Scattered Reflections on Scattering Parameters 1101

IEICE TRANS. ELECTRON., VOL.E99–C, NO.10 OCTOBER 2016 1100

INVITED PAPER Special Section on Microwave and Millimeter-Wave Technology Scattered Reﬂections on Scattering Parameters —Demystifying Complex-Referenced S Parameters—

Shuhei AMAKAWA†a), Member

SUMMARY The most commonly used scattering parameters (S parameters) are normalized to a real reference resistance, typically 50 Ω.In some cases, the use of S parameters normalized to some complex reference impedance is essential or convenient. But there are different definitions of complex-referenced S parameters that are incompatible with each other and serve different purposes. To make matters worse, different simulators implement different ones and which ones are implemented is rarely prop- erly documented. What are possible scenarios in which using the right one matters? This tutorial-style paper is meant as an informal and not overly technical exposition of some such confusing aspects of S parameters, for those who have a basic familiarity with the ordinary, real-referenced S pa- Fig. 1 A length, , of transmission line driven by a matched source rameters. (source impedance ZS = Z0) and terminated with a load impedance ZL. ffi ffi key words: S parameters, reflection coe cient, transmission coe cient, Z0 is the characteristic impedance of the line. traveling waves, pseudo waves, power waves, reference impedance, renormalization transformation somehow defined, can we define S parameters uniquely? 1. Introduction Not yet. We have a choice between defining S parameters, including reflection coefficients, based on voltages or cur- According to Carlin [1], the earliest article that dealt with rents. The choice may [3], [9]–[13] or may not [14] affect the scattering parameters or S parameters was [2], published the values of S parameters, depending on how you define in 1920. The first book [3] that gave extensive coverage of current scattering parameters. We opt for the voltage-based the subject was published in 1948 [1]. The S parameters definition, as is commonly practiced. described in this book are essentially the same S parameters If a transmission line is terminated with an impedance as those we most often (but not always!) use today. Those ZL as shown in Fig. 1, the voltage reflection coefficient at the are the real-referenced S parameters. terminating load is [11]–[13] To define S parameters, we must first define an effec- − tive voltage and an effective current from the electric and = ZL Z0 , S 11 + (1) magnetic fields in a waveguide, respectively. A “wave- ZL Z0 guide” here may refer to a (quasi-)TEM (transverse electro- where Z0 is the characteristic impedance of the line. Z0 magnetic) transmission line, a hollow metallic waveguide, appears in Eq. (1) because the reflection coefficient is a or some other form of waveguide. In the case of ideal TEM description of a 1-port in question (in this case, the load transmission lines, the mapping of electromagnetic (EM) impedance ZL) in terms of incident and reflected traveling fields to voltages and currents is unique. But in general, wave amplitudes in a one-dimensional medium (i.e. trans- there is some arbitrariness in the mapping. This arbitrariness mission line) that feeds the 1-port. Z0 is a physical property implies that there is arbitrariness in the definition of charac- of the one-dimensional medium, not of the 1-port. The stan- ffi teristic impedance, too. We don’t delve here into the di - dard assumption, often made implicitly, is that the transmis- cult and controversial problem of how the mapping should sion line is lossless, and therefore Z is real. Its standard ff 0 be done [4]–[8], and simply assume that e ective voltage value is 50 Ω [15], [16]. To emphasize the fact that the value and current have been defined appropriately. We will here- of S 11 depends on Z0, and that its value is real, it is more after refer to them simply as “voltage” and “current,” respec- appropriate to write instead tively. We will also assume that our waveguide is a quasi- − TEM transmission line. = ZL Rref . S 11 (Rref ) (2) Now that we have voltages and currents in waveguides ZL + Rref Manuscript received February 15, 2016. The above notation of explicitly showing the reference re- Manuscript revised June 27, 2016. † sistance Rref of an S parameter in parentheses was intro- The author is with the Graduate School of Advanced Sciences duced by Woods [17]. The notation is summarized in Ta- of Matter, Hiroshima University, Higashihiroshima-shi, 739–8530 Japan. ble 1. The standard choice of Rref is the characteristic resis- a) E-mail: [email protected] tance [18], [19] R0 of the lossless transmission line, which / DOI: 10.1587 transele.E99.C.1100 the load ZL terminates. Why don’t we simply write S 11 (R0)?

Copyright c 2016 The Institute of Electronics, Information and Communication Engineers AMAKAWA: SCATTERED REFLECTIONS ON SCATTERING PARAMETERS 1101

Table 1 Notation for showing reference impedances. Example Description

S ij(50 Ω) S parameter with ports i and j referenced to 50 Ω

S(Rref ) S matrix with all ports referenced to Rref

S 21 (Rref1, Rref2) Port 1 referenced to Rref1, port 2 referenced to Rref2

S(Zref1, Zref2) 2-port S matrix referenced to Zref1 and Zref2

S(Zref ) S matrix referenced to reference impedance matrix Z = diag(Z , Z , ···) ref ref1 ref2 Fig. 2 AThevenin´ signal source with an impedance ZS feeds a load impedance ZL. How can the power absorbed (dissipated) by the load be maximized? Well, we could, but we might want to use an Rref value ff R di erent than 0, which is a property of the transmission on the system under consideration, the appropriate one to line. In some situations, we might want to assign to Rref a use differs. The most appropriate value to use as the refer- value that is not a property of a physical object (transmis- ence impedance Z might be the characteristic impedance sion line) at hand. We will later see when such a need arises ref § Z0 that feeds the network, the source impedance ZS that di- ( 3.1). The characteristic resistance and the reference resis- rectly feeds the network, or some other value. You might tance should not be mixed up. possibly think that complex-referenced S parameters are a In the real world, all transmission lines are lossy, at matter of purely academic concern with little practical use. least to a small degree. Then, the characteristic impedance But that is not so. We work on millimeter-wave CMOS cir- Z 0 expressed in terms of the per-unit-length RLGC param- cuit design [22]–[29] and related measurements [30]–[42], eters, and regularly use both types of complex-referenced S parameters out of necessity. Situations in which using the right R + ωL = j , one would matter include millimeter-wave and terahertz on- Z0 + ω (3) G j C wafer measurements, where methods of vector network analyzer (VNA) calibration and de-embedding that work well assumes a complex value, unless the the distortionless con- at lower microwave frequencies fail. Also relevant would dition [9], [11], [20], be power transfer systems, in which long transmission lines R G are deployed and minimizing losses is imperative. It is un- = , (4) L C fortunate that, in spite of the practical importance of the subject, resources are largely limited to research papers scat- ffi is met. What happens, then, to the reflection coe cient? tered about everywhere (a recent exception is [43]), at times Can we keep using Eq. (1) or (2) with a complex Zref (refer- with somewhat biased views. ence impedance) substituted for Rref as follows, To make matters worse, microwave engineers are left − with microwave simulators, EM simulators and related = ZL Zref , S 11 (Zref ) (5) programs that are strangely silent about which complex- ZL + Zref referenced S parameters they implement (if they do), or or do we need something different? It is curious that most whether they do. It is practically very important that you microwave textbooks refer to the complex characteristic understand which S parameters you want to use and which impedance formula, Eq. (3), yet many of them are silent ones your simulator implements. I hope this article helps about how to define reflection coefficients and S param- develop practicing microwave engineers’ awareness of the eters when Z0 or Zref is complex. Even microwave metrol- potential dangers of using wrong S parameters in the wrong ogists don’t appear to have looked very seriously into the context. This article was derived from an article that I pre- issue [21], perhaps till mid 1980s. sented at MWE 2015 [44], which, in turn, was based upon a Another question that springs to mind in this connec- tutorial that I gave in 2011 [45]. Articles that discuss related tion is this: how does the definition of S 11 with a complex Z0 issues include [4], [17], [46]–[51]. relate to the well-known textbook problem (Fig. 2) of maxi- mizing the power absorbed (and dissipated) by a load ZL fed 2. Two Definitions of Reflection Coefficients by a signal source with an impedance ZS? We all know that, to maximize the power absorbed by the load in Fig. 2, the A reflection coefficient is the S parameter of a 1-port. We load impedance ZL and the source impedance ZS must be can learn a great deal about S parameters by looking at re- = ∗ = ffi complex conjugate of each other (ZL ZS). Does S 11 0 flection coe cients. (defined in what way?) imply that the power absorbed by the load in Fig. 1 is maximized? This question is not as trivial 2.1 Transmission Line and Reflection Coefficients as it might appear (§2.5). In this article, we look at complex-referenced S pa- If the far end of a length of lossy transmission line is ter- rameters. There are, at least, two distinct definitions of minated with its complex characteristic impedance Z0 as complex-referenced S parameters. They are incompatible shown in Fig. 3, the line looks as if it were infinitely long with each other and serve different purposes. Depending (Zin = Z0) as seen from the near end. It means that no IEICE TRANS. ELECTRON., VOL.E99–C, NO.10 OCTOBER 2016 1102

