Mixed-mode impedance and reflection coefficient of two-port devices

Progress In Electromagnetics Research, Vol. 130, 411–428, 2012 MIXED-MODE IMPEDANCE AND REFLECTION COEFFICIENT OF TWO-PORT DEVICES T. Carrasco1, *, J. Sieiro1, J. M. Lopez-Villegas1, N. Vidal1, R. Gonzalez-Echevarria2, and E. Roca2 1Department of Electronics, Faculty of Physics, University of Barcelona, Marti i Franques 1, Barcelona 08028, Spain 2Institute of Microelectronics of Seville, CSIC and University of Seville Avda. Americo Vespucio s/n, Isla de la Cartuja, Seville 41092, Spain Abstract—From the point of view of mixed-mode scattering parameters, Smm, a two-port device can be excited using different driving conditions. Each condition leads to a particular set of input reflection and input impedance coefficient definitions that should be carefully applied depending on the type of excitation and symmetry of the two-port device. Therefore, the aim of this paper is to explain the general analytic procedure for the evaluation of such reflection and impedance coefficients in terms of mixed-mode scattering parameters. Moreover, the driving of a two-port device as a one-port device is explained as a particular case of a two-port mixed-mode excitation using a given set of mixed-mode loads. The theory is applied to the evaluation of the quality factor, Q, of symmetrical and nonsymmetrical inductors. 1. INTRODUCTION Currently, most of the RFICs are designed to work in differential configuration due to noise immunity. In fact, the majority of the receiver chipsets available in today’s market, as well as some of the components that conform them, are fully differential [1, 2]. Obviously, it implies that the figures of merit of these devices are better expressed in terms of mixed-mode scattering parameters (Smm), which were first introduced by Bockelman and Eisenstadt [3, 4]. In spite of their widespread use in RFIC design, there is still some misunderstanding about how the differential (common)-mode input impedance Zd(Zc) Received 29 May 2012, Accepted 18 July 2012, Scheduled 20 August 2012 * Corresponding author: Carrillo Tomas Carrasco ([email protected]). 412 Carrasco et al. must be calculated in terms of Smm when a two-port is seen as a one-port device. Thus, it is usually found that the differential reflection coefficient Γdof a two-port device is assumed to be Sdd [5– 8]. Therefore, the differential input impedance is calculated using the following bilineal impedance transformation Zdd = 2Z0 1 + Sdd 1−Sdd (1) where 2Z0is the differential surge impedance. Certainly, Zdd in (1) coincides with the differential term of the mixed-mode Z-parameter matrix of a two-port device; however, as it has been previously mentioned in [9] and [10], a close look to (1) reveals that such expression only matches with Zd(i.e., Zd=Zdd) for fully symmetrical two-port devices. For non-symmetrical devices and taking into account the definition of Sdd, (1) neglects any conversion to a reflected commonmode power wave. Then, special care must be taken when using (1) as an equivalent expression to Zd. For instance, the evaluation of the quality factor Qusing the next definition Q=Im {Zdd} Re {Zdd}(2) should be only applied to symmetrical topologies. For non-symmetrical inductors, e.g., spiral inductors, (2) wrongly estimates Q, due to the fact that the component boundary conditions are wrongly set, thus the common-mode conversion is completely dismissed. A similar misunderstanding can be pointed out when Zcc is directly related to Zc; in this case, any conversion to differential-mode is not considered. To avoid the former problem, [10] and [11] transform the description of the two-port device from S-parameters to Z-parameters. Then, a floating current or voltage source is applied between the input ports of the device, instead of normalized power waves. These boundary conditions lead to a different input impedance definition which can be transformed back to S-parameters. Besides, an equivalent solution is found in [12] and [13] by means of applying the same floating signal sources to the two-port, but such boundary conditions are straightly expressed in terms of S-parameters, instead of transforming to Z-parameters. In both cases, the input impedance found shows the non-symmetrical response of the device. However, up to the author’s knowledge no procedure expressing Γd(Γc) or Zd(Zc) in terms of Smm has been yet proposed. Even more, all previous cases must be understood as particular cases of a general expression based on the definitions of mixed-mode scattering parameters and loads. Within this framework, the rest of this paper is devoted to extend the theory of mixed-mode scattering parameters not only to Progress In Electromagnetics Research, Vol. 130, 2012 413 symmetrical devices, but to non-symmetrical or actual devices. For this reason, in Section 2, a general expression of Γd(Γc) will be obtained which resembles the well-known expressions of Γin (Γout) for a single-ended two-port device. Such definitions will allow to obtain Zd(Zc) in terms of Smm. In Section 3, it is shown that former particular cases of the driving of a two-port device are reduced to the application of a short, open and matched mixed-mode load conditions on the general expression of Γd(Γc). As a practical case in Section 4, an adequate definition of Qwill be obtained, by means of Zd, that allows the direct comparison between symmetrical and non-symmetrical inductors. Finally, the conclusions of this work are presented in Section 5. 