Antennas WithNon - Foster Matching Networks

)Ra = dissipative loss resistance Xa = antenna reactance. (17) Since the radiation resistance represents power that is “delivered” by the antenna to the rest of the universe, we replace the radiation resistance with a transformer to the impedance of free space, or more conveniently, to any port impedance that we wish (such as 50 ). The turns-ratio of the transformer is given by N = √ Rr Z0 , (18) where Z0 is the desired port impedance. The resulting two-port representation of the antenna is shown in Fig. 8. At each frequency, a two-port representation of the form shown in Fig. 8 can be con- structed, and the two-port scattering matrix evaluated and written into an appropriate file for- mat (such as Touchstone) for use in a circuit simulator. Note that when port 2 of the two-port shown in Fig. 8 is terminated in the proper port impedance, the antenna’s input impedance is obtained as Za = Z0 1 + S111 − S11 (19) and its total efficiency is obtained as e tot = |S21| = ( 1 − |S11|2 ) e c d . (20) In some situations, it may not be practical (or even possible) to determine the radiation efficiency of the antenna. In this case, we can usually assume a radiation efficiency of 100% (as we have done for our example ESA). Despite this assumption, the proposed model still allows us to design P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 13 ajX lR 1:N0Z 0Z FIGURE 8: Two-port representation of an antenna (valid at a single frequency) a matching circuit with the advantage of monitoring and optimizing both return (match), insertion loss (total efficiency), and (in the case of an active matching network) the stability of the overall circuit. PERFORMANCE OF ESA WITH TRADITIONAL PASSIVE MATCHING NETWORK Any number of passive matching circuits can be used to provide a (theoretically) perfect match to our example ESA at 60 MHz. One of the most common ways to match such an antenna is to use an L-section consisting of two inductors as shown in Fig. 9. Using readily available design formulas for the L-section (e.g., from Chapter 5 of [2]), one obtains the following values for the inductors when designing for a perfect match at 60 MHz: L1 = 477 nH L2 = 51.9 nH. (21) The major disadvantage of using a passive matching network with an electrically small antenna is that any dissipative losses in the components of the matching network reduce the overall radiation efficiency. To examine this effect, let’s assume that each inductor has a Q of 100 at 60 MHz, which is reasonable for these inductance values in this frequency range. The combination of the matching network and two-port model of the antenna can be analyzed using an appropriate circuit simulator. Here we use Agilent advanced design system (ADS). P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 14 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS FIGURE 9: Schematic captured from Agilent ADS of ESA monopole with passive matching network The schematic of the antenna and its matching network captured from Agilent ADS is shown in Fig. 9. The computed return loss looking into the input of the matching network is shown in Fig. 10, and the total efficiency of the antenna/matching network combination is shown in Fig. 11. Of course, the return loss result could have been obtained readily without the proposed two-port model of the antenna. However, without the use of a rigorous two-port model of the antenna, the total efficiency result would have to be calculated outside of the circuit simulator. With the use of the two-port model for the antenna, it becomes possible, for example, to use the circuit simulator’s built-in optimization tools to maximize the overall radiation efficiency over commercially available inductor values, or to examine the effect of component tolerances using Monte-Carlo simulation. As is evident from the above example, the impedance bandwidth of our example ESA with a passive matching network is quite limited. In fact, with the passive matching network shown in Fig. 9, the half power (−3 dB efficiency) bandwidth is less than 3 MHz (agreeing with our calculation using the Bode–Fano limit). As a result it is likely that any reasonable component tolerances or environmental changes would cause the antenna to be de-tuned. The antenna system’s bandwidth can be increased by intentionally introducing loss into the passive matching network, but at the price of reduced maximum efficiency, the value of which can be readily evaluated inside of the circuit simulator using our approach. An interesting alternate approach that has been proposed recently is to use non-Foster reactances to provide a broadband match [3, 4]. P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 15 40 50 60 70 8030 90 -15 -10 -5 -20 0 m2 Return Loss (dB) dB (S (1 ,1 )) freq, MHz m2 freq=60.MHz dB(S(1,1))=-15.7 FIGURE 10: Return loss at input of passive matching network and antenna computed using Agilent ADS 40 50 60 70 8030 90 20 40 60 80 0 100 m1 Overall Efficiency (%) m1 freq=60.MHz mag(S(2,1))*100=83.3 m ag (S (2 ,1 )) *1 00 freq, MHz FIGURE 11: Overall efficiency (in percent) of passive matching network and antenna computed using Agilent ADS P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 16 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 0Z 0ZmL ( )ma LL +− a−C Active matching network Two-port model of antenna FIGURE 12: Antenna with active matching network using non-Foster reactances PERFORMANCE OF ESA WITH IDEAL NON-FOSTER MATCHING NETWORK A conceptual representation of the simplified ideal active matching network together with the two-port antenna model is shown conceptually in Fig. 12. The design equations for the components of the active matching network can be readily extracted from [3, 4] as follows. To design the active