Abstract:
An embodiment of the voltage controlled oscillator is provided. The oscillator comprises a first inductor set, a second inductor set, a second capacitor, a voltage source and a negative resistance element. The inductance of the second inductor set is k times the inductance of the first inductor set. The voltage source applies an ac voltage to the second inductor set. The negative resistance element is coupled to the second inductor set to provide a negative resistance to resonate the second capacitor at the second inductor set.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention relates to a voltage controlled oscillator, and more particularly to a voltage controlled oscillator with a transformer. 
     2. Description of the Related Art 
     The voltage controlled oscillator is a component used as a local oscillator in many wireless communication devices. With a voltage controlled oscillator circuit, a desired frequency can be easily obtained by adjusting the charging and discharging times of capacitors or capacitive elements. For this reason, a voltage-controlled oscillator circuit is normally used in an apparatus which requires different clock frequencies. Today, CMOS integrated circuits of low power consumption have become widespread. With this trend, various voltage controlled oscillator circuits of the CMOS configuration are being developed. However, an LC tank voltage controlled oscillator usually comprises a large capacitor and a small inductor. Meanwhile, parasitic capacitor effect increases due to the small inductor, resulting in phase noise in the output clock signal of the voltage controlled oscillator. 
       FIG. 1  is a circuit diagram of an LC oscillator. The inductor L 1  placed in parallel with a capacitor C 1  resonates at a frequency ω=1/√{square root over (L 1 C 1 )}. At this frequency, the impedances of the inductor and the capacitor are equal and opposite. However, in practice, inductors or/and capacitors suffer from resistive components. For example, the series resistance of the metal wire in the inductor can be modeled as the resistor R s  shown in  FIG. 1 . An infinite quality factor Q of the inductor L 1  is defined as L 1 ω/R s . In the  FIG. 1 , the equivalent impedance is given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         Z 
                         eq 
                       
                       ⁡ 
                       
                         ( 
                         s 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           R 
                           s 
                         
                         + 
                         
                           
                             L 
                             1 
                           
                           ⁢ 
                           s 
                         
                       
                       
                         1 
                         + 
                         
                           
                             L 
                             1 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                           ⁢ 
                           
                             s 
                             2 
                           
                         
                         + 
                         
                           
                             R 
                             s 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                           ⁢ 
                           s 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
     and hence, 
     
       
         
           
             
               
                 
                   
                     
                        
                       
                         
                           Z 
                           eq 
                         
                         ⁡ 
                         
                           ( 
                           
                             s 
                             = 
                             jω 
                           
                           ) 
                         
                       
                        
                     
                     2 
                   
                   = 
                   
                     
                       
                         
                           R 
                           s 
                           2 
                         
                         + 
                         
                           
                             L 
                             1 
                             2 
                           
                           ⁢ 
                           
                             ω 
                             2 
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   L 
                                   1 
                                 
                                 ⁢ 
                                 
                                   C 
                                   1 
                                 
                                 ⁢ 
                                 
                                   ω 
                                   2 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             R 
                             s 
                             2 
                           
