Patent Publication Number: US-9853648-B2

Title: Compensation apparatus and inductor-based apparatus

Description:
This application claims the benefit of U.S. provisional patent application No. 62/066,955, filed Oct. 22, 2014, the disclosure of which is incorporated by reference herein in its entirety. 
    
    
     BACKGROUND 
     Technical Field 
     The disclosure relates in general to a compensation apparatus and an inductor-based apparatus, and more particularly to a compensation apparatus and an inductor-based apparatus with temperature compensation function. 
     Description of the Related Art 
     Phase-locked loop (hereinafter, PLL) circuits are widely used for the distribution of clocks on microprocessors, digital signal processors (DSPs), field programmable gate arrays (FPGAs), etc. In a PLL circuit, a voltage-controlled oscillator (hereinafter, VCO) is the key analogue block to determine the frequency stability and the jitter performance. 
     In current technology, LC VCO is a widely used scheme for CMOS based VCO design. The oscillation frequency of a LC VCO is determined by the resonator inductor and capacitor. However, the oscillation frequency of the LC VCO is temperature dependent. Factors such as changes of inductance, changes of Q factor of the inductance, swing amplitude of the VCO itself, or non-linear capacitance etc. cause variations of the oscillation frequency of LC VCO over temperature. Consequentially, techniques for temperature compensation should be utilized to improve the accuracy of the oscillation frequency. 
     Techniques for compensating for the effects of temperature change on VCO frequency are developed. The conventional temperature compensation circuits utilize varactors to provide a negative temperature coefficient (TC) to the LC VCO. With the negative temperature coefficient, the frequency drift caused by temperature variation is eliminated. However, the varactors may occupy significant silicon area. Therefore, a more economic approach for TC compensation is required. 
     SUMMARY 
     The disclosure is directed to a compensation apparatus and an inductor-based apparatus for compensating frequency of an input signal. The input signal is outputted by an oscillator such as a voltage-controlled oscillator (VCO), a digitally controlled oscillator (DCO) or the like. The oscillator is a portion of a primary circuit, whose inductance and resistance are compensated by a current flowing through a compensation circuit. Consequentially, frequency of the input signal is adjusted. 
     According to one embodiment, a compensation apparatus including a primary circuit and a compensation circuit is provided. The primary circuit provides a first voltage, a second voltage, and a first current flowing through the first inductor. The primary circuit includes a function circuit generating an input signal and a first inductor. The first inductor is coupled between a first terminal with the first voltage and a second terminal with the second voltage. The compensation circuit includes a second inductor and a current source circuit. The second inductor is coupled between a third terminal with a third voltage and a fourth terminal with a fourth voltage. The current source circuit outputs a second current flowing through the second inductor. The current source circuit adjusts a frequency of the input signal. The primary circuit and the compensation circuit are coupled via the first inductor and the second inductor. 
     According to another embodiment, an inductor-based apparatus for adjusting frequency of an input signal is provided. The inductor-based apparatus includes a first inductor, a second inductor and a current source circuit. The first inductor is coupled between a first terminal with a first voltage and a second terminal with a second voltage. A first current flows through the first inductor. The second inductor is coupled between a third terminal with a third voltage and a fourth terminal with a fourth voltage. The current source circuit outputs a second current flowing through the second inductor to adjust a frequency of the input signal. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic diagram illustrating the primary circuit and the compensation circuit placed in the transformer-like arrangement. 
         FIG. 2  is a schematic diagram illustrating the compensation circuit capable of adjusting resistance and inductance of the primary circuit. 
         FIG. 3  is a schematic diagram illustrating the compensation circuit according to the present invention. 
         FIG. 4  is a schematic diagram illustrating the compensation circuit having only the inductance inducing circuit. 
         FIG. 5  is a schematic diagram illustrating the frequency variations of the VCO circuit in  FIG. 4  with temperature drifts between −20 to 80° C. 
         FIG. 6  is a schematic diagram illustrating the application to a digitally controlled oscillator. 
     
