Systematic Deembeding of the Transmission Line Parameters on High-density Substrates with Probe-tip Calibrations Janusz Grzyb, Didier Cottet, Gerhard Tröster, Electronics Lab, ETH Zurich, Gloriastrasse 35, 8092 Zurich, SWITZERLAND Phone: +41 1 632 79 21, Fax: +41 1 632 12 10, e-mail:
[email protected]
Abstract Three procedures allowing the systematic determination of the propagation characteristic of lines and deembedding the influence of feeding discontinuities are presented. All of them are based on the known measurement technique of two lines with different lengths on the measured substrate after earlier calibration with standard substrate. The first one doesn’t involve the knowledge of the reference planes location and their shift to the probe tips in the extraction process and allows arbitrary position of the probe tips along the lines (not necessarily located at a position approximately in front of the physical beginning of the lines). In effect equivalent element models of error boxes representing the feeding discontinuities don’t have to be lumped models. The second one is a modified procedure from [4], wherein the probe-tip discontinuity error box is modeled by RLCG element. This new formulation allows to omit one of the limitations of the S-matrix asymmetry of an arbitrary reciprocal junction (in our case probe-tip discontinuity). This limitation is a complex characteristic impedance of the meausred lines. The third procedure allows to determine the characterisitc impedance of the lines with the wider or narrower pitch than that of the used probes. The complex propagation constant and characteristic impedance have been determined for lines on fused silica substrate up to 110GHz and error box modesl have been verified up to this frequency range. I Introduction Precise S-Parameter Vector Network Analyzer measurements at the component level are an essential requirement to obtain designs operating within their target specifications. Before any accurate S-parameter measurements can be performed the measurement system needs to be calibrated to remove systematic errors resulting from different reflections. The non-idealities of the measurement system are quantified using mathematical error correction model. The aim of the calibration procedure is to determine the error coefficients in these models. This is done by measuring several “known” calibration standards. An accurate on-wafer fabrication of calibration standards is, however, difficult with the non-precision processing techniques (low-cost technologies) or if the goal is the characterization of the newly established technological process. Therefore, many practical measurements rely on off-wafer calibrations using a special standard calibration substrate. Usually both contact geometry and substrate (losses and permittivity) differ between calibration structures and the DUT. A technique is needed to account for these differences as they drastically deteriorate the
measurement accuracy [1,2] and have a large effect on the extracted characteristic impedance of the measured lines, esp. at the mm-wave frequencies. If these effects are compensated for, the accuracy is comparable to an on-wafer calibration [1]. We present three procedures allowing the systematic determination of the propagation characteristics of the lines and deembedding the influence of the feeding discontinuities. These procedures are based on measurement technique [4] of two lines with different lengths on the measured substrate after earlier calibration with a standard substrate. In contrast to the method in [4] we are able to omit the constraint of the error box S-matrix symmetry. Due to a new formulation of the problem we omit one of the limitations of the S-matrix asymmetry of an arbitrary reciprocal junction (in our case probe-tip discontinuity). This limitation is a complex characteristic impedance of the measured lines. The first method is a modified procedure from [4]. We model the probe-tip discontinuity error box by the same RLCG elements combination, but with our new formulation. The second procedure doesn’t involve the knowledge of the exact location of the reference planes in the extraction process (no shift of the reference plane to probe tips) and thus allows arbitrary position of the probe tips along the lines. In effect the error boxes and their equivalent model elements representing the feeding don’t have to be lumped. Therefore the method is able to take into consideration shunt admittance (open transmission line stub behind the probe-tip), series admittance at the probe tip and location of the reference plane (also represented by an equivalent transmission line). Such a formulation of the problem also allows the extraction of the equivalent position of the reference plane that is equal to its physical location in case of negligible influence of the series parasitic impedance (usually parasitic inductance). The third procedure allows determining the characteristic impedance of the lines with wider or narrower pitch than that of the used probes. As a feeding structure we use a piece of the line with matched probe-tip pitch with previously extracted propagation characteristics and a tapered line. The complex propagation constant and characteristic impedance of different line geometries have been determined for lines on fused-silica substrate up to 110GHz and error box models have been verified up to this frequency range. The extracted characteristic impedances have been compared with the simulation results from FEM 3D EM tool (Ansoft HFSS). They have showed very good agreement.
