Abstract:
A phase-locked loop (PLL) with a decreased frequency tuning gain K VCO  and a loop filter using capacitor multiplication technique to get high chip area efficiency. To get decreased frequency tuning gain, K VCO , a voltage to current converter in a voltage-controlled oscillator (VCO) in the PLL may comprise a first voltage to current converter and a second voltage to current converter. The trans-conductance of the first voltage to current converter is 1/β of that of the second voltage to current converter, wherein β&gt;1. The first voltage to current converter is controlled by an output voltage of a loop filter in the PLL, and the second voltage to current converter is controlled by a relative DC voltage, which may be the junction node between R1 and C1 in a loop filer of the PLL. Capacitor multiplication technique may use an auxiliary charge pump to charge or discharge the junction node between R1 and C1 inversely to the main charge pump. When the charge or discharge current unit of the auxiliary charge pump is α times of the main charge pump, the capacitance of C1 may be reduced to just (1−α) times of what it needed in a conventional loop stability compensation method, wherein α&lt;1.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of priority to previously filed U.S. provisional patent application Ser. No. 61/078,962, filed Jul. 8, 2008, entitled LOW-JITTER, RING BASED PHASE LOCKED LOOP WITH A VOLTAGE CONTROLLED OSCILLATOR STRUCTURE OF LOW FREQUENCY TUNING GAIN AND A LOOP FILTER STRUCTURE OF HIGH AREA EFFICIENCY. That provisional application is hereby incorporated by reference in its entirety. 
    
    
     BACKGROUND 
     1. Field of the Invention 
     The present invention relates generally to phase-locked loops (PLLs). 
     2. Description of Related Art 
     Phase-locked loops are widely used in electronic devices, such as computers, telecommunications equipment and radio. However, known PLLs have been insufficiently noise insensitive. 
       FIG. 1  illustrates a known PLL  100 , which is used to maintain an output frequency ω out  to be a multiple times of a reference frequency ω ref . The PLL  100  has a phase and frequency detector (PFD)  101 , a charge pump  102 , a loop filter  103 , a voltage-controlled oscillator (VCO)  104  and a divider  105 . The VCO  104  has a voltage supply  1041  which as shown may be, e.g., a high power supply rejection (PSR) regulator, a voltage to current converter  1042  and a current-controlled oscillator (ICO)  1043 . The ICO  1043  may be, e.g., a ring based oscillator. The voltage to current converter  1042 , having a voltage to current converting rate gm=I C /V ctrl , may generate a control current I C  in proportion with a control voltage V ctrl  from the loop filter  103 . The ICO  1043 , having a frequency tuning gain K ICO =ω out /I c , may output a frequency ω out  based on the control current I C . The divider  105  is on the feedback route from the output of the ICO  1043  to the input of the PFD  101 , and may divide the output frequency ω out  by an integer N and send a feedback frequency, ω fb =ω out /N, to the input of the PFD  101 . 
     The PFD  101  may compare the feedback frequency ω fb  with the reference frequency ω ref . When ω fb  is lower than the reference frequency ω ref , the PFD  101  may output switching signals PU and PD to the charge pump  102 , closing a switch  1021  and keeping a switch  1022  open, so as to charge a charge storage device C1 in the loop filter  103 . Consequently, the control voltage V ctrl  at the output of the loop filter  103  is up, the control current I C  supplied to the ICO  1043  is up, and the output frequency ω out  is up until it equals N ω ref . 
     When ω is higher than the reference frequency ω ref , the PFD  101  may generate switching signals PU and PD to open the switch  1021  and close the switch  1022 , so that the charge storage device C1 in the loop filter  103  may discharge via the switch  1022 . Consequently, the control voltage V ctrl  at the output of the loop filter  103  is down, the control current I C  is down, and the output frequency ω out  is down, until ω out =N ω ref . 
     When ω fb  equals ω ref , the PFD  101  may keep both switches  1021  and  1022  closed to maintain the relationship. 
     The pole position and zero position of the PLL  100  are: 
     
       
         
           
             
               
                 
