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SECTION I

THERE is a strong need for device integration and measurement techniques for submillimeter wave applications, such as earth observation instruments and Terahertz (THz) imaging systems [1], [2]. Device modeling and design verification require high quality $S$-parameter measurements. Vector network analyzers (VNAs) are now available up to THz frequencies [3], [4], [5].

For THz applications, membrane supported terahertz monolithic integrated circuits (TMICs) have proven advantageous from aspects such as low loss and low parasitic capacitances in devices [6], [7], [8]. For characterization of membrane supported circuits, traditional on-wafer probing techniques cannot be used, because the enclosing package environment defines the electrical properties. Therefore, a method was developed for $S$-parameter characterization of these circuits in the WR-03 frequency band (220 GHz–325 GHz) [9] based on thru-reflect-line (TRL) calibration [10].

However, initial measurements in [9], [11] showed large uncertainties and repeatability problems. It was unclear whether this was caused by drift in the VNA, repeatability in waveguide connections, assembly tolerances or problems with the grounding of membrane circuits. Thus, a sensitivity analysis is needed to aid the understanding of these issues.

The impact of waveguide flange misalignment on VNA measurements has been studied in [12], [13]. However that work focused on analyzing reflection errors. To the authors knowledge there is only one paper on the propagation of errors through the TRL algorithm [14]. However, that paper does not treat the situation where an embedding structure is included in the DUT and the TRL standards.

In this paper, we present a numerical sensitivity analysis on TRL calibrated $S$-parameter measurements of membrane circuits mounted in E-plane split waveguide blocks. The impact of waveguide and membrane circuit misalignment, and waveguide dimension mismatch is propagated through the TRL calibration to a set of residual errors. The errors with the largest impact are also investigated analytically.

SECTION II

The calibration was done with a membrane circuit TRL kit [9], [11].

Fig. 1 shows the waveguide to membrane circuit transition designed for the WR-03 waveguide band. The membrane circuits were fabricated on 3 $\mu$m thick GaAs [9]. The membrane circuit TRL standards are denoted m-thru, m-line and reflect. The nominal dimensions of the waveguide blocks are summarized in Table I.

Fig. 2 shows the E-plane split block with MIL-DTL-3922/67D waveguide flanges. The split block design in [9] was improved by recessing the area around the connecting surfaces. The recess improved the beam lead to ground connection as well as the yield of the circuit assembly.

Fig. 3(a) shows a photo of an E-plane type split block where a membrane circuit is mounted. The circuit is aligned in a predefined channel and is suspended in an air channel formed by two symmetrical block halves. The beam leads that are clamped in the blocks provide mechanical support as well as electrical ground connection. The split block assembly incorporates an aligning feature to aid the assembly. As shown in Fig. 3(a), with the membrane shape matching the channel in both $X$- and $Z$-directions. Fig. 3(b) shows a schematic of the shape of the semiconductor membrane circuit, the membrane circuit is slightly wider near the interface between the waveguide and the air channel which is indicated by the red rectangle. The channel was designed to be 20 $\mu$m wider than the membrane and 20 $\mu$m shorter than the distance between the alignment points, to accommodate for manufacturing tolerances of mechanics and circuits.

A test structure using two quarter wavelength shorted stubs,Fig. 4, was characterized to investigate the quality of the ground connections [11]. The structure was measured before and after recessing the block surface. Fig. 4 shows a better response for the case using a recessed block.

SECTION III

The sensitivity analysis is split in two parts. First, a single waveguide block carrying a membrane circuit thru calibration standard (m-thru) is simulated for a number of tolerance cases. Second, the tolerances (from Table II) are applied to all membrane circuit calibration standards and their $S$-parameters in the waveguide interface are calculated. These $S$-parameters are then used in a Monte Carlo analysis of a TRL calibration. The TRL calibration results in a calibration set with a reference plane in the membrane circuit. Each calibration set is compared to the calibration set for the nominal circuit to obtain the residual errors for each tolerance case.

To give an idea of the impact of the different tolerances the $S$-parameters of a single waveguide block carrying an m-thru were simulated for each tolerance as described in Section III.C. The $S$-parameters have the reference plane in the waveguide interface.

