Antennas with Non-Foster Matching Networks phần 3 docx

P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 16 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 0 Z 0 Z m L () 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 = R 0  ω ω 0  2 + j  ωL a − 1 ωC a  . (22) The parameters of the model may be obtained from the “actual” antenna impedance Z a (ob- tained from simulation or measurement) as R 0 = Re { Z a ( ω 0 ) } ⎡ ⎢ ⎢ ⎣ ω 1 −1 ω 1 ω 2 −1 ω 2 ⎤ ⎥ ⎥ ⎦ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ L a 1 C a ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ = ⎧ ⎪ ⎨ ⎪ ⎩ Im { Z a ( ω 1 ) } Im { Z a ( ω 2 ) } ⎫ ⎪ ⎬ ⎪ ⎭ . (23) whereω 0 is the design frequency (in radianspersecond),ω 1 and ω 2 define the band offrequencies over which the model is being applied, and Re ( Z a ) and Im ( Z a ) 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 L m = √ R 0 Z 0 ω 0 . (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 –L m in series and L m in shunt to match the real part of the antenna impedance (with its frequency-squared dependence) to the desired impedance level (Z 0 ) over a broadband. Using the above design equations with ω 1 = 2π × 50 MHz and ω 2 = 2π × 70 MHz, we obtain the following component values for the active matching network: C a = 8.657 pF L a = 188.6nH L m = 45.57 nH. (25) Fig. 13 showstheschematiccapturedfromAgilent ADS of thetwo-portantennamodeltogether 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 2 1 Ref C 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-Fosterreactancesarerealized using activecircuitscalled negative impedanceconvert- 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-Fosterbehavior 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 inthe 1920s asa means toneutralize 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 mag(S(2,1))*100 Overall Efficiency (%) FIGURE15: 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 Z L Z in =-kZ L (k>0) L =- L 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 11 h 1 i 1 v 22 h + 212 vh 121 ih - + - 2 i 2 v + - 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 (S 21 ) 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: h 11 = 0 h 22 = 0 h 12 · h 21 = 1. (26) Let’s consider two special cases of Eq. (24): first, h 12 = h 21 =−1 and second h 12 = h 21 = 1. The first case is called a voltage inversion NIC (VINIC) since v in = v 1 =−v 2 =−v L i in = i 1 =−i 2 = i L Z in = v in i in = −v L i L =−Z L . (27) The hybrid parameter model for the VINIC is shown in Fig. 18. The second case is called a current inversion NIC (CINIC) since v in = v 1 = v 2 = v L i in = i 1 = i 2 =−i L Z in = v in i in = v L −i L =−Z L . (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 1 ii in = 1 vv in = + 2 v 1 i - + - L ii = 2 L vv = 2 + - L Z 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 v in = v L v 3 = v in + Ri in = v L − Ri L ⇒ i in =−i L . (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 1 ii in = 1 vv in = + 2 v 1 i - + - L -ii = 2 L vv = 2 + - L Z 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 R R in i in v + - L Z - + L i L v + - 3 v 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, Z L , 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 v 3 = v 1 v  3 = v 2 i 3 =−i 1 = i 2 . (30) R R 1 i 1 v + − L Z − + 2 i 2 v + − 3 v − + R R 3 v 3 i 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 i 3 = v 3 − v  3 Z L i 2 =− v 2 Z L . (31) Combining Eqs. (30) and (31), we obtain v 3 − v  3 Z L =− v 2 Z L or v 1 − v 2 Z L =− v 2 Z L or v 1 = 0. (32) Thus, Z in =− v 1 i 1 = 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 theFNICcircuitshownin Fig.23, we replacethe bipolar junctiontransformers Q1and Q2 with thesmall-signalT-modelshown in Fig.24.Doing so,weobtainthe 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 1 i 1 v + − L Z 2 i 2 v + − 1 Q 2 Q 3 v 3 v' FIGURE 23: FNIC circuit using two transistors FNIC, assume that the same impedance that is to be inverted, Z L , 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 (v 1 , v 2 , v 3 , and v  3 )as 1 r e v 1 − 1 r e v  3 =−i 1 −  1 Z L + 1 r e  v 2 + 1 r e v 3 = 0  1 r e − g m  v 1 + g m v 2 +  1 Z L − g m  v 3 +  g m − 1 r e − 1 Z L  v  3 = 0 g m v 1 +  1 r e − g m  v 2 +  g m − 1 r e − 1 Z L  v 3 +  1 Z L − g m  v  3 = 0. (34) C r e g m v be + v be - 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 1 i 1 v + − L Z 2 i 2 v + − 3 v 3 v' r e + v be1 - r e + v be2 - 1bem vg 2bem vg FIGURE 25: Small-signal equivalent for FNIC circuit using two transistors Solution of the above system of equations yields Z in = v 1 i 1 = 2g m r e Z L − 2Z L − 2r e . (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 (g m ). Note that for large values of transconductance, we have r e = 1 g m . (36) Hence, for an ideal transistor (with infinite transconductance), Eq. (35) yields Z in −−→ g m →∞ 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 . January 19, 2007 17: 23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 23 We also have i 3 = v 3 − v  3 Z L i 2 =− v 2 Z L . (31 ) Combining Eqs. (30 ) and (31 ), we obtain v 3 − v  3 Z L =− v 2 Z L or v 1 −. 2007 17: 23 16 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 0 Z 0 Z m L () ma LL +− a −C Active matching network Two-port model of antenna FIGURE 12: Antenna with active matching network using non-Foster. RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17: 23 24 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 1 i 1 v + − L Z 2 i 2 v + − 1 Q 2 Q 3 v 3 v' FIGURE 23: FNIC circuit using two transistors FNIC,