                           ⁢ 
                           
                             C 
                             1 
                             2 
                           
                           ⁢ 
                           
                             ω 
                             2 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
     The magnitude of Z e q in eq. 2 reaches to a peak value in the vicinity of ω=1/√{square root over (L 1 C 1 )}, but the actual resonance frequency is still partially dependent on R s . 
     According to the described, controlling the resonance frequency by adjusting the capacitor C 1  and inductor L 1  should be achievable. However, in practice, the inductance of the inductor L 1  is not easily precisely adjusted, and the capacitor C 1  may suffer parasitic capacitor effect. Thus, the resonance frequency ω is unstable. 
     BRIEF SUMMARY OF THE INVENTION 
     An embodiment of the voltage controlled oscillator according to the invention is disclosed. The voltage controlled oscillator comprises a first capacitor, a first inductor, a second inductor, a third inductor, a fourth inductor, a voltage source and a negative resistance element. The first inductor comprises a first terminal coupled to the first terminal of the third inductor and a second terminal coupled to the first terminal of the first capacitor. The second inductor comprises a first terminal coupled to the first terminal of the fourth inductor and a second terminal coupled to the second terminal of the first capacitor. The second capacitor comprises a first terminal coupled to the second terminal of the third inductor and a second terminal coupled to the second terminal of the fourth inductor. The inductance of the third inductor or the fourth inductor is N times the inductance of the first inductor or the second inductor. The voltage source applies an ac voltage to the third and fourth inductors. The negative resistance element is coupled to the third and fourth inductors to provide a negative resistance. 
     Another embodiment of a phase locked loop device is provided. The PLL device comprises a PFD unit, a charging pump circuit, a loop filter, and a voltage controlled oscillator. The PFD unit measures a phase and a frequency difference between a reference clock signal and a feedback clock signal of the PLL device and the PFD unit outputs a difference signal UP and a difference signal DN. The charging pump circuit receives and transforms the difference signals UP and DN into a current. The loop filter receives and transforms the current into a voltage. The voltage controlled oscillator receives the voltage and outputs an output signal, wherein the voltage controlled oscillator comprises a first inductor set and a second inductor set, and the inductance of the second inductor set is k times the inductance of the first inductor set. 
     A detailed description is given in the following embodiments with reference to the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present invention can be more fully understood by reading the subsequent detailed description and examples with references made to the accompanying drawings, wherein: 
         FIG. 1  is a circuit diagram of an LC oscillator. 
         FIG. 2  is a circuit diagram of an oscillator based on an embodiment of the invention. 
         FIG. 3  is a circuit diagram of an oscillator based on another embodiment of the invention. 
         FIG. 4  is an equivalent circuit diagram of the voltage controlled oscillator shown in  FIG. 2 . 
         FIG. 5  is a schematic diagram of a phase locked loop device of an embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The following description is of the best-contemplated mode of carrying out the invention. This description is made for the purpose of illustrating the general principles of the invention and should not be taken in a limiting sense. The scope of the invention is best determined by reference to the appended claims. 
       FIG. 2  is a circuit diagram of an oscillator based on an embodiment of the invention. The LC circuit  21  comprises an inductor L 2a , an inductor L 2b  and a second capacitor C 2 . The inductor L 2a  comprises a second terminal and a first terminal coupled to one terminal of the second capacitor C 2 . The inductor L 2b  comprises a first terminal coupled to another terminal of the second capacitor C 2 , and a second terminal coupled to the second terminal of the inductor L 2 a. The inductor L 2a  is further connected to the inductor L 1a , and the inductor L 2b  is further connected to the inductor L 1b . The inductor L 1a  comprises a second terminal and a first terminal coupled to one terminal of the first capacitor C 1 . The inductor L 1b  comprises a first terminal coupled to another terminal of the first capacitor C 1 , and a second terminal coupled to the second terminal of the inductor L 1 a. In this embodiment, the inductance of the inductor L 1a  is N times the inductance of the second inductor L 2a , and the inductance of the inductor L 1b  is N times the inductance of the second inductor L 2b . In this embodiment, the inductors L 1a , L 1b , L 2a  and L 2b  receive the same voltage V DD . A negative resistance element  22  is coupled to the inductors L 1a  and L 1b  to provide a negative resistance to resonate the inductors L 1a , L 1b  and the first capacitor C 1 . The negative resistance element  22  comprises a first transistor and a second transistor. The first transistor T 1  comprises a first input terminal coupled to the first clock output terminal N 1 , a first output terminal, and a first control terminal coupled to the second clock output terminal N 2 . The second transistor T 2  comprises a second input terminal coupled to the second clock output terminal N 2 , a second output terminal coupled to the first output terminal, and a second control terminal coupled to the first clock output terminal N 1 . The negative resistance element  22  further comprises a resistor R coupled between the first output terminal and ground. In this embodiment, the clock output terminal N 1  and N 2  are respectively coupled to the buffer  23  and  24  to output the output differential clock signal V out+  and V out− . 
     Without the inductors L 2a  and L 2b  and the second capacitor C 2 , the resonance frequency is ω=1/√{square root over (L 1 (C 1 +C para ))}. With the inductors L 2a  and L 2b  and the second capacitor C 2 , the resonance frequency is as followed: 
     
       
         
           
             
               
                 
                   
                     ω 
                     1 
                   
                   = 
                   
                     1 
                     / 
                     
                       
                         
                           
                             L 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 C 
                                 1 
                               
                               + 
                               
                                 C 
                                 para 
                               
                             
                             ) 
                           
                         
                         
