Patent Publication Number: US-2019190149-A1

Title: Radio-Frequency Apparatus with Multi-Band Balun with Improved Performance and Associated Methods

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is related to, and hereby incorporates by reference in its entirety for all purposes, U.S. patent application Ser. No. ______, filed on ______, titled “Radio-Frequency Apparatus with Multi-Band Balun and Associated Methods,” attorney docket number SILA399. 
    
    
     TECHNICAL FIELD 
     The disclosure relates generally to radio-frequency (RF) apparatus and, more particularly, to apparatus for multi-band matching baluns with improved performance, and associated methods. 
     BACKGROUND 
     With the increasing proliferation of wireless technology, such as Wi-Fi, Bluetooth, and mobile or wireless Internet of things (IoT) devices, more devices or systems incorporate RF circuitry, such as receivers and/or transmitters. To reduce the cost, size, and bill of materials, and to increase the reliability of such devices or systems, various circuits or functions have been integrated into integrated circuits (ICs). For example, ICs typically include receiver and/or transmitter circuitry. 
     The RF ICs typically work with circuitry external to the IC to provide a wireless solution. Examples of the external circuitry include baluns, matching circuitry, antennas, filters, switches, and the like. 
     The description in this section and any corresponding figure(s) are included as background information materials. The materials in this section should not be considered as an admission that such materials constitute prior art to the present patent application. 
     SUMMARY 
     A variety of apparatus and associated methods are contemplated according to exemplary embodiments. According to one exemplary embodiment, an apparatus includes an RF apparatus, and a multi-band matching balun coupled to the RF apparatus. The multi-band matching balun includes at least one three-element frequency-dependent resonator (TEFDR) and at most three reactive elements. 
     According to another exemplary embodiment, an apparatus includes an RF apparatus, and a multi-band matching balun coupled to the RF apparatus. The multi-band matching balun includes at least one three-element frequency-dependent resonator and at most three lumped reactive elements (LREs). Neither the at least one three-element frequency-dependent resonator nor the at most three lumped reactive elements includes variable or tunable capacitors or inductors. 
     According to another exemplary embodiment, a method of operating an apparatus, that includes an RF apparatus, includes using a multi-band matching balun coupled to the RF apparatus to provide impedance matching and balun functionality. The multi-band matching balun includes at least one three-element frequency-dependent resonator (TEFDR) and at most three reactive elements. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The appended drawings illustrate only exemplary embodiments and therefore should not be considered as limiting the scope of the application or the claims. Persons of ordinary skill in the art will appreciate that the disclosed concepts lend themselves to other equally effective embodiments. In the drawings, the same numeral designators used in more than one drawing denote the same, similar, or equivalent functionality, components, or blocks. 
         FIG. 1  shows a circuit arrangement using a conventional balun. 
         FIG. 2  shows a circuit arrangement for an RF apparatus with a multi-band matching balun according to an exemplary embodiment. 
         FIG. 3  shows a circuit arrangement for an RF apparatus with a multi-band matching balun according to another exemplary embodiment. 
         FIG. 4  shows a circuit arrangement for an RF apparatus with a multi-band matching balun according to another exemplary embodiment. 
         FIG. 5  shows a circuit arrangement for an RF apparatus with a multi-band matching balun according to another exemplary embodiment. 
         FIG. 6  shows a three-element frequency-dependent resonator according to an exemplary embodiment. 
         FIG. 7  shows a three-element frequency-dependent resonator according to another exemplary embodiment. 
         FIG. 8  shows a three-element frequency-dependent resonator according to another exemplary embodiment. 
         FIG. 9  shows a three-element frequency-dependent resonator according to another exemplary embodiment. 
         FIG. 10  shows a circuit arrangement for a balun with lumped reactive elements. 
         FIG. 11  shows a circuit arrangement for a 12-element multi-band matching balun according to an exemplary embodiment. 
         FIG. 12  shows a circuit arrangement for a 12-element multi-band matching balun according to an exemplary embodiment. 
         FIG. 13  shows a circuit arrangement for a 10-element multi-band matching balun according to an exemplary embodiment. 
         FIG. 14  shows a circuit arrangement for a 10-element multi-band matching balun according to an exemplary embodiment. 
         FIG. 15  shows a circuit arrangement for an 8-element multi-band matching balun according to an exemplary embodiment. 
         FIG. 16  shows a circuit arrangement for an 8-element multi-band matching balun according to an exemplary embodiment. 
         FIG. 17  shows a circuit arrangement for a 6-element multi-band matching balun according to an exemplary embodiment. 
         FIG. 18  shows a circuit arrangement for a 6-element multi-band matching balun according to an exemplary embodiment. 
         FIG. 19  shows a circuit arrangement for a multi-band matching balun with a harmonic trap according to an exemplary embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     The disclosed concepts relate generally to RF apparatus and, more particularly, to multi-band matching baluns, i.e., apparatus that provides the combined functionality of impedance matching circuits (or impedance matching networks or impedance matching circuitry or matching circuit or matching networks or matching circuitry) and baluns, and associated methods. 
     Impedance matching or impedance transformation circuits are typically used in RF apparatus, such as receivers, transmitters, and/or transceivers, to provide an interface or match between circuitry that have different impedances. 
     More specifically, in the case of purely resistive impedances, maximum power transfer takes place when the output impedance of a source circuit equals the input impedance of a load circuit. In the case of complex impedances, maximum power transfer takes place when the input impedance of the load circuit is the complex conjugate of the output impedance of the source circuit. 
     As an example, consider an antenna with a 50-ohm impedance (R=50Ω) coupled to a receive or receiver (RX) circuit with a 50-ohm impedance. In this case, maximum power transfer takes place without the user of an impedance matching circuit because the output impedance of the antenna equals the input impedance of the RX circuit. 
     Now consider the situation where an antenna with a 50-ohm impedance (R=50Ω) coupled to an RX circuit with a 250-ohm impedance. In this case, because the respective impedances of the antenna and the RX circuit are not equal, maximum power transfer does not take place. 
     Use of an impedance matching circuit, however, can match the impedance of the antenna to the impedance of the RX circuit. As a result of using the impedance matching circuit, maximum power transfer from the antenna to the RX circuit takes place. 