matching network, we first fit the antenna impedance to a simple model. Since the antenna is an electrically small monopole, the real part of the antenna impedance is assumed to vary as the square of frequency, and the imaginary part is modeled as a series LC. This simple model predicts an impedance that is denoted as Z¯a and given by Z¯a = R0 ( ω ω0 )2 + j ( ωLa − 1 ωCa ) . (22) The parameters of the model may be obtained from the “actual” antenna impedance Za (ob- tained from simulation or measurement) as R0 = Re {Za (ω0)} ⎡ ⎢ ⎢ ⎣ ω1 −1 ω1 ω2 −1 ω2 ⎤ ⎥ ⎥ ⎦ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ La 1 Ca ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ = ⎧ ⎪ ⎨ ⎪ ⎩ Im {Za (ω1)} Im {Za (ω2)} ⎫ ⎪ ⎬ ⎪ ⎭ . (23) where ω0 is the design frequency (in radians per second), ω1 and ω2 define the band of frequencies over which the model is being applied, and Re (Za ) and Im (Za ) are the real and imaginary P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 17 parts of the antenna impedance respectively. The last of the necessary design equations is Lm = √ R0 Z0 ω0 . (24) Basically, the active matching network works by canceling the antenna’s reactance over a broad- band using negative impedance elements, and then using a transformer section consisting of –Lm in series and Lm in shunt to match the real part of the antenna impedance (with its frequency-squared dependence) to the desired impedance level (Z0) over a broadband. Using the above design equations with ω1 = 2π × 50 MHz and ω2 = 2π × 70 MHz, we obtain the following component values for the active matching network: Ca = 8.657 pF La = 188.6 nH Lm = 45.57 nH. (25) Fig. 13 shows the schematic captured from Agilent ADS of the two-port antenna model together with non-Foster matching network comprising an ideal negative inductor and capacitor. The return loss obtained from the simulation is shown in Fig. 14. Notice that the return loss is better than 10 dB from about 36 MHz to above 90 MHz, even though the antenna is electrically small. Fig. 15 shows the total efficiency of the antenna/matching network combination. Note that total efficiency better than 95% is achieved from about 36 MHz to above 90 MHz. It should also be noted that the total efficiency slightly exceeds 100% near 43 MHz. However, VAR VAR1 Cneg=8.657 Lneg=234.2 Lm=45.57 S_Param SP1 Step=1 MHz Stop=90 MHz Start=30 MHz S-PARAMETERS Zin Zin1 Zin1=zin(S11,PortZ1) Zin N Term Term2 Z=50 Ohm Num=2 S2P SNP1 21 RefC Cneg C=-Cneg pF L Lneg L=-Lneg nH L Lm L=Lm nH Term Term1 Z=50 Ohm Num=1 Two-port model of antenna FIGURE 13: Schematic captured from Agilent ADS of ESA monopole with idealized active matching network P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 18 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 40 50 60 70 8030 90 -40 -30 -20 -10 -50 0 freq, MHz dB (S (1 ,1 )) Return Loss (dB) FIGURE 14: Return loss at input of idealized active matching network and antenna computed using Agilent ADS conservation of power is not being violated because an active matching network requiring a DC power supply is implied. Non-Foster reactances are realized using active circuits called negative impedance convert- ers (NICs). NICs are intrinsically unstable (consider a negative resistor), and thus the stability of the combined matching network and antenna must be evaluated to ensure that the antenna does not radiate spuriously. As we shall see, the two-port antenna model allows us to readily evaluate small-signal stability measures using the circuit simulator. BASICS OF NEGATIVE IMPEDANCE CONVERTERS (NICS) Non-Foster behavior can be achieved by using active circuits called negative impedance convert- ers (NICs). An ideal NIC can be defined as an active two-port device in which the impedance (or admittance) at one terminal pair is the (possibly scaled by a positive constant) negative of the impedance (or admittance) connected to the other terminal pair. An ideal NIC is shown conceptually in Fig. 16. NICs originated in the 1920s as a means to neutralize resistive loss in circuits [5]. Accord- ing to Merill, negative impedance circuits were used to develop a new type of telephone repeater P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 19 40 50 60 70 8030 90 70 80 90 100 60 110 freq, MHz m ag (S (2 ,1 )) *1 00 Overall Efficiency (%) FIGURE 15: Overall efficiency (in percent) of idealized active matching network and antenna computed using Agilent ADS called the E1. This repeater employed a feedback amplifier to provide transmission gains of 10 dB in two-wire telephone systems with extremely low loss. Due to the operation of the neg- ative impedance circuit, the E1 repeater was able to amplify voice signals at a lower cost than conventional repeaters at the time. More recently, Yamaha incorporated negative impedance circuits in their Yamaha Servo Technology (YST) to compensate for resistive losses in the voice coil of a loudspeaker [6]. The minimization of resistive loss in the amplifier–speaker system eliminated inaccuracies in sound reproduction. Moreover, the NIC in the YST maintained Ideal NIC ZL Zin=-kZL (k>0) FIGURE 16: Conceptual representation of ideal NIC P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 20 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 11h 1i 1v 22h + 212 vh 121 ih - + - 2i 2v + - FIGURE 17: Hybrid parameter model for general two-port network better control of the speaker cone, which allowed more air to escape through desired output ports rather than through the cone itself, resulting in maximized sound quality. Although NICs have been proven useful at audio frequencies, they have high frequency applications as well. As described in [7], a negative resistance circuit can be employed to compensate for the parasitic losses in the pass-band of a passive