High Frequency Design From March 2006 High Frequency Electronics Copyright © 2006 Summit Technical Media MATCHING NETWORKS The Yin-Yang of Matching: Part 1—Basic Matching Concepts By Randy Rhea Agilent Technologies hen an electri- This two-part article covers cal signal prop- basic matching concepts Wagates through and matching network media, a portion of the topologies, emphasizing signal is reflected at the methods for obtaining interface between sec- the desired performance tions with differing with networks that are impedances. This is anal- realizeable in practice ogous 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 devel- oped by Phillip Smith [1]. Any impedance with a positive real part may be displayed on the standard, unity radius Smith chart. The hori- zontal line is pure resistance. Circles with a Figure 1 · The impedance Smith chart with center on this line are constant resistance. unity radius. 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 respect to the real axis on the right. An ideal sometimes normalized to the desired reference match has ⏐ρ⏐= 0 and total reflection from a impedance of the system such as 50 ohms. For pure reactive impedance at the circumference example, the inductive impedance 50 + j50 of the chart has ⏐ρ⏐ = 1. The reflection coeffi- ohms becomes 1 + j1 when normalized and is cient may be computed from plotted at the intersection of the circle labeled 1 and the top arc labeled 1. An admittance ver- ZZ− ρ= 0 (1) + sion of the chart is a left-right mirror of the ZZ0 impedance chart. A detailed description of the chart is given in Smith’s book and a CD-ROM where Z is the impedance and Z0 is the refer- tutorial by Glenn Parker [2]. ence impedance of the system. The center of the chart is the reference The magnitude of the reflection coefficient impedance of the system. An impedance at is a radial parameter. In other words, it is a this point represents an ideal match. The function of radial distance from the center of length of a vector from the center of the chart the chart. Other useful radial formats are to any impedance point is the magnitude of related to ρ. the reflection coefficient, ⏐ρ⏐. The angle of the =− ρ vector is from the center of the chart with RLdB 20 log (2) 16 High Frequency Electronics High Frequency Design MATCHING NETWORKS RLdB VSWR LA 0.025 32.0 1.051 0.003 0.050 26.0 1.105 0.011 0.075 22.5 1.162 0.024 0.100 20.0 1.222 0.044 0.125 18.1 1.286 0.068 0.150 16.5 1.353 0.099 0.175 15.1 1.424 0.135 0.200 14.0 1.500 0.177 0.225 13.0 1.581 0.226 Figure 2 · A load of 50 – j33.86 0.250 12.0 1.667 0.280 ohms matched to a 50 ohm source 0.300 10.5 1.857 0.410 using an inductor of reactance 0.350 9.1 2.077 0.568 +33.86 ohms. 0.400 8.0 2.333 0.757 0.450 6.9 2.636 0.983 0.500 6.0 3.000 1.249 53.89 nH inductor with a reactance 0.600 4.4 4.000 1.938 +33.86 ohms at 100 MHz. At this fre- Figure 3 · Return loss and ampli- 0.700 3.1 5.667 2.924 quency the inductor and load reac- tude transmission responses of the 0.800 1.9 9.000 4.437 tance series resonate effectively con- complex-conjugate matched load 0.900 0.9 19.000 7.212 necting the 50 ohm source directly to (solid traces) and the original load the 50 ohm load resistance. Given in without matching (dashed traces). Table 1 · Radial parameters. LA Figure 3 are the transmission and accounts only for reflection loss reflection responses of the original and not dissipation loss. load and the matched load. The L-Network Matching match is achieved only at the series This section introduces L-net- resonant frequency of the load capac- works for matching both real and itance and matching inductance. In complex impedances. At a single fre- 1 + ρ VSWR = (3) fact, the match is worse above 140 quency, any positive-real complex − ρ 1 MHz than without the matching net- impedance can be matched to any work. This is a characteristic of other positive-real complex imped- =− −ρ 2 LA 10log() 1 (4) matching networks; the presence of ance using no more than two reactive reactive elements often provides elements. For example, in a 50 ohm system, selectivity. Consider the common need to an impedance 50 + j50 