# EE246 Microwave Engineering 29 Oct Leeson H. O. #31 Autumn Impedance Matching

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2 TL = -10 log (1 - l l 2 ) db From this table, one can see that the reflected power from the mismatch becomes of the same order of magnitude as the incident power at SWR 1.5, although in many situations we require return loss greater than 30 db, which corresponds to SWR = 1.07, a real challenge over any bandwidth. SWR l l Return loss db 1-l l 2 Transmitted loss db Typical Matching Situations It's relatively easy to design resistive terminations for waveguide and TEM structures that provide extremely wideband loads with Z=Z o of the transmission line. For situations where the loads must dissipate substantial power, thermal considerations come into play, and some form of conductive or convective cooling is necessary to prevent destruction of the load. In the extreme case, a long enough length of terminated lossy transmission line will present a high-power matched load to a transmission line of the same dimensions. The going gets tough in cases where impedance matching is required to narrow-band elements such as antennas or to devices with substantial reactance and resistance levels that are much lower (or higher) than typical transmission lines. An arbitrary impedance can, in principle, be matched at a single frequency by adding sufficient transmission line to move the impedance around the Smith chart until it lies on an admittance circle that passes through the center of the chart (g=20 millimhos or millisiemens), then adding susceptance of the proper sign to move the combined admittance to the =0 point. The simplified Smith charts here show one of the two possible solutions for an arbitrary normalized impedance. The other solution is - 2 -

3 Although inconvenient to realize in transmission line format, there are two other solutions that are obtained by rotating the arbitrary impedance until it is on the 50 circle, then adding the proper series reactance to bring the resulting impedance to the 50 point. Recall that a useful expression for the impedance of a lossless transmission line of characteristic impedance Z o with an arbitrary load Z L and electrical length is Z = Z o Z L +jz o tan Z o +jz L tan This can be used in connection with a spreadsheet or other calculation aid to keep track of the real and imaginary parts with varying frequency. But these are single-frequency solutions to the impedance matching problem. Because one of the major advantages of microwave usage is the opportunity to transmit substantial bandwidth, and because in practice one would hope to avoid a requirement for a unique circuit for each of the many frequencies in a typical 40% waveguide band, broadband solutions to the matching problem are valuable and sought-after. Lumped LC Networks The transmission-line solutions illustrated above correspond approximately to a simple L- network in lumped circuit theory. Depending upon the specific frequency characteristics of the arbitrary impedance to be matched, one of the four approaches listed here will generally provide the greatest bandwidth. The same is true with transmission line matching networks. Shunt C Shunt L Series L Series C Series L Series C Shunt C Shunt L - 3 -

5 Y o Y L A lossless double stub tuner can match to line Y o any load whose conductive component at the plane of the closest stub is less than G = Y o (1+cot 2 ), where is the electrical length between the two stubs. This implies that values of = n (0, 180, etc.) are optimum since they will match infinite G, but the losses in a practical tuner defeat this concept. In practice, the stubs are spaced odd multiple of /8, which will match any admittance whose conductance is less than G = 2Y o at the plane of the stub nearest the load. By varying the line length between that stub and the load, almost any load can be matched at a single frequency. The Bode-Fano Limit There exists a general limit on the bandwidth over which an arbitrarily good impedance match can be obtained in the case of a complex load impedance. It is related to the ratio of reactance to resistance, and to the bandwidth over which we desire to match the load. Bode and Fano derived, for lumped circuits, a fundamental limitation that is expressed for a parallel RC load impedance as o ln 1 l ( )l d RC Since ln (1) = 0, there is no contribution to this integral over frequencies for which l l=1, so we can see that we want to have the maximum mismatch out of the band of interest. If we assume this condition, the integral is limited to the bandwidth of interest, and we can get an idea of how well we can match an arbitrary complex impedance over that bandwidth. Consider the idealized situation shown in the figure here, where the reflection coefficient is unity outside the band of interest and is m in the frequency range

6 1 m For this simplification, the integral becomes ln 1 m RC Recalling that ln (x) = -ln (1/x) and solving for m, we have m e-1/2 frc This can be solved for the maximum bandwidth for which a given RC product can be matched to within a given reflection coefficient or SWR. There are similar limitations on other forms of complex impedance, but the general nature of the limitation is the important issue. A general implication of the Bode-Fano limit is that one should not waste any match out-of-band, and that the best inband match is obtained with Tchebyscheff rather than maximally flat networks. The best broadband impedance matching practice incorporates the complex load impedance into a multisection filter structure with a design that includes the characteristics of the load. However, it is useful to understand some simpler methods as well. Quarter-Wave Transformers Quarter-wave transmission line transformers are a simple, but bulky, method of matching impedances. At band center, a quarter wave transformer has an input impedance that is Z = Z 01 2 /Z L, which is simply the general formula for input impedance with = /2. /4 Z 0 Z 01 Z L The bandwidth of such a matching transformer depends upon the ratio of Z L /Z o, which is the load impedance normalized to the desired input impedance. The reflection coefficient is approximately - 6 -

