Patent Publication Number: US-2012032751-A1

Title: Variable impedance matching circuit

Description:
TECHNICAL FIELD 
     The present invention relates to a variable impedance matching circuit used with a device such as an amplifier. 
     BACKGROUND ART 
     A power amplifier efficiently amplifies the power of a transmission signal to a power level required by a system. Generally, a radio frequency circuit containing a power amplifier is designed so as to match a certain load (impedance Z 0 ). However, a load impedance of a power amplifier especially in a mobile terminal varies according to changes of the electromagnetic environment around the antenna and therefore the output power and efficiency of the amplifier can decrease. There is an art in which a tuner is connected between a power amplifier and an antenna in order to reduce degradation due to variations in load. The tuner is made up of variable devices (variable inductive and capacitive elements). The simplest tuner circuit configurations may be combinations of three elements illustrated in  FIGS. 14A to 14D . Mathematically, the circuit configurations can deal with any variations in load. 
     SUMMARY OF THE INVENTION 
     A sufficiently wide variable range is demanded of a variable device in order to deal with load variations in a sufficiently wide range. However, while a variable inductive element is mathematically conceivable, no practical inductive element has been commercialized as of this writing. In practice, it is difficult to configure the circuits illustrated in  FIGS. 14A  to  14 D. Therefore, it has needed to take some measures to deal with load variations in a sufficiently wide range, such as increasing the number of elements used. 
     An object of the present invention is to provide a variable impedance matching circuit capable of adjusting impedance without using a variable inductive element as if the circuit were using a variable inductive element and accordingly capable of dealing with variations in load in a wide range with a small number of elements. 
     A variable impedance matching circuit of the present invention includes a series or parallel connection of a fixed inductive element and a first variable capacitive element and a second variable capacitive element connected in series with the series or parallel connection, wherein the susceptance of the circuit can be changed by changing the capacitance of each of the variable capacitive elements. 
     EFFECTS OF THE INVENTION 
     The variable impedance matching circuit of the present invention is capable of adjusting impedance without using a variable inductive element as if the circuit were using a variable inductive element. Therefore, the variable impedance matching circuit can deal with load variations in a wide range with a small number of elements. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram illustrating an exemplary configuration of a variable impedance matching circuit  100  of the present invention; 
         FIG. 2  is a diagram illustrating an exemplary configuration of the variable impedance matching circuit  100  of the present invention combined with a fixed capacitive element; 
         FIG. 3  is a diagram illustrating variable capacitance value versus absolute susceptance value characteristics in the variable impedance matching circuit  100  of the present invention; 
         FIG. 4  is a diagram illustrating an exemplary configuration of a variable impedance matching circuit based on the configuration in  FIG. 1  capable of supporting two frequency bands; 
         FIG. 5  is a diagram illustrating an exemplary configuration of a variable impedance matching circuit based on the configuration in  FIG. 2  capable of supporting two frequency bands; 
         FIG. 6  is a diagram illustrating variable capacitance value versus absolute susceptance value characteristics at an input signal frequency of 2 GHz when a switch in the variable impedance matching circuit in  FIG. 4  is turned to an L p1o     —     1  side; 
         FIG. 7  is a diagram illustrating variable capacitance value versus absolute susceptance value characteristics at an input signal frequency of 2 GHz when a switch in the variable impedance matching circuit in  FIG. 4  is turned to an L p1o     —     2  side; 
         FIG. 8  is a diagram illustrating an exemplary configuration of a variable impedance matching circuit  200  of the present invention; 
         FIG. 9  is a diagram illustrating an exemplary configuration of the variable impedance matching circuit  200  of the present invention combined with a fixed capacitive element; 
         FIG. 10  is a diagram illustrating variable capacitance value versus absolute susceptance value characteristics in the variable impedance matching circuit  200  of the present invention; 
         FIG. 11  is a diagram illustrating an exemplary configuration of a variable impedance matching circuit  300  of the present invention; 
         FIG. 12  is a diagram illustrating an exemplary configuration of the variable impedance matching circuit  300  of the present invention combined with a fixed capacitive element; 
         FIG. 13  is a diagram illustrating variable capacitance value versus reactance value characteristics in the variable impedance matching circuit  300  of the present invention; 
         FIG. 14A  is a diagram illustrating a first exemplary configuration of a background-art variable impedance matching circuit; 
         FIG. 14B  is a diagram illustrating a second exemplary configuration of a background-art variable impedance matching circuit; 
         FIG. 14C  is a diagram illustrating a third exemplary configuration of a background-art variable impedance matching circuit; and 
         FIG. 14D  is a diagram illustrating a fourth exemplary configuration of a background-art variable impedance matching circuit. 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     Embodiments of the present invention will be described below in detail. 
     First Embodiment 
       FIG. 1  illustrates an exemplary configuration of a variable impedance matching circuit  100  of the present invention. In the variable impedance matching circuit  100 , one fixed inductive element and two variable capacitive elements together act as the variable inductive element L p1  of the variable circuit in  FIG. 14A . 
     The variable impedance matching circuit  100  includes a series connection of a variable capacitive elements C s1  and C s2 , and a series connection between a series connection of a fixed inductive element L p1o  and a variable capacitive element C p1  and a variable capacitive element C p2 . Both ends of the series connection between the series connection of the fixed inductive element L p1o  and the variable capacitive element C p1  and the variable capacitive element C p2  are grounded. The connection point of the series connection of the variable capacitive elements C s1  and C s2  is connected to the connection point of the series connection between the series connection of the fixed inductive element L p1o  and the variable capacitive element C p1  and the variable capacitive element C p2 . 
     The fixed inductive element L p1o  is a fixed inductor having an inductance of L p1o . The variable capacitive elements C p1  and C p2  are variable capacitive elements having capacitances C p1  and C p2 , respectively. The variable capacitive elements may be implemented by semiconductor elements or implemented using MEMS technology, and may be manufactured and configured by any methods. 
     The admittance Y p1  of the series connection of the fixed inductive element L p1o  and the variable capacitive element C p1  is given by the following expression: 
     
