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2 Impedane Transformers X Impedane Transformers Vitaliy hurbenko, Viktor Krozer and Tonny ubæk Tehnial University of Denmark Denmark 1. Introdution Impedane mathing is an important aspet in the design of mirowave and millimeter wave iruitry sine impedane mismathes may severely deteriorate the overall performane of eletroni systems. In high-power appliations, the standing eletromagneti wave resulting from mismath in a transmission line is highly undesirable as it leads to amplitudes of voltage and urrent whih might be several times higher than those in a mathed line. This an lead to disruption or even damage of the dieletri in the transmission line. A refleted eletromagneti wave an also result in frequeny pulling of signal generators onneted to the mismathed transmission line, thereby shifting the osillation frequeny from the desired. In transeiver appliations, antenna mismath leads to signal power loss and lower signalto-noise ratio, thereby deteriorating the overall transmit or reeive performane. When designing low-noise amplifiers, it is often required to ontrol the input network mismath. Generally, it is not possible to design an amplifier whih has the optimum input impedane for minimum noise figure equal in value to the optimum impedane for maximum gain. The input network is then should be mismathed in order to provide a lownoise operation. Impedane transformers an also be effetively used to improve seletivity of resonant iruits and are very useful in filter design. Low values of soure and load impedane derease the loaded quality fator Q and inrease the bandwidth of a given resonant iruit. This makes it very diffiult to design even a basi LC high-q resonant iruit for use between two very low values of soure and load resistanes. A ommon method to overome this problem is to use impedane transforming iruits to present the resonant iruit with a soure or load resistane that is muh larger than what is atually present. Consequently, by utilizing impedane transformers, both the Q of the resonator and its seletivity an be inreased. Mathing a omplex impedane in a wide frequeny range is most ommonly ahieved by using one of the following tehniques: - passive two-port networks onsisting of reative omponents; - passive two-port networks onsisting of resistive omponents;

3 304 Passive Mirowave Components and Antennas Wideband mathing an also be ahieved by means of ferrite irulators in whih the refleted wave is guided to an absorbing load, and ferrite isolators in whih the transmission losses are different for the inident and refleted waves. For a wideband mathing, it is preferable to plae the mathing network as lose as possible to the load, as it is illustrated in Fig. 1. V V Generator 0 0 Mathing network Load L Generator Mathing network (a) 0 V V 0 Load L Fig. 1. Voltage standing wave patterns. Plaing of the mathing network with regard to the generator for wideband (a), and narrowband (b) mathing. This onept will be demonstrated in the later setion 3.1 by onsidering an example of mathing a omplex load using shunt stubs. In this hapter, different tehniques for wideband mathing are presented. Setions thru 4 briefly present some of the well-known mathing tehniques while the use of oupled transmission lines for wideband mathing is treated in depth in Setion 5. The first part of the hapter inludes a disussion of resistive and reative lumped elements in Setion, different types of stub mathing in Setion 3, and the use of series of transmission lines in Setion 4. Sine these tehniques are all thoroughly treated in the literature, only the designonsiderations relevant for applying the tehniques for wideband mathing are treated here while the reader is referred to the literature for speifis, suh as the relevant formulas for alulating the values of the different omponents. The use of oupled transmission lines for wideband impedane mathing is not as widely used as the tehniques desribed in setions thru 4. Hene, in Setion 5, a detailed presentation of this tehnique is given. (b)