II Existing methods versus our problem formulation The general two-port measurement problem is shown in Fig.1, where it is intended to measure the S-parameters of a two-port device at the indicated DUT reference planes.
The parameter " l " denotes the length difference between THRU and LINE standards, which is usually very accurately defined by the metalization process resolution. Knowing this value one is able to determine the precise value of γ. e −γ l =
2 2 2 L12 + T122 − (T11 − L11 ) 2 ± [ L12 + T122 − (T11 − L11 ) 2 ]2 − 4 L12 T122 2 L12T12
S22 =
T11 − L11 T12 − L12e −γ l
S11 = T11 − S 22T12
S12 S 21 = T12 (1 − S222 )
Fig.1 Two-port measurement problem Losses and phase delays caused by the connectors, cables, transitions and switching as well as isolation errors of the VNA have been accounted for by the initial on-wafer calibration. Hence the error boxes on both sides of the DUT represent the differences between the on-wafer substrate standards and their off-wafer counterparts. As the discontinuities on both sides of the DUT are equal, the measurement problem can be assumed to be symmetrical, such that only error box denoted as RA is to be determined. The calibration-comparison method [6] performs a full TRL off-wafer calibration. Similarly method [4] uses the same concept but it is based only on two transmission lines as offwafer standards. The reduction of the number of off-wafer standards was achieved under assumption that the reciprocity condition of a waveguide junction (probe-tip-line junction) forces the symmetry of their S-parameters. This condition is not necessarily true, as shown in [3]. The Lorentz reciprocity condition applied to junctions composed of reciprocal media that connect uniform but otherwise arbitrary waveguides shows that forward and reverse transmission coefficients do not have to be equal in the general case. Specifically for lossy hybrid modes, such as those found in CPW or microstrip lines, the effect of the complex characteristic impedance at low frequencies or even in the wide frequency range for strongly lossy substrates as low resistivity silicon cannot be neglected. This constraint does not influence the extraction of only the propagation constant because it involves a product of forward and reverse transmission coefficients of the error boxes. This constraint does, however, influences the characteristic impedance determination. With reference to Fig.1 the error boxes are characterized by Smatrix [S] and alternatively by the ABCD transmission matrix. To avoid confusion in notation we will denote the measured Sparameters for the Thru and Line connections as [T] and [L], respectively. We have assumed before that error boxes are identical and symmetrically arranged. By symmetry we have T11= T22, L11= L22 and by reciprocity T21= T12 , L21= L12. The reciprocity condition is valid in this case because the reference impedances of 50 Ω at both measurements ports are equal and real and the waveguides at both ports are identical. Solving for propagation constant in terms of the measured S-parameters we get the equations (1-4). The sign can be determined by the requirement that the real and imaginary parts of γ be positive.