+ transmission line that feeds the network), a1 and b1 (and V − 1 and V1 , too) are no longer directly related to the voltage- and current-traveling-wave amplitudes in the line. Only when Zref equals Z0 do a1 and b1 correspond to actual traveling- wave amplitudes in the transmission line. But hereafter, we conveniently forget the ﬁctitious nature of pseudo waves, and pretend that they are related to voltage- and current- Fig. 3 A length of transmission line terminated with its characteristic traveling-wave amplitudes. impedance Z0 at the far end looks as if the line were inﬁnitely long as seen The port voltage V1 and the port current I1 at the load from the near end. Z = Z and S = 0. in 0 11 (Z0) are related to the voltage- and current-traveling-wave amplitudes as

+ − waves reflect back when injected traveling waves reach the V1 = V + V , (9) ffi 1 1 far end. The reflection coe cient S 11 at the terminating = + + −. I1 I1 I1 (10) load, ZL = Z0, must be equal to 0. If ZL Z0, S 11 will = be nonzero. We, therefore, adopt Eq. (5) with Zref Z0 to The characteristic impedance relates the voltage- and ffi define the reflection coe cient of a 1-port ZL that termi- current-traveling-wave amplitudes traveling in the same di- nates the transmission line. If the terminating load has an rection: impedance ZL Z0, reflected waves come back to the near V+ V− end, and the line no longer appears infinitely long. The re- 1 = − 1 = Z (= Z ). (11) ffi + − 0 ref flection coe cient, defined as above, is a representation of I1 I1 a 1-port in terms of traveling-wave amplitudes that appear in a transmission line, through which the 1-port is excited. Z0 being complex (arg Z0 0) means that there is a phase difference between the voltage and current traveling waves. That’s why a property, Z0, of the transmission line enters Although a1 and b1 have the dimensions of square root the expression, Eq. (5), through Zref = Z0. The characteristic impedance of the line physically connected to the load is of power, they are just voltages multiplied by a real number, /| | the natural reference impedance (my preferred term) in this (Zref) Zref , as is clear from Eqs. (7) and (8). They are, ffi case. It is a property of the physical (as opposed to a virtual) therefore, essentially voltages, and the reflection coe cient, ffi environment in which the network in question is embedded. Eq. (6), should be understood as a voltage reflection coe - = = It is, however, not clear from Eq. (5) what the incident cient. If Zref is real (Zref (Zref) Rref), Eqs. (7) and (8) and reflected waves are. Equation (5) is a voltage reflection reduce to the widely known formulas: ffi § coe cient as noted in 1. Specifically, V+ + = 1 = + = 1 V1 Zref I1 , − a1 √ Rref I √ (12) V b R 1 R 2 S ≡ 1 = 1 , (6) ref ref 11 (Zref ) + V− − V1 a1 1 − 1 V1 Zref I1 b1 = √ = − Rref I = √ . (13) R 1 R 2 where V+ is the complex amplitude of the rightward- ref ref 1 − traveling wave incident upon the load, V is that of the a1 and b1 are often mixed up in the literature with power 1 + − leftward-traveling, reflected wave. While V1 and V1 are waves (Eqs. (19) and (20)) [54], which we will be discussing sufficient for defining a reflection coefficient, with a view in §2.3. While it is not incorrect to regard Eqs. (12) and (13) † to smooth extension to multiports , normalized wave ampli- as power waves, given the fact that Eqs. (19) and (20) reduce tudes, a1 and b1, are usually used [4]. to Eqs. (12) and (13) for real Zref, I would like to emphasize that a1 and b1 are voltage waves, expressed in square root of + ≡ (Zref) + = (Zref) V1 Zref I1 , watts. a1 V1 (7) |Zref| |Zref| 2 ffi (Z ) (Z ) V − Z I 2.2 Current Reflection Coe cients b ≡ ref V− = ref 1 ref 1 , (8) 1 |Z | 1 |Z | 2 ref ref The real-referenced current reflection coefficient of a load a1 and b1 are termed pseudo waves [4], because when ZL (Fig. 1) is given usually [3], [9]–[13] by ff Zref assumes a value di erent than the natural reference − = − = −ZL Rref . impedance (i.e. characteristic impedance Z0 of the physical S I11 (Rref ) S 11 (Rref ) (14) ZL + Rref †If a scattering matrix is defined using voltage amplitudes V+ − i and V j , a reciprocal network’s S matrix becomes symmetric only if This follows from reference resistances of all ports are equal [12], [52], [53].Ifai and − I b1 b j are used instead to define an S matrix, a reciprocal network’s S ≡ 1 = − , S I11 (Rref ) + (15) matrix becomes symmetric even when reference resistances are not I1 a1 all equal. But if reference impedances are complex, a reciprocal network’s S matrix may be asymmetric. where we used Eqs. (12) and (13). Its complex-referenced AMAKAWA: SCATTERED REFLECTIONS ON SCATTERING PARAMETERS 1103