       
         
           
             
               
                 
                   
                     Y 
                     
                       p 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   = 
                   
                     
                       j 
                        
                       
                           
                       
                        
                       ω 
                        
                       
                           
                       
                        
                       
                         C 
                         
                           p 
                            
                           
                               
                           
                            
                           1 
                         
                       
                     
                     
                       1 
                       - 
                       
                         
                           ω 
                           2 
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                          
                         
                           C 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where ω is the angular frequency of an input signal. 
     The admittance Y p2  of the variable capacitive element C p2  is given by the following expression: 
         Y   p2   =jωC   p2   (2)
 
     Therefore, the combined admittance Y p  of Y p1  and Y p2  is as given below: 
     
       
         
           
             
               
                 
                   
                     Y 
                     p 
                   
                   = 
                   
                     
                       
                         j 
                          
                         
                             
                         
                          
                         ω 
                          
                         
                             
                         
                          
                         
                           C 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                       
                         1 
                         - 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             L 
                             
                               p 
                                
                               
                                   
                               
                                
                               1 
                                
                               o 
                             
                           
                            
                           
                             C 
                             
                               p 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                       
                     
                     + 
                     
                       j 
                        
                       
                           
                       
                        
                       ω 
                        
                       
                           
                       
                        
                       
                         C 
                         
                           p 
                            
                           
                               
                           
                            
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Therefore, Y p  is inductive admittance when the following relational expression holds: 
       −∞&lt; Y   p ≦0  (4)
 
     Here, from Expressions (3) and (4), the following expressions can be obtained. 
     
       
         
           
             
               
                 
                   
                     1 
                     - 
                     
                       
                         ω 
                         2 
                       
                        
                       
                         L 
                         
                           p 
                            
                           
                               
                           
                            
                           1 
                            
                           o 
                         
                       
                        
                       
                         C 
                         
                           p 
                            
                           
                               
                           
                            
                           1 
                         
                       
                     
                   
                   ≤ 
                   0 
                 
               
               
                 
                   ( 
                   
                     5 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     - 
                     ∞ 
                   
                   &lt; 
                   
                     
                       
                         j 
                          
                         
                             
                         
                          
                         ω 
                          
                         
                             
                         
                          
                         
                           C 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                       
                         1 
                         - 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             L 
                             
                               p 
                                
                               
                                   
                               
                                
                               1 
                                
                               o 
                             
                           
                            
                           
                             C 
                             
                               p 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                       
                     
                     + 
                     
                       j 
                        
                       
                           
                       
                        
                       ω 
                        
                       
                           
                       
                        
                       
                         C 
                         
                           p 
                            
                           
                               
                           
                            
                           2 
                         
                       
                     
                   
                   ≤ 
                   0 
                 
               
               
                 
                   ( 
                   
                     5 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     Furthermore, from Expressions (5a) and (5b) the following expressions can be obtained. 
     