filter. The NIC helped to maximize the throughput (S21) of a narrowband band-pass filter with a center frequency of 14 GHz. Consider the general hybrid parameter model for a two-port network shown in Fig. 17. It is easy to show that for an ideal NIC (with k = 1), the following conditions must be met: h11 = 0 h22 = 0 h12 · h21 = 1. (26) Let’s consider two special cases of Eq. (24): first, h12 = h21 = −1 and second h12 = h21 = 1. The first case is called a voltage inversion NIC (VINIC) since vin = v1 = −v2 = −vL iin = i1 = −i2 = iL Zin = viniin = −vL iL = −ZL. (27) The hybrid parameter model for the VINIC is shown in Fig. 18. The second case is called a current inversion NIC (CINIC) since vin = v1 = v2 = vL iin = i1 = i2 = −iL Zin = viniin = vL −iL = −ZL. (28) The hybrid parameter model for the CINIC is shown in Fig. 19. P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 21 1iiin = 1vvin = + 2v 1i- + - Lii =2 Lvv =2 + - LZ FIGURE 18: Hybrid parameter model for VINIC The simplest practical implementation of an NIC makes use of an op-amp in the circuit shown in Fig. 20. Applying the “golden rules” of ideal op-amp analysis, we have vin = vL v3 = vin + Riin = vL − RiL ⇒ iin = −iL. (29) Thus, this simple op-amp circuit is a CINIC. Notice also that for this NIC, one side of the load is connected to ground. This type of circuit is called a grounded NIC (GNIC). The non-Foster matching circuit shown in Fig. 12 requires that the non-Foster circuit element (in that case a series negative 1iiin = 1vvin = + 2v 1i - + - L-ii =2 Lvv =2 + - LZ FIGURE 19: Hybrid parameter model for CINIC P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 22 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS RR ini inv + - LZ - + Li Lv + - 3v FIGURE 20: Basic op-amp NIC circuit LC) be floating—that is, not have either side connected to ground. This type of circuit element requires what we refer to as a floating NIC (FNIC). An FNIC can be realized using two op-amps as for example in the circuit shown in Fig. 21 [8]. To demonstrate that this circuit works as an FNIC, assume that the same impedance that is to be inverted, ZL, is also connected to port 2. If the circuit does indeed function as an FNIC, the input impedance looking into port 1 should be zero. Applying the “golden rules” of ideal op-amp analysis, we can show that v3 = v1 v′3 = v2 i3 = −i1 = i2. (30) R R 1i 1v + − LZ −+ 2i 2v + − 3v − + R R 3v 3i FIGURE 21: FNIC circuit using two op-amps P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 23 We also have i3 = v3 − v ′ 3 ZL i2 = − v2ZL. (31) Combining Eqs. (30) and (31), we obtain v3 − v′3 ZL = − v2 ZL or v1 − v2 ZL = − v2 ZL or v1 = 0. (32) Thus, Zin = −v1i1 = 0 (33) demonstrating that the circuit between terminals 1 and 2 acts as an FNIC. The simplified equivalent circuit of the ideal FNIC is shown in Fig. 22. In addition to realizing NICs with op-amps, the literature contains many examples of NICs that can be realized (at least theoretically) using two transistors. In [9], a catalog of all known two-transistor NIC designs is presented. One of the earliest proposed two-transistor NICs, and the most appropriate for active matching networks since it can realize an FNIC, is shown in Fig. 23. (Note that this schematic does not show the DC biasing of the devices. The exact biasing scheme can affect circuit performance especially stability.) To analyze the FNIC circuit shown in Fig. 23, we replace the bipolar junction transformers Q1 and Q2 with the small-signal T-model shown in Fig. 24. Doing so, we obtain the small-signal equivalent circuit for the FNIC shown in Fig. 25. To demonstrate that this circuit works as an L−Z FIGURE 22: Simplified equivalent circuit of ideal FNIC P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 24 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 1i 1v + − LZ 2i 2v + − 1Q 2Q 3v 3v' FIGURE 23: FNIC circuit using two transistors FNIC, assume that the same impedance that is to be inverted, ZL, is also connected to port 2. If the circuit does indeed function as an FNIC, the input impedance looking into port 1 should be zero. Utilizing nodal analysis, we can write the system of equations for the four unknown nodal voltages (v1, v2, v3, and v′3) as 1 re v1 − 1re v ′ 3 = −i1 − ( 1 ZL + 1 re ) v2 + 1re v3 = 0 ( 1 re − gm ) v1 + gmv2 + ( 1 ZL − gm ) v3 + ( gm − 1re − 1 ZL ) v′3 = 0 gmv1 + ( 1 re − gm ) v2 + ( gm − 1re − 1 ZL ) v3 + ( 1 ZL − gm ) v′3 = 0. (34) C re gm vbe + vbe - B E FIGURE 24: Small-signal T-model for BJT P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 25 1i 1v + − LZ 2i 2v + − 3v 3v' re + vbe1 - re + vbe2 - 1bemvg 2bemvg FIGURE 25: Small-signal equivalent for FNIC circuit using two transistors Solution of the above system of equations yields Zin = v1i1 = 2gmre ZL − 2ZL − 2re . (35) The general consensus in the literature seems to be that the best way (at least in theory) to realize the so-called two-transistor NICs is to replace each transistor with a kind of idealized “super transistor” called a second generation negative current conveyor (CCII-) [10]. We can think of a CCII- as a BJT with infinite transconductance (gm). Note that for large values of transconductance, we have re = 1gm . (36) Hence, for an ideal transistor (with infinite transconductance), Eq. (35) yields Zin −−→gm→∞ 0. (37) Thus, the circuit shown in Fig. 23 behaves as an FNIC provided the transistors have large enough transconductance. SIMULATED AND MEASURED NIC PERFORMANCE To date we have simulated a variety of NIC circuit realizations using both small-signal S- parameter and SPICE models of the active devices. We have also constructed and measured the P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 26 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 2.09 V -456 uV - -193 uV + -2.05 mV -5 