= − extension is S I11 (Zref ) S 11 (Zref ). Less common but another valid definition of current reflection coefficient is [14] − − I1 b1 ZL Rref S ≡ = = S = , (16) I 11 (Rref ) + 11 (Rref ) + I1 a1 ZL Rref − = − − = where, I1 I1 , and the port current is given by I1 I+ − I−, instead of Eq. (10). This amounts to accounting 1 1 − for the direction of reflected current “outside” I . This not Fig. 4 A length of transmission line terminated with the complex conju- 1 ∗ so popular definition is not completely worthless, because gate, Z0, of the characteristic impedance Z0. Eq. (16) is actually consistent with the power-wave reflection coefficient, Eq. (21), which is a current reflection coef- 1 2 2 ∗ (Zref) ficient (§2.7, §2.8). = |a1| −|b1| − 2(a b1) (23) 2 1 (Z ) ref | |2 2.3 Reflection Coefficient for Power Maximization = a1 − 2 − (Zref) . 1 S 11 (Zref ) 2 S 11 (Zref ) (24) 2 (Zref) Let’s get back to Fig. 2 and think about how a reflection coefficient should be defined if we want it to be zero when the Note that in this article V1 and I1 are amplitudes, not rms = ∗ values. If Zref is real, the last terms in Eqs. (23) and (24) power absorbed by the load is maximized. Since ZL ZS is the condition for power maximization, an appropriate defi- disappear, and we obtain the well-known result: nition of the reflection coefficient would be = 1 | |2 −| |2 = 1| |2 − 2 , ∗ PL a1 b1 a1 1 S 11 (Rref ) (25) ZL − Z 2 2 S = ref (17) P11 (Zref ) + ZL Zref where a1 and b1 are given by Eqs. (12) and (13). In Eq. (25), | |2/ | |2/ with Z = Z . A subscript ‘P’ is added to the left-hand a1 2 and b1 2 can be interpreted as incident and re- ref S | |2 side of Eq. (17) to make it distinguishable from Eq. (5). Re- flected powers, respectively, and S 11 can be understood ffi flection coefficients of this type can be traced back to [55]. as the reflection coe cient for power. In contrast, when Equation (17) reduces to Eq. (2) when Z is real. When Zref is complex, the last term in Eq. (24) kicks in, and ref 2 2 = |a1| /2 and |b1| /2 can no longer be interpreted as pow- S P11 (ZS) 0, the power absorbed by the load ZL is maximized, and the absorbed power equals the available power, ers [10], [52], [59]. This might appear undesirable properties of a1 and b1 as defined by Eqs. (7) and (8). Pavs, of the signal source. On the other hand, ap1 and bp1 are defined so that the | / |2 | |2 ≡ ES,rms 2 = ES . same form as Eq. (25) results even when Zref is complex: Pavs (18) (ZS) 8 (ZS) = 1 | |2 −| |2 = 1| |2 − 2 . PL ap1 bp1 ap1 1 S P11 (Zref ) (26) ES is the amplitude of the voltage source in Fig. 2, and ES,rms 2 2 is its root-mean-square (rms) value. The “waves” incident This is highly pleasing compared to the seemingly awkward upon the load and reflected back in Eq. (17) are [54], [56]– Eq. (24), and thereafter, power-wave S parameters became [58] network theorists’ favorite definition of complex-referenced 1 V + Z I S parameters [52], [60], [61]. Power-wave S parameters also a = 1 ref 1 , (19) p1 saw widespread adoption by microwave engineers, too [12], (Zref) 2 − ∗ [14], [62]–[65]. But as we will see, the pleasing property 1 V1 Zref I1 bp1 = , (20) comes at a price. At this point, I only point out the fact that (Zref) 2 the last term in Eq. (24) can’t be nulled out; it still lurks in bp1 Eq. (26). Otherwise, the conservation of energy would be S = . (21) P11 (Zref ) violated. In this sense, nothing is fundamentally wrong with ap1 Eq. (24). Also note that in Eq. (26) the reflection coefficient 2 ap1 and bp1 are usually referred to as the power waves [54], for power is the scalar quantity |S P11| , not the complex S P11. although they have the dimensions of square root of power. The physical meaning of its phase, arg S P11, is not as clear As mentioned earlier, Eqs. (7) and (8) are not power waves. as arg S 11 [48], [65].

2.4 Power Absorbed by a Load ∗ 2.5 Transmission Line Terminated with Z0 P What about the power, L, absorbed by the load in Fig. 1? What if the terminating load impedance in Fig. 1 is ZL = From Eqs. (6) through (10), we get ∗ Z0, as shown in Fig. 4? Looking leftward into the line from ∗ + − + − ∗ the load, the input impedance is Z0. The natural reference (V1I ) [(V + V )(I + I ) ] P = 1 = 1 1 1 1 (22) impedance there, therefore, is Z = Z for both S and L 2 2 ref 0 11 IEICE TRANS. ELECTRON., VOL.E99–C, NO.10 OCTOBER 2016 1104

Fig. 5 A length of transmission line terminated with its characteristic impedance Z0. Traveling waves emanating from the signal source are absorbed by the load Z0 without any reﬂection.

Fig. 7 A passive ZL’s range of arg ZL on an impedance plane. | arg ZL|≤ π/2.

∗ equal to Pavs? Perhaps by changing the load from Z0? What happens then to S 11 and S P11 at the right end of the line?

2.6 Moduli of Reﬂection Coeﬃcients

Most textbooks state that the modulus of a passive load’s ffi (ZL with (ZL) ≥ 0) reflection coefficient is at most unity. Fig. 6 Input reflection coe cients S 11 and S P11 at the left end of the line. This statement is valid for reflection coefficients defined by Eq. (2) or (17), but not for Eq. (5). If Zref is complex, |S | might become greater than unity (|S | > 1) S . Since Fig. 5 corresponds to the case where S = 0, 11 (Zref ) 11 (Zref ) P11 11 (Z0) even if Z is passive. In contrast, Eq. (17) always satis- S 0 in the case of Fig. 4. To be more specific, the L 11 (Z0) fies |S |≤1 for passive Z , which, again, might give voltage reflection coefficient of the load Z∗ is, from Eq. (5), P11 (Zref ) L 0 the impression that power waves, Eqs. (19) and (20), are su- ∗ Z − Z0 X perior to pseudo waves, Eqs. (7) and (8). S = 0 = −j 0 (Fig. 4), (27) 11 (Z0) ∗ + Let’s look more closely at what this is about [59].Let Z0 Z0 R0 ZL where Z = R + jX . This means that the line wouldn’t z ≡ . (29) 0 0 0 L Z appear infinitely long as seen from the signal source. How- ref ever, since the leftward and rightward input impedances at Then, from Eq. (5), ∗ the load are, respectively, Z0 and Z0, the power-wave reflec- − ffi = zL 1. tion coe cient of the load is, from Eq. (17), S 11 (Zref ) (30) zL + 1 ∗ − ∗ Z0 Z0 ≥ | |≤π/ S = = 0 (Fig. 4). (28) Since (ZL) 0 by passivity assumption, arg ZL 2, as P11 (Z0) Z∗ + Z 0 0 shown in Fig. 7. Let Zref = Z0, where Z0 is given by Eq. (3). Assuming that our transmission line is an ordinary right- This means that the power PL (Eq. (26)) absorbed by the , , , > > load is maximized. But wait. In Fig. 5 (not Fig. 4), all handed line [66] with R L G C 0, we have (Zref) 0. Since the complex square root function is given by traveling waves are absorbed by the load as suggested by S = 0. Equation (24) with S = 0 seems to suggest arg z 11 (Z0) 11 z1/2 = ± |z| exp j , (31) that PL is maximized in the case of Fig. 5, too. What’s going 2 on here? | arg Z | <π/4 as shown in Fig. 8. From Eq. (29) and Figs. 7 The short answer: |a |2/2 = P > |a |2/2; see the first ref p1 avs 1 and 8, we get terms of Eqs. (24) and (26). The power flowing out from the signal source in Fig. 5 is less than its available power, 3π | arg z | < , (32) Eq. (18). In this rough sketch, we pretended that → 0 L 4 and ignored the power dissipated by the lossy transmission as shown in Fig. 9. Let us now take a look at the numera- line itself. That must, of course, be taken into consider- tor and the denominator of Eq. (30) on a complex plane ation in practice in power transfer problems. What is the (Fig. 10). It is geometrically clear from Fig. 10 that |zL − 1| actual power available at the right end of the transmission can be greater than |zL + 1|, and hence |S 11| > 1 is possible. line in Fig. 4? It must be less than Pavs.Evenattheleft Analytically, end of the line in Fig. 4, the power that flows into the line − ∗ − | |2 + − should, in general, be less than Pavs due to the mismatch 2 zL 1 zL 1 zL 1 2 (zL) |S | = · = . (33) there (S 0 in Fig. 6). How can that power be made 11 (Zref ) + ∗ + | |2 + + P11 (ZS) zL 1 zL 1 zL 1 2 (zL) AMAKAWA: SCATTERED REFLECTIONS ON SCATTERING PARAMETERS 1105

Fig. 11 A Norton signal source with an admittance YS feeds a load impedance ZL.

| | low frequencies [67].AS 11 (Zref ) value signiﬁcantly greater than unity might, therefore, be an artifact of unrealistic sim- Fig. 8 The range of arg Zref on an impedance plane when Zref = Z0. | | <π/ ulation, for example. But any discomfort associated with arg Zref 4. | |≤ even the slightest deviation from S 11 (Zref ) 1 might as well be an “artifact” of human mind, because no laws of physics | |≤ demand that S 11 (Zref ) 1.