       
         
           
             
               
                 
                   
                     C 
                     
                       p 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   ≥ 
                   
                     1 
                     
                       
                         ω 
                         2 
                       
                        
                       
                         L 
                         
                           p 
                            
                           
                               
                           
                            
                           1 
                            
                           o 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     6 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     C 
                     
                       p 
                        
                       
                           
                       
                        
                       2 
                     
                   
                   ≤ 
                   
                     
                       C 
                       
                         p 
                          
                         
                             
                         
                          
                         1 
                       
                     
                     
                       
                         
                           ω 
                           2 
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                          
                         
                           C 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                       - 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   
                     6 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     Differentiating the right-hand side of Expression (6b) in C p1  yields the following expression. 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       
                         dC 
                         
                           p 
                            
                           
                               
                           
                            
                           1 
                         
                       
                     
                      
                     
                       { 
                       
                         
                           C 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                         
                           
                             
                               ω 
                               2 
                             
                              
                             
                               L 
                               
                                 p 
                                  
                                 
                                     
                                 
                                  
                                 1 
                                  
                                 o 
                               
                             
                              
                             
                               C 
                               
                                 p 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                           - 
                           1 
                         
                       
                       } 
                     
                   
                   = 
                   
                     
                       
                         - 
                         1 
                       
                       
                         
                           ( 
                           
                             
                               
                                 ω 
                                 2 
                               
                                
                               
                                 L 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                    
                                   o 
                                 
                               
                                
                               
                                 C 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                             - 
                             1 
                           
                           ) 
                         
                         2 
                       
                     
                     &lt; 
                     0 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Because the right-hand side of Expression (6b) monotonically decreases with respect to C p1 , the maximum value C p2max  of C p2  is a minimum when C p1  is its maximum value C p1max . Therefore, the required ranges of C p1  and C p2  are: 
     
       
         
           
             
               
                 
                   0 
                   ≤ 
                   
                     C 
                     
                       p 
                        
                       
                           
                       
                        
                       1 
                        
                       min 
                     
                   
                   ≤ 
                   
                     
                       1 
                       
                         
                           ω 
                           2 
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                       
                     
                      
                     
                         
                     
                      
                     and 
                      
                     
                         
                     
                      
                     
                       1 
                       
                         
                           ω 
                           2 
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                       
                     
                   
                   &lt; 
                   
                     C 
                     
                       p 
                        
                       
                           
                       
                        
                       1 
                        
                       max 
                     
                   
                 
               
               
                 
                   ( 
                   
                     8 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     C 
                     
                       p 
                        
                       
                           
                       
                        
                       2 
                        
                       min 
                     
                   
                   &lt; 
                   
                     
                       1 
                       
                         
                           ω 
                           2 
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                       
                     
                      
                     
                         
                     
                      
                     and 
                      
                     
                         
                     
                      
                     
                       1 
                       
                         
                           ω 
                           2 
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                       
                     
                   
                   ≤ 
                   
                     C 
                     
                       p 
                        
                       
                           
                       
                        
                       2 
                        
                       max 
                     
                   
                 
               
               
                 
                   ( 
                   
                     8 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     From the foregoing it follows that Y p  is inductive admittance when C p1  is in the range of Expression (8a) and C p2  is in the range of Expression (8b). Therefore, the set of the single fixed inductive element L p1o  and the two variable capacitive elements C p1  and C p2  can be caused to function as if the set were a variable inductive element. Thus, the set can act as the variable inductive element L p1  in the variable matching circuit in  FIG. 14A . A normal variable capacitive element has its specific variable capacitance range. Therefore, variable capacitive elements that have the variable capacitance ranges as given below may be used as C p1  and C p2 : 
         C   p1min   ≦C   p1   ≦C   p1min +Δ p1   =C   p1max   (9a)
 
         C   p2min   =C   p2max −Δ p2   ≦C   p2   ≦C   p2max   (9b)
 
     where Δ p1  and Δ p2  are the variable capacitance ranges. 
     The smaller the absolute value of the capacitance of a variable capacitive element, the smaller the size of the capacitive element. In order to reduce the absolute value of the capacitance, the variable capacitive element C p1  may be formed by a fixed capacitive element C p1o  (0&lt;C p1o ≦C p1min ) and a variable capacitive element C p1 ′ provided in parallel with the fixed capacitive element C p1o  and, similarly, the variable capacitive element C p2  may be formed by a fixed capacitive element C p2o  (0&lt;C p2o ≦C p2max −Δ p2 ) and a variable capacitive element C p2 ′ provided in parallel with the fixed capacitive element C p2o  as illustrated in  FIG. 2 . With this configuration, the absolute values of the capacitances of the variable capacitive elements C p1 ′ and C p2 ′ used can be reduced from the absolute values of the capacitances of C p1  and C p2  in Expressions (9a) and (9b) by C p1o  and C p2o , respectively, as shown by the expressions given below. With this configuration, smaller variable capacitive elements can be used. 
         C   p1min   −C   p1o   ≦C   p1   ′≦C   p1min   −C   p1o +Δ p1   (10a)
 
