Coupled Transmission Lines as Impedance Transformer

Transcription

1 Downloaded from orbit.dtu.dk on: Jul 3, 8 Coupled Transmission ines as Impedance Transformer Jensen, Thomas; hurbenko, Vitaliy; Krozer, Viktor; Meincke, Peter Published in: IEEE Transactions on Microwave Theory and Techniques ink to article, DOI:.9/TMTT Publication date: 7 Document Version Peer reviewed version ink back to DTU Orbit Citation (APA): Jensen, T., hurbenko, V., Krozer, V., & Meincke, P. (7). Coupled Transmission ines as Impedance Transformer. IEEE Transactions on Microwave Theory and Techniques, 55, DOI:.9/TMTT General rihts Copyriht and moral rihts for the publications made accessible in the public portal are retained by the authors and/or other copyriht owners and it is a condition of accessin publications that users reconise and abide by the leal requirements associated with these rihts. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-makin activity or commercial ain You may freely distribute the UR identifyin the publication in the public portal If you believe that this document breaches copyriht please contact us providin details, and we will remove access to the work immediately and investiate your claim.

2 Coupled Transmission ines as Impedance Transformer Thomas Jensen, Vitaliy hurbenko, Viktor Krozer, Peter Meincke Technical University of Denmark, Ørsted DTU, ElectroScience, Ørsteds Plads, Buildin 38, 8 Ks. ynby, Denmark, Phone: , Fa: , tje@oersted.dtu.dk Abstract A theoretical investiation of the use of a coupled line section as an impedance transformer is presented. It is shown how to properly select the terminations of the coupled line structures for effective matchin of real and comple loads in both a narrow and a wide frequency rane. The correspondin circuit confiurations and the desin procedures are proposed. Synthesis relations are derived and provided for efficient matchin circuit construction. Desin eamples are iven to demonstrate the fleibility and limitations of the desin methods and to show their validity for practical applications. Wideband matchin performance with relative bandwidth beyond % and return loss R > db is demonstrated both theoretically and eperimentally. Good areement is achieved between the measured and predicted performance of the coupled line transformer section. Inde Terms Coupled transmission lines, directional coupler, impedance matchin, impedance matri, microstrip lines, strip lines. I. TRODUCTION n recent years, coupled transmission lines have been I suested as a matchin element due to reater fleibility and compactness in comparison to quarter wavelenth transmission lines [-3]. It has been demonstrated that matchin real and comple loads with coupled lines leads to more compact realizations and could therefore become important at millimeter-wave frequencies for on-chip or ow Temperature Cofired Ceramics (TCC) matchin solutions. Another area where coupled line structures are useful is matchin of antenna array structures, as successfully demonstrated in []. The quarter-wave transformer is simple and easy to use, but it has no fleibility beyond the ability to provide a perfect match at the center frequency for a real-valued load, althouh a comple load of course can be matched by increasin the lenth of the quarter-wave transformer. The coupled line section provides a number of variables which can be utilized for matchin purposes. These variables are the even and odd mode impedances and loadin of the throuh and coupled ports. This loadin can be done in form of a feedback connection which provides additional zeros for broadband matchin. These variables can be chosen to provide a perfect match or any desired value of the reflection coefficient at the operatin frequency. The bandwidth of the coupled line transformer can be further increased in case of mismatch. In addition, as will be shown, it is also possible to match a comple load. This is a eneralization of matchin with a quarter-wave transmission line transformer. This paper focuses on developin the necessary formulas for applyin coupled line sections in matchin applications, as well as the appropriate basic analysis of the coupled line section. Establishin a desin framework will enable a widespread use of coupled line structures as novel impedance matchin elements, in addition to the standard lines and lumped elements. In the lower GHz rane the loadin of the throuh and coupled ports can be done with lumped elements which allows for easy matchin of both real and imainary impedance values. At hiher frequencies it is not possible to use lumped elements, but the difference between the even and odd mode impedances is a parameter which makes it possible to turn a mied real and imainary control load at the throuh port into a purely imainary one, which can be implemented with a transmission line stub. Equations for matchin purposes, which are based on controllable parameters of coupled transmission line sections, are presented for backward-wave couplers includin microstrip and stripline transmission line couplers. The couplin required for a iven application often becomes too tiht for a practical implementation. Therefore an investiation into the rane of load values that can realistically be matched with the coupled line section has been carried out. Finally loadin of the throuh and coupled port with an interconnectin transmission line is considered with the purpose to achieve a wide operatin frequency rane. In [5] a broadband impedance transformer based on coupled transmission lines is presented. The synthesis procedure for this circuit is eplained. By usin this procedure it is possible to shape the frequency response by placin transmission minima in the spectrum. II. THE USE OF A COUPED E SECTION AS AN IMPEDANCE TRANSFORMER Fi. shows the eneral coupled lines confiuration. There is no established terminoloy for use of the coupled line section as an impedance transformer, in this paper it is found useful to use the port names from directional coupler terminoloy in the discussion of the circuit, but the numeration used corresponds to filter desin with coupled line sections because the theory is developed from that point of view.

