I. INTRODUCTION
One of the most formidable challenges in the field of microwave telecommunications is designing tunable devices. In the near future, more and more applications will require systems that can operate over variable frequency ranges. Tunable impedance transformers are important in applications such as transistor impedance matching over a large tunable bandwidth for maximum gain or minimum noise factor purposes, in designing tunable power dividers, in characterizing Monolithic Microwave Integrated Circuit (MMIC) transistors, and in general matching networks.
Microwave impedance transformers and matching networks are based either on transmission-line sections having specific characteristic impedances, or on L, T, or Π structures. The most usual section type is the quarter-wave impedance transformer, while L, T, or Π structures can be realized by using lumped elements like CLC (capacitor–inductor–capacitor) devices [Reference Sun and Fidler1], or by using single- or double-stub structures.
The overall length of transmission-line-based structures can be reduced by loading a high impedance line with capacitors [Reference Hirota, Minakawa and Muraguchi2]. Such devices can also be made tunable if the fixed capacitors are replaced by tunable capacitors, such as varactor diodes or Micro Electro Mechanical Systems (MEMS) capacitors.
In the last decade, several authors have proposed the use of transmission lines loaded by switches in series or in parallel to realize tunable impedance transformers [Reference Bischof3, Reference Collins, Pollard and Miles4, Reference McIntosh, Pollard and Miles5]. These first devices demonstrated the impedance transformer principle but they cover a small part of the Smith chart. Based on a CLC structure and a quarter-wave transformer, a resonant cell topology [Reference Sinsky and Westgate6] has been used in several configurations [Reference Jung, Kan, Park, Chung, Kim and Kwon7, Reference Kim, Jung, Kang, Park, Kim and Kwon8, Reference De Mingo, Valdovinos, Crespo, Navarro and Garcia9]. In spite of a medium Smith chart coverage, the length of these devices is important because they are realized with a minimum of two quarter-wave transformers.
With the MEMS switches and MEMS varactors development, other designs based on single-stub [Reference Vaha-Heikkila, Varis, Tuovinen and Rebeiz10], double-stub [Reference Lange, Papapolymerou, Goldsmith, Malczewski and Kleber11, Reference Papapolymerou, Lange, Goldsmith, Malczewski and Kleber12, Reference Zheng, Kirby, Pajic, Pothier, Papapolymerou and Popovic13] or triple-stub topologies [Reference Vaha-Heikkila, Varis, Tuovinen and Rebeiz14, Reference Watley, Zhou and Medle15] have also been realized. Most of these devices need lots of MEMS switches or varactors complicating the bias commands. Moreover, these impedance transformers require large surface to cover, in general, small or medium part of the Smith chart.
Lots of these impedance transformers are demonstrated to be used as tuners and some of them are used to realize antenna [Reference De Mingo, Valdovinos, Crespo, Navarro and Garcia9] or transistor [Reference Qiao, Molfino, Lardizabal, Pillans, Asbeck and Jerinic16] matching. Recently, new and improved tunable impedance transformers have been described. A very compact lumped-element CLC impedance transformer with a ±36% tunable bandwidth for a 50 Ω load was demonstrated [Reference Jrad, Perrier, Bourtoutian, Duchamp and Ferrari17]. Some designs, based on transmission lines with variable characteristic impedance, were also presented [Reference Jeong, Kim, Chang and Kim18, Reference Chun and Hong19]. These devices are original but not lead a large coverage of the Smith chart. A double-slug impedance tuner, based on a distributed MEMS transmission line and employing 80 RF-MEMs switches, could produce 1954 different complex impedances around the center of the Smith chart [Reference Shen and Barker20]. This device is also original but we note the important number of MEMS switches witch complicate the bias commands. A MEMS impedance tuner was realized at 25 GHz [Reference Lu, Katehi and Peroulis21]. This design was used with MEMS varactors, and provided continuous impedance coverage. Compared to the other impedance transformers referenced in this paper, this MEMS impedance tuner is optimized in term of surface, variable element number, and Smith chart coverage.
