Composite Right- and Left-Handed Traveling-Wave Field-Effect Transistors

Abstract

We introduce a composite right- and left-handed travelling-wave field-effect transistor (CRLH TWFET) for developing large-scale platform to support left-handed waves. The device represents two electromagnetically coupled CRLH transmission lines by capacitance and FET transconductance. Owing to the couplings, two different modes can support waves in CRLH TWFETs. It was experimentally established that waves supported by one of the modes were amplified, while those supported by the other mode were significantly attenuated. To quantify the wave propagation in CRLH TWFETs, we developed a numerical model based on the transmission line theory that well simulated measured results. This paper discusses the results of numerical calculations that validate the design criteria of CRLH TWFETs.

1. Introduction

Composite right- and left-handed (CRLH) transmission lines (TLs) have been extensively studied and several important breakthroughs have been achieved in the management of electromagnetic continuous waves [1]. CRLH TLs have a single series and shunt inductance-capacitance (LC) pair in each unit cell. They exhibit lefthandedness; that is, at certain frequencies, their phase velocity has a sign opposite to that of their group velocity. Figure 1 shows the unit cell of a CRLH TL, where 𝐿_𝑅, 𝐶_𝐿, 𝐿_𝐿, and 𝐶_𝑅 are the series inductance, series capacitance, shunt inductance, and shunt capacitance, respectively. The series LC pair consists of 𝐿_𝑅 and 𝐶_𝐿, and the shunt pair consists of 𝐿_𝐿 and 𝐶_𝑅. These parameters define the upper and lower zero-wavelength frequencies of the CRLH TL, which are given by 𝜔_𝑢=max(1/(𝐶_𝐿𝐿_𝑅)^1/2,1/(𝐶_𝑅𝐿_𝐿)^1/2) and 𝜔_𝑙=min(1/(𝐶_𝐿𝐿_𝑅)^1/2,1/(𝐶_𝑅𝐿_𝐿)^1/2),   respectively. The line exhibits righthandedness at frequencies greater than 𝜔_𝑢 and left-handedness at frequencies lower than 𝜔_𝑙. A frequency band gap exists between 𝜔_𝑙 and 𝜔_𝑢 in which all supporting modes are evanescent.

Figure 1 

Unit cell of CRLH TL.

The use of CRLH TLs may sometimes be impractical because of their finite electrode resistance and substrate current leakage, and may require loss compensation schemes. Recently, several authors have investigated methods for amplifying LH waves in 1D using transistors [2–4].

In particular, we analyse the use of two CRLH TLs that interact continuously via transistor transconductance together with both capacitive and inductive couplings, which we call a CRLH travelling-wave field-effect transistor (TWFET), or CRLH TWFET for brevity.

There are two different schemes to achieve loss compensation or amplification of left-handed waves. One of the methods is to interconnect passive CRLH sections with amplifier section. Based on this methodology, Casares-Miranda et al. [2] fabricated a leaky-wave antenna, whose gain was much enhanced compared to its passive counterpart. Multiple cascades of the interconnecting sections enable extension of the effective propagation length of a LH wave. However, a phase jump occurs at the amplifier section. Another method is continuous amplification. Si et al. [3] proposed a microwave CRLH TL incorporating ideal gain devices in each unit cell. They demonstrated a simple circuit cell consisting of two FETs operated as such gain devices. On the other hand, Maezawa et al. [4] proposed a CRLH TL periodically loaded with resonant tunnel diode (RTD) pairs. The negative differential resistance (NDR) of RTDs can compensate the loss in passive CRLH TLs. It is suggested that the line can amplify signals traveling along the line by the NDR of the RTD. In either case, the unit-cell structure is much simpler than CRLH TWFETs. The CRLH TL has inductance-capacitance pairs in each unit cell. With the simple introduction of an NDR device, these pairs can exhibit resonant behaviour. We carried out the stability analysis with respect to CRLH TWFETs to establish the fact that they succeed in suppressing instability and amplifying the LH waves [5]. On the other hand, as far as we know, such stability analysis has not been carried out for the schemes that employ NDR device in each unit cell. Table 1 summarizes the merits and demerits of the above-mentioned three platforms for compensating attenuation of LH waves.

Table 1 

Figure of merits of amplifying LH waves.