3 Fi.. Coupled transmission lines. The port names are in areement with directional coupler terminoloy for a backward-wave coupler, while the numeration is the same as used in filter desin with coupled line sections. The impedance matri for the -port open-circuited coupled line section in Fi., where transverse electromanetic (TEM) wave propaation and symmetric transmission lines are assumed, can be epressed in terms of the even and odd mode impedances, e and o, and the electrical lenth, θ []: j ( ) cotθ, (a) j. 33 e o j ( ) cotθ, (b) 3 3 e o coupled input j ( ) cscθ, (c) 3 3 e o j ( ) cscθ, (d) 3 3 e o The input impedance of the coupled line section with the coupled and throuh port open-circuited and the isolated port terminated in the load impedance, can be calculated from the current and voltae at the input port V, () I where. In the followin derivations only the center frequency is considered and hence, θ π, cot θ, and csc θ. Furthermore the notation used. Then () simplifies to e o and e o is ( j ). (3) The reflection coefficient with the enerator impedance attached to port is iven by Γ. () For a perfect match the reflection coefficient has to be zero and thus Solvin for - leads to Coupled ines e, o, θ 3 e o. (5). (6) This establishes the condition for a perfect match at the isolated throuh operatin frequency in terms of the difference between the even and odd mode impedance. This condition is similar to the well-known relation, for a quarter-wave transmission line transformer, when the characteristic impedance is echaned with the difference between the even and odd mode impedances of the coupled lines. It is also possible to retain the reflection coefficient in the formula and use it as a parameter, yieldin Γ e o. (7) Γ The parameter - can be chosen within reasonable limits. A lare value of - implies a tiht couplin, where the couplin e o coefficient is iven by C. Tiht couplin implies e o a lare value of C and requires a lare difference between the even and odd mode impedances. A lare value of - necessitates a very narrow ap between ede-coupled transmission lines or a lare round plane spacin, e.. in stripline technoloy. From (7) it can be seen that transformin a lare-valued load requires a tihter couplin than a small-valued load, and even thouh a non-zero reflection coefficient can increase the bandwidth it also requires tihter couplin. The quarter-wave transformer has the same lenth as the coupled line impedance transformer for a iven application and requires only one transmission line, so the only obvious advantae of the latter is that it constitutes a perfect DC block. However, a load attached to the throuh port opens for improvement of the transformer characteristics for the coupled line section as will be shown in the net sections. A. oadin of the throuh port for etended matchin capabilities. Usin the input port as input and with the load to be transformed attached to the isolated port leaves two available ports (the throuh and coupled port) that can be eploited to improve the matchin capabilities. The loadin of the throuh port will be considered in this section. Port, coupled input Coupled ines e, o, θ 3 isolated throuh Port, Fi.. The -port coupled transmission line section reduced to a -port with the coupled port open-circuited and the throuh port terminated in load.

