Matching Network Yin-Yang - Part 1

This two-part article covers basic matching concepts and matching network topologies, emphasizing methods for obtaining the desired performance with networks that are realizeable in practice  

When an electrical signal propagates through media, a portion of the signal is reflected at the interface between sections with differing impedances. This is analogous to light reflection in optical systems. The reflected signal may pose problems, and the power in the reflected signal reduces the transmitted power. Figure 1 shows the impedance chart developed by Phillip Smith [1]. Any impedance with a positive real part may be displayed on the standard, unity radius Smith chart. The horizontal line is pure resistance. Circles with a center on this line are constant resistance. Arcs converging at center right are constant reactance. The top half of the chart is inductive and the bottom half is capacitive. The chart is sometimes normalized to the desired reference impedance of the system such as 50 ohms. For example, the inductive impedance 50 + j50 ohms becomes 1 + j1 when normalized and is plotted at the intersection of the circle labeled 1 and the top arc labeled 1. An admittance version of the chart is a left-right mirror of the impedance chart. A detailed description of the chart is given in Smith’s book and a CD-ROM tutorial by Glenn Parker [2]. The center of the chart is the reference impedance of the system. An impedance at this point represents an ideal match. The length of a vector from the center of the chart to any impedance point is the magnitude of the reflection coefficient, ⏐ñ⏐. The angle of the vector is from the center of the chart with respect to the real axis on the right. An ideal match has ⏐ñ⏐= 0 and total reflection from a pure reactive impedance at the circumference of the chart has ⏐ñ⏐ = 1  

Complex-Conjugate Match 

Consider a load impedance of ZL = RL – jXL at 100 MHz where RL = 50 ohms and XL = –33.86 ohms as shown in Figure 2. This load is matched to the 50 ohm source using a series 53.89 nH inductor with a reactance +33.86 ohms at 100 MHz. At this frequency the inductor and load reactance series resonate effectively connecting the 50 ohm source directly to the 50 ohm load resistance. Given in Figure 3 are the transmission and reflection responses of the original load and the matched load. The match is achieved only at the series resonant frequency of the load capacitance and matching inductance. In fact, the match is worse above 140 MHz than without the matching network. This is a characteristic of matching networks; the presence of reactive elements often provides selectivity. The impedance seen looking toward the source through the matching inductor is 50 + j33.86 ohms. This is the complex-conjugate of the load impedance. In fact, at any node in a lossless matching network, the impedance to the left is the complex conjugate of the impedance to the right. This is another characteristic of matching networks. If the load reactance is in parallel with the resistance, then a shunt element can cancel the load susceptance and provide a parallel resonant match. A single inductor or capacitor can provide a match only if the load and source resistance are equal, or near enough so to provide an adequate match. In general, we are not so fortunate and two matching elements are required. 

 L-Network Matching

 This section introduces L-networks for matching both real and complex impedances. At a single frequency, any positive-real complex impedance can be matched to any other positive-real complex impedance using no more than two reactive elements. Consider the common need to match a complex impedance to a real impedance such as 50 ohms. Given in Figure 4 are the eight unique topologies comprised of two L-C elements. Each topology is capable of matching certain complex load terminations on the network’s right to a real source resistance on the network’s left. The charts are normalized to the desired source resistance. The matchable space is enclosed by the green curves in Figure 4. Notice these curves are the familiar shape of the Chinese Yin-Yang for the four topologies that include both an inductor and a capacitor. I first became aware of these concepts when reading Smith’s book Electronic Applications of the Smith Chart [1]. Smith’s unique ability to graphically express important concepts encompasses yin-yang! 

 Next Issue

 In the next issue, this article concludes with Part 2, covering the topics of impedance transformation through transmission line sections, multiple-section transformers, absorbing reactance using filters, and analysis of load characteristics for selection of the preferred matching method.