In the following, we first clarify our design concept of CRLH TWFETs. We then discuss several fundamental properties of a CRLH TWFET including the device configuration, dispersion relationship, and design criteria for wave amplification and termination schemes. Next, we describe the numerical model to simulate the measured results. Then, we numerically confirm the validity of the design criteria of a CRLH TWFET.

2. Design Concept of CRLH TWFETs

Figure 2 illustrates our design concept of CRLH TWFETs. Owing to couplings, two different propagation modes, called the c- and 𝜋-modes, can support waves in CRLH TWFETs. Each mode has its own speed, characteristic impedance, and the voltage fraction between the gate and drain lines. In general, the c-mode waves have the even parity between the gate and drain lines. On the other hand, the 𝜋-mode waves have the odd parity between the gate and drain lines.

Figure 2 

Design concept of CRLH TWFETs.

Our design concept stems from the fact that a CRLH TWFET can amplify the waves carried by one of two modes, and attenuate ones carried by the other mode significantly. We can exclude the influences caused by waves carried by the attenuating mode to suppress modal dispersions. Moreover, wave reflection at the ends can be minimized by the matched terminations with the characteristic impedances corresponding to the amplifying mode.

At present, we consider the case where the 𝜋-mode is the unique one to be amplified. Then, the terminal resistances 𝑅_Tg and 𝑅_Td are set equal to the 𝜋-mode characteristic impedances for the gate and drain lines, respectively. In order to develop the waves carried by the 𝜋-mode, the input ports 𝑉_in,𝑔 and 𝑉_in,𝑑 are excited by the waves satisfying the 𝜋-mode voltage fraction. In this study, we describe the results of numerical calculations using a model of a CRLH TWFET based on the elemental transmission line theory. The model well simulates the measured properties of the device and can evaluate the contributions of the finite series inductors to both LH and RH waves travelling in a line. The operation bandwidth is limited by the FET cutoff frequency. Moreover, the influences of parasitic inductance and capacitance, which cannot be assumed a priori, increase at high frequencies. However, as far as the line is based on the print-circuit-board (PCB) technologies, we can well suppress the influences of parasitic elements, whose operation bandwidth presently extends the microwave frequencies.

3. CRLH TWFETS

Figure 3 shows the circuit configuration of the unit cell of a CRLH TWFET. Each unit cell includes an FET. For the gate (drain) line, 𝐿_{𝑅𝑔(𝑑),}𝐶_{𝐿𝑔(𝑑)}, 𝐿_{𝐿𝑔(𝑑)}, and 𝐶_{𝑅𝑔(𝑑)} represent the series inductor, series capacitor, shunt inductor, and shunt capacitor, respectively. The gate line is connected with the gate electrode of the FET, whose drain electrode is connected with the drain line. The source electrode is grounded. Note that the biasing voltages to the gate (𝑉_𝐺) and drain (𝑉_𝐷) of the FET can be rendered via 𝐿_𝐿𝑔 and 𝐿_𝐿𝑑, respectively. Every inductor accompanies finite parasitic resistance, which is denoted as 𝑅_𝐺, 𝑅_𝐷, 𝑅_in,𝑔, and 𝑅_in,𝑑 for 𝐿_𝑅𝑔, 𝐿_𝑅𝑑, 𝐿_𝐿𝑔, and 𝐿_𝐿𝑑, respectively. The drain-source current of an FET is modelled as 𝐼_𝑑𝑠(𝑉_𝐺)=𝛽(𝑉_𝐺−𝑉_th)², where 𝛽, 𝑉_𝑡ℎ, and V are the transconductance coefficient, threshold voltage, and the gate line voltage, respectively. For analytical evaluations, we consider the small-signal counterpart of 𝐼_𝑑𝑠, that is, 𝐼_𝑑𝑠(𝑉_𝐺)=𝐺_𝑚(𝑉_𝐺−𝑉_th), where 𝐺_𝑚 is the transconductance given by 2 𝛽(𝑉_𝐺−𝑉_th). Moreover, an FET includes the gate-source capacitance 𝐶_𝑔𝑠, gate-drain capacitance 𝐶_𝑔𝑑, and drain-source capacitance 𝐶_𝑑𝑠. These FET parasitic capacitances can be embedded in elements in Figure 2. Concretely, 𝐶_𝑔𝑠, 𝐶_𝑑𝑠, and 𝐶_𝑔𝑑 can be included in 𝐶_𝑅𝑔, 𝐶_𝑅𝑑, and 𝐶_𝑚, respectively.

