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Coupled Transmission Lines as Impedance

Jensen, Thomas; Zhurbenko, Vitaliy; Krozer, Viktor; Meincke, Peter

Published in: IEEE Transactions on Microwave Theory and Techniques

Link to article, DOI: 10.1109/TMTT.2007.909617

Publication date: 2007

Document Version Peer reviewed version

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Citation (APA): Jensen, T., Zhurbenko, V., Krozer, V., & Meincke, P. (2007). Coupled Transmission Lines as Impedance Transformer. IEEE Transactions on Microwave Theory and Techniques, 55(12), 2957-2965. https://doi.org/10.1109/TMTT.2007.909617

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1 Coupled Transmission Lines as Impedance Transformer Thomas Jensen, Vitaliy Zhurbenko, Viktor Krozer, Peter Meincke Technical University of Denmark, Ørsted•DTU, ElectroScience, Ørsteds Plads, Building 348, 2800 Kgs. Lyngby, Denmark, Phone:+45-45253861, Fax: +45-45931634, E-mail: [email protected]

Abstract— A theoretical investigation of the use of a coupled line transformer. line section as an impedance transformer is presented. It is shown This paper focuses on developing the necessary formulas for how to properly select the terminations of the coupled line applying coupled line sections in matching applications, as structures for effective matching of real and complex loads in well as the appropriate basic analysis of the coupled line both a narrow and a wide range. The corresponding circuit configurations and the design procedures are proposed. section. Establishing a design framework will enable a Synthesis relations are derived and provided for efficient widespread use of coupled line structures as novel circuit construction. Design examples are given to matching elements, in addition to the standard lines and demonstrate the flexibility and limitations of the design methods lumped elements. and to show their validity for practical applications. Wideband In the lower GHz range the loading of the through and matching performance with relative bandwidth beyond 100% and coupled ports can be done with lumped elements which allows return loss RL > 20 dB is demonstrated both theoretically and experimentally. Good agreement is achieved between the for easy matching of both real and imaginary impedance measured and predicted performance of the coupled line values. At higher it is not possible to use lumped transformer section. elements, but the difference between the even and odd mode Index Terms—Coupled transmission lines, directional coupler, impedances is a parameter which makes it possible to turn a impedance matching, impedance matrix, microstrip lines, strip mixed real and imaginary control load at the through into lines. a purely imaginary one, which can be implemented with a . I. INTRODUCTION Equations for matching purposes, which are based on n recent years, coupled transmission lines have been controllable parameters of coupled transmission line sections, I suggested as a matching element due to greater flexibility are presented for backward-wave couplers including and compactness in comparison to quarter wavelength microstrip and stripline transmission line couplers. transmission lines [1-3]. It has been demonstrated that The required for a given application often becomes matching real and complex loads with coupled lines leads to too tight for a practical implementation. Therefore an more compact realizations and could therefore become investigation into the range of load values that can realistically important at millimeter-wave frequencies for on-chip or Low be matched with the coupled line section has been carried out. Temperature Cofired Ceramics (LTCC) matching solutions. Finally loading of the through and coupled port with an Another area where coupled line structures are useful is interconnecting transmission line is considered with the matching of array structures, as successfully purpose to achieve a wide operating frequency range. In [5] a demonstrated in [2]. broadband impedance transformer based on coupled The quarter-wave transformer is simple and easy to use, but transmission lines is presented. The synthesis procedure for it has no flexibility beyond the ability to provide a perfect this circuit is explained. By using this procedure it is possible match at the center frequency for a real-valued load, although to shape the frequency response by placing transmission a complex load of course can be matched by increasing the minima in the spectrum. length of the quarter-wave transformer. The coupled line section provides a number of variables which can be utilized II. THE USE OF A COUPLED LINE SECTION AS AN IMPEDANCE for matching purposes. These variables are the even and odd TRANSFORMER mode impedances and loading of the through and coupled Fig. 1 shows the general coupled lines configuration. There ports. This loading can be done in form of a feedback is no established terminology for use of the coupled line connection which provides additional zeros for broadband section as an impedance transformer, in this paper it is found matching. useful to use the port names from directional coupler These variables can be chosen to provide a perfect match or terminology in the discussion of the circuit, but the numeration any desired value of the coefficient at the operating used corresponds to filter design with coupled line sections frequency. The bandwidth of the coupled line transformer can because the theory is developed from that point of view. be further increased in case of mismatch. In addition, as will be shown, it is also possible to match a complex load. This is a generalization of matching with a quarter-wave transmission 2

