S Matrix versus ABCD Chain Matrix Formulation in Probe-tip Calibrations Janusz Grzyb, 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 We present some advantageous features of the chain ABCD matrix formulation in comparison with S-matrix formulation in characteristic impedance extraction with probetip calibrations. The procedure is based on the known measurement technique of two different on-wafer line standards [1]. An initial off-wafer LRM or TRL calibration with standard calibration substrate is assumed. The first feature is the formulation of the whole extraction problem in terms of the ABCD chain matrix. It allows us to omit one of the limitations of the true traveling waves based Smatrix [3] asymmetry of a general reciprocal transition between two different waveguides, in particular probe-tip-line junction. This limitation is a difference between complex characteristic impedances of both waveguides. On the contrary, the symmetry of its admittance matrix or equivalently the determinant AD-BC of its chain matrix is not influenced by this effect. If one models the probe-tip-line transition by use of only a shunt admittance or cascade of a shunt admittance and a series impedance, the A element of chain matrix is not influenced by these parasitics and is equal to one. This allows immediate characteristic impedance extraction without analyzing any of these transition models. Moreover, one is able to choose the valid model and verify if any of the possible equivalent parasitic elements can be neglected. The next feature is that the chain matrix formulation modified in a specific way allows the extraction of the equivalent position of the reference planes at the probe-tips. Equivalent means that their locations are equal to their physical locations in case of negligible influence of the series parasitic impedance (usually parasitic inductance). This is very interesting property if we take into consideration that the exact position of the probe tips and reference planes are not known. The last feature allows us to extract the characteristic impedance of the lines without modeling the transition structure under assumption that the condition A=D of its chain ABCD matrix (the model of a reciprocal structure is symmetric) is approximated with good accuracy. Such a formulation takes automatically into consideration even a distributed nature of the transition, which can be of importance at mm-wave frequencies. All the equivalent elements values of the models are extracted from analytical equations for every frequency point. Thus their frequency behavior can be investigated and possible validity of the model (constant equivalent element values) proved. I General Waveguide Theory With the increasing use of planar transmission lines, junctions between waveguides supporting lossy hybrid modes have become common. In this case the classical microwave circuit theories fail. The classical waveguide theory fails to
appreciate the subtleties of the S matrix, which relates the traveling wave intensities instead of the waveguide voltages and currents. The VNA calibrated with TRL method measures the unique S matrix, which relates the actual traveling waves, not some arbitrary voltage or current quantities [3]. In effect this S-matrix is normalized to the characteristic impedance at the test ports. Power wave based S-matrix can be very often found in the literature. These are mathematical artifacts, which are different from actual traveling waves propagating in waveguides. VNA measurements do not determine relations between power waves. The traveling wave intensities are independent of the transverse electric modal field magnitude. Only their phases depend on the modal field phase. In that they are distinct from classical unnormalized voltage S-matrix. With such a definition the S-matrix of a 2-port with reciprocal impedance matrix normalized to two different real reference impedances still shows symmetry. This on the contrary to the classical unnormalized voltage S-matrix, which does not show any symmetry. In effect we can conclude that such defined intensities have a physical meaning, not only mathematical. In case of complex reference impedances only the phase differences of the reference impedances influence the symmetry of the S-matrix. Usually characteristic impedances found on lossy planar transmission lines are complex. In effect unique traveling wave S-matrix of any general structure with reciprocal impedance matrix but connected to different waveguides is not symmetric. In the course of the paper reading we will always refer to the mentioned definition of S-matrix. II 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. 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 off-wafer calibration. Hence the error boxes on both sides of the DUT represent artificial junctions which reflect the differences between two probe-tipline junctions for both off-wafer and on-wafer calibrations. (trans-wafer error boxes). 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.