Table 1 resumes the impedance transformers topology and performance. Lot of these examples required more than three varactors, complicating the circuit with numerous bias voltages. Those based on quarter-wave transmission lines result in physically long structures with narrow bandwidths.
Impedance transformers topology and performance.
Moreover, many of these designs were characterized only for a 50 Ω load, so that the insertion loss was known only for that particular case. In some instances, only S 11 was measured, resulting in no information on insertion loss.
This article describes how a tunable and compact impedance transformer, using only two pairs of varactors, was designed. Two different methods for the optimization and experimental characterization of the circuit were developed. The first and the simplest method was based on the synthesized impedances representation with a 50 Ω load. In this case, the synthesized impedance was extracted from the S 22 parameter. The insertion loss was not measured for complex loads.
The second method was much more complete: the return loss and the insertion loss of the transformer, loaded by a complex impedance, were calculated and measured using an external tuner. We emphasize that the first method is accurate only for lossless devices, as it cannot extract the characteristics of lossy circuits.
In a previous study [Reference Perrier, Ferrari, Duchamp and Vincent24, Reference Pistono, Perrier, Bourtoutian, Kaddour, Jrad, Duchamp, Duvillaret, Vincent, Vilcot and Ferrari25], we demonstrated the ability of a new topology to design a tunable complex-impedance transformer, operating at 5 GHz. Its length was ~λ/3 and it needed only two varactors. Its key features were a large matching impedance range, from 5 to 300 Ω, and a tunable frequency range of ±15%. However, this prototype exhibited significant insertion loss, from 2 to 6 dB, owing to the use of low-Q varactors, resulting in a nonoptimized design.
This article describes the design and performance of a new, more compact, and better optimized configuration than the one presented earlier [Reference Pienkowski and Wiatr24]. The design procedure leads to a small λ/10 long transformer with reduced insertion loss.
This article is organized into six sections. Section II details the principle of the impedance transformer. The transformations from 50 Ω to possible complex impedances are illustrated on Smith charts. Section III presents the development of two different simulation approaches: “synthesized impedance” method, and the “matching load” method. The two methods are compared in the case of lossy and lossless circuits. Section IV discusses the design of an impedance transformer for a proof of concept. This device is realized in a printed board hybrid technology. The simulated and measured results obtained by the two different methods are compared in Section V. Finally, Section VI contains concluding remarks.
II. PRINCIPLE
The equivalent electrical circuit of the complex-impedance transformer is shown in Fig. 1. Its total electrical length is θ, and it consists of three transmission-line sections of equal characteristic impedance Z c, and different electrical lengths θ 1, θ 2, and θ 3. The transmission line is loaded by two tunable capacitors C v1 and C v2.
Equivalent electrical circuit of the impedance transformer.
Assuming that the electrical length θ i (i = 1, 2, or 3) of each section is small as compared to the electrical wavelength (λ° = 2π), a section can be replaced by its lumped-element equivalent circuit, as shown in Fig. 2(b). The characteristic impedance Z c and phase velocity v ϕ of the transmission line are defined thus
where L is the inductance and C the capacitance per unit length of the transmission line.
Equivalent circuits of one section of the impedance transformer.
From the lumped equivalent circuit of a section (see Fig. 2(b)), Z ci and v ϕι can also be expressed thus
where L′i = Ll i and C i′ = Cl i. The quantities, L′i is the equivalent inductance and C i′ the equivalent capacitance of a section of physical length l i:
where c is the light celerity and ε r eff the effective dielectric constant.
Each section of the impedance transformer is equivalent to a transmission line with a tunable characteristic impedance Z ci and phase velocity v ϕ i (see Fig. 2(c)). The maximum tunability of Z ci and v ϕ i was obtained when C i′ ≪ C vi. This condition was satisfied when Z c was large, leading to the simplified equivalent electrical circuit of Fig. 2(d).
