 1.5 dB to 9 GHz, 2.5 dB to 20 GHz, and 4.5 dB to 50 GHz. Nevertheless, the reduction in reflection dynamic range is insignificant because the change in impedance caused by the different states of the switch are masked by the isolation of the signal separation devices and the match that is provided by the switch. Therefore, a load match provided by this embodiment of the MTS does not significantly change by multiplexing test channels 202, 203 to the DUT terminals with switches 200, 201.

[0088] In addition, it is to be appreciated that one advantage of this configuration of the MTS of FIG. 3 is that it is scalable to any number of ports, without added complexity to its systematic error correction methodology. For example, FIG. 4 illustrates a 9-port MTS 20 that comprises four total channels, one reference channel 102 and three test channels 27, 28 and 29. The three test channels are multiplied by a plurality of single-pole, three-throw, absorptive-type switches 21, 22, 23, 24, 25, 26 and 48 to ports 1-9 of the MTS 20 and thus to terminals 1-9 of DUT 49. Thus, the 9-port MTS of FIG. 4 comprises shared test channels 27, 28, and 29. This embodiment of the MTS also comprises the signal generator 100 that provides the test signal for testing the DUT 49, and a coupler 101 that couples the test signal as a reference signal to the reference channel 102.

[0089] With this arrangement, ports 1-3, 4-6, and 7-9 of the MTS share a respective receiver 27-29. According to this embodiment a combination of single-pole, three-throw absorptive-type switch 21 and single-pole, three-throw, absorptive-type switches 22, 23 and 24 multiplex the signal source 100 to corresponding ports 1-3, 4-6 and 7-9 of the MTS and to the corresponding terminals of the DUT. Each port of the MTS has its own signal separation device 11, 12, 13, 14, 15, 16, 17, 18, and 19 that is multiplexed to one of the three test channels 27, 28 and 29, by corresponding single-pole, three-throw, absorptive-type switches 25, 26 and 48.

[0090] With this embodiment 20 of the 9-port MTS of FIG. 4, at each sweep of the test signal, as set of three test-to-reference-channel measurements can be made by the MTS. In other words, at each sweep of the test signal, the ratios A/R. B/R and C/R can be determined over the frequency range of interest of the test signal. As is understood by one of skill in the art, a 9-port DUT comprises eighty-one total S-parameters that characterize the 9-port DUT. Accordingly, with this arrangement of the 9-port MTS, all eighty-one S-parameters of the 9-port DUT can be measured with a total of twenty-seven sweeps of the test signal.

[0091] It is to be appreciated that the systematic error calibration methodology of the embodiments of FIG. 3 can be repeated to accommodate the additional ports of the 9-port embodiment of the MTS of FIG. 4. In other words, the same calibration methodology can be used without any additional calibration methodology to accommodate different switch configurations that have been added by the 9-port embodiment of the MTS.

[0092] It is also to be appreciated that according to another aspect of some embodiments of the MTS apparatus and method of this disclosure, a plurality of optimized MTS configurations can be provided for any DUT having any number of terminals. According to this aspect, an optimized embodiment of an MTS exists if a number of terminals of the DUT is divisible by a number of test channels of the MTS and an equal number of terminals of the DUT are allocated to each test channel of the MTS.

[0093] For example, a DUT having six terminals can be measured by an optimum MTS of this disclosure having any of two, three or six test channels. Referring to FIG. 2, there is illustrated one optimum embodiment of an MTS of this disclosure. For example, the DUT 128 of FIG. 2 can be a 6-terminal DUT, and that the MTS can be a 6-pot MTS. Each terminal of the DUT is coupled to a respective port of the MTS and thus to a respective test channel receiver 104-106, 115-113, by a respective signal separation device 107-109, and 118-116. It is understood by one of skill in the art that a 6-port DUT has a total of N² or in other words, a total of thirty-six S-parameters that can be determined to fully characterize the 6-port DUT. With the optimum embodiment of the MTS of FIG. 2, it is to be appreciated that for each sweep of the test signal provide by the source 101, a total of six measurements of the test channel to the reference channel or, in other words, measurement ratios T₁/R, T₂/R, T₃/R, T₄/R, T₅/R and T₆/R, can be determined for each sweep of the test signal. Accordingly, by appropriate switching of switches 103, 110-112 and 121-119, a total of six sweeps of the test signal can be provided to a measure all thirty-six S-parameters of the 6-port DUT.

[0094] Referring to FIG. 5(a), there is illustrated another embodiment of an optimum 6-port MTS 40 that can be used to measure a 6-terminal DUT. This embodiment of an optimized 6-port MTS 40 comprises one reference channel 102, two test channels 46 and 47, and single-pole, double-throw absorptive-type switch 41 that switches the test signal output by source 100 to one of the two single-pole, three-throw absorptive-type switches 51, 42. The single-pole, three-throw absorptive-type switches 51, 42 switch the test signal to one of the 6-ports of the 6-port MTS that are coupled to respective terminals of the 6-pot DUT 45 (in FIG. 5(a) the test signal is represented as provided to port one of the MTS and terminal 1 of the DUT). It is to be understood that with this embodiment, for each sweep of the test signal provided by the signal source 100, with the appropriate switching of single-pole, three-throw, absorptive-type switches 51, 42, and single-pole, three-throw absorptive-type switches 43 and 44, a ratio of two test channel measurements (A/R and B/R) can be determined. Accordingly, it is to be understood that for each sweep of the test signal, two S-parameters of the DUT can be determined. Therefore a total of eighteen weeps of the test signal can be provided by this embodiment of the MTS to measure all thirty-six S-parameters of the DUT.

[0095] Referring now to FIG. 5b), there is illustrated another embodiment 30 of an optimum 6-port MTS that can be used to measure a 6-terminal DUT. This embodiment of an optimized 6-port MTS 30 comprises one reference channel 102, three test channels 27, 28, and 29, a single-pole, three-throw absorptive-type switch 31 that routes a test signal output by source 100 through three single-pole, two-throw, absorptive-type switches 32, 33 and 34. The single-pole, two-throw, absorptive-type switch in conjunction with single-pole, three-throw switch 31 routes the signal source 100 to any of the ports 1-6 of the MTS which are coupled to corresponding terminals 1-6 of the DUT 45. It is to be understood that with this embodiment of the MTS of the disclosure, for each sweep of a test signal provided by the signal source 100, three measurements of ratios of the test channels 27, 28 and 29 to the reference channel 102 (A/R, B/R, and C/R) can be provided by appropriate switching of the single-pole, two-throw, absorptive-type switches 32-34 and single-pole, two-throw, absorptive-type switches 35, 36 and 37. Accordingly, with this 6-port MTS, for each sweep of the test signal provided by the source 100, there can be measured three S-parameters of the DUT 45. Therefore, with this embodiment of an optimized 6-port MTS, a total of twelve measurements can be used to determine all of the S-parameters of the DUT.

[0096] Accordingly, it is to be appreciated that there are a plurality of embodiments of an optimized MTS that can be used to measure any N-terminal DUT. For the 6-terminal DUT illustrated above, there are, for example, the 6-test channel receiver illustrated in FIG. 2 which can measure all of the S-parameters of a 6-terminal DUT with six frequency sweeps of the test signal, the three test channel embodiment of the MTS as illustrated in FIG. 5b) where all of the S-parameters of a 6-terminal DUT can be measured with twelve sweeps of the test signal, and the 2-test channel MTS illustrated in FIG. 5a) where all the S-parameters of the 6-teminal DUT can be measured with eighteen sweeps of the test signal. It is to be appreciated that criteria to provide the carious optimum embodiments of an MTS are maximizing the speed of the measurement by providing as many test channel receivers as feasible, and providing a number of test channels such that each test channel is used for each sweep of the test signal. In other words, providing a number of test channels so that no test channel is unused during any sweep of the test signal or in other words, providing a number of test channels that is divisible into the number of terminals of the DUT.

[0097] U.S. Pat. No. 5,578,932 (hereinafter the '932 patent), herein incorporated by reference in its entirety, discloses an embodiment of an N-port calibration device. FIG. 6 of this disclosure illustrates one embodiment of such an N-port calibration device. It is to be appreciated that according to one aspect of some embodiments of the method and MTS apparatus of this disclosure, an N-port calibration device can simply be coupled by a one-time connection of the N-port calibration device to an N-port MTS, such as is illustrated in FIG. 2, and the N-port calibration device can be used to completely characterize the N-port MTS. It is also to be appreciated that according to one aspect of some embodiments of the method and MTS apparatus of this disclosure, it is not necessary to have any pattern of connection between the MTS and the N-port calibration device. In other words, any port of the N-port MTS can be connected to any port of the N-port calibration device. In addition, according to another aspect of some embodiments of the method and MTS apparatus of this disclosure, a metrology laboratory can characterize the N-port calibration device into a collection of one and 2-port transfer standards to accurately characterize the N-port calibration device prior to its being used to calibrate an N-port MTS.

[0098] Referring to FIG. 6, there is illustrated an N-port automatic calibration device as disclosed in the '932 patent. The 1-port error coefficients for each port of the N-port MTS, can be determined by presenting three reflection standards with the N-port calibration device to each port of the N-port MTS. The three reflection standard may be, for example, a short, an open and a load reflection standard and may be presented with single-pole, four-throw switches 301, 302, 303, 304 . . . 305 and 306 that comprises the N-port calibration device. The automatic calibration device further comprises a single-pole, multi-throw switch 310 comprising a single pole 307 and N1 throws 317, 318, 319, 320 . . . 321. An additional throw (a thru connection) 311, 312, 313, 314, 315 . . . 316 of each switch 301 . . . 306 can be coupled to any of the single pole 307 and N1 throws 317-321 of the single-pole, multi-pole, multi-throw switch 310. For example, as illustrated in FIG. 6, the throw 31 lof single-pole, four throw switch 301 can be coupled to the pole 307 of single-pole, N1 throw switch 310, and the remainder of the throws 312-316 can be couple to respective throws 317, 318, 319, 320 . . . 321 of single-pole, N1 throw switch 310. It is to be understood that according to one aspect of some embodiments of the automatic calibration device of this disclosure, the throws of the various single-pole, four-throw switches need not be coupled to any particular pole of throw of the single-pole, N1 throw switch 310. In other words, any through path throw of any of the single-pole, four-throw switches 311-316 can be coupled to the single pole 307 or any of the throws 317-321 of the single-pole, N1 throw switch.