     More specifically, the impedance matching circuit is coupled between the antenna and the RX circuit. The impedance matching circuit has two ports, with one port coupled to the antenna, and another port coupled to the RX circuit, respectively. 
     At the port coupled to the antenna, the impedance matching circuit ideally presents a 50-ohm impedance to the antenna. As a result, maximum power transfer takes place between the antenna and the impedance matching circuit. 
     Conversely, at the port coupled to the RX circuit, the impedance matching circuit presents a 250-ohm impedance to the RX circuit. Consequently, maximum power transfer takes place between the impedance matching circuit and the RX circuit. 
     In practice, the impedance matching circuit often fails to perfectly match the impedances. In other words, signal transmission from one network to another is not perfect and 100% of the signal power is not transmitted. As a result, reflection occurs at the interface between circuits or networks with imperfectly matched impedances. 
     The reflection coefficient, S 11 , may serve as one measure or figure of merit for the level of impedance matching. A lower S 11  denotes better power transmission (better impedance matching), and vice-versa. 
     As noted, the optimum load impedance (e.g., input impedance of receive circuitry, such as the input impedance of a low-noise amplifier (LNA)) for matching purposes would be the complex conjugate of the source impedance (e.g., an antenna). However, due to the relatively high-Q (quality factor) of the input impedance of the receive circuitry, conjugate impedance match might prove relatively difficult or even impossible. As a compromise, impedances may be matched for maximum voltage gain, i.e., highest impedance where the input capacitance of the LNA (CLNA) is resonated out by the impedance matching circuitry (multi-band matching balun). 
     Baluns provide a way of interfacing a balanced circuit (e.g., differential input or output) with an unbalanced circuit (e.g., single-ended input or output). Baluns are typically used to interface RX circuits or transmit or transmitter (TX) circuit with differential inputs or outputs, respectively, to a single-ended antenna. 
       FIG. 1  shows a circuit arrangement using a conventional balun  2  coupled to an IC  5 . Balun  2  includes four components or elements, i.e., inductor L 1 , inductor L 12 , capacitor C 1 , and capacitor C 12 . Balun  2  is typically coupled to RX circuitry (not shown) in IC  5 . Balun  2  matches the impedance at the input (say, 50Ω) to the input impedance of the circuitry in IC  5 , such as the input impedance of RX circuitry, typically 100-200Ω. Thus, for circuitry with higher input impedance, use of balun  2  results in relatively high impedance mismatch, which in turns results in power loss. 
     As known to persons of ordinary skill in the art, balun  2  constitutes a single-frequency balun. In other words, balun  2  provides a reasonable S 11  value (say, −10 dB) at a single frequency, or within a single relatively narrow frequency band. 
     Various embodiments according to the disclosure combine the functionality of multi-band matching circuits and the functionality of baluns, i.e., the provide matching baluns. Matching baluns according to various embodiments provide not only impedance matching functionality, but also balun functionality in multiple frequency bands. 
     In exemplary embodiments, matching baluns and associated methods are disclosed. The matching baluns are relatively low cost, may be used with RF receivers, RF transmitters, and/or RF transceivers. Matching baluns according to various embodiments have relatively high Q (quality factor), and differential TX and/or RX ports. 
     Furthermore, matching baluns according to various embodiments may be adapted to various operating frequency ranges, power levels, and RX circuit or RX and TX circuit impedances. In addition, matching baluns according to various embodiments may be used with a variety of RX or RX and TX circuit configurations, as desired. 
     As noted above, matching baluns according to various embodiments realize both the balun and impedance-matching functions (e.g., 50-ohm single-ended to 750-ohm differential) in multiple frequency bands, i.e., the matching baluns can work well simultaneously in two separate bands. In some embodiments, the frequency bands might include 310-370 MHz±10%, 370-434 MHz±10%, and 868-928 MHz±10%. 
     As persons of ordinary skill in the art will understand, however, the disclosed concepts may be used to provide multi-band matching baluns for other frequency bands. Other frequency bands may be accommodated by making appropriate modifications to the component values used in the multi-band matching baluns, as persons of ordinary skill in the art will understand. 
     Multi-band matching baluns according to various embodiments use 12, 10, 8, or 6 elements or components (capacitors, inductors), such as lumped surface mount device (SMD) components (or other lumped components). The components are fixed-value components, i.e., they are not and do not include tunable or variable components (i.e., no inductor or capacitor whose inductance or capacitance, respectively, may be varied or tuned), nor are they switchable components, nor do they use multiple paths (i.e., use of more than one path in the RF front-end matching circuit and, thus, multiple inputs and/or outputs for different frequency bands (plus applying couplers, splitters, diplexers, and/or multiplexers) or use of RF switches), as are used conventionally. 
     In some embodiments, rather than using lumped components, distributed components may be used to realize matching baluns, as desired, and as persons of ordinary skill in the art will understand. Multi-band matching baluns intended for relatively high frequencies, such as over a gigahertz or other desired frequency value, may be realized using distributed components, as persons of ordinary skill in the art will understand. 
     Some exemplary embodiments are described with component values and/or impedance values and/or configurations for particular frequency bands and/or for particular RX and/or TX circuitry. Such embodiments are merely illustrative and are not intended and should not be construed as limiting the disclosed concepts. 
     As persons of ordinary skill in the art will understand, the concepts for multi-band matching baluns are not limited to those exemplary or illustrative frequency values or impedance levels (e.g., input impedance of RX circuitry). Multi-band matching baluns that accommodate other frequency bands and/or impedance values may be designed and realized by making appropriate modifications or designing appropriate multi-band matching baluns, as persons of ordinary skill in the art will understand. 
     As noted above, multi-band matching baluns according to various embodiments may be used in a variety of apparatus.  FIG. 2  shows a circuit arrangement for an RF apparatus with a multi-band matching balun  30  according to an exemplary embodiment. 
     More specifically, The embodiment in  FIG. 2  shows an RF apparatus  35  that has RX functionality, i.e., by using RX circuitry  40 . Antenna  15  receives RF signals and provides the signals to filter  20 . Filter  20  is coupled to matching balun  30 . 
     Multi-band matching balun  30  matches the output impedance of filter  20  to the input impedance of RX circuitry  40 . Multi-band matching balun  30  also provides balun functionality, as described above. More specifically, multi-band matching balun  30  couples or interfaces the single-ended output of filter  20  to the differential input of RX circuitry  40 . 