4 Impedane Transformers 305. Mathing Using esistive and eative Lumped Elements esistive elements or attenuators an be effetively used to lower the level of the refleted signal from the load in a very wide frequeny range. It should be noted, though, that the effiieny of suh mathing networks is low beause they attenuate not only the refleted but also the inident wave. Another type of mathing network uses lumped reative omponents to math a omplex load impedane to a desired omplex impedane. For moderate bandwidths, the omponent values of two-element mathing networks an be found relatively easy by first hoosing a pair of initial values on the basis of the Smith Chart and then applying omputational optimization. To inrease the bandwidth, more than two reative elements are required. The synthesis and optimization of multi-element wideband mathing iruits an be aomplished by means of software tools, whih are urrently available in a wide variety. The implementation of this type of transformers in mirowave and millimeter wave range is limited due to the low Q-fator of lumped omponents. Therefore, lumped element mathing is usually employed only at low frequenies, or in appliations where ompat size is a major requirement, e.g., in monolithi mirowave integrated iruits design (Kinayman & Aksun, 005). 3. Stub mathing This setion is dediated to mathing iruits that use open-iruited or short-iruited transmission line setions, onneted in parallel with the load or transmission feed line. This is a well developed mathing tehnique whih is often used in mirowave and millimeter wave iruits. In this setion, some of the important operational priniples and properties of shunt stub mathing iruits are disussed. More detailed analysis of this type of mathing tehnique is available in the literature (Pozar, 1998), (Kinayman & Aksun, 005). 3.1 Single-Stub Mathing This is one of the most simple and onvenient ways of mathing a transmission line with a load whih has real or omplex impedane. This method was developed by Tatarinov V. V. in 1931 and is widely used for narrow-band mathing in mirowave and millimeter wave appliations. It onsists of a short iruited or open iruited stub and a piee of transmission line between the load and the stub. An example of the single-stub mathing iruit is shown in Fig.. There are several hoies of eletri distane θ d from the load to the mathing stub. In the first ase (Fig. (a)), the distane between the load and mathing stub is hosen as short as possible while this distane is hosen to be several times longer in the seond ase (Fig. (b)). The responses of these two mathing iruits are shown in Fig. 3. The 10 db refletion loss bandwidth of the iruit in Fig. (a) is 10.3 % while the same parameter for the iruit in Fig. (b) is equal to 1.9 %. Thus, by using θ d = instead of θ d = 8.05, the bandwidth is inreased by more than a fator of 5. There is also a differene in the wideband response of the mathing iruits. The iruit with long distane between the load and the mathing stub demonstrates more passbands in the same frequeny range.

5 306 Passive Mirowave Components and Antennas 3. Double-Stub Mathing Single-stub mathing an math any load impedane, but it requires a variable eletri length of the transmission line between the load and the stub. This poses pratial diffiulties for adjustable tuners. θ d = pf θ d = pf 50 Ω 50 Ω 70 Ω 50 Ω 50 Ω 70 Ω 50 Ω 50 Ω θ l = 0.5 θ l = (a) Fig.. Two single-stub mathing solutions. (a) wideband, (b) narrowband. The load is mathed at f 0 = 1 GHz. 0 (b) -10 (a) -0 Magnitude of S (db) (f-fo)/fo (%) Fig. 3. Magnitude of S versus offset frequeny for the mathing iruits in Fig.. Here, f 0 = 1 GHz is the enter frequeny of operation. Therefore, it would be more useful to have the length fixed and still be able to math a wide range of load impedanes. This an be ahieved with double-stub mathing, as shown in Fig. 4, whih allows for an arbitrary eletri distane between the load and the stub. (b)

6 Impedane Transformers 307 λ g /8 0 jb 0 jb 1 0 L Open or short l Open or short l 1 Fig. 4. Double-stub mathing. The first stub an be plaed at arbitrary distane from the load. It should be noted that stub spaings near 0 or λ g / (where λ g is the guided wavelength) lead to mathing networks that are very frequeny sensitive (Pozar, 1998), and onsequently, very narrowband. In pratie, stub spaings are usually hosen as odd number of λ g /8, for example λ g /8, 3λ g /8 or 5λ g / Triple-Stub Mathing The double-stub mathing iruit an not math all load impedanes. For a speified distane between two stubs, the mathing is possible only for limited values of loads, whih depend on amplitude and phase of the standing wave. This limitation arises from the fat, that the stub itself an not hange the real part of the impedane at the point of onnetion to the transmission line. This limitation an be overomed by using a triple-stub mathing as the one shown in Fig. 5. λ g /4 λ g /4 0 jb 0 3 jb 0 jb 1 0 L Open or short l 3 Open or short l Open or short l 1 Fig. 5. Triple-stub mathing. The first stub an be plaed at arbitrary distane from the load.