[0099] As disclosed in the '932 patent, the 1-port error coefficients at each port comprise a directivity coefficient E_DI, a source match coefficient E_SI and a reflection tracking coefficient E_RI, where I1, 2 . . . N and refers to the port number of the MTS. It is to be appreciated that according to one aspect of some embodiments of the MTS and automatic calibration device of this disclosure, the three reflection standards provided by the automatic calibration device need not be a short, an open and a load. In contrast, the three reflection coefficients can be any reflection value so long as the three reflection coefficients are well distributed throughout the complex reflection plane.

[0100] For example, referring to FIG. 7, there is illustrated one embodiment of a single-pole, double-throw switching device 53 can be provided in one embodiment of an N-port automatic calibration device and method of this disclosure, to provide three reflection coefficients as well as a through condition. In other words, FIG. 7 illustrates an embodiment of a single-pole, double-throw switch 53 that can replace any of the single-pole, four-throw switches 301-306 of FIG. 6. The single-pole, double-throw switch 53 comprises a common arm 54 and two output arms 55, 56. The single-pole, double-throw switch further comprises a plurality of FET devices 59, 60, 61, 62, 63, and 64. The single-pole, double-throw switch of FIG. 7 is normally operated in one of two conditions. In the first condition, a control voltage of 10V can be applied to control terminal 57 and a control voltage of 10V can be applied to control arm 58. In this configuration, FETs 62, 64, and 60 are biased to an off state and FETs 61, 63, and 59 are biased to an on state. In this arrangement, the common arm 54 is coupled to the second output arm 56 as illustrated in FIG. 8(a). In one embodiment of the single-pole, double-throw switch 53, such that a termination impedance can be presented by the switch to the common arm 54 of the switch. The N-port calibration device operating in this condition provides one of the three reflection coefficients to be presented by the N-port automatic calibration device.

[0101] Referring again to FIG. 7, in a second condition of operation of the switch 53, a control voltage of 10V can be applied to the first control terminal 57 of the switch 53 and a control voltage of 10V can be applied to the second control terminal 58 of the switch 53. With this arrangements, FETs 60, 64, and 62 are biased-on and FETs 59, 63, and 61 are biased-off. With this arrangement, the common arm 54 of the switch 53 is couple to the first output arm 55 of the switch as is illustrated in FIG. 8(b). According to one embodiment of an N-port calibration device comprising the single-pole, double-throw switch 53, the output arm 55 can be coupled to any of the pole 307 or any of the throws 317-321 of the single-pole, N1 throw switch 310 of the N-port calibration device of FIG. 6. Accordingly, an N-port calibration device of FIG. 6 comprising a plurality of switches 53 operating in the second condition can provide a through connection between ay two ports of the N-port calibration device.

[0102] Another aspect of an N-port calibration device comprising the single-pole, double-throw switch 53 of FIG. 7, is that the single-pole, double-throw switch 53 can be operated in a third condition. In the third condition, a bias voltage of 10V can be applied to both of control terminals 57 and 58. With this arrangement, all of FETs 59, 60, 61, 62, 63 and 64 are biased to an on condition. Accordingly, with this arrangement the common arm 54 is coupled to both the first output arm 55 and to the second output arm 56 as is illustrated in FIG. 8(c). In other words, short circuit is provided between the first output arm and the second output arm of the single-pole, double-throw switch of FIG. 7. With this arrangement, a short circuit can be provided to the common arm 54 of the switch 53. Accordingly, an N-port calibration comprising switches 53 operating in a third condition can provide a second reflection coefficient of the three reflection to be presented by the N-port automatic calibration device.

[0103] Another aspect of an N-port calibration device comprising the single-pole, double-throw switch 53 of FIG. 7, is that the single-pole, double-throw switch 53 can also be operated in a fourth condition, wherein a control voltage of 10V can be applied to both the first terminal 57 and the second terminal 58 of the switch. With this arrangement, each of FETs 59-64 will be biased to an open state so that the common arm 54 is isolated from each of the first output arm 55 and the second output arm 56, as is illustrated in FIG. 8(d). In other words, an open circuit is provided between the first output arm 55 and the second output arm 56. Accordingly, an N-port calibration device comprising switches 53 and operating in the fourth condition can provide a third reflection coefficient of the three reflection coefficients to be presented by the N-port calibration device.

[0104] It is to be appreciated that one advantage of an N-port calibration device comprising a plurality of the single-pole, double-throw switches of 53 of FIG. 7 that replace each of the single-pole, four0throw switches 301-306 of FIG. 6, is that three reflection standards provided by the N-port calibration device to each port of the N-port calibration device, can be well distributed throughout the complex impedance plane. FIG. 9 illustrates a difference in phase presented by the single-pole, double-throw switches 53 at the RF common input arm 54 when operating in the third and fourth conditions or, in other words, between providing the short and open conditions described above. It can be seen from FIG. 9, that a phase difference of substantially 180 exists over the frequency range 30 kHz to 6 GHz. Accordingly, with this arrangement, there exist a well distributed phase difference between the short condition and the open condition provided by the single-pole, double-throw switch 53 and by this embodiment of an N-port calibration device of this disclosure. In addition, the third reflection standard is the matched termination provided when the switch is operated in condition one described above, which will lie on a Smith chart mid way between the open and short impedances. Therefore, three reflection standards well distributed throughout the complex impedance plane can be provided by the switch 53 of FIG. 7 incorporated into the N-port calibration device of FIG. 6 as described above.

[0105] In addition, it is to be appreciated that an improved reflection loss and an improved amount of loss in the through state can be provided by the single-pole, double-throw switch 53 of FIG. 7. For example, the single-pole, double-throw switch of FIG. 7 will have a lower loss in the through condition (condition 2) as described above than a single-pole, four-throw switch such as illustrated in FIG. 6. For example, FIG. 10 illustrates a reflection loss of the single-pole, double-throw switch operating in the open fourth condition. In contrast, the reflection loss of a single-pole for throw switch typically has increased loss, for example, because additional FET devices connected to the common arm than the single-pole, double throw switch of FIG. 7.

[0106] According to another aspect of one embodiment of the single-pole, double-throw switch 53 of FIG. 7, it is to be appreciated that it has been observed that when operating the switch in the fourth condition by applying 10V to both control terminals 57 and 58, a 2V operating voltage appears at the common arm 54 of the switch. Accordingly, according to one embodiment of the switch 53 and an N-port calibration device of this disclosure, a blocking capacitor 66 can be provided at the common input port 54 so as to protect any circuitry that the N-port calibration device may be coupled to. In addition, it is to be appreciated that according to another aspect of the single-pole, double-throw switch 53 of FIG. 7, it has been observed that when operating the switch in the fourth condition or, in other words, the open circuit condition, circuit performance repeatability has been observed to be improved by providing a resistor 65 having a value of 47K between the common input arm 54 and ground. According to this embodiment of this disclosure, the switch53 can be provided with the resistor 65 so as to improve repeatability of the switch with little effect on transmission losses provided by the single-pole, double throw switch.

[0107] As disclosed in the '932 patent, a load match of each port of the N-port MTS can be determined from one of the S-parameters of the N(N1)/2 through paths of the N-ports of the calibration device, and the three 1-port error coefficients directivity E_DI, source match E_SI, and reflection tracking E_RI calculated at each port of the MTS. In addition, the '932 patent discloses that an isolation error coefficient for a twelve-term error correction model can also be determined. However, it is to be appreciated that according to some embodiments of an N-port calibration device of this disclosure, the isolation coefficient error term is omitted. While the isolation coefficient has been useful, for example, with VNAs which have limited switch isolation, with newer VNAs that have excellent switch isolation, this isolation error coefficient can no longer be used and can actually introduce more errors in determining an isolation error coefficient. In addition, it is to be appreciate that according to some embodiments of an N-port calibration device of this disclosure, only N1 through measurements of the N-port calibration device can be made to determine the load match error coefficients for each port of the N-port MTS.

[0108] It is to be appreciated that in order to determine systematic error coefficients of an N-port MTS, according to one aspect of some embodiments of an MTS of this disclosure the N-ports of the MTS can be considered as combination of N(N-1)/2 two port paths. For example, for the four-port MTS illustrated in FIG. 3, the MTS has six 2-port signal paths between ports: 1-2, 1-3, 1-4, 2-3, 2-4, and 3-4. As known to those skilled in the art and as discussed, for example, in U.S. Pat. No. 5,578,932 herein incorporated by reference in its entirety, the standard 2-port S-parameters can be obtained for each of the six two port signal paths and a standard twelve-term error correction can be applied to any measurements made with the MTS to remove test system errors. However, it is to be appreciated that although this twelve-term error correction corrects for any mismatches between the respective two terminals of the DUT and corresponding two ports of the MTS, it does not correct for any mismatches between the same two terminals of the DUT and the remaining ports of the MTS. Thus, one aspect of some embodiments of the MTS of this disclosure is that the reflection coefficients of each MTS port can be obtained during calibration of the MTS, and can be used in conjunction with the reflection coefficients of the DUT to correct for all mismatches between the two terminals of the DUT and any port of the MTS. It is also to be appreciated that in the measurement of the N(N1)/2 2-port paths, there are several redundant systematic error coefficients that can be determined. According to another aspect of some embodiments of the MTS of this disclosure, these redundant systematic error coefficients can be eliminated. In addition, according to still another aspect of some embodiments of the MTS of this disclosure, some of the systematic error coefficients can be determined according to an algorithmic proves of this disclosure instead of measuring the systematic error coefficients.