Fig.1 Two-port measurement problem In most cases it is assumed that the electromagnetic reciprocity condition of a waveguide junction forces its
reciprocity ratio equal to one (defined as a ratio of forward and reverse transmission coefficients). This condition is not necessarily true, as shown in [2]. It does not influence the extraction of the propagation constant only because it involves a product of forward and reverse transmission coefficients of the junction. It does, however, influence the characteristic impedance determination. The reciprocity ratio of our artificial junction is expressed by a ratio of both reciprocity ratios for real off-wafer and on-wafer junctions and involves phases of complex characteristic impedances (unique traveling waves Smatrix) and reciprocity factors of both on- and off-wafer waveguides. In lossless waveguides reciprocities factors and characteristic impedances are real, so the S-matrix is symmetric. For lossy hybrid modes, such as those found in CPW or microstrip lines, the effect of phases of the complex characteristic impedances at low frequencies or even in the wide frequency range for strongly lossy substrates (low resistivity silicon) cannot be neglected. The symmetry of the error box admittance matrix or equivalently determinant ADBC of its chain matrix are not influenced by the differences in their complex impedances. Only the ratio of the reciprocity factors for both on- and off-wafer waveguides affects this property but in most common planar transmission lines [2] it can be assumed to be equal to 1. Thus the problem formulation in terms of chain matrices simplifies the procedure. The resulting S-matrix of the error box can be synthesized from the extracted chain matrix and characteristic impedance of the measured lines. The formulation concerns the whole measured standards at a time and not the probe-tip-line discontinuity only. The lack of assumption of the error box S-matrix symmetry does not allow us to analyze it separately. An alternative solution to omit the general S-matrix asymmetry of our junctions is to normalize it to real reference impedances at both ports (pseudo-waves S-matrix). It demands the complex impedance transform to be calculated, which cannot be represented by a classical “turns ratio” of a conventional impedance transformer. This, in turn, will substantionally complicate the next step of the procedure, namely equivalent circuit modeling of the error box of our transitions represented by an impedance transformed pseudo-wave S-matrix. II.A Calibration to the probe-tips Using the ABCD representation of the THRU and LINE standards one is able to solve for Ar2 Z o product: Ar2 Z o =
Br2
Cr =
) −∆ )
2 χ ± χ 2 − ∆2
o
χ2
2
(1) (2)
BL sinh γ lTHRU − BT sinh γ lLINE =∆ cosh γ lLINE ⋅ sinh γ lTHRU − cosh γ lTHRU ⋅ sinh γ lLINE
(3)
B BL cosh γ lTHRU − BT cosh γ lLINE = =χ Zo cosh γ lTHRU ⋅ sinh γ lLINE − cosh γ lLINE ⋅ sinh γ lTHRU
(4)
AL − cosh γ lLINE − B sinh γ lLINE /(Z o A) 2 B cosh γ lLINE + AZ o sinh γ lLINE + B 2 sinh γ lLINE /(Z o A)
(5)
2 Ar Br =
Ar2 Zo +
( Z = (χ ± 2
∆2
2 r
Zo denotes the characteristic impedances of the lines being measured. AT BT CT DT and AL BL CL DL and Ar Br Cr Dr represent the ABCD matrices of the measured THRU and LINE standards, and error box, respectively. Equations (1-5) are general equations describing a reciprocal junction matrix
ABCD in terms of two measured lines (THRU and LINE). They are independent of any discontinuity model. Now, let us assume that we use the general CGLR error box model of the probe-tip discontinuity as in Fig.2 [1].
Cp
Gp
Rp
Lp
Fig.2 Lumped model of the probe-tip-line discontinuity
From the ArBrCrDr matrix representations of the CGLR error box model one always has Ar=1, which allows immediate extraction of the measured line characteristic impedance from (1) regardless of which parasitic elements of the model really exist. The equations for the equivalent parasitic series inductance and resistance and shunt capacitance and conductance are as follows: Lp = Im R p = Re
C p = Im
∆ 4π f
(6)
∆ 2
(7)
Cr 2π f
(8)
G p = Re ( Cr )
(9)
With such a formulation one can automatically decide which model to use: simple shunt admittance or cascade of shunt admittance and series impedance. II.B Calibration without reference plane shift Typical calibration procedures calibrate the measured standards to the probe-tips and then model the probe-tip-line discontinuity by a lumped model. This involves a reference plane shift from the initial position in the middle of the THRU line to the probe-tips. The location of the probe tips and reference planes is not known exactly because of the limited precision of probe tips positioning and distributed nature of the probe-tip-line discontinuity. We present the procedure using a different formulation of the problem that does not involve the reference plane shift and does not need the relative position between probe-tips and the middle of a THRU line in the extraction process. 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 parasitic series impedance at the probe-tips can be neglected. Otherwise this extracted relative position is different from its physical counterpart by some transmission line length, which models this effect. Such a formulation allows additionally an arbitrary position of the probe-tips on the on-wafer 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 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.3. ‘d1’ denotes a quasidistance between quasi-probe-tip reference plane and the middle of the THRU line. ‘d2’ denotes a quasi-length of the open-ended stub connected in front of quasi-probe-tip
reference plane. It reflects the difference in shunt admittances of the on- and off-wafer open stubs seen at the probe-tips. Quasi Probe-Tip Ref. Plane
THRU line Ref. Plane
Zo, γ, d1 Zo, γ, d2 Fig.3 The equivalent circuit model of the error box
The quasi-length d1 reflects series connection of a physical length of the measured line 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 2 Ar 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
(10) (11)
The parameter " l " denotes the length difference between THRU and LINE standards. AT BT CT DT and AL BL CL DL represent the ABCD matrices of the measured THRU and LINE standards, respectively. Denoting Ar2 Z o product as ψ we arrive at two equations for Br and Cr matrix elements: Br =
Cr =
BT 2
Zo
(12)
ψ
AT − 1 ψ BT Zo
(13)
Ar2 Z o product in case of our error box model is equal to Z o cosh 2 γ d1 and allows to determine the proper choice of the
root in Eq. (1). This choice is made by demanding Re ( A2 Z o ) ≥ 0 . Calculating the ratio Br/Ar leads to the solution
for distance d1: tanh γ d1 =
BT 2ψ
(14)
Then Z o and d2 take the forms as in (15-16): Zo =
tanh γ d2 = − jZo
BT 2 cosh γ d1 ⋅ sinh γ d1
AT −1 B A −1 tanh γ d1 − 2 T 2 = − jZo T − BT 2Zo cosh γ d1 BT Zo
(15) (16)
As an example we demonstrate the procedure for the measurements of 2 different CPW line geometries (Width/Space/GND_Width SET1=100/14/200µm, SET2= 150/20/300µm) on fused silica substrate up to 110GHz by means of HP 8510XF VNA with waveguide coplanar probes (pitch 150µm) and alumina calibration substrate from Cascade Microtech. The probes have been positioned to 25µm beyond the physical beginning of the lines. The length of the THRU and LINE standards are 2mm and 11.5mm, respectively. The extracted lengths d1, d2 for both line sets are shown in Fig.4. They are approximately constant over the whole frequency range.