If Z c is large and if the total electrical length θ is small as compared to the wavelength, the equivalent circuit of the complex-impedance transformer can be simplified, as shown in Fig. 3. This simplified equivalent electrical circuit will be used to explain the principle of the complex-impedance transformer. Each section has now been replaced by its equivalent inductance L i′. The quantities Z L1, Z LC1, Z L2, Z LC2, and Z out are the output impedances as seen at different transverse planes indicated by the dotted lines, when the input port is terminated by 50 Ω. The Smith charts of Fig. 4 show the principle of all the impedance transformations, starting from the 50-Ω input port in (a) to the output impedance Z out in (e). These transformations assume a fixed frequency and no losses. In the impedance chart of Fig. 4(a), one can see that the inductance L′1 transforms the 50 Ω input impedance to Z L1 at the output end of inductor L′1. In the admittance chart of Fig. 4(b), it can be seen that the variable capacitive admittance ωC v1 transforms Z L1 to a range of admittances Y LC1. The Y LC1 output admittance values are part of the circle defined by ωC v1min and ωC v1max, where C v1min and C v1max are the minimum and maximum capacitance values. A second L′C v section is necessary to transform this circular arc into a surface, as shown in the impedance and admittance charts (Figs 4(c) and 4(d)). The final inductance L′3 allows the achievable Z out area on the Smith chart to be rotated. This area corresponds to a value of S 22 when the output is terminated by 50 Ω.
Simplified equivalent electrical circuit of the complex-impedance transformer.
Principle of impedance transformations. The impedances (a) Z L1, (b) Z LC1, (c) Z L2, (d) Z LC2, and (e) Z out are synthesized at the transverse planes of the impedance transformer, as shown in Fig. 3.
III. OPTIMIZATION METHOD
The Smith charts given in Section II show that the Z out impedance area depends on the three inductances L′1, L′2, and L′3, the minimum and maximum values C min and C max of the two capacitors, and the operating frequency f. All these parameters need to be optimized according to the tuner application. In this section we develop and compare two different methods for optimizing an impedance transformer. The first method is based on the display of the Z out area (see Fig. 4(e)) using the S 22 parameter for a 50 Ω load. Examples of synthesized impedances obtained by this method are shown in Subsection IIIA below. The second method calculates the S 11 and S 21 parameters when the impedance transformer is loaded with complex impedances. The principle of this “matching load” method is given in Subsection IIIB. In Subsection IIIC we compare the two methods with a typical example. The comparison is made both by including and omitting the losses to determine their effect on the second optimization method.
In practice, not all the elements constituting the impedance transformer are ideal. In the optimization process, the complete equivalent electrical circuit of a commercial varactor diode (Fig. 5) was considered. M/A-COM™ varactors (MA4ST-1240) with a series inductance L s = 1.2 nH, a series resistance R s = 1.6 Ω, a case capacitance C c = 0.11 pF, and a tunable capacitance C(V) ranging from 1.5 to 8.6 pF were used. A single varactor was used to realize each tunable capacitor in Fig. 1. To compare the two approaches, we specified that the impedance transformer should have the parameters Z C = 200 Ω, θ 1 = 15°, θ 2 = 15°, and θ 3 = 8°. Ideal transmission lines were assumed for the simulations.
Equivalent circuit of the reverse-biased varactor diode.
A) First approach: the “synthesized impedance” method
The first approach, the “synthesized impedance” method, is based on the display of the S 22 parameter when the impedance transformer is loaded by 50 Ω. Many state of the art tuners are just characterized this way, and so insertion loss versus the load, is not known.
Figure 6 shows the synthesized impedances of the impedance transformer at (a) 0.5 GHz, (b) 1 GHz, and (c) 1.5 GHz. The thick lines correspond to a fixed minimum or maximum value for the capacitance of one varactor and a complete variation of the other. In some cases, this area is not sufficient to show all possible synthesized impedances. With intermediate values of the two capacitors, all the shaded area shown in Fig. 7 can be covered by this impedance transformer. The synthesized impedance area in Fig. 7(b) is larger than the area in Fig. 6(b). In this section, simulation results are shown as in Fig. 7. In Section V, measured results are presented as in Fig. 6.