7 308 Passive Mirowave Components and Antennas It allows for an arbitrary distane between the load and the stub and also allows to math arbitrary load impedane. The operation of triple-stub mathing iruit an be treated as a ombination of two double-stub mathing iruits and stub spaings are usually hosen as λ g /4. 4. Series Transmission Line Mathing This setion is dediated to mathing iruits that use series transmission lines, suh as single setion quarter-wave transformer, multisetion transformers, and tapered transmission lines. 4.1 The Quarter-Wave Transformer The quarter-wave transformer is one of the most simple and pratial iruits for impedane mathing, espeially for mathing of real load impedanes. It is also possible to math a omplex load using the quarter-wave transformer, but this requires an additional length of transmission line between the load and the quarter-wave transformer to transform the omplex load impedane into a real impedane. A iruit employing a quarter-wave transformer is shown in Fig. 6. λ g / Fig. 6. A single setion quarter-wave mathing transformer. One of the main drawbaks of this transformer is the requirement to have available a transmission line with an impedane of In some ases, e.g., mathing with oaxial able, the required quarter wave transmission line alls for a nonstandard value of the harateristi impedane. 4.. Transformers with Fixed Values of Charateristi Impedane Another useful type of series transformers are those whih are based on transmission lines with the same harateristi impedanes as the lines whih should be mathed. Suh transformers are onvenient for interonnetion of standard lines as well as transmission lines with different geometry, where realization of transmission lines of arbitrary harateristi impedane involve diffiulties. The simplest realization of suh transformer is shown in Fig. 7 (a) and desribed in detail by (Aizenberg et al., 1985).

8 Impedane Transformers 309 l l (a) l l 1 l 1 l Fig. 7. Transformers with fixed values of harateristi impedane onsisting of (a) two setions and (b) four setions. (b) This transformer onsists of two transmission line setions. The harateristi impedanes of these lines are the same as impedanes of lines to be mathed. The length of one setion is l g g atan 1 n 1 1 n, (1) where n = 0 / 01 is the transformation ratio. For small values of n, the value of l approahes λ g /1, implying that the total length of the transformer approahes λ g /6. For inreasing n, the value of l approahes 0. The operating frequeny band of the desribed transformer is about 5 % narrower in omparison to the quarter-wave transformer, and its length for pratial values of n is times shorter. The response of the transformer in Fig. 7 (a) is shown in Fig. 8 (urve (a)) and ompared to the response of the onventional quarter-wave transformer (Fig. 8 urve (b)). For transformation ratio :1 the eletrial length of the setion in Fig. 7 (a) is equal to 8.1. The ahieved for this ratio bandwidth at 0 db return loss level is 31 %.

9 310 Passive Mirowave Components and Antennas Magnitude of S (db) (a) λ g /6 transformer in Fig. 7 (a); (b) quarter-wave transformer; (f-fo)/fo (%) ()transformer in Fig. 7(b). Fig. 8. Comparison of mathing harateristi of quarter-wave transformer (Fig. 6) and transformers with fixed values of harateristi impedane (Fig. 7). The transformation ratio is :1. Here, f 0 is the enter frequeny of operation. A more broadband stepped impedane transformer is shown in Fig. 7 (b)). It onsists of four setions with the length of the outermost setions being shorter than the length of setions in the middle. Fig. 8 (urve ()) shows the magnitude of S for a transformer with the fallowing parameters: the transformation ratio is :1; θ 1 /θ = l 1 /l = Here θ 1 = π l 1 / λ g, and θ 1 = π l 1 / λ g () are the eletrial lengths of the setions in Fig. 7 (b). The total length of the transformer is l 1 + l = 0.346λ g. The ahieved bandwidth at 0 db return loss level is 71 %. The bandwidth and inband refletion level of this type of transformer depend on length of the setions (Aizenberg et al., 1985). (b) (a) () 4.3 Tapered Transmission Lines As desribed above, the bandwidth of the quarter-wave transformer is limited. In order to extend its operating frequeny band, multisetion transformers, with different harateristi impedane in eah setion, may be used. In ontrast to the transformers desribed in the previous setion, the lengths of the setions used in the multisetion transformer an be hosen equal to eah other. The desired refletion oeffiient response as a funtion of frequeny an be ahieved by properly hoosing the harateristi impedane of the transmission line setions. In the limit of an infinite number of setions, the multisetion transformer beomes a ontinuously tapered line. There are many ways to hoose the taper profile. By hanging the type of taper, one an obtain different passband harateristis. Several taper profiles may be onsidered: linear, exponential, triangular, and so on.