The TRL sensitivity analysis is performed by running the TRL algorithm with simulated data. The membrane circuit calibration standards including the waveguide blocks were simulated in an electromagnetic (EM) simulator. Perturbations were applied to the dimensions and alignment of the waveguide and membrane circuits. The simulations were partitioned in three parts: the waveguide flange interface with the major part of the waveguide, the waveguide to membrane circuit transition, and the membrane circuit TRL standards. All perturbations described in Section III-C were applied to each TRL standard, one perturbation at a time. Each perturbation was sampled at $0, \pm u_{x}$ where is the value of the perturbation in Table II. Only the waveguide width was sampled differently with samples every $\mu$m because [15] indicates that it can easily be the most important source of uncertainty for transmission and reflection phase. Each perturbation was applied to each of the calibration standards while keeping the other standards nominal. Giving three groups of calibration sets where each standard was perturbed.

An approach based on [16] was used to obtain residual errors. However, instead of evaluating in terms of $T$-parameters, we chose $S$-parameters to get results as residual errors on $S$-parameter form. Also the TRL algorithm is on $S$-parameter form [17]. The procedure to obtain the residual errors is illustrated in Fig. 5, where $e_{a}$ and $e_{b}$ are error boxes representing the nominal configuration, $\mathhat{e}_{a}$ and $\mathhat{e}_{b}$ are the perturbed error boxes, and $\Delta e_{a}$ and $\Delta e_{b}$ are the residual errors. Thus $\Delta e_{x}$ are obtained by deembedding [18] $e_{x}$ from $\mathhat{e}_{x}$. The terms in $\mathhat{e}_{x}$ are often called residual errors [19] and given names that correspond to the model parameters of a reflectometer, directivity, match and tracking.

The residual directivity at each port is given by TeX Source $$D_{1} = \Delta e_{a, 11}\ {\hbox{and}}\ D_{2} = \Delta e_{b,22}. \eqno{\hbox{(1)}}$$ The residual match is given by TeX Source $$M_{1} = \Delta e_{a,22} \ {\hbox{and}}\ M_{2} = \Delta e_{b,11}. \eqno{\hbox{(2)}}$$ Reflection tracking is caused by changes in the product of the transmission terms in $e_{a}$ and $e_{b}$ TeX Source $$R_{1} = \Delta e_{a,12} \Delta e_{a,21}\ {\hbox{and}}\ R_{2} = \Delta e_{b,12} \Delta e_{b,21} \eqno{\hbox{(3)}}$$ and transmission tracking TeX Source $$T_{1} = \Delta e_{a,21} \Delta e_{b,21}\ {\hbox{and}}\ T_{2} = \Delta e_{a,12} \Delta e_{b,12}. \eqno{\hbox{(4)}}$$

Due to symmetry in the TRL algorithm we define the transmission tracking as $T = T_{1} = T_{2}$.

To obtain the total residual errors we cascade the residual errors for each simulated TRL calibration. First by combining all combinations for each kind of perturbation TeX Source $$E_{a, {\rm Y}} = e_{a, {\rm Y}, {\rm thru}} \otimes e_{a, {\rm Y}, {\rm reflect}} \otimes e_{a, {\rm Y}, {\rm line}} \eqno{\hbox{(5)}}$$ where $\otimes$ indicates cascade connection, Y indicates which kind of perturbation and for $e_{a, {\rm Y}, {\rm x}} {\rm x}$ indicates that the $x$ standard was varied according to $Y$ and the other standards kept at nominal in the simulated TRL calibration. To obtain the total residual error, all the $E_{a, {\rm Y}}$ were randomly sampled 500 times and cascaded TeX Source $$E_{a, {\rm tot}} = E_{a, \Delta X_{\rm flange}} \otimes E_{a, \Delta Y_{\rm flange}} \otimes \cdots. \eqno{\hbox{(6)}}$$ giving 500 samples for each residual error.

This section lists the tolerances accounted for in the sensitivity analysis. All tolerances are summarized in Table II. The orientation of the $X$-, $Y$-, and $Z$-directions, as discussed below, are indicated in Figs. 2 and 3.

Misalignments in the $X$- and $Y$-directions are considered. Angular misalignment was not considered because rectangular waveguide interfaces are tolerant to angular misalignment [14]. The specification of the MIL-DTL-3922/67D interface allows as much as 63.5 $\mu$m deviation due to the combination of the alignment hole and the alignment pin tolerances [20]. However, for this work tighter tolerances were used for the alignment pins, a misalignment of ${\pm} {\hbox{50}}\ \mu$m was used for both X- and Y-directions.

Based on our experience the misalignment between the membrane circuit and channel is considered to be 5 $\mu$m in the $X$- and $Z$-directions.