Coupled Lines operating frequency in terms of the difference between the even and odd mode impedance. This condition is similar to the well-known relation Z = Z Z , for a quarter-wave Z0e, Z0o, θ 0 g L coupled isolated 2 3 transmission line transformer, when the is exchanged with the difference between the even input through and odd mode impedances of the coupled lines. 1 4 It is also possible to retain the in the formula and use it as a parameter, yielding Fig. 1. Coupled transmission lines. The port names are in agreement with directional coupler terminology for a backward-wave coupler, while the 1+ ΓIN numeration is the same as used in filter design with coupled line sections. Z − Z = 2 Z Z . (7) 0e 0o g L 1− Γ IN The impedance matrix for the 4-port open-circuited coupled The parameter Z- can be chosen within reasonable limits. A line section in Fig. 1, where transverse electromagnetic (TEM) large value of Z- implies a tight coupling, where the coupling wave propagation and symmetric transmission lines are Z − Z coefficient is given by C = 0e 0o . Tight coupling implies assumed, can be expressed in terms of the even and odd mode Z0e + Z0o impedances, Z0e and Z0o, and the , θ [4]: a large value of C and requires a large difference between the Z = Z = Z = Z = − j 2(Z + Z )cotθ , (1a) 11 22 33 44 0e 0o even and odd mode impedances. A large value of Z- necessitates a very narrow gap between edge-coupled Z = Z = Z = Z = − j 2(Z − Z )cotθ , (1b) 12 21 34 43 0e 0o transmission lines or a large ground plane spacing, e.g. in

Z13 = Z31 = Z24 = Z42 = − j 2(Z0e − Z0o )cscθ , (1c) stripline technology. From (7) it can be seen that transforming a large-valued = = = = − ( + ) θ , (1d) Z14 Z41 Z23 Z32 j 2 Z0e Z0o csc load requires a tighter coupling than a small-valued load, and even though a non-zero reflection coefficient can increase the j = −1 . bandwidth it also requires tighter coupling. The of the coupled line section with the The quarter-wave transformer has the same length as the coupled and through port open-circuited and the isolated port coupled line impedance transformer for a given application terminated in the load impedance ZL, can be calculated from and requires only one transmission line, so the only obvious the current and at the input port advantage of the latter is that it constitutes a perfect DC block. V Z Z + Z However, a load attached to the through port opens for Z = 1 = 11 L , (2) IN improvement of the transformer characteristics for the coupled I1 ZL + Z22 line section as will be shown in the next sections. where Z = Z11Z22 − Z21Z12 . In the following derivations only the center frequency is A. Loading of the through port for extended matching capabilities. considered and hence, θ = π 2 , cotθ = 0 , and cscθ = 1. Using the input port as input and with the load to be Furthermore the notation Z = Z + Z and Z = Z − Z is + 0e 0o − 0e 0o transformed attached to the isolated port leaves two available used. Then (2) simplifies to ports (the through and coupled port) that can be exploited to 2 2 2 − Z21Z12 − (− j 2) Z− Z− improve the matching capabilities. The loading of the through Z IN = = = . (3) Z L Z L 4Z L port will be considered in this section.

The reflection coefficient with the generator impedance Zg attached to port 1 is given by Coupled Lines 2 Z− − Z g Z − Z Z0e, Z0o, θ IN g 4Z L coupled isolated Port 2, ZL ΓIN = = . (4) Z + Z Z 2 2 3 IN g − + Z 4Z g L Port 1, Zg input through For a perfect match the reflection coefficient has to be zero 1 4 and thus 2 Z− ZIN − Z g = 0 . (5) Z4 4Z L

Solving for Z- leads to

Z0e − Z0o = 2 Z g Z L . (6) Fig. 2. The 4-port coupled transmission line section reduced to a 2-port with This establishes the condition for a perfect match at the the coupled port open-circuited and the through port terminated in load Z4. 3

0 The open-circuited impedance matrix [Z] for a two-port that ) is based on the four-port coupled line section in Fig. 2, where x x dB x ( x the coupled port is left open-circuited and the through port is x 21

S x loaded with an arbitrary impedance Z4, that will be used as a -10 x x matching parameter, can be derived as (see appendix II) and

2 11 x x  Z+ − j  S  Z−  4Z 2 -20 x [Z] =  4  . (8) S...... 11 Z0e =150Ω   − x = Ω  j  S...... 21  Z0o 50   Z− 0    S11 Z0e =125Ω 2 x ......   -30 x S...... 21  Z0o = 25Ω 