10 Impedane Transformers 3 For a given taper length, the Klopfenstein taper has been shown to be optimum in the sense that the refletion oeffiient is minimum over the passband (Pozar, 1998). Alternatively, for a speified level of refletion oeffiient, the Klopfenstein taper yields the shortest mathing setion. However, it should be noted that the response of this taper has equal level of ripples in its passband. In many ases, the relation between the physial dimensions and the harateristi impedane of a guiding struture is ompliated and the generation of an optimal tapering onfiguration is thus not a trivial task. This implies that a linear or exponential tapering of the physial dimensions of the transmission line is often hosen for pratial implementations. 5. Coupled Line Transformers In reent years, oupled transmission lines have been suggested as a mathing element due to greater flexibility and ompatness in omparison to quarter wavelength transmission lines (Jensen et al., 007). It has been demonstrated that mathing real and omplex loads with oupled lines leads to more ompat realizations and ould therefore beome important at millimeter-wave frequenies for on-hip mathing solutions. Another area where oupled line strutures are useful is mathing of antenna array strutures, as suessfully demonstrated by (Jaworski & Krozer, 004). As it was shown above, the quarter-wave transformer is simple and easy to use, but it has no flexibility beyond the ability to provide a perfet math at the enter frequeny for a realvalued load, although a omplex load of ourse an be mathed by inreasing the overall length of the transformer. The oupled line setion provides a number of variables whih an be utilized for mathing purposes. These variables are the even and odd mode impedanes and loads of the through and oupled ports. This loading an be done in form of a feedbak onnetion whih provides additional zeros for broadband mathing. These variables an be hosen to provide a perfet math or any desired value of the refletion oeffiient at the operating frequeny. The bandwidth of the oupled line transformer an be further inreased in ase of mismath. In addition, it is also possible to math a omplex load. In the lower GHz range the loading of the through and oupled ports an be done with lumped elements whih allows for easy mathing of both real and imaginary impedane values. At higher frequenies it is not possible to use lumped elements, but the differene between the even and odd mode impedanes is a parameter whih makes it possible to turn a mixed real and imaginary ontrol load at the through port into a purely imaginary one, whih an be implemented with a transmission line stub. 5.1 Symmetri Coupled Line Setion Coupled line impedane transformers are very useful at millimetre wave frequenies where they suessfully perform diret urrent bloking and an handle large impedane transformation avoiding transverse resonanes whih our in a onventional low impedane quarter-wave transformer. The most ommon onfiguration of the transformer is shown in Fig. 9.

11 31 Passive Mirowave Components and Antennas L g Fig. 9. Symmetri oupled transmission line transformer. In this onfiguration, the diagonal terminals of the oupled line setion are loaded with generator ( g ) and load ( L ) impedanes. The opposite terminals are open iruited. In this standard onfiguration however, the eletrial performane of the oupled lines transformer in terms of insertion loss and bandwidth an not ompete with performane of the orresponding quarter-wave transformer (Mongia et al., 1999). 5. Asymmetri Coupled Line Setion Symmetri oupled lines represent a restrited onfiguration of the more general lass of oupled lines. They allow for a simpler analysis, however, for wideband appliations asymmetri oupled lines are preferable. For example, the bandwidth of a forward-wave diretional oupler using asymmetri oupled transmission lines is greater than the one formed using symmetri ones (Jones & Bolljahn, 1956). In this setion the design of a wideband impedane transformer based on asymmetri oupled lines is desribed. The onsidered wideband impedane transformer is based on asymmetri, uniform oupled lines in nonhomogeneous medium. A mirostrip line is one of the most ommonly used lasses of transmission lines in nonhomogeneous medium. Edge-oupled mirostrip lines are shown in Fig. 10. Condutor 1 (4) (3) Condutor l ε r (1) () Fig. 10. A oupled mirostrip line four-port.

12 Impedane Transformers 313 For the purpose of analysis, this oupled line four-port is transformed to a two-port network with arbitrary load using impedane matrix representation. The investigations presented in this book are only for the most ommonly used onfiguration, when diagonal terminals of the oupled lines are loaded with generator and load impedanes. Thus, the entire iruit an be represented as a two-port network, whih performs impedane transformation between a generator impedane g onneted to a port 1 and a load impedane L onneted to a port 3, as shown in Fig.. ['] (1") () (3) L ["] (") (4) [] (1) g Fig.. Two-port network representation for the oupled line impedane transformer. As it an be seen in Fig., the network onsists of the oupled line four-port network desribed by an impedane matrix [] and arbitrary load matrix at opposite terminals desribed by matrix ["]. In pratie, ports and 4 are in general either short-iruited or open-iruited with a orresponding representation of the two-port network ["]. The magnitude of S is equal to S db 0 log IN IN ij, ij, L g ij, ij, L g, (3) where IN is the input impedane of the transformer, whih is a funtion of the load impedane L, impedane matrix elements of oupled lines ij and the arbitrary load (i and j are the indexes of the matrix elements). Using the general impedane matrix representation for oupled lines (Tripathi, 1975) and boundary onditions at ports and 4 the input impedane is expressed by ij IN 13 1 a 14 b 1 a1 14 b1, (4) 33 3 a 34 b L where a , (5a)