Synthesized impedances at (a) 0.5 GHz, (b) 1 GHz, and (c) 1.5 GHz.
Total area of the synthesized impedances at (a) 0.5 GHz, (b) 1 GHz, and (c) 1.5 GHz.
B) Second approach: the “matching load” method
For the second method, the impedance transformer was loaded by complex impedance. The setup shown in Fig. 8 corresponds to a typical real working configuration of the transformer, for example when it is used as a matching network for a transistor. Here, the input return loss S 11 and insertion loss S 21 of the impedance transformer were investigated. In the following discussion, we refer to this approach as the “matching load” method.
Experimental setup for the measurement of S 11 and S 21.
The simulations were done by a Mathematica [26] program, developed to automatically calculate the Smith chart coverage for the entire range of varactor capacitances. A flowchart of the program is given in Fig. 9.
Flowchart for the Mathematica simulation program.
Figure 10 shows all the complex loads that were tested by the program. For each load, the cascade ABCD and S matrices of the impedance transformer were calculated, with two criteria |S 11|max and |S 21|min applied to the |S 11| and |S 21| parameters. The two capacitor values, C v1 and C v2 that satisfy the two criteria were extracted. Then the “matching load” area was plotted on the Smith chart. Equations for calculating S parameters in the case of a complex load are [Reference Frickey27]:
where Re(Z in) and Re(Z out) are the real parts of the input and output impedances, respectively.
Complex loads generated by the program.
Comment: The simplified equivalent circuit of Fig. 3 is used in order to understand and easily visualize all the impedance transformations on the Smith chart. However, in the simulation process, for the “synthesized impedance” and “matching load” methods, we have compared equivalent electrical circuit of Figs 1 and 3 by using ABCD transmission-line matrix and ABCD inductance matrix, respectively. We have demonstrated that results are identical for small length impedance transformers and similar for longer devices. In this paper all the simulations shown in the figures have carried out with the equivalent electrical circuit of Fig. 1 i.e. with the used of real transmission lines.
C) Comparison between the two methods
In this subsection, simulation results of the impedance transformer obtained by the two different methods and described at the beginning of Section III are compared. Results are shown for the 1 GHz center frequency. In Sub-subsections 1), 2), and 3), the two methods are compared with the varactor R s as a parameter. In Sub-subsection 4), we demonstrate that the results obtained when an impedance transformer was optimized using the “synthesized impedance” method, without considering losses, can be quite different from those obtained when the varactor losses were included.
1) LOSSLESS VARACTORS
In lossless microwave devices, the S-parameter moduli are related by
For a matching criterion |S 11| < −20 dB, relation (8) leads to |S 21| > −0.04 dB. With the criteria |S 11|max = −20 dB and |S 21|min = −0.04 dB, Fig. 11 shows the “conjugate synthesized impedances” and the “matching loads” obtained from the two different methods. In this lossless case, the results of the two methods are perfectly superposed.
Simulated coverage areas according to the “conjugate synthesized impedances (S 22*)” approach (⚊) and the “matching loads” approach (•••), at 1 GHz without varactor's series resistance, i.e., lossless varactors.
2) LOSSY VARACTORS
In this sub-subsection, the varactor series resistance R s was considered. Figure 12 shows the results obtained from the two different methods when |S 11|max = −20 dB, for several |S 21|max values. Because of R s, the “matching load” method gives no possible loads when |S 21|min = −0.1 dB; for |S 21| < −2 dB, the covered areas remain small. In this case, a perfect superposition of the results obtained by the two methods is never obtained. These results show that for lossy varactors, the “conjugate synthesized impedance” method, which is much simpler for experimental characterization, fails to give the correct covered area. Figure 12 proves the importance of the “matching load” method to know the insertion loss of the device versus this complex load. So, a 50 Ω measurement is not sufficient to characterize the tuner insertion loss.