[0109] The systematic error coefficients of a 2-port VNA have been described, for example, with respect to FIG. 7 of U.S. Pat. No. 5,434,511 (hereinafter the '511 patent), which is herein incorporated by reference in its entirety. In addition, the '511 patent discloses how the systematic error coefficients can be determined for a 2-port VNA with the aid of an automatic calibration device. It is to be appreciated that in accordance with some embodiments of the method and apparatus of this disclosure, an automatic calibration device can be used to calibrate an N-port MTS, instead of the mechanical standards typically used in the art (as discussed in the 511 patent).

[0110] It is also to be appreciated that according to one aspect of some embodiments of the method and apparatus of this disclosure, an automatic calibration device is particularly suited to embodiments of an MTS of this disclosure, where an increase in the in the number of ports from 2-ports to N-ports makes it impractical to carry out a manual calibration using mechanical standards. In addition, additional benefits of using an automatic calibration device with embodiments of an MTS of this disclosure comprise: that all of the systematic error coefficients of the MTS can be determined by connecting an automatic calibration device once to the MTS; a calibration accuracy of the MTS can be increased due to a reduced amount of connections and disconnections of calibration device to the MTS; a calibration time is lessened since the calibration is automatic and needs little operator involvement; there is less damage or wear to interface hardware of the MTS and to the automatic calibration device, such as cables and connectors, due to fewer connections that may be made; and the calibration process is less prone to operator error and does not require a high skill set.

[0111] According to another aspect of some embodiments of the method and apparatus of this disclosure, it is to be appreciated that the transmission tracking error coefficients (some of the overall systematic error coefficients) can be determined by N1 through measurements, instead of the N(N1)/2 through measurements that have been done in the prior art, as was discussed above. For example, for a four-terminal DUT, only three through measurements may be made with certain conditions between any of the six possible 2-terminal signal paths, and all six 2-terminal signal paths 1-2, 1-3, 1-4, 2-3, 2-4, and 3-4 need not be measured.

[0112] One condition for measuring all of the transmission tracking error coefficients of an N-port DUT with N1 through measurements is the selection fN1 2-terminal paths of the DUT that includes all of the terminals. For example, for the four-terminal DUT, 2-terminal through paths that include all four terminals such as, 1-2, 1-3, and 1-4 can be selected. In contrast, 2-terminal paths 2-3, 2-4 and 3-4 may not be selected if all of the transmission tracking error coefficients are to be measured by N1 through measurements, because terminal 2 of the DUT has been omitted from the selected two terminal paths.

[0113] Another possible condition for measuring all of the transmission tracking error coefficients of an N-port DUT with N1 through measurements (for DUTs with more than four terminals) is that a bridging 2-terminal path is selected. For example, referring to FIG. 14 there is illustrated an eight terminal DUT. According to this aspect of this disclosure, an eight terminal DUT can be characterized with seven 2-terminal path measurements. It is to be appreciated that the eight terminal DUT can be considered two four-terminal devices, with a first device comprising terminals 1, 2, 5 and 6 and a second device comprising terminals 3, 4, 7 and 8 as illustrated in FIG. 14. According to the first condition discussed above, the 2-terminal paths 1-2, 1-5, and 1-6 can be selected for the first device and the 2-terminal paths 3-4, 3-7, and 3-8 can be selected for the second device, to provide six of the seven 2-terminal paths. One additional bridging 2-terminal path between the first device and the second device can also be selected. For example, a briding 2-terminal path can be any one of two terminal paths 1-3, 1-4, 1-7, 1-8, 2-3, 2-4, 2-7, 2-8, 3-5, 3-6, 4-5, 4-6, 5-7, 5-8, 6-7, and 6-8 between the first device and the second device of FIG. 14.

[0114] Thus, one advantage of some embodiments of the MTS and automatic calibration device of this disclosure is that a full calibration of the MTS can be accomplished with fewer through measurements than has previously been known. This advantage can be especially important as the number of ports of the MTS system increases. For example, it is to be appreciated that a full through calibration of an eight-port MTS with twenty-eight possible 2-port signal through paths can be done according to some embodiments of this disclosure, with only seven through measurements of the twenty-eight two terminal through paths. Similarly, for a 16-port MTS, a full through calibration can be done with only fifteen through measurements of a possible one hundred and twenty 2-terminal through paths.

[0115] Accordingly, it is to be appreciated that according to one aspect of some embodiments of this disclosure, the reflection tracking term error coefficients for the N(N1)/2 2-port paths can be calculated from N1 through measurements. For example, a four-port calibration device has a total of six 2-port paths. However, only three through paths can be measured instead of the six, and all of the calibration coefficients of the MTS can still be determined. For a four-port MTS as illustrated in FIG. 3, there are six 2-port paths which are 1-2, 1-3, 1-4, 2-3, 2-4, and 3-4. However, it is to be appreciated that according to some embodiments of the MTS and automatic calibration device of this disclosure, three of the six through paths, such as for example, through paths 1-2, 1-3, and 1-4 can be measured. Referring to equations (2)-(7), a complex variable K_ij between two ports of the six possible through paths of the MTS can be defined as: $\begin{array}{cc}{K}_{12}=\frac{E_{\mathrm{R1}}}{E_{\mathrm{T12}}}& \left(2\right)\\ K_{13}=\frac{E_{\mathrm{R1}}}{E_{\mathrm{T13}}}& \left(3\right)\\ K_{14}=\frac{E_{\mathrm{R1}}}{E_{\mathrm{T14}}}& \left(4\right)\\ K_{23}=\frac{E_{\mathrm{R2}}}{E_{\mathrm{T23}}}& \left(5\right)\\ K_{24}=\frac{E_{\mathrm{R2}}}{E_{\mathrm{T24}}}& \left(6\right)\\ K_{34}=\frac{E_{\mathrm{R3}}}{E_{\mathrm{T34}}}& \left(7\right)\end{array}$

[0116] In equations (2)-(7), the term E_TPQ is the forward transmission tracking coefficients between ports P-Q and the term E_TQP is the reverse transmission tracking coefficient between ports P-Q. In addition, the term E_RI, where I refers to a port number of the MTS, is the reflection tracking coefficient for the corresponding port I1, 2, 3, 4. It is to be appreciated that all of the parameters on the right side of the equations (2)-(4) can be known where the through paths 1-2, 1-3, and 1-4 have been measured with the four-port calibration device coupled to the four-port MTS. It is also to be appreciated that according to some embodiments of the method and apparatus of this disclosure, three of the six through paths for a four-port MTS can be measured to fully calibrate the MTS. Thus, it is to be appreciated that one advantage of some embodiments of the method and apparatus of this disclosure is that for a four-port MT, three through calibrations can be used to determine all six forward and reverse tracking coefficients for the four-port MTS.

[0117] Referring to equation (8), the complex variable K_ij between ports 2 and 3 (K₂₃) can be calculated from equations (2) and (3). $\begin{array}{cc}{K}_{23}=\frac{K_{13}}{K_{12}}=\frac{E_{\mathrm{T12}}}{E_{\mathrm{T13}}}& \left(8\right)\end{array}$

[0118] Referring now to equation (9), from equations (5) and (8), the forward transmission tracking coefficient between ports 2 and 3, ET₂₃ is given by: $\begin{array}{cc}{E}_{\mathrm{T23}}=\frac{E_{\mathrm{R2}}{E}_{\mathrm{T13}}}{E_{\mathrm{T12}}}& \left(9\right)\end{array}$

[0119] Referring to equation (10), the reverse transmission tracking coefficient between ports 2 and 3 is given by: $\begin{array}{cc}{E}_{\mathrm{T32}}=\frac{E_{\mathrm{R2}}{E}_{\mathrm{R3}}}{E_{\mathrm{T23}}}& \left(10\right)\end{array}$

[0120] Referring to equation (11), the complex variable K_ij between ports 2 and 4 (K₂₄) can be calculated from equations (2) and (4). $\begin{array}{cc}{K}_{24}=\frac{K_{14}}{K_{12}}=\frac{E_{\mathrm{T12}}}{E_{\mathrm{T14}}}& \left(11\right)\end{array}$

[0121] Referring to equation (1), from equations (6) and (11), the forward transmission tracking coefficients between ports 2 and 4 (ET₂₄) is given by: $\begin{array}{cc}{E}_{\mathrm{T24}}=\frac{E_{\mathrm{R2}}{E}_{\mathrm{T14}}}{E_{\mathrm{T12}}}& \left(12\right)\end{array}$

[0122] Referring to equation (13), the reverse transmission tracking coefficient between ports 2-4 is given by: $\begin{array}{cc}{E}_{\mathrm{T42}}=\frac{E_{\mathrm{R2}}{E}_{\mathrm{R4}}}{E_{\mathrm{T24}}}& \left(13\right)\end{array}$

[0123] Referring to equation (14), from equations (5), (6) and (7), the forward transmission tracking coefficient ET₃₄ is given by: $\begin{array}{cc}{E}_{\mathrm{T34}}=\frac{E_{\mathrm{R3}}{E}_{\mathrm{T23}}}{E_{\mathrm{T24}}}& \left(14\right)\end{array}$

[0124] Referring to equation (15), the reverse transmission tracking coefficient between ports 3-4 is given by: $\begin{array}{cc}{E}_{\mathrm{T43}}=\frac{E_{\mathrm{R3}}{E}_{\mathrm{R4}}}{E_{\mathrm{T34}}}& \left(15\right)\end{array}$

[0125] Thus, all of the forward and reverse transmission tracking terms for the 2-port paths 2-3, 2-4, and 3-4 can be calculated from the 1-port reflection tracking coefficients at each port of the MTS and from the forward and reverse transmission tracking coefficients of the three 2-port paths 1-2, 1-3, and 1-4.