has a reflec- The impedance seen looking match a complex impedance to a real tion coefficient of 0.4472 at 63.43°, a toward the source through the impedance such as 50 ohms. Given in return loss of 7.0 dB, a VSWR of matching inductor is 50 + j33.86 Figure 4 are the eight unique topolo- 2.618 and an insertion loss of 0.9691 ohms. This is the complex-conjugate gies comprised of two L-C elements. dB. The insertion loss, LA, is due to of the load impedance. In fact, at any Each topology is capable of matching reflection. It does not account for dis- node in a lossless matching network, certain complex load terminations on sipation loss that is present with the impedance to the left is the com- the network’s right to a real source lossy components or resistance in the plex conjugate of the impedance to resistance on the network’s left. The circuit. Table 1 lists radial parame- the right. This is another characteris- charts are normalized to the desired ters. This table is useful for disci- tic of matching networks. source resistance. The matchable plines other than matching. For If the load reactance is in parallel space is enclosed by the green curves example, a filter designed for 0.177 with the resistance, then a shunt ele- in Figure 4. Notice these curves are dB passband ripple has a passband ment can cancel the load susceptance the familiar shape of the Chinese return loss of 15 dB. and provide a parallel resonant Yin-Yang for the four topologies that match. A single inductor or capacitor include both an inductor and a capac- Complex-Conjugate Match can provide a match only if the load itor. I first became aware of these con- Consider a load impedance of ZL = and source resistance are equal, or cepts when reading Smith’s book RL – jXL at 100 MHz where RL = 50 near enough so to provide an ade- Electronic Applications of the Smith ohms and XL = –33.86 ohms as shown quate match. In general, we are not Chart [1]. Smith’s unique ability to in Figure 2. This load is matched to so fortunate and two matching ele- graphically express important con- the 50 ohm source using a series ments are required. cepts encompasses yin-yang! 18 High Frequency Electronics Certain impedances are matched works by using graphical techniques. by only two topologies. For example, Here are formulas for finding the ele- impedances with real components ment values analytically. For a refer- greater than the reference imped- ence impedance Z0 = R0 – j0 , and a ance are matched by only type 1 and load termination impedance ZL = RL 2 networks. Other impedances are – jXL, the load admittance is (eq. 5) matched by four topologies. For R X example, Z = 0.5 + j1 is matched by Y =+GjB = L − j L LL L 22+ 22+ types 1, 4, 5 or 7. RXLLRXLL Phil’s book describes how to find the element values for these L-net- then setting the following: Figure 4 · Diagrams of matchable impedance space for L-networks of type 1 through 8. High Frequency Design MATCHING NETWORKS Figure 5 · Frequency domain responses of network types 1, 4, 5 and 7 that match 25 + j43.33 ohms to 50 ohms. Figure 6 · Agilent GENESYS software screen depicting the design and responses of all eight L-network topologies. G =−L 2 A GL (6) R 2 0 ()RCX2 ++( ) 2 ++( )2 LL ()RXCLL L =− (15) L =− (24) 4 2πfC( + X) 8b π ( + ) =−()− L 2 fXL C B RRLLL0 R X (7) =− 1 C4 (16) Example =−() − − 2πfC CRRRXLLL0 (8) Consider a load termination impedance of 25 + j43.33 ohms at 100 GA22+ C = L (17) MHz. The return loss is 4.8 dB and 5a π we then have 2 fA the transmission loss is 1.76 dB. Referring to Figure 2 we see that AB− C =− L (18) type 1, 4, 5 or 7 is used to match this = A 5b 2πf L1 (9) load. Eqs. 5 through 10 are used to 2πfG()22+ A L find the element values for type 1: L1 A L = (19) is 79.66 nH and C1 is 43.47 pF. AB− 6a 2πfG()22+ A C = L (10) L Which of these topologies is best? 1 π 2 f The amplitude transmission respons- 1 L =− (20) es are given in Figure 5 as computed 1 6b 2πfA( − B) L = (11) L and displayed by the Agilent 2 π ( + ) 2 fA BL 1 GENESYS program suite [3]. Notice C =− (21) that all four provide full transmission GA22+ 7a 2πfB C = L (12) at the design frequency, 100 MHz.