Figure 3 

Unit cell of CRLH TWFET. Both the gate and drain lines have the CRLH TL structures. These two lines are coupled via the mutual inductance 𝐿𝑚, mutual capacitance 𝐶𝑚, and FET contributions.

Owing to the presence of the passive coupling between the gate and drain lines, two different propagation modes can carry the waves [6].

Each mode has its own dispersion, characteristic impedance, and the voltage fraction between the lines. Figure 4 shows a typical dispersion relationship of each mode. To obtain this, we set 𝐿_𝑅𝑔, 𝐶_𝐿𝑔, 𝐿_𝐿𝑔, and 𝐶_𝑅𝑔 to 0.75 nH, 1.5 pF, 1.25 nH, and 1.0 pF, respectively. Moreover, 𝐿_𝑅𝑑, 𝐶_𝐿𝑑, 𝐿_𝐿𝑑, 𝐶_𝑅𝑑, and 𝐶_𝑚 are set to 1.0 nH, 2.0 pF, 0.5 nH, 1.5 pF, and 0.2 pF, respectively. There are mostly two different modes for a given frequency, although only the unique mode is allowed in the filled frequencies in Figure 4. For definiteness, we call the fast and slow modes the 𝜋 and c, respectively. Accordingly, the solid and dotted curves correspond to the 𝑐- and 𝜋- modes, respectively. The upper (lower) two exhibit RH (LH) properties.

Figure 4 

Dispersive property of CRLH TWFET. The filled frequency regions are sandwiched by zero-wavelength frequencies, which are 3.6, 4.1, 4.7, and 5.6 GHz in ascending order.

By inputting the voltage signals having the c- (𝜋-) mode voltage fraction between the lines, they are carried by the c- (𝜋-) mode. In addition, the matched termination with the c- (𝜋-) mode characteristic impedance suppresses the multiple reflections that terribly distort the waves in the line. When the voltage fraction is positive (negative), the corresponding mode becomes in phase (out of phase).

We consider the cases when a small 𝐺_𝑚 is introduced. We can then obtain the c- (𝜋-) mode per-unit-cell gain 𝛼_𝑐(𝜋) by series-expanding the complex dispersion with respect to 𝐺_𝑚, which is given by 𝛼_𝑐=−𝜐^𝑐𝐿_𝑚𝐺_𝑚41/𝜐^2𝑐−1/𝜐^2𝜋𝜐1^2𝑐−1𝜐^2𝜋+1𝛼_𝜋𝜐^2𝑚,=−𝜐^𝜋𝐿_𝑚𝐺_𝑚41/𝜐^2𝑐−1/𝜐^2𝜋𝜐1^2𝑐−1𝜐^2𝜋−1𝜐^2𝑚,(1) where 𝜐_𝑐(𝜋) represents the phase velocity of the c- (𝜋-) mode. Moreover, 𝜐_𝑚 is defined by 1𝜐^2𝑚=−2𝐶^𝑚𝐶_𝐿𝑔𝐶_𝐿𝑑𝐿_𝑚𝜔⁴𝜔𝜔^22seg−1𝜔²𝜔²_sed+𝐶_𝑚−1𝐶1_𝐿𝑔𝜔1²_seg−1𝜔²+𝐶1_𝐿𝑑𝜔1²_sed−1+𝜔𝜔²_2𝐿𝑔𝜔⁴1−𝜔𝜔²²_sed−𝜔²𝜔²_shg+𝜔𝜔^2𝐿𝑑41−𝜔𝜔²²_sed−𝜔²+1𝜔²_shd𝜔^2𝑅𝑔+1𝜔^2𝑅𝑑,(2) where 𝜔_𝑅𝑔=1/(𝐿_𝑅𝑔𝐶_𝑅𝑔)^1/2, 𝜔_𝑅𝑑=1/(𝐿_𝑅𝑑𝐶_𝑅𝑑)^1/2,𝜔_𝐿𝑔=1/(𝐿_𝐿𝑔𝐶_𝐿𝑔)^1/2, 𝜔_𝐿𝑑=1/(𝐿_𝐿𝑑𝐶_𝐿𝑑)^1/2, 𝜔_seg=1/(𝐿_𝑅𝑔𝐶_𝐿𝑔)^1/2, 𝜔_sed=1/(𝐿_𝑅𝑑𝐶_𝐿𝑑)^1/2, 𝜔_shg=1/(𝐿_𝐿𝑔𝐶_𝑅𝑔)^1/2, and 𝜔_shd=1/(𝐿_𝐿𝑑𝐶_𝑅𝑑)^1/2. Note that 𝜐_𝑚 can become imaginary for frequencies for which the RHS of (3) becomes negative. Moreover, (3) becomes singular for 𝐿_𝑚=0. In such a situation, 𝛼_𝑐(𝜋) should be written as 𝛼_𝑐=−𝜐^𝑐𝐶_𝑚𝐺_𝑚1−𝜔²/𝜔²_sed1−𝜔²/𝜔²_seg𝛼2𝜔⁴𝐶_𝐿𝑔𝐶_𝐿𝑑1/𝜐^2𝑐−1/𝜐^2𝜋,(3)_𝜋=−𝜐^𝜋𝐶_𝑚𝐺_𝑚1−𝜔²/𝜔²_sed1−𝜔²/𝜔²_seg2𝜔⁴𝐶_𝐿𝑔𝐶_𝐿𝑑1/𝜐^2𝑐−1/𝜐^2𝜋.(4)

