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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. It is also valid in 11+-QZS Z +-QZS ^^hhthe case when a capacitive impedance is to be matched to a pure real impedance such as 50 X, regardless of the

The meaning of this result is that for each set of source value of RL and RS (no solution is available if both of and load impedances (defined by their quality factors) the impedances are inductive). Given only the quality

we can solve (3) and find the required transformer factors QS and QL of the source and load impedances definable by its L (normalized by the source resistance (at the frequency interest), one can graphically find

and the frequency of operation for an assumed cou- the required values of QXL1 and QXL2 . Then, given the

pling factor k). The problem of finding the transformer actual Re ZRLL= and Re ZRSS= , the actual val- " , " , to match ZS to ZL is then solved. However, we would ues of L1 and L2 can be determined, which eventually like to provide a graphical tool that solves this match- forms the transformer at the frequency of interest. ing problem that is general enough to provide a solution independent of parameters such as the frequency and Matching Example source or load resistances. To do that, instead of solv- The general algorithm for using the charts in Figure 3 ing directly for the primary and secondary inductor is as follows:

coils L1 and L2, we define normalized parameters that ••For a given ZS and ZL , calculate QS and QL at the allow a general graphical representation of the solu- frequency of interest.

tion. We found that defining cross-quality factors for ••Find the crossing point of QS and QL on the

the two coils does just that: QLXL11= ~ /Re ZS and matching chart. ^^hh" ,

102 November/December 2014 Matching Chart Derivation

The ideal N:1 transformer reflects the load impedance and solving it in respect to Zu, one obtains two multiplied by N2, preserving its quality factor, as solutions: described by (S1). Shifting to an admittance notation u 2aQQSS++QL leads to (S2). Z12, = 21a + 22 2 ^ h ZN2 ==ZNLLRj--NQLLRR/ 22jQL R , (S1) 2 2 2 24aaQQSS++QQL -+a 1 + S " ^ h ^ h^ h . 111 +jQL 21a + Y2 ==$ 2 / Gj22+ QGL ;.G2 2 0 (S2) ^ h ZR22 1 +QL (S7) To satisfy impedance matching, one shall equate u ZYii= 1/ (Figure 2) while expressing Zi in terms of Equation (S7) suggests that the value of Z ) ZS to Yi in terms of Y2: depends solely on the quality factors of the source

) and load impedances QS and QL, and on the ZZi =-S jL~a =+RjSSQRS -~aL ^h magnetic coupling factor, k. The actual values = 11= . (S3) 1 Yi of the source and the load impedance does Y2 + jL~ not play any role. Moreover, for a given Zu, the

value of L1 can be extracted relying on the value Substituting Y2 (S2) into (S3) and equating the real of RS and is independent of RL . Recalling that and the imaginary parts yields: u 2 ZL==~a //RkSS~ 1 - LR1 yields (S8). ^h Z RS ]G2 = (S4) 2 2 u ] RQS +-SSRL~a ~L1 Z [ ^ h = 2 / Q XL1 . (S8) -+ RS 1 -k -=1 QRSS ~aL (S5) ]QGL 2 2 2 . ] ~L RQS +-SSRL~a \ ^ h The next step is to find the value of L2 . Using 2 Substituting G2 from (S4) into (S5) and multiplying GR22=+11/ QL from (S2) on (S4), while ^^ hh u both hands of the equation by RS yields a quadratic keeping in mind that QQiS=-Z, (S9) is derived

equation in terms of L, (S6). It is interesting to note 2 u R2 1 +Qi that the expression ZL/ ~a /RS is the normalized = 2 . (S9) RS 1 +QL impedance of the inductor aL on a Smith chart with ) 22 ZR0 = S, where the normalized ZS is located on the Replacing R2 by its definition: RN2 ==RkLLRL12/L u ) u ) 2 2 Re ZS = 1 circle, and Im ZQS = S . (S1), yields RR2 //SL==kL12LR$$//RkSS~LR1 $ RL / " , " , ~L2 . ^h ~aL As the value of ~LR1 / s has been already - QS - QL RS c RS m calculated in (S8) while the ratio RR2 / S is given by 22-=a ~aL ~aL ~aL (S9), it is possible to express the value ~LR2 / L as a 11+ QS --+ QS c RS m c RS m function of the load and the source quality factor and (S6) a magnetic coupling coefficient

