Proceedings of the World Congress on Engineering and Computer Science 2019 WCECS 2019, October 22-24, 2019, San Francisco, USA Balun Modeling for Differential Amplifiers Kun Chen, Zheng Liu, Xuelin Hong, Ruinan Chang, and Weimin Sun Abstract—This work proposes an improved lumped element model. Particular for this core is an additional element which model for accurately characterizing transformer based baluns. will be shown later to play a critical role in modeling The model can accurately describe balun behaviors for both common mode behavior. Section III will derive the formulae differential and common modes. Lumped elements extraction from Y parameter is presented, and the relationship between for extracting the model elements from the Y parameter. In the proposed model and the classic compact transformer model Section IV, the differential, common and divider modes will is derived. Numerical results will be presented to validate the be analyzed separately, where we will justify the addition of accuracy of the proposed model. the extra element in the transformer core. Section V will Index Terms—balun modeling, transformer modeling, push- discuss the relationship of the proposed transformer core pull amplifiers, power amplifier (PA), compact model, differen- model with the popular compact model that is based on ideal tial mode, common mode, mutual inductance. transformers and leakage and magnitization inductances. Section VI will discuss the physics and modeling of the I. INTRODUCTION common mode mutual inductance. And in Section VII, some RANSFORMERS are important passive components straight-forward adaptations of the transformer core for more T which have various applications such as broadband accurate modeling of actual transformer baluns, followed by impedance matching, power combining and dividing. They numerical results in Section VIII. Finally, conclusions will are found in many RF circuits and systems, and could be on- be drawn in Section IX. chip, on-die, or in package. Particularly, in the design of high performance differential (push-pull) power amplifiers (PAs), II. EXTENDED BALUN MODEL transformers serve as both inster-stage and output baluns. The use of transformer baluns enhances PA performance in the sense that the loadline is doubled for a given output power compared to single-ended PA, hence reducing loss due � /2 � � /2 to a smaller impedance transformation ratio. Furthermore, it " 1 ) also offers significant common mode rejection, which will P �/2 P be beneficial for improving even order harmonic rejections. , / Numerous models have been proposed to characterize �"/2 �)/2 transformers, like the compact model [1], Y-network model �& [2], transmission line model [3], frequency dependent trans- �/2 �" �) former model [4], etc. However, little attention is paid to P. common mode modeling. In [5], a scalable model is proposed which takes into account both differential and common �"/2 �)/2 modes by adopting six mutual inductances, and physical and P- P0 empirical based functions are used in determining the circuit parameters; however, one note for this model is that the �"/2 �1 �)/2 phase imbalance prediction bandwidth is somewhat limited for multi-turn baluns. The purpose of this paper is not an extensive scalable model that depicts different configurations for the balun; Fig. 1: Extended Balun Model instead, the interest is in a simple and accurate lumped element model whose core elements (L, R, C) are explicitly extracted by its Y parameter which can be easily simulated Fig. 1 shows the schematic of the proposed 5-port ex- by electromagnetic (EM) solver and measured by vector tended balun model. In this model, the primary inductor network analyzer (VNA) nowadays. In this way, one can has series inductance Lp, series resistance Rp, and self- then iterate the balun designs efficiently. Also, a new way capacitance Cp, while the secondary inductor has Ls, Rs and of accounting for the common mode response is proposed, Cs. The primary and secondary inductors have a mutual in- which models the common mode mutual inductance in an ductance of M, and the mutual capacitances between primary explicit and accurate manner. and secondary inductor are denoted by Cm. For convenience, The paper is organized as follows: In Section II, we will we have assumed that the inductors are equally split into present the schematic and analysis of a transformer balun two, and the polarity signs are on top side for all inductors