In the above, 𝛼_𝑐(𝜋) is defined to be negative for amplifying RH waves. Correspondingly, the amplitude of the wave increases if 𝛼_𝑐(𝜋) becomes positive for LH waves, because the phase velocity has the opposite sign from the group velocity. By definitions, 𝜐_𝜋 is always greater than 𝜐_𝑐; therefore, 𝛼_𝑐𝛼_𝜋<0 for any frequencies if 𝐿_𝑚 is negligible. It is concluded that one of the two modes can be uniquely amplified. More specifically, the c- (𝜋-) mode can be uniquely amplified for LH (RH) waves if 𝜔<𝜔_sel or 𝜔>𝜔_seu, where 𝜔_seu=max(𝜔_seg,𝜔_sed) and 𝜔_sel=min(𝜔_seg,𝜔_sed). On the other hand, the c- (𝜋-) mode can be uniquely amplified for RH (LH) waves if 𝜔_sel<𝜔<𝜔_seu.

By the introduction of any small but finite resistance R, we obtain the form 𝛼_𝑐,𝜋=𝑝_𝑐,𝜋𝐺_𝑚+𝑞_𝑐,𝜋𝑅 with some coefficients 𝑝_𝑐,𝜋 and 𝑞_𝑐,𝜋 for complex dispersion. Irrespective of 𝑞_𝑐,𝜋, only one of the two modes is possibly amplified, if 𝑝_𝑐𝑝_𝜋<0. Thus, the key criteria that a CRLH TWFET can be designed to satisfy the condition for the unique mode to be amplified are equally valuable even with any loss or leakage.

In general, the resistance becomes larger for higher frequencies. It tends to reach a saturated value. Instead of modelling this frequency dependence, we employ the saturated resistance for all the frequencies for overestimating. We assigned the values of parasitic resistance to be proportional to the inductance, such that the resistance for 1.0-nH inductor becomes 0.15 Ω. By adding the linearized contribution of resistances in per-unit-cell gain, we obtain Figures 5 and 6, setting 𝐺_𝑚 to 2.0 mS. Figure 5 shows the per-unit-cell gain for low frequencies corresponding to the LH branches. The amplified waves are expected only for the 𝜋-mode at the frequencies between 3.0 and 3.3 GHz. The gain becomes maximal at 3.2 GHz, where the discrepancy between 𝜐_𝑐 and 𝜐_𝜋 becomes minimal. Figure 6 is for high frequencies corresponding to the RH branches. The amplified waves are expected only for the c-mode at the frequencies greater than 4.95 GHz.

Figure 5 

Per-unit-cell gain for LH frequencies. The solid and dashed curves represent the 𝜋- and c-modes, respectively. The filled region corresponds to amplification.