The impedance matrix is symmetric and therefore the circuit of Magnitude λ S11{ trf } is reciprocal, but Z4 should be chosen imaginary or with a ...... 4 large real part to avoid excessive power dissipation. The input 0 0.5 1 1.5 2 impedance derived from (2) is then Frequency (GHz) 2 2 Fig. 4. Response of the circuit in Fig. 3. |S11| is -30 dB at the design frequency Z+ Z− as desired. Note that the insertion loss is non-zero because power is dissipated Z IN = + . (9) in the . The response with tighter coupling (Z0e=125Ω, Z0o=25Ω) and a 4Z4 4Z L comparision with a quarter-wave transformer is also included in the plot. Substituting this expression in the formula for the input reflection coefficient (4) and solving for Z4 gives If the coupling is increased, setting Z0e = 125 Ω and 2 Z (1− Γ ) Z0o = 25 Ω, results in Z4 = 199 Ω and an increased bandwidth Z = + IN , (10) 4 2 of 44% for the same matching conditions. The physical Z− 4Z g (1+ ΓIN )− (1− ΓIN ) dimensions for the first case in edge-coupled stripline Z L technology with εr = 3.38 and a ground plane spacing of 5 mm which determines the value of Z4 for a given value of Zg, ZL, are a conductor width of 0.55 mm and a conductor separation Z+, Z-, and ΓIN. The condition for Z 4 = ∞ , i.e. open-circuited, of 0.21 mm. is only fulfilled for Z− = 2 Z g Z Lγ . This demonstrates that B. Achieving purely imaginary control loads the coupled line section impedance transformer is indeed a An interesting property of the coupled line section as a generalization of the standard transmission line transformer, matching element is the ability to match a complex load. where the impedance Z4 represents an additional degree of Splitting Z4 in a real and an imaginary part gives freedom. 2 γ 2 − 2 2 2 Z+ (4Zg ZL Z− Re{ZL}) Z− Z+ Im{ZL} Z4 = − j ∆ ∆ , (11) Matching example with a real-valued load 2 2 2 2 2 The ability of the coupled line section to match a real- ∆ = 16Zgγ ZL + Z− (Z− − 8Zgγ Re{ZL}), valued load is illustrated with a simple example. 1+ ΓIN where γ = , Zg and γ are assumed real-valued. The following parameters are given: Zg = 50 Ω, ZL = 100 Ω, 1− ΓIN Z+ = 200 Ω, Z- = 100 Ω, and ΓIN = 0.032. From this expression it can be seen that it is possible to use Z- Using (10) results in Z4 = 353 Ω. The circuit configuration to make the real part of Z4 equal to zero, which establishes a is shown in Fig. 3 and the frequency response plotted at an condition for a purely imaginary value of Z4 arbitrary design frequency of 1 GHz in Fig. 4. 2 2 2 Z+ (4Z gγ Z L − Z− Re{Z L})= 0 , (12) Z0e, Z0o, θ Port 2, ZL Z gγ Z− = 2 Z . (13) L Re{Z } R L Port 1, Zg This is especially useful for high frequency applications, where the matching load to be attached to Z4 can simply be realized as a transmission line stub of a specified length. An added Fig. 3. Matching configuration example. The response with Zg = 50 Ω, ZL = advantage is that a purely reactive load is in principle lossless. 100 Ω, Z+ = 200 Ω, and Z- = 100 Ω is shown in Fig. 4.

A reflection coefficient of 0.032 corresponds to -30 dB. In Matching example with a complex-valued load this example a reflection coefficient magnitude < -20 dB spans The following parameters are given: Zg = 50 Ω, a bandwidth of 22%. ZL = 40 + j20 Ω, Z+ = 200 Ω, and ΓIN = 0. Z- is found to be 100 Ω and Z4 = -j400 Ω which can be implemented as a short transmission line stub with a length of

12.1º and a characteristic impedance of Z0eZ0o = 86.6 Ω . 4

Z0e, Z0o, θ significantly worse, see Fig. 6. Port 2, ZL The limitation of the coupled line section becomes apparent when Z4 from (10) is plotted against a range of possible load Z0, θStub Port 1, Zg values. Fig. 7 shows the required value of Z4 for a perfect match plotted versus the real part of ZL. The figure depicts two sets of curves, one for the real part of Z4 and one for the imaginary part of Z4, for imaginary load values, Im{ZL} of 20, 40 and 60 Ω, respectively, Z0e = 125 Ω, Z0o = 25 Ω, and Zg = Port 2, ZL 50 Ω.