                   
                     ω 
                     z 
                   
                   = 
                   
                     1 
                     
                       
                         R 
                         1 
                       
                       ⁢ 
                       
                         C 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
             
               
                 
                   
                     ω 
                     p 
                   
                   = 
                   
                     
                       
                         1 
                         
                           
                             R 
                             1 
                           
                           ⁢ 
                           
                             C 
                             2 
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               C 
                               2 
                             
                             
                               C 
                               1 
                             
                           
                         
                         ) 
                       
                     
                     ≈ 
                     
                       1 
                       
                         
                           R 
                           1 
                         
                         ⁢ 
                         
                           C 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     So the ratio between the pole and zero position is: 
     
       
         
           
             
               
                 
                   
                     
                       ω 
                       p 
                     
                     
                       ω 
                       z 
                     
                   
                   = 
                   
                     
                       
                         C 
                         1 
                       
                       + 
                       
                         C 
                         2 
                       
                     
                     
                       C 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     The gain bandwidth of the PLL  100  is: 
     
       
         
           
             
               
                 
                   
                     ω 
                     c 
                   
                   ≈ 
                   
                     
                       
                         I 
                         cp 
                       
                       
                         2 
                         * 
                         π 
                       
                     
                     * 
                     
                       R 
                       1 
                     
                     * 
                     
                       g 
                       m 
                     
                     * 
                     
                       K 
                       ICO 
                     
                     * 
                     
                       1 
                       N 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     For the PLL  100 , the output phase noise at the output frequency ω out  contributed by R1 is: 
     
       
         
           
             
               
                 
                   
                     ϕ 
                     n 
                     2 
                   
                   = 
                   
                     4 
                     * 
                     K 
                     * 
                     T 
                     * 
                     