2.7 Smith Chart and Reﬂection Coeﬃcients

The result of §2.6 immediately leads to a disturbing conclu-

sion: If Zref is complex, a locus of a passive load’s S 11 (Zref ) can stray out of the unit circle on a Zref-centered Smith chart [69]. Note also that Zref = Z0 is usually frequency- dependent, which might be another nuisance. | |≤ Given the fact that S P11 (Zref ) 1 for passive loads,

you might expect S P11 (Zref ) helps here too. But it’s not that simple. Recall that the Smith chart is derived from Fig. 9 The range of arg z , defined by Eq. (29), on a complex plane. L Eq. (5). Since Eq. (17) is different from Eq. (5), we can- | arg zL| < 3π/4. not plot S P11 (Zref ) on a Smith chart in the usual way we know. Let’s consider an ideal “short” (ZL = 0). We all know that short’s reflection coefficient is S 11 = −1. This = − (S 11 (Zref ) 1) is valid whether or not Zref is real in Eq. (5). But we get from Eq. (17) and ZL = 0 Z∗ = − ref . S P11 (Zref ) (short) (34) Zref − Evidently, S P11 (Zref ) 1 unless Zref is real. This appears to go against microwave engineer’s common sense. Now, how

can we plot S P11 (Zref ) on a Smith chart? Note that [54], [64] [R + j(X + X )] − R Z − Rref S = L L ref ref = L , (35) Fig. 10 Eq. (30)’s numerator z − 1 and the denominator z + 1ona P11 (Zref ) L L [RL + j(XL + Xref)] + Rref Z + Rref complex plane. L Zref = Rref + jXref, (36) ≡ + + . ZL RL j(XL Xref) (37) (zL) in Eq. (33) can be positive or negative because of | | > < Eq. (32). S 11 (Zref ) 1 results if (zL) 0. Equation (35) has the same form as Eq. (5). By plotting ZL | | The fact that S 11 (Zref ) can exceed unity even for a pas- instead of ZL on an Rref-centered Smith chart, we can find sive load has long been known [10], [11], [59], [67]–[70], the absolute value and the argument of S P11 (Zref ). Note that but unfortunately, only to not so many of those who have Rref will often be frequency-dependent. known it. It is also well understood that no laws of physics But this is not the whole story. The above is valid if | | > are violated even when S 11 (Zref ) 1, however uncomfort- the signal source is a Thevenin´ equivalent as in Fig. 2. But able you might feel with it. The√ theoretical maximum value if you start the theoretical development from a Norton-type | | + . ff of S 11 (Zref ) is stupendous 1 2 2 41 [59]! In reality, signal source (Fig. 11), you can get a di erent conclusion! It |(Z0)| is usually a small fraction of (Z0)(> 0), and the can be shown that (see [48] and Appendix F of [65]) another | | > values of S 11 (Z0) ( 1) we encounter in real life (measure- possible and perfectly valid definition of the power-wave re- ments especially) will be fairly close to unity except at very flection coefficient is IEICE TRANS. ELECTRON., VOL.E99–C, NO.10 OCTOBER 2016 1106

∗ Z ZL − Z Z S = ref · ref = ref S . (38) Pseudo waves, Eqs. (7) and (8), are fictitious in that PV11 (Zref ) ∗ + ∗ P11 (Zref ) Zref ZL Zref Zref they represent traveling waves that would be present in the transmission line if its characteristic impedance were The ‘V’ in the subscript indicates that this reflection coef- Z ( Z ). Power waves are still more fictitious in that ffi ref 0 ficient is a voltage reflection coe cient. Although I didn’t their phases are not measurable and can be dictated arbi- S explain this, contrary to popular belief, P11 (Zref ) should be trarily, usually following the convention started in the early ffi § understood as a current reflection coe cient [48] ( 2.8). days [56]–[58]. Is it possible that the phases of power waves Equation (17) reduces to Eq. (16) when Zref is real. Equa- will one day turn out physically significant somehow, just as ff † tion (38) follows from a somewhat di erent definition of the phase of the wave function turned out to have observ- power waves than Eqs. (19) and (20): able significance in quantum mechanics [72]? I prefer to doubt that. The difference between Eqs. (17) and (38) arises 1 I1 + YrefV1 a = , (39) only for a mathematical reason: p1 2 (Yref) ∗ −1 −1 1 I1 − Y V1 z (z) , (43) b = ref , (40) p1 2 (Yref) where z is a complex number. For example, if bp1 S ≡ . (41) 1 PV11 (Yref ) Z = R + jX = , (44) ap1 S S S YS = / Yref 1 Zref is the reference admittance. From Eq. (38), then short’s reflection coefficient is S = −1! This might ap- PV11 1 R X pear more desirable than Eq. (34). However, the definition Y = = S − j S . (45) S Z 2 + 2 2 + 2 Eq. (41) is rarely adopted in practice. S RS XS RS XS In any case, why does such arbitrariness in the defi- The available power of a signal source can be written as nition of power waves and associated SP parameters arise? Thevenin´ and Norton signal sources are just two equiva- 1 P = (V I∗) (46) lent representations of the same physical source. Note that avs 2 i i | | = | | S P11 (Zref ) S PV11 (Yref ) and that the arbitrariness is in the 1 ∗ 1 ∗ § = (Z )I I = a a (Fig. 2) (47) phase. This is related to the remark made at the end of 2.4. 2 S i i 2 p1 p1 Other definitions than Eqs. (21) and (41) are also possible. 1 1 = (Y )V V∗ = a a∗ (Fig. 11), (48) For example, yet another valid current-based definition of 2 S i i 2 p1 p1 power-wave reflection coefficient is where − ∗ ZL Zref S = −S = − . (42) ES PI11 (Zref ) P11 (Zref ) ≡ , ZL + Zref Ii (incident current) (49) 2(ZS) Equation (42) reduces to Eq. (14) when Z is real. It can I ref V ≡ S (incident voltage), (50) further be shown that [51] the choice of phases of power i 2 (YS) waves is arbitrary. What does this physically mean? a ≡ (Z )I (Eq. (19)), (51) p1 S i ≡ . 2.8 Measurable Waves and S Parameters ap1 (YS)Vi (Eq. (39)) (52)