    
    
     In the following detailed description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the disclosed embodiments. It will be apparent, however, that one or more embodiments may be practiced without these specific details. In other instances, well-known structures and devices are schematically shown in order to simplify the drawing. 
     DETAILED DESCRIPTION 
     The present disclosure directly improves the stability of oscillation frequency of an output of an oscillator, such as the LC VCO, an DCO or the like. Instead of providing a compensation circuit having the negative temperature coefficient, the compensation circuit according to the present disclosure shrinks a value of the temperature coefficient by reducing deviation of the oscillation frequency. 
       FIG. 1  is a schematic diagram of a compensation apparatus  1  according to an embodiment of the present disclosure. The compensation apparatus  1  includes a primary circuit  11  and a compensation circuit  13 . The primary circuit  11  and the compensation circuit  13  are coupled via a transformer  10  including a primary inductor L 1  and a compensation inductor L 2 . The primary circuit  11  includes the primary inductor L 1  and a function circuit  11   a . The function circuit  11   a  includes a negative resistor −R and a capacitor C. The primary inductor L 1  is coupled to the function circuit  11   a  through a first terminal s 1  and a second primary terminal s 2 . The first terminal s 1  is with a first voltage, and the second terminal s 2  is with a second voltage. The voltage difference between the first terminal s 1  and the second terminal s 2  is defined as a first voltage difference V 1 . A first current I 1  flows through the primary inductor L 1 . 
     The compensation circuit  13  includes a compensation inductor L 2  and a current source circuit  13   a , and the current source circuit  13   a  outputs a current equal to 
             (       β     j   ⁢           ⁢   ω       ·   Gm   ·     V   1       )         
flowing through the compensation inductor L 2 . The Gm represents the large signal transconductance, and represents a first amplifying parameter. The compensation inductor L 2  is coupled between a third terminal s 3  with a third voltage, and a fourth terminal s 4  with a fourth voltage. The voltage difference between the third terminal s 3  and the fourth terminal s 4  is defined as a second voltage difference V 2 . The current flowing through the compensation inductor L 2  is defined as a second current I 2 . The relative position between the primary inductor L 1  and the compensation inductor L 2  is defined as spacing s, and the mutual inductance between the primary inductor L 1  and the compensation inductor L 2  is defined as M.
 
     In  FIG. 1 , the first voltage difference V 1  and the second voltage difference V 2  can be represented in equation (1). 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             V 
                             1 
                           
                           = 
                           
                             
                               
                                 sL 
                                 1 
                               
                               ⁢ 
                               
                                 I 
                                 1 
                               
                             
                             + 
                             
                               sMI 
                               2 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             V 
                             2 
                           
                           = 
                           
                             
                               sMI 
                               1 
                             
                             + 
                             
                               
                                 sL 
                                 2 
                               
                               ⁢ 
                               
                                 I 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     1 
                     ) 
                   
                 
               
             
           
         
       
     
     As the first current I 1  flows through the primary inductor L 1 , variance of the first current I 1  creates a varying magnetic field on the secondary inductor L 2  and the varying magnetic field causes changes of the second current I 2 . Furthermore, the first current I 1  is determined by the first voltage difference V 1 . Consequentially, the second current I 2  is related to the first voltage difference V 1  and can be represented as shown in equation (2). In equation (2), β represents a second amplifying parameter, Gm represents the large signal transconductance, and j is the imaginary unit. 
     
       
         
           
             
               
                 
                   
                     I 
                     2 
                   
                   = 
                   
                     
                       β 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                       
                     
                     · 
                     
                       G 
                       m 
                     
                     · 
                     
                       V 
                       1 
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     2 
                     ) 
                   
                 
               
             
           
         
       
     
     According to equation (1) and equation (2), the first voltage difference V 1  can be further represented as equation (3). 
     
       
         
           
             
               
                 
                   
                     V 
                     1 
                   
                   = 
                   
                     
                       
                         sL 
                         1 
                       
                       ⁢ 
                       
                         I 
                         1 
                       
                     
                     
                       ( 
                       
                         1 
                         - 
                         
                           β 
                           · 
                           M 
                           · 
                           
                             G 
                             m 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     3 
                     ) 
                   
                 
               
             
           
         
       
     
     A simplified equation, x=y·(1+z) −1 ≈y·(1−z) is valid if z&lt;&lt;1. Since (β·G m ·M)&lt;&lt;1 in equation (3), the first voltage difference V 1  shown in equation (3) can be simplified and represented in equation (4).
 
 V   1   ≈s·L   1 (1+β· M·G   m )· I   1   equation (4)
 
     According to equation (4), the effective inductance of the primary inductor L 1  can be defined as L 1 (1−β·M·G m ) 
     As s·L 1 (1+β·M·G m ) is imaginary, and the transconductance G m  changes with temperature, the effective inductance of the primary inductor L 1  can be used for compensating the frequency drift in accordance with temperature change. 
     In addition, by changing the second current I 2 , the impedance looking into the primary circuit  11  will change accordingly. Change of the impedance looking into the primary circuit  11  is represented as equation (5). In equation (5), w is the oscillation frequency.
 
