Patent Publication Number: US-2019190482-A1

Title: Radio-Frequency Apparatus with Multi-Band Balun 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 “Apparatus for Balun with Improved Performance and Associated Methods,” attorney docket number SILA402. 
    
    
     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 a plurality of capacitors and a plurality of inductors. None of the plurality of capacitors and none of the plurality of inductors is variable or tunable. 
     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. 
     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 a plurality of capacitors and a plurality of inductors. None of the plurality of capacitors and none of the plurality of inductors is variable or tunable. 
    
    
     
       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 7-element multi-band matching balun according to an exemplary embodiment. 
         FIG. 11  shows a circuit arrangement for a 7-element multi-band matching balun according to another exemplary embodiment. 
         FIG. 12  shows a circuit arrangement for a 5-element multi-band matching balun 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 11 , inductor L 12 , capacitor C 11 , 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. 
     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. 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 5 or 7 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 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 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. 
     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. 
       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 ω 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. X1 denotes the desired inductance at f 1 , and X2 denotes the desired inductance at f 2 , where f 1 &lt;f 2 , X1&gt;X2, and Z=jωX, and where ω l &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 ω 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. X1 denotes the desired inductance at f 1 , and X2 denotes the desired inductance at f 2 , where f 1 &lt;f 2 , X1&gt;X2, and Z=jωX, and where ω l &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): 
     
       
         
           
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                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               C 
                
               
                   
               
                
               1 
             
             = 
             
               
                 B 
                  
                 
                     
                 
                  
                 2 
               
               - 
               
                 C 
                  
                 
                     
                 
                  
                 3 
                 * 
                 
                   
                     ω 
                     0 
                     2 
                   
                   
                     
                       ω 
                       0 
                       2 
                     
                     - 
                     
                       ω 
                       2 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               L 
                
               
                   
               
                
               3 
             
             = 
             
               1 
               
                 
                   ω 
                   0 
                   2 
                 
                 * 
                 C 
                  
                 
                     
                 
                  
                 3 
               
             
           
         
       
     
     where Y, j, f, C 1 , L 3 , C 3 , and ω 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 l &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 
                
               
                   
               
                
               f 
               * 
               C 
                
               
                   
               
                
               1 
               * 
               
                 
                   1 
                   - 
                   
                     
                       
                         ( 
                         
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           f 
                         
                         ) 
                       
                       2 
                     
                     * 
                     L 
                      
                     
                         
                     
                      
                     3 
                     * 
                     C 
                      
                     
                         
                     
                      
                     3 
                   
                 
                 
                   1 
                   - 
                   
                     
                       
                         ( 
                         
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           f 
                         
                         ) 
                       
                       2 
                     
                     * 
                     L 
                      
                     
                         
                     
                      
                     3 
                     * 
                     
                       ( 
                       
                         
                           C 
                            
                           
                               
                           
                            
                           1 
                         
                         + 
                         
                           C 
                            
                           
                               
                           
                            
                           3 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               ω 
               x 
             
             = 
             
               2 
                
               π 
                
               
                   
               
                
               
                 f 
                 x 
               
             
           
         
       
       
         
           
             
               C 
                
               
                   
               
                
               3 
             
             = 
             
               
                 
                   B 
                    
                   
                       
                   
                    
                   1 
                   * 
                   B 
                    
                   
                       
                   
                    
                   2 
                 
                 
                   
                     B 
                      
                     
                         
                     
                      
                     1 
                   
                   - 
                   
                     B 
                      
                     
                         
                     
                      
                     2 
                   
                 
               
               * 
               
                 
                   
                     ω 
                     1 
                     2 
                   
                   - 
                   
                     ω 
                     2 
                     2 
                   
                 
                 
                   
                     ω 
                     0 
                     2 
                   
                   - 
                   
                     ω 
                     1 
                     2 
                   
                   - 
                   
                     ω 
                     2 
                     2 
                   
                   + 
                   
                     
                       
                         ω 
                         1 
                         2 
                       
                        
                       
                         ω 
                         2 
                         2 
                       
                     
                     
                       ω 
                       0 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               C 
                
               
                   
               
                
               1 
             
             = 
             
               B 
                
               
                   
               
                
               1 
               * 
               
                 
                   
                     ω 
                     0 
                     2 
                   
                   - 
                   
                     ω 
                     1 
                     2 
                   
                 
                 
                   
                     ω 
                     0 
                     2 
                   
                   - 
                   
                     ω 
                     1 
                     2 
                   
                   + 
                   
                     
                       
                         ω 
                         1 
                         2 
                       
                       * 
                       B 
                        
                       
                           
                       
                        
                       1 
                     
                     
                       C 
                        
                       
                           
                       
                        
                       3 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               L 
                
               
                   
               
                
               3 
             
             = 
             
               1 
               
                 
                   ω 
                   0 
                   2 
                 
                 * 
                 C 
                  
                 
                     
                 
                  
                 3 
               
             
           