4 3 The open-circuited impedance matri [] for a two-port that is based on the four-port coupled line section in Fi., where the coupled port is left open-circuited and the throuh port is loaded with an arbitrary impedance, that will be used as a matchin parameter, can be derived as (see appendi II) j [ ]. (8) j The impedance matri is symmetric and therefore the circuit is reciprocal, but should be chosen imainary or with a lare real part to avoid ecessive power dissipation. The input impedance derived from () is then. (9) Substitutin this epression in the formula for the input reflection coefficient () and solvin for ives ( Γ ), () ( Γ ) ( Γ ) which determines the value of for a iven value of,,, -, and Γ. The condition for, i.e. open-circuited, is only fulfilled for γ. This demonstrates that the coupled line section impedance transformer is indeed a eneralization of the standard transmission line transformer, where the impedance represents an additional deree of freedom. Matchin eample with a real-valued load The ability of the coupled line section to match a realvalued load is illustrated with a simple eample. The followin parameters are iven: 5 Ω, Ω, Ω, - Ω, and Γ.3. Usin () results in 353 Ω. The circuit confiuration is shown in Fi. 3 and the frequency response plotted at an arbitrary desin frequency of GHz in Fi.. Port, e, o, θ Port, Fi. 3. Matchin confiuration eample. The response with 5 Ω, Ω, Ω, and - Ω is shown in Fi.. A reflection coefficient of.3 corresponds to -3 db. In this eample a reflection coefficient manitude < - db spans a bandwidth of %. R Manitude of S and S (db) - - S... e 5Ω S... o 5Ω S... e 5Ω -3 S... o 5Ω S... { λ trf }.5.5 Frequency (GHz) Fi.. Response of the circuit in Fi. 3. S is -3 db at the desin frequency as desired. Note that the insertion loss is non-zero because power is dissipated in the resistor. The response with tihter couplin (e5ω, o5ω) and a comparision with a quarter-wave transformer is also included in the plot. If the couplin is increased, settin e 5 Ω and o 5 Ω, results in 99 Ω and an increased bandwidth of % for the same matchin conditions. The physical dimensions for the first case in ede-coupled stripline technoloy with ε r 3.38 and a round plane spacin of 5 mm are a conductor width of.55 mm and a conductor separation of. mm. B. Achievin purely imainary control loads An interestin property of the coupled line section as a matchin element is the ability to match a comple load. Splittin in a real and an imainary part ives ( Re{ }) γ Im{ } j, () 6γ ( 8γ Re{ } ), Γ where γ, and γ are assumed real-valued. Γ From this epression it can be seen that it is possible to use - to make the real part of equal to zero, which establishes a condition for a purely imainary value of γ Re, ( { }) γ. (3) Re { } This is especially useful for hih frequency applications, where the matchin load to be attached to can simply be realized as a transmission line stub of a specified lenth. An added advantae is that a purely reactive load is in principle lossless. Matchin eample with a comple-valued load The followin parameters are iven: 5 Ω, j Ω, Ω, and Γ. - is found to be Ω and -j Ω which can be implemented as a short transmission line stub with a lenth of.º and a characteristic impedance of eo Ω.

5 Port, e, o, θ Port,, θ Stub Port, sinificantly worse, see Fi. 6. The limitation of the coupled line section becomes apparent when from () is plotted aainst a rane of possible load values. Fi. 7 shows the required value of for a perfect match plotted versus the real part of. The fiure depicts two sets of curves, one for the real part of and one for the imainary part of, for imainary load values, Im{ } of, and 6 Ω, respectively, e 5 Ω, o 5 Ω, and 5 Ω. e5 Ω, o 5 Ω, 5 Ω, θ stub e, o, θ (Ω) Thick lines: Re() Thin lines: Im() Fi. 5. Matchin confiuration for a comple load. Schematic drawin above and possible stripline realization below. The response with the parameters 5 Ω, j Ω, Ω, - Ω, 86.6 Ω, and θstub.º is shown in Fi. 6. The characteristic impedance of the stub can be chosen to ive a line width equal to the transmission lines in the coupled lines section in order to avoid discontinuities. Here e o is used for the stub line characteristic impedance, but it is not a necessary condition. Manitude of S and S (db) - Port, -... S coupled... S lines... S { λ trf }... S Frequency (GHz) Fi. 6. Response of the circuit in Fi. 5 and compared with a quarter-wave transformer. Note that this matchin confiuration is lossless because is purely imainary. The response in Fi. 6 shows a perfect match at the center frequency and a - db bandwidth of 3%. The lenth of the matchin stub is very short compared to traditional transmission line matchin circuits and will only slihtly increase the circuit size. The load used in this eample can also be matched with a quarter-wave transformer and an etra lenth of transmission line to account for the imainary part of the load, but the bandwidth achieved is smaller and out of band rejection is Real part of (Ω) Im() Ω Im() Ω Im()6 Ω Im() Ω Im() Ω Im()6 Ω Fi. 7. The arrows indicate the eneral direction of the curves when the couplin is increased, i.e. the required value of the real part of becomes more positive and the imainary part becomes more neative. Note that for this choice of the even and odd impedance, a neative real part is required for when Im{} Ω and Re{}< Ω (thick blue curve below ). To decrease the power dissipation a lare value of is desirable, but this implies stroner couplin. Decreasin the couplin will lower the real part of towards the value of with the result that most of the input power is dissipated in. If the condition for a purely imainary value of, Eq. (3), is applied then Fi. 8 shows that tiht couplin values are required for lare-valued loads. e o (Ω) Real part of (Ω) Re() Γ Im() Ω Im() Ω Im()6 Ω Fi. 8. Values of e - o when is required to be purely imainary and Γ (perfect match).