2. INPUT REFLECTION COEFFICIENT OF A TWO-PORT DEVICE When attempting to calculate Γd(Γc), it is useful to keep in mind the definition of the input(output) reflection coefficient Γin (Γout) of a two-port device. As it can be seen in Fig. 1, Γin is defined as the quotient between the incident power wave a1and the reflected power wave b1at the input port P1, when a load ZLhas been connected to the output port. Besides, Γout is the reflection coefficient towards the output port P2, when a source impedance ZShas been connected to the input port [14, 15]. Due to the fact that there is a linear transformation between Sand Smm, i.e., S=M−1SmmM[16], the two-port network in Fig. 1 can be represented as a two-port device where the input port and the output port have been substituted by a differential and commonmode ports. At this point, Γd(Γc) can be correctly defined by analogy to Γin (Γout). Consequently, Γd(Γc) is the input reflection coefficient of a two-port device when exciting with a differential (common)-mode power wave meanwhile the two-port device is loaded with a common Sstd ZL a2 a1 b1b2 in P 2 P1 Γ Figure 1. Input reflection coefficient of a two-port device in standard S-parameters. 414 Carrasco et al. (differential)-mode load. 2.1. Differential-mode Input Reflection Coefficient, Γd Figure 2 shows how a differential power wave adis launched towards a two port device meanwhile a common-mode impedance Zc Lis connected to the common-mode port. Smm relates the incident and reflected differential and common-mode power waves by µbd bc¶=µSdd Sdc Scd Scc ¶µ ad ac¶.(3) In this case, the two-port scatters back two power waves bdand bc. The common-mode reflected wave bcreaches the common-mode load Zc L, which reflects a common-mode wave ac. Thereby, it can be written the following relation ac= Γc Lbc(4) where Γc Lis the reflection coefficient associated with Zc L. Replacing (4) in (3) and after some algebra, Γdis expressed as follows Γd=Sdd −|S|Γc L 1−SccΓc L .(5) Notice that, whenever a two-port is completely symmetric and balanced (i.e., S11 =S22 and S12 =S21, which lead to Sdc =Scd = 0 and |S|=SddScc), Γdis equal to Sdd irrespective of the connected load Zc L. In this case, as it has been previously mentioned, Zdmatches Zdd. It is also interesting to rewrite (4) as a function of the incident power wave a1and a2at each port referred to the common ground. From [3], ad(c)and bd(c)read as ad(c)=1 √2(a1∓a2) bd(c)=1 √2(b1∓b2) (6) ZL ac ad bdbc Smm PdPc c d Γ Figure 2. Differential-mode reflection coefficient of a two-port device in mixed-mode S-parameters. Progress In Electromagnetics Research, Vol. 130, 2012 415 where the upper and lower signs hold for the differential and commonmode, respectively. Substituting (6) in (4), the following relation is obtained a1=−a2+√2Γc Lbc.(7) Note that even though a differential power wave adis launched through the two-port device, a1equals −a2only in two cases: 1) Γc L= 0, i.e., the common-mode load is a matched load; 2) bc= 0, i.e., the two-port is purely balanced. For the remaining cases, a1differs from −a2due to the fact that a common-mode power wave acis scattered back by the common-mode load. Keeping in mind the existing linear transformation between Sand Smm parameters, notice the duality of (5) when it is compared with the Γin (Γout) expression of the single-ended analysis of a two-port device Γin =S11 −|S|ΓL 1−S22ΓL Γout =S22 −|S|ΓS 1−S11ΓS . (8) By means of this comparison, one realizes that, as in the single-ended case, three standard loads can be defined: a matched load (Γc L= 0), an open-circuit (Γc L= 1) and a short-circuit (Γc L=−1). Notice that, each of these cases results in a different boundary condition when substituting Γc L= 0, 1, −1 in (4) as it will be analyzed in Section 3. 2.2. Common-mode Input Reflection Coefficient, Γc Whenever a common-mode power wave acis launched towards a twoport device, as it can be seen in Fig. 3, two power waves, bdand bc are scattered back. Now, bdreaches the differential load, Zd L, which reflects a differential-mode normalized power wave ad. The relationship established between adand bdthrough Zd Lis ad bd Smm PdPc ac bc ZL d c Γ Figure 3. Common-mode reflection coefficient of a two-port device in mixed-mode S-parameters. 