An Efficient Transformer Modeling Approach for mm-Wave Circuit Design

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ “This is an Accepted Manuscript of an article published by Elsevier in: AEU - INTERNATIONAL JOURNAL OF ELECTRONICS AND COMMUNICATIONS on January 2021, available at: https://doi.org/10.1016/j.aeue.2020.153496 .” Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 1 An Efficient Transformer Modeling Approach for mm-Wave Circuit Design Fabio Passosa * , Elisenda Rocaa, Javier Sieirob, Rafael Castro-Lopeza, and Francisco V. Fernandeza a Instituto de Microelectrónica de Sevilla, CSIC and Universidad de Sevilla. C/Americo Vespucio, 28. 41092 Sevilla (SPAIN). E-mail: [email protected], [email protected].es, [email protected], [email protected] b Dept. d’Enginyeria Electrònica i Biomèdica, Universitat de Barcelona, Martí i Franquès, 1, 08028 Barcelona (SPAIN). E-mail: [email protected] Corresponding Author: Elisenda Roca. E-mail: [email protected] Abstract: In this paper, a Gaussian-process surrogate modeling methodology is used to accurately and efficiently model transformers, which are still a bottleneck in radio-frequency and millimeterwave circuit design. The proposed model is useful for a wide range of frequencies from DC up to the millimeter-wave range (over 100 GHz). The technique is statistically validated against fullwave electromagnetic simulations. The efficient model evaluation enables its exploitation in iterative user-driven design approaches, as well as automated design exploration involving thousands of simulations. As experimental results, the model is used in several scenarios, such as the design of an inter-stage amplifier operating at 60 GHz, where the model assisted in the simulation of the transformers and baluns used, and the design of individual transformers and a matching network. Keywords: transformers, machine learning, metamodeling, millimeter-waves integrated circuits, design methodology. * Present address: Instituto de Telecomunicações, - Torre Norte - Piso 10, Av. Rovisco Pais, 1, 1049 - 001 Lisboa, Portugal. E-mail: [email protected] Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 2 1. INTRODUCTION With Today´s demand for high data rate communications, the need for millimeter-wave (mm-Wave) circuits operating in the multi-gigahertz regime is increasing [1]. However, the design of passive devices, such as transformers, used for single-ended to differential conversion (e.g., using baluns) or power transfer and matching between stages (e.g., using inter-stage transformers with center taps) [2]-[6], is still a major bottleneck in mm-Wave circuit design. Their design is still highly dependent on electromagnetic (EM) simulators and designers usually perform iterative EM simulations in order to reach a satisfactory design  a time-consuming process that severely impacts the total design time of these circuits. This high computational cost also prevents the development of circuit optimization strategies, which require a massive number of simulations. To shorten these design times, several lumped-element analytical models that replace the typically lengthy EM transformer simulations have been proposed [7]-[13]. For instance, the lumped-element model based on four scalable inductor -models reported in [7] can predict the performance of transformers with acceptable accuracy. However, it was validated just up to 5 GHz and for 4-port transformers only, such as the one presented in Fig. 1(a). Other works have also used the popular analytical -model for transformers, but only up to 10 GHz and for square transformers [8],[9]. A more complex 2-model was also applied to transformers in the mm-Wave range [10]. Nevertheless, the parameters of this model were derived from both analytical equations and fitting parameters, and, therefore, it is not easily scalable. Moreover, the above-mentioned works [7]-[10] modeled transformers as 4-port devices with two terminals for the primary and two for the secondary, and none included the ability to model transformers with center-tap, like the topologies shown in Fig. 1(b) and Fig. 1(c). However, in RF and mm-Wave circuits, transformers are more frequently used differentially and with center-taps on the primary and/or the secondary. Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 3 Also, the center-tap is strictly necessary when the transformer is used as bias supply. Extracting an analytical model for 6-port symmetric transformers (including the center-taps on the primary and secondary) is more complicated due to the many interactions between the primary and secondary coils. Although lumped-element models that can predict the performance of 6-port transformers in the mm-Wave range have been reported [11]-[13], either they do not take into account some magnetic coupling effects between the windings (hence losing accuracy) or they need prior fabrication to extract the model parameters (hence more difficult to build). Moreover, the accuracy of these models is always evaluated against a few transformer samples and without a detailed error analysis. Also, most analytical models are only valid in a small part of the design space, compromising simulation accuracy outside those regions. Similarly, lumped-element analytical models struggle to be wideband (e.g., from around DC up to multi-GHz), and, therefore, their usage in mm-Wave circuit design may be limited. Therefore, when considering the current state-of-the-art, it is possible to conclude that there is still a need for a fast-to-evaluate modeling technique that is:  highly accurate (with accuracy similar to EM simulation);  valid in the design space of interest;  wideband (e.g., valid from DC up to the mm-Wave regime);  valid for different transformer topologies (e.g., baluns and transformers with 4and 6ports); Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 4  usable with any transformer configuration (e.g., any port connected to ground). This modeling technique would provide not only a fast evaluation of the transformers, but it will enable circuit simulation with accuracy similar to EM simulation for these elements from the early stages of the design process, therefore improving the exploration of transformer’s design space during circuit design and reducing total design time. This work proposes a methodology based on Gaussian-processes that meets the five criteria mentioned above. Gaussian-process metamodeling techniques have been exploited in different fields of electronic engineering. In particular, they have been successfully used for the design of inductors, providing accurate simulation results [14][15] that have also been experimentally validated [16]. As with inductors, one key element to obtain accurate models able to cover the complete design space of transformers, is the development of specific modeling strategies adapted to the intrinsic properties of transformers. The modeling methodology presented here allows a very efficient and accurate modeling of the transformer’s S-parameters. By modeling S-parameters instead of using equations to calculate the component values of a lumped-element model, the modeling technique is equation-independent and, therefore, can be used for transformers with any number of ports and any configuration. Furthermore, as it will be shown, the model is built from accurate full-wave EM simulations. The model is also wideband and valid in the design space of interest, and it can be easily incorporated in the electrical simulations of the circuits. (a) (b) (c) Fig. 1. Different transformer configurations. (a) 4-port transformer. (b) 6-port balun. (c) 6-port inter-stage transformer. Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 5 The rest of this paper is organized as follows. Section 2 briefly presents the main transformer design parameters and performances. In Section 3, the basics of Gaussian-process surrogate modeling is presented, and the specific modeling strategy to be followed to generate accurate and efficient transformer models is discussed. Section 4 experimentally demonstrates the construction and validation of a transformer model in a 65-nm CMOS technology. In Section 5, the suitability of the model to the design of transformers, matching networks and amplifiers is demonstrated, as well as its use in transformer and circuit optimization processes. Finally, in Section 6, conclusions are drawn. 2. TRANSFORMER PARAMETERS AND PERFORMANCES Transformers are devices composed of two coils: primary and secondary. Typically, in the mmWave range, these components are formed using one-turn coils, which are built using the two upper metal layers with an intermediate dielectric layer. 2.1.1 Geometric Parameters Fig. 2 shows the top-view of a 6-port octagonal symmetrical transformer. The geometry of this transformer is usually defined by four geometric parameters: the inner diameter of the primary (DinP) and secondary (DinS) and turn width of the primary (wP) and secondary (wS) (with the number of turns of primary (NP) and secondary (NS) being usually one). Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 6 2.1.2 Coil Performances Some of the most relevant transformer performances are the inductance of the primary and secondary coils, LP and LS, respectively, and the quality factor of the primary and secondary coils, QP and QS, respectively. In this work, the performance parameters of the primary coil of 6-port structures, as the ones in Fig. 1(b) and Fig. 1(c), are calculated as [17]: 13 23 31 32 11 12 21 22 33 ( )·( ) () 2 2·(1 ) diffP S S S S S S S S SS       (1) 0 1 1 diff diff P dP P iff S ZS Z   (2) Im( ) 2 diffP P Z Lf   () Re( ) diffP P diffP Im Z QZ  (3) while for the secondary coil are calculated as, 44 45 54 55 46 56 64 65 66 ( ) ( )·( ) 2 2·(1 ) diffS S S S S S S S S SS        (4) 0 1 1 diff diff S dS S iff S ZS Z   (5) Im( ) 2 diffS S Z Lf   () Re( ) diffS diff S S Im QZ Z  (6) Fig. 2. Example of a 6-port inter-stage octagonal symmetrical transformer illustrating the location of each port. Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 7 where Syy are the S-parameter components of the 6-port structure (being y the port number), Z0 is the adopted port impedance and ƒ is the frequency. 2.1.3 Transformer Performances Other important performance parameters of transformers are the transmission gain and the coupling factor (k), which can be described as the ability of the transformer to transfer power and its magnetic strength, respectively. The transmission gain is given by S21 when the transformer is in a 2-port configuration (S41 in the 6-port transformer of Fig. 2). The mutual reactive (kim) and mutual resistive (kre) coupling factors are given by [18]: 14 41 11 44 Im( )Im( ) Im( )Im( ) im ZZ kZZ  14 41 11 44 Re( )Re( ) Re( )Re( ) re ZZ kZZ  (7) where Zyy represent the impedance parameters, which can be easily calculated from the Sparameters. 3. TRANSFORMER MODELING Surrogate modeling is an engineering method used to model complex systems that cannot be easily measured either by experiments or simulations [19],[20]. Therefore, an approximate model of the outcome, based on the output response to a selected number of input samples, is used instead. In this work, this response is constructed with Gaussian Process (GP) models. Therefore, Section 3.1 will briefly present the fundamentals of this technique. Then Section 3.2 describes the basic steps that need to be followed to generate the surrogate models. However, in order to obtain highly accurate model for transformers using a reduced number of samples, different modeling strategies have been developed that exploit the knowledge of the underlying mathematical method and the electromagnetic problem itself. The complete methodology including all the modeling strategies is presented in Section 3.3. Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 8 3.1 Brief introduction to Gaussian Process Models This subsection presents a brief mathematical introduction to GP models; much deeper details can be found in the specialized literature, e.g., [22]. Assuming a set of samples 𝒙=(𝒙1,⋯,𝒙𝑛) and observed responses 𝒚=(𝒚1,⋯,𝒚𝑛), GP predicts a function value at a given design point 𝒙∗ by modeling 𝒚(𝒙∗) as a stochastic process with mean μ and Gaussian distributed error 𝜀(𝒙) with variance 𝜎2. Assuming continuity in the function, the values 𝒚(⋅) of two points 𝒙𝑖 and 𝒙𝑗 should be close if they are highly correlated. In this work, we use the Gaussian correlation function R to describe the correlation between two variables. The correlation between the errors at 𝑥𝑖 and 𝒙𝒋 are given by 𝐶𝑜𝑟𝑟(𝜀(𝒙𝒊),𝜀(𝒙𝒋))=exp (−∑ 𝜃𝑙|𝑥𝑖𝑙 −𝑥𝑗𝑙|𝑝𝑙) 𝑑 𝑙=1 (8) where 𝜃𝑙>0 is the correlation parameter in the l direction, pl determines the smoothness in the l direction, and d is the dimension of x. The values of μ, σ,  l and pl are determined by maximizing the likelihood function that equals 𝒚=𝒚𝑖 at 𝒙=𝒙𝑖 for 𝑖 =1,2,… ,𝑛. The optimal values of μ and σ can be found by maximizing the likelihood function: ℎ= 1 (2𝜋)𝑛 2 ⁄(𝜎2)𝑛 2 ⁄|𝑹|1 2 ⁄𝑒𝑥𝑝(− 1 2𝜎2(𝒚−𝟏𝜇)𝑇𝑹−1(𝒚 − 𝟏𝜇)) (9) where the μ and σ values that maximize the likelihood function are: 𝜇 =(𝟏𝑻𝑹−1𝒚)−1(𝟏𝑇𝑹−1𝟏) (10) 𝜎2=(𝒚−𝟏𝜇)𝑇𝑹−1(𝒚−𝟏𝜇) 𝑛(11) where 1 is an all-ones 𝑛×1 vector. Each element in the 𝑛×𝑛 matrix R is given by equation (8). Once the values of  l and pl that maximize (9) are found, the estimated values of μ and 𝜎2 are obtained from (10) and (11). Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 15 describe the 6-port structure (a matrix of 6x6 S-parameters). Although a considerable number of models needs to be created, the time needed for the generation of the 72 models is only 21 seconds for the transformer model and 4.5 seconds for the balun model (for each frequency point). The difference between the model creation times comes from the exponential complexity growth of GP models with the number of training samples. Besides, these models have to be created only once for each WF and, afterwards, can be saved for further usage (see Fig. 3). Furthermore, evaluation times of the stored models will be negligible (in the order of ms or less), thus, being extremely beneficial for improving the efficiency of mm-Wave circuit design simulation. The model was created to operate up to 150 GHz (the SRFx filtering was set at 200 GHz). The model validation was performed on the previously generated test sets and at three different (a) (b) (c) (d) Fig. 4. Analysis of the model error of the inductance and quality factor for the transformers at three different frequencies using box-and-whisker plots. (a) (b) (c) (d) Fig. 5. Analysis of the model error of the inductance and quality factor for the baluns at three different frequencies using box-and-whisker plots. Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 16 frequencies (1 GHz, 60 GHz and 150 GHz). The error statistics of the results can be observed in Fig. 4 and Fig. 5. Although the model predicts S-parameters, in order to reduce the number of plots, these representations are limited to the inductance and quality factor, obtained from (3) and (6) and not to each component of the S-parameters. For a better statistical assessment of the inductance and quality factor errors provided by the surrogate model, box-and-whisker plots are presented for both the primary and secondary coils. For the transformers, Fig. 4(a) and Fig. 4(b) show the relative error in percentage for the inductance of the primary and secondary, respectively. For all tested frequencies, the mean relative error (given by the red horizontal line in the box-andwhisker plots) is lower than 1% when compared to full-wave EM simulations, which is extremely accurate for such high frequency modeling. Regarding the quality factor, Fig. 4(c) and Fig. 4(d) show that the mean relative errors are always below 5%. However, some samples achieve higher errors (14 transformers have errors higher than 20% at 150 GHz). Nevertheless, it should be taken into account that, at 150 GHz, the quality factor of these transformers is quite low and both, Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 17 inductance and quality factor, change quickly with frequency. Therefore, the relative error in percentage increases and may be perceived as high whereas indeed it can be neglected, especially because such transformers will never be used at that frequency. For the baluns, the same considerations can be made from Fig. 5. The error for inductance is always lower than 1% and the error for the quality factor is usually less than 4%. (a) (b) Fig. 6. Comparison between full-wave EM simulation (LxEM and QxEM) and the model (LxM and QxM) for inductance and quality factor for two transformers. Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 18 In Fig. 6(a) it is possible to observe a plot of Lx and Qx versus frequency for a randomly chosen transformer and Fig. 6(b) shows the transformer with the highest error from the statistical validation analysis. As previously said, although the error at 150 GHz is more than 30%, it can be seen in Fig. 6(b) that it may be neglected because the transformer will preferably not be used at that frequency since Q is around zero. The same has been done in Fig. 7 for two baluns, observing a very good matching over the entire frequency range. 5. MODEL APPLICATION TO CIRCUIT DESIGN 5.1 mm-Wave Circuit Design In this Section, the model is used to efficiently and accurately design transformers during the design of an inter-stage amplifier operating at 60 GHz. The simplified schematic of the amplifier is shown in Fig. 8. The chosen amplifier topology has two stages and uses three transformers in (a) (b) Fig. 7. Comparison between full-wave EM simulation (LxEM and QxEM) and the model (LxM and QxM) for inductance and quality factor for two baluns. Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 19 three broadband matching networks: the input-balun T1, the inter-stage transformer T2 and the output-transformer T3. Each gain stage (Gm1 and Gm2) is built with two transistors connected in common-source (CS) configuration for signal amplification (named M1 for stage 1 and M2 for stage 2). The gate-drain capacitance CGD is responsible for reverse isolation and stability degradation of the CS amplifiers. Therefore, in order to improve stability and achieve stable designs, a MIM capacitance is connected between the gate and drain of each transistor in order to neutralize the CGD parasitic capacitance [24]. These capacitances are called stability capacitances (Cstab), and are included in both gain stages. This configuration enables unconditional stability in differential mode, high reverse isolation and high gain at mm-Wave [24]. The gate biasing of each stage is achieved through the center-tap of the secondary coil of the transformer. The circuit is operating with a supply voltage VDD=1.2 V. The first step to design the amplifier is to set the active stages for optimal power consumption, and, afterwards the input and output impedances of each stage can be simulated. Next, the impedance matching networks are designed using the model instead of EM simulations, which leads to a more efficient design process. The performances of the designed amplifier can be seen in Fig.9, whereas the design variables values are given in Table 1. The circuit was electrically simulated using SpectreRF, but the transformers were simulated with the models developed for each topology (i.e., 6-port balun model for T1 and 6-port transformer model for T2 and T3) and included in the simulation as S-parameter Fig. 8. Simplified schematic of the 2-stage inter-stage amplifier. The first stage of the amplifier is sourcedegenerated for better input-matching. Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 20 files. The amplifier achieved 21 12.79 dB,S 11 10.68 dBS and 22 11.16 dBS at 60 GHz. The output-referred compression point is 1dB 6.93 dBm.P In order to inspect the amplifier performance deviations due to the model errors, all the transformers were EM simulated and the amplifier was re-simulated. The performance deviations between using the model or using a full-wave EM simulator are depicted in Fig. 9 and are detailed in Table 2 for the frequency of 60 GHz. These deviations are very small (below 2%), demonstrating that the new modeling approach provides a very fast and accurate simulation model for circuit design. Table 1. Values of the Circuit Components for the Designed Amplifier. C1 C2 C3 C4 C5 C6 Cstab1 Cstab2 R1 R2 64.8fF 53.9fF 9.47fF 14.3fF 20.2fF 167fF 3.48pF 53.9fF 1.15k 556.8 LG1 , LG2 LS M1 M2 T1 T2 T3 L=48pH Q=28.69 L=62pH Q=21.3 w=0.7µm l=60nm nf=26 m=2 w=1.5µm l=80nm nf=16 m=6 LP=75.5pH LS=79.8pH QP=6.24 QS=5.98 LP=131.1pH LS=74.8pH QP=43.46 QS=20.09 LP=68.4pH LS=50.4pH QP=28.04 QS=14.03 (a) (b) Fig. 9. Performance comparison of he designed amplifier: (a) Gain and input matching. S11M, S21M and S22M represent the performances of the amplifier with the transformers simulated with the model and S11EM, S21EM and S22EM represent the performances of the amplifier with the transformers electromagnetically simulated. (b) Output power (POUT) as a function of the input power (PIN). Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 21 5.2 Transformer Synthesis The potential of the modeling approach presented in Section 3 can also be demonstrated when a given transformer is needed for an application. The model can then be used with SpectreRF (used to simulate the transformer with the desired configuration) connected to an optimization algorithm. Furthermore, by simulating the transformer in SpectreRF, the desired matching loads can be set and, therefore, parameters such as impedance transformation can be accurately considered during simulation. As an example, an optimization was performed using the single-objective stochastic optimization algorithm Particle Swarm Optimization (PSO) [25], with the objective of designing a transformer for the automotive radar E-Band (77-81 GHz), with the following characteristics: 10, 90 pH 110 pH, 120 GHz 10, 75 pH 85 pH, 120 GHz P P P S S S Q L SRF Q L SRF         (14) and the optimization objective was to maximize the sum of QP and QS. The optimization was performed with 40 particles and 50 iterations and took 2.75 minutes of CPU time. The achieved transformer has DinP =50 µm, DinS =26 µm, wP=10 µm and wS=5 µm. To assess the accuracy of the model, the obtained transformer was EM simulated and its performance was compared to the model. The comparison of the frequency behavior between fullwave EM simulation and the model can be seen in Fig. 10, where it is possible to observe that the model accurately predicts the inductance and quality factor of the synthesized inductor over the entire simulated frequency range (1 to 150 GHz). A more detailed analysis of the model error at Table 2. Performances of the Designed Amplifier Using the Transformer Models vs. EM Simulation S11M (dB) S11EM (dB) S11 (%) S21M (dB) S21EM (dB) S21 (%) S22M (dB) S22EM (dB) S22 (%) -10.68 -10.57 1.07 12.79 12.86 0.49 -11.16 -11.35 1.73 Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 22 77 GHz is shown in Table 3, where it is observed that the errors are less than 2% for the inductance of both the primary and the secondary and less than 3.5% for the quality factor. Hence, once again, it can be concluded that the model is accurate and proves to be a very useful resource for mmWave designers. 