[0126] It is to be appreciated that according to some embodiments of the method and apparatus of this disclosure, the number of through measurements is minimized to N1 through measurements in order to fully characterize the N-port MTS. For example, for an eight-port MTS which has a total of twenty-eight 2-port through paths, there can be seven through measurements, and three reflection standard measurements at each of the ports to obtain all error correction coefficients for the eight-port MTS including all twenty-eight forward and reverse transmission tracking coefficients. Similarly, a 16-port MTS has a total of one hundred and twenty possible 2-port paths. However, according to some embodiments of the method and apparatus of this disclosure, only fifteen through calibration path measurements as well as three reflection standard measurements at each port, can be measured to obtain all of the error correction coefficients including one hundred and twenty forward and reverse transmission-tracking coefficients. It is further to be appreciated that according to some embodiments of the method and apparatus of this disclosure, N1 of the possible N(N1)/2 2-port through paths can be measured. Therefore, it is an advantage of some of the embodiments of the method and apparatus of this disclosure that the complexity to calibrate an N-port MTS is greatly reduced.

[0127] According to some embodiments of the method and apparatus of this disclosure, a method of calibrating an N-port MTS comprises connecting each port of an N-port calibration device to a port of the N-ports MTS, and presenting a plurality of calibration standards to the MTS with the automatic calibration device. For example, a plurality of reflection coefficients comprising three reflection standards are presented by the N-port calibration device to each port of the MTS. From these measurements, a directivity E_DI, a source math E_SI and a reflection tracking coefficient E_RI can be determined for each port of the MTS. In addition, a minimum of N1 2-port through conditions are presented by the N-port calibration device to the N-port MTS and are measured by the MTS. From these measurements, the load match coefficient for each port of the N-port of the MTS can be determined. For each of a possible the N(N1)/2 possible 2-port paths of the N-port MTS, twelve-term systematic error coefficients are calculated. Further, N1 forward and reverse transmission tracking coefficients can be determined. From the reflection tracking coefficients of each port and the N1 forward and reverse transmission tracking coefficients, the forward and reverse transmission tracking coefficients for the remainder of the N(N1)/2 2-port paths can be determined. For each possible N((N1)/2 2-port paths of the N-port MTS, twelve-term systematic error coefficients are calculated.

[0128] After the N-port MTS has been calibrated with the N-port calibration device as a discussed above, the N-port calibration device can be removed from its connection to the MTS, and a DUT to be measure can be couple to the MTS.

[0129] As known to those of skill in the art, for an N-terminal DUT a minimum of N² measurements can be performed to characterize the N-terminal DUT. For each of a possible N(N1)/2 2-terminal paths of the N-terminal DUT, the corresponding twelve-term systematic error coefficients of the MTS can be used to correct the measurement of each 2-terminal DUT path. As known to those of skill in the art, the N-terminal DUT will have a total of N(N1)/2 2-terminal S-parameter subsets. Each 2-terminal S-parameter subset has corresponding four S-parameter elements as illustrated in FIG. 1(a), which are a subset of an N-by-N S-parameter matrix of the N-terminal DUT. In other words, the N by N S-parameter matrix that characterized the N-terminal DUT can be broken down into N(N1)/2 subset 2-terminal S-parameter matrices. For example, a 4-terminal DUT has a sixteen-element S-parameter matrix that can also be represented by six 2-terminal matrices that total twenty-four measured S-parameters. It is to be appreciated that the six 2-terminal matrices have more S-parameters than the sixteen S-parameters that make up the 4-terminal S-parameter matrix, because there can be redundant measurements made of the reflection elements. For example, for the 4-terminal DUT, S₁₁, S₂₂, S₃₃, and S₄₄ can be measured two additional times in the six 2-terminal subset matrices.

[0130] However, it is to be appreciated that according to some embodiments of the method and apparatus of this disclosure, during N(N1)/2 2-terminal measurements of an N-terminal DUT, all redundant reflection measurements need not be made. Thus, the measured S-parameters of the N-terminal DUT is equal to N².

[0131] Nevertheless, it is to be appreciated that it is not always possible to provide a reflection less termination to each terminal of an N-terminal DUT, during the N(N1)/2 2-terminal measurements (without the redundant measurements). Therefore the 2-terminal measured S-parameters for the DUT may be different from the actual N-by-N S-parameter matrix. For example, the measured S-parameters can be corrupted by arbitrary terminations presented by the N-port MTS. However, as will be discussed in detail infra, it is to be appreciated that according to some embodiments of the method and apparatus of this disclosure, since the reflection coefficient of each port of the MTS is known, the measured S-parameters of the DUT can be normalized by their corresponding known reflection coefficients.

[0132] Referring to equation (16), for each set of N((N1)/2 2-port S-parameters which are connected by their corresponding 12-term error correction algorithm, the normalized S-parameters (hereinafter S_nj) terminated in an arbitrary reflection coefficient _S and _L is given by:

[0133] where $=\left[\begin{array}{cc}{}_S& 0\\ 0& {}_L\end{array}\right],$

[0134] * is complex conjugate of and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

[0135] is the 2 by 2 identity matrix

[0136] The N(N1)/2 normalized S-parameter matrices can be grouped into the N-by-N S-parameter matrix. As discussed supra, according to one embodiment of the method and apparatus of this disclosure, the redundant reflection coefficients can be measured only once. For example, from (16) the six normalized matrices of a 4-terminal DUT between the six 2-terminal paths 1-2, 1-3, 1-4, 2-3, 2-4 and 3-4 are given by (17): $\begin{array}{cc}\left[\begin{array}{cc}{S}_{11}& S_{12}\\ S_{12}& S_{22}\end{array}\right],\left[\begin{array}{cc}{S}_{11}& S_{13}\\ S_{31}& S_{33}\end{array}\right],\left[\begin{array}{cc}{S}_{11}& S_{14}\\ S_{41}& S_{44}\end{array}\right],\left[\begin{array}{cc}{S}_{22}& S_{23}\\ S_{32}& S_{33}\end{array}\right],\left[\begin{array}{cc}{S}_{22}& S_{24}\\ S_{42}& S_{44}\end{array}\right]\mathrm{and}\left[\begin{array}{cc}{S}_{33}& S_{34}\\ S_{43}& S_{44}\end{array}\right]& \left(17\right)\end{array}$

[0137] Also, the load reflection coefficients presented by the MTS for the six 2-terminal through paths are given by (18): $\begin{array}{cc}\left[\begin{array}{cc}{}_1& 0\\ 0& {}_2\end{array}\right],\left[\begin{array}{cc}{}_1& 0\\ 0& {}_3\end{array}\right],\left[\begin{array}{cc}{}_1& 0\\ 0& {}_4\end{array}\right],\left[\begin{array}{cc}{}_2& 0\\ 0& {}_3\end{array}\right],\left[\begin{array}{cc}{}_2& 0\\ 0& {}_4\end{array}\right]\mathrm{and}\left[\begin{array}{cc}{}_3& 0\\ 0& {}_4\end{array}\right]& \left(18\right)\end{array}$

[0138] The six 2-terminal matrices of normalized S-parameters can be grouped into one normalized four 4 by 4 S-parameter matrix S_n in (19): $\begin{array}{cc}Sn=\left[\begin{array}{cccc}{S}_{11}& S_{12}& S_{13}& S_{14}\\ S_{21}& S_{22}& S_{23}& S_{24}\\ S_{31}& S_{32}& S_{33}& S_{34}\\ S_{41}& S_{42}& S_{43}& S_{44}\end{array}\right]& \left(19\right)\end{array}$

[0139] Also, the six 2-terminal matrices of load reflection coefficients can be grouped into one 4 by 4 load reflection matrix in (20): $\begin{array}{cc}=\left[\begin{array}{cccc}{}_1& 0& 0& 0\\ 0& {}_2& 0& 0\\ 0& 0& {}_3& 0\\ 0& 0& 0& {}_4\end{array}\right]& \left(20\right)\end{array}$

[0140] The standard of an N-terminal DUT normalized to a given reference impedance such as 50 can be determined from the normalized S, matrix, the load reflection coefficient matrix F and identity matrix 1. Referring to equation (21), the standard S-parameters S of an N-terminal device can be given by (21):

[0141] where $\begin{array}{c}Sn=\left[\begin{array}{cccccc}{S}_{11}& S_{12}& & & & S_{1j}\\ S_{21}& S_{22}& & & & S_{2j}\\ \begin{array}{c}\\ \\ \end{array}& \begin{array}{c}\begin{array}{c}\\ \end{array}\\ \end{array}& \begin{array}{c}\begin{array}{c}\\ \end{array}\\ \end{array}& \begin{array}{c}\begin{array}{c}\\ \end{array}\\ \end{array}& \begin{array}{c}\begin{array}{c}\\ \end{array}\\ \end{array}& \begin{array}{c}\begin{array}{c}\\ \end{array}\\ \end{array}\\ S_{\mathrm{i1}}& S_{\mathrm{i2}}& & & & S_{ij}\end{array}\right]i=1,2,,N\mathrm{and}j=1,2,,N.\\ =\left[\begin{array}{cccccc}{}_1& 0& & & & 0\\ 0& {}_2& & & & 0\\ \begin{array}{c}\\ \\ \end{array}& \begin{array}{c}\\ \\ \end{array}& \begin{array}{c}\begin{array}{c}\\ \end{array}\\ \end{array}& \begin{array}{c}\begin{array}{c}\\ \end{array}\\ \end{array}& \begin{array}{c}\begin{array}{c}\\ \end{array}\\ \end{array}& \begin{array}{c}\begin{array}{c}\\ \end{array}\\ \end{array}\\ 0& 0& & & & {}_{ij}\end{array}\right]={}_i\mathrm{for}i=j\mathrm{and}=0\mathrm{for}ij\\ I=\left[\begin{array}{cccccc}1& 0& & & & 0\\ 0& 1& & & & 0\\ & & & & & \\ & & & & & \\ & & & & & \\ 0& 0& & & & 1\end{array}\right]I=1\mathrm{for}i=j\mathrm{and}I=0\mathrm{for}ij\end{array}$

[0142] It is to be appreciated that there are presently no traceable calibration artifacts from any standards laboratory for greater than a 2-terminl device. Thus, according to one aspect of some embodiments of the MTS of this disclosure, an N-terminal DUT can be broken down into N(N1)/2 2-terminal measurements to comply with available calibration artifacts. It is also to be appreciated that according to another aspect of some embodiments of the MTS of this disclosure, redundant measurements of the DUT can be eliminated. It is further to be appreciated that according to another aspect of some embodiments of the MTS of this disclosure, measurements of the DUT with the MTS can be modified to correct for imperfect terminations presented to terminals of the DUT by he MTS.