                       R 
                       1 
                     
                     * 
                     
                       
                         ( 
                         
                           
                             g 
                             m 
                           
                           * 
                           
                             
                               K 
                               ICO 
                             
                             s 
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     wherein K is the Boltzmann constant, T is a temperature value, and s=jω, which is a variable in frequency domain. 
     One limitation of the known PLL  100  is its high frequency tuning gain (K VCO ), which is the frequency/voltage gain of the VCO. The high frequency tuning gain, together with a wide loop bandwidth used to suppress phase noise, may make the PLL more sensitive to the noise from the PFD, the charge pump and the loop filter. Therefore, it may be desirable to provide a PLL which may have a decreased frequency tuning gain K VCO  and the wide loop bandwidth. 
     Another limitation of the known PLL  100  is that the loop filter  103  may occupy a large chip area when being integrated on a chip. One known solution uses an active loop filter, but the active device may bring additional noise. Another known solution uses a passive feed forward loop filter with noiseless resistor multiplication, but it may increase sensitivity to switching glitches of the charge pump. Therefore, it may be desirable to provide a PLL with a loop filter which has high chip area efficiency but low noise and switching glitch sensitivity. 
     SUMMARY 
     A voltage-controlled oscillator (VCO), comprising: a voltage supply; a voltage to current converter for converting a voltage signal from the voltage supply into a control current, the voltage to current converter comprising a first voltage to current converter and a second voltage to current converter coupled in parallel to produce the control current; and a current-controlled oscillator (ICO), generating an output frequency in response to the control current. 
     A phase-locked loop (PLL), comprising: a phase and frequency detector (PFD) for comparing a feedback frequency and a reference frequency and generating a first switching signal and a second switching signal; a first charge pump, comprising a first current source coupled in series with a first switch, a second switch and a second current source, wherein the first switch is controlled by the first switching signal, the second switch is controlled by the second switching signal and the first current source provides a current I CP ; a loop filter, comprising a first branch and a second branch coupled in parallel, wherein the first branch comprises a resistance device coupled in series with a first charge storage device, and the second branch comprises a second charge storage device; and a voltage controlled oscillator (VCO) providing an output frequency, wherein the output frequency is sent to the PFD as the feedback frequency. The VCO comprise: a voltage supply; a voltage to current converter for converting a voltage signal from the voltage supply into a control current, the voltage to current converter comprising a first voltage to current converter and a second voltage to current converter coupled in parallel to produce the control current; and a current-controlled oscillator (ICO), generating the output frequency in response to the control current. 
     A capacitor multiplication structure may be used in the PLL to increase the capacitance of the first charge storage device without increasing its size. The PLL may further comprise a second charge pump coupled in parallel with the first charge pump, wherein the second charge pump comprises a third current source, a third switch, a fourth switch and a fourth current source coupled in series, to charge or discharge the junction of the resistance device and the first charge storage device. The third current source provides a current α*I CP  to decrease voltage variations over the first charge storage device, and wherein α&lt;1. The third switch is turned on and off approximately simultaneously with the first switch, but in the opposite direction. The fourth switch is turned on and off approximately simultaneously with the second switch, but in the opposite direction. 
     A method for controlling a voltage-controlled oscillator (VCO), comprising: converting a voltage signal into a first control current with a first voltage to current converter; converting a voltage signal from the voltage supply into a second control current with a second voltage to current converter; combining the first control current and the second control current into a control current; and generating an output frequency in response to the control current with a current-controlled oscillator (ICO). 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWING FIGURES 
       Embodiments are described herein with reference to the accompanying drawings, similar reference numbers being used to indicate functionally similar elements. 
         FIG. 1  illustrates a known PLL. 
         FIG. 2  is a circuit schematic depicting a VCO with a low frequency tuning gain according to one embodiment. 
         FIG. 3  is a circuit schematic depicting a PLL having a VCO with a low frequency tuning gain according to one embodiment. 
         FIG. 4  illustrates an exemplary circuit implementation of the PLL in  FIG. 3 . 
         FIG. 5  illustrates an example of the PSR performance (power supply noise rejection from V DD     —     VCO  to V C ) of the PLL in  FIG. 4 . 
         FIG. 6  illustrates an exemplary circuit implementation of the PLL in  FIG. 3 . 
         FIG. 7  illustrates an exemplary circuit implementation of the PLL in  FIG. 3 . 
         FIG. 8  illustrates a transient simulation result for control voltages V ctrl     —     A  and V ctrl     —     B  in  FIG. 3  when the PLL  300  is acquiring lock. 
         FIGS. 9A ,  9 B,  9 C and  9 D illustrate exemplary relationships between K VCO ′ and V ctrl     —     B  in the PLL of  FIG. 3 . 
         FIG. 10  is a circuit schematic depicting a PLL with the VCO in  FIG. 2  and a passive loop filter with capacitor multiplication according to one embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     The following discussion describes a PLL with an oscillator structure having a low frequency tuning gain K VCO  and a passive loop filter structure with noiseless capacitor multiplication. 
       FIG. 2  is a circuit schematic depicting a VCO with a low frequency tuning gain according to one embodiment. 
     A VCO  204  may include a voltage supply such as a high PSR regulator  2041 , a voltage to current converter  2042  and an ICO  2043 . The voltage to current converter  2042  may include two voltage to current converters,  2042 A and  2042 B, coupled in parallel between the voltage supply  2041  and the ICO  2043 . The converter  2042 A may be controlled by a control voltage V ctrl     —     A  from a voltage source A, which may be, e.g., the output of a loop filter in a PLL. The converter  2042 B may be controlled by a control voltage V ctrl     —     B  from a voltage source B, which may be, e.g., a relative DC voltage. The converter  2042 A may have a voltage to current converting rate, or trans-conductance, g m     —     A , and generate a control current I A  based on the control voltage V ctrl     —     A . The converter  2042 B may have a trans-conductance g m     —     B , and generate a control current I B  based on the control voltage V ctrl     —     B . Control currents I A  and I B  may be added at C and a combined control current I C  may be provided to the ICO  2043 . 
     In one embodiment, g m     —     A :g m     —     B =1: β, and β&gt;1. Thus, in the voltage to current converter  2042 :
 
 g   m   =g   m     —     A   +g   m     —     B   =g   m     —     A (1+β)= g   m     —     B (1/β+1);   (6)
 
 g   m     —     A   =g   m /(1+β);   (7)
 
 g   m     —     B   =β*g   m /(β+1);   (8)
 