The misalignment between the two halves of the blocks can be in either the $X$- or $Z$-direction. The misalignment tolerance was set to ${\pm} {\hbox{10}}\ \mu$ m.

The manufacturing of membrane circuits has good repeatability in dimensions where the resolution is set by the optical lithography process. All the membrane circuit TRL standards and membrane circuits were fabricated in the same batch to minimize variations between circuits. Therefore we do not include these tolerances in the analysis.

To estimate the manufacturing tolerances of the block the waveguide aperture was visually inspected under microscope after assembly of the membrane circuits. Fig. 6 shows four conditions: (a) the rectangular waveguide were slightly deformed in the bottom half block, where the width was narrower on the top; (b) good condition; (c) a clear scratch was observed on the surface and the waveguide aperture was damaged; (d) the two half blocks were misaligned horizontally.

The width $(w_{\rm wg})$ and height $(h_{\rm wg})$ of the waveguide apertures were measured on four blocks with a measurement microscope. $h_{wg}$ was smaller than the nominal value and varied from 410 to 425 $\mu$ m. Meanwhile, $w_{\rm wg}$ was larger than the nominal value of 864 $\mu$m and varied from 880 to 890 $\mu$ m. Fig. 7 shows a microscope picture of the backshort and the channel. The width of the channel $w_{\rm ch}$ was approximately 293 $\mu$m (designed for 300 $\mu$ m).

The effect of dimensional errors of the waveguide and channel on VNA measurement is considered. Based on the actual measured dimensions, the tolerance in channel width was set to ${\pm} {\hbox{10}}\ \mu$m compared to the nominal design, and combined with the tolerance of ${\pm} {\hbox{5}}\ \mu$m in both width and height of the waveguide. The height and width tolerance for the waveguides used in the simulations were set to ${\pm} {\hbox{5}}\ \mu$m because the variation measured between the blocks were of this magnitude and for our purposes it is the variation between the blocks that causes problems during measurement. Because the calibration procedure only requires the blocks are identical not that they exactly conform to the waveguide interface standard.

SECTION IV

Fig. 8 shows simulations for three different cases: well aligned waveguide apertures, the apertures having 50 $\mu$m misalignment at two ports in $X$- and $Y$-directions, respectively. Ripples, due to flange to flange reflections, were observed in the two misaligned cases. The amplitude of the ripples is larger in the case of misalignment in $Y$-direction, especially at lower frequencies.

The misalignment in $X$- and $Z$-directions of the membrane circuit in the mechanical block was simulated and compared to simulations of a nominal circuit. Fig. 9. shows fairly small changes in the return loss.

Fig. 10 shows the simulation results of the misalignment between two split block halves in both $X$- and $Z$-direction. There is a larger influence on the return loss for the X-misalignment.

The method discussed in Section III-A was used to estimate the residual error caused by membrane circuit and waveguide misalignment using the sensitivity parameters in Table II. Figs. 12 and 13 show the residual directivity and residual match, respectively. The residual directivity and match are small complex numbers causing errors in a way that makes the phase component less interesting than the magnitude, thus we only plot the magnitude. The line representing the 95% percentile for the residual directivity and match ranges between 0.05 and 0.2. In logarithmic scale this corresponds to ${-}{\hbox{25}}$ dB to ${-}{\hbox{15}}$ dB.

Figs. 14 and 15 show that the uncertainty in tracking is larger in the angular direction than in amplitude. The angular uncertainty is higher at low frequency. This is because the phase is more sensitive to waveguide width changes close to the cut-off frequency.

The dominating contributor to the residual tracking errors is the waveguide width error. If we ignore the width error we get a residual tracking error varying approximately ${\pm} 5^{\circ}$ instead of ${\pm} {\hbox{30}}^{\circ}$ at the low frequency end, see Fig. 16 corresponding to approximately ${\pm}{\hbox{1.5}}^{\circ}$/mm of the waveguide section. We also see that the remaining errors cause a bias and are not centered about 0°. This is expected because many of the remaining errors have a quadratic characteristic, i.e., only the magnitude of the deviation is important and not its sign. We would have to improve the waveguide width tolerance about a factor of 6 to get equal impact on the tracking error from the waveguide width error as the other contributions. Alternatively we would have to use a more compact block design by reducing the length of the waveguide sections by a factor of 6 to get the same effect.

