Beyond the Smith Chart: A Universal Graphical Tool for Impedance Matching Using Transformers

Abstract

T he topic of impedance transformation and matching is one of the well-established and essential aspects of microwave engineering. A few decades ago, when discrete radio-frequency (RF) design was dominant, impedance matching was mainly performed using transmission-lines techniques that were practical due to the relatively large design size. As microwave design became possible using integrated on-chip components, area constraints made L C section matching (using lumped passive elements) more practical than transmission line matching. Both techniques are conveniently visualized and accomplished using the well-known graphical tool, the Smith chart.Since CMOS technology was primarily and initially developed for digital purposes, the lack of high-quality passive components made it practically useless for RF design. The device speed was also far inferior to established III-V technologies such as GaAs heterojunction bipolar transistors (HBTs) and high-electron mobility transistors (HEMTs). The first RF CMOS receiver was constructed at 1989 [1], but it would take another several years for a fully integrated CMOS RF receiver to be presented. The scaling trend of CMOS in the past two decades improved the transistors' speed exponentially, which provided more gain at RF frequencies and also enabled operation at millimeter-wave (mm-wave) frequencies. The

3D Smith Chart Constant Quality Factor Semi-Circles Contours for Positive and Negative Resistance Circuits

“…The quality factor of an impedance Z or admittance Y can be defined as (1) [12][13][14][15][16][17], (this is denoted also as nodal quality factor in [15][16][17]) where X represents its reactance, B its susceptance, R resistance and G conductance as defined in (1) where Z and Y can be related through (2). Other authors skip the absolute value sign in (1) [8][9][10][11], however this doesn't change anything in respect to their geometry, only to the sign labelling convention. Using the sign conventions [12][13][14][15][16][17] (as for example at p.102 in [13]), we do not allow negative Q values for passive circuits with positive resistances.…”

“…1 (b) shows the family of the radial lines obtained for various values of ≠ 0. Imposing now Q constant and positive in (3) one gets the contours obtained in [8][9][10][11][12][13][14][15][16][17] irrespective of the presence of the absolute value in (1). However, by using inversive geometry [18], it can be proven that imposing (5) in (3), the set of radial constant Q lines is proven to generate a family of coaxal circles as r and x are swept from -∞ to +∞.…”

“…Table III summarizes the results generated by (10). 5 shows the constant Q semi-circles on the 3D Smith chart together with the cutting fold planes given by equation (10). A more detailed view on the rendered 3D Smith chart is given in Fig.…”

“…Constant quality factor (Q) representations on the Smith chart are a visual way to determine the quality factor of various passive microwave circuits, being mostly known in the microwave frequency range community [8][9][10][11][12][13][14][15][16][17]. These constant Q shapes are often denoted as contours or curves, [8][9][10][11][12] but rarely as circles [11]. In this work we first prove that the constant Q curves represent circle arcs mapped on coaxal circle families [18] while providing for the first time (to the best of our knowledge) their equations, i.e.…”

“…The Smith chart [1][2], proposed by Philip Hagar Smith in 1939, survived the passing of years, becoming an icon of microwave engineering [3], being nowadays still used in the design and measurement stage of various radio frequency or microwave range devices [4][5][6][7], for plotting a variety of frequency dependent parameters. Constant quality factor (Q) representations on the Smith chart are a visual way to determine the quality factor of various passive microwave circuits, being mostly known in the microwave frequency range community [8][9][10][11][12][13][14][15][16][17]. These constant Q shapes are often denoted as contours or curves, [8][9][10][11][12] while extremely seldom as circles [11].…”