                           
                             refered 
                             ⁢ 
                             _ 
                             ⁢ 
                             to 
                           
                           ⁢ 
                           
                             _ 
                             ⁢ 
                             first 
                           
                           ⁢ 
                           
                             _ 
                             ⁢ 
                             inducto 
                             ⁢ 
                             r 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       1 
                       / 
                       
                         L 
                         1 
                       
                     
                     · 
                     
                       
                         ( 
                         
                           
                             
                               C 
                               para 
                             
                             + 
                             
                               C 
                               1 
                             
                           
                           
                             N 
                             2 
                           
                         
                         ) 
                       
                       
                         
                           refered 
                           ⁢ 
                           _ 
                           ⁢ 
                           to 
                         
                         ⁢ 
                         
                           _ 
                           ⁢ 
                           second 
                         
                         ⁢ 
                         
                           _ 
                           ⁢ 
                           inductor 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     The parasitic capacitor C para  is generated at the input terminal of the first transistor T 1 . In this embodiment, only the parasitic capacitor C para  at the input terminal of the first transistor T 1  is described for illustration, while in practice, the parasitic capacitor is also generated at the input terminal of the second transistor T 2 . According to the resonance frequency ω 1 , the parasitic capacitor effect caused by the parasitic capacitor C para  can be significantly reduced. Furthermore, inductors L 1a , L 1b  and the capacitor C 1  adjustments are easier. 
       FIG. 3  is a circuit diagram of an oscillator based on another embodiment of the invention. The second inductor L 2  is further connected to a first inductor L 1 . The first inductor L 1  is parallel connected to a first capacitor C 1 . In this embodiment, the inductance of the first inductor L 1  is N times the inductance of the second inductor L 2 . In this embodiment, the first inductor L 1  receives the voltage V DD , and the second inductor L 2  receives the voltage V 2 . In this embodiment, the capacitors C 1  and C 2  are voltage controlled capacitors and their capacitance can be adjusted respectively by the voltages V DD  and V 2 . The first transistor T 1  and a second transistor T 2  form a negative resistance element  22  which is coupled to the first inductor L 1  to provide a negative resistance to resonate the first inductor L 1  and the first capacitor C 1 . The first transistor T 1  comprises a first input terminal coupled to the first clock output terminal N 1 , a first output terminal, and a first control terminal coupled to the second clock output terminal N 2 . The second transistor T 2  comprises a second input terminal coupled to the second clock output terminal N 2 , a second output terminal coupled to the first output terminal, and a second control terminal coupled to the first clock output terminal N 1 . The negative resistance element  22  further comprises a resistor R coupled between the first output terminal and ground. In this embodiment, the clock output terminal N 1  and N 2  are respectively coupled to the buffer  23  and  24  to output the output differential clock signal V out+  and V out− . 
       FIG. 4  is a schematic diagram of the input impedance to the voltage controlled transformer with transformer according to the invention. The input impedance Z IN  is derived as following. 
     
       
         
           
             
               
                 
                   
                     Z 
                     IN 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             sL 
                             2 
                           
                           ⁢ 
                           
                              
                              
                           
                           ⁢ 
                           
                             1 
                             
                               sC 
                               2 
                             
                           
                         
                         + 
                         
                           sL 
                           1 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                        
                        
                     
                     ⁢ 
                     
                       1 
                       
                         sC 
                         1 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       ( 
                       
                         
                           
                             sL 
                             2 
                           
                           
                             
                               
                                 sL 
                                 2 
                               
                               · 
                               
                                 sC 
                                 2 
                               
                             
                             + 
                             1 
                           
                         
                         + 
                         
                           sL 
                           1 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                        
                        
                     
                     ⁢ 
                     
                       1 
                       
                         sC 
                         1 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       ( 
                       
                         
                           
                             sL 
                             2 
                           
                           
                             1 
                             + 
                             
                               
                                 S 
                                 2 
                               
                               ⁢ 
                               
                                 L 
                                 2 
                               
                               ⁢ 
                               
                                 C 
                                 2 
                               
                             
                           
                         
                         + 
                         
                           sL 
                           1 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                        
                        
                     
                     ⁢ 
                     
                       1 
                       
                         sC 
                         1 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         
                           sL 
                           2 
                         