5.3 Filter Synthesis for Broadband Matching Network Apart from the transformer-based resonators used for matching, one of the most common gainbandwidth enhancement techniques in mm-Wave is the usage of high order filters [24]. There, a matching network synthesis method was proposed based on analytical equations. However, after synthesizing the network, some series inductors or shunt capacitors still needed to be added during circuit design because the analytical equations do not provide accurate modeling. In this section, the design of the fourth-order matching network shown in Fig. 11 is considered, using the proposed modeling technique for transformers. In this case, the multi-objective optimization algorithm NSGA-II [26] is used to explore the design trade-offs of this filter topology. Fig.10. Comparison between full-wave EM simulation (LxEM and QxEM) and the model (LxM and QxM). Table 3. Performances of the Synthesized Transformer at 77 GHz. EM-Simulation vs. Model Parameters (m) LPM (pH) LPEM (pH) L (%) QPM QPEM Q (%) DinP=50 wP=10 DinS=26 wS=5 109.5 110 0.45 31.43 32.51 3.33 LSM (pH) LSEM (pH) L (%) QSM QSEM Q (%) 78.52 77.01 1.96 15.46 15.00 3.06 Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 23 For this example, since a single-to-differential transformation is desired, the balun is the preferred topology. Nevertheless, in order to demonstrate that the optimization can be performed with any other structure, an additional optimization is performed using the transformer topology. Both optimizations were performed with the objective of achieving filters operating around 60 GHz and with a bandwidth (BW) of 10 GHz, with the following constraints: 11 10 dB 10 GHz 100 GHz 100 GHz PS S BW SRF SRF     (15) The optimization objectives were: 21 maximize @60 minimize S Area GHz (16) For this specific example, the network is designed for an impedance transformation from 50 to 400 of resistance and 20fF of reactance. The optimizations used 40 individuals and 50 generations and lasted 23 minutes of CPU time. Fig. 12 shows the results of both optimizations, where each symbol represents a fully-designed fourth-order network. It is possible to conclude that matching networks using baluns are able to achieve lower areas, however with higher losses. This result is not surprising because, in order to minimize area, smaller baluns are used, which have lower quality factors and, therefore, higher losses. In order to achieve the desired performances, transformers need to be larger (therefore, Fig. 11. Broadband input matching using a fourth-order filter for bandwidth enhancement of mm-Wave circuits. Manuscript ID AEUE_153496 DOI:10.1016/j.aeue.2020.153496. 24 larger filter area), but have lower losses (by increasing the distance between coils). For illustration’s sake, the performances of the matching network using the balun that achieved the lowest area are illustrated in Fig. 13. The balun was electromagnetically simulated and the filter was re-simulated to assess the performance deviations due to the usage of the model (represented in Fig. 13 and detailed data at 60 GHz in Table 4). It is possible to conclude, that due to the outstanding accuracy of the balun model, negligible deviations are achieved when compared to a full-wave EM simulator (less than 0.1% error). 6. CONCLUSIONS A surrogate modeling strategy has been proposed and exploited for the modeling of several transformer topologies operating at the mm-Wave regime. The models have been statistically Fig. 12. Trade-off S21 vs. √Area for the filter topology in Figure 11. Fig. 13. Performances of the synthesized fourth-order wideband matching network which achieved lowest area using a balun. S11M and S21M represent the performances of the matching network with the transformer simulated with the model and S11EM and S21EM represent the performances of the matching network with the electromagnetically simulated transformer.