         
       
     
     where Y, j, f, C 1 , L 3 , C 3 , and ω 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 l &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, three-element frequency-dependent resonators  60  may be used to implement multi-band matching baluns.  FIG. 10  shows a circuit arrangement for a 7-element multi-band matching balun according to an exemplary embodiment. 
     More specifically, 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.,  750   n ) 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, multi-band matching balun  30  includes 7 components, shown as inductor L 109 , inductor L 101 , capacitor C 101 , capacitor C 103 , inductor L 102 , inductor L 103 , and capacitor C 102 . Capacitor C 101 , inductor L 101 , and inductor L 109  form a three-element frequency-dependent resonator  60  that represents a frequency-dependent inductor, as described above in connection with  FIG. 6 . 
     As noted above, three-element frequency-dependent resonators  60  may be used to implement or realize multi-band matching baluns, e.g., by using three-element frequency-dependent resonators  60  in baluns. For instance, by replacing inductor L 11  in  FIG. 1  with a three-element frequency-dependent resonator  60  formed by capacitor C 101 , inductor L 101 , and inductor L 109 , the single-frequency balun in  FIG. 1  is transformed into multi-band matching balun  30  in  FIG. 10 . As a result, CLNA is now simultaneously resonated out in two frequency bands, thus implementing multi-band impedance matching. 
     Also, an additional inductor L 103  is coupled between port P 1  and “internal” node of the three-element frequency-dependent resonator formed by capacitor C 101 , inductor L 101 , and inductor L 109 . The addition of inductor L 103  create another 4-element balun that includes L 101 , L 103 , and C 101 ×C 103  within an original 4-element discrete balun (similar to balun  2  in  FIG. 1 ) that includes L 109 , L 102 , C 103 , and C 102  in  FIG. 7 , where the notation “C 101 ×C 103 ” denotes the series combination of capacitors C 101  and C 103 , i.e., C 101 ×C 103 /(C 101 +C 103 ). Consequently, multi-band matching balun  30  in  FIG. 10  provides impedance matching and balun functionality in multiple frequency bands. 
     Using the values L 109 =1 nH, L 101 =3 nH, L 103 =4 nH, L  102 =2 nH, C 102 =2 pF, C 101 =3 pF, and C 103 =1 pF, multi-band matching balun  30  provides impedance matching and balun functionality in the frequency band from ˜310 MHz to ˜434 MHz (realized by two resonances with slightly different frequencies) and also in the frequency band from ˜868 MHz to ˜928 MHz (realized by one resonance). As persons of ordinary skill in the art will understand, however, other component values may be used to provide multi-band matching balun functionality in other frequency bands, as desired. 
       FIG. 11  shows a circuit arrangement for a 7-element multi-band matching balun according to an exemplary embodiment. More specifically, 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., 75Ω) 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, multi-band matching balun  30  includes 7 components, shown as inductor L 111 , inductor L 114 , capacitor C 111 , capacitor C 113 , inductor L 112 , inductor L 113 , and capacitor C 112 . Capacitor C 111 , inductor L 111 , and inductor L 114  form a three-element frequency-dependent resonator that represents a frequency-dependent inductor, as described above in connection with  FIG. 7 . Thus, inductor L 11  in  FIG. 1  is replaced by the three-element frequency-dependent resonator formed by capacitor C 111 , inductor L 111 , and inductor L 114 . Otherwise, the circuit operates similarly to the circuit in  FIG. 10 , described in detail above. 
       FIG. 12  shows a circuit arrangement for a 5-element multi-band matching balun  30  according to an exemplary embodiment. More specifically, 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, multi-band matching balun  30  includes 5 components, shown as inductor L 121 , inductor L 122 , capacitor C 121 , capacitor C 122 , and inductor L 123 . Multi-band matching balun  30  in  FIG. 12  is derived from the 7-element multi-band matching balun  30  in  FIG. 10 . More specifically, two components have been removed from the 7-element multi-band matching balun  30  in  FIG. 10 : L 109  (which has a relatively large value), and capacitor C 102  (which has a relatively small value). 
     The five elements or components cause three resonances that cover the frequency bands for the sub-GHz ISM region (310-434 MHz and 868-928 MHz). Two of the three resonances are relatively close to each other in the lower frequency bands. Using component values of C 122 =10 pF, C 121 =2 pF, L 121 =55 nH, L 122 =28 nH, and L 123 =30 nH provides impedance matching and balun functionality in the frequency band from ˜310 MHz to ˜434 MHz and also in the frequency band from ˜868 MHz to ˜928 MHz. As persons of ordinary skill in the art will understand, however, other component values may be used to provide multi-band matching balun functionality in other frequency bands, as desired. 
     Note that 5-element multi-band matching balun  30  represents an optimized version (fewer components or elements, for a lower-count/lower-cost bill of materials) of the 7-element multi-band matching balun described above. Thus, 5-element multi-band matching balun  30  does not include any three-element frequency-dependent resonators. The resonators in 5-element multi-band matching balun  30 , however, are cross-coupled to each other, and one element or component is part of more than resonators (for example, L 121 −L 123 +L 122 −C 121  and L 123 −L 122 −C 121 ). Note further that 5-element multi-band matching balun  30  represents three resonances, where L 122  and C 122  components are mostly responsible for the lowest frequency resonance, while L 123 , L 121  and C 121  elements give rise to the two upper frequency resonances. 
     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.