                         
                           1 
                           + 
                           
                             
                               S 
                               2 
                             
                             ⁢ 
                             
                               L 
                               2 
                             
                             ⁢ 
                             
                               C 
                               2 
                             
                           
                         
                       
                       + 
                       
                         sL 
                         1 
                       
                     
                     
                       1 
                       + 
                       
                         
                           sC 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 sL 
                                 2 
                               
                               
                                 1 
                                 + 
                                 
                                   
                                     S 
                                     2 
                                   
                                   ⁢ 
                                   
                                     L 
                                     2 
                                   
                                   ⁢ 
                                   
                                     C 
                                     2 
                                   
                                 
                               
                             
                             + 
                             
                               sL 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       ( 
                       
                         
                           sL 
                           2 
                         
                         + 
                         
                           
                             sL 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   s 
                                   2 
                                 
                                 ⁢ 
                                 
                                   L 
                                   2 
                                 
                                 ⁢ 
                                 
                                   C 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               s 
                               2 
                             
                             ⁢ 
                             
                               L 
                               2 
                             
                             ⁢ 
                             
                               C 
                               2 
                             
                           
                         
                         ) 
                       
                       + 
                       
                         
                           sC 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               sL 
                               2 
                             
                             + 
                             
                               
                                 sL 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       s 
                                       2 
                                     
                                     ⁢ 
                                     
                                       L 
                                       2 
                                     
                                     ⁢ 
                                     
                                       C 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     The denominator part: 
     
       
         
           
             
               
                 ( 
                 
                   1 
                   + 
                   
                     
                       s 
                       2 
                     
                     ⁢ 
                     
                       L 
                       2 
                     
                     ⁢ 
                     
                       C 
                       2 
                     
                   
                 
                 ) 
               
               + 
               
                 
                   sC 
                   1 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       sL 
                       2 
                     
                     + 
                     
                       
                         sL 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               s 
                               2 
                             
                             ⁢ 
                             
                               L 
                               2 
                             
                             ⁢ 
                             
                               C 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 1 
                 + 
                 
                   
                     s 
                     2 
                   
                   ⁢ 
                   
                     L 
                     2 
                   
                   ⁢ 
                   
                     C 
                     2 
                   
                 
                 + 
                 
                   
                     
                       s 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           L 
                           2 
                         
                         + 
                         
                           L 
                           1 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     C 
                     1 
                   
                 
                 + 
                 
                   
                     s 
                     4 
                   
                   ⁢ 
                   
                     L 
                     1 
                   
                   ⁢ 
                   
                     L 
                     2 
                   
                   ⁢ 
                   
                     C 
                     1 
                   
                   ⁢ 
                   
                     C 
                     2 
                   
                 
               
               = 
               
                 
                   1 
                   + 
                   
                     
                       s 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             L 
                             2 
                           
                           ⁢ 
                           
                             C 
                             2 
                           
                         
                         + 
                         
                           
                             L 
                             2 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                         + 
                         
                           
                             L 
                             1 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       s 
                       4 
                     
                     ⁢ 
                     
                       L 
                       1 
                     
                     ⁢ 
                     
                       L 
                       2 
                     
                     ⁢ 
                     
                       C 
                       1 
                     
                     ⁢ 
                     
                       C 
                       2 
                     
                   
                 
                 = 
                 
                   1 
                   - 
                   
                     
                       ω 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             L 
                             2 
                           
                           ⁢ 
                           
                             C 
                             2 
                           
                         
                         + 
                         
                           
                             L 
                             2 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                         + 
                         
                           
                             L 
                             1 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       ω 
                       4 
                     
                     ⁢ 
                     
                       L 
                       1 
                     
                     ⁢ 
                     
                       L 
                       2 
                     
                     ⁢ 
                     
                       C 
                       1 
                     
                     ⁢ 
                     
                       C 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               ω 
               2 
             
             = 
             
               
                 1 
                 2 
               
               ⁢ 
               
                 { 
                 
                   ( 
                   
                     
                       
                         L 
                         2 
                       
                       ⁢ 
                       
                         C 
                         2 
                       
                     
                     + 
                     
                       
                         L 
                         2 
                       
                       ⁢ 
                       
                         C 
                         1 
                       
                     
                     + 
                     
                       
                         
                           L 
                           1 
                         
                         ⁢ 
                         
                           C 
                           1 
                         
                       
                       ± 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     L 
                                     2 
                                   