     Note that filter  20  is optional in various embodiments, and may be omitted. More specifically, filter  20  is typically used if higher selectivity or blocking is desired.  FIG. 3  shows a circuit arrangement for an RF apparatus with a multi-band matching balun  30  according to another exemplary embodiment. In this exemplary embodiment, filter  20  is omitted, and antenna  15  is coupled to multi-band matching balun  30 . 
     Multi-band matching balun  30  matches the impedance of antenna  15  (typically 50Ω) to the input impedance of RX circuitry  40 . Multi-band matching balun  30  also provides balun functionality, as described above. More specifically, multi-band matching balun  30  couples or interfaces the single-ended output of antenna  15  to the differential input of RX circuitry  40 . 
     As noted above, multi-band matching baluns according to various embodiments may also be used in RF apparatus that include TX functionality.  FIG. 4  shows a circuit arrangement for an RF apparatus that includes TX circuitry  45  and multi-band matching balun  30  according to another exemplary embodiment. 
     TX circuitry  45  provides RF signals to be transmitted (typically through a power amplifier (not shown) to multi-band matching balun  30 . Multi-band matching balun  30  is coupled to filter  20 . Filter  20  filters the RF signals, and provides the filtered RF signals to antenna  15 . The filtered RF signals are transmitted via antenna  15 . 
     Multi-band matching balun  30  matches the output impedance of TX circuitry  45  to the input impedance of filter  20 . Multi-band matching balun  30  also provides balun functionality, as described above. More specifically, multi-band matching balun  30  couples or interfaces the single-ended input of filter  20  to the differential output of TX circuitry  45 . 
     Note that filter  20  is optional in various embodiments, and may be omitted. More specifically, filter  20  is typically used if the TX mode of operation generates higher harmonics than allowed by the applicable or desired standards or are desired. 
       FIG. 5  shows a circuit arrangement for an RF apparatus with a multi-band matching balun  30  according to another exemplary embodiment. In this exemplary embodiment, filter  20  is omitted, and antenna  15  is coupled to multi-band matching balun  30 . 
     Multi-band matching balun  30  matches the impedance of antenna  15  (typically 50Ω) to the output impedance of TX circuitry  45 . Multi-band matching balun  30  also provides balun functionality, as described above. More specifically, multi-band matching balun  30  couples or interfaces the single-ended input of antenna  15  to the differential output of TX circuitry  45 . 
     Note that multi-band matching balun  30  may also be used in RF apparatus that have both RX and TX functionality, i.e., transceivers. In such embodiments, various techniques, such as RX-TX switches and/or direct coupling of RX circuitry and TX circuitry to balun/impedance matching circuitry may be used, as persons of ordinary skill in the art will understand. One or more multi-band matching baluns  30  may be used to provide impedance-matching and balun functionality, as described above. 
     One aspect of the disclosure relates to the use of three-element frequency-dependent resonators (TEFDRs) to realize multi-band matching baluns. In various embodiments, one or more three-element frequency-dependent resonators may be used to replace corresponding component(s) in a balun. 
     Generally speaking, and as described in more details in connection with the exemplary embodiments shown in the figures, multi-band matching baluns according to various embodiments may use one, two, three, or four TEFDRs, or at least one TEFDR. In addition, the multi-band matching baluns may use zero, one, two, or three reactive lumped elements (LREs), such as inductors or capacitors (e.g., balun  2  in  FIG. 1  uses four LREs, i.e., two capacitors, and two inductors), or at most three LREs. 
     The three-element frequency-dependent resonators use fixed-value (fixed capacitance or inductance value, not variable, tunable, or switchable) components, such as lumped capacitors and/or inductors. The three-element frequency-dependent resonators provide the functionality of an inductor or capacitor. Unlike a fixed-value inductor or capacitor, however, the inductance or capacitance of three-element frequency-dependent resonators varies as a function of frequency without the use of variable, tunable, or switchable inductors or capacitors. 
     Note that, as described above, rather than using exclusively lumped elements, other arrangements may be used to realize reactive components (inductors, capacitors) in multi-band matching baluns according to various embodiments. Thus, in some embodiments, distributed reactive elements (DREs), such as distributed inductors and/or capacitors, may be used in multi-band matching baluns instead of LREs. In some embodiments, a combination of LREs and DREs may be used, as desired, to implement multi-band matching baluns. Note further that the reactive components or elements may be used either individually (e.g., a capacitor or inductor in a multi-band matching balun), or in combination with other reactive elements to realize TEFDRs (e.g., a capacitor and two inductors used to realize a TEFDR that is used in a multi-band matching balun). 
       FIGS. 6-9  show three-element frequency-dependent resonators according to exemplary embodiments. More specifically,  FIGS. 6-7  show three-element frequency-dependents that may be used to replace an inductor in a balun. Conversely,  FIGS. 8-9  show three-element frequency-dependents that may be used to replace a capacitor in a balun, such as balun  2  shown in  FIG. 1 , in order to realize multi-band matching baluns. 
     The three-element frequency-dependent resonators can be used as frequency-dependent components (with lower inductance or capacitance at a higher band) and thus they are able to serve as building blocks of more complex circuits, which resonate simultaneously in multiple frequency bands. As a result, by using three-element frequency-dependent resonators, multi-band matching baluns may be realized, as described below in detail. 
     Referring to  FIG. 6 , a three-element frequency-dependent resonator  60  is shown coupled between points A and B. Three-element frequency-dependent resonator  60  includes capacitor C 61 , inductor L 61 , and inductor L 62 , all of which have fixed (not variable, tunable, or switchable) capacitance and inductance values, respectively, and may be realized by using lumped components, as described above. 
     Capacitor C 61  is coupled to point A and also in series with inductor L 61 . Inductor L 61  is also coupled to point B. Inductor L 62  is coupled between points A and B, i.e., in parallel with the series combination of capacitor C 61  and inductor L 61 . 
     As shown, the three-element frequency-dependent resonator  60  is equivalent to an inductor L(f), where “(f)” denotes dependence of inductance on frequency, i.e., the inductance of L(f) is a function of frequency. Unlike a traditional inductor, the inductance of three-element frequency-dependent resonator  60  varies as a function of frequency. The following formulas provide a technique for calculating the values of the components in three-element frequency-dependent resonator  60  (with L 1  corresponding to L 62 , C 3  corresponding to C 61 , and L 3  corresponding to L 61  in the figure): 
     