         C   p2max −Δ p2   −C   p2o   ≦C   p2   ′≦C   p2max   −C   p2o   (10b)
 
     A variable susceptance range that can be obtained by changing C p1  and C p2  when the set of the single fixed inductive element L p1o  and the two variable capacitive elements C p1  and C p2  in the configuration in  FIG. 1  is caused to function as if the set were a variable inductive element will be calculated. Then, the inductance range equivalent to the variable susceptance range that would be achieved only by an inductor will be determined. 
     By way of illustration, required C p1  and C p2  when an input signal frequency is 1 GHz and L p1o  is 2 nH will be calculated. From Expression (8a), C p1min  is approximately 12.7 pF. For simplicity, assume that C p1min  is 12 pF and Δ p1  is 9 pF. Then 12≦C p1 ≦21 pF. Here, since C p1max  is 21 pF, C p2max  is 31.9 pF from Expression (8b). For simplicity, assume that C p2max  is 32 pF and Δ p2  is 9 pF. Then 23 C p2 ≦32 pF.  FIG. 3  illustrates plots of the absolute value of susceptance in the configuration in  FIG. 2  in which C p1  in  FIG. 1  is divided into two, C p1 ′ and C p1o , and C p2  in  FIG. 1  is divided into two, C p2 ′ and C p2o , where C p1o =12 pF, C p2o =23 pF, and variable capacitance values C p1 ′ and C p2 ′ are changed in the ranges Δ p1  and Δ p2  (0 to 9 pF), respectively (variable capacitance value versus absolute susceptance value characteristics). The filled circles represent a plot obtained by changing C p1 ′ while C p2  is fixed at C p2min  (=23 pF) and filled squares represent a plot obtained by changing C p2 ′ while C p1  is fixed at C p1max  (=21 pF). The solid curve without circles nor squares represents the absolute value of susceptance obtained by changing the inductance value equivalent to the variable inductive element L p1  in  FIG. 14A  in the range of 0 to 10 nH. It can be seen from  FIG. 3  that susceptance value adjustment in a range equivalent to a range achievable by changing the inductance value L p1  by 10 nH or more can be achieved by changing the values of C p1  and C p2 , when the input signal frequency=1 GHz, L p1o =2 nH, C p1 =12 to 21 pF and C p2 =23 to 32 pF. 
     The configuration in  FIG. 14A  has been changed to a configuration that does not use a variable inductive element in the first embodiment described above. The configuration in  FIG. 14B  also can be changed to a configuration that does not use a variable inductive element in a way similar to that in the first embodiment. 
     In this way, the variable impedance matching circuit  100  of the present invention is capable of adjusting impedance without using a variable inductive element as if the circuit  100  were using a variable inductive element. Accordingly, the variable impedance matching circuit  100  is capable of dealing with variations in load in a wide range with a small number of elements. 
     [Variation] 
     In the variable impedance matching circuit  100  of the present invention, the fixed inductive element L p1o  and the fixed capacitive elements C p1o  and C p2o  are optimized for different frequency bands used and are allowed to be alternately selected by a switch, thereby a variable impedance matching circuit that can be used with multiple frequency bands can be configured.  FIGS. 4 and 5  illustrate exemplary configurations of a variable impedance matching circuit  150  based on the configurations in  FIGS. 1 and 2 , respectively, that can be used with two frequency bands. In the configuration in  FIG. 4 , fixed inductive elements L p1o     —     1  and L p1o     —     2  can be alternately selected by two SPDT switches according to a frequency band used. In the configuration in  FIG. 5 , a pair of fixed inductive elements L p1o     —     1  and L p1o     —     2 , a pair of fixed capacitive elements C p1o     —     1  and C p10     —     2 , and a pair of fixed capacitive elements C p2o     —     1  and C p2o     —     2  can be alternately selected by two SPDT switches according to a frequency band used. 
     Variable capacitance value versus absolute susceptance value characteristics obtained when the capacitance value of each variable capacitive element in the configuration in  FIG. 4  is changed in the same way as in  FIG. 3  will be determined. Here, L p1o     —     1  is 2 nH and L p1o     —     2  is 0.5 nH. When the switches are turned to the L p1o     —     1  side, the same configuration as that in  FIG. 1  is provided. Accordingly, the variable capacitance value versus susceptance absolute value characteristics as illustrated in  FIG. 3  are obtained when a signal of a frequency of 1 GHz is input. When a signal of a frequency of 2 GHz is input, the variable capacitance value versus susceptance absolute value characteristics illustrated in  FIG. 6  are obtained. It can be seen from  FIG. 6  that the range of the susceptance absolute values covered is narrower than the range covered when the 1-GHz signal is input.  FIG. 7  illustrates variable capacitance value versus susceptance absolute value characteristics obtained when the switches are turned to the L p1o     —     2  side to input a signal of a frequency of 2 GHz. It can be seen from  FIG. 7  that susceptance absolute values equivalent to the susceptance values obtained with 1 GHz can be obtained with 2 GHz by using L p1o     —     2  optimized for the input signal of 2 GHz. 
     Second Embodiment 
       FIG. 8  illustrates an exemplary configuration of a variable impedance matching circuit  200  of the present invention. The variable impedance matching circuit  200  has another configuration in which one fixed inductive element and two variable capacitive elements together act as the variable inductive element L p1  in the variable matching circuit in  FIG. 14A , as in the first embodiment. 
     The variable impedance matching circuit  200  includes a series connection of a variable capacitive elements C s1  and C s2 , a series connection between a parallel connection of a fixed inductive element L p1o  and a variable capacitive element C p1  and a variable capacitive element C p2 . One end of the series connection between the parallel connection of the fixed inductive element L p1o  and the variable capacitive element C p1  and the variable capacitive element C p2  is connected to the connection point between the variable capacitive elements C s1  and C s2  and the other end is grounded. 
     The fixed inductive element L p1o  is a fixed inductor with an inductance of L p1o . The variable capacitive elements C p1  and C p2  are variable capacitive elements having capacitances of C p1  and C p2 , respectively. The variable capacitive elements may be implemented by semiconductor elements or implemented using MEMS technology, and may be manufactured and configured by any methods. 
     The impedance Z p1  of the parallel connection of the fixed inductive element L p1o  and the variable capacitive element C p1  can be given by the following expression. 
     