V V- 5 V V+ 5 V V+ -5 V V- 2.09 nA R R1 R=1 GOhm 1.53 mA V_DC SRC3 Vdc=-5 V -1.56 mA V_DC SRC4 Vdc=5 V Port P2 Num=2 0 A C C8 C=0.1 uF 0 A C C7 C=0.1 uF 0 A C C9 C=0.1 uF P ort P 3 Num=3 P ort P 1 Num=1 -2.09 nA -3.57 uA -3.33 uA 13.8 uA 1.56 mA -1.53 mA opa690 OP A1 FIGURE 26: Schematic of OPA690 for simulation in Agilent ADS obtained by using the SPICE model and the data sheet for the device provided by TI performance of several of these NIC circuits. Unfortunately, successful simulation of an NIC circuit has not always led us to a successful physical implementation. One reason for this is that all NIC circuits are only conditionally stable—that is certain auxiliary conditions must be met for the circuit to be stable. In this section we will review our progress in physically realizing NIC circuits for use in active non-Foster matching networks. The reader should be aware that this topic is one for which a great deal of work remains to be done. It is this author’s opinion that the major advances in this area will be made by analog circuit designers who have been convinced by antenna engineers of the rewards to be reaped in pursuing the development of high frequency NICs. NIC ZL Rin Signal Generator Vin Vneg Iin -ZL Vg Rg Zin FIGURE 27: Circuit for evaluating the performance of a grounded negative impedance P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 27 The first NIC circuit that we consider is a grounded negative resistor (GNR) realized using the OPA690 op-amp from Texas Instruments (TI). The OPA690 is a wideband, voltage- feedback op-amp with a unity gain bandwidth of 500 MHz. Using the SPICE model for the device and the data sheet [11] provided by TI, an Agilent ADS model of the OPA690 can be created as shown in Fig. 26. In this circuit, port 1 is the noninverting input, port 2 is the inverting input, and port 3 is the single-ended output port. The 0.1 uF capacitors are used to RF bypass both the +5 V and −5 V power supplies, and the 1 G resistor is used to simulate an open circuit for the disable pin of the OPA690 for normal operation [11]. Fig. 26 also shows the results of the DC analysis of the Agilent ADS model of the OPA690. From this analysis, we see that the overall power consumption is approximately 15.5 mW, which can be considered low power for a discrete circuit design. To characterize the behavior of the grounded negative impedance, the circuit shown in Fig. 27 is used. Fig. 28 illustrates an Agilent ADS schematic for time-domain simulation of the OPA690 GNR test circuit. The overall stability of this circuit Vg Vi n Vneg Vt Sine Vg Phase = 0 Damping = 0 Delay = 0 n sec Freq = 0.5 MHz Amplitude = 100 mV Vd c = 0 m V Tran Tran1 Max Time Step = 0.5 n sec Stop Time=5 usec TRANSIENT R Rin R = 100 Ohm R R7 R = R scale OPA690_port X1 R R1 0 R = 50 Ohm R R3 R = R scale 2 VAR VA R 1 Rscale 2 = 250 Rscale = 250 Eq n Va r R Rg R = 50 Ohm FIGURE 28: Schematic captured from Agilent ADS of the circuit for evaluating the performance of the OPA690 NIC P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 28 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 1 2 3 40 5 -40 -20 0 20 40 -60 60 time, usec m2 m5 m2 time = 500.1nsec Vin = 0.050 m5 time = 1.500 usec Vneg = 0.049 V ne g, m V V in , m V FIGURE 29: Agilent ADS simulated waveforms Vin and Vneg waveforms at 0.5 MHz for the circuit shown in Fig. 27 must be carefully considered. For high frequency, internally compensated op amps such as the OPA690, the gain as a function of frequency can be represented by [12] A(s ) = A0ωb s , (38) where A0 represents the DC gain of the op amp and ωb represents the op amp’s 3 dB fre- quency. Using this gain model for the op amp, the overall transfer function T (s )of the OPA690 evaluation circuit (without the generator) can be computed (employing the golden rules of op-amps) as T (s ) = 1ZL−Rin ZL+R − sA0ωb ( 1 + RinR ) . (39) It is well known that it is necessary for the poles of T (s ) to lie in the left-half of the s -plane in order for the system to be stable. Consequently, the input resistor Rin must be greater than the load impedance ZL. One clever way, proposed in [9], to both ensure stability and evaluate the performance of the grounded negative impedance is to set the condition that Rin − ZL = 50 . (40) This choice allows us to evaluate performance in terms of return loss in a 50  system using a vector network analyzer (VNA). P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 29 FIGURE 30: Photograph of fabricated OPA690 NIC evaluation board If the GNR in the circuit of Fig. 27 is functioning properly, then ideally we should have Vneg = −Vin. (41) Results of the time-domain simulation performed in Agilent ADS for the circuit of Fig. 28 are shown in Fig. 29. Clearly, the condition given in (41) is satisfied almost exactly and the GNR functions properly at 500 kHz. Because of the excellent simulation results, a printed circuit board (PCB) implementation of the GNR test circuit shown in Fig. 28 was realized using readily available FR4 copper laminate and surface mount device (SMD) resistors and capacitors. Fig. 30 shows the assembled OPA690 GNR evaluation board. The simulated and measured return losses are compared in Fig. 31. In general there is excellent agreement between simulation and measurement. However, for frequencies less than 2 MHz, the measured return loss deviates somewhat from the simulation. The main cause of this discrepancy is attributed to low frequency calibration error of the VNA cables. If the 20 dB return loss bandwidth is taken to be the figure-of-merit, then the bandwidth of the OPA690 GNR is about 5 MHz. If this specification is relaxed to the 15 dB return loss bandwidth, then the bandwidth of the GNR increases to about 10 MHz. In either case, these results confirm that conventional