 V   1   =j ω( L   1   ·I   1   +M·I   2 )  equation (5)
 
     The second current I 2  can be represented in parameters of the first voltage difference V 1 . That is, I 2 =γ·V 1 −j·κ·V 1  The coefficients of the first voltage difference V 1  are further defined as a trans-conductance γ and a trans-susceptance κ. Therefore, equation (5) can be represented as equation (6). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           V 
                           1 
                         
                         = 
                           
                         ⁢ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   L 
                                   1 
                                 
                                 · 
                                 
                                   I 
                                   1 
                                 
                               
                               + 
                               
                                 M 
                                 · 
                                 
                                   I 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ω 
                               · 
                               
                                 L 
                                 1 
                               
                               · 
                               
                                 I 
                                 1 
                               
                             
                           
                           + 
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ω 
                               · 
                               M 
                               · 
                               
                                 ( 
                                 
                                   
                                     γ 
                                     · 
                                     
                                       V 
                                       1 
                                     
                                   
                                   - 
                                   
                                     j 
                                     · 
                                     κ 
                                     · 
                                     
                                       V 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ω 
                               · 
                               
                                 L 
                                 1 
                               
                               · 
                               
                                 I 
                                 1 
                               
                             
                           
                           + 
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ω 
                               · 
                               M 
                               · 
                               γ 
                               · 
                               
                                 V 
                                 1 
                               
                             
                           
                           + 
                           
                             ω 
                             · 
                             M 
                             · 
                             κ 
                             · 
                             
                               V 
                               1 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
           
         
       
     
     Furthermore, the first voltage difference V 1  can be represented in equation (7). 
     
       
         
           
             
               
                 
                   
                     V 
                     1 
                   
                   = 
                   
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ω 
                         · 
                         
                           L 
                           1 
                         
                         · 
                         
                           I 
                           1 
                         
                       
                     
                     
                       1 
                       - 
                       
                         ( 
                         
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ω 
                               · 
                               M 
                               · 
                               γ 
                             
                           
                           + 
                           
                             ω 
                             · 
                             M 
                             · 
                             κ 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     7 
                     ) 
                   
                 
               
             
           
         
       
     
     It is assumed, jω·M·γ&lt;&lt;1 and ω·M·κ&lt;&lt;1. Therefore, the relationship between the first voltage difference V 1  and the first current I 1  can be further simplified as equation (8). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           V 
                           1 
                         
                         ≈ 
                           
                         ⁢ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ω 
                             · 
                             
                               L 
                               1 
                             
                             · 
                             
                               
                                 I 
                                 1 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   1 
                                   + 
                                   
                                     ( 
                                     
                                       
                                         j 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           ω 
                                           · 
                                           M 
                                           · 
                                           γ 
                                         
                                       
                                       + 
                                       
                                         ω 
                                         · 
                                         M 
                                         · 
                                         κ 
                                       
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ω 
                               · 
                               
                                 L 
                                 1 
                               
                               · 
                               
                                 I 
                                 1 
                               
                             
                           
                           - 
                           
                             
                               ω 
                               2 
                             
                             · 
                             
                               L 
                               1 
                             
                             · 
                             
                               I 
                               1 
                             
                             · 
                             M 
                             · 
                             γ 
                           
                           + 
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ω 
                                 2 
                               
                               · 
                               
                                 L 
                                 1 
                               
                               · 
                               
                                 I 
                                 1 
                               
                               · 
                               M 
                               · 
                               γ 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ω 
                               · 
                               
                                 
                                   L 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       ω 
                                       · 
                                       M 
                                       · 
                                       γ 
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 I 
                                 1 
                               
                             
                           
                           - 
                           
                             
                               ω 
                               2 
                             
                             · 
                             M 
                             · 
                             γ 
                             · 
                             
                               L 
                               1 
                             
                             · 
                             
                               I 
                               1 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     8 
                     ) 
                   
                 
               
             
           
         
       