Measurement is the act of sensing some physical quantities. As noted in §2.1, Eqs. (7) and (8) with Zref = Z0 are physi- 3. Scattering Matrices (S Matrices) cal voltage traveling waves (multiplied by a real constant), and therefore these are measurable complex (phasor) quan- We have already learned significantly about 1-port S param- tities. Receivers in a calibrated VNA sense mathematically eters (reflection coefficients). It is straightforward to extend transformed version of these waves. Modern VNAs adopt them to multiports. analog-to-digital converters for voltage measurements [71]. What about the power waves, Eqs. (19) and (20) (or 3.1 Definitions and Uses Eqs. (39) and (40))? Power, too, is a physical and measurable quantity, but it’s a scalar quantity. Power measure- Multiport extension of Eqs. (6) through (8) are ments, therefore, reveal only |ap1| and |bp1|.ButSP pa- φ (Zrefi) Vi + ZrefiIi rameters given by Eq. (21) are complex quantities. How a = ej i (port i), (53) i (Zrefi) |Z | can we measure arg ap1 and arg bp1? Well, perhaps you refi 2 can’t [17], [46], [47]. If the arguments of power waves are †In quantum mechanics, probability is given by |ψ|2,whereψ not measurable quantities, that explains their arbitrariness is the complex wave function. Obviously, arg ψ doesn’t affect |ψ|2, (§2.7). at least in most elementary problems. AMAKAWA: SCATTERED REFLECTIONS ON SCATTERING PARAMETERS 1107 ⎡ ⎤ ··· ⎢ ⎥ ⎢ . . ⎥ ⎢ . .. ⎥ ⎢ . S Pij(Z , Z ) ⎥ S = ⎢ refi ref j ⎥ . (64) P(Zref ) ⎢ .. . ⎥ ⎢ , . . ⎥ ⎣⎢ S P ji (Zrefi Zref j) ⎦⎥ ··· Fig. 12 Series reactance as a 2-port. Equations (63) and (64) are also known as the generalized S parameter and the generalized S matrix, respectively [12], φ (Zref j) V j − Zref jI j b = ej j (port j), (54) [14], [63]. j (Zref j) |Z | 2 ref j For example, the SP parameters of a series reactance b j (Z ) (Fig. 12) with Z = Z and Z = Z are given by [14] = ref j , ref1 S ref2 L S ji (Zrefi, Zref j) (S parameter) (55) ai (Zrefi) ⎡ ⎤ jX + ZS + ZL − 2(ZS) ⎢ ··· ⎥ S P11 (Z , Z ) = , (65) ⎢ ⎥ S L jX + Z + Z ⎢ . . ⎥ S L ⎢ . .. ⎥ ⎢ . S ij(Z , Z ) ⎥ S = ⎢ refi ref j ⎥ . 2 (ZS) (ZL) (Zref ) ⎢ . . ⎥ (56) S , = S , = , (66) ⎢ . . ⎥ P21 (ZS ZL) P12 (ZS ZL) + + ⎢ , . . ⎥ jX ZS ZL ⎣⎢ S ji (Zrefi Zref j) ⎦⎥ ··· jX + ZS + ZL − 2(ZL) S , = . (67) P22 (ZS ZL) jX + Z + Z φ φ S L ej i and ej j in Eqs. (53) and (54) are phase factors that may be needed depending on the mapping mentioned in §1 [4]. Since They can often be made to disappear in Eq. (55), which (ZS) + (ZL) always does in Eq. (6). Z in Eq. (56) is the reference (ZS)(ZL) ≤ , (68) ref 2 impedance matrix: we can conclude from Eq. (66) that Zref = diag(Zref1, Zref2, ···). (57) ≤ S P21 (ZS, ZL) 1 (69) The moduli of reflection coefficients of a passive network may exceed unity if Zref is complex (§2.6). Likewise, as anticipated. the moduli of transmission coefficients of a passive network In majority of textbooks that cover complex-referenced may exceed unity if Zref is complex. For example, the S S matrices, the SP matrix is the complex-referenced S ma- matrix of a series reactance (Fig. 12) is given by [12] trix. But that doesn’t mean the pseudo-wave S matrices are unimportant. On the contrary, they are indispen- = 1 jX 2Zref . sable for microwave and millimeter-wave metrology, be- S(Zref ) (58) jX + 2Zref 2Zref jX cause some fundamental VNA calibration algorithms, such as thru-reflect-line (TRL) [4], [21], [73], are formulated us- It follows that ing complex-referenced S matrices (not SP matrices) [4], | |2 [74], [75]. Like it or not, you get complex-referenced S 2 4 Zref S Z = . (59) parameters (referenced to the natural reference impedance) 21 ( ref ) X2 + 4X(Z ) + 4|Z |2 ref ref from measurements, at least in some situations, and you −j(π/4) have no choice but to work with them. When you do get If X = 1 and Zref = e (Fig. 8), them, you will most likely want to convert them to 50-Ω- 2 4 referenced S parameters for further manipulation and sav- S 21 (Z ) = 1.84 > 1. (60) ref 12 − 4 · √1 + 4 ·|1|2 ing into files, because results like Eq. (60) are, at best, 2 confusing. Some simulators and file formats don’t sup-

This, of course, doesn’t violate any laws of physics, because port S(Zref ). The mathematical operation of changing refer- | |2 ence impedances is called the renormalization transforma- Eq. (60) isn’t power gain. Recall that in Eq. (24), S 11 (Zref ) isn’t a power reflection coefficient, either. tion [17], [47], [76], [77]. It can be done either by using a ↔ Similarly, Eqs. (19) through (21) can be extended to direct S(Zref ) S conversion formula [47], [76], [77],or (Zref ) multiports. by cascading appropriate conversion networks [4]. On the other hand, the use of complex-referenced SP 1 V + Z I = i refi i , matrices is not mandatory. They can be quite useful, for api (Zrefi) (incident on port i) (61) (Zrefi) 2 example, for amplifier design especially when lengths of − ∗ interconnecting transmission lines (not including intentional 1 V j Zref jI j bp j (Z ) = (out of port j), (62) stubs and delay lines) can be ignored. However, every- ref j (Z ) 2 ref j thing (including SP parameters) can be expressed in terms

bp j (Zref j) of real-referenced S parameters [63], possibly derived from S , = (S parameter), (63) P ji (Zrefi Zref j) P measured, complex-referenced pseudo-wave S parameters; api (Zrefi) IEICE TRANS. ELECTRON., VOL.E99–C, NO.10 OCTOBER 2016 1108 0e−γ S = −γ , (74) (Z0) e 0

regardless of the value of . This property is used in the formulation of TRL [4], [21], [73]. It also is the reason for the line’s (generally unknown) Z0 becoming the reference impedance (Zref = Z0) of the new reference planes after per- Fig. 13 Cascading is possible only if bp2 = ap3 and ap2 = bp3 are satis- fied. forming TRL calibration. More generally, a pair of reference impedances Zi1 and Zi2 that makes S 11 = S 22 = 0 are known as the image so you don’t have to use SP parameters if you don’t want to. impedances [78] of the 2-port. If You use them if you find them useful and/or less disturbing S S because moduli of passive SP parameters are guaranteed to = 11 12 , S(Zref1, Zref2) (75) be less than or equal to unity. It is advisable in this case S 21 S 22 too that you perform S → S before saving your P(Zref ) (50 Ω) then data. Make sure to use the right formula [54] (different from −θ → i12 S(Zref ) S(50 Ω)) for the conversion. 0e S , = −θ , (76) In any case, you must be clear about which type of S (Zi1 Zi2) e i21 0 parameter you are dealing with. Conversion formulas for θ θ Z ↔ S and Z ↔ S , for example, are different. Note where i21 and i12 are the image propagation parameters. (Zref ) P(Zref ) = = θ = θ = γ also that SP parameters have further unusual properties. Obviously, Zi1 Zi2 Z0 and i21 i12 in the case of transmission lines. 3.2 Cascading The matrix elements of the complex-referenced SP matrix of a length of transmission line are Let us introduce the power-wave cascading matrix TP 2 − ∗ γ + − ∗ (Z0 Zref1Zref2) tanh Z0(Zref2 Zref1) (Fig. 13). S , = , P11 (Zref1 Zref2) 2 + γ + + (Z0 Zref1Zref2) tanh Z0(Zref1 Zref2) bp1 ap2 TP11 TP12 ap2 (77) = TP = . (70) ap1 bp2 TP21 TP22 bp2 2 − ∗ γ + − ∗ (Z0 Zref1Zref2) tanh Z0(Zref1 Zref2) S , = , P22 (Zref1 Zref2) 2 + γ + + In terms of the elements of SP, (Z0 Zref1Zref2) tanh Z0(Zref1 Zref2) (78) 1 S S − S S S T = P12 P21 P11 P22 P11 . (71) = P − S P21 (Zref1, Zref2) S P12 (Zref1, Zref2) S P21 S P22 1 2Z0 (Zref1)(Zref2)/ cosh γ We want = . (79) 2 + γ + + (Z0 Zref1Zref2) tanh Z0(Zref1 Zref2) bp1 = ap2 = ap4 TP1 TP1TP2 (72) Note that Z = Z = Z doesn’t make S = S = 0. ap1 bp2 bp4 ref1 ref2 0 P11 P22 In this sense, SP matrices of lossy transmission lines are not to be valid. This requires that bp2 = ap3 and ap2 = bp3 be sat- terribly useful. = isfied at the interconnecting plane (Fig. 13). Since V2 = V3 A pair of reference impedances that makes S P11 = − = ∗ S = 0 is called the conjugate image impedances [55]. and I2 I3 hold there, Zref2 Zref3 follows from Eqs. (61) P22 and (62) as a requirement [46]. This is in stark contrast with S P11 = S P22 = 0 means that simultaneous conjugate match- the ordinary T matrices, for which Zref2 = Zref3 is required. ing is achieved at the input and output ports. Since a trans- So be extra careful when doing cascading operations with mission line is a symmetric 2-port, the conjugate image SP parameters or TP matrices. impedances are the same for both ports. Unlike the image impedance Z0, the conjugate image impedance depends on 3.3 S Matrices of a Length of Transmission Line . This implies that it will be difficult to formulate VNA calibration algorithms based on power waves and to meas- The complex-referenced S matrix of a length, , of transmis- ure SP parameters directly. sion line is 3.4 Amplifier Gains 1 S = (Zref ) 2 + 2 + γ Z Z 2Z0Zref coth( ) The use of S , instead of S , can be beneficial for 0 ref P11 (Zref ) 11 (Zref ) 2 − 2 / γ reasons explained in §2.6. What about the use of 2-port SP × Z0 Zref 2Z0Zref sinh( ) . 2 2 (73) 2Z0Zref/ sinh(γ) Z − Z parameters? Suppose you are designing a multi-stage am- 0 ref plifier. Consider a single stage within it (Fig. 14). Its 50-Ω- 2 With Zref = Z0, Eq. (73) reduces to referenced power gain is |S 21 (50 Ω)| . What is its power gain AMAKAWA: SCATTERED REFLECTIONS ON SCATTERING PARAMETERS 1109