- u u + 2 The expression QZS is Qi, the quality factor ~L2 = Z $ 1 QL / u 2 Q XL2 . (S10) of Zi . Modifying (S6) by replacing ~aLR/ S = Z RL a 1 +Qi

••Extract the values of QXL1 and QXL2 curves meet- source and load impedances are QS = 3 and QL = 2, ing at the crossing point. respectively.

••Finally, obtain the required L1 and L2 that form By finding the intersection of QS = 3 and QL = 2

the transformer using the frequency of interest ~ lines on the chart of Figure 4(b), values of QXL1 = 22.

and the source and load resistances RS and RL: and QXL2 = 12. are extracted (solution #1). Based on

LQ11= XL RS /,~ LQ22= XL RL /~. these values, the source and load resistances, together To demonstrate the practical applicability of the with the operating frequency, the magnitudes of the matching chart, a numerical matching example is given. transformer inductors, L1 = 580 pH and L1 = 160 pH

In the example, a source impedance of ZjS =-100 300 are found. In the same manner, using solution #2 [(Fig- is to be matched to a load impedance of ZjL =-50 100 ure 4(a)], values of L1 = 35.nH and L2 = 16.nH are at f = 60 GHz. The quality factors calculated from the obtained; values much larger than those obtained by

November/December 2014 103 The return loss for both the solutions was also sim- 100 ulated under slight deviations of k. Plots for k = 07., 250 k = 09., in addition to k = 08., while using the original 120 inductor values are presented on Figure 5. It can be 60 seen that, with the transformer designed by solution #1, the resonance frequency shifts by only 5% from 10 30 250 its original value, while with solution #2, it shifts by 15 more than 50%. Since the proposed method aims to L

Q 8 supply a starting point for the design flow, solution 120 #2 again proves to be less practical. For these reasons, 4 60 1 30 only solution #1 will be used. 15 8 4 Equalization of Inductors ~L1/Real(ZS) Implementation of integrated transformers requires ~L2/Real(ZL) exact tuning of the inductors to achieve the correct 0.1 winding ratio, quality factor, magnetic coupling, 0.1 110 100 high self-resonance frequency (SRF), and inductors QS sizing. All this is done under a limitation of finite (a) metal layers via parasitics and by inherited physi- 100 cal asymmetry resulting from large winding ratios 0.02 60 and the use of underpasses with lower metal lay- 0.03 0.05 30 ers. Large inductors, with large winding ratio, N, 0.1 15 result in additional design difficulties such as deg- 0.2 8 radation of the transformer quality factor [9] and the 10 4 0.5 SRF [11]. To simplify the design flow and increase 2 1 the accuracy of the transformer, it is often desir-

L 1 0.5 Q able to use small inductors with low winding ratio. 2 0.2 4 0.1 The proposed matching chart can be used to easily 0.05 1 8 review the design possibilities and the modifications 15 30 0.03 required to balance (or equalize) the transformer (i.e., LL12= ). For LL12= : 0.02 ~L1/Real(ZS) ~L2/Real(ZL) QXL1 ~LZ1 / Re S Re ZL 60 / " , = " , . (6) 0.1 QXL2 ~LZ2 / Re L Re ZS 0.11 10 100 " , " , QS According to (6) and based on the values of the pre- (b) vious example, Re ZL / Re ZS = 05.. This ratio will " , " , be used in the following discussion. Figure 3. Matching chart: ~LZ1S/Re and ~LZ2L/Re " , " , The set of points with QQXL12/.XL = 05 is marked contours versus QL and QS for k = 0.8. (a) Solution #2 of (S7). (b) Solution #1 of (S7). on the matching chart (Figure 6) by a gray line. To equalize the inductors, the original point must be shifted toward any location on the gray line by

means of alternating QS a nd/or QL without chang- solution #1 and not very practical in an IC at 60 GHz ing the real part of the source/load impedances. Two

and so solution #1 will be used. cases are demonstrated: option A—increasing QL to

