Antennas with Non-Foster Matching Networks phần 2 ppt

P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 6 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS FIGURE 2: Geometry of monopole antenna as modeled in Antenna Model software. The monopole is a copper cylinder 0.6 m in length and 0.010 meters in diameter, mounted on an infinite perfect ground plane total quality factor of the antenna/matching network combination is given by Q tot = 1 1 Q r + 1 Q m , (8) where Q m is the quality factor of the matching network. However, the loss in the matching network reduces the total efficiency of the system resulting in less total energy being coupled into free space. P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 7 FIGURE 3: Real part of input impedance of the ESA monopole obtained from simulation FIGURE 4: Imaginary part of input impedance of the ESA monopole obtained from simulation P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 8 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS FIGURE 5: Radiation quality factor of the ESA monopole obtained from simulation FOSTER’S REACTANCE THEOREM AND NON-FOSTER CIRCUIT ELEMENTS Foster’s reactance theorem is a consequence of conservation of energy and states that for a lossless passive two-terminal device, the slope of its reactance (and susceptance) plotted versus frequency must be strictly positive, i.e., ∂X ( ω ) ∂ω > 0 and ∂B ( ω ) ∂ω > 0. (9) A device is called passive if it is not connected to a power supply other than the signal source. Such a two-terminal device (or one-port network) can be realized by ideal inductors, ideal capacitors, or a combination thereof. It turns out that a corollary that follows from Foster’s reactance theorem is even more important than the theorem itself. The corollary states that the poles and zeros of the reac- tance (and susceptance) function must alternate. By analytic continuity, we can generalize this corollary of Foster’s reactance theorem to state the following about immittance (impedance and P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 9 admittance) functions for a passive one-port network comprising lumped circuit elements: 1. The immittance function can be written as the ratio of two polynomial functions of the Laplace variable s = σ + jω: Z ( s ) = N ( s ) D ( s ) . (10) 2. The coefficients of the polynomials N(s ) and D(s ) are positive and real. 3. The difference in the orders of N(s ) and D(s ) is either zero or 1. As two examples of the above, consider the following: A) Capacitor. The impedance function is given by Z ( s ) = 1 sC . (11) B) Series RLC. The impedance function is given by Z ( s ) = R +sL + 1 sC = s 2 LC + sCR + 1 sC . (12) If a two-terminal device has an immittance function that does not obey any of the three conse- quences of Foster’s reactance theorem listed above, then it is called a “non-Foster” element. A non-Foster element must be an active component in the sense that it consumes energy from a power supply other than the signal source. Two canonical non-Foster elements are the negative capacitor and the negative inductor. These circuit elements violate the second consequence of Foster’s reactance theorem in the list. A) Negative capacitor. The impedance function of a negative capacitor of value −C (with C > 0) is given by Z ( s ) = −1 sC . (13) B) Negative inductor. The impedance function of a negative inductor of value −L (with L > 0) is given by Z ( s ) =−sL. (14) BASIC CONCEPTS OF MATCHING AND BODE–FANO LIMIT It is well known from basic electrical circuit theory that maximum power transfer from a source to a load is achieved when the load is impedance matched to the source, that is when the load impedance is the complex conjugate of the source impedance. Matching between source and P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 10 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS L Z 0 Z L Matching Network FIGURE 6: Matching network concept load is also important so as to minimize reflections which can result in signal dispersion or even cause damage to the source. In general the load impedance is not the same as the source impedance, and a matching networkis required to provide amatch between the two impedances. The basic matching network concept is illustrated in Fig. 6. Ideally, a matching network would be lossless and provide a match between the source and load over all frequencies. This is theoretically possible only if both the source and load impedances are real and the matching network is an ideal transformer. In most situations, the source impedance is real (often 50 ) and the load impedance is a complex quantity which varies with