13 314 Passive Mirowave Components and Antennas a , (5b) b a b, (5) 4 1. (5d) 43 a A total number of six quantities is required to desribe asymmetri oupled lines (Mongia et al., 1999), being: 1 and π1, whih are, respetively, the harateristi impedanes of line 1 for and π modes of propagation; γ and γ π, the propagation onstants of and π modes; and π, the ratios of the voltages on the two lines for and π modes. Thus, the elements of the impedane matrix are given by 44 l oth l 1 oth 1 1 1, (6a) l oth l oth l sh l 1sh l sh l 1sh 1 1 1, (6b), (6), (6d) 33 1 oth 1 l oth l 1 1, (6e)

14 Impedane Transformers sh 1 l sh l 1 1, (6f) where l is the length of the oupled line setion, as it is shown in Fig. 10. These relations are substituted into (5) and (4) to alulate the input impedane and finally the refletion oeffiient of the transformer. From relation (3) it an be seen that the mathing properties of the transformer depend not only on oupled line parameters, but also on load of ports and 4, whih are desribed by elements. This dependene introdues additional degree of freedom during design ij proedure and an be used to expand the bandwidth of the impedane transformer, as shown below. Loading With Transmission Line As an example, terminals and 4 an be loaded with a mirostrip transmission line. The impedane matrix of the transmission line with harateristi impedane 0, length l, and propagation onstant γ is given by 0 oth l 0 sinh l The transformer onfiguration is shown in Fig sinh l 0 oth l. (7) 0, θ IN Line Line 1 1" " OUT 1, π1,, π,, π, θ=(θ + θ π )/ Fig. 1. Shemati illustration of the transformer based on oupled line setion and a transmission line load.

15 316 Passive Mirowave Components and Antennas In order to simplify further alulations, the transmission lines are onsidered to be lossless, and eletrial lengths of the oupled line setion (θ + θ π )/ and the mirostrip transmission line θ are assumed equal, resulting in γl = jβl = jθ, γ l = jθ, γ π l = jθ π, (8) θ = (θ +θ π )/, (9) where θ and θ π are the eletrial lengths of the oupled line setion for and π mode respetively. θ is a funtion of frequeny and an be used for the analysis of the spetrum of the transformer refletion oeffiient. The response (3) for the transformer of Fig. 1 is shown in Fig Magnitude of S (db) Eletrial length (deg) Fig. 13. esponse of transformer shown in Fig. 1. The transformation ratio is 1:. As it an be seen in Fig. 13, this transformer onfiguration exhibits an additional minimum in the magnitude of S in omparison to the traditional impedane transformer based on oupled line setion with open-iruited terminals (Kajfez, 1981). These minima are nonuniformly distributed in the frequeny domain. This is due to the differenes in eletrial lengths θ and θ π for two oupled line modes in nonhomogeneous medium. For the ase of homogeneous medium the propagation onstants for the two modes are equal, γ = γ π, and hene the eletrial lengths for the two propagating modes are also equal. It is therefore possible to obtain three equidistant refletion zeros in the spetrum of the refletion oeffiient. Beause transmission lines in a homogeneous medium are a speial ase of transmission lines in a nonhomogeneous medium the expressions given above are also valid for response alulations. It an be depited from the data in Fig. 14 that the transformer provides wideband operation with uniformly distributed refletion zeros in the frequeny domain. In addition, the distane between the zero loations an be varied by adjusting the parameters of the struture.