on ΓS, and the operating gain (or power gain) GP, which depends only on ΓL, can be written concisely as S P21 with appropriate reference impedances (Fig. 15).

Γ ≡ Pavn = Γ , Γ∗ GA( S) GT( S out) (85) Pavs Fig. 14 A single amplifying stage within a multi-stage ampliﬁer. 1 −|Γ |2 1 = S ·|S |2 · (86) |1 − S Γ |2 21 1 −|Γ |2 11 S out 2 S P21 (ZS,ZL) 2 ∗ = = S P21 (Z ,Z ) , (87) − 2 S out 1 S P22 (ZS,ZL)

S 21ΓSS 12 Γout = S 22 + , (88) 1 − S 22ΓS

Γ ≡ PL = Γ∗ , Γ GP( L) GT( in L) (89) Pin 1 1 −|Γ |2 = ·|S |2 · L (90) 1 −|Γ |2 21 |1 − S Γ |2 in 22 L 2 S P21 (ZS,ZL) 2 Fig. 15 Power gains of a 2-port. Γin: Input reflection coefficient. Γout: = = S ∗ , , (91) 2 P21 (Zin ZL) Output reflection coefficient. P : Power available from the source. P : − avs in 1 S P11 (ZS,ZL) Power absorbed by the 2-port network. Pavn: Power available from the 2-port network. Pin: Power absorbed by the load. ITN: Impedance trans- forming network. S 12ΓLS 21 Γin = S 11 + . (92) 1 − S 22ΓL under the operating condition (mismatches with the preced- Equations (87) and (91) are easier to grasp than Eqs. (86) ing and following stages included)? and (90). This conceptual gain is the power of using SP pa- The transducer gain GT is the gain that includes the rameters. Since GA and GP are the gains when conjugately mismatches with the “source” and the “load” (Fig. 15). GT, matched on the load side and the source side, respectively, ffi therefore, depends both on the source reflection coe cient G (Γ , Γ ) ≤ G (Γ ) and G (Γ , Γ ) ≤ G (Γ ) hold. Γ ffi Γ T S L A S T S L P L S and the load reflection coe cient L. When the 2-port in question is unconditionally stable, Z − 50 the maximum possible value of GT equals the maximum Γ = S , (80) S Z + 50 available gain GMA. Since GMA is a property of a 2-port, S Γ Γ Z − 50 it depends neither on S nor on L. Γ = L , (81) L + ZL 50 ≡ Γ , Γ = 2 , GMA GT( ci1 ci2) S P21 (Zci1,Zci2) (93) Γ , Γ ≡ PL GT( S L) (82) 1 + Γci j Pavs Z = ( j = 1, 2). (94) ci j 1 − Γ (1 −|Γ |2)(1 −|Γ |2)|S |2 ci j = L S 21 . (83) | − Γ − Γ − Γ Γ |2 (1 S 22 L)(1 S 11 S) S 12S 21 S L Zci1 and Zci2 are the conjugate image impedances [55] mentioned in §3.3. Γ = Γ∗ and Γ = Γ∗ hold. While Eq. (83) already answers the question about the gain in ci1 out ci2 in terms of 50-Ω-referenced S parameters, it is worth men- 4. Concluding Remarks tioning that GT can also be written concisely as follows: 2 In this article we looked at two types of complex-referenced GT(ΓS, ΓL) = S P21 (Z ,Z ) . (84) S L S parameters. Both show somewhat weird properties Although Eqs. (83) and (84) are mathematically completely compared to the familiar 50-Ω-referenced S parameters the same, the latter should be much easier to understand in- and serve different purposes. Pseudo-wave S parameters tuitively. When, for example, writing a program for design (Eq. (55)) are indispensable for millimeter-wave metrology optimization, if a library of functions are available for ma- because your measurement reference planes might end up nipulating SP parameters, it is not only conceptually easier having complex reference impedances. In such a case, to use Eq. (84) but also less error-prone than writing Eq. (83) you have no choice but to deal with them. The most im- in the program. portant thing to do about them is to convert them to real- Likewise, the available gain GA, which depends only referenced S parameters by renormalization transformation IEICE TRANS. ELECTRON., VOL.E99–C, NO.10 OCTOBER 2016 1110