7 l l cos Z L /Z o Z L /Z o To determine the bandwidth for a given impedance ratio and maximum SWR, solve for each band edge in terms of the band-edge reflection coefficient for that SWR. As an example, for maximum SWR of 1.5 and an impedance ratio of 4, the bandwidth for 1.5 SWR is approximately 34% (±17%). at Multiple Quarter-Wave and Tapered Transformers The bandwidth limitation of the single quarter-wave section can be overcome by using multiple sections, with impedances chosen to provide maximally flat or Tchebyscheff passbands. In either case, the improvement in bandwidth carries a cost of substantial physical size, which is not compatible with the current trend toward application of miniature semiconductors and integrated circuits. It is also possible to use a gradual taper rather than multiple impedance steps, but again the taper must be gradual so the physical length is considerable for a well-designed tapered line. In both cases there is substantial literature devoted to the design of multisection and tapered transmission line transformers, but these have become a specialized application because of their physical size. Short Transformers Short transmission line sections can be used as matching networks in many current circuit examples. If the load is complex, a matching section shorter than /4 can be used for impedance matching under certain conditions. 1 Z 0 Z 01 Z L For Z L = R + jx, we can obtain a match with the above circuit under the following conditions: Z 01 = RZ o - X2 Z o Z o - R and tan 1 = Z 01 Z o - R XZ o - 7 -

8 This requires R Z o and X 2 < R(Z o - R). As an example, if R = 20 and Zo = 50, the load can be matched if lxl With the right combination of load reactance and resistance, this matching section can be shorter than /4. Transmission Lines as LC Circuits 1 2 Z 0 Z 01 Z 02 Z L A more generally useful matching network can be realized by the use of two short sections of transmission line, of differing impedances. For the case of Z 02 substantially higher than Z L (say, Z 02 >3lZ L l) and 2 small compare to a wavelength ( 2 < /6), the higher impedance section can be approximated to a series inductive reactance of X L Z 02 2 Likewise, if Y 01 is large compared with ly L l and 1 is small compare to a wavelength, the lower impedance section can be approximated by a shunt capacitive susceptance of B C Y 01 1 Thus short narrow and wide sections of transmission line can be used to approximate the lumped-element L-C networks shown earlier. The Series Section Transformer 2 An interesting special case of this transformer occurs for Z 02 = Z o. This implementation, called the series-section transformer, can be used to match a range of impedances with very short sections. In order to match a load impedance Z L = R + jx, Z 01 must be greater than Z o SWR or less than Z o / SWR. If we define n = Z 01 /Z o, and normalize R and X to Z o to define r = R/Z o and x = X/Z o, 1 = tan -1 ± (r - 1) 2 + x 2 r (n - 1/n) 2 - (r - 1) 2 - x 2, and 2 = tan -1 (n - r/n) tan 1 + x r + xn tan See Regier, F., "Impedance Matching with a Series Transmission Line Section," Proc. IEEE, Vol. 59, No. 7, July 1971, pp

9 The benefit of this type of matching transformer can be seen in the following example of matching a 75 load to a 50 line using a section of 75 transmission line for Z 01. In this case, the SWR relative to Z o is 75/50 = 1.5, so the matching section impedance must be greater than = 61, so 75 cable will work. We have n = 1.5, r = 1.5 and x = 0, so tan 1 = ( ) = 0.56 and tan 2 = (1.5-1) = For each case, = 0.08, so the total length of 0.16 is substantially shorter than might be expected. For the simple case of n = r and x = 0, both line lengths are equal and tan = n(n - 1) n 3-1. This design can be verified by use of the Smith chart. Resonant Matching of Resonant Elements One last interesting matching network is particularly applicable in the case of a load impedance that has the characteristics of a series RLC network. This is typical of antennas, and is shown schematically here. Z L The impedance plot of this type of load, in this case for R>Z o, is shown in the first Smith chart plot here: - 9 -

10 In order to reduce the SWR over the bandwidth shown, it is possible to place a resonant circuit across the terminals of the impedance, or to feed the impedance through a half-wave line of suitable Z o. In either case, the operation of the matching network is to close up the arc of the impedance plot, resulting in a smaller range of SWR. /2 Z 01 Z L A final step is to use a matching network such as an LC network, its transmission line equivalent or the series section network to bring the impedance plot to the center of the Smith chart. This type of matching is not widely known, but is highly effective in the case of impedances of the form of a loaded series resonant circuit. More than one halfwavelength can be used if the line impedance is not optimum, and a short addition or subtraction can be made from the line length to "center" an impedance plot that is not symmetrical. For the specific impedance plot shown here, we require Z 01 <Zo. A more detailed description of this matching concept is found in Rizzi 3. Other similar examples are found in the MIT Rad Lab series 4. Other Matching Methods In addition to the transmission line reactance methods outlined here, more powerful methods exist for tough problems. A family of broadband transformers has been described by Ruthroff, and extremely broadband transmission line transformers have been developed by Sevick 5. These can be applied at the lower microwave frequencies. Finally, for the cases in which extremely low return loss is required over a substantial bandwidth, nonreciprocal ferrite circulators and isolators are widely applied to obtain very low SWR. These two related techniques will be the subject of future lectures. 3 Rizzi, P. Microwave Engineering - Passive Circuits, Prentice Hall, 1988, pg Montgomery, Dicke & Purcell, Principles of Microwave Circuits, MIT Rad Lab Series #8, 1947, pg Sevick, J., Transmission Line Transformers, 2nd Edition, ARRL,

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