                                   ⁢ 
                                   
                                     C 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     L 
                                     2 
                                   
                                   ⁢ 
                                   
                                     C 
                                     1 
                                   
                                 
                                 + 
                                 
                                   
                                     L 
                                     1 
                                   
                                   ⁢ 
                                   
                                     C 
                                     1 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             4 
                             ⁢ 
                             
                               L 
                               1 
                             
                             ⁢ 
                             
                               L 
                               2 
                             
                             ⁢ 
                             
                               C 
                               1 
                             
                             ⁢ 
                             
                               C 
                               2 
                             
                           
                         
                       
                     
                   
                   ) 
                 
                 } 
               
             
           
         
       
     
     In this embodiment, we only consider the high frequency response portion, i.e. with “+” sign, because the inductor Q corresponding to the low frequency is too low and the voltage controlled oscillator will still be oscillated. 
     
       
         
           
             
               ω 
               2 
             
             = 
             
               
                 
                   
                     
                       L 
                       2 
                     
                     ⁢ 
                     
                       C 
                       2 
                     
                   
                   + 
                   
                     
                       L 
                       2 
                     
                     ⁢ 
                     
                       C 
                       1 
                     
                   
                   + 
                   
                     
                       L 
                       1 
                     
                     ⁢ 
                     
                       C 
                       1 
                     
                   
                 
                 2 
               
               ⁢ 
               
                 { 
                 
                   1 
                   + 
                   
                     
                       1 
                       - 
                       
                         
                           4 
                           ⁢ 
                           
                             L 
                             1 
                           
                           ⁢ 
                           
                             L 
                             2 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                           ⁢ 
                           
                             C 
                             2 
                           
                         
                         
                           
                             ( 
                             
                               
                                 
                                   L 
                                   2 
                                 
                                 ⁢ 
                                 
                                   C 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   L 
                                   2 
                                 
                                 ⁢ 
                                 
                                   C 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   L 
                                   1 
                                 
                                 ⁢ 
                                 
                                   C 
                                   1 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
                 } 
               
             
           
         
       
     
     Since √{square root over (1+x)}≅1+0.5x, thus, 
     
       
         
           
             
               
                 
                   
                     ω 
                     2 
                   
                   = 
                   
                     
                       
                         
                           
                             L 
                             2 
                           
                           ⁢ 
                           
                             C 
                             2 
                           
                         
                         + 
                         
                           
                             L 
                             2 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                         + 
                         
                           
                             L 
                             1 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                       
                       2 
                     
                     ⁢ 
                     
                       { 
                       
                         1 
                         + 
                         1 
                         - 
                         
                           
                             2 
                             ⁢ 
                             
                               L 
                               1 
                             
                             ⁢ 
                             
                               L 
                               2 
                             
                             ⁢ 
                             
                               C 
                               1 
                             
                             ⁢ 
                             
                               C 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   
                                     L 
                                     2 
                                   
                                   ⁢ 
                                   
                                     C 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     L 
                                     2 
                                   
                                   ⁢ 
                                   
                                     C 
                                     1 
                                   
                                 
                                 + 
                                 
                                   
                                     L 
                                     1 
                                   
                                   ⁢ 
                                   
                                     C 
                                     1 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       ( 
                       
                         
                           
                             L 
                             2 
                           
                           ⁢ 
                           
                             C 
                             2 
                           
                         
                         + 
                         
                           
                             L 
                             2 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                         + 
                         
                           
                             L 
                             1 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       { 
                       
                         1 
                         - 
                         
                           
                             
                               L 
                               1 
                             
                             ⁢ 
                             
                               L 
                               2 
                             
                             ⁢ 
                             
                               C 
                               1 
                             
                             ⁢ 
                             
                               C 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   
                                     L 
                                     2 
                                   
                                   ⁢ 
                                   
                                     C 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     L 
                                     2 
                                   
                                   ⁢ 
                                   
                                     C 
                                     1 
                                   
                                 
                                 + 
                                 
                                   
                                     L 
                                     1 
                                   
                                   ⁢ 
                                   
                                     C 
                                     1 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       } 
                     
                   
                 
               
             
           
         
       
     
     Assuming C 2 &lt;&lt;C 1   
     
       
         
           
             
               
                 
                   
                     ω 
                     2 
                   
                   ≅ 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           
                             L 
                             2 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                         + 
                         