     
     According to equation (8), the coefficient of the imaginary part of the first current I 1 , that is L 1 (1+ω·M·γ), can be considered as an inductance looking into the primary circuit  11 . Similarly, the coefficient of the real part of the first current I 1  that is (−ω 2 ·m·γ·L 1 ), can be considered as a negative resistance looking into the primary circuit  11 . 
     Accordingly, the compensation circuit  13  is capable of influencing the first voltage difference V 1  through the inductance and the negative resistance as mentioned. The negative resistance and the inductance are further utilized to compensate the loss in the crystal and the non-linear capacitance. 
       FIG. 2  is a schematic diagram of a compensation apparatus  2  according to an embodiment of the present disclosure. The compensation apparatus  2  can be considered as a combination of a function circuit  22  and an inductor-based apparatus  20 . The function circuit  22  generates an input signal outputted at the first terminal s 1  and the second terminal s 2 . The inductor-based apparatus  20  includes a primary inductor L 1 , a compensation inductor L 2 , and a current source circuit  230 . 
     Alternately, the compensation apparatus  2  can be considered as a combination of a compensation circuit  23  and a primary circuit  21 . The compensation circuit  23  is capable of adjusting the negative resistance and inductance of the primary circuit  21  through the primary inductor L 1  and the compensation inductor L 2 . The compensation circuit  23  includes the current source circuit  230  and the compensation inductor L 2 . 
     In one embedment, the current source circuit  230  includes only one of the inductance inducing circuit  233  and the resistance inducing circuit  231 . The inductance inducing circuit  233  may cause an inductance change of the primary circuit  21 , and the resistance inducing circuit  231  causes a resistance change of the primary circuit  21 . That is, the inductance inducing circuit  233  or the resistance inducing circuit  231  will correspondingly and independently cause an electric property change of the primary circuit  21 . Furthermore, the compensation circuit  23  may also include plural inductance inducing circuits  233  and/or plural resistance inducing circuits  231 . 
       FIG. 3  is a schematic diagram of a compensation apparatus  3  according to an embodiment of the present disclosure. The compensation apparatus  3  includes a compensation circuit  33  and a primary circuit  31 . The primary circuit  31  includes a function circuit  32  and the primary inductor L 1 . The compensation circuit  33  includes the compensation inductor L 2  and a current source circuit  330 . The current source circuit  30  further includes the resistance inducing circuit  331 , and the inductance inducing circuit  333 , which are electrically connected to the third terminal s 3  and the fourth terminal s 4  of the compensation inductor L 2  in parallel. The resistance inducing circuit  331  outputs a sub-current I 22  to the compensation inductor L 2 , and the inductance inducing circuit  333  outputs a sub-current I 21  to the compensation inductor L 2 . 
     The resistance inducing circuit  331  includes a current source  3311 , two switches (PMOS transistors Mr 1  and Mr 2 ), and two compensation modules  331   a ,  331   b . The current source  3311  is electrically connected between the supply voltage Vcc and sources of the PMOS transistors Mr 1  and Mr 2 . The current source  3311  outputs a source current. 
     The two compensation modules  331   a  and  331   b  are symmetric to each other. In  FIG. 3 , the two compensation modules  331   a  and  331   b  are amplifier circuits providing a first amplifying parameter α. The compensation module  331   a  is electrically connected between the second terminal s 2  with voltage VPR and gate of the PMOS transistor Mr 1 . Drain of the PMOS transistor Mr 1  is electrically connected to the third terminal s 3 . The compensation module  331   b  is electrically connected between the first terminal s 1  with voltage VPL and gate of the PMOS transistor Mr 2 . Drain of the PMOS transistor Mr 2  is electrically connected to the fourth terminal s 4 . 
     Since the primary circuit  31  provides differential signals at the first terminal s 1  and the second terminal s 2 , the polarity of the first voltage difference V 1  changes in turn. Accordingly, flowing direction of both the first current I 1  and the second current I 2  may vary. When the second current I 2  flows in a downward direction, the current source  3311  contributes a sub-current I 22  flowing through the PMOS transistor Mr 1 . When the second current I 2  flows in an upward direction, the current source  3311  contributes a sub-current I 22  flowing through the PMOS transistor Mr 2 . That is, the PMOS transistors Mr 1 , Mr 2  are complementary conducted and the current source  3311  alternatively induces a change of the resistance at the primary circuit  31  through conduction of the PMOS transistors Mr 1  and Mr 2 . 
     The inductance inducing circuit  333  includes a current source  3331 , two switches (PMOS transistors MI 1  and MI 2 ), and two compensation modules  333   a ,  333   b . The current source  3331  is electrically connected between a supply voltage Vcc and sources of the PMOS transistors MI 1  and MI 2 . The current source  3331  outputs a source current. 
     Structure and connection of the two compensation modules  333   a ,  333   b  are symmetric to each other. Each of the two compensation modules  333   a ,  333   b  provides the second amplifying parameter β and a phase shift. The compensation module  333   a  is electrically connected between the second terminal s 2  with voltage VPR and gate of the PMOS transistor MI 1 . Drain of the PMOS transistor MI 1  is electrically connected to the third terminal s 3 . The compensation module  333   b  is electrically connected between the first terminal s 1  with voltage VPL and gate of the PMOS transistor MI 2 . Drain of the PMOS transistor MI 2  is electrically connected to the fourth terminal s 4 . 
     As mentioned above, flowing direction of the second current I 2  is varied with the polarity change of the primary circuit  31 . When the second current I 2  flows in a downward direction, the current source  3331  contributes a sub-current I 21  flowing through the PMOS transistor MI 1 . When the second current I 2  flows in an upward direction, the current source  3331  contributes a sub-current I 21  flowing through the PMOS transistor MI 2 . That is, the PMOS transistors MI 1  and MI 2  are complementary conducted and the current source  3331  alternatively induces a change of the inductance of the primary inductor L 1  through conduction of the PMOS transistors MI 1  and MI 2 . 
     With the sub-current I 22  outputted by the resistance inducing circuit  331 , and the sub-current I 21  outputted by the inductance inducing circuit  333 , the second current I 2  in  FIG. 3  can be represented as summation of the two sub-currents I 21  and I 22 , that is I 2 =I 21 +I 22 . Therefore, the first voltage difference V 1  according to equation (6) can be further conducted in equation (9). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           V 
                           1 
                         
