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Beyond the Smith Chart

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Beyond the Smith Chart IMAGE LICENSED BY GRAPHIC STOCK Beyond the Smith Chart Eli Bloch and Eran Socher he topic of impedance transformation and Since CMOS technology was primarily and ini- matching is one of the well-established tially developed for digital purposes, the lack of and essential aspects of microwave engi- high-quality passive components made it practically neering. A few decades ago, when discrete useless for RF design. The device speed was also radio-frequency (RF) design was domi- far inferior to established III-V technologies such as Tnant, impedance matching was mainly performed us- GaAs heterojunction bipolar transistors (HBTs) and ing transmission-lines techniques that were practical high-electron mobility transistors (HEMTs). The first due to the relatively large design size. As microwave RF CMOS receiver was constructed at 1989 [1], but design became possible using integrated on-chip com- it would take another several years for a fully inte- ponents, area constraints made LC- section match- grated CMOS RF receiver to be presented. The scal- ing (using lumped passive elements) more practical ing trend of CMOS in the past two decades improved than transmission line matching. Both techniques are the transistors’ speed exponentially, which provided conveniently visualized and accomplished using the more gain at RF frequencies and also enabled opera- well-known graphical tool, the Smith chart. tion at millimeter-wave (mm-wave) frequencies. The Eli Bloch ([email protected]) is with the Department of Electrical Engineering, Technion— Israel Institute of Technology, Haifa 32000, Israel. Eran Socher ([email protected]) is with the School of Electrical Engineering, Tel-Aviv University, 69978, Israel. Digital Object Identifier 10.1109/MMM.2014.2356114 Date of publication: 12 November 2014 100 1527-3342/14©2014IEEE November/December 2014 use of copper metallization and the increased num- ber of metal layers [up to ten layers for some CMOS k I1 I2 integrated circuit (IC) processes] also improved the V1 V2 integrated passive devices in CMOS for RF circuits. L1 L2 The ability to integrate RF circuits with mixed signal and digital circuits on the same chip generated the motivation to use CMOS for RF applications. By 2005, (a) CMOS was already the dominant technology in most 2 RF applications below 10 GHz [2]. (1-k )L1 2 Monolithic inductors play a critical role in RF k L1 components, but in marked contrast to capacitors, N:1 they exhibit much lower quality factors, particularly at low frequencies and on silicon conductive sub- (b) strates. The evolution of on-chip inductors has also made considerable progress since the 1960s and 1970s Figure 1. (a) A first-order transformer model, with 11 when it was widely claimed that integrated induc- -11k as the magnetic coupling coefficient. (b) An tors with reasonable Q were practically unrealizable equivalent circuit with an ideal N:1 transformer. and should be placed externally. As a result, induc- tors were implemented using bondwires and package sizing and winding ratio for exact conjugate matching pins. Only with the appearance of an accurate ana- given a load and source impedance in the manner lytical model did integrated spiral inductors become used in the case of lumped inductors and capacitors or popular. With the increasing operating frequency transmission lines. of narrowband ICs, monolithic inductors and trans- This article is focused on proposing and demon- formers became more popular. In current mm-wave strating a universal graphical tool (a nomogram) for ICs, their physical dimensions become comparable conjugate impedance matching using transformers. to the size of active blocks. As passive components The tool not only offers direct determination of trans- have a significant impact on the system performance, former parameters but also provides the designer with extensive work on silicon integrated inductors and insight into design tradeoffs and alternatives such as transformers modeling has been performed, offering transformer sizing, matching bandwidth, and various an extensive analysis of various transformer topolo- winding ratios, thus easily enabling a design starting gies, layout geometries, substrate impact, and layout point and leading to an optimized solution. parasitics [3]–[6]. In contrast with nonintegrated RF circuits, trans- Transformer Impedance Matching formers (made of coupled inductors) play a key role The graphical tool developed in this work is based on in CMOS-based RFICs [7], [8] and became the obvious a first-order transformer approximation of a nonideal choice when it comes to impedance matching and cas- transformer [Figure 1(a)] represented by two ideal caded blocks in CMOS RF and mm-wave