Figure 6 

Per-unit-cell gain for RH frequencies. The solid and dashed curves represent the 𝜋- and c-modes, respectively. The filled region corresponds to amplification.

4. Numerical Simulation of Test Line

Recently, we experimentally characterized a CRLH TWFET using discrete components on a breadboard [7]. We built a 90-section device. For convenience, all the series inductances were eliminated. As a result, the test line exhibited LH property for all the frequencies allowed to travel. The transistors used were Toshiba 2SK30A, whose typical threshold voltage was −1.8 V. The inductors and capacitors were implemented using the TDK SPT0406 series and the TDK FK series, respectively. The values of 𝐶_𝐿𝑔, 𝐶_𝑅𝑔, 𝐶_𝐿𝑑, 𝐶_𝑅𝑑, and 𝐶_𝑚 were set to 470, 47, 470, 470, and 47 pF, respectively. Moreover, the values of 𝐿_𝐿𝑔 and 𝐿_𝐿𝑑 were set to 10.0 and 4.7 μH, respectively. The gate and drain lines were fed by the envelope pulses generated by an NF WF1974 two-channel function generator. The waveforms were detected by Agilent 10073C passive probes and monitored in the time domain by using an Agilent DSO90254A oscilloscope. Through the measurements, we found that the 𝜋-mode gains amplitudes, while the c-mode loses them. Moreover, when the test line is terminated with the 𝜋-mode characteristic impedance, the multiple reflections of waves could be effectively suppressed.

To reinforce these observations, we simulate the measured results by solving the transmission equations of the test line using finite-difference time-domain method. The equations are given by 𝐿_𝑅𝑔𝑑²𝐼_𝑛𝑑𝑡²+𝐿_𝑚𝑑²𝐽_𝑛𝑑𝑡²=−𝐼^𝑛𝐶_𝐿𝑔−𝑅_𝑔𝑑𝐼_𝑛𝑑𝑡−𝑑𝑑𝑡𝑉^𝑛𝐿_𝑅𝑑−𝑉_𝑛−1,𝑑²𝐽_𝑛𝑑𝑡²+𝐿_𝑚𝑑²𝐼_𝑛𝑑𝑡²=−𝐽^𝑛𝐶_𝐿𝑑−𝑅_𝑑𝑑𝐽_𝑛𝑑𝑡−𝑑𝑑𝑡𝑊^𝑛𝑊_𝑛−𝑊_𝑛−1,−𝑉_𝐷𝐿_𝐿𝑑=𝑑𝑑𝑡+𝑅𝐿^in,𝑑_𝐿𝑑−𝐶^𝑅𝑑+𝐶_𝑚𝑑𝑊^𝑛+𝐶_𝑚𝑑𝑡𝑑𝑉_𝑛𝑑𝑡+𝐽^𝑛𝑉_𝑛−𝐽_𝑛+1−𝐼_𝑑𝑠𝑉_𝑛,−𝑉_𝐺𝐿_𝐿𝑔=𝑑×−𝐶𝑑𝑡_𝑅𝑔+𝑅𝐿^in,𝑔_𝐿𝑔+𝐶_𝑚𝑑𝑉^𝑛𝑑𝑡+𝐼^𝑛−𝐼_𝑛+1𝐶_𝑚𝑑𝑊_𝑛𝑑𝑡,(5) where 𝑉_𝑛 and 𝑊_𝑛 represent the gate and drain line voltages at the nth cell, respectively. Moreover, 𝐼_𝑛 and 𝑗_𝑛 represent the gate and drain line current flowing into the nth cell, respectively. Figures 7(a) and 7(b) show the measured and calculated waveforms, respectively, when gaussian envelope pulses with a carrier frequency of 2.5 MHz were input at one of the ends with the 𝜋-mode voltage fraction between the gate and drain lines. The upper and lower waveforms were monitored at the 10th and 40th cells, respectively. We can clearly see that the waves gain amplitude during propagation in Figure 7(a). Similar comparison between the measured and calculated waveforms is carried out for the c-mode envelope pulses in Figure 8. The upper and lower waveforms were monitored for the carrier frequency of 2.5 MHz at the 5th and 10th cells, respectively. In contrast to the 𝜋-mode waves, the c-mode ones lose amplitude during propagation. In both cases, the calculated waveforms well simulate the measured ones, including the amplitude gain and dispersive waveform distortions. By varying the carrier frequency of the input envelope pulses and voltage fractions, we obtain the frequency dependence of the per-unit-cell gain in Figure 9. Circles and squares represent the measured and calculated results for the 𝜋-mode inputs, respectively. The similarity between the measured and calculated results is established, so that we conclude that the numerical evaluations predict sufficiently the quantitative performance of CRLH TWFETs.