                         = 
                           
                         ⁢ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ω 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     L 
                                     1 
                                   
                                   · 
                                   
                                     I 
                                     1 
                                   
                                 
                                 + 
                                 
                                   M 
                                   · 
                                   
                                     I 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 L 
                                 1 
                               
                               · 
                               
                                 I 
                                 1 
                               
                               · 
                               M 
                               · 
                               
                                 ( 
                                 
                                   
                                     I 
                                     21 
                                   
                                   + 
                                   
                                     I 
                                     22 
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     9 
                     ) 
                   
                 
               
             
           
         
       
     
     In other words, the first voltage difference V 1  is affected by the sub-currents I 21  and I 22 , which are provided by the inductance inducing circuit  333  and the resistance inducing circuit  331 . The sub-current I 21  outputted by the inductance inducing circuit  333 , and the second sub-current I 22  outputted by the resistance inducing circuit  331  can be represented as equation (10). 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             I 
                             21 
                           
                           = 
                           
                             
                               ( 
                               
                                 - 
                                 
                                   V 
                                   1 
                                 
                               
                               ) 
                             
                             · 
                             
                               β 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                               
                             
                             · 
                             
                               ( 
                               
                                 - 
                                 
                                   G 
                                   
                                     m 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             I 
                             22 
                           
                           = 
                           
                             
                               ( 
                               
                                 - 
                                 
                                   V 
                                   1 
                                 
                               
                               ) 
                             
                             · 
                             
                               ( 
                               
                                 - 
                                 
                                   G 
                                   
                                     m 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                               
                               ) 
                             
                             · 
                             α 
                           
                         
                       
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     10 
                     ) 
                   
                 
               
             
           
         
       
     
     The equation (10) is used to replace the sub-currents I 21 , and I 22  in equation (9), and the first voltage difference V 1  can be further conducted as equation (11). 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             V 
                             1 
                           
                           = 
                             
                           ⁢ 
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ω 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       L 
                                       1 
                                     
                                     · 
                                     
                                       I 
                                       1 
                                     
                                   
                                   + 
                                   
                                     M 
                                     · 
                                     
                                       ( 
                                       
                                         
                                           I 
                                           21 
                                         
                                         + 
                                         
                                           I 
                                           22 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   L 
                                   1 
                                 
                                 · 
                                 
                                   I 
                                   1 
                                 
                               
                             
                             + 
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 M 
                                 · 
                                 
                                   [ 
                                   
                                     
                                       
                                         - 
                                         
                                           V 
                                           1 
                                         
                                       
                                       · 
                                       
                                         β 
                                         
                                           j 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           ω 
                                         
                                       
                                       · 
                                       
                                         ( 
                                         
                                           - 
                                           
                                             G 
                                             
                                               m 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               1 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           - 
                                           
                                             V 
                                             1 
                                           
                                         
                                         ) 
                                       
                                       · 
                                       
                                         ( 
                                         
                                           - 
                                           
                                             G 
                                             
                                               m 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               2 
                                             
                                           
                                         
                                         ) 
                                       
                                       · 
                                       α 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   L 
                                   1 
                                 
                                 · 
                                 
                                   I 
                                   1 
                                 
                               
                             
                             + 
                             
                               
                                 V 
                                 1 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     β 
                                     · 
                                     M 
                                     · 
                                     