       
         
           
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     where Z, j, f, L 1 , L 3 , C 3 , and w denote, respectively, impedance, the imaginary unit, frequency, the inductance of the L 1  component, the inductance of the L 3  component, the capacitance of the C 3  component, and the angular frequency. X 1  denotes the desired inductance at f 1 , and X 2  denotes the desired inductance at f 2 , where f 1 &lt;f 2 , X 1 &gt;X 2 , and Z=jωX, and where ω 1 &lt;ω o &lt;ω 2 . The quantity f o  is the series resonant point, which should be at the frequency (f 1 +f 2 )/2, but other values—between f 1  and f 2 —may be used, as desired. 
     Referring to  FIG. 7 , a three-element frequency-dependent resonator  60  is shown coupled between points A and B. Three-element frequency-dependent resonator  60  includes capacitor C 71 , inductor L 71 , and inductor L 72 , all of which have fixed (not variable, tunable, or switchable) capacitance and inductance values, respectively, and may be realized by using lumped components, as described above. 
     The parallel combination of capacitor C 71  and inductor L 71  is coupled to inductor L 72  and to point B. Inductor L 72  is also coupled to point A. 
     As shown, the three-element frequency-dependent resonator is equivalent to an inductor L(f), where “(f)” denotes dependence of inductance on frequency, i.e., the inductance of L(f) is a function of frequency. Unlike a traditional inductor, the inductance of three-element frequency-dependent resonator  60  varies as a function of frequency. The following formulas provide a technique for calculating the values of the components in three-element frequency-dependent resonator  60  (with L 1  corresponding to L 72 , C 3  corresponding to C 71 , and L 3  corresponding to L 71  in the figure): 
     
       
         
           
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     where Z, j, f, L 1 , L 3 , C 3 , and w denote, respectively, impedance, the imaginary unit, frequency, the inductance of the L 1  component, the inductance of the L 3  component, the capacitance of the C 3  component, and the angular frequency. X 1  denotes the desired inductance at f 1 , and X 2  denotes the desired inductance at f 2 , where f 1 &lt;f 2 , X 1 &gt;X 2 , and Z=jωX, and where ω 1 &lt;ω o &lt;ω 2 . The quantity f o  is the parallel resonant point, which should be at the frequency (f 1 +f 2 )/2, but other values—between f 1  and f 2 —may be used, as desired. 
     As noted above,  FIGS. 8-9  show three-element frequency-dependents that may be used to replace a capacitor, e.g., a capacitor used in a balun, such as balun  2  in  FIG. 1 . Referring to  FIG. 8 , a three-element frequency-dependent resonator  60  is shown coupled between points A and B. Three-element frequency-dependent resonator  60  includes capacitor C 81 , inductor L 81 , and capacitor C 82 , all of which have fixed (not variable, tunable, or switchable) capacitance and inductance values, respectively, and may be realized by using lumped components, as described above. 
     Capacitor C 81  is coupled to point A and also in series with inductor L 81 . Inductor L 81  is also coupled to point B. Capacitor C 82  is coupled between points A and B, i.e., in parallel with the series combination of capacitor C 81  and inductor L 81 . 
     As shown, three-element frequency-dependent resonator  60  is equivalent to a capacitor C. However, unlike a traditional capacitor, the capacitance of the three-element frequency-dependent resonator varies as a function of frequency. The values of capacitor C 81 , inductor L 81 , and capacitor C 82  may be calculated as follows (with C 1  corresponding to C 82 , C 3  corresponding to C 81 , and L 3  corresponding to L 81  in the figure): 
     
       
         
           
             Y 
             = 
             
               j 
               * 
               2 
                
               π 
                
               
                   
               
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               f 
               * 
               
                 ( 
                 
                   
                     C 
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                     1 
                   
                   + 
                   
                     
                       C 
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                       1 
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                             ( 
                             
                               2 
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                                
                               f 
                             
                             ) 
                           
                           2 
                         
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                         L 
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                         3 
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                         C 
                          
                         
                             
                         
                          
                         3 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               ω 
               x 
             
             = 
             
               2 
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               π 
                
               
                   
               
                
               
                 f 
                 x 
               
             
           
         
       
       
         
           
             
               C 
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             = 
             
               
                 
                   B 
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                 - 
                 B2 
               
               
                 
                   
                     ω 
                     0 
                     2 
                   
                   
                     
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                       ω 
                       1 
                       2 
                     
                   
                 
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                     ω 
                     0 
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                       ω 
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                       ω 
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               C 
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             = 
             
               
                 B 
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                 C 
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                 3 
                 * 
                 
                   
                     ω 
                     0 
                     2 
                   
                   
                     
                       ω 
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                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               L 
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               3 
             
             = 
             
               1 
               
                 
                   ω 
                   0 
                   2 
                 
                 * 
                 C 
                  
                 
                     
                 
                  
                 3 
               
             
           