       
         
           
             
               
                 
                   
                     Z 
                     
                       p 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   = 
                   
                     
                       jω 
                        
                       
                           
                       
                        
                       
                         L 
                         
                           p 
                            
                           
                               
                           
                            
                           1 
                            
                           o 
                         
                       
                     
                     
                       1 
                       - 
                       
                         
                           ω 
                           2 
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                          
                         
                           C 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     The impedance Z p2  of the variable capacitive element C p2  can be given by the following expression. 
     
       
         
           
             
               
                 
                   
                     Z 
                     
                       p 
                        
                       
                           
                       
                        
                       2 
                     
                   
                   = 
                   
                     1 
                     
                       jω 
                        
                       
                           
                       
                        
                       
                         C 
                         
                           p 
                            
                           
                               
                           
                            
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Therefore, the combined impedance Z p  of Z p1  and Z p2  is as given below. 
     
       
         
           
             
               
                 
                   
                     Z 
                     p 
                   
                   = 
                   
                     
                       
                         jω 
                          
                         
                             
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                       
                       
                         1 
                         - 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             L 
                             
                               p 
                                
                               
                                   
                               
                                
                               1 
                                
                               o 
                             
                           
                            
                           
                             C 
                             
                               p 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                       
                     
                     + 
                     
                       1 
                       
                         jω 
                          
                         
                             
                         
                          
                         
                           C 
                           
                             p 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Therefore, Z p  is inductive impedance when the following relational expression holds: 
         0 ≦ Z   p &lt;∞  (14)
 
     Here, the following expressions can be obtained from Expressions (13) and (14). 
     
       
         
           
             
               
                 
                   
                     1 
                     - 
                     
                       
                         ω 
                         2 
                       
                        
                       
                         L 
                         
                           p 
                            
                           
                               
                           
                            
                           1 
                            
                           o 
                         
                       
                        
                       
                         C 
                         
                           p 
                            
                           
                               
                           
                            
                           1 
                         
                       
                     
                   
                   ≥ 
                   0 
                 
               
               
                 
                   ( 
                   
                     15 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   0 
                   ≤ 
                   
                     
                       
                         ω 
                          
                         
                             
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                       
                       
                         1 
                         - 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             L 
                             
                               p 
                                
                               
                                   
                               
                                
                               1 
                                
                               o 
                             
                           
                            
                           
                             C 
                             
                               p 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                       
                     
                     - 
                     
                       1 
                       
                         ω 
                          
                         
                             
                         
                          
                         
                           C 
                           
                             p 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                     
                   
                   &lt; 
                   ∞ 
                 
               
               
                 
                   ( 
                   
                     15 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     Furthermore, the following expressions can be obtained from Expressions (15a) and (15b). 
     