model, which is an extension of the classic mutual inductance unless otherwise specified, and parasitic capacitances other than Cp, Cs and Cm won’t be considered until later sections. Manuscript received July 22, 2019; revised July 29, 2019. Compared to conventional balun model, the proposed model K. Chen, Z. Liu, X. Hong and W. Sun are with Skyworks Solutions, Inc., Newbury Park, CA, 91362 USA. e-mail: [email protected]. has an added impedance Zc at the center tap. In balun design, R. Chang is an electrical engineer and has a PhD from UCSD. a proper center tap loading is usually desired for optimal ISBN: 978-988-14048-7-9 WCECS 2019 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering and Computer Science 2019 WCECS 2019, October 22-24, 2019, San Francisco, USA performance. But here we would consider Zc not as an Plugging Eq. (18-19) into Eq. (4) and simplifying, we yield external load, but as part of the balun model, and the value j and meaning of it will be revealed later. In this model, mutual M = : ![Y − Y −1(Y + 1 Y + j!C )(Y + j!C )] resistance is not included since it is usually negligible. Also, 14 14 12 2 15 p 34 s (20) we only consider center tap at the primary, and it is trivial to generalize to the case with a center tap at the secondary. With Zp, Zs and M, the coupling factor K and turn ratio T Using Kirchhoff’s current law and the definition of mutual are defined as inductance, one can find the following Y parameters of the p balun model as K = M= LpLs (21) q Z 1 T = Ls=Lp: (22) Y = j!(C + C ) + s + (1) 11 p m Z Z + !2M 2 Z + 4Z p s p c For Z , one can easily write that Z 1 c Y = −j!C − s + (2) 12 p Z Z + !2M 2 Z + 4Z 1 1 p s p c Zc = − Zp (23) j!M Y55 4 Y13 = −j!Cm − 2 2 (3) ZpZs + ! M In the above extraction, Zp, Zs, Zc and M depend on j!M Cp and Cs, which are usually much smaller than Cm, and Y = (4) 14 2 2 only play a noticeable role at very high frequencies. In most ZpZs + ! M 2 cases, we can directly put Cp = Cs = 0 and still maintain Y = − (5) 15 Z + 4Z good accuracy. However, instead of regarding Cp and Cs as p c nil, it makes more sense to get an approximate expression Z Y = j!(C + C ) + p (6) for them. 33 s m Z Z + !2M 2 p s One can write that Z Y = −j!C − p (7) 1 Z 34 s Z Z + !2M 2 s p s Y12 − Y55 = −j!Cp − 2 2 : (24) 4 ZpZs + ! M Y35 = 0 (8) 4 For Zt, t 2 fp; sg, using Taylor expansion for the resistive Y = (9) 55 part Rt, we have Zp + 4Zc Zt ≈ Rt;0 + !(Rt;!;0 + jLt) (25) where Zt = Rt + j!Lt; t 2 fp; sg. The rest of the Y parameters can be found by using reciprocity and symmetry where Rt;0 = Rtj!=0, Rt;!;0 = @Rt=@!j!=0. When Rt;0 of the network: !jRt;!;0 + jLtj, which is the case when the quality factor Q of the inductor is not very small, or when the frequency Y = Y ; i 6= j (10) t ij ji is high enough, we have, from Eq. (24), Y11 = Y22 (11) @!(4Y12 − Y55) Y13 = Y24 (12) C ≈ − : (26) p 8j!@! Y14 = Y23 (13) Note that in the above, the right hand side is a complex Y = Y (14) 15 25 number, while the left hand side should be real. The inter- Y33 = Y44 (15) pretation is that the imaginary part of the right hand side is Y35 = Y45: (16) much smaller than the real part and shall be ignored. This convention is assumed through this work. Similarly, one obtains III. PARAMETER EXTRACTION AND FITTING @!(2Y + Y + Y ) By extraction, here it refers to representing the lumped C ≈ − 34 35 45 (27) s 4j!@! elements in the model as a function of the Y parameters; while by fitting, it means using simple empirical functions where Y35 and Y45 are added to make it more numerically to extrapolate a wideband behavior from just a few frequency robust. As mentioned previously, only the resistance Rp and points. For the proposed model, we will regard all com- Rs have strong frequency dependence, which is mostly due ponents as constant across frequencies except Rp and Rs, to skin effect and proximity effect. The resistance model in because for most applications, the dependence of inductance [6] can be adopted: and capacitance on frequency is very weak. 2 0 (f=f0t) Looking at Y11 and Y12, one notices that Rt = Rt;dc(ξt coth ξt) · (1 + rt;rf 2 ) (28) 1 + (f=f0t) 1 C = (Y + Y + Y ) (17) where (·)0 means taking the real part, and ξ = (1 + j) tt m j! 11 12 15 t 2δ with δ being the skin depth, and tt the thickness of the From Y34, Y12, and Y15, we get metal, while rt;rf is a constant determined by geometry and technology, and f0t is a frequency factor which can be Z = −j!M(Y + j!C )Y −1 (18) p 34 s 14 determined by some fitting.