         
       
     
     where Y, j, f, C 1 , L 3 , C 3 , and w denote, respectively, admittance, the imaginary unit, frequency, the capacitance of the C 1  component, the inductance of the L 3  component, the capacitance of the C 3  component, and the angular frequency. B 1  denotes the desired capacitance at f 1 , and B 2  denotes the desired capacitance at f 2 , where f 1 &lt;f 2 , B 1 &gt;B 2 , and Y=jωB, and where ω 1 &lt;ω o &lt;ω 2 . The quantity f o  is the series resonant point, which should be at the frequency (f 1 +f 2 )/2, but other values—between f 1  and f 2 —may be used, as desired. 
     Referring to  FIG. 9 , a three-element frequency-dependent resonator  60  is shown coupled between points A and B. Three-element frequency-dependent resonator  60  includes capacitor C 91 , inductor L 91 , and capacitor C 92 , all of which have fixed (not variable, tunable, or switchable) capacitance and inductance values, respectively, and may be realized by using lumped components, as described above. 
     The parallel combination of capacitor C 92  and inductor L 91  is coupled to capacitor C 91  and to point B. Capacitor C 91  is also coupled to point A. 
     As shown, three-element frequency-dependent resonator  60  is equivalent to a capacitor C(f), where “(f)” denotes dependence of capacitance on frequency, i.e., the capacitance of C(f) is a function of frequency. Unlike a traditional capacitor, the capacitance of three-element frequency-dependent resonator  60  varies as a function of frequency. The values of capacitor C 91 , inductor L 91 , and capacitor C 92  may be calculated as follows (with C 1  corresponding to C 91 , C 3  corresponding to C 92 , and L 3  corresponding to L 91  in the figure): 
     
       
         
           
             Y 
             = 
             
               j 
               * 
               2 
                
               π 
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                
               f 
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               C 
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               1 
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                         ( 
                         
                           2 
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                           π 
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                            
                           f 
                         
                         ) 
                       
                       2 
                     
                     * 
                     L 
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                      
                     3 
                     * 
                     C 
                      
                     
                         
                     
                      
                     3 
                   
                 
                 
                   1 
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                         ( 
                         
                           2 
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                           π 
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                            
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                         ) 
                       
                       2 
                     
                     * 
                     L 
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                      
                     3 
                     * 
                     
                       ( 
                       
                         
                           C 
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                            
                           1 
                         
                         + 
                         
                           C 
                            
                           
                               
                           
                            
                           3 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               ω 
               x 
             
             = 
             
               2 
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               π 
                
               
                   
               
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                 f 
                 x 
               
             
           
         
       
       
         
           
             
               C 
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               3 
             
             = 
             
               
                 
                   
                     B 
                      
                     
                         
                     
                      
                     1 
                   
                   - 
                   B2 
                 
                 
                   
                     B 
                      
                     
                         
                     
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                     ω 
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                     ω 
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                   + 
                   
                     
                       
                         ω 
                         1 
                         2 
                       
                        
                       
                         ω 
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                         2 
                       
                     
                     
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               C 
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                     ω 
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                   + 
                   
                     
                       
                         ω 
                         1 
                         2 
                       
                       * 
                       B 
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                        
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               L 
                
               
                   
               
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               3 
             
             = 
             
               1 
               
                 
                   ω 
                   0 
                   2 
                 
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                 C 
                  
                 
                     
                 
                  
                 3 
               
             
           
         
       
