A.-L Perrier

P Ferrari
email: ferrari@univ-savoie.fr

J.-M Duchamp

D Vincent

A Varactor Tunable Complex Impedance Transformer

We have conceived and realized a microwave tunable transformer. The tunability is obtained by loading a transmission line by reverse biased Schottky diodes acting as voltage variable capacitances. Only two diodes pairs are necessary to obtain impedance tunability between 5 Ω to 300 Ω, with an operating frequency varying of more than ± 10 %. Complex loads can also be matched with a wide covering of the Smith chart, but small bandwidth of about 2 %. The device length is around a third of the guided wavelength.

I INTRODUCTION

Tunable devices are of great interest in the field of microwave telecommunications. Tunable filters allow the use of only one filter for the purpose of emission and reception [START_REF] Abbaspour-Tamijani | Miniature and Tunable Filters Using MEMS Capacitors[END_REF], [START_REF] Pistono | Tunable band-pass microwave filters based on defect commandable photonic band gap waveguides[END_REF], leading to smaller emitter -receiver systems. In the same manner, a tunable complex impedance transformer is interesting for the matching of transistors with a large bandwidth (more than ± 10%) purpose. Impedance transformers can also be used to realize tunable power dividers useful in the field of beam steering antenna arrays. In this paper, we present an original topology for a tunable complex impedance transformer. It is realized by loading a transmission line with variable capacitors. Variable capacitors are obtained with reverse biased Schottky diodes. This approach can be seen as similar to nonlinear transmission lines used as tunable delay lines [START_REF] Rodwell | Active and nonlinear wave propagation in ultrafast electronics and optoelectronics[END_REF], [START_REF] Duchamp | Comparison of Fully Distributed and Periodically Loaded Nonlinear Transmission Lines[END_REF]. However, in our case, the design is very different owing the fact that not only the phase has been tuned, but also the characteristic impedance. The varactors are used in small signal around a tunable DC bias, leading to small signal tunable capacitances. A tunable impedance transformer was already carried out in 1997 by Sinsky and Westgate [START_REF] Sinsky | Design of an electronically tunable microwave impedance transformer[END_REF]. Their device was realized in microstrip technology for a freuency around 2.25GHz. The principle was to use a quarter-wave impedance transformer to achieve a simple tunable ladder network. This conception leads to λ/2 length devices and needs at least three varactors. In our approach, only two varactors are required (two pairs for a CPW technology without via). Furthermore the device length is around λ/3, leading to smaller insertion losses. Moreover, another interesting feature is that complex load impedances can be matched and the bandwidth is not limited by quarter wavelength transformers. We show simulations and measurements of a first prototype of tunable complex impedance transformer using a coplanar waveguide topology (CPW) on a FR4 substrate. Next simulations of a second design are presented on a RT Duroïd substrate with more efficient varactors at 10 GHz. In these two cases the device length is about 35 % of the guided wavelength. Fig. 1 shows the topology of the complex impedance transformer. A CPW transmission line of length d is loaded by two pairs of diodes, the distance between these two diode's pairs is optimized in term of return loss. With the CPW technology the surface mounted Schottky diodes can be easily soldered. The two diode's pairs are independently biased (voltages V 0 and V 1 ), leading to a much more efficient tunability compared to a single bias. Two air gaps are realized in the ground plane to allow different biases.

II PRINCIPLE

Is we take a series L C model to simplify for the varactor, neglecting the series resistance, the effective capacitance can be written as :

2 1         - = -off cut eff C C ω ω (1)
where

LC off cut 1 = - ω is the case cut-off frequency.
So, when L increases the effective capacitance increases leading to a frequency shift to ward low frequencies for the same matching point. Indeed, probably the parasitic inductance is lower in practice than indicated by the manufacturer. From relation [START_REF] Abbaspour-Tamijani | Miniature and Tunable Filters Using MEMS Capacitors[END_REF] it is also easy to see that the varactor effective capacitance is frequency dependent and

d V 0 V 0 V 1 V 1 air gaps
Surface mounted capacities increases with frequency, for off cut-

> ω ω

, the varactor has an inductance behaviour.