416 Carrasco et al. ad= Γd Lbd(9) wherein Γd Lis the reflection coefficient of the differential-mode load. A dual expression of (5) can be written when substituting (9) in (3), Γc=Scc −|S|Γd L 1−SddΓd L .(10) Notice that, when the two-port is symmetric and balanced (i.e., Sdc =Scd = 0 and |S|=SddScc), Γcequals Scc irrespective of the connected load. As it has been previously mentioned, only in this case Zcmatches Zcc. By replacing (6) in (9), a1relates to a2as follows a1=a2+√2Γd Lbd.(11) Therefore, even when a common-mode power wave acis launched through the two-port device, a1equals a2only in two cases: 1) Γd L= 0, i.e., the differential-mode load is a matched load; or 2) bd= 0, i.e., the two port is ideally balanced. Otherwise, a1differs from a2. This is due to the fact that a differential-mode power wave adis reflected back by the differential-mode load. Equation (10) represents the dual case of (5). Therefore, three differential mixed-mode load conditions can be defined by means of Γd L= 0, 1, −1. 3. MIXED-MODE DRIVING CONDITIONS Three driving conditions can be defined which lead to different boundary conditions for a two-port device and different expressions of Γd(Γc). In order to explore these driving conditions, it is very illustrative to think about the theoretical realization of a true mixedmode VNA as the one in Fig. 4. Notice that, in contrast to the two-port device in Fig. 3, the device represented in Fig. 4 is a physical realization where the two input ports and the existing common-ground can be associated to either a single-ended or a mixed-mode representation. Actually, the physical realization of a pure-mode VNA (PMVNA) is rather difficult and, even though some works have been conduced toward its consecution [17, 18], current multi-port VNAs implement Bockelman’s formulation to display Smm. In fact, a commercial PMVNA is not yet available. It is also important to notice that the DUT is normally connected to the PMVNA by means of a pair of coupled transmission lines and a ground reference which allows the propagation of differential and common-mode power waves. However, as it has been previously Progress In Electromagnetics Research, Vol. 130, 2012 417 o 180 0o o 180 0o bi 0o o 180 aiDUT Port1 Port2 Z0 Z0Z0 Z0Z0 Z0 VS reflected incident coupler coupler Σ Σ Σ ∆∆ ∆ Figure 4. Pure-mode vector network analyzer. demonstrated in [3], if the even and odd-mode characteristic impedance are chosen to be equal, such reference coupled lines can be considered uncoupled transmission lines. Even more, as it is also mention, there is not restriction for the length of the reference lines, thus zero length transmission lines can be defined and (6) still holds on. 3.1. Matched Load Γd(c) L=0 Whenever a power wave is launched from the Vsgenerator in Fig. 4, a switchable 0◦/180◦hybrid generates either a differential ador commonmode acpower waves. These incident waves can be measured by setting accordingly the 0◦/180◦input at the incident wave hybrid. When the selected power wave reaches the DUT, the incident power wave is scattered back, as well as an opposite mode wave is generated due to the asymmetry of the two-port device. Both modes can be measured by setting the switchable 0◦/180◦reflected wave hybrid. It is important to notice that whenever one mode is selected, the opposite mode is connected to its surge impedance 2Z0or Z0/2 through each hybrid, thus non scattered wave from the loads is allowed. This assumption is similar to connect Z0at the opposite port with respect to the one that is being measured when the measure of S11 or S22 is done by means of a classical VNA. Assuming the condition that the incoming wave is ad, and keeping in mind that Γc L= 0, from (5) Γdreads as Γd=Sdd.(12) In this case, from (4) acis equal to zero. Thus, a1equals −a2. Only when the reflected power wave is absorbed in the common-mode load, the normalized power waves ingoing into the two-port device are equal in magnitude and opposite sign, thus they are pure differential signals. Besides, whenever it is assumed that a1=−a2, this condition always leads to Γd=Sdd, even though acdiffers actually from zero in an actual measurement setup. Then, it is not surprising that, when attempting to calculate Γdby means of a 0◦/180◦hybrid or an equivalent device 418 Carrasco et al. which supposedly generates the boundary condition a1=−a2, Γdis misunderstood as Sdd and any common-mode conversion is directly dismissed. In order to calculate Zd, the bilineal transformation (1) is valid and Zdis equal to Zdd. Likewise, if acis launched by the power generator Vs, and Γd L= 0 is selected at the hybrids, from (10) Γcreads as Γc=Scc.