     
     where Y, j, f, C 1 , L 3 , C 3 , and w denote, respectively, admittance, the imaginary unit, frequency, the capacitance of the C 1  component, the inductance of the L 3  component, the capacitance of the C 3  component, and the angular frequency. B 1  denotes the desired capacitance at f 1 , and B 2  denotes the desired capacitance at f 2 , where f 1 &lt;f 2 , B 1 &gt;B 2 , and Y=jωB, and where w 1 &lt;w o &lt;ω 2 . The quantity f o  is the parallel resonant point, which should be at the frequency (f 1 +f 2 )/2, but other values—between f 1  and f 2 —may be used, as desired. 
     As noted above, three-element frequency-dependent resonators  60  may be used to implement multi-band matching baluns. As noted above, in various embodiments, one through four TEFDRs and zero through three LREs (or reactive elements or components, generally) are used to realize multi-band matching baluns. 
     Generally speaking, multi-band matching baluns according to various embodiments are obtained by replacing one or more LREs in the circuit shown in  FIG. 10 . More specifically, balun  2  in  FIG. 10  illustrates the balun of  FIG. 1 , but with the individual capacitors and inductors replaced with LREs. By replacing one, two, three, or four of the LREs with TEFDRs, multi-band matching baluns according to various embodiments are obtained. 
     One aspect of the disclosure relates to 12-element multi-band matching baluns. Twelve-element multi-band matching baluns are obtained by replacing all four LREs in  FIG. 10  with TEFDRs. The TEFDRs may be any of the TEFDRs shown in  FIGS. 6-9 . Depending on the choice of TEFDRs to replace the LREs in  FIG. 10 , multi-band matching baluns with different and/or desired characteristics (e.g., frequency bands of operation) may be obtained. 
       FIG. 11  shows a circuit arrangement for a 12-element multi-band matching balun  30  according to an exemplary embodiment. In this embodiment, all four LREs (i.e., LRE 1 -LRE 4 ) in  FIG. 10  are replaced with four corresponding TEFDRs (i.e., TEFDR 1 -TEFDR 4 ). 
     Referring to  FIG. 11 , multi-band matching balun  30  is coupled to port P 1  (e.g., single-ended antenna, filter, etc., with a given impedance, such as 50Ω). Multi-band matching balun  30  is also coupled to RF apparatus  35 , more specifically, to the input of RX circuitry  40 . 
     In the embodiment shown, the input of RX circuitry  40  includes LNA  50 . The input circuit of LNA  50  is represented by a resistor RLNA (e.g., 750Ω) in parallel with a capacitor CLNA (e.g., 1.1 pF). Thus, multi-band matching balun  30  provides impedance matching between the impedance presented at port P 1  and the input impedance of LNA  50 . 
     As noted, twelve-element multi-band matching baluns are obtained by replacing all four LREs in  FIG. 10  with TEFDRs to realize the circuit in  FIG. 11 . The TEFDRs may be any of the TEFDRs shown in  FIGS. 6-9 . Depending on the choice of TEFDRs to replace the LREs in  FIG. 10 , multi-band matching baluns with different and/or desired characteristics (e.g., frequency bands of operation) may be obtained. 
     As merely one example,  FIG. 12  shows a circuit arrangement for a 12-element multi-band matching balun  30  according to an exemplary embodiment. The multi-band matching balun shown in the exemplary embodiment of  FIG. 12  is obtained by: (i) replacing LRE 1  in  FIG. 10  (or inductor L 1  in  FIG. 1 ) with TEFDR  60  shown in  FIG. 6 ; (ii) replacing LRE 2  in  FIG. 10  (or capacitor C 1  in  FIG. 1 ) with TEFDR  60  shown in  FIG. 8 ; (iii) replacing LRE 3  in  FIG. 10  (or inductor L 2  in  FIG. 1 ) with TEFDR  60  shown in  FIG. 6 ; and (iv) replacing LRE 4  in  FIG. 10  (or capacitor C 2  in  FIG. 1 ) with TEFDR  60  shown in  FIG. 8 . 
     Multi-band matching balun  30  in  FIG. 12  illustrates merely one example of a 12-element multi-band matching balun, which uses TEFDRs  60  shown in  FIGS. 6 and 8 , but not TEFDRs  60  shown in  FIGS. 7 and 9 . Other possibilities exist and are contemplated for realizing 12-element multi-band matching baluns. For example, in some embodiments, rather than using TEFDR  60  shown in  FIG. 6  to realize inductor L 1  (and/or inductor L 2 ) in  FIG. 1 , TEFDR  60  shown in  FIG. 7  may be used. As another example, in some embodiments, TEFDR  60  shown in  FIG. 6  may be used to realize inductor L 1  in  FIG. 1 , and TEFDR  60  shown in  FIG. 7  may be used to realize inductor L 2 . 
     Similar options exist with respect to realizing (or replacing) capacitors. For example, in some embodiments, rather than using TEFDR  60  shown in  FIG. 8  to realize inductor C 1  (and/or inductor L 2 ) in  FIG. 1 , TEFDR  60  shown in  FIG. 9  may be used. As another example, in some embodiments, TEFDR  60  shown in  FIG. 8  may be used to realize capacitor C 1  in  FIG. 1 , and TEFDR  60  shown in  FIG. 8  may be used to realize capacitor C 2 . Similar possibilities exist and are contemplated with respect to 10-element, 8-element, and 6-element multi-band baluns, described below. 
     One aspect of the disclosure relates to 10-element multi-band matching baluns. Ten-element multi-band matching baluns are obtained by replacing three of four LREs in  FIG. 10  with three corresponding TEFDRs. The TEFDRs may be any of the TEFDRs shown in  FIGS. 6-9 . Depending on the choice of TEFDRs to replace an LRE in  FIG. 10 , multi-band matching baluns with different and/or desired characteristics (e.g., frequency bands of operation) may be obtained. 
       FIG. 13  shows a circuit arrangement for a 10-element multi-band matching balun  30  according to an exemplary embodiment. In this embodiment, LRE 1  is not replaced with a corresponding TEFDR. The remaining LREs, i.e., LRE 2 -LRE 4  in  FIG. 10 , are replaced with three corresponding TEFDRs (i.e., TEFDR 2 -TEFDR 4 ) to realize multi-band matching balun  30  in  FIG. 13 . 
     Referring to  FIG. 13 , multi-band matching balun  30  is coupled to port P 1  (e.g., single-ended antenna, filter, etc., with a given impedance, such as 50Ω). Multi-band matching balun  30  is also coupled to RF apparatus  35 , more specifically, to the input of RX circuitry  40 . 
     In the embodiment shown, the input of RX circuitry  40  includes LNA  50 . The input circuit of LNA  50  is represented by a resistor RLNA (e.g., 750Ω) in parallel with a capacitor CLNA (e.g., 1.1 pF). Thus, multi-band matching balun  30  provides impedance matching between the impedance presented at port P 1  and the input impedance of LNA  50 . 
     As noted, 10-element multi-band matching baluns are obtained by replacing three of the four LREs in  FIG. 10  with TEFDRs to realize the circuit in  FIG. 13 . The TEFDRs may be any of the TEFDRs shown in  FIGS. 6-9 . Depending on the choice of TEFDRs to replace the LREs in  FIG. 10 , multi-band matching baluns with different and/or desired characteristics (e.g., frequency bands of operation) may be obtained. 
     As merely one example,  FIG. 14  shows a circuit arrangement for a 10-element multi-band matching balun  30  according to an exemplary embodiment. The multi-band matching balun shown in the exemplary embodiment of  FIG. 14  is obtained by: (i) replacing LRE 2  in  FIG. 10  (or capacitor C 1  in  FIG. 1 ) with TEFDR  60  shown in  FIG. 8 ; (ii) replacing LRE 3  in  FIG. 10  (or inductor L 2  in  FIG. 1 ) with TEFDR  60  shown in  FIG. 6 ; and (ii) replacing LRE 4  in  FIG. 10  (or capacitor C 2  in  FIG. 1 ) with TEFDR  60  shown in  FIG. 8 . Note that inductor L 1  (LRE 1 ) has not been replaced with a TEFDR. 
     Multi-band matching balun  30  in  FIG. 14  illustrates merely one example of a 10-element multi-band matching balun, which uses TEFDRs  60  shown in  FIGS. 6 and 8 , but not TEFDRs  60  shown in  FIGS. 7 and 9 . Other possibilities and varieties exist and are contemplated for realizing 10-element multi-band matching baluns, as described above with respect to 12-element multi-band matching baluns. 
     Generally speaking, 10-element multi-band matching baluns may be realized as follows (references are to the reactive components in  FIG. 1 ):
         L 1  is kept as a single reactive component, and C 1 , L 2 , and C 2  are replaced with corresponding TEFDRs selected from the embodiments shown in  FIGS. 6-9 ;   L 2  is kept as a single reactive component, and C 1 , L 1 , and C 2  are replaced with corresponding TEFDRs selected from the embodiments shown in  FIGS. 6-9 ;   C 1  is kept as a single reactive component, and L 1 , L 2 , and C 2  are replaced with corresponding TEFDRs selected from the embodiments shown in  FIGS. 6-9 ; or   C 2  is kept as a single reactive component, and C 1 , L 2 , and L 1  are replaced with corresponding TEFDRs selected from the embodiments shown in  FIGS. 6-9 .
 