     According to equation (7), when β is considerably greater than 1, the trans-conductance g m     —     A  of the converter  2042 A may be only a small part of the trans-conductance g m  of the voltage to current converter  2042 . 
     When V ctrl     —     B  is stable, I B  may be stable, and l C  may only change with V ctrl     —     A . In a small signal model, the equivalent resistor of the ICO  2043  may be much larger than 1/g m     —     A , and
 
 V   C   =V   ctrl     —     A ( g   m     —     A /( g   m     —     A   +g   m     —     B )= V   ctrl     —     A /(1+β)   (9)
 
     Assuming that K VCO ′ is the tuning gain from V ctrl     —     A  to the output frequency, and K VCO  is the tuning gain from V c  to the output frequency, then: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           K 
                           VCO 
                           ′ 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             ω 
                             out 
                           
                           / 
                           
                             V 
                             ctrl_A 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ω 
                             out 
                           
                           / 
                           
                             ( 
                             
                               
                                 V 
                                 c 
                               
                               * 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   β 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             K 
                             VCO 
                           
                           / 
                           
                             ( 
                             
                               1 
                               + 
                               β 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     When the trans-conductance of the VCO  204  is similar to the trans-conductance of the VCO  104  in the known PLL  100 , by using two converters  2042 A and  2042 B and controlling the converter  2042 B, whose trans-conductance is β times the trans-conductance of  2042 A, with the control voltage V ctrl     —     B  which is relatively stable and is separated from the control voltage V ctrl     —     A  for the converter  2042 A, the tuning gain of the VCO  204  may be reduced to about 1/(1+β) of the tuning gain of the VCO  104  in the known PLL  100 . When β=4, for example, the tuning gain of the VCO  204  may be only about ⅕ of that of the VCO  104 . By adjusting the value of β, the tuning gain of the VCO  204  may be further decreased. The decrease may only need minimal circuit changes, but may make the VCO  204  significantly less sensitive to the noise from other parts in a PLL. 
       FIG. 3  is a circuit schematic depicting a PLL with the VCO in  FIG. 2  according to one embodiment. A PLL  300  may have a PFD  301 , a charge pump  302 , a loop filter  303 , and a VCO  304  (corresponding to the VCO  204  in  FIG. 2 ). 
     The charge pump  302  may have a current source  3021 , switches  3022  and  3023 , and a current source  3024  coupled in series between a fixed voltage AVDD and a fixed voltage P. Switches  3022  and  3023  may be turned on and off by switching signals PU and PD from the PFD  301  respectively, and may be, e.g., transistors. The current sources  3021  and  3024  may provide a current I CP . 
     In the loop filter  303 , a charge storage device C1 and a resistance device R1 may be coupled in series between the output of the charge pump  302  and the fixed voltage P, and a charge storage device C2 may be coupled between the output of the charge pump  302  and the fixed voltage P as well, in parallel with the circuit branch including the charge storage device C1 and the resistance device R1. The charge storage devices C1 and C2 may be, e.g., capacitors, and the resistance device R1 may be, e.g., a resistor. A voltage to current converter  3042 A may take its control voltage V ctrl     —     A  from the junction of R1 and C2, and a voltage to current converter  3042 B may take its control voltage V ctrl     —     B  from the junction of R1 and C1. 
     The current source  3021  may charge the charge storage device C1 via the switch  3022  and the resistance device R1 to raise the control voltages V ctrl     —     A  and V ctrl     —     B  when the switch  3022  is closed and the switch  3023  is open. Or the charge storage device C1 may discharge via the resistance device R1 and the switch  3023  to lower the control voltages V ctrl     —     A  and V ctrl     —     B  when the switch  3022  is open and the switch  3023  is closed. 
     As discussed above with reference to  FIG. 2 , the tuning gain of the VCO  304  may be about 1/(1+β) of that of the known VCO  104 . The decrease in tuning gain may make the PLL  300  less sensitive to switching glitches from the charge pump  302 , and the noise from the PFD  301 , the charge pump  302 , and the loop filter  303 . 
       FIGS. 4 ,  6  and  7  illustrate exemplary circuit implementations of the PLL in  FIG. 3 . 
     In the circuit shown in  FIG. 4 , N-type field effect transistors (FETs) MNA and MNB may be used as the voltage to current converters  3042 A and  3042 B in  FIG. 3 , and the fixed voltage P may be ground. The trans-conductance of MNA is g m     —     A , the trans-conductance of MNB is g m     —     B , and
 