       
         
           
             
               
                 
                   
                     C 
                     
                       p 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   ≤ 
                   
                     1 
                     
                       
                         ω 
                         2 
                       
                        
                       
                         L 
                         
                           p 
                            
                           
                               
                           
                            
                           1 
                            
                           o 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     16 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     C 
                     
                       p 
                        
                       
                           
                       
                        
                       2 
                     
                   
                   ≥ 
                   
                     
                       1 
                       - 
                       
                         
                           ω 
                           2 
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                          
                         
                           C 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                     
                     
                       
                         
                           ω 
                           2 
                         
                          
                         L 
                       
                       - 
                       
                         L 
                         
                           p 
                            
                           
                               
                           
                            
                           1 
                            
                           o 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     16 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     Here, differentiating the right-hand side of Expression (16b) in C p1  yields the following expression: 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       
                         dC 
                         
                           p 
                            
                           
                               
                           
                            
                           1 
                         
                       
                     
                      
                     
                       { 
                       
                         
                           1 
                           - 
                           
                             
                               ω 
                               2 
                             
                              
                             
                               L 
                               
                                 p 
                                  
                                 
                                     
                                 
                                  
                                 1 
                                  
                                 o 
                               
                             
                              
                             
                               C 
                               
                                 p 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             ω 
                             2 
                           
                            
                           
                             L 
                             
                               p 
                                
                               
                                   
                               
                                
                               1 
                                
                               o 
                             
                           
                         
                       
                       } 
                     
                   
                   = 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Because the right-hand side of Expression (16b) monotonically decreases with respect to C p1 , the minimum value C p2min  of C p2  is a maximum when C p1  is its minimum value C p1min . Therefore, the required ranges of C p1  and C p2  are: 
     
       
         
           
             
               
                 
                   
                     C 
                     
                       p 
                        
                       
                           
                       
                        
                       1 
                        
                       min 
                     
                   
                   &lt; 
                   
                     
                       1 
                       
                         
                           ω 
                           2 
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                       
                     
                      
                     
                         
                     
                      
                     and 
                      
                     
                         
                     
                      
                     
                       1 
                       
                         
                           ω 
                           2 
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                       
                     
                   
                   ≤ 
                   
                     C 
                     
                       p 
                        
                       
                           
                       
                        
                       1 
                        
                       max 
                     
                   
                 
               
               
                 
                   ( 
                   
                     18 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     0 
                     ≤ 
                     
                       C 
                       
                         p 
                          
                         
                             
                         
                          
                         2 
                          
                         min 
                       
                     
                     ≤ 
                     
                       
                         1 
                         - 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             L 
                             
                               p 
                                
                               
                                   
                               
                                
                               1 
                                
                               o 
                             
                           
                            
                           
                             C 
                             
                               p 
                                
                               
                                   
                               
                                
                               1 
                                
                               min 
                             
                           
                         
                       
                       
                         
                           ω 
                           2 
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                    
                   
                     
 
                   
                    
                   
                     
                       
                         1 
                         - 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             L 
                             
                               p 
                                
                               
                                   
                               
                                
                               1 
                                
                               o 
                             
                           
                            
                           
                             C 
                             
                               p 
                                
                               
                                   
                               
                                
                               1 
                                
                               min 
                             
                           
                         
                       
                       
                         
                           ω 
                           2 
                         
                          
                         
                           L 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                       
                     
                     &lt; 
                     
                       C 
                       
                         p 
                          
                         
                             
                         
                          
                         2 
                          
                         max 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     18 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     From the foregoing it follows that Z p  is inductive impedance when C p1  is in the range of Expression (18a) and C p2  is in the range of Expression (18b). Therefore, the set of the single fixed inductive element L p1o  and the two variable capacitive elements C p1  and C p2  can be caused to function as if the set were a variable inductive element. Thus, the set can act as the variable inductive element L p1  in the variable matching circuit in  FIG. 14A . A normal variable capacitive element has its specific variable capacitance range. Therefore, variable capacitive elements that have the variable capacitance ranges as given below may be used as C p1  and C p2 : 
         C   p1min   =C   p1max −Δ p1   ≦C   p1   ≦C   p1max   (19a)
 