III RESULTS

A. First Prototype

V V V V 0 min 1 0 = = = and V V V V 40 max 1 0 = = =
for measurements. The tuning bandwidth, calculated for a return loss lower than -20 dB, is about ± 7% around 5 GHz for simulations and measurements. The same bandwidth is obtained for load impedances varying from 5 Ω to 300 Ω. A frequency shift can be noted between measurement and simulation results, the matching is between 4.49 GHz to 5.3 GHz for simulations, and 4.73 GHz to 5.46 GHz for measurements. This shift can be due to a bad estimation of the case parasitic inductance given by the manufacturer. The influence of the case parasitic inductance has an important effect on the varactor effective capacitance [START_REF] Abbaspour-Tamijani | Miniature and Tunable Filters Using MEMS Capacitors[END_REF]. Measurements for a mean value of the frequency (5 GHz) have been reported in Fig. 2 to show that the return loss can be very small (50 dB in that case).

S 11 and S 21 modulus for the extreme frequencies have been reported in Table 1. Simulations and measurements are in good agreement, except the frequency shift mentioned above. Measurement results show more losses than simulations (between 0.8 to 1.5 dB more). This is due to the air gaps with surface mounted capacitor residual losses which were not taken into account in the simulations. Excepted air gap losses witch could be minimized with a finer technology, losses are mostly due to the diodes series resistance and substrate losses. We will see in the next section that they can be lower than 0.5 dB if we use more efficient substrate and diodes are used.

Complex loads can also be matched with this impedance transformer. Fig. 3 shows the simulation of the Smith Chart coverage for a return loss better than 20 dB, at 4.8 GHz in (a) and 5 GHz in (b). The coverage is very large. This is very interesting in the field of amplifier design. The complex impedance transformer can be used instead of a quarter-wave transformer associated with stubs in the design of the matching networks. However a large coverage is only obtained for a small bandwidth, about 2 %.

B Second Prototype

The first prototype has been realized and measured to show the feasibility of complex impedance transformers. It was realized on FR4 with low quality diodes, leading to significant insertion losses. The second prototype shown in this section is realized on a RT Duroïd 5880 substrate ( . At 10 GHz the transformer is 6.06 mm length, i.e. 32 % of the wavelength. Figure 4 shows the simulation results. The tunable bandwidth is always ± 10 % for 25 Ω to 100 Ω loads.

IV CONCLUSION

A tunable complex impedance transformer has been designed and measured. The tunability is available for a wide range of complex loads and broad bandwidth. Losses can be reduced to less than 0.5 dB by using high quality factor diodes with a low losses dielectric substrate. The large Smith Chart coverage leads to a device that can be used efficiently instead of a classical quarter-wave transformer associated with stubs for the design of microwave amplifiers matching networks. The wide bandwidth (more than ± 10 %) allows this impedance transformer to be used in the context of wideband amplifiers. Furthermore it could also be used to address the problem of tunable or reconfigurable power dividers.

Fig 1 .

 1 Fig 1. Topology of the tunable device.

First.

  measurements have been carried out with less efficient Schottky diodes (Philips BAT62-03W) on a FR4 substrate ( 4 = r ε). Characteristics of these diodes are as follows : 350 fF to 220 fF tunable capacitance for a 0-40V reverse bias, a 6.6 Ω series resistance, a case parasitic capacitance The transformer is 15.6 mm long (38% of the wavelength) for a 5 GHz operating frequency. The CPW was realized with a mechanical machining, leading to 160 µm air gaps. To minimize insertion losses, surface mounted capacitances have been connected on the air gaps, as shown in Fig.1.

Fig. 2 .

 2 Fig. 2. Measurement and simulation of |S 11 | and |S 21 |.

Fig. 2

 2 Fig. 2 shows simulations (a) and measurements (b) when the impedance transformer is first loaded by 50 Ω. The extreme matching points leading to a 20 dB return loss have been reported. These points correspond to a variable capacitance of fF C 350 max = and fF C 220 min = for

Frequency

Fig. 3 .

 3 Fig. 3. Smith Chart coverage for complex loads matching.

  high quality factor Schottky diodes (Agilent HSCH-5314). The model given by Agilent for theses diodes leads to a 130 fF to 50 fF tunable capacitance for a 0-4V reverse bias, a 2.8 Ω series resistance, a case parasitic capacitance fF

Fig. 4 .