(13) From (9) adis equal to zero, thus a1equals a2. Therefore, if the boundary condition a1=a2is assumed, it directly leads to obtain that Γc=Scc, although adcould actually differ from zero. Then, when Γcis calculated by means of a device which supposedly generates a1=a2, Γcis misunderstood as Scc and any differential-mode conversion is dismissed. In this case, Zccan be calculated by means of the bilineal transformation Zc=Z0 2 1 + Scc 1−Scc (14) where Z0/2 is the surge impedance for the common-mode. Although the normal operation of a PMVNA is the one previously described, the ports Σ and ∆ at the hybrids, where the differential or common-mode load are connected, can be left open or shorted. In these cases, a scattered wave is allowed and Γd(c) Lequals 1 or −1 respectively. 3.2. Γdwhen Γc L=1 Replacing Γc L= 1 in (5), Γdresults as follows Γd=Sdd −|S| 1−Scc .(15) As it has been previously mention, if the device is symmetric (i.e., Scd =Sdc = 0 and |S|=SddScc), Γdequals Sdd. It is also interesting to notice that if the device is also floating, as the balanced antenna discussed in [10, 19] (i.e., Scc = 1 and |S|=Sdd), Γdcalculated by means of (15) results in an indetermination. This result was previously mentioned in [20], but now by using (5) the indetermination is naturally solved and Γdresults in Sdd. Once Γdhas been obtained, Zdcan be easily calculated by means of the bilineal transformation Zd= 2Z0 1+Γd 1−Γd .(16) However, it is also very illustrative to obtain Zdby using circuit theory, i.e., writing the voltage and current signals at each port in Progress In Electromagnetics Research, Vol. 130, 2012 419 terms of mixed-mode S-parameters. Proceeding in this way, both Smm and circuit theory are shown to be mathematically equivalent and, additionally, the goodness of the bilineal impedance transformation is highlighted. Thus, when Γc L= 1, (7) can be rewritten as a1=−a2+√2bc.(17) As it has been previously mentioned, a1differs from a2except when the two port is purely symmetric. Otherwise, replacing bcby (6) and by means of the following expressions [3] Vi=pZ0(ai+bi) Ii=1 √Z0 (ai−bi),(18) (17) can be written as I1=−I2. It means that the normalized power waves can be replaced by a floating current source as it is shown in Fig. 5 and the result is equivalent to (15). Consistently, the condition I1=−I2implies that there is no common-mode current flowing into the device; thus, from the point of view of the common-mode, it can be seen as an open load, i.e., Γc L= 1. In order to calculate Zd, both currents can be expressed in terms of S-parameters as follows I1=1 √Z0 (a1−b1) = 1 √Z0 [(1 −S11)a1−S12a2] I2=1 √Z0 (a2−b2) = 1 √Z0 [−S21a1+ (1 −S22)a2]. (19) Thus, a2can be written in terms of a1: a2=−a1(−1 + S11 +S21) −1 + S22 +S12 =−a1 Scc +Scd −1 Scc −Scd −1.(20) Note that for a symmetric and reciprocal two-port device (i.e., Sdc = Scd = 0), (20) results in a2=−a1. Otherwise, the ingoing waves are different at each port, and such difference depends on Scc and Scd which convey a common-mode conversion. Now, by means of (20), the voltage and currents at each node of the DUT can be rewritten as a function of a1as Vi=pZ0a1½±Sdd −Scc + 1 −|S| 1−Scc +Scd +2Scd 1−Scc +Scd ¾(21) I1=−I2=1 √Z0 a1 Sdd −Scc −2Scd + 1 −|S| 1−Scc +Scd (22) 426 Carrasco et al. 5. CONCLUSIONS The theory of mixed-mode scattering parameters has been extended not only to symmetrical devices, but to non-symmetrical or actual devices, finding a general expression for Γd(Γc) that resembles the wellknown expressions of Γin (Γout) for a single-ended two-port device. It has also been shown that the former particular cases of the driving of a two-port device are reduced to the application of a short, open and matched mixed-mode load conditions on the general expression of Γd(Γc). Moreover, such definitions allow to obtain Zd(Zc) in terms of Smm. An analytic connection between scattering-parameters description in both versions, through the use of S-parameters and Smm, and lumped elements description has naturally been used toward the consecution of these expressions. As a practical case, the Qvalue for symmetrical and non-symmetrical inductors has been obtained using the definition of Zd, illustrating the differences when considering different boundary conditions. ACKNOWLEDGMENT This work was supported in part by the Spanish Ministry of Innovation and Science (with support from the European Regional Development Fund) under contract TEC2010-14825/MIC and TEC2010-21484; and in part by the Andalusian Regional Council of Innovation, Science and Enterprise under contract TIC-2532 and with the support of the Department of University, Research and society of information of Government of Catalonia. T. Carrasco would like to thank Prof. Mohammed Ismail from Ohio State University, for his warm hospitality during his visit to the ElectroScience Laboratory in OSU. REFERENCES 1. Amin, Y. and H. Tenhunen, “Development and analysis of exible UHF RFID antennas for “green” electronics,” Progress In Electromagnetics Research, Vol. 130, 1–15, 2012. 2. Wu, S.-M., C.-T. Kuo, P.-Y. Lyu, Y. L. Shen, and C.-I. 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