→ [5] D.F. Williams, L.A. Hayden, and R.B. Marks, “A complete multi- S(Zref ) S(50 Ω). Power-wave S parameters (Eq. (63)) can be useful for amplifier design and evaluation of antennas, but mode equivalent-circuit theory for electrical design,” J. Res. Natl. Inst. Stand. Technol., vol.102, no.4, pp.405–423, 1997. their use is not required because SP parameters are derived quantities from measurable S parameters (§2.8). Either way, [6] D.F. Williams and B.K. Alpert, “Causality and waveguide circuit theory,” IEEE Trans. Microw. Theory Tech., vol.49, no.4, it is essential that you clarify which variant of complex- pp.615–623, April 2001. referenced S parameters you are using and what the values [7] F. Olyslager, D. De Zutter, and A.T. de Hoop, “New reciprocal cir- of your reference impedances are. cuit model for lossy waveguide structures based on the orthogonal- One very practically important point that I haven’t been ity of the eigenmodes,” IEEE Trans. Microw. Theory Tech., vol.42, able to discuss is simulators and related programs. We no.12, pp.2261–2269, Dec. 1994. [8] F. Olyslager, Electromagnetic Waveguides and Transmission Lines, need simulators written by somebody else because we can’t Oxford University Press, 1999. write everything ourselves. But the confusion surround- [9] W.C. Johnson, Transmission Lines and Networks, McGraw-Hill, ing complex-referenced S parameters seen in the literature 1950. is, unfortunately and understandably, reflected in simulators [10] N. Marcuvitz, editor, Waveguide Handbook, Dover, 1965; Republi- and their documents, too. Very often, complex-referenced cation of McGraw-Hill, 1951. S parameters are poorly or not at all documented, even if [11] R.W.P. King, Transmission-Line Theory, McGraw-Hill, 1955. [12] R.E. Collin, Foundations for Microwave Engineering, 2nd ed., they are implemented somehow. If you have a support con- Wiley-Interscience, 2001. tract with your simulator vendor, ask your support engineer. [13] R. Collier, Transmission Lines: Equivalent Circuits, Electro- You are lucky if your support engineer doesn’t share the said magnetic Theory, and Photons, Cambridge University Press, 2013. “misfortune.” If you are not so lucky, you must somehow [14] R. Mavaddat, Network Scattering Parameters, World Scientific, figure out which ones are implemented in your simulator. 1996. [15] T.H. Lee, Planar Microwave Engineering: A Practical Guide to The outcome might not be as simple as “this simulator im- Theory, Measurement, and Circuits, Cambridge University Press, plements which variant of S parameters.” Some inconsis- 2004. tency could possibly exist in integrated simulation environ- [16] E. Bogatin, Signal and Power Integrity—Simplified, 2nd ed., Pren- ments. Some commercial microwave simulators implement tice Hall, 2009. [17] D. Woods, “Multiport-network analysis by matrix renormalisation SP(Z ) in their circuit simulation environment. But in EM ref employing voltage-wave S -parameters with complex normalisa- simulation, more suitable one to use would be S(Zref ).How ff tion,” Proc. IEE, vol.124, no.3, pp.198–204, March 1977. are di erent parts of an integrated environment interfaced [18] L.N. Dworsky, Modern Transmission Line Theory and Applications, with each other when reference impedances are complex? Wiley-Interscience, 1979. What can we do with poorly documented simulators? [19] G. Miano and A. Maffucci, Transmission Lines and Lumped Cir- We must at least be vigilant and be very clear about what cuits, Academic Press, 2001. we want to do using which kind of S parameter. Perhaps, [20] J.C. Freeman, Fundamentals of Microwave Transmission Lines, we should also be putting more effort into application en- Wiley-Interscience, 1996. [21] G.F. Engen, Microwave Circuit Theory and Foundations of Micro- gineering, at least in the short term, till awareness of the wave Metrology, Peter Peregrinus, 1992. importance of the issue grows in the field. It seems to me [22] M. Fujishima, S. Amakawa, K. Takano, K. Katayama, and T. that the significance of application engineering in a situation Yoshida, “Tehrahertz CMOS design for low-power and high-speed like this is grossly underappreciated. See [79], an excellent wireless communication,” IEICE Trans. Electron., vol.E98-C, no.12, application-engineering paper on a different but related sub- pp.1091–1104, Dec. 2015. [23] S. Hara, K. Katayama, K. Takano, I. Watanabe, N. Sekine, A. ject. Kasamatsu, T. Yoshida, S. Amakawa, and M. Fujishima, “Compact 160-GHz amplifier with 15-dB peak gain and 41-GHz 3-dB band- Acknowledgments width,” Radio Freq. Integrated Circuits Symp., pp.7–10, May 2015. [24] S. Hara, I. Watanabe, N. Sekine, A. Kasamatsu, K. Katayama, I would like to thank Professor Kiyomichi Araki and Pro- K. Takano, T. Yoshida, S. Amakawa, and M. Fujishima, “Com- fessor Atsushi Sanada for giving me the opportunity to pact 138-GHz amplifier with 18-dB peak gain and 27-GHz 3-dB present the earlier version of this article at MWE 2015 [44]. bandwidth,” IEEE Int. Symp. Radio-Freq. Integration Technol., pp.55–57, Aug. 2015. This work was supported in part by Fujikura Ltd. and [25] K. Takano, K. Katayama, S. Amakawa, T. Yoshida, and M. JSPS.KAKENHI. Fujishima, “Wireless digital data transmission from a 300-GHz CMOS transmitter,” Electron. Lett., vol.52, no.15, pp.1353–1355, References 2016. [26] K. Katayama, K. Takano, S. Amakawa, S. Hara, A. Kasamatsu, K. [1] H.J. Carlin, “The scattering matrix in network theory,” IRE Trans. Mizuno, K. Takahashi, T. Yoshida, and M. Fujishima, “A 300GHz Circuit Theory, vol.3, no.2, pp.88–97, June 1956. 40nm CMOS transmitter with 32-QAM 17.5Gb/s/ch capability over [2] G.A. Campbell and R.M. Foster, “Maximum output networks for 6 channels,” Int. Solid-State Circuits Conf., pp.342–343, Feb. 2016. telephone substation and repeater circuits,” Trans. American Insti- [27] K. Takano, K. Katayama, T. Yoshida, S. Amakawa, and M. tute of Electrical Engineers, vol.39, no.1, pp.231–290, 1920. Fujishima, “124-GHz CMOS quadrature voltage-controlled oscilla- [3] C.G. Montgomery, R.H. Dicke, and E.M. Purcell, editors, Principles tor with fundamental injection locking,” Asian Solid-State Circuits of Microwave Circuits, Dover, 1965; Republication of McGraw- Conf., pp.77–80, Nov. 2015. Hill, 1948. [28] S. Mizukusa, K. Takano, K. Katayama, S. Amakawa, T. Yoshida, [4] R.B. Marks and D.F. Williams, “A general waveguide circuit theory,” and M. Fujishima, “Analytical design of small-signal amplifier with J. Res. Natl. Inst. Stand. Technol., vol.97, no.5, pp.533–562, 1992. maximum gain in conditionally stable region,” Asia-Pacific Micro- AMAKAWA: SCATTERED REFLECTIONS ON SCATTERING PARAMETERS 1111