Fig.4 Extracted d1nad d2 transmission line lengths
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 the effective permittivity of the alumina is considerably higher than that of fused silica. Comparing d1 length with its physical counterpart (975µm) one can see that they differ only by 3µm 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=-2.56pF and C=-1.4pF for SET1 and SET2 lines, respectively. For comparison purposes we have performed similar extraction procedure using the lumped CGLR model from Fig.2 [1] but with our new chain matrix formulation. The extracted averaged parasitic elements 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 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. 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 nature. So, our quasilength d1 also accounts for some amount of the shunt capacitance at the probe tips and explains the differences between extracted shunt capacitance of the lumped model and quasi-shunt capacitance of the distributed model. The determined characteristic impedance, effective εr of both line sets are shown in Fig.5. For comparison purposes, the
HFFS simulated Zo is also drawn. Local non-monotonical behavior of the extracted quasi-lengths d1 and d2 and characteristic impedances can be seen as the influence of mainly probe placement inaccuracy, half wavelength difference between LINE and THRU standard at some frequencies.
Fig.5 Extracted characteristic impedance and effective εr
II.C A=D assumption of the error box chain matrix The last feature allows us to extract the characteristic impedance of the lines without modeling the internal transition structure, even of distributed nature, under assumption that the condition A=D of its chain ABCD matrix (the model of a reciprocal structure is symmetric) is approximated with good accuracy. Using this property and combining it with the equations (1), (2) and (5) one is able to extract the characteristic impedance of the measured lines. The final equation for the characteristic impedance of the lines is as follows: 1 2η ( AL − cosh γ lline − sinh γ lline ⋅η / ∆ ) = 1+ Zc ∆2 ( 2cosh γ lline + sinh γ lline ⋅ (η / ∆ + ∆ / η ) )
(17)
η = χ ± χ 2 − ∆2
(18)
Proper choice of the sing in assumption: Re( Z c ) > 0 .
η
is provided by the following
We arrive at a very important general conclusion: one is able to extract the characteristic impedance of the lines embedded in reciprocal measurement feeding structures based on two line measurements without using of any fixed model of this feeding structure if it is well approximated by a symmetrical model (ABCD matrix fulfill the equation A=D). Such a formulation takes automatically into consideration even a distributed nature of the transition, which can be of importance at mm-wave frequencies. This property can be used to extract the characteristic impedances of microstrip and taper-fed CPW lines with probe tips. III Conclusions Some advantageous features of the chain ABCD matrix formulation in comparison with S-matrix formulation in characteristic impedance extraction with probe-tip calibrations have been presented. Due to formulation in terms of the chain matrix, complex characteristic impedance dependence of the error box reciprocity ratio (or equivalently S-matrix
asymmetry) is easily omitted. It has been shown that in the general CGRL equivalent model of the probe-tip-line transition, A element of its chain matrix is always equal to one and not disturbed by any of these parasitics. Moreover, one is able to verify if any of possible equivalent parasitic elements can be neglected. The modified chain matrix based procedure does not involve the reference plane shift to the probe tips in the extraction process. On the contrary, it enables an extraction of the equivalent position of the probe-tips along the line. This is a very interesting property if we take into consideration that the exact position of the probe tips and reference planes are not known. It has also been demonstrated that the symmetry of the model of the reciprocal feeding structure allows to extract the characteristic impedance of the lines without using of any fixed transition model. Such a formulation takes automatically into consideration even a distributed nature of the transition, which can be of importance at mm-wave frequencies. This property can be successfully used to extract the characteristic impedances of microstrip and taper-fed CPW lines with probe tips. References [1] 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. [2] D.F. Williams, R.B. Marks, "Reciprocity Relations in Waveguide Junctions," IEEE Trans. MTT, vol. 41, pp. 1105-1110, June/July 1993 [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.