ICs. Trans- inductors (the ideality assumption will be reviewed formers enable symmetric differential operation, with and justified later on) LL12, coupled with a mutual a virtual ground along the symmetry line. They pro- magnetic coupling coefficient k. To use this representa- vide dc separation between the stages and easy bias- tion for circuit analysis purposes, an equivalent scheme ing through the center tap. In addition, transformers [Figure 1(b)] is presented by [20], comprising two ideal 2 2 enable high voltage swings with low-voltage head- leakage inductors 1 -kL1, kL1 and an ideal N:1 ^h room and are used in implementations of power com- transformer, with Nk= LL12/. bining [9], [10], resonance loading [11]–[13], bandwidth In cases when the transformer is used as a match- peaking, low-noise feedback [14], and baluns and serve ing network between two stages, the values of L1 and as a key component in active building blocks such as L2 should be determined to produce a conjugate match power amplifiers [15]–[17] and low-noise amplifiers between a source and a load. [14], [18], [19]. For the convenience of the derivation process, the In contrast to the more traditional LC- imped- following notations are used: ance matching (L-sections, r-sections, etc.) performed using a Smith chart, no similar method exists to per- 1 -k2 a / (1) form exact conjugate impedance matching using k2 2 transformers. The method usually used for matching Lk/ L1 . (2) is adding additional parallel and series capacitors to resonate the transformer’s residual inductances [9]. To Using (1) and (2) with the equivalent scheme yields the best of our knowledge, no straightforward tool has the scheme shown in Figure 2, with inductor values been developed to determine the required transformer replacing aL and L. As the typical value of k for November/December 2014 101 QLXL22= ~ /.Re ZL These cross-quality factors 2 ^^hh" , u ZS* Zi,Yi Z2 = ZLN can be calculated from the solution of Z using (3) with aL Zu QXL1 = , (4) 1 -k2 Z 2 Z L L u 1 +QL S = Z $ QXL2 u 2 . (5) a 1 +-QZS N:1 ^h The result is that it is possible now to find normal- Figure 2. A modified representation of the transformer, ized solutions (QXL1 and QXL2 ) for the inductance of including nodes impedances notations. both coils, assuming a desired transformer coupling factor k and depending only on the source and load impedance quality factors QS and QL . The details of monolithic transformers is usually in a range of these calculations are shown in the “Matching Chart 07..-08, a will be consequently equal to 10- ..5 Derivation.” Without a loss of generality, it is assumed that both the source impedance, ZS, and the load impedance, Graphical Tool ZL, are capacitive. This is a practical assumption For QXL1 and QXL2 , (4) and (5) form the foundations when active stages are involved. To define an unam- of the matching chart proposed in this study. As biguous quality factor for both capacitive and induc- shown by (3), Zu has no direct dependence on the tive impedances, the definition QZ=-Im / Re Z frequency of interest or on the actual source and "",, is used. Using this definition, Q 2 0 for capacitive load impedance values. Since the cross-quality fac- u impedances and Q 1 0 for inductive ones. One can tors QXL1 and QXL2 are functions of Z, QS, QL , and also express both the load and the source impedances k [as shown in (4) and (5)], they also have normal- in terms of its quality factor, i.e., ZRLL=-jQLLR , and ized values, and it is possible to plot the QXL1 and ZRSS=-jQSSR with RRLS,.2 0 the QXL2 contours as a function of QL and QS for a The purpose of using the transformer is to conju- given (or assumed) value of k (Figure 3). The induc- gate match the impedance so that the transformer tor values L1 and L2 derived from the chosen QXL1 ) would show an impedance of ZS to the source imped- and QXL2 would of course be frequency dependent, ance ZS . Alternatively, the same condition holds at according to RQSXL1 /~ and RQL XL2 /,~ respectively. other points along the connection between the source The matching chart in Figure 3 is a contour plot of and the load, e.g., at the intermediate impedance point QQXL1 SL,Q and QQXL2 SL,Q values based on the ) ^h ^h of Zi . Expressing Zi in terms of ZS and Yi in terms substitution of the solution of (3) into (4) and (5) (for of ZL and equating ZYii= 1/ yields a quadratic equa- a typical coupling value of k = 08. ), suitable for any u tion in L or in its normalized value of ZL= ~a /RS (see capacitive source and load impedances (as plotted for “Matching Chart Derivation”) QQSL, 2 0). As a result of the QXL1 and QXL2 indepen- dency on the actual source and load impedances and on u the frequency of operation, the chart is universal and QL --()QZS -=a u 22u u . (3) suitable for any matching application.
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