(1) (2) (3) (4)
Now, we formulate the problem in terms of ABCD matrices. Moreover, the formulation concerns the whole measured standards at a time and not the probe-tip discontinuity only. The lack of assumption of the error box Smatrix symmetry doesn’t allow us to analyze it separately as we are able to generate nor the full S-matrix neither the full ABCD matrix of the error box without looking at the whole measured standards. This formulation is advantageous for impedance accuracy extraction because we are able to omit one of the limitations of the S-parameter matrix asymmetry, namely complex characteristic impedance of the measured waveguide. On the contrary, the complex characteristic impedance does not restrict the symmetry of our error box impedance or ABCD matrices and allows smaller extraction error than under assumption of the S-parameter matrix symmetry. The asymmetry of the impedance or ABCD matrices representing probe-tip discontinuity between reciprocal media in most common cases can be neglected [7]. Additional advantage of such an approach is that the resulting S-matrix of the error box can be synthesized from the extracted ABCD matrix and characteristic impedance of the measured lines without breaking the constraint of the error box S-matrix asymmetry for complex characteristic impedance of the measured lines. II.A Existing probe-tip error box models In order to determine the characteristic impedance of the measured lines one has to model the probe-tip error box. The calibration comparison method introduced in [5] for the first time attempted to account for these discontinuities effects in characteristic impedance extraction process. A treatment of the error boxes measured by the calibration comparison method was presented in [6] and determines the characteristic impedance accounting for only a shunt contactpad capacitance and conductance. A model, which accounts for the presence of a possible parasitic inductance due to the step-in-width between probe and line, was presented in [4]. The error box model consists of a shunt admittance and series impedance. The parameters of the model are extracted based on the measurements of two lines with different lengths. III Proposed procedures and models III.A Distributed model Typical approach calibration procedures calibrate the measured standards to the probe-tips and then model the
probe-tip-line discontinuity by a lumped model. In result the equivalent models are limited to models representing “lumped” error boxes. This also involves a reference plane shift from the initial position in the middle of the THRU line to the probe-tips. This, in turn, demands the knowledge of the exact relative position between the middle of the THRU line and the probe-tips and complicates the procedure. The location of the probe tips is also not known exactly because of limited precision of probe tips positioning. We present the procedure using different formulation of the problem that does not involve the reference plane shift and doesn’t need the relative position between probe-tips and the middle of a THRU line. On the contrary, it enables the extraction of the equivalent probe-tip position. ‘Equivalent’ means that it is the exact physical position of the probe tips if the inductive series discontinuity can be neglected. Otherwise this extracted relative position is different from its physical counterpart by some transmission line length, which models this inductive effect. Such formulation allows an arbitrary position of the probe-tips on the line, not necessarily close to the physical beginning of the lines (typically 25µm), and still properly accounts for the differences between reference calibration and off-wafer calibration. The first step of the procedure treats the whole THRU line standard as a connection of only two error boxes with their ABCD matrices denoted as ArBrCrDr . The equivalent circuit model of the error box is shown in Fig.2. THRU line Ref. Plane
Quasi Probe-Tip Ref. Plane
Zo, γ, d1
Zr
Zo
Zo, γ, d2
parasitic series impedance can be neglected. ‘d2’ denotes a quasi-length of the open-ended stub connected before quasiprobe-tip reference plane. The quasi-length d1 reflects series connection of a physical length of the measured line between the middle of the THRU standard and parasitic series impedance (in particular series parasitic inductance and resistance of the step-in-width between probe and strip). The presented model is therefore able to take into consideration the combination of shunt admittance, series impedance and shift in the reference plane location. Using the ABCD representation of the THRU and LINE standards one is able to solve for A2 Z o product: Ar2 Z o =
−4(cosh γ l ⋅ BT − BL ) ± ∆ 8sinh γ l
∆ = 16 (cosh γ l ⋅ BT − BL )2 − sinh 2 γ l ⋅ BT2