 4 Fig. 4. Simulated |S 11 | and |S 21 |.

Table 1 .

 1 Summary of |S 11 | and |S 21 | for the first impedance transformer.


J M Duchamp

A L Perrier

P Ferrari

MS, CPW and Fin-line attenuation and dispersion effects on microwave and millimeter wave PBG structure parameters

This paper presents a simple analytical method to describe both effects of linear transmission line losses and, for the first time, frequency dispersion on photonic band-gap (PBG) propagation parameters. Applied to a coplanar waveguide PBG structure, this approach points out the attenuation difference between the two frequently used electrical models : one with fully distributed parameters and one with semi distributed parameters. Moreover, a comparison of the same PBG design with three technologies of transmission lines (CPW, MS and Fin-line) gives very interesting results on the Bragg frequency value and the PBG attenuation.

INTRODUCTION

Recently, the microwave and millimeter wave domains have shown a great interest in photonic band-gap (PBG) structures. PBG are periodical structures enhancing passbands and stop-bands in the frequency response which are interesting for filters [1]- [START_REF] Sor | Miniature Low-Loss CPW Periodic Structures for Filter Applications[END_REF] or non linear transmission line (NLTL) [START_REF] Duchamp | Comparison of Fully Distributed and Periodically Loaded Nonlinear Transmission Lines[END_REF]- [START_REF] Hollung | A Distributed Heterostructure Barrier Varactor Frequency Triplei[END_REF] designs. PBG can be realized easily with transmission lines periodically loaded with shunt-connected reactive elements (PL-PBG).

The propagation constant and the characteristic impedance of PL-PBG have already been studied [START_REF] Zhu | Guided-wave characteristics of periodic coplanar waveguides with inductive loading-unit-length transmission parameters[END_REF] but no analytical method of losses study is yet available. Two types of electrical losses models are available : distributed ones and lumped ones. For example, coplanar wave-guide (CPW) dielectric (a,J and radiative (oL.J Complex propagation parameters of one PL-PBG section (Fig. 1) are obtained from ABCD matrix :

The PBG attenuation, propagation constant and characteristic impedance are defined by : A. Fully Distributed Mudel parameter equations are extracted. Then losses results obtained on a on a CPW periodical structure, from the two models ( a fully distributed one and a semi distributed one) are compared. The last part points out By identification between one PBG section (Fig. 1-a) and the fully distributed parameter model (Fig. I-b), real and imaginary parts of [START_REF] Sor | Miniature Low-Loss CPW Periodic Structures for Filter Applications[END_REF] give these two equations :

.

-

the effect of dispersion on PL-PBG parameters by comoarinr the same PL-PBG desirns with three different

L.YLoad c o s b , . d ) ) ( 5 )

In the case of a loss-less transmission line (aL equal to a, . < + a L . d + a,-,), equation %e @becomes equal to [START_REF] Simons | Coplanar Waveguide Circuits[END_REF].

It still being true if linear losses are small, the fxst order Taylor formula of R, 6) and 3, ,

-

The various Bragg frequencies fB,i are obtained from relation [START_REF] Simons | Coplanar Waveguide Circuits[END_REF] when cos(bpBGd) =-I.

We notice that Bragg frequencies are not modified by small losses (no a, dependency).

A lumped approximation of fSl (8) is used to define the section length d.

f B , l = (CO .~,_m, ) / k . G z c -L . d ) (8)
The PBG attenuation apBG is given by relation [START_REF] Frankel | Terahertz Attenuation and Dispersion Characteristics of Coplanar Transmission Lines[END_REF]. Two characteristic limits of apBG are pointed out :

At low frequency, PL-PBG losses are equal to linear transmission line losses and at Bragg frequencies, when propagation is forbidden, equivalent PBG losses become infinite.

B. Semi Distributed Model

The second model has the propagation parameter bL distributed (instead of lumped when LC parameters are used) and losses are described by two lumped resistors (Fig. 2).

C. CPW Model Comparison

In this part, modelling results of an example structure will be given. A CPW transmission line periodically loaded by capacitances, illustrates the differences of these two approachs.