Simulated coverage areas at 1 GHz according to the “conjugate synthesized impedance” (S 22*) method (⚊) and the “matching load” method (•••), when R s = 1.6 Ω. Results are plotted for |S 21|min values from −0.1 to −20 dB.
For further simulations, the criterion |S 21|min = −2 dB was applied.
3) INFLUENCE OF THE VARACTOR SERIES RESISTANCE R s
In this sub-subsection the complex-impedance coverage was investigated by varying the series resistance R s of the varactor as a parameter. Figure 13 compares the results obtained from the “synthesized impedance” and “matching load” methods, with |S 11|max = −20 dB and |S 21|min = −2 dB. For each Smith chart, the areas obtained by the two methods decrease as R s increases.
Simulated 1 GHz coverage areas according to the “conjugate synthesized impedance” (S 22*) method (⚊) and the “matching load” method (•••). Here the varactor series resistance R s is varied.
The difference of area between the two methods increase with R s so for a fixed complex load the insertion loss increases with R s.
4) APPLICATION EXAMPLE
To investigate the impact of choosing the design method, an impedance transformer was designed to cover the largest possible area, while using the “synthesized impedance” method. The same MA4ST-1240 M/A-COM™ varactor (R s = 1.6 Ω) was used and the characteristic impedance of the line was also fixed to 200 Ω. The maximum “conjugate synthesized impedance” area was obtained when θ 1 = 40°, θ 2 = 15°, and θ 3 = 8°. The same transformer was then simulated using the “matching load” method, with the criteria |S 11| < −20 dB and |S 21| > −2 dB. Results are compared in Fig. 14. A few loads allowing |S 11| < −20 dB and |S 21| > −2 dB were found at 1 GHz (see Fig. 14(a)), but no loads were found at 1.5 GHz (see Fig. 14(b)). However, a large area was obtained with the “conjugate synthesized impedance” method.
Simulated coverage areas according to the “conjugate synthesized impedance” (S 22*) method (⚊) and the “matching load” method (•••), at (a) 1 GHz, and (b) 1.5 GHz for a 40°–15°–8° impedance transformer, with R s = 1.6 Ω.
These typical results bring to the fore the importance of the design method. For lossy varactors the “synthesized impedance” method, which is used by several researchers in the field, can be very inaccurate because insertion loss information cannot be obtained.
The same impedance transformer (θ 1 = 40°, θ 2 = 15°, θ 3 = 8°, and Z c = 200 Ω) was simulated by the two methods assuming R s = 0.5 Ω (see Fig. 15). The covered areas of the “matching load” are very different compared to the results with R s = 1.6 Ω (Fig. 14).
Simulated coverage areas according to the “conjugate synthesized impedances” (S 22*) method (⚊) and the “matching load” method (•••), at (a) 1 GHz, and (b) 1.5 GHz for a 40°–15°–8° impedance transformer, with R s = 0.5 Ω.
To bring to the fore the importance of the “matching load” method, we can compare Fig. 15(a) with Fig. 13 (R s = 0.5 Ω) and Fig. 14(a) with Fig. 13 (R s = 1.5 Ω). These results obtained with the same varactors give totally different covered area. So an important difference between the two different methods of simulation is clearly pointed out. We note that insertion loss is more critical for this topology (θ 1 = 40°, θ 2 = 15°, and θ 3 = 8°) than for the first topology (θ 1 = 15°, θ 2 = 15°, and θ 3 = 8°).
It is obvious that in the case of lossy varactors, i.e. the reality in most cases, the “matching load” method has to be used in order to calculate and optimize by simulation the different parameters as the electrical length and the characteristic impedance of each transmission line, in order to achieve a maximum covered area in the Smith chart.