These results are verified using a CAD simulation the value of +46. and option B—decreasing QS to the

tool. The transformer was modeled by ideal coupled value of +12.. To increase QL from 2 to 4.6 (Figure 6

inductors, with a coupling coefficient k, loaded by ZL option A) at 60 GHz without changing the real part

and driven by a source ZS [Figure 5(a)]. The return of ZL, an additional capacitor of 21 fF is added in a loss is plotted for both solutions #1 and #2 [Figure 5(b) series to the load. Having done that, the new values of

and(c)]. For k = 08. (the value of k used for the match- the cross-quality factor of L1 and L2 are QXL1 = 15. 5

ing chart on Figure 3), an exact matching is achieved and QXL2 = 31., leading to LL12==410 pH. Decreas-

at 60 GHz. However, since prior to the actual trans- ing QS without changing the real part is possible former design, the exact value of k cannot be accu- only by adding a series inductor or an inductive

rately predicted, it is valuable to see the sensitivity transmission line. To reduce QS to the value of 1.2 of the solution to variations in the magnetic coupling (option B in Figure 6), a series inductor of 480 pH coefficient. is required. Based on that option, the new values of

104 November/December 2014 100 250 S 11 k 120 ZS = 100 – 300j Z L = 50 – 100 60 L1 L2 10 30 Solution #2: 250 QXL1 = 12 15 j L QXL2 = 11 Q 8 (a) 4 120 60 0 1 30 15 8 k = 0.8 (dB) -25 4 k = 0.7 11

~L1/Real(ZS) S k = 0.9 ~L2/Real(ZL) -50 30 40 50 60 70 80 90 100 110 0.1 f (GHz) 0.1 1 10 100 (b) QS (a) 0

100 0.02 60 0.03 (dB) -25

0.05 11

30 S 0.1 15 -50 0.2 8 30 40 50 60 70 80 90 100 110 10 0.5 4 f (GHz) Solution #1: 2 Q = 2.2 1 (c) XL1 1

L QXL2 = 1.2 0.5

Q 20 0.2 4 0.1 0.05 Figure 5. (a) Transformer test bench schematics. 1 8 = = 15 30 0.03 (b) S11 for solution #1. Lp1 580 H, Lp2 160 H with different values of k. (c) S11 for solution #2. Ln1 = 3.5 H, 0.02 Ln2 = 1.6 H with different values of k. ~L1/Real(ZS) ~L2/Real(ZL) 60 0.1 0.1 1 10 100 QS 100 (b) 0.02 Points with 60 0.03 ~L1 Re{Z } Figure 4. An example of the matching procedure for 0.05 S =30 0.5 ~L20.1Re{Z } ZjS =-100 300 and ZjL =-50 100 . (a) Solution #2. L 0.1 15 (b) Solution #1. 0.2 8 10 4 0.5 2 1 1 L Option B 0.5 = = Q Option A cross-quality factors are QXL1 09. and QXL2 18. , 2 0.2 leading to LL12==240 pH. 4 0.1 0.05 The plot of the return loss of the equalized trans- 1 Original8 Point 15 0.03 former is presented in Figure 7. Both options yield match- 30 ing at 60 GHz. Option B requires a smaller transformer ~L1/Real(ZS) 0.02 at the expense of an additional inductor, while option A ~L2/Real(ZL) required only a small additional capacitor. It is interest- 60 ing to note that the matching bandwidth of option B is 0.1 0.1 110 100 larger due to the smaller quality factors involved. This QS technique allows the designer to control not only the transformer sizing and the winding ratio but also the desired matching bandwidth. Figure 6. The inductors equalization process.