op-amps can be used to construct NICs, but faithful negative impedance will exist only to about 10 MHz or so. The use of op-amp-based NICs at higher frequencies must await the development of op-amps with significantly higher unity gain bandwidths than are currently available. Moreover, the parasitics of the device and circuit board will have to be minimized as much as possible. P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 30 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 2 4 6 8 10 12 14 16 18 200 -40 -30 -20 -10 -50 0 freq, MHz dB (R et ur n_ L os s_ S im ul at ed ) dB (R et ur n_ L os s_ M ea su re d) FIGURE 31: Simulated and measured return loss for the OPA690 NIC evaluation circuit Because an op-amp’s gain-bandwidth product severely limits the upper frequency at which negative impedance conversion can occur, we next focus on NIC realizations using current feedback amplifiers (CFAs) whose performance is (theoretically) not limited by their gain-bandwidth products, but mostly by their internal parasitic elements. Consequently, NICs employing these amplifiers should be more broadband in nature. To investigate this possibility, the MAX435 wideband operational transconductance amplifier (WOTA) manufactured by Maxim was selected as the NIC’s active device used to realize a GNR. This device was chosen because of its simplicity, versatility, fully differential operation, and extremely wideband behavior. The current of the device is set by an external resistor Rset (normally 5.9 k [13]), and the voltage gain of the MAX435 WOTA is set by the current gain of the device (approximately 4), the transconductance element value (Zt), and the load resistor value (ZL) as [13] Av = Ai ZLZt = 4 ZL Zt . (42) This voltage gain Av of the MAX435 was set as high as possible without its internal parasitics severely limiting the bandwidth of the amplifier. For a typical application, the load impedance ZL must be chosen to be a finite value (usually 25  or 50 ) [13]. A SPICE model for the MAX435 was obtained from Maxim IC’s website and configured as a fully differential amplifier for simulation in Agilent ADS as shown in Fig. 32. It was found through measurement that if Zt was less than 5 , then the gain of the amplifier rolled off very quickly because a pole was introduced in the pass-band of the device. This phenomenon was modeled as an effective output P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 31 5 V +V -5 V -V 17.8 m V 17.8 m V 3.70 V 3.70 V 3.70 V -5 V -V 0 V -7.66 uV -5 V -V 5 V +V 5 V +V -3.69 uA R R39 R=ZL Ohm 3.40 uA R R38 R=ZL Ohm Port P4 Num =4 0 A C C24 C=CL pF 0 A C C25 C=CL pF VAR VAR1 CL=250 ZL=50 Zt=5 Eqn Va r -1.53 uA R R37 R=Zt Ohm -1.47 m A s r_da l_RCWP_540_F_19950814 R36 PART_NUM=RCWP5405901F 5.90 kOhm Port P2 Num =2 Port P1 Num =1 Port P3 Num =3 -34.5 m A V_DC SRC2 Vdc=5 V -33.0 m A V_DC SRC1 Vdc=5 V -1.47 m A-4.71 uA 1.47 m A 997 pA -1.53 uA1.53 uA 7.55 uA -33.0 m A 33.0 m A MAX435_1 X3 0 A C C21 C=200 nF 0 A C C22 C=200 nF 0 A C C23 C=200 nF FIGURE 32: Schematic of MAX435 for simulation in Agilent ADS obtained by using the SPICE model and augmenting it to match experimental results capacitance CL and included in the analysis of the device. Ports 1 and 2 are the noninverting and inverting inputs, respectively, while ports 3 and 4 are the noninverting and inverting outputs, respectively. Included with the SPICE model are the external elements Zt, ZL, CL, and Rset along with power supply decoupling capacitors. The overall power consumption of the WOTA in simulation is the sum of the power of the dual supplies, which is approximately 340 mW. Fig. 33 shows the MAX435 as a differential amplifier being used in an NIC evaluation circuit for a grounded negative resistor. The NIC topology used has been cataloged as topology IIIa in [6]. The MAX435 replaces both of the BJTs (or CCII-s) in the topology, thus simplifying the design and minimizing component count. Hence, a two-transistor NIC circuit can be simply constructed employing a single active device. Another distinct advantage of using the MAX435 is that no RF chokes are needed to bias the device, which allows for more compact layout schemes and reduced loss. Ideally, the input impedance of the evaluation circuit should be 50  over all frequencies resulting in a reflection coefficient of zero. As a quick proof-of-concept, the MAX435 GNR was breadboarded using a MAX435 in a 14-pin dual in-line package and surface mount discrete components. Wires with small diameters were used in some cases to create short circuits. In addition, copper tape strips were P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 32 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS R ZL R=ZL Ohm VAR VAR1 ZL=50 Rin=100 Rs ca le 2=1000 Rs ca le =1000 Eqn Va r R Rin R=Rin Ohm DC DC1 DC MAX_435_port_wo_TLs X1 S_Pa ra m SP1 Ste p= Stop=200 MHz Sta rt=.3 MHz S-PARAMETERS Zin Zin1 Zin1=zin(S11,PortZ1) Zin N Te rm Te rm 1 Z=50 Ohm Num =1 R Rs ca le 2 R=Rs ca le 2 Ohm R Rs ca le R=Rs ca le Ohm FIGURE 33: Schematic captured from Agilent ADS of the circuit for evaluation of the MAX435 NIC used to create a good ground plane for the device as recommended in [13]. Fig. 34 shows the assembled MAX435 GNR evaluation board. The simulated and measured return losses are compared in Fig. 35. In general there is good agreement between simulation and measurement. If the 15 dB return loss bandwidth is taken to be the figure-of-merit, then the bandwidth of the MAX435 GNR is about 18 MHz. We made a couple of unsuccessful attempts to increase the bandwidth of the MAX435 GNR circuit. In our first attempt, we replaced the MAX435 in DIP-14 package and breadboard construction with an unpackaged MAX435 and professional wirebond and PCB construction. FIGURE 34: Photograph of fabricated