[0143] According to some embodiments of the method and MTS apparatus of this disclosure, characterization of a multi-terminal device can be provided for balanced circuits or mixed-mode multi-terminal devices. As was discussed supra, some of the benefits of using differential circuits in the high-speed digital communications environment in comparison to single-ended devices, are immunity to electromagnetic interference, suppression of even-order harmonics and higher tolerance to non-ideal grounds. With today's digital communication systems, data transfer rates are increasing drastically. In addition, digital data streams for video, HDTV, graphics, serial interface and many other applications are resulting in higher and high bandwidth. These digital boards and systems. It is to be appreciated that although the data transmitted is digital, the transmission media that the digital data travels along may be analog with a differential circuit topology. Thus, differential circuit topology is extensively used in high-speed digital data communication applications.

[0144] According to some embodiments of the method and apparatus of this disclosure, the measurement and analysis of differential circuits can be achieved by transformation of standard single-ended S-parameters into mixed-mode S-parameters S_mm. As was discussed supra and Bockelman IEEE Article, mixed-mode S-parameters of a linear balanced devices corresponding to waves propagating in several coupled modes can be determined from a standard S-parameters matrix, and comprise:

[0145] S_dd a pure differential-mode;

[0146] S_cc a pure common-mode;

[0147] S_dc conversion from the common-mode to the differential-mode; and

[0148] S_cd conversion from the differential-mode to the common-mode.

[0149] However, it is to be appreciated that with today's communication systems, there may be single-ended devices that may interface with a balanced device. Also, there may be DUTs where both single-ended and balanced topologies are integrated together as one DUT or system. Thus, according to some embodiments of the method and apparatus of this disclosure, there is provided a general solution to determine the mixed-mode S-parameter matrix S_mm for any multiple single-ended and multiple balanced topologies for any N-terminal DUT. Additional elements of the mixed-mode S-parameters matrix of this disclosure include purely single ended parameters as well as parameters that represent a conversion of single-ended to balanced and balanced to single-ended modes. The mixed mode S-parameters matrix S_mm can be determined from a linear combination of the standard multi-port S-parameters matrix S.

[0150] According to some embodiments of the method and apparatus of this disclosure, there is provided mixed-mode S-parameters matrix S_mm that can represent a mixture of single-ended and balanced parameters. These parameters comprise:

[0151] S_ss which is a pure single-ended parameter;

[0152] S_ds which is a conversion of a single-ended parameter to a differential parameter;

[0153] S_sd which is a conversion of a differential parameter to a single-ended parameter;

[0154] S_cs which is a conversion of a single-ended parameter to a common-mode parameter;

[0155] and

[0156] S_sc which is a conversion of a common-mode parameter to a single-ended parameter.

[0157] According to some embodiments of the method and apparatus of this disclosure, the standard S-parameter matrix Scan be transformed into the mixed-mode S-parameter matrix S_mm through a similarity transformation with use of a scalar orthogonal matrix M. The S and the S_mm have the same eigenvalues and different eigenvectors.

[0158] Referring to equation (22), it is to be appreciated that according to some embodiments of the method and apparatus of this disclosure, the single-ended matrix S can be mapped into the mixed-mode S-parameters matrix S_mm by Equation (22).

[0159] Where

[0160] S_mm is the mixed-mode S-parameter matrix, S is the standard unbalanced S-parameter matrix characterized by some embodiments of the MTS of this disclosure and M is the scalar orthogonal matrix. It is to be appreciated that none of the elements of the M are complex. It is also to be appreciated that according to some embodiments of the method and apparatus of this disclosure, the columns of the scalar orthogonal matrix M describe the physical terminals of the DUT and the rows of the M describe scalar coefficients corresponding to a specific mode of operation of the DUT. Further, a remainder of the elements of the scalar orthogonal matrix M can be set to 0.

[0161] According to some embodiments of the method and apparatus of this disclosure, a method of assigning the scalar coefficients to the scalar orthogonal matrix M corresponding to specific modes of the DUT, comprises: assigning a value of 1 for each column of the M and corresponding row for each single-ended input and output terminal of the DUT; assigning a value of 1/{square root}{square root over (2)} and 1/{square root}{square root over (2)} for each column of the M and a corresponding row for each terminal of a balanced input and output pair of the DUT which can be operated in a differential mode; and assigning a value of 1/{square root}{square root over (2)} and 1/{square root}{square root over (2)} for each column of the M and a corresponding row for each terminal of a balanced input and output pair of the DUT which can be operated in a common-mode.

[0162] Referring to FIG. 11, there is illustrated a schematic representation of a combined single-ended transmission line 50 and a balanced transmission line 52. The mixed-mode S-parameters of the combined single-ended transmission line and balanced transmission line can be determined by measuring a standard 66 S-parameter matrix for the six terminals of the DUT illustrated in FIG. 11, with MTS of this disclosure. A 66 S-parameter matrix S can be written as represented by (23): $\begin{array}{cc}S=\left[\begin{array}{cccccc}{S}_{11}& S_{12}& S_{13}& S_{14}& S_{15}& S_{16}\\ S_{21}& S_{22}& S_{23}& S_{24}& S_{25}& S_{26}\\ S_{31}& S_{32}& S_{33}& S_{34}& S_{35}& S_{36}\\ S_{41}& S_{42}& S_{43}& S_{44}& S_{45}& S_{46}\\ S_{51}& S_{52}& S_{53}& S_{54}& S_{55}& S_{56}\\ S_{61}& S_{62}& S_{63}& S_{64}& S_{65}& S_{66}\end{array}\right]& \left(23\right)\end{array}$

[0163] Referring to the above-described coefficients for the scalar orthogonal matrix M and to FIG. 11, there is now illustrated an example of assigning scalar coefficients of a scalar orthogonal matrix to the illustrated DUT. Terminal 1 of the DUT of FIG. 11 is a single-ended parameter. In one embodiment, a first row and a second row of the M can correspond to single-ended operation of the DUT. Therefore, the first row, first column of the M can be assigned a value of 1. In addition, a remainder of elements of the first row can be set to a value of zero. Since terminal 4 of the DUT of FIG. 11 is also a single-ended parameter, the second row, fourth column of the scalar orthogonal matrix M can also be set to a value of 1. The remainder of elements of the second row can also be set to zero.

[0164] In one embodiment, the third and fourth rows of the M can correspond to a differential operation of the DUT. Referring to FIG. 11, terminals 2 and 3 of the DUT can constitute a differential input pair for the DUT, and therefore a third row, second column of the scalar orthogonal matrix M can be set to 1/{square root}{square root over (2)} an the third row, third column of the scalar orthogonal matrix M can be set to 1/{square root}{square root over (2)}. In addition, a remainder of elements of the third row can be set to zero. Since terminals 5 and 6 of the DUT can constitute a differential output pair of the DUT, the fourth row, fifth column of the scalar orthogonal matrix M can be set to 1/{square root}{square root over (2)} and the fourth row, sixth column of the scalar orthogonal matrix M can be set to 1/{square root}{square root over (2)}. A remainder of the elements of the fourth row can also be set to zero.

[0165] In one embodiment, the fifth and sixth row of the M can correspond to common mode operation of the DUT. Referring again to FIG. 11, since terminals 2 and 3 of the DUT can also constitute a common input pair of the DUT, a fifth row, second column of the scalar orthogonal matrix M can be set to 1/{square root}{square root over (2)} and a fifth row, third column of the M can also be set to 1/{square root}{square root over (2)}. A remainder of the elements of the fifth row can be set to zero. Similarly, since terminals 5 and 6 of the DUT can also constitute a common output pair of the DUT, a sixth row, fifth column of the scalar orthogonal matrix M can be set to 1/{square root}{square root over (2)} and a sixth row, sixth column of the M can be set to 1/{square root}{square root over (2)}. The remainder of elements of the sixth row are also set to zero. With the above-illustrated exemplary embodiment for assigning coefficients to the scalar orthogonal matrix, the following scalar orthogonal matrix M can be provided by (24): $\begin{array}{cc}M=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]& \left(24\right)\end{array}$