(a)

(b)

Figure 7 

Time-domain waveforms in test CRLH TWFET. (a) Measured and (b) calculated ones for the 𝜋-mode envelope pulses.

(a)

(b)

Figure 8 

Time-domain waveforms in test CRLH TWFET. (a) Measured and (b) calculated ones for the c-mode envelope pulses.

Figure 9 

Frequency dependence of per-unit-cell gain. Circles and squares represent the measured and numerical dependences, respectively. Positive gains correspond to wave amplification. The solid curve plots (5) for reference.

To discriminate the c- and 𝜋-modes clearly, we carried out calculations of a large-size line. We set the total cell number to 500 and an envelope pulse with the carrier frequency of 2.3 MHz was input to the 1st cell of the gate line. The pulse was split into the c- and 𝜋-mode components, which were well separated in space after the long-length propagation. Figures 10(a) and 10(b) show the calculated results for 𝑉_𝐺<𝑉_th and 𝑉_𝐺>𝑉_th, respectively. Upper and lower waveforms correspond to the pulses before and after the long-length propagation, respectively. The 𝜋-mode components preceded the c-mode ones and was amplified when𝑉_𝐺>𝑉_th. On the other hand, it is clearly seen that the c-mode components was more attenuated for 𝑉_𝐺>𝑉_th than for 𝑉_𝐺<𝑉_th.

(a)

(b)

Figure 10 

Numerically obtained envelope pulses in large CRLH TWFET. The total cell number is set to 500 without altering line parameters that were used to obtain Figures 6(b) and 7(b).

5. Numerical Performance of CRLH TWFETs

We evaluate numerically the performance of a CRLH TWFET including series inductors. We employed the same reactance values as those used to obtain Figures 5 and 6. The parasitic resistance and the gate bias voltage were tuned to clearly see the contributions of FET gain. It is expected that the line can amplify both the LH and RH waves, which are supported by the 𝜋- and c-modes, respectively. The numerical model we developed is equally applied not only to the MHz circuits but also to the microwave circuits, when the print-circuit-board techniques are effectively utilized; therefore, the results obtained by our model can be considered to be sufficiently reliable for these parameter values. Responses to the sinusoidal inputs were evaluated, considering practical situations where CRLH TWFETs would be applied. The thick and thin waveforms in Figure 11(a) correspond to the steady-state LH sinusoidal waves for 𝑉_𝐺>𝑉_th and for 𝑉_𝐺<𝑉_th, respectively. The frequency was set to 3.2 GHz, where the 𝜋-mode was supposed to be uniquely amplified. We thus designed terminal resistances to be matched with the 𝜋-mode characteristic impedances. Wave attenuation was well compensated by the FET contribution.

(a)

(b)

Figure 11 

Steady-state waveforms in a CRLH TWFET. (a) RH and (b) LH waves. The thin and thick waveforms correspond to 𝑉𝐺<𝑉th and 𝑉𝐺>𝑉th, respectively.

Finally, the properties of steady-state RH sinusoidal waves are shown in Figure 11(b). The frequency was set to 5.8 GHz. According to Figure 6, the c-mode is uniquely amplified, so that we set the terminations to be matched with the c-mode impedance. Compensation of attenuation was also established for the RH waves.

In both cases, it was confirmed that development of standing waves was well suppressed by the matched terminations.

6. Conclusions

We characterize waves in CRLH TWFETs. We found that one of the possible two propagation modes is uniquely amplified for wide range of frequency. By the terminations with the characteristic impedance corresponding to the amplifying mode, the multiple reflections of waves can be effectively suppressed. These design criteria may be practically useful to produce large-scale platform to support both the LH and RH waves.