IV. PROTOTYPE DESIGN
To demonstrate the principles developed in Sections II and III, a tunable impedance transformer proof of concept was designed for a 1-GHz working frequency. The circuit was optimized with a Mathematica program using the “matching load” method. Commercial varactors (M/A-COM™ type MA4ST-1240) were used. Their L s = 1.8 nH, R s = 1.6 Ω, and C c = 0.11 pF. The C(V) range, extracted from experimental results, was 1.0–8.6 pF for a bias voltage V range from 12 to 0 V. Coplanar waveguide (CPW) was used, the prototype being fabricated on a Rogers™ RO4003 substrate (εr = 3.36, tan(δ) = 0.0035, dielectric thickness 0.813 mm, and copper thickness 35 µm). The transmission-line characteristic impedance was set at 200 Ω, leading to a CPW central conductor width of 250 µm and a gap of 2.8 mm. Two varactors were used in parallel to realize the tunable capacitors. This is necessary for CPW symmetry and to lower effective series resistance.
The fabricated proof of concept is shown in Fig. 16. By providing an air gap in the ground plane, separate reverse biases V 1 and V 2 can be applied to the two pairs of diodes. Surface mounted capacitors were used to ensure ground continuity for the RF signal.
Photograph of the impedance transformer.
The overall electrical length was 38° (θ 1 = 15°, θ 2 = 15°, and θ 3 = 8°), corresponding to ~λ/10. The effective εr was 1.75, leading to l 1 = 9.4 mm, l 2 = 9.4 mm, and l 3 = 5 mm.
V. RESULTS
In this section, the simulated and measured results obtained by the two methods are compared.
In Subsection A, the tunable frequency range of the impedance transformer loaded by 50 Ω is shown. In Subsections B and C, the simulated and measured results obtained from the “synthesized impedance” and “matching load” methods are compared.
For the simulations, lossless transmission lines were assumed but all parasitic elements of the diodes were considered. Measurements were made using a Wiltron 360 vector network analyzer (VNA).
A) Tunable frequency range for a 50-Ω load
An initial measurement using a 50-Ω load was made to extract the tunable bandwidth and to confirm the varactor's equivalent electrical model.
Figure 17 shows the frequency tunability of the complex-impedance transformer. Simulated and measured results for the parameters S 11 and S 21, obtained for the extreme tunable frequencies, are shown. These correspond to the extreme capacitances of the variable capacitors. These results show that the transformer can be continuously tuned from 0.6 to 1.6 GHz, that is, ±60% around 1 GHz, with |S 11| <−20 dB and |S 21| > −2 dB. Good agreement between simulated and measured results was obtained for the whole tunable frequency range.
Simulated (– –) and measured (—) frequency tuning ranges for a fixed 50-Ω load.
B) “Synthesized impedance” method
In the experimental setup shown in Fig. 18, the impedance transformer was inserted between the two ports of the VNA, and a coaxial SOLT (short-open-load-through) calibration procedure was applied between the calibration plans P 1 and P 2. The S 22 parameter was measured at the impedance transformer output. The phase shift ϕ due to the SMA connector used in the prototype of Fig. 16 is given by
where f is the frequency in GHz. This phase shift is taken into account in measurement results.
Experimental setup for the measurement of S 22.
Figure 19 compares the simulated and measured results from 0.5 to 2 GHz. The conditions are the same as in Fig. 6, with one varactor bias being fixed at its minimum or maximum value and the other being varied over its full range.
Simulated S 22 values (⚊) compared with measured results (•••).
The tunable S 22 area changes with the working frequency, and maximum coverage was obtained between 1.0 and 1.2 GHz. A good agreement between simulations and mea- surements was obtained for all frequencies.
The area of impedances covered by our device is comparable to best results obtained in the literature [Reference Vaha-Heikkila, Varis, Tuovinen and Rebeiz10, Reference Lu, Katehi and Peroulis21] measured in this way, with a 50 Ω load. However, our device is much more simpler.
C) “Matching load” method
With the two criteria |S 11| < −20 dB and |S 21| > −2 dB, Fig. 20 shows the simulation results obtained when the operating frequency was varied from 0.5 to 2.0 GHz.
Simulation results for the “matching loads” method with the criteria |S 11| < −20 dB and |S 21| > −2 dB versus frequency.