MAX435 NIC evaluation board P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 33 20 40 60 80 100 120 140 160 1800 200 -20 -15 -10 -5 -25 0 freq, MHz dB (S im ul at ed _R et ur n_ L os s) dB (M ea su re d_ R et ur n_ L os s) FIGURE 35: Simulated and measured return loss for the MAX435 NIC evaluation circuit Our hope was that the new construction would greatly reduce parasitics resulting in an increase in bandwidth. Unfortunately this was not the case as the measured results for the new device were virtually identical to those of the original crude breadboard construction. In our second attempt, based on a suggestion from Maxim, we used the OPA690 as a gain-boosting stage for the WOTA. Simulations showed that this circuit should exhibit substantially improved bandwidth. Unfortunately the measured results were no better than the results we achieved with the MAX435 by itself. The third NIC circuit considered makes use of TI’s THS3202 CFA which possesses a 2 GHz unity gain bandwidth. Two amplifiers are contained within a single package. By combining the high speed of bipolar technology and all the benefits of complementary metal oxide semiconductor (CMOS) technology (low power, low noise, packing density), this amplifier is able to perform extremely well over a very large bandwidth. A SPICE model for the THS3202 can be downloaded from TI’s website and was implemented in Agilent ADS as shown in Fig. 36. The inductor and capacitor form a low-pass filter to prevent AC ripple on the power supply line. The THS3202 can be configured as a GNR much like the OPA690 GNR previously considered. Following the design guidelines in [14], the scaling resistors Rs 1 and Rs 2 were chosen to be 200  to maximize the gain and minimize the overall noise figure of the amplifier. Physical realizations of THS3202 GNR circuits were implemented using an evaluation module (THS3202 EVM) that was purchased through TI and shown in Fig. 37. This board was modified to realize a GNR. The simulated and measured return losses are compared in Fig. 38. If the 20 dB return loss P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 34 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS V- V+ V+ V- L FB2 R=.035 L=130 nH L FB1 R=.035 L=130 nH V_DC SRC3 Vdc=5 V C C6 C=22 uF C C7 C=22 uF V_DC SRC4 Vdc=-5 V Port P3 Num=3 C C9 C=0.1 uF C C8 C=100 pF C C3 C=0.1 uF C C5 C=100 pF Port P1 Num=1 ths 3202 X1 Port P2 Num=2 FIGURE 36: Agilent ADS model of the THS3202 with supply bypassing bandwidth is taken to be the figure-of-merit, then the simulation bandwidth of the THS3202 negative resistor evaluation circuit is about 120 MHz. Unfortunately, the measured bandwidth is only about 50 MHz. Nevertheless, the measured results for the THS3202 GNR are still significantly greater than the results obtained using either the OPA690 or the MAX435 as the NIC’s active devices. In the simulation, the measured input resistance of the THS3202 GNR FIGURE 37: Photograph of THS3202 evaluation board (THS3202 EVM) purchased from TI and modified to form an NIC evaluation circuit P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 35 50 100 150 200 250 300 350 400 4500 500 -40 -30 -20 -10 -50 0 freq, MHz m2m1 m1 fre q= dB(Re turn_Los s _Me as ure d)=-20.041 52.18MHz m2 fre q= dB(Re turn_Los s _S imulate d)=-20.012 118.4MHz dB (R et ur n_ L os s_ M ea su re d) dB (R et ur n_ L os s_ Si m ul at ed ) FIGURE 38: Simulated and measured return loss for the THS3202 NIC evaluation circuit is very nearly equal to –50  to frequencies greater than 500 MHz. However, the reactance of the THS3202 GNR is nonzero and behaves like a parasitic inductance. Thus, potentially we may be able to compensate for it and extend the bandwidth of the circuit. Having had some success in fabricating GNRs, we turned our attention to floating neg- ative resistors (FNRs). This work is still in its early stages, and only simulation results are presented here. To implement an FNIC, two THS3202 amplifiers (in the same package) can be used to realize the circuit shown in Fig. 21. The schematic of the FNIC captured from Agilent ADS is shown in Fig. 39. As with all the NIC circuits, particular attention needs to be paid to stability. Each of the GNR circuits previously considered is a one-port device that can be stabilized by employing a series resistor Rin that also allowed evaluation of the overall reflection coefficient S11 in a 50  system. The return loss of the resulting one-port was used as a figure-of-merit for the bandwidth of the GNR. To assess the performance of a floating negative impedance circuit, we can construct a so-called all-pass two-port network using the circuit shown in Fig. 40. Not only does this approach allow evaluation of the input return loss and the insertion loss as figures-of- merit, it also allows one to evaluate the small-signal stability of the network using conventional two-port measures. For the circuits that we consider here, the FNR has (ideally) an equivalent series resistance of −50  that negates a series 50  resistor. As a result, both the input and output impedances of the circuit should be 50 . In Fig. 41, a schematic captured from Agilent ADS shows the THS3202 FNIC configured as a –50  FNR and placed into an all-pass system configuration with a load impedance RL = 50 across ports 3 and 4. Notice in the schematic the presence of the μ′ token which allows the assessment of the small-signal stability of the network. Simulated results for return loss and small-signal stability of the THS3202 FNR in the all-pass network are shown in Fig. 42. Although the −20 dB return loss bandwidth is P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 36 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS R R3 R=Rscale Ohm R R4 R=Rscale Ohm Port P2 Num=2 Port P4 Num=4 ths3202_port X2 