6 5 It can be depicted that very tiht couplin is required for even moderate impedance loads. Relain the condition for the reflection coefficient, Γ, moves the curves towards hiher values and will therefore require even tihter couplin. A couplin coefficient larer than - db is difficult to achieve with microstrip or stripline ede-coupled lines, but is possible with a thick substrate or with broadside coupled transmission lines. III. WIDEBAND IMPEDANCE TRANSFORMATION. A. Derivation of analysis formulas The impedance transformers considered above are based on transmission lines in homoeneous medium, e.. striplines. They allow for a simpler analysis, however, in many practical cases, for eample in surface mount technoloy, it is more useful to deal with microstrip structures. The wideband impedance transformer proposed in this section is derived on the basis of asymmetric, uniformly coupled lines in a nonhomoeneous medium [5]. A microstrip line is one of the most commonly used classes of transmission lines in a nonhomoeneous medium. The proposed confiuration is a quarter wavelenth lon and provides three times wider operatin frequency rane in comparison to the traditional quarter-wave transformer. As discussed above the matchin properties of the transformer depend not only on coupled line parameters, but also on the loads at ports and in Fi.. This dependence introduces additional derees of freedom durin the desin procedure and is used here to sinificantly epand the bandwidth of the impedance transformer. The confiurations shown above use loadin of terminal and with terminal open-circuited. In the circuit considered below, both terminals are loaded usin an interconnectin microstrip stepped impedance transmission line as shown in Fi. 9. Port, θ/, θ/ " " ine θ ine 3 θ/ Port OUT c, π, c, π, R c, R π, θ(θ c θ π)/ Fi.9. The quarter-wave wideband impedance transformer. Schematic drawin above and possible microstrip realization below. OUT (3) The transformer consists of asymmetric coupled lines described by the electrical parameters c, c, π, π, which are, respectively, the characteristic impedances of line and for the c and π modes of propaation; θ c and θ π, the electrical lenths for the c and π modes; R c and R π, the ratios of the voltaes on the two lines for the c and π modes. The stepped impedance transmission line consists of two equal lenth transmission lines with characteristic impedances and, as shown in Fi. 9. The electrical lenth of each transmission line is set to be half the electrical lenth of the coupled line section to reduce the number of desin parameters. For the purpose of analysis, this structure is transformed to a two-port network with arbitrary load usin an impedance matri representation. Thus, the entire circuit can be represented as a two-port network, which performs impedance transformation between a enerator impedance connected to port and a load impedance connected to port 3, as shown in Fi.. [] () () [] (") (") ["] Fi.. Two-port network representation for the coupled line impedance transformer. From [5]. The model in Fi. consists of the coupled line four-port network described by an impedance matri [] and arbitrary load matri at opposite terminals described by matri ["]. In practice, ports and are in eneral either short-circuited or open-circuited with a correspondin representation of the twoport network ["]. The manitude of the reflection coefficient at port is equal to ( ) ( ) ij, ij, ij, ij, S lo, () db where is the input impedance of the transformer, which is a function of the load impedance, impedance matri elements of coupled lines ij and the arbitrary load ij (i and j are the indees of matri elements). The input impedance is calculated usin relations derived in [5] toether with the followin elements of the impedance matri ["] for the stepped impedance transmission line θ coth, (5a) ( ) sinh( θ ) (3)