Z o and Z r denote the characteristic impedances of the lines being measured and reference one, respectively. The transformer element in the equivalent circuit appears because the reference impedance was transformed to the characteristic impedance of the measured lines during the initial part of the off-wafer calibration. The transformer is supposed to be used only in a final S matrix representation including impedance mismatch between reference and measured lines and its parameters are calculated as a final independent step after the extraction procedure. In general, it isn’t a conventional impedance transformer. Its turns ratio reflects the complex impedance transform. When Z o (reference impedance Z r is real) is real, both matrices of a classical impedance and our equivalent impedance transformers are identical. ‘d1’ denotes a quasi-distance between quasi-probe-tip reference plane and the middle of the THRU line and is equal to its physical distance between physical position of the probetip and middle of the THRU line when the influence of the
(6)
AT BT CT DT and AL BL CL DL represent the ABCD matrices of the measured THRU and LINE standards, respectively. The parameter " l " denotes the length difference between THRU and LINE standards. Denoting Ar2 Z o product as ψ we result in two equations for Br and Cr matrix elements: Br = Cr =
BT 2
Zo
(7)
ψ
AT − 1 ψ BT Zo
(8)
Ar2 Z o product in case of our error box model is equal
Z o cosh 2 γ d1 and allows determining the proper choice of the root in Eq.(5). This choice is made by demanding Re ( Ar2 Z o ) ≥ 0 . Calculating the ratio Br/Ar leads to the solution
for distance d1: tanh γ d1 =
Fig.2 The equivalent circuit model of the error box
(5)
BT 2ψ
(9)
Then Z o and d2 take the forms as in (10-11): Zo =
BT 2 cosh γ d1 ⋅ sinh γ d1
A −1 AT − 1 tanh γ d1 B − 2 T 2 − tanh γ d2 = − jZo T = − jZo 2Zo cosh γ d1 Zo BT BT
(10) (11)
III.B Lumped error box model In this case we calibrate the measured standards to the probe-tips and then model the probe-tip-line discontinuity by a lumped RLCG model from [4] shown in Fig.3. However we use our novel formulation without the assumption of the error box S-matrix symmetry. The model accounts for any arbitrary series impedance (spec. series inductance and resistance) and shunt admittance (spec. parallel capacitance and conductance). We have performed similar logical extraction procedure as for previous distributed model directly on the ABCD
formulation of the whole measured standards without any earlier assumptions on error boxes S-matrix symmetry.
Gp
Cp
Lp
Rp
Fig.3 Lumped model of the probe-tip-line discontinuity In this case the ArBrCrDr matrix represents the error box of only probe-tip –line discontinuity as the reference planes have been shifted to the probe-tips using their relative location towards the center of the THRU line. Once more solving for Ar2 Z o product we result in: Ar2 Z o =
Br2
Feed Line
∆2
) −∆ )
2 χ ± χ −∆
o
2
χ2
2
2
BL cosh γ lTHRU − BT cosh γ lLINE Br2 = =χ Zo cosh γ lTHRU ⋅ sinh γ lLINE − cosh γ lLINE ⋅ sinh γ lTHRU
(15)
AL − cosh γ lLINE − Br sinh γ lLINE /( Zo Ar ) 2 2 Br cosh γ lLINE + Ar Zo sinh γ lLINE + Br sinh γ lLINE /( Z o Ar )
(16)
As before ATBTCTDT and ALBLCLDL represent the ABCD matrices of the measured THRU and LINE standards, respectively. The parameters " lTHRU " and " lLINE " represent the line lengths between both probe-tips for THRU and LINE standards, respectively. They fulfill the relation l = lLINE − lTHRU , wherein " l " is the previously introduced parameter and denotes the length difference between LINE and THRU standards. From ArBrCrDr matrix representations of the LRCG error box model one has Ar=1, which allows extraction of the measured line characteristic impedance from (12). Proper choice between two roots is provided by the following assumption: Re( Z o ) > 0 . Then B takes the form B = ∆ / 2 for
A = 1 which leads to the formulations for equivalent parasitic inductance and resistance of the model: ∆ Lp = Im 4π f ∆ R p = Re 2
(17) (18)
Determination of the equivalent parasitic capacitance and conductance demands the equation (16) for Cr. C C p = Im r 2π f G p = Re ( Cr )
III.C Zo of the lines with probe-tip pitch mismatch
Meas. Line
(13) (14)
Ar2 Zo +
Taper
(12)
BL sinh γ lTHRU − BT sinh γ lLINE =∆ cosh γ lLINE ⋅ sinh γ lTHRU − cosh γ lTHRU ⋅ sinh γ lLINE
2 Ar Br =
Cr =
( Z = (χ ± 2
The situation wherein the dimensions of the lines being measured exceed the pitch of the used probe-tips occurs quite often in high-density packaging, esp. if one intends to minimize conduction losses of the lines. Also in case of MMIC’s one often needs to measure the lines, which are much narrower than the probe-tips pitch. The third presented procedure allows determining characteristic impedance of the lines with the wider or narrower pitch than that of the used probes. As a feeding structure we use a piece of the line with matched probe-tip pitch and a tapered line. The layout of this configuration is shown on Fig.4. Thus the first step of the calibration procedure is the extraction of the complex propagation constant, characteristic impedance of the feed line and probe-tip discontinuity model by means of procedure form section III.B.