November/December 2014 105 In contrast to the more traditional The demand that QQLX11& L can be relaxed at the mm-wave regime as the quality factor of a capaci- LC- impedance matching performed tive impedance decreases with frequency. At lower using a Smith chart, no similar frequencies, where the assumption of QQLX11& L method exists to perform exact is not necessarily valid, a two-step iteration process can be used. The first iteration step is performed conjugate impedance matching using assuming ideal inductors. QS and QL are calculated transformers. based on the original source and load impedance values, and QXL1 and QXL2 are extracted from the

matching chart. The values of L1 and L2 are then

Transformer Parasitics calculated. The values of L12, together with the typi-

Since the proposed method was based on ideal induc- cal QL12, is used to estimate the inductor series resis-

tors, it is important to review the impact of a finite tance RL12, , which in turn, will be added in series to

inductor quality factor QL1 and QL2 . For example, an the source and the load resistance, changing QSL,

L1 inductor with a finite QL1 contributes a series resis- to QQllSL,,=+SLRRSL,,//SL RQLS12,,= LSRR,,LSL . The ^h tance RL1 connected in series to RS . To neglect this new values of QSL, , i.e., QlSL, , are now used to find

resistance, RS should be much greater than RL1, or the updated values of QXL12, , i.e., QlXL12, , and, conse-

QQLX11& L . quently, the updated inductors Ll12, . Silicon spiral inductors have two main loss mecha- The convergence process can be demonstrated by a nisms—resistive loss in the inductor trace metal and numerical example. In this example, the two-iteration

conductive loss in the silicon substrate. At low giga- method is used to match a QS = 60 source impedance

hertz frequencies, the substrate loss is significant and to QL = 60 load impedance by a transformer designed may even dominate the achievable quality factor [7], using a process with a typical inductor quality factor

[21]. As frequencies increase into the mm-wave region of QL12, . 10. In the first iteration, ideal inductors are

and typical inductors are smaller in value and area, assumed. QSL, = 60 yields QXL12, = 40 (Figure 3). Con-

substrate loss becomes less compared with metal sequently, ~LQ12,,= XL12$ RSL, . Based on the inductor

resistive loss, especially when including the skin quality factor, the parasitic resistance RL12, is calcu-

effect. Extensive work has been done on the optimi- lated: RLLL12,,==~ 12//QQ12,,XL12$ RQSL,,LS12= 4R ,L . zation of the quality factor and the SRF [8], [22], [23], This resistance is assumed to be absorbed into the offering methods such as decreasing the turns num- source and the load to maintain the ideal inductors

ber [9], [24], differential topologies [25], [26], thicker regime, yielding the new QSL, , i.e, metals [11], [27], and substrate shielding [18] to pre- vent the inductor quality factor degradation. Typical QQllSL,,=+SLRRSL,,//SL RQLS12,,= LS$ RR,,LSL ^h values of inductor quality factors feasible using a sili- ==02..QSL, 12 (7) con process are about 10–15 in the lower GHz range

[28], maintaining a similar order of magnitude up to Using QlSL, from (7) on the matching chart produc-

110 GHz [13]. A quality factor above 30 at 60 GHz es QlXL12, = 7 and, consequently, ~LQll12,,==XL12RlSL,

has been also reported [18]. Although the quality 74/ 0 QXL12, and 57RLSL,,= /.8 ~ 12 The example sug- factor and the SRF of a transformer are higher than gests an error of just 12.5% in the inductors values

stand-alone inductors [19], low Q values can degrade between the first iteration, based on infinite QL12, , the insertion loss, approximated by the expression and the second iteration. The reader is encouraged to 2 IL . 12- /,kQLL12Q [28]. Typical inductor param- perform a third iteration to verify that a convergence

eters of k = 08. and QQLL12==10would yield a 1-dB has been achieved. insertion loss. Admittance Notation In addition to the impedance matching functional- ity, a transformer is often used to provide dc bias to 0 the stage. In such cases, a series capacitor cannot be used to increase the port quality factor as it acts as