                                       G 
                                       
                                         m 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                     
                                   
                                   + 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ω 
                                       · 
                                       M 
                                       · 
                                       
                                         G 
                                         
                                           m 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                         
                                       
                                       · 
                                       α 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   L 
                                   1 
                                 
                                 · 
                                 
                                   I 
                                   1 
                                 
                               
                             
                             
                               1 
                               - 
                               
                                 [ 
                                 
                                   
                                     β 
                                     · 
                                     M 
                                     · 
                                     
                                       G 
                                       
                                         m 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                     
                                   
                                   + 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ω 
                                       · 
                                       M 
                                       · 
                                       
                                         G 
                                         
                                           m 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                         
                                       
                                       · 
                                       α 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     11 
                     ) 
                   
                 
               
             
           
         
       
     
     Similarly, with the assumption [β·M·G m1 +jω·M·G m2 ·α]&lt;&lt;1 the first voltage difference V 1  can be further simplified and represented as equation (12). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           V 
                           1 
                         
                         ≈ 
                           
                         ⁢ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               L 
                               1 
                             
                             · 
                             
                               I 
                               1 
                             
                             · 
                             
                               [ 
                               
                                 1 
                                 + 
                                 
                                   β 
                                   · 
                                   M 
                                   · 
                                   
                                     G 
                                     
                                       m 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     · 
                                     M 
                                     · 
                                     
                                       G 
                                       
                                         m 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                     
                                     · 
                                     α 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ω 
                               · 
                               
                                 I 
                                 1 
                               
                               · 
                               
                                 
                                   L 
                                   1 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     1 
                                     + 
                                     
                                       β 
                                       · 
                                       M 
                                       · 
                                       
                                         G 
                                         
                                           m 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 - 
                                 
                                   ω 
                                   2 
                                 
                               
                               ) 
                             
                             · 
                             M 
                             · 
                             
                               G 
                               
                                 m 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             · 
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                             · 
                             
                               L 
                               1 
                             
                             · 
                             
                               I 
                               1 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       12 
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                 
               
             
           
         
       
     
     According to equation (12), the coefficient of the imaginary part L 1 ·[1−β·M 10 G m1 ] can be considered as the inductance looking into the primary circuit  31 , and the coefficient of the real part, (−ω 2 )·M·G m2 ·α·L 1 , is considered as the negative resistance looking into the primary circuit  31 . 
     Since the first voltage difference V 1  can be represented in either equation (8) or equation (12), the coefficients of the imaginary part in both equations should be equivalent. That is, equation (13) can be conducted.
 
κ·ω· M=β·M·G   m1   equation (13)
 
     Accordingly, the second amplifying parameter β provided by the compensation modules  333   a ,  333   b  of the inductance inducing circuit  333  can be represented as 
     
       
         
           
             β 
             = 
             
               
                 
                   κ 
                   · 
                   ω 
                 
                 
                   G 
                   
                     m 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
               
               . 
             
           
         
       
     
     Since the first voltage difference V 1  can be represented in either equation (8) of equation (12), the coefficients of the real part in both equations should be equivalent. That is, equation (14) can be conducted.
 
−ω 2   ·M·γ·L   1 =(−ω 2 )· M·G   m2   ·α·L   1   equation (14)
 
     Accordingly, the first amplifying parameter α of the compensation module  331   a  and  331   b  of the resistance inducing circuit  331  can be represented as: 
     
       
         
           
             α 
             = 
             
               
                 γ 
                 
                   G 
                   
                     m 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                 
               
               . 
             
           
         
       
     