frequency. As a result, it is impossible to achieve an exact match (using a passive matching network) except at a single frequency (or more generally at a finite number of discrete frequencies), and the match quality degrades as frequency deviates away from this frequency. The measure of match quality is the reflection coefficient at the input of the matching network. Most commonly, the value of the reflection coefficient is represented in terms of return loss in decibels (dB). Return loss in dB is defined as RL =−20 log 10 (  ) , (15) where  is the reflection coefficient at the input of the matching network. Typically, return loss values of greater than 10 dB are considered acceptable. The Bode–Fano criterion provides us with a theoretical limit on the maximum bandwidth that can be achieved over which a lossless passive matching network can provide a specified maximum reflection coefficient given the quality factor of the load to be matched. It should be noted that in practice a given matching network will usually provide a bandwidth that is significantly less than the maximum possible bandwidth predicted by the Bode–Fano criterion. P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 11 f f + 00 f f ΔΔ L L − 0 f f m 1 FIGURE 7: Idealized reflection coefficient response for applying Bode–Fano criterion The most useful form of the Bode–Fano criterion may be stated as  f f 0 ≤ π Q 0 · ln  1   m  , (16) where f 0 is the center frequency of the match,  f is the frequency range of the match, Q 0 is the quality factor of the load at f 0 , and  m is the maximum reflection coefficient within the frequency range of the match. Equation (16) is derived assuming the reflection coefficient versus frequency response shown in Fig. 7, and that the fractional bandwidth of the match is small, i.e., f << f 0 . For our example ESA, Eq. (16) predicts that the fractional half-power bandwidth ( ≤  m = 0.7071) achievable at 60 MHz with an ideal passive matching network is 0.042 80. This fractional bandwidth corresponds to an absolute bandwidth of about 2.6 MHz at a center frequency of 60 MHz TWO-PORT MODEL OF AN ANTENNA In many situations it is desirable to model an antenna as a two-port network. Such a model can be used in circuit simulations to compute the overall efficiency of the antenna with a lossy passive matching network, as well as for evaluating the stability of the network that results P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 2007 17:23 12 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS when an antenna is connected to a matching circuit containing non-Foster elements. The model discussed here can be applied to any antenna (with a single feed point) and does not require the assumption of a particular equivalent circuit. Given the input impedance and radiation efficiency of an antenna at a specified frequency (from either simulation or measurements), a two-port representation of the antenna can be derived as follows. Let the complex input impedance of the antenna be denoted by Z a , and the radiation efficiency (as a dimensionless quantity between 0 and 1) be denoted by e cd . Then, we have the equivalent circuit (valid at that specific frequency) shown in Fig. 1 where Z a = R a + jX a = R r + R l + jX a R r = e cd R a = radiation resistance R l = (1 −e cd )R a = dissipative loss resistance X a = 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 =  R r Z 0 , (18) where Z 0 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 Z a = Z 0 1 + S 11 1 − S 11 (19) and its total efficiency is obtained as e tot = | S 21 | =  1 − | S 11 | 2  e cd . (20) In some situations, it may not be practical (or even possible) to determine the radiation efficiency of the antenna.Inthis case,we can usually assume aradiationefficiency of100% (as wehavedone 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 a jX l R 1:N 0 Z 0 Z 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: L 1 = 477 nH L 2 = 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 80 30 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 mag(S(2,1))*100 freq, MHz FIGURE 11: Overall efficiency (in percent) of passive matching network and antenna computed using Agilent ADS . impedance. Matching between source and P1: RVM MOBK060-01 MOBK060-Aberle.cls January 19, 20 07 17 :23 10 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS L Z 0 Z L Matching Network FIGURE 6: Matching network. 19, 20 07 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. January 19, 20 07 17 :23 ANTENNAS WITH NON-FOSTER MATCHING NETWORKS 13 a jX l R 1:N 0 Z 0 Z FIGURE 8: Two-port representation of an antenna (valid at a single frequency) a matching circuit with the