7 6, (5b) ( ) sinh( θ ) θ coth. (5c) ( ) sinh( θ ) The derivation of (5) is iven in Appendi I. Thus, the analysis of the structure can be performed usin this analytical representation. The electrical lenth is a function of frequency and is used here for the analysis of the spectrum of the transformer reflection coefficient. It can be depicted from the calculated response in Fi. that the transformer provides wideband operation, and the electrical lenth of the transformer is equal to a quarter wavelenth at the center frequency. Manitude of of S Sand S S, (db) db Fi.. Response of the 5- Ω impedance transformer shown in Fi. 9. In addition, the distance between the minima locations Δθ can be varied by adjustin the parameters of the structure. This distance Δθ characterizes operatin frequency bandwidth of the transformer. The characteristics of the transformer for three different values of Δθ are shown in Fi.. As it can be seen, the in-band level of the reflection coefficient depends on parameter Δθ. The estimation of the maimum level of the reflection coefficient between minima for different transformation ratios can be found usin the data shown in Fi.. Inband reflections (db) Δθ - Δθ3 Δθ / Fi.. The maimum level of the reflection coefficient between minima in Fi.. Δθ 5 Δθ Δθ 3 Electrical lenth, (de) As epected, the level of in-band reflections for the transformer decreases with reducin of transformation ratio, and reaches the absolute minimum at /. B. Synthesis of the wideband transformer Equation () was solved numerically for the transformer in Fi. 9 with respect to the desin parameters. Based on these solutions, desin curves for this transformer are obtained and shown in Fi Δθ3 Δθ Δθ5 c /.5 c/, π / c/, π / Rc θc/θ, θπ /θ.5 π / / (3a) Δθ3 Δθ Δθ5 3 d c / 8 6 π / / (3b) Δθ3 Δθ Δθ / (3c) θ c /θ Δθ3 Δθ Δθ d θ π /θ / (3d)

8 7 /, / Fi. 3. Desin curves for the impedance transformer in Fi. 9. The desin curve for the parameter R π is not presented here, because it can be easily found from the ratio [7] c π RcRπ. (6) c π The impedance transformer in Fi. 9 can be synthesized usin the desin curves in Fi. 3. It is interestin to note that the electrical lenths for both modes are essentially independent of the loadin condition and can be adjusted for convenience. Furthermore, for lare values of the transformation ratio / >, R c does not depend on the load impedance and is only slihtly dependent on Δθ. Finally, it can be concluded from Fi. 3 that and can be represented in the form, / m, /, where is the constant and m, is the slope. Therefore, the transformer can be efficiently synthesized usin only the parameters c, c, π, π, Δθ, and m,. C. Desin eample Consider the desin of a 5- Ω impedance transformer with the center frequency. GHz and Δθ. For this : transformer the desin parameters are chosen from Fi. 3 and listed in Table. Table. Electrical parameters of the transformer. / θ, (f.ghz) c/ c/ π/ π/ c, Ω c, Ω Δθ3 Δθ Δθ5 π,ω π, Ω, Ω / /, Ω / / / Rc Rc θc/θ θc, (f.ghz) θπ/θ θπ, (f.ghz) Based on these data the physical parameters of the circuit components are synthesized. The parameters iven in Table correspond to a transformer based on a substrate with a dielectric constant ε r 3.38 and thickness h.8 mm. The coupled line width is.37 mm for the input terminal and.5 mm for the output terminal. The ap between the coupled lines is. mm. Microstrip line widths are. mm and.5 mm for the transmission lines denoted and in Fi. 9, respectively. The physical lenth of the transformer is 3. mm. Based on these physical parameters the simulation of the transformer can be performed usin any freely or commercially available software circuit simulators, which contain the models for asymmetric coupled lines and microstrip transmission lines. The matchin circuit desin eample has been fabricated and measured. A photoraph of the fabricated 5 Ω transformer is shown in Fi.. In this eample the input transmission line is connected usin an air bride transition. Interconnectin line Fi.. Wideband quarter wavelenth impedance transformer. The microwave realization of the circuit in Fi. 9. The simulated and measured characteristics of the matchin circuit are iven in Fi. 5 and compared to the characteristics of the conventional quarter-wave transformer. Manitude of S (db) Air bride ( Frequency (GHz) simulated measured traditional quarter-wave transformer Fi. 5. Simulated and measured characteristics of the synthesized transformer; compared with the conventional quarter-wave transformer. As it can be seen from the simulated data, three minima in the reflection coefficient spectrum of the synthesized transformer are achieved as epected. The distance between minima corresponds to Δθ. The achieved bandwidth at db reflection level is 3 times larer as compared with a standard quarter-wave matchin circuit. Althouh the measured characteristics differ from the simulation at low manitude of the reflection coefficient, it is deemed suitable for most practical applications. The measured fractional bandwidth for this transformer confiuration is more than % for - db reflection coefficient level. IV. CONCUSIONS It is shown that coupled transmission lines is an attractive component for compact impedance transformer desin. The capabilities of the coupled line transformer are etended with θ 9 Coupled lines