(dB) -20 a dc block (though in principle a series capacitance

11 Option A S can be implemented as a series inductor above its Option B SRF [29]). A parallel reactive element can also be -40 50 55 60 65 70 used to change the port quality factor, but it does f (GHz) not naturally fit into our impedance-based matching methodology. To extend the chart to handle parallel Figure 7. The return loss for matching with option A and elements, an admittance-based representation has option B inductors equalization. also been developed.

106 November/December 2014 For a given source impedance ZRSS=-jQSSR , the 2 We would like to provide a graphical real part of its admittance is Re YRSS=+11/.QS " , ^^ hh Hence tool that solves this matching problem that is general enough to provide a 1 ~LY11$$Re SX= Q L 2 , (8) solution independent of parameters " , 1 +QS such as the frequency and source or and, similarly, load resistances. 1 ~LY22$$Re L = QXL 2 . (9) " , 1 +QL visualize the impedance matching process with a trans- Equations (8) and (9) are used to plot an equivalent former on a Smith chart and understand its limitations. admittance notation matching chart as shown in Fig- The operation of the coupled inductors transformer ure 8. The contour values suggest that large ratios of can be viewed according to the block representation of

~LY1 Re S to ~LY2 Re L cannot be achieved. This Figure 2. First, the load impedance is transformed using " , " , means that additional series components cannot be an ideal N:1 transformer. This transformation does not used to equalize the transformer, but they can be used change the quality factor of the reflected ZL, so N sim- to modify the QS and QL and thus modify the match- ply moves the load impedance along the corresponding ing bandwidth and the transformer sizing. equi-Q curve. Inductors aL and L are set to bring Y2

The quality factor of the load and source stay (ZL after N:1 transformation) from the quality-factor the same whether they are treated as impedance or line QL (in the bottom half of the Smith chart, i.e., +QL ) admittances. As a result, the matching charts for to the quality factor line -QL (i.e., QS at the upper half, ) impedance [Figure 3(a)] and admittance (Figure 8) corresponding to ZS ). As shown in “Matching Chart have the same axis and are easily interchangeable Derivation,” the value of the normalized inductor u u and easy to use when adding both parallel and serial impedances ZL= ~a /RS and ZL//a~= RS are deter- elements to control the transformer sizing and match- mined solely by QQLS, (for a given k). Based on that, ing bandwidth. the proposed matching process also can be viewed on a Smith chart. Using the Smith chart for transformer View of Matching on a Smith Chart impedance matching purposes, however, is not a one- For a community that is very familiar with the Smith step solution as with the proposed graphical tool and chart as a tool for impedance matching, it is interesting to requires a more iterative approach.

Choosing ZR0 = S, the normalized conjugate of the u ) source impedance is located at ZjS =+1 QS (Figure 9). u ) The algorithm states: ZS is marked on a unity resistance

100 circle with imaginary value of QS . For an arbitrary 0.01 u point ZZi = iS/R on a unity impedance circle, the dis- 0.02 u ) u ) u 0.01 tance between ZS and Zi equals QQSi-=~aLR/ S = Z u ) 0.04 (red arrows path), while the distance between Zi and u 0.08 0.02 the crossing point of the constant Re Yi circle with 10 " , 0.15 0.04

L 0.3 Q 0.08 0.6 i ~ Q Z

1 0.15 S Q 0.3 ~L1*Real(YS) Q S ~ ~ * Z i Z S 0.6 Q i ~L2*Real(YL) 1 a ~ 0.1 Z 0.1 110 100 Q L QS