CPW transmission line is printed on a GaAs substrate with Z , = 120 i2 @ 60 GHz (Fig. 3) and capacitance loads C , are chosen to achieve Z , , , G = 50 i2 (15). [START_REF] Simons | Coplanar Waveguide Circuits[END_REF][START_REF] Frankel | Terahertz Attenuation and Dispersion Characteristics of Coplanar Transmission Lines[END_REF][8] models are used to describe CPW losses and frequency dispersion parameters.

,,I tl G = 97 i m 4;' H=400# !E,, -12.9 Fig. 3 CPW geometrical and substrate parameters.

From relation (3) and for loss-less PBG structure, characteristic impedance is only real until the frequency reaches the Bragg value (in pass-hand frequency range). At fE,j ,Z,,,, is null. Then above fBi (in stop-band frequency ranges) Z,,,, becomes purely imaginary.

When the line is lossy, characteristic impedance Z,,,, is a complex value and Z , can be defined by ___ > 0 Vf .

I -I

At low frequency, a large signal equivalent impedance Models give equivalent results until 80% of f8,,.

The frequency varies from 10 GHz to 100 GHz. Fig. 5 compares the PBG attenuation apR& and the real part of characteristic impedance versus the frequency for these two models. We also add CPW attenuation aLv). Near the Bragg frequency f8,, (85 GHz for d = 189 p), the two aP8, models raise quickly (10). Although at low frequencies the semi-lumped model does not reach the expected limit : a,. A fully distributed electrical model has been used to describe each PBG section and the capacitance loads are calculated to achieve Z<-p8, = SO a.

Simulation results shown on Fig. 7 and Fig. 8 point out the effect of linear line dispersion.

It confirms that lumped propagation model (LC model) for Bragg frequency gives an underestimated fa, value. In our example, the fust Bragg frequency of CPW PBG structure, which is a low frequency disperse structure (less than 1% on the frequency range) is equal to 85 GHz instead of 80 GHz given by (8).

For frequency disperse lines, like MS or Fin-line, the PBG attenuation of one PBG section for MS, CPW,

IV. CONCLUSIONS

In this paper, we describe a new and simple method to take into account all losses, including radiative ones, and the frequency parameters dispersion in PL-PBG structures. A fully distributed approach seems the most accurate one and our measures will confirm it soon. Furthermore a really interesting result is drawn out when we compare the transmission line dispersion effects. A PL-PBG based on Fin-line exhibits periodically low losses at fmin and k.f,,.. This result can be used to design, for example, NLTL frequency multipliers in which the pump and generated frequencies are adjusted to reach the minimum attenuation at k.f,,.

F&FERENCES

Fig. 1

 1 Fig. 1 Periodical loaded structure with fully distributed parameters (a) and an electrical model of one equivalent section (b).

( 4 )

 4 Coplanar Wave-guide (CPW) and Fin-line. 34" European Microwave Conference -Amsterdam, 2004 3, G)= sin&,, .d)sin(PpBG .d) = sinhbL.d)(rinb,.d)+TZ, 1

  sinbL.d))

Fig. 2

 2 Fig. 2 Semi-distributed electrical model of one section As previously, identification of Fig. 1-b, with Fig. 2, in the case of low losses gives two new equations (1 1) for s e @ ) and(12)for3,6):

Fig. 4 showsFig. 4

 44 Fig.4shows Re

loo 7 -Fig. 5 andk, = 129 H=400p 4 4Fig. 6 MS

 7546 Fig.5and semi distributed models for the CPW example.Attenuation versus frequency for the fully distributed

Fig. 7

 7 Fig. 7 Real pan of the impedance characteristic of one PBG section for MS, CPW, Fin-line transmission lines.

Fig. 8

 8 Fig.8shows PBG attenuation of one PBG section versus frequency. MS and CPW line have the highest attenuation and still close on all the frequency range.However the most interesting results are about the finline attenuation which exhibits the periodical minimum values at f,,. = 28.5 GHz and at 2 fmk = 57 GHz.

  Fig. 8 Fin-line transmission lines.

34" European Microwave Conference -Amsterdam, 2004