(19) (20)
Fig.4 The wide line measurement configuration The influence of the probe-tip discontinuity and of a piece of feeding line can now be easily deembedded from the measurements of the THRU and LINE standards as the ABCD matrices of the probe-tip discontinuity and feed line have been extracted in the first step of the calibration and are known. The only element left to be deembedded is a tapered line. If the geometry of the wider (or narrower) lines is chosen such that the characteristic parameter ‘K’ of the CPW feed line and the line being measured does not change, the characteristic impedance of both lines are theoretically the same. ‘K’ is defined as a ratio of the center strip width and ground-toground spacing. In reality the characteristic impedances of both lines will be close to each other but not the same. The difference will come from the influence of the finite metalization thickness, metalization profile, degree of planarization and processing inaccuracies. Definitely, the final verification of the measured characteristic impedance should come from measurements, even if their geometric parameter ‘K’ does not change. As the characteristic impedances of both lines are close to each other, the tapered line can be modeled as a piece of uniform transmission line almost perfectly. Almost perfectly, because the current flow changes its direction along the taper and influences its frequency behavior. This effect can be modeled by a cascade of uniformly distributed series inductances and parallel capacitances along the taper. It results in equalities (21-22) for ATapBTapCTapDTap matrix of the taper. ATap = DTap CTap =
2 ATap −1
BTap
(21) (22)
Now, combining the equations (12), (13), (16) from the former section describing Ar, Br and Cr elements of the error box and Eq. (22) one is able to extract the characteristic impedance of the measured lines (wider or narrower). The parameters " lTHRU " and " lLINE " now represent the lengths of only the lines being measured (without feeding structures) for THRU and LINE standards, respectively. Equations (12-16) are general equations describing a reciprocal junction matrix ABCD in terms of two measured lines (THRU and LINE) and they are not only valid for the model in section III.B. They are independent of any discontinuity model. Thus, in case of our taper: Ar = ATap ,
Br = BTap , Cr = CTap , Dr = DTap and ATBTCTDT and ALBLCLDL matrices refer to the measured THRU and LINE standards after deembeding of the tip-probe discontinuity and length of feed line influence. The final equation for the characteristic impedance of the lines is as follows:
( AL − cosh γ lline − sinh γ lline ⋅η / ∆ ) 1 2η = 2 1 + Z c ∆ ( 2 cosh γ lline + sinh γ lline ⋅ (η / ∆ + ∆ / η ) )
(23)
η = χ ± χ 2 − ∆2
(24)
Proper choice of the sing in following assumption: Re( Z c ) > 0 .