[0166] It is to be appreciated that a transpose of the scalar orthogonal matrix M is equal to its inverse. Therefore, by taking the transpose of the M and from equation (22) above, the mixed-mode S-parameter matrix M of the DUT of FIG. 11 can be determined as (25): $\begin{array}{cc}\left[\begin{array}{cccccc}{S}_{11}& S_{14}& \frac{S_{12}-S_{13}}{\sqrt{2}}& \frac{S_{15}-S_{16}}{\sqrt{2}}& \frac{S_{12}+S_{13}}{\sqrt{2}}& \frac{S_{15}+S_{16}}{\sqrt{2}}\\ S_{41}& S_{44}& \frac{S_{42}-S_{43}}{\sqrt{2}}& \frac{S_{45}-S_{46}}{\sqrt{2}}& \frac{S_{42}+S_{43}}{\sqrt{2}}& \frac{S_{45}+S_{46}}{\sqrt{2}}\\ \frac{S_{21}-S_{31}}{\sqrt{2}}& \frac{S_{42}-S_{43}}{\sqrt{2}}& \frac{1}{2}\left(S_{22}-S_{23}-S_{32}+S_{33}\right)& \frac{1}{2}\left(S_{25}-S_{26}-S_{35}+S_{36}\right)& \frac{1}{2}\left(S_{22}+S_{23}-S_{32}-S_{33}\right)& \frac{1}{2}\left(S_{25}+S_{26}-S_{35}-S_{36}\right)\\ \frac{S_{51}-S_{61}}{\sqrt{2}}& \frac{S_{54}-S_{64}}{\sqrt{2}}& \frac{1}{2}\left(S_{52}-S_{53}-S_{62}+S_{63}\right)& \frac{1}{2}\left(S_{55}-S_{56}-S_{65}+S_{66}\right)& \frac{1}{2}\left(S_{52}+S_{53}-S_{62}-S_{63}\right)& \frac{1}{2}\left(S_{55}+S_{56}-S_{65}-S_{66}\right)\\ \frac{S_{21}+S_{31}}{\sqrt{2}}& \frac{S_{42}+S_{43}}{\sqrt{2}}& \frac{1}{2}\left(S_{22}-S_{23}+S_{32}-S_{33}\right)& \frac{1}{2}\left(S_{25}-S_{26}+S_{35}-S_{36}\right)& \frac{1}{2}\left(S_{22}+S_{23}+S_{32}+S_{33}\right)& \frac{1}{2}\left(S_{25}+S_{26}+S_{35}+S_{36}\right)\\ \frac{S_{51}+S_{61}}{\sqrt{2}}& \frac{S_{54}+S_{64}}{\sqrt{2}}& \frac{1}{2}\left(S_{52}-S_{53}+S_{62}-S_{63}\right)& \frac{1}{2}\left(S_{55}-S_{56}+S_{65}-S_{66}\right)& \frac{1}{2}\left(S_{52}+S_{53}+S_{62}+S_{63}\right)& \frac{1}{2}\left(S_{55}+S_{56}+S_{65}+S_{66}\right)\end{array}\right]& \left(25\right)\end{array}$

[0167] Referring again to FIG. 11, six single-ended ports MTS can be coupled to the six terminals of the DUT. The mixed-mode S-parameters for the DUT of FIG. 11 can be defined as single ended and balanced ports. For example, Port 1 and Port 2 are single ended ports and Port 3 and Port 4 are balanced ports. For this mixed mode port configuration of the DUT, the mixed-mode S-parameter matrix of the DUT can be summarized as (26): $\begin{array}{cc}S=\left[\begin{array}{cccccc}{S}_{\mathrm{ss11}}& S_{\mathrm{ss12}}& S_{\mathrm{sd13}}& S_{\mathrm{sd14}}& S_{\mathrm{sc13}}& S_{\mathrm{sc14}}\\ S_{\mathrm{ss21}}& S_{\mathrm{ss22}}& S_{\mathrm{sd23}}& S_{\mathrm{sd24}}& S_{\mathrm{sc23}}& S_{\mathrm{sc24}}\\ S_{\mathrm{ds31}}& S_{\mathrm{ds32}}& S_{\mathrm{dd33}}& S_{\mathrm{dd34}}& S_{\mathrm{dc33}}& S_{\mathrm{dc34}}\\ S_{\mathrm{ds41}}& S_{\mathrm{ds42}}& S_{\mathrm{dd43}}& S_{\mathrm{dd44}}& S_{\mathrm{dc43}}& S_{\mathrm{dc44}}\\ S_{\mathrm{cs31}}& S_{\mathrm{cs32}}& S_{\mathrm{cd33}}& S_{\mathrm{cd34}}& S_{\mathrm{cc33}}& S_{\mathrm{cc34}}\\ S_{\mathrm{cs41}}& S_{\mathrm{cs42}}& S_{\mathrm{cd43}}& S_{\mathrm{cd44}}& S_{\mathrm{cc43}}& S_{\mathrm{cc44}}\end{array}\right]& \left(26\right)\end{array}$

[0168] For the mixed mode S-parameter matrix (26) for the DUT of FIG. 11: S_ss11 is the single ended reflection coefficient of Port 1 of the mixed mode DUT; S_ss22 is the single ended reflection coefficient of Port 2 of the mixed mode DUT; S_ss21 is the single ended transmission coefficient from Port 1 to Port 2 of the mixed mode DUT; and S_ss12 is the single ended transmission coefficient from Port 2 and Port 1 of the mixed mode DUT.

[0169] In addition, S_sd13 is the conversion from a differential parameter to a single ended parameter between Port 3 and Port 1 of the mixed mode DUT; S_sd14 is the conversion from a differential parameter to a single ended parameter between Port 4 and Port 1 of the mixed mode DUT; S_sd23 is the conversion from a differential parameter to a single ended parameter between Port 3 and Port 2 of the mixed mode DUT; and S_sds24 is the conversion from a differential parameter to a single ended parameter between Port 4 and Port 2 of the mixed mode DUT.

[0170] Further, S_sc13 is the conversion from a common mode parameter to a single ended parameter between Port 3 and Port 1 of the mixed mode DUT; S_sc23 is the conversion from a common mode parameter to a single ended parameter between Port 3 and Port 2 of the mixed mode DUT; S_sc14 is the conversion from a common mode parameter to a single ended parameter between Port 4 and Port 1 of the mixed mode DUT; and S_sc24 is the conversion from a common mode parameter to a single ended parameter between Port 4 and Port 2 of the mixed mode DUT.

[0171] Still further, S_ds31 is the conversion from a single ended parameter to a differential a single ended parameter to a differential parameter between Port 2 and Port 3 of the mixed mode DUT; S_ds41 is the conversion from a single ended parameter to a differential parameter between Port 1 and Port 4 of the mixed mode DUT; and S_ds42 is the conversion from a single ended parameter to a differential parameter between Port 2 and Port 4 of the mixed mode DUT.

[0172] Also, S_cs31 is the conversion from a single ended parameter to a common mode parameter between Port 1 and Port 3 of the mixed mode DUT; S_cs32 is the conversion from a single ended parameter to a common mode parameter between Port 2 and Port 3 of the mixed mode DUT; S_cs41 is the conversion from a single ended parameter to a common mode parameter between Port 1 and Port 4 of the mixed mode DUT; and S_cs42 is the conversion from a single ended parameter to a common mode parameter between Port 2 and Port 4 of the mixed mode DUT.

[0173] It is to be appreciated that in the mixed mode S-parameter matrix (26) for the DUT of FIG. 11 a remainder of the parameters represent the parameters discussed in the Bockelman IEEE Article. Parameters S_dd33, S_dd34, S_dd43 and S_dd44 represent the behavior of the DUT when it is stimulated with a differential-mode signal and a differential-mode response is observed. There are four parameters including reflection parameters S_dd33, S_dd44 and transmission parameters S_dd43, S_dd34 for ports three and four of the DUT. In addition, parameters S_cc33, S_cc34, S_cc43 and S_cc44 represent the behavior of the DUT when it is stimulated with a common-mode signal and a common-mode response is observed. There are four parameters including reflection parameters S_cc33, S_cc44 and transmission parameters S_cc43, S_cc34 for ports three and four of the DUT.

[0174] Further, the mode conversion parameters S_cd33, S_cd34, S_cd43 and S_cd44 represent the behavior of the DUT when it is stimulated with a differential-mode signal and a common-mode response is observed. Still further, parameters S_dc33, S_dc34, S_dc43 and S_dc44 represent the behavior of the DUT when it is stimulated with a common-mode signal and a differential-mode response is observed.

[0175] There will now be described another embodiment of a method of determining the mixed mode S-parameters S_mm of a DUT. Referring to FIG. 12, there is illustrated a schematic of the DUT of FIG. 11 with the single-ended cable 50 coupled to ports 3, 6 of an embodiment of a 6-port MTS and with the differential cable 52 coupled to ports 1, 2, 4 and 5 of the 6-port MTS. According to the above-described embodiment, the mixed-mode S-parameter matrix S_mm of the DUT of FIG. 12 can be determined as (27): $\begin{array}{cc}\left[\begin{array}{cccccc}\frac{1}{2}\left(s_{11}-s_{12}-s_{21}+s_{22}\right)& \frac{1}{2}\left(s_{14}-s_{15}-s_{24}+s_{25}\right)& \frac{1}{2}\left(s_{11}-s_{12}-s_{21}+s_{22}\right)& \frac{1}{2}\left(s_{14}-s_{15}-s_{24}+s_{25}\right)& \frac{s_{13}-s_{23}}{\sqrt{2}}& \frac{s_{16}-s_{26}}{\sqrt{2}}\\ \frac{1}{2}\left(s_{41}-s_{42}-s_{51}+s_{52}\right)& \frac{1}{2}\left(s_{44}-s_{45}-s_{54}+s_{55}\right)& \frac{1}{2}\left(s_{41}-s_{42}-s_{51}+s_{52}\right)& \frac{1}{2}\left(s_{44}-s_{45}-s_{54}+s_{55}\right)& \frac{s_{43}-s_{53}}{\sqrt{2}}& \frac{s_{46}-s_{56}}{\sqrt{2}}\\ \frac{1}{2}\left(s_{11}-s_{12}-s_{21}+s_{22}\right)& \frac{1}{2}\left(s_{14}-s_{15}-s_{24}+s_{25}\right)& \frac{1}{2}\left(s_{11}-s_{12}-s_{21}+s_{22}\right)& \frac{1}{2}\left(s_{14}-s_{15}-s_{24}+s_{25}\right)& \frac{s_{13}-s_{23}}{\sqrt{2}}& \frac{s_{16}-s_{26}}{\sqrt{2}}\\ \frac{1}{2}\left(s_{41}-s_{42}-s_{51}+s_{52}\right)& \frac{1}{2}\left(s_{44}-s_{45}-s_{54}+s_{55}\right)& \frac{1}{2}\left(s_{41}-s_{42}-s_{51}+s_{52}\right)& \frac{1}{2}\left(s_{44}-s_{45}-s_{54}+s_{55}\right)& \frac{s_{43}-s_{53}}{\sqrt{2}}& \frac{s_{46}-s_{56}}{\sqrt{2}}\\ \frac{s_{31}-s_{32}}{\sqrt{2}}& \frac{s_{34}-s_{35}}{\sqrt{2}}& \frac{s_{31}-s_{32}}{\sqrt{2}}& \frac{s_{34}-s_{35}}{\sqrt{2}}& s_{33}& s_{36}\\ \frac{s_{61}-s_{62}}{\sqrt{2}}& \frac{s_{64}-s_{65}}{\sqrt{2}}& \frac{s_{61}-s_{62}}{\sqrt{2}}& \frac{s_{64}-s_{65}}{\sqrt{2}}& s_{63}& s_{66}\end{array}\right]& \left(27\right)\end{array}$