     Based on the above conduction, the first amplifying parameter α and the second amplifying parameter β can be properly selected and designed to compensate the electric property of the primary circuit  31 . Selection of the first amplifying parameter α affects the resistance of the primary circuit  31 , and the amplitude of the first voltage difference V 1 . Selection of the second amplifying parameter β affects the inductance of the primary inductor circuit, and the non-linear capacitance of the primary circuit  31 . 
     Depending on which electric property of the primary circuit  31  is targeted to be changed, the compensation circuit may selectively include the resistance inducing circuit  331  and/or the inductance inducing circuit  333 . In a case that only resistance of the primary circuit  31  is an issue to be improved, only the resistance inducing circuit  331  is used. Similarly, in a case that only inductance of the primary circuit  31  is an issue to be improved, only the inductance inducing circuit  333  is used. In one embodiment, a plurality of resistance inducing circuits  331  with various settings of the first amplifying parameters a and/or a plurality of inductance inducing circuits  333  with various settings of the second amplifying parameter β are collaboratively used. 
       FIG. 4  is a schematic diagram of a compensation apparatus  4  according to an embodiment of the present disclosure. The compensation apparatus  4  includes a compensation circuit  43  and a primary circuit  42 . In the compensation apparatus  4 , the compensation circuit  43  includes the inductance inducing circuit  433  and no resistance inducing circuit. In one embodiment, the function circuit  421  is a VCO circuit. As a differential pair used for compensating the inductance of the primary circuit  42 , the inductance inducing circuit  433  includes a current source  4331 , two switches (PMOS transistors MI 1  and MI 2 ), and two compensation modules  433   a  and  433   b . The current source  4331  is electrically connected between a supply voltage Vcc and sources of the PMOS transistors MI 1 , MI 2 . The current outputted by the current source  4311  is represented as CTAT, whose current value is complementary to the absolute temperature. When the current CTAT varies, the transconductance Gm=Gm 1  can be adjusted accordingly. 
     The structures of the compensation modules  433   a  and  433   b  are symmetric. Each of the compensation module  433   a  and the compensation module  433   b  includes a RC network, that is, a capacitor C and a resistor R. 
     The resistor R of the compensation module  433   a  is electrically connected between the second terminal s 2  with voltage level VPR and gate of the PMOS transistor MI 1 . The capacitor C of the compensation module  433   a  is electrically connected between the gate of the PMOS transistor MI 1  and a ground. 
     The resistor R of the second compensation module  433   b  is electrically connected between the first terminal s 1  with voltage level VPL and gate of the PMOS transistor MI 2 . The capacitor C of the second compensation module  433   b  is electrically connected between the gate of the PMOS transistor MI 2  and the ground. 
     The current source  4331  outputs a predetermined current CTAT to the compensation inductor L 2  through the PMOS transistors MI 1 , MI 2 , and the compensation inductor L 2  correspondingly induces an inductance change of the primary inductor L 1 . Since the gate of the PMOS transistors MI 1  and MI 2  are controlled by the voltage level VPR and VPL of the two terminals of the primary inductor L 1 , the second current I 2  can reflect the condition of the primary circuit  42 . 
     The second current I 2  in  FIG. 4  can be represented shown in equation (15). 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       2 
                     
                     ≈ 
                     
                       
                         ( 
                         
                           - 
                           
                             V 
                             1 
                           
                         
                         ) 
                       
                       * 
                       
                         1 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           CR 
                         
                       
                       * 
                       
                         ( 
                         
                           - 
                           
                             Gm 
                             1 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       V 
                       1 
                     
                     * 
                     
                       
                         Gm 
                         1 
                       
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         CR 
                       
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     15 
                     ) 
                   
                 
               
             
           
         
       
     
     The first voltage difference V 1  at the primary circuit  42  can be represented as the following equation. Accordingly, the inductance looking into the primary circuit  42  is equivalent to 
     
       
         
           
             
               
                 ( 
                 
                   
                     L 
                     1 
                   
                   + 
                   
                     
                       M 
                       * 
                       
                         Gm 
                         1 
                       
                       * 
                       
                         L 
                         1 
                       
                     
                     CR 
                   
                 
                 ) 
               
               . 
               
                 
 
               
               ⁢ 
               
                 V 
                 1 
               
             
             = 
             
               
                 j 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 ω 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         L 
                         1 
                       
                       * 
                       
                         I 
                         1 
                       
                     
                     + 
                     
                       M 
                       * 
                       
                         I 
                         2 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 
                   j 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ω 
                     ⁡ 
                     
                       ( 
                       
                         
                           L 
                           1 
                         
                         + 
                         
                           
                             M 
                             * 
                             
                               I 
                               2 
                             
                           
                           
                             I 
                             1 
                           
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     I 
                     1 
                   
                 
                 ≈ 
                 
                   j 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   ω 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       I 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           L 
                           1 
                         