η
approximately constant over the whole frequency range. Their equivalent average values in the measured band are d1=972µm, d2=-12.6µm , and d1=954µm, d2=-8.9µm for SET1 and SET2, respectively. The extracted d2 values are negative because effective permittivity of the alumina is considerably higher that that of fused silica. Comparing d1 length with its physical counterpart (975um) one can see that they differ only by 2um for SET1. It means that the parasitic series inductance is negligible in this case. The same length for SET2 differs considerably from the physical one and shows importance of the parasitic series inductance. As the probes have been located close to the physical beginning of the lines, the transmission line length d2 can be seen as the shunt capacitance at the probe tips. Its equivalent average value in the measured band is C=-2.56pFand C=-1.4pF for SET1 and SET2 lines, respectively. For comparison purposes we have performed similar extraction procedure using the lumped RLGC model from [4] but with our new formulation form section III.B. The extracted averaged parasitic parameters for both line sets are as follows: SET1 – R=-0.81Ω, L=-0.48pH, G=50uS, C=-2.8pF, SET2 – R=-0.57Ω, L=-3.95pH, G=21uS, C=-3.11pF
is provided by the
This way we have shown that we can extract the characteristic impedance of the lines exceeding the probe-tip pitch without using of any taper model in the calculation process. This is general very important conclusion: one is able to extract the characteristic impedance of the lines embedded in a reciprocal measurement feeding structures based on two line measurements without looking into the feeding structure if its ABCD matrix fulfill the equation A=D (the model of a reciprocal structure is symmetric). IV Measurement results We will demonstrate the presented procedures based on the measurements of only 3 different CPW line geometries (5µm metallization thickness) for simplicity of analysis. The lines have been produced on on fused silica substrate and measured up to 110GHz by means of HP 8510XF VNA with waveguide coplanar probes (pitch 150um) and alumina calibration substrate from Cascade Microtech. The probes have been positioned to 25um beyond the physical beginning of the lines. The length of the THRU standard (2mm) has been chosen long enough so that it enables the single-mode calibration condition. Longer THRU standard is also important for the extraction accuracy because it limits the influence of the phase instabilities, esp. at lower frequencies. The length of the LINE standard (11.5mm) is chosen to allow THRU-LINE phase difference of 30° at about 2GHz. Multi-line procedure, of course, would increase the accuracy of the extraction. IV.A Simple lines without taper The line geometries of the two simple CPW line sets (without taper) are as follows: (Width/Space/GND_Width) SET1=100/14/200um, SET2= 150/20/300um. The extracted (by procedure from section III.A) lengths d1, d2 for both line sets are shown in Fig.3. They are
Fig.3. Extracted d1nad d2 transmission line lengths The extracted L values are in good agreement with the distributed error model results. The determined parasitic inductance for the line set SET1 is very small but its corresponding value in case of line set SET2 is about 8 times larger and justifies the quasi-reference plane shift in the distributed element model. The origin of the parasitic inductance can be regarded as the probe-tip position difference between calibration and measured substrates referenced to the inner edges of the line and ground strips or in other words as the difference between the ground-to-ground spacing of the substrate and the measured line geometries. Comparing the extracted parasitic capacitance values for both models one notices the differences in their determined values. Where does this difference come from? The parasitic series inductance represented in distributed model by some
added transmission line length in the quasi-length d1 cannot be considered separately without noticing that a piece of transmission line also represents some shunt capacitance according to its LC model. So, our quasi-length d1 also accounts for some amount of the shunt capacitance at the probe tips and justifies the differences between extracted shunt capacitance of the lumped model and quasi-shunt capacitance of the distributed model The determined characteristic impedance, effective εr and insertion loss of both line sets are shown in Fig.4. For comparison purposes, the HFFS simulated Zo is also drawn. HFSS metal layers were modelled as 3D geometries, which is especially important for very narrow slots between lines. In simulations surface impedance formulation for conductors was used. This approach is fully justified, as the skin depth of the 5um thick silver line is 1.4um at 2 GHz. Local nonmonotonical behaviour of the extracted quasi-lengths d1 and d2 and characteristic impedances can be seen as the influence of mainly probe placement accuracy, half wavelength difference between LINE and THRU standard at some frequencies.
following 240/30/400µm. The extracted insertion loss and effective εr are shown on Fig.5. Analyzing the frequency behavior of the losses one can observe its instabilities beyond 50GHz. This is the effect of higher-order mode launching.