The residual errors can be used in an uncertainty budget [19]. However for a complete uncertainty budget additional uncertainty contributions must be analyzed and included, e.g., linearity, frequency error, repeatability. Fig. 17 shows the result of applying the simulated residual errors, using (17) and (18) in [19], to the measurements in Fig. 4. The magnitude uncertainty for $S_{21}$ is large at the band edges where mismatch errors are large. At approximately 260 GHz where the circuit is well matched the uncertainty is very low. The error sources considered in the simulation cause small magnitude variations which would be larger if uncertainty in losses were included in the simulation. The uncertainties in $S_{11}$ look more reasonable but there is a shift in frequency, compared to simulations.

SECTION V

To further understand the impact of the waveguide width and length on the reflection and transmission tracking we have derived an analytical expression for their impact on the phase error since this seems to be the major problem for our structure.

For the derivation we assumed that each calibration standard and the DUT was embedded in a reflection less waveguide transmission line structure with a transmission coefficient given by TeX Source $$\eqalignno{\beta L_{x, y} &= {L_{x,y} \pi\over w_{x,y}} \sqrt{\left({2w_{x,y} f\over C_{0}}\right)^{2}} -1 = {L_{x,y} \pi \over w_{x,y}} \underbrace{\sqrt{\left({f\over f_{c}}\right)^{2}-1}}_{F}\cr &&\hbox{(7)} }$$ where $L_{x,y}$ is the physical length and $w_{x,y}$ is the width of the waveguide section for object $x$ on port $y$, and $f_{c}$ is the cutoff frequency.

By performing the TRL algorithm analytically using nominal values for the embedding structures except for one of the objects we obtain the residual reflection and transmission tracking similarly as for the numerical case. We then combine the contributions for uncertainties caused by each object and obtain the following by linearizing the exponents to get TeX Source $$\eqalignno{{\rm R}_{1} & \approx \exp \left({{\rm i}\pi L_{2}\over w^{2}F} \delta {\rm w}_{r, 2} - {{\rm i}\pi L_{2}\over a^{2}F} \delta {\rm w}_{t, 2} - {2{\rm i}\pi L_{1}\over w^{2}F} \delta {\rm w}_{d,1}\right. \cr & \quad - {{\rm i}\pi L_{1}\over w^{2}F} \delta {\rm w}_{r, 1} - {{\rm i}\pi L_{1}\over w^{2}F} \delta {\rm w}_{t, 1} \cr & \quad -{2{\rm i}\pi F\over w} \delta L_{d, 1} - {{\rm i}\pi F\over w} \delta {\rm L}_{r, 1} + {{\rm i}\pi F\over w} \delta {\rm L}_{r, 2} \cr & \quad \left. - {{\rm i}\pi F\over w} \delta L_{t, 1} - {{\rm i}\pi F\over w} \delta {\rm L}_{t, 2}\right) & \hbox{(8)} \cr \noalign{\hbox{and}} \cr {\rm T}_{1} & \approx \exp \left(- {i\pi L_{2}\over w^{2}F} \delta {\rm w}_{d,2} - {i\pi L_{2}\over w^{2}F} \delta {\rm w}_{t,2} - {i\pi L_{1}\over w^{2}F} \delta {\rm w}_{d,1}\right. \cr & \quad -{i \pi L_{1}\over w^{2}F} \delta {\rm w}_{t,1} - {i\pi F\over w} \delta {\rm L}_{d,1} - {i\pi F\over w}\delta {\rm L}_{d,2} \cr & \quad \left. - {i\pi F\over w} \delta {\rm L}_{t,1} - {i\pi F\over w} \delta {\rm L}_{t,2} \right) & \hbox{(9)}}$$ where $F$ approximately varies between 0.8 and 1.6. Assuming that $\delta L_{x} \approx \delta {\rm w}_{x}$ the terms associated with $\delta {\rm L}_{x,y}$ can be neglected as long as $L_{x} \gg w_{x}$. Giving the expressions TeX Source $$\eqalignno{R_{1} & \approx \exp \left(- {2i\pi L_{1}\over w^{2}F} \delta {\rm w}_{d,1} - {i\pi L_{1}\over w^{2}F} \delta {\rm w}_{r,1} + {i\pi L_{2}\over w^{2}F} \delta {\rm w}_{r,2}\right.\cr & \quad \left. - {i\pi L_{1}\over w^{2}F} \delta {\rm w}_{t,1} - {i\pi L_{2}\over w^{2}F} \delta {\rm w}_{t,2} \right) & \hbox{(10)}\cr \noalign{\hbox{and}} {\rm T}_{1} & \approx \exp \left(- {i\pi L_{1}\over w^{2}F} \delta {\rm w}_{d,1} - {i\pi L_{2}\over w^{2}F} \delta {\rm w}_{d,2} - {i\pi L_{1}\over w^{2}F} \delta {\rm w}_{t,1}\right.\cr & \quad \left. - {i\pi L_{2}\over w^{2}F} \delta {\rm w}_{t,2}\right). & \hbox{(11)} }$$