         C   p2min   ≦C   p2   ≦C   p2min +Δ p2   =C   p2max   (19b)
 
     where Δ p1  and Δ p2  are the variable capacitance ranges. 
     The smaller the absolute value of the capacitance of a variable capacitive element, the smaller the size of the capacitive element. In order to reduce the absolute value of the capacitance, the variable capacitive element C p1  is formed by a fixed capacitive element C p1o  (0&lt;C p1o ≦C p1max −Δ p1 ) and a variable capacitive element C p1 ′ provided in parallel with the fixed capacitive element C p1o  as illustrated in  FIG. 9 . Similarly, the variable capacitive element C p2  may be formed by a fixed capacitive element C p2o  (0&lt;C p2o ≦C p2min ) and a variable capacitive element C p2 ′ provided in parallel with the fixed capacitive element C p2o . With this configuration, the absolute values of the capacitances of the variable capacitive elements C p1 ′ and C p2 ′ used can be reduced from the absolute values of the capacitances of C p1  and C p2  in Equations (19a) and (19b) by C p1o  and C p2o , respectively, as shown by Expressions given below. Accordingly, smaller variable capacitive elements can be used. 
         C   p1max   −C   p1o   −Δ≦C   p1   ′≦C   p1max   −C   p1o   (20a)
 
         C   p2min   −C   p2o   ≦C   p2   ′≦C   p2min +Δ p2   −C   p2o   (20b)
 