Fig.5 Extracted effective εr and insertion loss of wide line The same effect can be observed in frequency characteristics of the imaginary part of the extracted characteristic impedance, which is directly responsible for the losses and. It is shown on Fig.6a. Launching of the higherorder modes makes the calibration process improper. In result the extracted characteristic impedance of the wide lines is valid only up to this frequency. Its real part is shown on Fig.6b.
Fig.4 Extracted characteristic impedance, effective εr and insertion loss of both line sets IV.B Lines with taper We analyze the method from section III.C based on only one line set for simplicity. The length of the THRU and LINE standards are the same as in case of simple CPW lines from former section, but additionally they are fed by a cascade of 300µm long 150/20/300µm (SET 2) lines and 350µm long taper. The dimensions of the wide lines being extracted are the
Fig.6 Extracted imaginary and real parts of the wide lines characteristic impedance V Conclusions Three procedures allowing the systematic determination of the line parameters, including final analytical formulation,
have been presented. The formulation of the problem is in terms of ABCD matrices. More important, the formulation concerns the whole measured standards at a time and not the probe-tip discontinuity only. It allows omitting the assumption that the reciprocity condition of a waveguide junction (probetip-line junction) forces the symmetry of its S-parameters. The first method is the modified procedure from [4] modeling the probe-tip discontinuity error box by RLCG elements combination, but with the new formulation. The second procedure doesn’t involve the knowledge of the exact location of the reference planes in the extraction process (no shift of the reference plane to probe tips). In effect error boxes and their equivalent model elements representing the feeding don’t have to be lumped. The method is able to take into consideration shunt admittance (open transmission line stub behind the probe-tip), series admittance at the probe tip and location of the reference plane (also represented by an equivalent transmission line). Such a formulation of the problem also allows the extraction of the equivalent position of the reference plane that is equal to its physical location in case of negligible influence of the series parasitic impedance (usually parasitic inductance). The third procedure allows determining characteristic impedance of the lines with a wider or narrower pitch than that of the used probes. As a feeding structure we use a piece of the line with matched probe-tip pitch with previously extracted propagation characteristics and a tapered line. The complex propagation constant and characteristic impedance of different line geometries have been determined based on the presented methods for lines on fused-silica substrate up to 110GHz and error box models have been verified up to this frequency range. The extracted characteristic impedances have been compared with the simulation results from FEM 3D EM tool (Ansoft HFSS) and have showed very good agreement. Acknowlwdgenments The authors wish to acknowledge the measurement support of Hansruedi Benedickter. References [1] D.F. Williams, R.B. Marks, "Compensation for Substrate Permittivity in Probe-Tip Calibrations,"44th ARFTG Conf. Dig. , pp. 20-30, Dec. 1994. [2] D.K. Walker, D.F. Williams, "Compensation for Geometrical Variations in Coplanar Waveguide Probe-Tip Calibrations," IEEE Microwave and Guided Wave Letters, vol. 7, pp. 97-99, April 1997. [3] R.B. Marks, D.F. Williams, "A General Waveguide Circuit Theory," J. Res. Natl. Inst Stand. Technol., vol. 97, pp. 533-561, Sept-Oct. 1992. [4] G. Carchon, B. Nauwelaers, W. de Raedt, D. Schreurs and S. Vandenberghe, "Characterising differences between measurement and calibration wafer in probe-tip calibrations," Electronic Letters, vol. 35, pp. 1087-1088, June, 1999.
[5] D.F. Williams, R.B. Marks, "Accurate Transmission Line Characterization," IEEE Microwave and Guided Wave Letters, vol. 3, pp.247-249, August 1993. [6] D.F. Williams, U. Arzt, H. Grabinski, "Accurate Characteristic Impedance Measurement on Silicon," IEEE MTT-S Symp. Dig., pp. 1917-1920, June 1998. [7] D.F. Williams, R.B. Marks, "Reciprocity Relations in Waveguide Junctions," IEEE Trans. MTT, vol. 41, pp. 1105-1110, June/July 1993