ths3202_port X3 Port P3 Num=3 VAR VAR1 Rscale=200 Eqn Va r Port P1 Num=1 R R2 R=Rscale Ohm R R1 R=Rscale Ohm FIGURE 39: Schematic captured from Agilent ADS of the THS3202 FNIC circuit L−Ζ 0Z 0Z L−Ζ FIGURE 40: All-pass circuit for evaluating the performance of a floating negative impedance P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 37 R R5 R=R_L Ohm MuPrime MuPrime 1 MuPrime 1=mu_prime(S) MuPrime VAR VAR1 Rin=50 R_L=50 Eqn Var S_Param SP1 Step=1000 kHz Stop=500 MHz Sta rt=10 MHz S-PARAMETERS R R10 R=Rin Ohm Term Term2 Z=50 Ohm Num=2 Floa ting_NIC_Antoniou_1a_THS_port X1Term Term1 Z=50 Ohm Num=1 FIGURE 41: Schematic captured from Agilent ADS of the THS3202 FNIC of Fig. 38 configured as a FNR and installed in the all-pass evaluation circuit broadband (approximately 100 MHz), the circuit is unconditionally stable only for frequencies less than 50 MHz. In an attempt to create an FNR with greater small-signal stability, we arranged two THS3202 GNRs back-to-back as shown in Fig. 43. Analyzing the circuit assuming ideal op-amps, we find that the equivalent resistance seen between ports 1 and 2 is given by Rin = R3 R1R1 − R2 . (43) Consequently, for the input resistance Rin to be the negative of the load impedance R3, the following relationship between R1 and R2 must be chosen as R2 = −2R1. (44) P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 38 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 50 100 150 200 250 300 350 400 4500 500 -30 -20 -10 -40 0 freq, MHz m3 m3 freq = 104.0MHz dB(S(1,1)) = -20.022 50 100 150 200 250 300 350 400 4500 500 0.2 0.4 0.6 0.8 1.0 0.0 1.2 freq, MHz (a) (b) dB (S (1 ,1 )) M uP ri m e1 FIGURE 42: Simulated (a) return loss and (b) stability of the all-pass test circuit for the THS3202 FNR An all-pass implementation simulation in Agilent ADS with R3 = 50 is depicted in Fig. 44 where the FNR is placed inside a two-port data item box. To minimize noise and maximize gain, R1 and R2 are chosen to be as small as possible (200  and 400 , respectively) without affecting the performance of the FNR. The two 25  resistors on each side of the FNR complete the all-pass test circuit. The simulation results of Fig. 45 show that the −20 dB return loss bandwidth is only about 30 MHz, but the network is close to being unconditionally stable over almost the entire frequency range. We found that the input reactance Xin is negative and so might be compensated over a limited frequency range using a series inductor. By trial and P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 39 Port P2 Num=2 Port P1 Num=1 ths3202_port X1 ths3202_port X2 R R5 R=R3 Ohm R R6 R=R3 Ohm R R2 R=R1 Ohm R R4 R=R2 Ohm R R3 R=R2 Ohm R R1 R=R1 Ohm FIGURE 43: Schematic captured from Agilent ADS of the THS3202 FNR circuit formed by two back-to-back GNRs S_Param SP 1 Step = 1000 kHz Stop = 300 MHz Start = 10 MHz S-PARAMETERS MuPrime MuPrime 1 MuPrime 1 = mu_prime (S) MuPrime R R9 R=25 Ohm R R10 R=25 Ohm Floating_NIC_Back_to_Back_port X1 Term Term1 Z=50 Ohm Num=1 Term Term2 Z=50 Ohm Num=2 0 FIGURE 44: Schematic captured from Agilent ADS of the THS3202 FNR of Fig. 42 installed in the all-pass evaluation circuit P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 40 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS m1 freq=33.00MHz Simulated_Return_Loss=-20.066 50 100 150 200 2500 300 -30 -25 -20 -15 -10 -35 -5 freq, MHz m1 50 100 150 200 2500 300 1.0 1.1 1.2 1.3 0.9 1.4 freq, MHz (a) (b) S im ul at ed _R et ur n_ L os s M uP ri m e1 FIGURE 45: Simulated (a) return loss and (b) stability of the all-pass test circuit for the THS3202 FNR formed by two back-to-back GNRs error, we found that placing an inductance of 45 nH in series with the FNR maximized the return loss bandwidth and stability of the all-pass test circuit as shown in Fig. 46. The simulated –15 dB return loss bandwidth is expanded to greater than 250 MHz. Unfortunately, the circuit is not unconditionally stable for frequencies less than 125 MHz, but may be relatively easy to stabilize since μ′ is so close to unity. Another way to implement an FNIC is to use two BJTs to realize the circuit shown in Fig. 23. Following the work reported in [15] and [16], we use the NE85630 NPN silicon P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 41 (a) (b) 50 100 150 200 2500 -35 -30 -25 -20 -15 -40 -10 freq, MHz 50 100 150 200 2500 300 300 1.0 1.2 1.4 1.6 0.8 1.8 freq, MHz S im ul at ed _R et ur n_ L os s M uP ri m e1 FIGURE 46: Simulated (a) return loss and (b) stability of the all-pass test circuit for the THS3202 FNR formed by two back-to-back GNRs with a 45 nH series inductor RF transistor from NEC. The schematic of the FNR all-pass test circuit using these devices captured from Agilent ADS is shown in Fig. 47. The simulated performance of this FNR test circuit is shown in Fig. 48. As can be seen, the −20 dB return loss bandwidth approaches 200 MHz, and the circuit is unconditionally stable at all simulation frequencies. It should be noted that the simulation is performed using only the S-parameters of the NE85630 (rather than a SPICE model) valid under a specified bias condition.2 The exact details of the biasing circuit are 2S-parameters for the NE85630 device are provided from 50 MHz to 3.6 GHz. Since we are simulating our circuits below 50 MHz, we are also relying on an accurate extrapolation of the S-parameters. P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 42 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS F IG U R E 47 : Sc he m at ic ca pt ur ed fr om A gi le nt A D S of th e al l- pa ss te st ci rc ui tf or th e N E 85 63 0 FN R P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 43 50 100 150 200 2500 (b) (a) 300 -28 -26 -24 -22 -20 -18 -30 -16 freq, MHz d B (S (1 ,1 )) 50 100 150 200 2500 300 0.95 1.00 1.05 0.90 1.10 freq, MHz