[0176] There will now be described another embodiment of a method of determining the mixed-mode S-parameter S_mm of a DUT. Referring now to FIG. 13, there is illustrated a schematic diagram of a balun. As known to those of skill in the art baluns provide an interface from a single-ended circuit at terminal 1 of the balun to a balanced circuit at terminals 2 and 3 of the balun. The mixed-mode S-parameters S_mm of the balun can be determined according to the above-described methodology and for the terminal connections of the balun to a 3-port MTS of this disclosure, as illustrated in FIG. 13, with port 1,2, and 3 of the MTS coupled to terminals 1, 2, and 3 of the balun. For this arrangement, the S, the M and the S_mm matrices can be given by (28): $\begin{array}{cc}\left(28\right)S=\left[\begin{array}{ccc}{S}_{11}& S_{12}& S_{13}\\ S_{21}& S_{22}& S_{23}\\ S_{31}& S_{32}& S_{33}\end{array}\right]M=\left[\begin{array}{ccc}1& 0& 0\\ 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]S_{mm}=\left[\begin{array}{ccc}{S}_{11}& \frac{S_{12}-S_{13}}{\sqrt{2}}& \frac{S_{12}+S_{13}}{\sqrt{2}}\\ \frac{S_{21}-S_{31}}{\sqrt{2}}& \frac{1}{2}\left(S_{22}-S_{23}-S_{32}+S_{33}\right)& \frac{1}{2}\left(S_{22}+S_{23}-S_{32}-S_{33}\right)\\ & \frac{1}{2}\left(S_{22}-S_{23}+S_{32}-S_{33}\right)& \frac{1}{2}\left(S_{22}+S_{23}+S_{32}+S_{33}\right)\end{array}\right]& \end{array}$

[0177] where $\left[\begin{array}{ccc}{S}_{\mathrm{ss11}}& S_{\mathrm{sd12}}& S_{\mathrm{sc12}}\\ S_{\mathrm{ds21}}& S_{\mathrm{dd22}}& S_{\mathrm{dc22}}\\ S_{\mathrm{cs21}}& S_{\mathrm{cd22}}& S_{\mathrm{cc22}}\end{array}\right]$

[0178] There will now be described another embodiment of a method of determining the mixed-mode S-parameters S_mm of a DUT. Referring now to FIG. 14, there is illustrated a schematic diagram of two differential transmission line 80, 82, which can comprise part of a high-speed serial data bus that can be used, for example, to transport digital data between computers and peripheral devices at transfer rates of, for example, one Gigabit per second (Gb/s). The mixed-mode S-parameters of the two differential transmission lines can be determined from the above-described methodology. The S, M, and S_mm matrices can be given by (29): $\begin{array}{cc}S=\left[\begin{array}{cccccccc}{S}_{11}& S_{12}& S_{13}& S_{14}& S_{15}& S_{16}& S_{17}& S_{18}\\ S_{21}& S_{22}& S_{23}& S_{24}& S_{25}& S_{26}& S_{27}& S_{28}\\ S_{31}& S_{32}& S_{33}& S_{34}& S_{35}& S_{36}& S_{37}& S_{38}\\ S_{41}& S_{42}& S_{43}& S_{44}& S_{45}& S_{46}& S_{47}& S_{48}\\ S_{51}& S_{52}& S_{53}& S_{54}& S_{55}& S_{56}& S_{57}& S_{58}\\ S_{61}& S_{62}& S_{63}& S_{64}& S_{65}& S_{66}& S_{67}& S_{68}\\ S_{71}& S_{72}& S_{73}& S_{74}& S_{75}& S_{76}& S_{77}& S_{78}\\ S_{81}& S_{82}& S_{83}& S_{84}& S_{85}& S_{86}& S_{87}& S_{88}\end{array}\right]& \left(29\right)\\ M=\left[\begin{array}{cccccccc}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0& 0\\ 0& 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0& 0\\ 0& 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]& \end{array}$

[0179] and $S_{mm}=\left[\begin{array}{cc}{S}_{\mathrm{dd}}& S_{dc}\\ S_{cd}& S_{\mathrm{cc}}\end{array}\right]$

[0180] where $\begin{array}{c}{S}_{\mathrm{dd}}=\left[\begin{array}{cccc}\frac{1}{2}\left(S_{11}-S_{12}-S_{21}+S_{22}\right)& \frac{1}{2}\left(S_{15}-S_{16}-S_{25}+S_{26}\right)& \frac{1}{2}\left(S_{13}-S_{14}-S_{23}+S_{24}\right)& \frac{1}{2}\left(S_{17}-S_{18}-S_{27}+S_{28}\right)\\ \frac{1}{2}\left(S_{51}-S_{52}-S_{61}+S_{62}\right)& \frac{1}{2}\left(S_{55}-S_{56}-S_{65}+S_{66}\right)& \frac{1}{2}\left(S_{53}-S_{54}-S_{63}+S_{64}\right)& \frac{1}{2}\left(S_{57}-S_{58}-S_{67}+S_{68}\right)\\ \frac{1}{2}\left(S_{31}-S_{32}-S_{41}+S_{42}\right)& \frac{1}{2}\left(S_{35}-S_{36}-S_{45}+S_{46}\right)& \frac{1}{2}\left(S_{33}-S_{34}-S_{43}+S_{44}\right)& \frac{1}{2}\left(S_{37}-S_{38}-S_{47}+S_{48}\right)\\ \frac{1}{2}\left(S_{71}-S_{72}-S_{81}+S_{82}\right)& \frac{1}{2}\left(S_{75}-S_{76}-S_{85}+S_{86}\right)& \frac{1}{2}\left(S_{73}-S_{74}-S_{83}+S_{84}\right)& \frac{1}{2}\left(S_{77}-S_{78}-S_{87}+S_{88}\right)\end{array}\right]\\ \\ S_{dc}=\left[\begin{array}{cccc}\frac{1}{2}\left(S_{11}+S_{12}-S_{21}-S_{22}\right)& \frac{1}{2}\left(S_{15}+S_{16}-S_{25}-S_{26}\right)& \frac{1}{2}\left(S_{13}+S_{14}-S_{23}-S_{24}\right)& \frac{1}{2}\left(S_{17}+S_{18}-S_{27}-S_{28}\right)\\ \frac{1}{2}\left(S_{51}+S_{52}-S_{61}-S_{62}\right)& \frac{1}{2}\left(S_{55}+S_{56}-S_{65}-S_{66}\right)& \frac{1}{2}\left(S_{53}+S_{54}-S_{63}-S_{64}\right)& \frac{1}{2}\left(S_{57}+S_{58}-S_{67}-S_{68}\right)\\ \frac{1}{2}\left(S_{31}+S_{32}-S_{41}-S_{42}\right)& \frac{1}{2}\left(S_{35}+S_{36}-S_{45}-S_{46}\right)& \frac{1}{2}\left(S_{33}+S_{34}-S_{43}-S_{44}\right)& \frac{1}{2}\left(S_{37}+S_{38}-S_{47}-S_{48}\right)\\ \frac{1}{2}\left(S_{71}+S_{72}-S_{81}-S_{82}\right)& \frac{1}{2}\left(S_{75}+S_{76}-S_{85}-S_{86}\right)& \frac{1}{2}\left(S_{73}+S_{74}-S_{83}-S_{84}\right)& \frac{1}{2}\left(S_{77}+S_{78}-S_{87}-S_{88}\right)\end{array}\right]\\ \\ S_{cd}=\left[\begin{array}{cccc}\frac{1}{2}\left(S_{11}-S_{12}+S_{21}-S_{22}\right)& \frac{1}{2}\left(S_{15}-S_{16}+S_{25}-S_{26}\right)& \frac{1}{2}\left(S_{13}-S_{14}+S_{23}-S_{24}\right)& \frac{1}{2}\left(S_{17}-S_{18}+S_{27}-S_{28}\right)\\ \frac{1}{2}\left(S_{51}-S_{52}+S_{61}-S_{62}\right)& \frac{1}{2}\left(S_{55}-S_{56}+S_{65}-S_{66}\right)& \frac{1}{2}\left(S_{53}-S_{54}+S_{63}-S_{64}\right)& \frac{1}{2}\left(S_{57}-S_{58}+S_{67}-S_{68}\right)\\ \frac{1}{2}\left(S_{31}-S_{32}+S_{41}-S_{42}\right)& \frac{1}{2}\left(S_{35}-S_{36}+S_{45}-S_{46}\right)& \frac{1}{2}\left(S_{33}-S_{34}+S_{43}-S_{44}\right)& \frac{1}{2}\left(S_{37}-S_{38}+S_{47}-S_{48}\right)\\ \frac{1}{2}\left(S_{71}-S_{72}+S_{81}-S_{82}\right)& \frac{1}{2}\left(S_{75}-S_{76}+S_{85}-S_{86}\right)& \frac{1}{2}\left(S_{73}-S_{74}+S_{83}-S_{84}\right)& \frac{1}{2}\left(S_{77}-S_{78}+S_{87}-S_{88}\right)\end{array}\right]\\ \\ S_{dc}=\left[\begin{array}{cccc}\frac{1}{2}\left(S_{11}+S_{12}+S_{21}+S_{22}\right)& \frac{1}{2}\left(S_{15}+S_{16}+S_{25}+S_{26}\right)& \frac{1}{2}\left(S_{13}+S_{14}+S_{23}+S_{24}\right)& \frac{1}{2}\left(S_{17}+S_{18}+S_{27}+S_{28}\right)\\ \frac{1}{2}\left(S_{51}+S_{52}+S_{61}+S_{62}\right)& \frac{1}{2}\left(S_{55}+S_{56}+S_{65}+S_{66}\right)& \frac{1}{2}\left(S_{53}+S_{54}-S_{63}+S_{64}\right)& \frac{1}{2}\left(S_{57}+S_{58}+S_{67}+S_{68}\right)\\ \frac{1}{2}\left(S_{31}+S_{32}+S_{41}+S_{42}\right)& \frac{1}{2}\left(S_{35}+S_{36}+S_{45}+S_{46}\right)& \frac{1}{2}\left(S_{33}+S_{34}-S_{43}+S_{44}\right)& \frac{1}{2}\left(S_{37}+S_{38}+S_{47}+S_{48}\right)\\ \frac{1}{2}\left(S_{71}+S_{72}+S_{81}+S_{82}\right)& \frac{1}{2}\left(S_{75}+S_{76}+S_{85}+S_{86}\right)& \frac{1}{2}\left(S_{73}+S_{74}-S_{83}+S_{84}\right)& \frac{1}{2}\left(S_{77}+S_{78}+S_{87}+S_{88}\right)\end{array}\right]\end{array}$