Z0e=125 Ω, Z0o = 25 Ω, Zg = 50 Ω

Z0, θstub

Z0e, Z0o, θ

4 Thick lines: Re(Z ) Thin lines: Im(Z4) (Ω)

4 Z

Im(ZL)=20 Ω Im(ZL)=40 Ω Port 1, Zg Im(ZL)=60 Ω Fig. 5. Matching configuration for a complex load. Schematic drawing above Im(ZL)=20 Ω and possible stripline realization below. The response with the parameters Zg Im(ZL)=40 Ω = 50 Ω, ZL = 40 + j20 Ω, Z+ = 200 Ω, Z- = 100 Ω, Z0 = 86.6 Ω, and θStub = Im(ZL)=60 Ω 12.1º is shown in Fig. 6.

Real part of ZL (Ω) The characteristic impedance of the stub can be chosen to give Fig. 7. The arrows indicate the general direction of the curves when the a line width equal to the transmission lines in the coupled lines coupling is increased, i.e. the required value of the real part of Z4 becomes section in order to avoid discontinuities. Here Z Z is more positive and the imaginary part becomes more negative. Note that for 0e 0o this choice of the even and odd impedance, a negative real part is required for used for the stub line characteristic impedance, but it is not a Z4 when Im{ZL}=20 Ω and Re{ZL}<40 Ω (thick blue curve below 0). necessary condition. To decrease the power dissipation a large value of Z4 is 0 x x x x x ) desirable, but this implies stronger coupling. Decreasing the dB

( coupling will lower the real part of Z4 towards the value of Zg

21 with the result that most of the input power is dissipated in Z4. S -10 x x x x If the condition for a purely imaginary value of Z4, Eq. (13), is and and applied then Fig. 8 shows that tight coupling values are 11 S required for large-valued loads. x x -20 ...... S11 coupled Re(Z4) = 0   ...... S21  lines  ΓIN = 0 x S ...... 11 {λ trf }

Magnitude of of Magnitude 4 x S21 ...... -30 0 0.5 1 1.5 2 (Ω)

Frequency (GHz) o

0

Fig. 6. Response of the circuit in Fig. 5 and compared with a quarter-wave Z

– transformer. Note that this matching configuration is lossless because Z4 is e purely imaginary. 0 Z

Im(ZL)=20 Ω The response in Fig. 6 shows a perfect match at the center Im(ZL)=40 Ω frequency and a -20 dB bandwidth of 31%. The length of the Im(ZL)=60 Ω matching stub is very short compared to traditional transmission line matching circuits and will only slightly increase the circuit size. The load used in this example can also be matched with a Real part of ZL (Ω) quarter-wave transformer and an extra length of transmission Fig. 8. Values of Z0e - Z0o when Z4 is required to be purely imaginary and line to account for the imaginary part of the load, but the ΓIN = 0 (perfect match). bandwidth achieved is smaller and out of band rejection is 5

It can be depicted that very tight coupling is required for The transformer consists of asymmetric coupled lines even moderate impedance loads. Relaxing the condition for the described by the electrical parameters Zc1, Zc2, Zπ1, Zπ2, which reflection coefficient, ΓIN, moves the curves towards higher are, respectively, the characteristic impedances of line 1 and 2 values and will therefore require even tighter coupling. for the c and π modes of propagation; θc and θπ, the electrical A coupling coefficient larger than -10 dB is difficult to lengths for the c and π modes; Rc and Rπ, the ratios of the achieve with microstrip or stripline edge-coupled lines, but is on the two lines for the c and π modes. The stepped possible with a thick substrate or with broadside coupled impedance transmission line consists of two equal length transmission lines. transmission lines with characteristic impedances Z01 and Z02, as shown in Fig. 9. The electrical length of each transmission III. WIDEBAND IMPEDANCE TRANSFORMATION. line is set to be half the electrical length of the coupled line section to reduce the number of design parameters. A. Derivation of analysis formulas For the purpose of analysis, this structure is transformed to a The impedance considered above are based on two-port network with arbitrary load using an impedance transmission lines in homogeneous medium, e.g. striplines. matrix representation. Thus, the entire circuit can be They allow for a simpler analysis, however, in many practical represented as a two-port network, which performs impedance cases, for example in surface mount technology, it is more transformation between a generator impedance Zg connected to useful to deal with microstrip structures. port 1 and a load impedance ZL connected to port 3, as shown in Fig. 10. The wideband impedance transformer proposed in this section is derived on the basis of asymmetric, uniformly [Z'] coupled lines in a nonhomogeneous medium [5]. A microstrip line is one of the most commonly used classes of transmission (2) (1") lines in a nonhomogeneous medium. The proposed (4) (2") [Z"] configuration is a quarter wavelength long and provides three times wider operating frequency range in comparison to the [Z] ZL (3) traditional quarter-wave transformer. As discussed above the matching properties of the Zg transformer depend not only on coupled line parameters, but (1) also on the loads at ports 2 and 4 in Fig. 2. This dependence introduces additional degrees of freedom during the design Fig. 10. Two-port network representation for the coupled line impedance procedure and is used here to significantly expand the transformer. From [5]. bandwidth of the impedance transformer. The configurations shown above use loading of terminal 4 and with terminal 2 The model in Fig. 10 consists of the coupled line four-port open-circuited. In the circuit considered below, both terminals network described by an impedance matrix [Z] and arbitrary are loaded using an interconnecting microstrip stepped load matrix at opposite terminals described by matrix [Z"]. In impedance transmission line as shown in Fig. 9. practice, ports 2 and 4 are in general either short-circuited or open-circuited with a corresponding representation of the two-