16 Impedane Transformers Magnitude of S (db) Fig. 14. esponse of the transformer in homogeneous medium ase. The eletrial length of the transformer is equal to a quarter wavelength at the enter frequeny. Comparing the results in Fig. 13 and Fig. 14 it an be dedued that the impedane transformer in nonhomogeneous medium has approximately the same bandwidth as the one in homogenous medium. However, in many ases, like for example in surfae mount tehnology, it is more useful to deal with mirostrip strutures. Loading With Stepped Impedane Transmission Line Eletrial length (deg) The differenes in eletrial lengths of the oupled lines in nonhomogeneous medium an be ompensated by introduing a stepped impedane transmission line, as it is shown in Fig , θ/ 0, θ/ 1" " IN Line 3 Line OUT 1, π1,, π,, π, θ=(θ + θ π )/ Fig. 15. Shemati illustration of the wideband impedane transformer. The transformer onsists of asymmetri oupled lines desribed by the eletrial parameters 1,, π1, π, whih are, respetively, the harateristi impedanes of line 1 and for the and π modes of propagation; θ and θ π, the eletrial lengths for the and π modes; and π, the ratios of the voltages on the two lines for the and π modes. The stepped impedane transmission line onsists of two equal length transmission lines with harateristi

17 318 Passive Mirowave Components and Antennas impedanes 01 and 0, as shown in Fig. 15. The eletrial length of eah transmission line is set to be half the eletrial length of the oupled line setion to redue the number of design parameters. For the purpose of analysis, this struture is transformed into a two-port network with arbitrary load using an impedane matrix representation. Thus, the entire iruit an be represented as a two-port network, whih performs impedane transformation between a generator impedane g onneted to the port 1 and a load impedane L onneted to the port 3, as shown in Fig.. The magnitude of S at the port 1 is defined by (3). The input impedane IN in (3) is alulated using relations (4)-(6) together with the orresponding elements of the impedane matrix ["] for the stepped impedane transmission line. A series onnetion of two transmission lines shown in Fig. 16 an be desribed as a onnetion of two two-port networks. 01, l 1, γ 1 0, l, γ (1") (") Fig. 16. Series onnetion of transmission lines. The impedane matries of the transmission lines with harateristi impedanes 01, 0, lengths l 1, l, and propagation onstants γ 1, γ are given by (1) () (1) (1) 1 () () 1 (1) 1 (1) () 1 () 01 oth 1l 01 sinh 1l1 0 oth l 0 sinh l 1 01 sinh l oth l 0 sinh l 1 1 oth l, (10). () Impedane matrix for the overall iruit in Fig. 16 is derived using boundary onditions at the ommon terminal. At this terminal the voltages of two two-ports are equal, and urrents are equal and oppositely direted. Thus, impedane matrix elements are found to be: 01 (1) oth l 1 1 (1) 1 () (1) 01 oth oth sinh, l l l (1a)

18 Impedane Transformers l oth l sinh l sinh l, (1) 1 () () 1 (1) oth () oth l () 1 () (1) 0 oth oth sinh. l l l (1b) (1) In ase of transmission lines with equal eletrial length θ/ (1) an be rewritten as 01 oth sinh , (13a) , (13b) sinh 0 oth sinh (13) These equations are used for the alulation of elements of the matrix ["] in Fig.. Thus, the analysis of the struture now an be performed using (3). It an be depited from the data in Fig. 17 that the transformer provides wideband operation, and the eletrial length of the transformer is equal to a quarter wavelength at the enter frequeny. Magnitude of Sand S, db (db) Δθ = 50 Δθ = 40 Δθ = Eletrial length, (deg) Fig. 17. esponse of the Ω impedane transformer shown in Fig. 15.

19 30 Passive Mirowave Components and Antennas In addition, the distane between the minima loations Δθ an be varied by adjusting the parameters of the struture. This distane Δθ haraterizes operating frequeny bandwidth of the transformer. The harateristis of the transformer for three different values of Δθ are shown in Fig. 17. As it an be seen, the in-band level of the refletion oeffiient depends on parameter Δθ. The estimation of the maximum level of the return loss between minima for different transformation ratios an be found using the data shown in Fig Inband eturn refletions loss (db) (db) Δθ=40 Δθ=30 Δθ= L / g Fig. 18. The minimum level of the return loss between minima in Fig. 17. As expeted, the level of in-band return loss for the transformer inreases with reduing of transformation ratio, and reahes the absolute maximum at L / g = Multisetion Coupled Line Transformers To further inrease the bandwidth, it is possible to reate an impedane transformer using more oupled line setions onneted in series. The example of a mirostrip two setion impedane transformer is shown in Fig. 19. IN interonneting transmission lines λ/ OUT oupled line setions Fig. 19. Layout of the Ω multi setion impedane transformer.