[0181] Referring again to FIG. 14, eight single-ended ports of an 8-port MTS can be coupled to the eight terminals of the DUT. The mixed-mode S-parameters for the DUT of FIG. 14 can be defined as balanced ports. For example, Port 1, Port 2, Port 3, and Port 4 are balanced ports. For this mixed mode port configuration of the DUT, the mixed-mode S-parameter matrix of the DUT can be summarized as (30): $\begin{array}{cc}\left[\begin{array}{cccccccc}{S}_{\mathrm{dd11}}& S_{\mathrm{dd12}}& S_{\mathrm{dd13}}& S_{\mathrm{dd14}}& S_{\mathrm{dc11}}& S_{\mathrm{dc12}}& S_{\mathrm{dc13}}& S_{\mathrm{dc14}}\\ S_{\mathrm{dd21}}& S_{\mathrm{dd22}}& S_{\mathrm{dd23}}& S_{\mathrm{dd24}}& S_{\mathrm{dc21}}& S_{\mathrm{dc22}}& S_{\mathrm{dc23}}& S_{\mathrm{dc24}}\\ S_{\mathrm{dd31}}& S_{\mathrm{dd32}}& S_{\mathrm{dd33}}& S_{\mathrm{dd34}}& S_{\mathrm{dc31}}& S_{\mathrm{dc32}}& S_{\mathrm{dc33}}& S_{\mathrm{dc34}}\\ S_{\mathrm{dd41}}& S_{\mathrm{dd42}}& S_{\mathrm{dd43}}& S_{\mathrm{dd44}}& S_{\mathrm{dc41}}& S_{\mathrm{dc42}}& S_{\mathrm{dc43}}& S_{\mathrm{dc44}}\\ S_{\mathrm{cd11}}& S_{\mathrm{cd12}}& S_{\mathrm{cd13}}& S_{\mathrm{cd14}}& S_{\mathrm{cc11}}& S_{\mathrm{cc12}}& S_{\mathrm{cc13}}& S_{\mathrm{cc14}}\\ S_{\mathrm{cd21}}& S_{\mathrm{cd22}}& S_{\mathrm{cd23}}& S_{\mathrm{cd24}}& S_{\mathrm{cc21}}& S_{\mathrm{cc22}}& S_{\mathrm{cc23}}& S_{\mathrm{cc24}}\\ S_{\mathrm{cd31}}& S_{\mathrm{cd32}}& S_{\mathrm{cd33}}& S_{\mathrm{cd34}}& S_{\mathrm{cc31}}& S_{\mathrm{cc32}}& S_{\mathrm{cc33}}& S_{\mathrm{cc34}}\\ S_{\mathrm{cd41}}& S_{\mathrm{cd42}}& S_{\mathrm{cd43}}& S_{\mathrm{cd44}}& S_{\mathrm{cc41}}& S_{\mathrm{cc42}}& S_{\mathrm{cc43}}& S_{\mathrm{cc44}}\end{array}\right]& \left(30\right)\end{array}$

[0182] There will now be described another embodiment of a method of determining the mixed-mode S-parameters S_mm of a DUT Referring now to FIG. 15, there is illustrated four pairs of differential transmission line 92, 94, 96, and 98 which can be used, for example, in a local area network as category 5, category 6, and category type cables. The mixed-mode S-parameters S_mm of the four pairs of differential transmission lines can be determined according to the above-described embodiment. The S, M S_mm can be given by (31): $\begin{array}{cc}\begin{array}{c}S=\left(\begin{array}{cccccccccccccccc}{S}_{11}& S_{12}& S_{13}& S_{14}& S_{15}& S_{16}& S_{17}& S_{18}& S_{19}& S_{110}& S_{111}& S_{112}& S_{113}& S_{114}& S_{115}& S_{116}\\ S_{21}& S_{22}& S_{23}& S_{24}& S_{25}& S_{26}& S_{27}& S_{28}& S_{29}& S_{210}& S_{211}& S_{212}& S_{213}& S_{214}& S_{215}& S_{216}\\ S_{31}& S_{32}& S_{33}& S_{34}& S_{35}& S_{36}& S_{37}& S_{38}& S_{39}& S_{310}& S_{311}& S_{312}& S_{313}& S_{314}& S_{315}& S_{316}\\ S_{41}& S_{42}& S_{43}& S_{44}& S_{45}& S_{46}& S_{47}& S_{48}& S_{49}& S_{410}& S_{411}& S_{412}& S_{413}& S_{414}& S_{415}& S_{416}\\ S_{51}& S_{52}& S_{53}& S_{54}& S_{55}& S_{56}& S_{57}& S_{58}& S_{59}& S_{510}& S_{511}& S_{512}& S_{513}& S_{514}& S_{515}& S_{516}\\ S_{61}& S_{62}& S_{63}& S_{64}& S_{65}& S_{66}& S_{67}& S_{68}& S_{69}& S_{610}& S_{611}& S_{612}& S_{613}& S_{614}& S_{615}& S_{616}\\ S_{71}& S_{72}& S_{73}& S_{74}& S_{75}& S_{76}& S_{77}& S_{78}& S_{79}& S_{710}& S_{711}& S_{712}& S_{713}& S_{714}& S_{715}& S_{716}\\ S_{81}& S_{82}& S_{83}& S_{84}& S_{85}& S_{86}& S_{87}& S_{88}& S_{89}& S_{810}& S_{811}& S_{812}& S_{813}& S_{814}& S_{815}& S_{816}\\ S_{91}& S_{92}& S_{93}& S_{94}& S_{95}& S_{96}& S_{97}& S_{98}& S_{99}& S_{910}& S_{911}& S_{912}& S_{913}& S_{914}& S_{915}& S_{916}\\ S_{101}& S_{102}& S_{103}& S_{104}& S_{105}& S_{106}& S_{107}& S_{108}& S_{109}& S_{1010}& S_{1011}& S_{1012}& S_{1013}& S_{1014}& S_{1015}& S_{1016}\\ S_{111}& S_{112}& S_{113}& S_{114}& S_{115}& S_{116}& S_{117}& S_{118}& S_{119}& S_{1110}& S_{1111}& S_{1112}& S_{1113}& S_{1114}& S_{1115}& S_{1116}\\ S_{121}& S_{122}& S_{123}& S_{124}& S_{125}& S_{126}& S_{127}& S_{128}& S_{129}& S_{1210}& S_{1211}& S_{1212}& S_{1213}& S_{1214}& S_{1215}& S_{1216}\\ S_{131}& S_{132}& S_{133}& S_{134}& S_{135}& S_{136}& S_{137}& S_{138}& S_{139}& S_{1310}& S_{1311}& S_{1312}& S_{1313}& S_{1314}& S_{1315}& S_{1316}\\ S_{141}& S_{142}& S_{143}& S_{144}& S_{145}& S_{146}& S_{147}& S_{148}& S_{149}& S_{1410}& S_{1411}& S_{1412}& S_{1413}& S_{1414}& S_{1415}& S_{1416}\\ S_{151}& S_{152}& S_{153}& S_{154}& S_{155}& S_{156}& S_{157}& S_{158}& S_{159}& S_{1510}& S_{1511}& S_{1512}& S_{1513}& S_{1514}& S_{1515}& S_{1516}\\ S_{161}& S_{162}& S_{163}& S_{164}& S_{165}& S_{166}& S_{167}& S_{168}& S_{169}& S_{1610}& S_{1611}& S_{1612}& S_{1613}& S_{1614}& S_{1615}& S_{1616}\end{array}\right)\\ M=\left(\begin{array}{cccccccccccccccc}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0& 0& 0& 0& 0& 0\\ 0& 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0& 0\\ 0& 0& 0& 0& 0& 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0& 0& 0& 0& 0& 0\\ 0& 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0& 0\\ 0& 0& 0& 0& 0& 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)\end{array}& \left(31\right)\end{array}$