Measurements were carried out using an experimental approach similar to that used for the simulations. A mechanical tuner [28] was used as a complex load for the tunable impedance transformer under test. The measurement steps for the calibration are detailed in Fig. 21. First, a coaxial SOLT calibration was used to define reference plans P 1 and P 2 (see Fig. 21(a)). Then the input impedance of the mechanical tuner was measured, as shown in Fig. 21(b). Finally, the |S 11| and |S 21| parameters of the tunable transformer, loaded by the mechanical tuner, were measured (see Fig. 21(c)). We assume that the insertion loss of the mechanical tuner was negligible for the extraction of the insertion loss |S 21| of the impedance transformer. The phase shifts of the SMA connectors are taken into account in measurement results.
Principle of the setups for the measurement of S 11 and S 21: (a) calibration planes, (b) load impedance measurement, and (c) measurement of S parameters.
The measured results are shown in Fig. 22 for three different frequencies: 0.8, 1.0, and 1.5 GHz. The lowest frequency was 0.8 GHz owing to the limited mechanical tuner bandwidth. The Smith charts show all the points for which the criteria |S 11| < −20 dB and |S 21| > −2 dB were satisfied. Each of these complex loads was associated with a pair of bias voltages (V 1, V 2), corresponding to two varactor capacitance values (C v1, C v2).
Measured matched complex loads, under the conditions −|S 11| <−20 dB and |S 21| > −2 dB, at three frequencies: (a) 0.8 GHz, (b) 1.0 GHz, and (c) 1.5 GHz. Simulations extracted from Fig. 20 in the same conditions: (d) 0.8 GHz, (e) 1.0 GHz, and (f) 1.5 GHz.
As this measurement procedure is much more time consuming than the “synthesized impedance” method, fewer measured load points were obtained, resulting in a “matching area” that is not so well defined as the simulated area.
The agreement between the measured results of Fig. 22 (a, b, c) and the simulated points in Fig. 22 (d, e, f) is good. The measurements show a large “matching area” that is slightly smaller than the simulated area.
Figure 23 shows typical measured results obtained at 1 GHz for three different complex loads.
Measured results for three different complex loads at 1 GHz, with the criteria |S 11| < −20 dB and |S 21| > −2 dB.
Conclusions and prospects: We believe that the totality of the results presented here validates our approach to both the design and the measurement methods.
VI. CONCLUSIONS
A principle for designing a compact tunable impedance transformer, based on a single transmission line loaded by only two pairs of varactors, has been proposed. The length of the transformer is only λ/10. A prototype with a 1 GHz center working frequency has been realized using commercial varactor diodes.
A good agreement has been obtained between the simulations and measurements and, as expected, the network provided a large coverage of the Smith chart (real part from 20 to 90 Ω at 0.8 GHz and from 30 to 170 Ω at 1.5 GHz), with a range of tunable working frequencies over ±40%.
Two different approaches to the design and measurements have been investigated. It is shown that an external tuner is necessary for accurate determination of the Smith chart coverage.
A MMIC prototype, in a 0.35 µm BiCMOS technology, is under development. It is believed that such an impedance transformer can be a good candidate for tunable matching of an amplifier embedded in a reconfigurable front-end.
ACKNOWLEDGEMENT
This work was supported by the Région Rhône-Alpes.
APPENDIX
Figure 24 shows a simplified RC equivalent circuit of a single varactor diode inserted between a source impedance Z in and an output load impedance Z out.
Input impedance Zs in seen from Z in.
We denote Zs in the input impedance as seen from Z in. In admittance form this is
where Z out = Re(Z out)+jIm(Z out). The input can be matched when Zs in and Z in are complex conjugates, that is when Zs in = Z in*, or equivalently, Ys in = Y in*, leading to
where Z in = Re (Z in)+jIm (Z in). Thus the output admittance that can be matched is
This corresponds to the impedance determined from the measurement of S 11 and S 21, as shown in Subection IIIB.