Z01, θ/2 Z02, θ/2 port network [Z"]. The magnitude of the reflection coefficient at port 1 is equal 1" 2" to Line 2 Port 2   Z IN (Zij , Zij′′, ZL )− Z g OUT S = 20 log  , (14) 2 3 11 dB  ′′ +   Z IN (Zij , Zij , ZL ) Z g  Port 1 Line 1 Zc1, Zπ1, Zc2, Zπ2, IN Rc, Rπ, θ=(θc + θπ)/2 where ZIN is the input impedance of the transformer, which is a 1 4 function of the load impedance ZL, impedance matrix elements

of coupled lines Zij and the arbitrary load Zij′′ (i and j are the OUT indexes of matrix elements). (3) The input impedance ZIN is calculated using relations derived in [5] together with the following elements of the Z02 θ/2 impedance matrix [Z"] for the stepped impedance transmission θ line Z01 2 θ  2Z01 (1) Z11′′ = Z01 coth  − , (15a)  2  (Z01 + Z02 )⋅ sinh(θ ) IN Fig.9. The quarter-wave wideband impedance transformer. Schematic drawing above and possible microstrip realization below. 6

2Z Z As expected, the level of in-band reflections for the Z′′ = Z′′ = 01 02 , (15b) 12 21 transformer decreases with reducing of transformation ratio, (Z01 + Z02 )⋅sinh(θ ) and reaches the absolute minimum at ZL/Zg = 1.

θ  2Z 2 B. Synthesis of the wideband transformer Z′′ = Z coth  − 02 . (15c) 22 02 Equation (14) was solved numerically for the transformer in  2  (Z01 + Z02 )⋅sinh(θ ) Fig. 9 with respect to the design parameters. Based on these The derivation of (15) is given in Appendix I. solutions, design curves for this transformer are obtained and Thus, the analysis of the structure can be performed using shown in Fig. 13. this analytical representation. 2.5 The electrical length is a function of frequency and is used Δθ=30° Δθ=40° here for the analysis of the spectrum of the transformer g 2 Δθ=50° reflection coefficient. It can be depicted from the calculated /Z Z c1/Z g response in Fig. 11 that the transformer provides wideband π1

, Z 1.5 operation, and the electrical length of the transformer is equal g /Z

to a quarter wavelength at the center frequency. c1 Z 1

0 Z π1/Z g Δθ = 50° 0.5 1 2 3 4 5 6 7 8 9 10 (dB) -10 Δθ = 40° 2 Δθ = 30° ZL/Zg (13a) -20 and S and

14

11 Δθ=30° -30 12 Δθ=40°

g Δθ=50°

/Z 10 Z 1 30 d -40 Z c2/Z g π2 8 , Z g Magnitude of S11and S22, dB -50 6 /Z

Magnitude of S of Magnitude 0 20 40 60 80 100 120 140 160 180 c2

Z 4 Electrical length,( deg) 2 Z π2/Z g 0 Fig. 11. Response of the 50-100 Ω impedance transformer shown in Fig. 9. 1 2 3 4 5 6 7 8 9 10 Z /Z In addition, the distance between the minima locations Δθ L g (13b) can be varied by adjusting the parameters of the structure. This 1 distance Δθ characterizes operating frequency bandwidth of the transformer. The characteristics of the transformer for three Δθ=30° 0.98 Δθ=40° different values of Δθ are shown in Fig. 11. As it can be seen, Δθ=50° the in-band level of the reflection coefficient depends on 0.96

parameter Δθ. The estimation of the maximum level of the Rc reflection coefficient between minima for different 0.94 transformation ratios can be found using the data shown in 0.92 Fig. 12. 0.9 0 1 2 3 4 5 6 7 8 9 10 ZL/Zg -10 (13c)