Since the embedding waveguides at both ports are milled at the same time it makes sense to assume that $\delta {\rm w}_{x,1}$ and $\delta {\rm w}_{x,2}$ are correlated, in the present case $L = L_{1} = L_{2}$ nominally which means the expressions simplify to TeX Source $$\eqalignno{R_{1} & \approx \exp \left(- {2i\pi L\over w^{2}F} \delta w_{d} - {2i\pi L\over w^{2}F} \delta w_{t} \right) & \hbox{(12)}\cr \noalign{\hbox{and}} {\rm T}_{1} & \approx \exp \left(- {2i\pi L\over w^{2}F} \delta {\rm w}_{d}- {2i\pi L\over w^{2}F} \delta {\rm w}_{t}\right). & \hbox{(13)} }$$ Assuming $\delta {\rm a}_{x}$ are uncorrelated we get TeX Source $$\eqalignno{\angle R_{1} & \approx - {2\sqrt 2 \pi {\rm L}\over w^{2}F} \delta{\rm w} & \hbox{(14)}\cr \noalign{\hbox{and}} \angle T_{1} & \approx - {2\sqrt 2 \pi {\rm L}\over w^{2}F} \delta{\rm w}. & \hbox{(15)} }$$ In order to compare to the numerical results only the $\delta w_{t}$ term in (12) and (13) is applicable because the DUT term was not included in the numerical simulations. Using the tolerances in Table II this gives a phase error of approximately 37° at the low frequency end which agrees well with the numerical results.

SECTION VI

We have presented a detailed numerical sensitivity analysis of the TRL calibration in planar membrane circuits. The work was done in the WR-03 frequency band. The analysis considered the impact of waveguide flange misalignment, machining, and assembly tolerances on the TRL calibrated measurements. The flange tolerances introduce ripples in the measurements due to flange to flange reflections. In particular the tracking errors show large sensitivity to variations in waveguide width. There is larger phase sensitivity than magnitude sensitivity in the residual tracking errors.

A simple analytical expression that describes the dominating contributors to the tracking errors has also been derived. Thus we would have to reduce the width tolerance or the length of the waveguide section by about a factor of 6 to get the uncertainty contribution of the waveguide width to be approximately equal to the other uncertainty contributions.

The analytical expression can be used to explore how the errors will scale with wavelength and waveguide size. The waveguide width is approximately proportional to the wavelength in the waveguide, the achievable mechanical tolerances are constant, and the length $(L)$ of the embedding structure is governed by mechanical constraints and is not easily reduced. Thus the phase error $\langle T \propto (1/\lambda^{2})$, which can be useful to estimate if a certain block design with its tolerances, can be scaled to a smaller waveguide dimension with acceptable phase errors.

The authors thank the co-workers at the Department of Microtechnology and Nanoscience at Chalmers University of Technology, especially C.-M. Kihlman for machining the waveguide blocks, Dr. T. Bryllert, Robin Dahlbäck, and N. Wadefalk for helpful discussions.

This work was carried out in the GigaHertz Centre in a joint project supported in part by the Swedish Governmental Agency of Innovation Systems (VINNOVA), Chalmers University of Technology, Wasa Millimeter Wave AB, Omnisys Instruments AB, and SP Technical Research Institute of Sweden, and also supported by the European Community's Seventh Framework Programme [FP7/2007–2013] under Grant Agreement 242424.

T. N. T. Do, H. Zhao, A.-Y. Tang, and J. Stake are with the GigaHertz Centre, Terahertz and Millimetre Wave Laboratory at the Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-41296 Göteborg, Sweden.

P. Sobis is with Omnisys Instruments AB, SE-42132 Västra Frölunda, Sweden, and also with the GigaHertz Centre, Chalmers University of Technology, SE-41296 Göteborg, Sweden.

K. Yhland and J. Stenarson are with SP Technical Research Institute of Sweden, SE-50462 Borås, Sweden, and also with the GigaHertz Centre, Chalmers University of Technology, SE-41296 Göteborg, Sweden (e-mail: jorgen.stenarson@sp.se).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

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