                           
                             L 
                             1 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       { 
                       
                         1 
                         - 
                         
                           
                             
                               L 
                               1 
                             
                             ⁢ 
                             
                               L 
                               2 
                             
                             ⁢ 
                             
                               C 
                               1 
                             
                             ⁢ 
                             
                               C 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   
                                     L 
                                     2 
                                   
                                   ⁢ 
                                   
                                     C 
                                     1 
                                   
                                 
                                 + 
                                 
                                   
                                     L 
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                                   ⁢ 
                                   
                                     C 
                                     1 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       L 
                       1 
                     
                     ⁢ 
                     
                       
                         C 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
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                           N 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         1 
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                               L 
                               1 
                             
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                               L 
                               2 
                             
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                               C 
                               1 
                             
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                                 ( 
                                 
                                   
                                     L 
                                     1 
                                   
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                                     C 
                                     1 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   1 
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                                   N 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   ≅ 
                     
                   ⁢ 
                   
                     
                       L 
                       1 
                     
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                         C 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
                           + 
                           N 
                         
                         ) 
                       
                     
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                       { 
                       
                         1 
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                                 L 
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                                 L 
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                             1 
                             
                               
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                                   1 
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                                   N 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   ≅ 
                     
                   ⁢ 
                   
                     
                       
                         L 
                         1 
                       
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                           C 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             1 
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                           ) 
                         
                       
                     
                     - 
                     
                       
                         
                           L 
                           2 
                         
                         ⁢ 
                         
                           C 
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                           ( 
                           
                             1 
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                           ) 
                         
                         2 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ∂ 
                 
                   ω 
                   2 
                 
               
               
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                   C 
                   1 
                 
               
             
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                 L 
                 1 
               
               ⁡ 
               
                 ( 
                 
                   1 
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                   N 
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 
                   ∂ 
                   
                     ω 
                     2 
                   
                 
                 
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                     2 
                   
                 
               
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                       ) 
                     
                     2 
                   
                 
               
             
             = 
             
               
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                       ) 
                     
                     2 
                   
                 
               
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                 L 
                 1 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     ∂ 
                     
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                       2 
                     
                   
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                       2 
                     
                   
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                     L 
                     1 
                   
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                     ( 
                     
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                     ) 
                   
                 
                 
                   
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                       N 
                       
                         
                           ( 
                           
                             1 
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                             N 
                           
                           ) 
                         
                         2 
                       
                     
                   
                   ⁢ 
                   L 
                 
               
             
             = 
             
               
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                     ( 
                     
                       1 
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                     ) 
                   
                   
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                         ( 
                         
                           1 
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                           N 
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               ≅ 
               
                 
                   ( 
                   
                     1 
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                   ) 
                 
                 2 
               
             
           
         
       
     
       FIG. 5  is a schematic diagram of a phase locked loop device of an embodiment of the invention. PFD unit  51  receives a reference clock signal REF_CK and a feedback clock signal FBK_CK and measures the phase and frequency difference therebetween to output phase difference signals, UP and DN. Charging pump circuit  52  receives and transforms the phase difference signals UP and DN into a current to charge loop filter  53 . The loop filter  53  receives the current from charging pump circuit to limit the rate of change of a capacitor voltage, VCON, resulting in slow rising or falling voltage corresponding to the phase and frequency difference. The voltage controlled oscillator (VCO)  54  generates an output clock signal according to the voltage VCON. The voltage controlled oscillator  54  comprises a first inductor and a second inductor, and the inductance of the first inductor is N times the inductance of the second inductor. Two embodiments of the voltage controlled oscillator  54  are illustrated with the oscillators shown in the  FIG. 2  and  FIG. 3 . Feedback divider  55  has a parameter M to generate the feedback clock signal FBK_CK with wider range frequency, wherein the frequency of the feedback clock signal FBK_CK is M times the frequency of the output clock signal. In an ideal situation, when the PLL is in an in-lock state, the phase difference signal UP synchronizes to the phase difference signal DN. 
     While the invention has been described by way of example and in terms of preferred embodiment, it is to be understood that the invention is not limited thereto. To the contrary, it is intended to cover various modifications and similar arrangements (as would be apparent to those skilled in the art). Therefore, the scope of the appended claims should be accorded the broadest interpretation so as to encompass all such modifications and similar arrangements.