-20 1.08 1.06 -30 1.04

Δθ=40° /θ θ c/θ -40 Δθ=30° π 1.02 Δθ=30° Δθ=40° Δθ=40°

Inband reflections (dB) 1 Δθ=50° /θ, θ /θ, -50 c 0.98 30 d θ π/θ θ 1 2 3 4 5 6 7 8 9 10 0.96 ZL/Zg 0.94 Fig. 12. The maximum level of the reflection coefficient between minima in 0.92 Fig. 11. 1 2 3 4 5 6 7 8 9 10 ZL/Zg (13d) 7

2.6 2.5 mm for the transmission lines denoted Z01 and Z02 in Fig. 9, Δθ=30° respectively. The physical length of the transformer is Δθ=40° Z 01/Z g Δθ=50° 43.1 mm. Based on these physical parameters the simulation of 2.1 /Z the transformer can be performed using any freely or 02 commercially available software circuit simulators, which 1.6

/Z, Z /Z, contain the models for asymmetric coupled lines and 01

Z Z 02/Z g microstrip transmission lines. The matching circuit design 1.1 example has been fabricated and measured. A photograph of the fabricated 50 – 100 Ω transformer is shown in Fig. 14. In 0.6 this example the input transmission line is connected using an 1 2 3 4 5 6 7 8 9 10 air bridge transition. ZL/Zg Fig. 13. Design curves for the impedance transformer in Fig. 9.

Interconnecting line θ = 90° The design curve for the parameter Rπ is not presented here, because it can be easily found from the ratio [7]

Zc2 Zπ 2 = = −R Rπ . (16) Z Z c Air bridge c1 π1 Coupled lines The impedance transformer in Fig. 9 can be synthesized Fig. 14. Wideband quarter wavelength impedance transformer. The using the design curves in Fig. 13. It is interesting to note that microwave realization of the circuit in Fig. 9. the electrical lengths for both modes are essentially independent of the loading condition and can be adjusted for The simulated and measured characteristics of the matching convenience. Furthermore, for large values of the circuit are given in Fig. 15 and compared to the characteristics transformation ratio ZL / Zg > 2, Rc does not depend on the load of the conventional quarter-wave transformer. impedance and is only slightly dependent on Δθ. Finally, it can be concluded from Fig. 13 that Z01 and Z02 can be represented 0 in the form Z01,02 / Zg = Z00 + m1,2 ZL / Zg, where Z00 is the constant and m1,2 is the slope. Therefore, the transformer can (1x x (dB) be efficiently synthesized using only the parameters Zc1, Zc2, -20

11 x x Zπ1, Zπ2, Δθ, and m1,2.

C. Design example -40 simulated Consider the design of a 50-100 Ω impedance transformer measured with the center frequency 1.1 GHz and Δθ = 40°. x traditional quarter-wave

Magnitude of S of Magnitude transformer For this 1:2 transformer the design parameters are chosen from -60 Fig. 13 and listed in Table 1. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Frequency (GHz) Table 1. Electrical parameters of the transformer. Fig. 15. Simulated and measured characteristics of the synthesized

transformer; compared with the conventional quarter-wave transformer.

g g g g g g

g

θ θ c /Z /Z /Z /Z / /Z /Z / ,° /Z c 1 2 π R L θ c1 c2 π π 01 02 θ θ As it can be seen from the simulated data, three minima in Z Z Z Z Z Z Z the reflection coefficient spectrum of the synthesized (f=1.1GHz) 2 90 1.35 2.5 0.8 1.5 0.9 0.8 0.93 1.042 0.958 transformer are achieved as expected. The distance between minima corresponds to Δθ = 40°. The achieved bandwidth at –20 dB reflection level is 3 times larger as compared with a

Ω Ω Ω Ω Ω

Ω , c ,° ,° , , , , , standard quarter-wave matching circuit. Although the c π R π1 c1 c2 π2 01 02 θ θ Z Z Z Z Z Z measured characteristics differ from the simulation at low (f=1.1GHz) (f=1.1GHz) magnitude of the reflection coefficient, it is deemed suitable 67.5 125 40 75 45 40 0.93 93.78 86.22 for most practical applications. The measured fractional bandwidth for this transformer configuration is more than Based on these data the physical parameters of the circuit 120 % for -20 dB reflection coefficient level. components are synthesized. The parameters given in Table 1 correspond to a transformer based on a substrate with a IV. CONCLUSIONS dielectric constant εr = 3.38 and thickness h = 0.8 mm. The It is shown that coupled transmission lines is an attractive coupled line width is 1.37 mm for the input terminal and component for compact impedance transformer design. The 0.54 mm for the output terminal. The gap between the coupled capabilities of the coupled line transformer are extended with lines is 0.41 mm. Microstrip line widths are 2.22 mm and 8