~ Y 2 Figure 8. An admittance matching chart: ~LY1S$ Re " , and ~LY2L$ Re contours versus QL and QS for k = 0.8 [solution" , #1 of (3)]. For a given source impedance ZRSS=-jQSSR , the real part of its admittance is Figure 9. The transformer matching process using a 2 Re YRSS=+11/.QS Smith chart. " , ^^ hh

November/December 2014 107 u The proposed presents the designer To find Zi , an initial guess is required, followed by several correction steps until final convergence. Once the full span of design possibilities, completed, the value of Zu is extracted and the value of u 2 tradeoffs, and alternatives, providing L1 is calculated using LZ1 =-RkS /.~ 1 ^h2 To find L2 , one must first find NZ= Re 2 / control on transformer sizing, winding " , Re ZL (ZY22= 1/ is graphically obtained from the " , u ratio, and matching bandwidth. Smith chart after successfully finding Zi ). Finally, 2 2 LK2 = LN1 / completes the process. It is therefore clear that matching using the Smith chart is poten- u the QL line (green arrows path), namely Y2, equals tially possible but not very easy or intuitive, which u uu RLS //~a= Z (Figure 9). Since ZZ$ aa/,= and for really motivated this work. u k = 08., a = 05., a point Zi must be found in a way to satisfy the product of the two paths (the black and the Practical Transformer Verification gray one in Figure 9) to be equal to 0.5. In this section, a practical case study is examined. Two CMOS 65-nm differential buffers are matched using a transformer at a 120-GHz frequency. The trans- former was designed based on the inductor param-

S11 eters extracted from the matching chart and verified Zout = 10–55j = ZS k Zin = 20–104j = ZL using an Agilent Momentum electromagnetic simula- tor. Later, the matching was validated again using the L1 L2 VDD transformer full electromagnetic model to assess the Buffer 1 Buffer 2 accuracy of the design. L L Bias 1 2 At this current example, a source of ZZout1 ==S

10 -55j is matched to a ZZin2 ==L 20 -140j at a k frequency of 120 GHz [Figure 10(a)]. Those values (a) represent the single-ended impedance (half of the dif-

ferential one), leading to quality factors of QS = 55. L1 = 43 PA, Q1 = 11.5 at 120 GHz L2 = 80 pH, Q2 = 10 at 120 GHz and QL = 52.. Using the matching chart of Figure 3 k = 0.725 at 120 GHz (solution #1), one can obtain the normalized induc- SRF = 182 GHz Metal 9 tor impedances of QXL1 = 33. and QXL2 = 3. Extract- Outp ing the inductor values leads to L1 = 43.p7 H and

L1 = 79.p6 H, single-ended values (half of the trans- Outn former), and a coupling factor K = 0..725 Metal 8 A monolithic transformer was designed using a CMOS 65-nm process and verified using an Agi- lent Momentum simulator [Figure 10(b)]. The top

thick metal (ME9) was used to implement L1 in a single symmetrical loop, and ME8 was used to imple-

ment L2 with an additional smaller loop to achieve In p the larger inductance. The transformer parameters, Inn extracted from the transformer Z-matrix, were com- (b) pared with the target values for validation. The final design demonstrates the desired inductors values 0 with a magnetic coupling of K = 0.725 [Figure 10(b)], -5 slightly lower then K = 08. used for the matching chart. Finally, the matching was verified by measur- (dB) -10 11 ing the return loss as seen by the output of Buffer 1 S [Figure 10(c)]. The return loss plot demonstrates about -15 123 GHz -17 dB matching at 123 GHz, a 2.5% deviation from 100 110 120 130 140 the target frequency, mostly attributed to the differ- f (GHz) ent than assumed coupling factor, which could also (c) be refined by designing more symmetrical trans- formers with lower winding ratios. Figure 10. Amplifier interstage impedance matching using a practical transformer. (a) System blocks schematics, Conclusion (b) transformer layout and parameters, and (c) return loss In this study, a novel graphical tool for impedance after matching. matching using transformers was developed and

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