                         + 
                         
                           
                             M 
                             * 
                             
                               Gm 
                               1 
                             
                             * 
                             
                               L 
                               1 
                             
                           
                           CR 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     As mentioned above, minimizing unexpected frequency drift accompanied by the temperature change is an important issue, and techniques for minimizing temperature coefficient (TC) is necessary. Instead of providing a compensation circuit having negative temperature coefficient (TC), the proposed compensation circuit actively reduces the intrinsic temperature coefficient (TC) of the primary circuit through the present disclosure. With the compensation apparatus according to the present disclosure, the oscillation frequency of the input signal provided by the primary circuit will be relatively independent of the temperature change. 
       FIG. 5  is a schematic diagram illustrating the frequency variations of the VCO circuit in  FIG. 4  with temperature drifts between −20 to 80° C. The vertical axis represents the frequency in MHz and the horizontal axis represents the temperature change. In  FIG. 5 , the frequency is a function of temperature for a fixed reference input voltage. 
     Referring to  FIG. 5 , in a case that no predetermined compensation current is outputted by the current source  4331  (that is, the curve annotated with CTAT=0), the oscillation frequency of the VCO circuit drifts from 2820 MHz to 2805 MHz within a range of temperature −20 to 80° C. After normalization and conversion, the temperature coefficient TC associated with the VCO circuit is represented as 59 ppm/° C. within the range −20 to 80° C. 
     As shown in  FIG. 5 , the slope of the curves decreases when the current source  4331  outputs the predetermined compensation current CTAT. With proper setting of CTAT, the frequency drift of the first voltage difference V 1  decreases and the oscillation frequency stability of the first voltage difference V 1  can be improved. In practical applications, the range of the predetermined current may vary between several μA to mA. 
     For example, the curves corresponding to CTAT=5, 6 or 7 μA are relatively flat in  FIG. 5 , meaning that the output frequency deviation of curves corresponding to the current source  4331  outputting current value of 5, 6 or 7 mA are flatter. In other words, after the compensation circuit is used, the temperature coefficient of the primary circuit is relatively close to a constant. 
     When the predetermined compensation current CTAT is 5 μA, the frequency of the first voltage difference V 1  drifts between 2800 MHz to 2975 MHz. When the predetermined compensation current CTAT is 6 μA, the frequency of the first voltage difference V 1  drifts between 2795 MHz to 2793 MHz. When the predetermined compensation current CTAT is 7 μA, the frequency of the first voltage difference V 1  drifts between 2793 MHz to 2792 MHz. When the predetermined compensation current CTAT is 15 μA, the frequency of the first voltage difference V 1  roughly drifts in between 2783 MHz to 2776 MHz. 
     In conclusion, once the current source  4331  in  FIG. 4  outputs a predetermined compensation current, the variance range of the frequency of the primary circuit is reduced. Consequentially, the VCO circuit with the inductance inducing circuit becomes more robust to temperature variation. 
     In one embodiment, the primary circuit can be an output of an oscillator, an output of voltage-controlled oscillator (VCO), or an output of digitally controlled oscillator (DCO).  FIG. 6  is a schematic diagram illustrating the application to a function circuit including a digitally controlled oscillator. The function circuit  60  includes a thermometer and an analog-to-digital converter (ADC)  61 , a DSP  63 , a sigma delta modulator (hereinafter, SDM)  65  and the DCO  67 . The thermometer and ADC  61  are coupled to the DSP  63 . The DSP  63  is coupled to the SDM  65  and the DCO  67 . The DSP  63  and the SDM  65  use switched capacitors (not shown) to fine tune the output of the DCO  67  respectively through a first fine tune path  62 , and a second fine tune path  64 . The output of the DCO  67  can be compensated by an inductor-based apparatus  69 . Details of which are not illustrated to avoid redundancy. 
     The proposed compensation circuit induces a change of resistance and/or inductance at the primary circuit. With such change of resistance and/or inductance, the temperature coefficient of the primary circuit is accordingly reduced, and the compensation circuit according to the present disclosure is capable of reducing the temperature coefficient. In other words, the compensation circuit adjusts the temperature coefficient of the primary circuit. 
     Depending on the required accuracy of the frequency drift, the temperature compensation circuit of the present disclosure may be used alone, or collaboratively used with other temperature compensation circuits. In some situations, the frequency deviation of the primary circuit is acceptable after the usage of the temperature compensation circuit according to the present disclosure. Thus, only the temperature compensation circuit according to the present disclosure needs to be adopted. 
     In some situations that a more limited deviation of the frequency drift is required, the present disclosure is capable of providing a coarse adjustment of the temperature coefficient, and fine adjustment of temperature coefficient can be collaboratively provided by other types of temperature coefficient compensation circuit. With the compensation circuit of the present disclosure, the temperature coefficient of the primary circuit has been shrunk. Therefore, even if the previously mentioned varactor based temperature compensation circuits need to be used together, the extra area occupied by the varactors can be accordingly reduced. 
     It will be apparent to those skilled in the art that various modifications and variations can be made to the disclosed embodiments. It is intended that the specification and examples be considered as exemplary only, with a true scope of the disclosure being indicated by the following claims and their equivalents.