Let us now calculate the output admittance Ys out as seen from Z out, as shown in Fig. 25. This is the impedance that is extracted from the measurement of S 22, as in Subection IIIA:
Output impedance Zs out seen from Z out.
At this point, it becomes obvious that Y out and Ys out* are different. This is because R s is not equal to zero. These equations explain why the two measurement approaches investigated in Subections IIIA and B do not lead to the same results when the series resistance of the varactors is considered.
Anne-Laure Perrier was born in France in 1980. She received the M.Sc. degree in Optics, Optoelectronics, and Microwaves from the INPG (“Institut National Polytechnique de Grenoble”), Grenoble, France, in 2003. She received the Ph.D. degree in 2006 from the Laboratory of Microwaves and Characterization (LAHC), University of Savoie, France. Her research interests include the theory, design, and realization of tunable-impedance transformers.
She has been an Assistant Professor since September 2008 at Claude Bernard University (Lyon, France), where she teaches electronics and signal processing. She continues her research at the Research Center on Medical Imaging (Creatis-LRMN). She designs and realizes RF sensors for MRI (Magnetic Resonance Imaging) applications.
Jean-Marc Duchamp was born in Lyon, France, on April 10, 1965. He received the M.Sc. degree from the University of Orsay, (France), in 1988 and the Engineer degree in 1990 from Supelec. He has been a Research Engineer at Techmeta (France) from 1991 to 1996. He received the Ph.D. degree in 2004 from LAHC, University of Savoie (France). He has been an Assistant Professor since 2005 at J. Fourier University (Grenoble, France), where he teaches electronics and telecommunications. His current research interests include the analysis and design of nonlinear microwave and millimeter-wave circuits, such as nonlinear transmission lines, periodic structures, and tunable impedance transformers.
Olivier Exshaw was born in France in 1973. He received the Engineer degree in the field of microelectronics from ENSERG/INPG, Grenoble, France, in 2003. He was with the Ultra- High-Frequency and Optoelectronic Characterization Laboratory until the end of 2005. From 2006, he has been Electronic Service Project Manager at the Research Center for Very Low Temperatures (CRTBT) at CNRS.
Robert G. Harrison (M'82) received the B.A. and M.A. (Eng.) degrees from Cambridge University, UK, in 1956 and 1960, respectively, and the Ph.D. and D.I.C. degrees from the University of London, UK, in 1964.
From 1964 to 1976, he was with the Research Laboratories of RCA Ltd., Ste-Anne-de-Bellevue, QC, Canada, In 1977, he became Director of Research at Com Dev Ltd., Dorval, QC, where he worked on nonlinear microwave networks. From 1979 to 1980, he designed spread-spectrum systems at Canadian Marconi Company, Montreal, QC, Canada. From 1980, he has been a Professor in the Department of Electronics, Carleton University, Ottawa, ON, Canada. His research interests include the modeling of nonlinear microwave device/circuit interactions by a combination of analytical and numerical techniques. More recently, he has been developing new analytical models of ferromagnetic phenomena based on both quantum-mechanical and classical physics. He has authored or coauthored more than 60 technical papers, mostly in the area of nonlinear microwave circuits, as well as several book chapters on microwave solid-state circuit design. He holds several patents on microwave frequency-division devices. He became a Distinguished Research Professor of Carleton University in 2005.
Dr. Harrison received the “Inventor” award from Canadian Patents and Development in 1978.
Philippe Ferrari was born in France in 1966. He received the B.Sc. degree in Electrical Engineering in 1988 and the Ph.D. degree from the INPG (“Institut National Polytechnique de Grenoble”), France, in 1992.
From September 2004, he has been a Professor at the University Joseph Fourier at Grenoble, France, while he continues his research at the Institute of Microelectronics, Electromagnetism, and Photonics (IMEP) at INPG.
His main research interest is the design and realization of tunable and miniaturized RF and millimeter wave devices, transmission lines, filters, phase shifters, power dividers, tuners, in PCB and RFIC technologies.
He is also involved in the development of time-domain techniques for the measurement of passive microwave devices and for soil moisture content.