2 the help of different kinds of auxiliary loads, connected to Z (1) ′′ (1) ( 12 ) diagonally opposite terminals. Using this concept different Z11 = Z11 − = Z01 coth(γ 1l1 ) Z (2) + Z (1) circuits have been proposed for matching real and complex 11 11 (19a) loads in a narrow and wide frequency range. It is demonstrated Z 2 − 01 , theoretically and experimentally that it is possible to improve 2 (Z01 coth(γ 1l1 )+ Z02 coth(γ 2l2 ))⋅sinh (γ 1l1 ) the fractional matching bandwidth beyond 100% at –20 dB reflection level by introducing an interconnecting transmission (1) (2) line. Although the proposed structure is still a quarter Z12 Z12 Z12′′ = Z21′′ = wavelength long, it provides a three times wider operating Z (2) + Z (1) 11 11 (19b) frequency range in comparison to the traditional quarter-wave Z Z = 01 02 , transformer. (Z coth(γ l ) + Z coth(γ l ))⋅ sinh(γ l )⋅ sinh(γ l ) A general model for such a configuration of the transformer 01 1 1 02 2 2 1 1 2 2 was developed based on mode characteristics. This general 2  (2)  model establishes the design curves for the impedance  Z12  ′′ = (2) −   = γ transformer. Based on the analysis of this model different load Z22 Z11 Z02 coth( 2l2 ) Z (2) + Z (1) configurations at the free terminals are proposed resulting in 11 11 (19c) improved matching characteristics of the overall circuit. Z 2 − 02 . 2 The considered examples demonstrate matching between (Z01 coth(γ1l1)+ Z02 coth(γ 2l2 ))⋅sinh (γ 2l2 ) real and complex impedances in a narrow and a wide frequency range. In case of transmission lines with equal electrical length θ/2 (19) can be rewritten as

APPENDIX I 2 θ  2Z ′′ = − 01 A series connection of the transmission lines shown in Z11 Z01 coth  , (20a)  2  (Z01 + Z02 )⋅ sinh(θ ) Fig. 16 can be described as a connection of two two-port networks. 2Z Z Z′′ = Z′′ = 01 02 , (20b) 12 21 (Z + Z )⋅sinh(θ ) (1") Z01, l1, γ1 Z02, l2, γ2 (2") 01 02

θ  2Z 2 Z′′ = Z coth  − 02 . (20c) Fig. 16. Series connection of transmission lines. 22 02  2  (Z01 + Z02 )⋅sinh(θ )

The impedance matrices of the transmission lines with characteristic impedances Z01, Z02, lengths l1, l2, and These equations are used for the calculation of elements of propagation constants γ1 , γ2 are given by the matrix [Z"] in Fig. 10.

 Z 01  (1) (1) Z 01 coth(γ 1l1 )  Z Z  (γ ) (1) = 11 12 =  sinh 1l1  APPENDIX II [Z ]  (1) (1)  , (17)  Z 01  Z 21 Z 22  γ A schematical drawing showing a coupled line section  Z 01 coth( 1l1 )  sinh(γ 1l1 )  with ports 2 and 4 terminated in arbitrary loads is shown in Fig. 17.  Z  γ 02  (2) (2)  Z 02 coth( 2l2 )  [Z] (2) Z Z sinh(γ l ) = 11 12 =  2 2  Z2 [Z ]  (2) (2)  .(18) I´2  Z 0  Z 21 Z 22  γ  Z 02 coth( 2l2 ) I´1 2 3  sinh(γ 2l2 )  V´2 1 4 V´1 Impedance matrix for the overall circuit in Fig. 16 is derived Z4 using boundary conditions at the common terminal. At this terminal the voltages of two two-ports are equal, and currents are equal and oppositely directed. Fig. 17. 2-port circuit consisting of a coupled line section with ports 2 and 4 Thus, impedance matrix elements are found to be: terminated in arbitrary impedances. The Z matrix for the 4-port coupled line section is based on Eq. 1.