wave Conf., pp.774–776, Nov. 2014. [45] S. Amakawa, “Yet another introduction to S parameters,” 27th [29] S. Amakawa, “Theory of gain and stability of small-signal amplifiers Workshop on Silicon Analog & RF Electronics, Tutorial, Nov. 2011 with lossless reciprocal feedback,” Asia-Pacific Microwave Conf., (in Japanese). pp.1184–1186, Nov. 2014. [46] R.B. Marks and D.F. Williams, Comments on “Conversions between [30] S. Amakawa, A. Orii, K. Katayama, K. Takano, M. Motoyoshi, T. S , Z, Y, H, ABCD,andT parameters which are valid for complex Yoshida, and M. Fujishima, “Design of well-behaved low-loss mil- source and load impedances,” IEEE Trans. Microw. Theory Tech., limetre-wave CMOS transmission lines,” IEEE Workshop on Signal vol.43, no.4, p.914, April 1995. and Power Integrity, pp.1–4, May 2014. [47] H. Dropkin, Comments on “A rigorous technique for measuring the [31] K. Takano, S. Amakawa, K. Katayama, M. Motoyoshi, and M. scattering matrix of a multiport device with a two-port network ana- Fujishima, “Modeling of short-millimeter-wave CMOS transmis- lyzer,” and reply by J.C. Tippet and R.A. Speciale, IEEE Trans. sion line with lossy dielectrics with specific absorption spectrum,” Microw. Theory Tech., vol.31, no.1, pp.79–81, Jan. 1983. IEICE Trans. Electron., vol.E96-C, no.10, pp.1311–1318, Oct. 2013. [48] J.C. Freeman, “On the interpretation of scattering parameters,” [32] K. Takano, S. Amakawa, K. Katayama, M. Motoyoshi, and M. NASA Technical Documents, nasa techdoc 20000056853, NASA, Fujishima, “Characteristic impedance determination technique for 1999. CMOS on-wafer transmission line with large substrate loss,” 79th [49] D. Williams, “Traveling waves and power waves,” IEEE Microw. Automatic RF Techniques Group (ARFTG) Conf., pp.1–4, June Mag., vol.14, no.7, pp.38–45, 2013. 2012. [50] S. Llorente-Romano, A. Garca-Lamperez,´ T.K. Sarkar, and M. [33] S. Amakawa, K. Katayama, K. Takano, T. Yoshida, and M. Salazar-Palma, “An exposition on the choice of the proper S Fujishima, “Comparative analysis of on-chip transmission line parameters in characterizing devices including transmission lines de-embedding techniques,” IEEE Int. Symp. Radio-Freq. Integration with complex reference impedances and a general methodology Technol., pp.91–93, Aug. 2015. for computing them,” IEEE Antennas Propag. Mag., vol.55, no.4, [34] K. Katayama, S. Amakawa, K. Takano, and M. Fujishima, pp.94–112, Aug. 2013. “300-GHz MOSFET model extracted by an accurate cold-bias [51] T. Hirano, “Review and another derivation of the power wave,” de-embedding technique,” IEEE MTT-S Int. Microw. Symp., Microw. Opt. Technol. Lett., vol.57, no.1, pp.26–28, Jan. 2015. pp.1–4, May 2015. [52] H.J. Carlin and A.B. Giordano, Network Theory: An Introduction to [35] R. Goda, S. Amakawa, K. Katayama, K. Takano, T. Yoshida, and Reciprocal and Nonreciprocal Circuits, Prentice-Hall, 1964. M. Fujishima, “Modeling of wideband decoupling power line for [53] T.K. Ishii, Microwave Engineering, 2nd ed., Harcourt Brace Jo- millimeter-wave CMOS circuits,” IEEE Int. Symp. Radio-Freq. In- vanovich, 1989. tegration Technol., pp.151–153, Aug. 2015. [54] K. Kurokawa, “Power waves and the scattering matrix,” IEEE Trans. [36] S. Amakawa, R. Goda, K. Katayama, K. Takano, T. Yoshida, and M. Microw. Theory Tech., vol.13, no.2, pp.194–202, March 1965. Fujishima, “Wideband CMOS decoupling power line for millime- [55] S. Roberts, “Conjugate-image impedances,” Proc. IRE, vol.34, no.4, ter-wave applications,” IEEE MTT-S Int. Microw. Symp., pp.1–4, pp.198P–204P, April 1946. May 2015. [56] P. Penfield, Jr., “Noise in negative-resistance amplifiers,” IRE Trans. [37] R. Goda, S. Amakawa, K. Katayama, K. Takano, T. Yoshida, and Circuit Theory, vol.7, no.2, pp.166–170, June 1960. M. Fujishima, “Characterization of wideband decoupling power [57] D.C. Youla, “On scattering matrices normalized to complex port line with extremely low characteristic impedance for millime- numbers,” Proc. IRE, vol.49, no.7, p.1221, July 1961. ter-wave CMOS circuits,” Int. Conf. Microelectronic Test Struct., [58] K. Kurokawa, “Actual noise measure of linear amplifiers,” Proc. pp.220–223, March 2015. IRE, vol.9, no.9, pp.1391–1397, Sept. 1961. [38] A. Orii, M. Suizu, S. Amakawa, K. Katayama, K. Takano, M. [59] R.B. Adler, L.J. Chu, and R.M. Fano, Electromagnetic Energy Motoyoshi, T. Yoshida, and M. Fujishima, “On the length of thru Transmission and Radiation, Wiley, 1960. standard for TRL de-embedding on Si substrate above 110 GHz,” [60] V. Belevitch, Classical Network Theory, Holden-Day, 1968. Int. Conf. Microelectronic Test Struct., pp.81–86, March 2013. [61] N. Balabanian and T.A. Bickart, Electrical Network Theory, Wiley, [39] K. Takano, M. Motoyoshi, T. Yoshida, K. Katayama, S. Amakawa, 1969. and M. Fujishima, “Evaluation of CMOS differential transmission [62] K. Kurokawa, An Introduction to the Theory of Microwave Circuits, lines as two-port networks with on-chip baluns in millimeter-wave Academic Press, 1969. band,” 83rd Automatic RF Techniques Group (ARFTG) Conf., [63] G. Gonzalez, Microwave Transistor Amplifiers: Analysis and De- pp.1–5, June 2014. sign, 2nd ed., Prentice Hall, 1996. [40] K. Takano, K. Katayama, T. Yoshida, S. Amakawa, M. Fujishima, [64] M.W. Medley, Jr., Microwave and RF Circuits: Analysis, Synthesis S. Hara, and A. Kasamatsu, “Calibration of process parameters for and Design, Artech House, 1992. electromagnetic field analysis of CMOS devices up to 330 GHz,” [65] R.J. Weber, Introduction to Microwave Circuits: Radio Frequency IEEE Int. Symp. Radio-Freq. Integration Technol., pp.94–96, Aug. and Design Applications, IEEE Press, 2001. 2015. [66] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmis- [41] K. Takano, K. Katayama, S. Mizukusa, S. Amakawa, T. Yoshida, sion Line Theory and Microwave Applications, Wiley-Interscience, and M. Fujishima, “Systematic calibration procedure of process 2006. parameters for electromagnetic field analysis of millimeter-wave [67] R.A. Chipman, Theory and Problems of Transmission Lines, CMOS devices,” Int. Conf. Microelectronic Test Struct., pp.1–5, McGraw-Hill, 1968. March 2015. [68] R.J. Vernon and S.R. Seshadri, “Reflection coefficient and reflected [42] S. Amakawa, A. Orii, K. Katayama, K. Takano, M. Motoyoshi, T. power on a lossy transmission line,” Proc. IEEE, vol.57, no.1, Yoshida, and M. Fujishima, “Process parameter calibration for mil- pp.101–102, Jan. 1969. limeter-wave CMOS back-end device design with electromagnetic [69] J. Kretzschmar and D. Schoonaert, “Smith chart for lossy transmis- field analysis,” Int. Conf. Microelectronic Test Struct., pp.182–187, sion lines,” Proc. IEEE, vol.57, no.9, pp.1658–1660, Sept. 1969. March 2014. [70] S.C. Dutta Roy, “Some little-known facts about transmission lines [43] J.A. Dobrowolski, Microwave Network Design Using the Scattering and some new results,” IEEE Trans. Education, vol.53, no.4, Matrix, Artech House, 2010. pp.556–561, Nov. 2010. [44] S. Amakawa, “Demystifying S parameters: confusion surround- [71] J.P. Dunsmore, Handbook of Microwave Component Measurements ing S-parameter definitions,” Microwave Workshops & Exhibition, with Advanced VNA Techniques, Wiley, 2012. FR6A-1, pp.1–6, Nov. 2015 (in Japanese). [72] J.J. Sakurai, Modern Quantum Mechanics, revised edition, Addison- IEICE TRANS. ELECTRON., VOL.E99–C, NO.10 OCTOBER 2016 1112

Wesley, 1994. [73] G.F. Engen and C.A. Hoer, “ ‘Through-reﬂect-line’: An im- proved technique for calibrating the dual six-port automatic network analyzer,” IEEE Trans. Microw. Theory Tech., vol.27, no.12, pp.987–993, Dec. 1979. [74] H.-J. Eul and B. Schiek, “A generalized theory and new calibration procedures for network analyzer self-calibration,” IEEE Trans. Microw. Theory Tech., vol.39, no.4, pp.724–731, April 1991. [75] A. Rumiantsev and N. Ridler, “VNA calibration,” IEEE Microw. Mag., vol.9, no.3, pp.86–99, June 2008. [76] J.C. Tippet and R.A. Speciale, “A rigorous technique for measuring the scattering matrix of a multiport device with a 2-port network analyzer,” IEEE Trans. Microw. Theory Tech., vol.30, no.5, pp.661–666, May 1982. [77] E. Van Lil, Comments on “A rigorous technique for measuring the scattering matrix of a multiport device with a two-port network analyzer,” IEEE Trans. Microw. Theory Tech., vol.33, no.3, pp.286–287, March 1985. [78] G. Matthaei, L. Young, and E.M.T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, Artech House, Norwood, Massachusetts, 1980; Republication of McGraw- Hill, 1964. [79] E. Episkopou, S. Papantonis, W.J. Otter, and S. Lucyszyn, “Deﬁn- ing material parameters in commercial EM solvers for arbitrary met- al-based THz structures,” IEEE Trans. Terahertz Sci. Technol., vol.2, no.5, pp.513–524, Sept. 2012.

Shuhei Amakawa received B.S., M.S., and Ph.D. degrees in engineering from the Univer- sity of Tokyo in 1995, 1997, and 2001, respectively. He also received an MPhil degree in physics from the University of Cambridge. He was a research fellow at the Cavendish Labo- ratory, University of Cambridge from 2001 to 2004. After working for a couple of electronic design automation (EDA) companies, he joined the Integrated Research Institute, Tokyo Insti- tute of Technology in 2006. Since 2010, he has been with the Graduate School of Advanced Sciences of Matter, Hiroshima University, where he is currently an associate professor. He serves as Asso- ciate Editor of Electronics Letters since 2015. His research interests include modeling and simulation of nanoelectronic devices and systems, design of RF circuits and interconnects, and microwave theory and measurement.