As noted above, the TEFDRs selected to replace the corresponding LREs may be selected from TEFDRs  60  shown in  FIGS. 6-9 . Depending on the choice of TEFDRs, various multi-band matching balun with different characteristics (e.g., frequency bands of operation) may be obtained.
       

     One aspect of the disclosure relates to 8-element multi-band matching baluns. Eight-element multi-band matching baluns are obtained by replacing two of four LREs in  FIG. 10  with two corresponding TEFDRs. The TEFDRs may be any of the TEFDRs shown in  FIGS. 6-9 . Depending on the choice of TEFDRs to replace an LRE in  FIG. 10 , multi-band matching baluns with different and/or desired characteristics (e.g., frequency bands of operation) may be obtained. 
       FIG. 15  shows a circuit arrangement for an 8-element multi-band matching balun  30  according to an exemplary embodiment. In this embodiment, LRE 3  and LRE 4  are not replaced with corresponding TEFDRs. The remaining LREs, i.e., LRE 1 -LRE 2  in  FIG. 10 , are replaced with four corresponding TEFDRs (i.e., TEFDR 2 -TEFDR 4 ) to realize multi-band matching balun  30  in  FIG. 15 . 
     Referring to  FIG. 15 , multi-band matching balun  30  is coupled to port P 1  (e.g., single-ended antenna, filter, etc., with a given impedance, such as 50Ω). Multi-band matching balun  30  is also coupled to RF apparatus  35 , more specifically, to the input of RX circuitry  40 . 
     In the embodiment shown, the input of RX circuitry  40  includes LNA  50 . The input circuit of LNA  50  is represented by a resistor RLNA (e.g., 750Ω) in parallel with a capacitor CLNA (e.g., 1.1 pF). Thus, multi-band matching balun  30  provides impedance matching between the impedance presented at port P 1  and the input impedance of LNA  50 . 
     As noted, 8-element multi-band matching baluns are obtained by replacing two of the four LREs in  FIG. 10  with TEFDRs to realize the circuit in  FIG. 15 . The TEFDRs may be any of the TEFDRs shown in  FIGS. 6-9 . Depending on the choice of TEFDRs to replace the LREs in  FIG. 10 , multi-band matching baluns with different and/or desired characteristics (e.g., frequency bands of operation) may be obtained. 
     As merely one example,  FIG. 16  shows a circuit arrangement for an 8-element multi-band matching balun  30  according to an exemplary embodiment. The multi-band matching balun shown in the exemplary embodiment of  FIG. 16  is obtained by: (i) replacing LRE 1  in  FIG. 10  (or inductor L 1  in  FIG. 1 ) with TEFDR  60  shown in  FIG. 6 ; (ii) replacing LRE 2  in  FIG. 10  (or capacitor C 1  in  FIG. 1 ) with TEFDR  60  shown in  FIG. 8 . Note that inductor L 2  (LRE 3 ) and C 2  (LRE 4 ) have not been replaced with TEFDRs. 
     Multi-band matching balun  30  in  FIG. 16  illustrates merely one example of an 8-element multi-band matching balun, which uses TEFDRs  60  shown in  FIGS. 6 and 8 , but not TEFDRs  60  shown in  FIGS. 7 and 9 . Other possibilities and varieties exist and are contemplated for realizing 8-element multi-band matching baluns, as described above with respect to 12-element multi-band matching baluns. 
     Generally speaking, 8-element multi-band matching baluns may be realized as follows (references are to the reactive components in  FIG. 1 ):
         L 1  and C 1  are kept as single reactive components, and L 2  and C 2  are replaced with corresponding TEFDRs selected from the embodiments shown in  FIGS. 6-9 ;   L 1  and C 2  are kept as single reactive components, and L 2  and C 1  are replaced with corresponding TEFDRs selected from the embodiments shown in  FIGS. 6-9 ;   L 1  and L 2  are kept as single reactive components, and C 1  and C 2  are replaced with corresponding TEFDRs selected from the embodiments shown in  FIGS. 6-9 ;   L 2  and C 1  are kept as single reactive components, and L 1  and C 2  are replaced with corresponding TEFDRs selected from the embodiments shown in  FIGS. 6-9 ;   L 2  and C 2  are kept as single reactive components, and L 1  and C 1  are replaced with corresponding TEFDRs selected from the embodiments shown in  FIGS. 6-9 ; or   C 1  and C 2  are kept as single reactive components, and L 1  and L 2  are replaced with corresponding TEFDRs selected from the embodiments shown in  FIGS. 6-9 .
 
As noted above, the TEFDRs selected to replace the corresponding LREs may be selected from TEFDRs  60  shown in  FIGS. 6-9 . Depending on the choice of TEFDRs, various multi-band matching balun with different characteristics (e.g., frequency bands of operation) may be obtained.
       