     A variable susceptance range that can be obtained by changing C p1  and C p2  when the set of the single fixed inductive element L p1o  and the two variable capacitive elements C p1  and C p2  in the configuration in  FIG. 8  is caused to function as if the set were a variable inductive element will be calculated. Then, the inductance range equivalent to the variable susceptance range that would be achieved only by an inductor will be determined. 
     By way of illustration, required C p1  and C p2  when an input signal frequency is 1 GHz and L p1o  is 2 nH will be calculated. From Expression (18a), C p1max  is approximately 12.7 pF. For simplicity, assume that C p1max  is 13 pF and Δ p1  is 9 pF. Then 4≦C p1 ≦C p1 ≦13 pF. Here, since C p1min  is 4 pF, C p2min  is 8.7 pF or more from Expression (18b). For simplicity, assume that C p2mm  is 8 pF and Δ p2  is 9 pF. Then 8≦ p2 ≦17 pF.  FIG. 10  illustrates plots of the absolute value of susceptance in the configuration in  FIG. 9  in which C p1  in  FIG. 8  is divided into two, C p1o ′ and C p1o , and C p2  in  FIG. 8  is divided into two, C p2 ′ and C p2o , where C p1o =4 pF, C p2o =8 pF, and variable capacitance values C p1 ′ and C p2 ′ are changed in the ranges Δ p1  and Δ p2  (0 to 9 pF), respectively (variable capacitance value versus absolute susceptance value characteristics). The filled circles represent a plot obtained by changing C p1 ′ while C p2  is fixed at C p2max  (=17 pF) and filled squares represent a plot obtained by changing C p2 ′ while C p1  is fixed at C p1min  (=4 pF). The solid curve without circles nor squares represents the absolute value of susceptance obtained by changing the inductance value equivalent to the variable inductive element L p1  in  FIG. 14A  in the range of 0 to 10 nH. It can be seen from  FIG. 10  that susceptance value adjustment in a range equivalent to a range achievable by changing the inductance value L p1  by 10 nH or more can be achieved by changing the values of C p1  and C p2 , when the input signal frequency=1 GHz, L p1o =2 nH, C p1 =4 to 13 pF and C p2 =8 to 17 pF. 
     The configuration in  FIG. 14A  has been changed to a configuration that does not use a variable inductive element in the second embodiment described above. The configuration in  FIG. 14B  also can be changed to a configuration that does not use a variable inductive element in a way similar to that in the second embodiment. 
     In this way, the variable impedance matching circuit  200  of the present invention is capable of adjusting impedance without using a variable inductive element as if the circuit  200  were using a variable inductive element. Accordingly, the variable impedance matching circuit is capable of dealing with variations in load in a wide range with a small number of elements. If required susceptance values are within a more limited range, C p2  may be replaced with a fixed capacitance. Furthermore, the configurations of the variation of the first embodiment can be used in the second embodiment to configure a variable impedance matching circuit that can be used with multiple frequency bands. 
     Third Embodiment 
       FIG. 11  illustrates an exemplary configuration of a variable impedance matching circuit  300  of the present invention. The variable impedance matching circuit  300  has a configuration in which one fixed inductive element and two variable capacitive elements together act as the variable inductive element L s1  in the variable matching circuit in  FIG. 14D . 
     The variable impedance matching circuit  300  includes a series connection between a parallel connection of a fixed inductive element L s10  and a variable capacitive element C s1  and a variable capacitive element C s2 . 
     The variable impedance matching circuit  300  also includes a variable capacitive element C p1  one end of which is connected to one end of the series connection and the other end of which is grounded, and a variable capacitive element C p2  one end of which is connected to the other end of the series connection and the other end of which is grounded. 
     The fixed inductive element L s1o  is a fixed inductor with an inductance of L s1o . The variable capacitive elements C s1  and C s2  are variable capacitive elements having capacitances of C s1  and C s2 , respectively. The conditions of the elements are the same as the conditions in the second embodiment, except that the fixed inductive element L p1o  in the second embodiment is replaced with the fixed inductive element L s1o , the variable capacitive element C p1  is replaced with the variable capacitive element C s1  and the variable capacitive element C p2  is replaced with the variable capacitive element C s2 . Alternatively, the variable capacitive element C s1  may be formed by a parallel connection of a fixed capacitive element C s1o  and a variable capacitive element C s1 ′ having a smaller capacitance and the variable capacitive element C s2  may be formed by a parallel connection of a fixed capacitive element C s2o  and a variable capacitive element C s2 ′ having a smaller capacitance, thereby smaller variable capacitive elements can be used. In this case, the capacitances of the fixed capacitance elements C s1o  and C s2o  and the variable capacitive elements C s1 ′ and C s2 ′ that correspond to the variable capacitive elements C s1  and C s2 , respectively, can be calculated by replacing C p1 , C p2 , C p1o , C p2o , C p1 ′ and C p2 ′ with C s1 , C s2 , C s1o , C s2o , C s1 ′ and C s2 ′, respectively, in the method calculating C p1o , C p1 ′ and C p2o . C p2 ′ that correspond to C p1  and C p2 , respectively, described in the second embodiment. 
     The variable capacitive elements may be implemented by semiconductor elements or may be implemented using MEMS technology and may be manufactured and configured by any methods. 
     A variable reactance range that can be obtained by changing C s1  and C s2  when the set of the single fixed inductive element L s1o  and the two variable capacitive elements C s1  and C s2  in the configuration in  FIG. 11  is caused to function as if the set were a variable inductive element will be calculated. Then, the inductance range equivalent to the variable reactance range that would be achieved only by an inductor will be determined. 
     By way of illustration, required C s1  and C s2  when an input signal frequency is 1 GHz and L s1o  is 2 nH will be calculated. From Expression (18a), C s1max  is approximately 12.7 pF. For simplicity, assume that C s1max  is 13 pF and Δ s1  is 9 pF. Then 4≦C s1 ≦13 pF. Here, since C s1min  is 4 pF, C s2  is 8.7 pF or more from Expression (18b). For simplicity, assume that C s2min  is 8 pF and Δ s2  is 9 pF. Then 8≦C s2 ≦17 pF.  FIG. 13  illustrates plots of reactance values in the configuration in  FIG. 12  in which C s1  in  FIG. 11  is divided into two, C s1 ′ and C s1o , and C s2  in  FIG. 11  is divided into two, C s2 ′ and C s2o , where C s1o =4 pF, C s2o =8 pF, and variable capacitance values C s1 ′ and C s2 ′ are changed in the ranges Δ s1  and Δ s2  (0 to 9 pF), respectively (variable capacitance value versus reactance value characteristics). The filled circles represent a plot obtained by changing C s1 ′ while C s2  is fixed at C s2max  (=17 pF) and filled squares represent a plot obtained by changing C s2 ′ while C s1  is fixed at C s1min  (=4 pF). The solid curve without circles nor squares represents a reactance value obtained by changing the inductance value equivalent to the variable inductive element L s1  in  FIG. 14D  in the range of 0 to 10 nH. It can be seen from  FIG. 13  that reactance value adjustment in a range equivalent to a range achievable by changing the inductance value L s1  by 10 nH or more can be achieved by changing the values of C s1  and C s2 , when the input signal frequency=1 GHz, L s1o =2 nH, C s1 =4 to 13 pF and C s2 =8 to 17 pF. 
     The configuration in  FIG. 14D  has been changed to a configuration that does not use a variable inductive element in the third embodiment described above. The configuration in  FIG. 14C  also can be changed to a configuration that does not use a variable inductive element in a way similar to that in the third embodiment. 
     In this way, the variable impedance matching circuit  300  of the present invention is capable of adjusting impedance without using a variable inductive element as if the circuit  300  were using a variable inductive element. Accordingly, the variable impedance matching circuit  300  is capable of dealing with variations in load in a wide range with a small number of elements. 
     The allocations of functions of the components of the variable impedance matching circuits  100 ,  150 ,  200  and  300  of the present invention described above are not limited to those described in the embodiments. Changes can be made to the allocations as appropriate without departing from the scope of the present invention.