The impedance matrix for the 2-port based on this configuration can be written in the following form: 9

V '  Z ' Z ' I '  1 = 11 12 1 . (21)  '   ' '  '  V2  Z 21 Z 22 I 2  REFERENCES

Where the elements of the Z ' -matrix can be obtained with [1] Kian Sen Ang, Chee How Lee, and Yoke Choy Leong, “Analysis and design of coupled line impedance transformers,” basic circuit theory and gives 2004 IEEE MTT-S Int. Microwave Symp. Dig., vol. 3, ' (−Z4 − Z44)⋅ Z21 ⋅ Z12 + Z24 ⋅ Z41 ⋅ Z12 Z11 = Z11 + pp. 1951-1954, 2004. ∆ (23a) [2] G. Jaworski, and V. Krozer, “Broadband matching of dual-linear (− Z − Z )⋅ Z ⋅ Z + Z ⋅ Z ⋅ Z + 2 22 41 14 42 21 14 , polarization stacked probe-fed microstrip patch antenna,” ∆ Electronics Letters, vol. 40, no. 4, pp. 221-222, 2004. ' (−Z4 − Z44)⋅ Z23 ⋅ Z12 + Z24 ⋅ Z43 ⋅ Z12 [3] S. P. Liu, “ transformer using coupled Z12 = Z13 + ∆ (23b) microstrip lines,” IEEE MTT-S Int. Microwave Symp. Dig., (− Z − Z )⋅ Z ⋅ Z + Z ⋅ Z ⋅ Z + 2 22 43 14 42 23 14 , vol. 2, pp. 789–792, 1998. ∆ [4] E.M.T. Jones and J.T. Bolljahn, “Coupled Strip Transmission ' (−Z4 − Z44)⋅ Z21 ⋅ Z32 + Z24 ⋅ Z41 ⋅ Z32 Line Filters and Directional Couplers,” IRE Trans. Microwave Z21 = Z31 + ∆ (24c) Theory & Tech., vol. MTT-4, pp. 78-81, April 1956. (− Z − Z )⋅ Z ⋅ Z + Z ⋅ Z ⋅ Z [5] V. Zhurbenko, V. Krozer, P. Meincke, “Broadband Impedance + 2 22 41 34 42 21 34 , ∆ Transformer Based on Asymmetric Coupled Transmission Lines ' (−Z4 − Z44 )⋅ Z23 ⋅ Z32 + Z24 ⋅ Z43 ⋅ Z32 in Nonhomogeneous Medium,” IEEE MTT-S Int. Microwave Z22 = Z33 + Symp., 2007. ∆ , (24d) (− Z − Z )⋅ Z ⋅ Z + Z ⋅ Z ⋅ Z [6] D. Kajfez, S. Bokka, and C. E. Smith, “Asymmetric microstrip + 2 22 43 34 42 23 34 ∆ DC blocks with rippled response,” 1987 IEEE MTT-S Int. Microwave Symp. Dig., pp. 301–303, 1981. where ∆ = (−Z − Z )(−Z − Z )− Z Z . (22) 4 44 2 22 42 24 [7] V. K. Tripathi, “Asymmetric coupled transmission lines in an π inhomogeneous medium,” IEEE Trans. Microwave Theory & Whenθ = ; Z11 = Z 22 = Z 33 = Z 44 = Z 21 = Z12 = Z 34 = Z 43 = 0 . 2 Tech., vol. 23, no. 9, pp. 734-739, September 1975. This reduces most terms to 0 and we end up with

' −Z 2 ⋅ Z 41 ⋅ Z14 Z11 = . (25) (− Z 4 )(− Z 2 )− Z 42 Z 24

But if we take Z 2 to be open-circuited (= ∞) then 2 ' − ∞ ⋅ Z41 ⋅ Z14 − Z41 ⋅ Z14 Z+ Z11 = = = , (26a) (− Z4 )(− ∞)− Z42Z24 Z4 4Z4 and similarily for the other elements of the matrix

' Z42 ⋅ Z23 ⋅ Z14 Z12 = Z13 + (− Z − Z )(− Z − Z )− Z Z 4 44 2 22 42 24 , (26b) Z42 ⋅ Z23 ⋅ Z14 j = Z13 + = Z13 = − Z− (− Z4)(− ∞)− Z42Z24 2

' Z24 ⋅ Z41 ⋅ Z32 Z21 = Z31 + (− Z − Z )(− Z − Z )− Z Z 4 44 2 22 42 24 , (26c) Z24 ⋅ Z41 ⋅ Z32 j = Z31 + = Z31 = − Z− (− Z4 )(− ∞)− Z42Z24 2

' (−Z4 − Z44)⋅ Z23 ⋅ Z32 Z22 = (− Z − Z )(− Z − Z )− Z Z 4 44 2 22 42 24 . (26d) (− Z )⋅ Z ⋅ Z = 4 23 32 = 0 (− Z4)(− ∞)− Z42Z24

Expressed as the Z ' matrix for the reduced 2-port coupled lines circuit 2  Z + − j   Z −  ' [Z ]=  4Z 4 2  . (27) − j   Z − 0   2  π (2-port open-circuited impedance matrix at θ = and 2 expressed in terms of the even and odd mode impedances, Z4 as a parameter and Z2 = ∞ O.C stub).