     One aspect of the disclosure relates to 6-element multi-band matching baluns. Six-element multi-band matching baluns are obtained by replacing one of four LREs in  FIG. 10  with a corresponding TEFDR. The TEFDR may be any of the TEFDRs shown in  FIGS. 6-9 . Depending on the choice of TEFDR to replace an LRE in  FIG. 10 , multi-band matching baluns with different and/or desired characteristics (e.g., frequency bands of operation) may be obtained. 
       FIG. 17  shows a circuit arrangement for a 6-element multi-band matching balun  30  according to an exemplary embodiment. In this embodiment, LRE 2 , LRE 3 , and LRE 4  are not replaced with corresponding TEFDRs. The remaining LRE, i.e., LRE 1  in  FIG. 10 , is replaced with a corresponding TEFDR 1  to realize multi-band matching balun  30  in  FIG. 17 . 
     Referring to  FIG. 17 , multi-band matching balun  30  is coupled to port P 1  (e.g., single-ended antenna, filter, etc., with a given impedance, such as 50Ω). Multi-band matching balun  30  is also coupled to RF apparatus  35 , more specifically, to the input of RX circuitry  40 . 
     In the embodiment shown, the input of RX circuitry  40  includes LNA  50 . The input circuit of LNA  50  is represented by a resistor RLNA (e.g., 750Ω) in parallel with a capacitor CLNA (e.g., 1.1 pF). Thus, multi-band matching balun  30  provides impedance matching between the impedance presented at port P 1  and the input impedance of LNA  50 . 
     As noted, 6-element multi-band matching baluns are obtained by replacing one of the four LREs in  FIG. 10  with a corresponding TEFDR to realize the circuit in  FIG. 17 . The TEFDR may be any of the TEFDRs shown in  FIGS. 6-9 . Depending on the choice of TEFDR to replace the LRE in  FIG. 10 , multi-band matching baluns with different and/or desired characteristics (e.g., frequency bands of operation) may be obtained. 
     As merely one example,  FIG. 18  shows a circuit arrangement for a 6-element multi-band matching balun  30  according to an exemplary embodiment. The multi-band matching balun shown in the exemplary embodiment of  FIG. 18  is obtained by replacing LRE 1  in  FIG. 10  (or inductor L 1  in  FIG. 1 ) with TEFDR  60  shown in  FIG. 6 . Note that capacitor C 1  (LRE 2 ), inductor L 2  (LRE 3 ), and capacitor C 2  (LRE 4 ) have not been replaced with TEFDRs. 
     Multi-band matching balun  30  in  FIG. 18  illustrates merely one example of a 6-element multi-band matching balun, which uses TEFDRs  60  shown in  FIG. 6 , but not TEFDRs  60  shown in  FIGS. 7, 8, and 9 . Other possibilities and varieties exist and are contemplated for realizing 6-element multi-band matching baluns, as described above with respect to 12-element multi-band matching baluns. 
     Generally speaking, 6-element multi-band matching baluns may be realized as follows (references are to the reactive components in  FIG. 1 ):
         L 1 , L 2 , and C 1  are kept as single reactive components, and C 2  is replaced with a corresponding TEFDR selected from the embodiments shown in  FIGS. 8-9 ;   L 1 , L 2 , and C 2  are kept as single reactive components, and C 1  is replaced with a corresponding TEFDR selected from the embodiments shown in  FIGS. 8-9 ;   L 1 , C 2 , and C 1  are kept as single reactive components, and L 2  is replaced with a corresponding TEFDR selected from the embodiments shown in  FIGS. 6-7 ; or   L 2 , C 2 , and C 1  are kept as single reactive components, and L 1  is replaced with a corresponding TEFDR selected from the embodiments shown in  FIGS. 6-7 .
 
As noted above, the TEFDR selected to replace the corresponding LRE may be selected from TEFDRs  60  shown in  FIGS. 6-9 . Depending on the choice of TEFDR, various multi-band matching balun with different characteristics (e.g., frequency bands of operation) may be obtained.
       

     One aspect of the disclosure relates to harmonic traps, specifically, harmonic traps added to multi-band matching baluns.  FIG. 19  shows a circuit arrangement for a multi-band matching balun  30  with a harmonic trap according to an exemplary embodiment. The multi-band matching balun is obtained by replacing LRE 1  in  FIG. 10  (or inductor L 1  in  FIG. 1 ) with TEFDR  60  (in this example, using TEFDR  60  shown in  FIG. 6 ). 
     Referring again to  FIG. 19 , capacitor  192  is also coupled between an internal node of TEFDR  60  and circuit ground. The internal node of TEFDR  60  is a node obtained by breaking a single inductor in TEFDR  60  into two series-coupled inductors, i.e., inductor L 191  and inductor L 192  (typically, inductors L 191  and L 192  have the same inductance). TEFDR  60  also includes inductor L 193  and capacitor C 191 . 
     Breaking or dividing the single inductor in TEFDR  60  into inductor L 191  and L 192  allows obtaining a symmetric point. The symmetric point serves as virtual ground reference for the TEFDR load. 
     The combination L 191 -C 192  and the combination L 192 -C 192  have equal tuned resonances at the second-harmonic of the middle frequency of the higher frequency band of operation of the multi-band matching balun. As a result, the common-mode 2nd-harmonic signals is shunted to circuit ground. Consequently, common-mode suppression of multi-band matching balun and, therefore, the fundamental operation in the higher band of frequency for the multi-band matching balun is improved. 
     Referring to the figures, persons of ordinary skill in the art will note that the various blocks shown might depict mainly the conceptual functions and signal flow. The actual circuit implementation might or might not contain separately identifiable hardware for the various functional blocks and might or might not use the particular circuitry shown. For example, one may combine the functionality of various blocks into one circuit block, as desired. Furthermore, one may realize the functionality of a single block in several circuit blocks, as desired. The choice of circuit implementation depends on various factors, such as particular design and performance specifications for a given implementation. Other modifications and alternative embodiments in addition to the embodiments in the disclosure will be apparent to persons of ordinary skill in the art. Accordingly, the disclosure teaches those skilled in the art the manner of carrying out the disclosed concepts according to exemplary embodiments, and is to be construed as illustrative only. Where applicable, the figures might or might not be drawn to scale, as persons of ordinary skill in the art will understand. 
     The particular forms and embodiments shown and described constitute merely exemplary embodiments. Persons skilled in the art may make various changes in the shape, size and arrangement of parts without departing from the scope of the disclosure. For example, persons skilled in the art may substitute equivalent elements for the elements illustrated and described. Moreover, persons skilled in the art may use certain features of the disclosed concepts independently of the use of other features, without departing from the scope of the disclosure.