20 Impedane Transformers 31 The total eletrial length of the transformer is equal to half a wavelength at the enter frequeny. The response of the transformer is shown in Fig Magnitude of S (db) Frequeny (GHz) Fig. 0. esponse of the Ω impedane transformer shown in Fig. 19. The transformer exhibits six minima in the spetrum of refletion oeffiient. The ahieved frational mathing bandwidth is beyond a deade at -10 db refletion oeffiient level. The distane between the minima loations an be varied by adjusting the parameters of the struture. 6. eferenes Aizenberg G.., Belousov S. P., hurbenko E. M., Kliger G. A. & Kurashov A. G. (1985). Korotkovolnovye antenny, nd ed., Mosow, adio i Svaz (in ussian). Jaworski G. & Krozer V. (004). Broadband mathing of dual-linear polarization staked probe-fed mirostrip path antenna, Eletronis Letters, vol. 40, no. 4, pp. 1-. Jensen T., hurbenko V., Krozer V. & Meinke P. (007). Coupled Transmission Lines as Impedane Transformer, IEEE Transations On Mirowave Theory And Tehniques, vol. 55, no. 1, pp Jones E. M. T., & Bolljahn J. T. (1956). Coupled Strip Transmission Line Filters and Diretional Couplers, IE Trans. Mirowave Theory & Teh., vol. MTT-4, pp Kajfez D., Bokka S. & Smith C. E. (1981). Asymmetri mirostrip d bloks with rippled response, IEEE MTT-S Int. Mirowave Symp. Dig., pp Kinayman N. & Aksun M. I. (005). Modern Mirowave Ciruits. Arteh House, In. Mongia., Bahl I., Bhartia P.( 1999). F and mirowave oupled line iruits. Norwood: Arteh House mirowave library. Pozar D. M. (1998). Mirowave Engineering. Wiley. Tatarinov V. V. (1931). O pitanii begushei volnoi korotkovolnovyh antenn i ob opredelenii ih soprotivlenia, Vestnik elektrotehniki, no. 1 (in ussian).

21 3 Passive Mirowave Components and Antennas Tripathi V. K. (1975). Asymmetri oupled transmission lines in an inhomogeneous medium, IEEE Trans. Mirowave Theory & Teh., vol. 3, no. 9, pp

22 Passive Mirowave Components and Antennas Edited by Vitaliy hurbenko ISBN Hard over, 556 pages Publisher InTeh Published online 01, April, 010 Published in print edition April, 010 Modelling and omputations in eletromagnetis is a quite fast-growing researh area. The reent interest in this field is aused by the inreased demand for designing omplex mirowave omponents, modeling eletromagneti materials, and rapid inrease in omputational power for alulation of omplex eletromagneti problems. The first part of this book is devoted to the advanes in the analysis tehniques suh as method of moments, finite-differene time- domain method, boundary perturbation theory, Fourier analysis, mode-mathing method, and analysis based on iruit theory. These tehniques are onsidered with regard to several hallenging tehnologial appliations suh as those related to eletrially large devies, sattering in layered strutures, photoni rystals, and artifiial materials. The seond part of the book deals with waveguides, transmission lines and transitions. This inludes mirostrip lines (MSL), slot waveguides, substrate integrated waveguides (SIW), vertial transmission lines in multilayer media as well as MSL to SIW and MSL to slot line transitions. How to referene In order to orretly referene this sholarly work, feel free to opy and paste the following: Vitaliy hurbenko, Viktor Krozer and Tonny ubaek (010). Impedane Transformers, Passive Mirowave Components and Antennas, Vitaliy hurbenko (Ed.), ISBN: , InTeh, Available from: InTeh Europe University Campus STeP i Slavka Krautzeka 83/A ijeka, Croatia Phone: +385 (51) Fax: +385 (51) InTeh China Unit 405, Offie Blok, Hotel Equatorial Shanghai No.65, Yan An oad (West), Shanghai, 00040, China Phone: Fax:

23 010 The Author(s). Liensee IntehOpen. This hapter is distributed under the terms of the Creative Commons Attribution-NonCommerial- ShareAlike-3.0 Liense, whih permits use, distribution and reprodution for non-ommerial purposes, provided the original is properly ited and derivative works building on this ontent are distributed under the same liense.