M u P ri m e 1 FIGURE 48: Simulated (a) return loss and (b) stability of the all-pass test circuit for the NE85630 FNR neglected here, but do affect the circuit performance especially stability. The simulated results for the NE85630 are the best FNR results that we obtained. Thus, the NE85630 FNIC is used in the next section for the floating non-Foster reactance used in the active matching network for our ESA monopole. In addition to the NIC circuits discussed in detail in this section, we also made considerable effort trying to realize NIC circuits that utilized CCII- blocks implemented as cascades of GaAs PHEMT devices. We simulated these circuits extensively and were able to obtain excellent performance in simulation with bandwidths greater than 1 GHz. Unfortunately, our attempts to physically implement these designs have all ended in failure. Other researchers have also reported a lack of success using this approach [16], and so we have abandoned it for the present. P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 44 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS F IG U R E 49 : Sc he m at ic ca pt ur ed fr om A gi le nt A D S of V H F m on op ol e w ith ac tiv e m at ch in g ne tw or k P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 45 40 50 60 70 8030 90 -25 -20 -15 -30 -10 freq, MHz dB (S (1 ,1 )) Return Loss (dB) FIGURE 50: Return loss at input of optimized active matching network and antenna computed using Agilent ADS SIMULATED PERFORMANCE OF ESA WITH A PRACTICAL NON-FOSTER MATCHING NETWORK To illustrate the potential of non-Foster matching networks for ESAs, we designed and opti- mized in Agilent ADS a practical implementation of the active matching network shown in Fig. 12 for our ESA monopole antenna. We used a single FNIC of the form shown in Fig. 23 to implement the non-Foster series reactance consisting of − (La + Lm) in series with −Ca . The 40 50 60 70 8030 90 75 80 85 90 70 95 freq, MHz m ag (S (2 ,1 )) *1 00 Overall Efficiency (%) FIGURE 51: Overall efficiency (in percent) of optimized active matching network and antenna com- puted using Agilent ADS P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 46 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 40 50 60 70 8030 90 0.98 0.99 1.00 0.97 1.01 freq, MHz M u 1 M uP rim e 1 FIGURE 52: Small-signal geometrically derived stability factor for the optimized active matching net- work and antenna computed using Agilent ADS active devices (NE85630 silicon bipolar NPN transistors) were modeled using the S-parameter library in Agilent ADS. Not surprisingly, we found that the simulated NIC performance was far from ideal. Nevertheless, using the gradient optimizer in Agilent ADS, we were able to adjust the values of the capacitor and inductors in the matching network to achieve remark- able broadband performance from the ESA monopole. The schematic of the two-port antenna model and active matching network captured from Agilent ADS is shown in Fig. 49. Note the presence of the measurement component for the small-signal geometrically derived stability factors μ and μ′. The computed return loss looking into the input of the matching network is shown in Fig. 50, and the total efficiency of the antenna together with the active matching network is shown in Fig. 51. Note that an extremely broadband and highly efficient match has been achieved. The geometrically derived stability factors as a function of frequency are shown in Fig. 52. These factors must be strictly greater than 1 for the circuit to be unconditionally stable. Note that below about 31 MHz, the overall circuit is not unconditionally stable. This situation should ultimately be remedied to avoid spurious radiation from the antenna. CONCLUSIONS In this lecture, we discussed an exciting new area of research in antenna technology, namely, the use of non-Foster circuit elements in the matching network of an electrically small antenna. The contributions of this lecture were to summarize the current state-of-the-art in this subject, and to introduce some new theoretical and practical tools for helping others to continue the advancement of this technology. The new contributions include a rigorous method for gener- ating a two-port model for an antenna, an all-pass test circuit for evaluating the performance of P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 47 floating negative impedances, and a new kind of floating negative impedance converter formed from two back-to-back grounded negative impedance converters. REFERENCES [1] C. A. Balanis, Antenna Theory: Analysis and Design. 3rd ed. New York: John Wiley and Sons, Inc., 2005. [2] D. M. 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Sussman-Fort, “Gyrator-based biquad filters and negative impedance converters for microwaves,” Int. J. RF Microwave CAE Vol. 8, pp. 86–101, 1998. [10] A. Sedra, G. Roberts, and F. Gohh, “The current conveyor: history, progress, and new results,” IEEE Proc. G, Vol. 137, No. 2, pp. 78–87, Apr. 1990. [11] Texas Instruments, OPA690 Wideband Voltage-Feedback Operational Amplifier with Disable, 2005. [12] A. Sedra and K. C. Smith, Microelectronic Circuits, 4th ed., New York: Oxford University Press, 1998. [13] Maxim, MAX435/MAX436 Wideband Transconductance Amplifiers, 1993. [14] Texas Instruments, THS3202 Low Distortion, 2 GHz, Current Feedback Amplifier, 2004. [15] S. E. Sussman-Fort, “Matching network design using non-Foster impedances,” IEEE Long Island Section, Circuits and Systems Society [Online]. Available: cas/index.htm [Accessed: Dec. 6, 2005]. [16] S. E. Sussman-Fort and R. M. Rudish, “Progress in use of non-Foster impedances to match electrically-small antennas and arrays,” Antenna Applications Symposium, Allerton, 2005. P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 48