9 8 the help of different kinds of auiliary loads, connected to diaonally opposite terminals. Usin this concept different circuits have been proposed for matchin real and comple loads in a narrow and wide frequency rane. It is demonstrated theoretically and eperimentally that it is possible to improve the fractional matchin bandwidth beyond % at db reflection level by introducin an interconnectin transmission line. Althouh the proposed structure is still a quarter wavelenth lon, it provides a three times wider operatin frequency rane in comparison to the traditional quarter-wave transformer. A eneral model for such a confiuration of the transformer was developed based on mode characteristics. This eneral model establishes the desin curves for the impedance transformer. Based on the analysis of this model different load confiurations at the free terminals are proposed resultin in improved matchin characteristics of the overall circuit. The considered eamples demonstrate matchin between real and comple impedances in a narrow and a wide frequency rane. APPENDIX I A series connection of the transmission lines shown in Fi. 6 can be described as a connection of two two-port networks. Fi. 6. Series connection of transmission lines. The impedance matrices of the transmission lines with characteristic impedances,, lenths l, l, and propaation constants γ, γ are iven by [ ] () [ ], l, γ, l, γ (") (") () () () () coth sinh coth sinh ( γ l ) ( γ l ) ( γ l ) ( γ l ) sinh ( γ l ) ( ) γ l coth sinh, (7) ( γ l ) ( ) γ l coth.(8) Impedance matri for the overall circuit in Fi. 6 is derived usin boundary conditions at the common terminal. At this terminal the voltaes of two two-ports are equal, and currents are equal and oppositely directed. Thus, impedance matri elements are found to be: ( ) coth( γ l ) ( coth( ) coth( )) sinh ( ), γ l γ l γ l () () () coth ( ( γ l ) coth( γ l )) sinh( γ l ) sinh( γ l ), () () coth () ( γ l ) ( ( ) ( )) ( ). coth γ l coth γ l sinh γ l (9a) (9b) (9c) In case of transmission lines with equal electrical lenth θ/ (9) can be rewritten as θ coth, (a) ( ) sinh( θ ), (b) ( ) sinh( θ ) θ coth. (c) ( ) sinh( θ ) These equations are used for the calculation of elements of the matri ["] in Fi.. APPENDIX II A schematical drawin showin a coupled line section with ports and terminated in arbitrary loads is shown in Fi. 7. V I Fi. 7. -port circuit consistin of a coupled line section with ports and terminated in arbitrary impedances. The matri for the -port coupled line section is based on Eq.. [ ] 3 I V The impedance matri for the -port based on this confiuration can be written in the followin form:

10 9 V V I I. Where the elements of the -matri can be obtained with basic circuit theory and ives ( ) (3a) ( ), ( ) (3b) ( ) 3 3, ( ) (c) ( ) 3 3, ( ) , (d) ( ) where ( )( ). () π Whenθ ; This reduces most terms to and we end up with. (5) But if we take ( )( ) to be open-circuited ( ) then, (6a) ( )( ) and similarily for the other elements of the matri REFERENCES [] Kian Sen An, Chee How ee, and Yoke Choy eon, Analysis and desin of coupled line impedance transformers, IEEE MTT-S Int. Microwave Symp. Di., vol. 3, pp ,. [] G. Jaworski, and V. Krozer, Broadband matchin of dual-linear polarization stacked probe-fed microstrip patch antenna, Electronics etters, vol., no., pp. -,. [3] S. P. iu, Planar transmission line transformer usin coupled microstrip lines, IEEE MTT-S Int. Microwave Symp. Di., vol., pp , 998. [] E.M.T. Jones and J.T. Bolljahn, Coupled Strip Transmission ine Filters and Directional Couplers, IRE Trans. Microwave Theory & Tech., vol. MTT-, pp. 78-8, April 956. [5] V. hurbenko, V. Krozer, P. Meincke, Broadband Impedance Transformer Based on Asymmetric Coupled Transmission ines in Nonhomoeneous Medium, IEEE MTT-S Int. Microwave Symp., 7. [6] D. Kajfez, S. Bokka, and C. E. Smith, Asymmetric microstrip DC blocks with rippled response, 987 IEEE MTT-S Int. Microwave Symp. Di., pp. 3 33, 98. [7] V. K. Tripathi, Asymmetric coupled transmission lines in an inhomoeneous medium, IEEE Trans. Microwave Theory & Tech., vol. 3, no. 9, pp , September ( )( ), (6b) j ( )( ) 3 3 ( )( ), (6c) j ( )( ) ( ) 3 ( )( ) ( ) 3 3 ( )( ) 3. (6d) Epressed as the matri for the reduced -port coupled lines circuit j [ ]. (7) j (-port open-circuited impedance matri at π θ and epressed in terms of the even and odd mode impedances, as a parameter and O.C stub).