 g   m     —     A   :g   m     —     B =1:β, and β&gt;1
 
     In particular, the gate of MNA may receive the control voltage V ctrl     —     A , and the gate of MNB may receive the control voltage V ctrl     —     B . The drains of MNA and MNB may be coupled to a voltage supply such as a high PSR regulator  4041 , and the sources of MNA and MNB may be coupled to an ICO  4043 . The control voltage V ctrl     —     A  may control the current I A  flowing through MNA, and the control voltage V ctrl     —     B  may control the current I B  flowing through MNB. A control current l C , which is the sum of I A  and I B , may control the ICO  4043 . In one embodiment, MNA and MNB may be native devices, and an output noise of about −25 dB PSR from V DD     —     VCO  at the output of the voltage supply to V C  at the input of the ICO  4043  may be achieved, as shown in  FIG. 5 . 
     In the circuit shown in  FIG. 6 , P-type FETs MPA and MPB may be used as the voltage to current converters  3042 A and  3042 B in  FIG. 3 , and the fixed voltage P may be V DD     —     VCO  at the output of a voltage supply such as a high PSR regulator  6041 . The trans-conductance of MPA is g m     —     A , the trans-conductance of MPB is g m     —     B , and
 
 g   m     —     A   :g   m     —     B =1:β, and β&gt;1
 
     In particular, the gate of MPA may receive the control voltage V ctrl     —     A  from the junction of C2 and R1, and the gate of MPB may receive the control voltage V ctrl     —     B  from the junction of C1 and R1. The source of MPA may be coupled to the voltage supply  6041  via a resistor R a , and the source of MPB may be coupled to the voltage supply  6041  via a resistor R b . In one embodiment, R b =R a /β. The drains of MPA and MPB may be coupled to an ICO  6043 . Thus, the control voltage V ctrl     —     A  may control the current I A  flowing through MPA, and the control voltage V ctrl     —     B  may control the current I B  flowing through MPB. A control current I C , which is the sum of I A  and I B , may control the ICO  6043 . 
     In the circuit of  FIG. 7 , N-type FETs MNA and MNB may be used as the voltage to current converters  3042 A and  3042 B in  FIG. 3 , and the fixed voltage P may be the ground. The trans-conductance of MNA is g m     —     A , the trans-conductance of MNB is g m     —     B , and 
     
       
         
           
             
               
                 g 
                 m_A 
               
               ⁢ 
               
                 : 
               
               ⁢ 
               
                 g 
                 m_B 
               
             
             = 
             
               
                 
                   g 
                   m 
                 
                 
                   β 
                   + 
                   1 
                 
               
               ⁢ 
               
                 
                   : 
                 
               
               ⁢ 
               
                 
                   
                     g 
                     m 
                   
                   * 
                   β 
                 
                 
                   β 
                   + 
                   1 
                 
               
             
           
         
       
       
         
           
             β 
             &gt; 
             1 
           
         
       
     
     In particular, the gate of MNA may receive the control voltage V ctrl     —     A , and the gate of MNB may receive the control voltage V ctrl     —     B . The source of MNA may be grounded via a resistor R a , and the source of MNB may be grounded via a resistor R b . In one embodiment, R b =R a /β. The drains of MNA and MNB may be coupled to a voltage supply such as a high PSR regulator  7041  via a current mirror  7044 . 
     The current mirror  7044  may have P-type FETs MPP1 and MPP2. The junction of the gates of MPP1 and MPP2 and the drain of MPP1 may be coupled to the junction of the drains of MNA and MNB. The source of MPP1 may be coupled to the voltage supply  7041  via a resistor R c , and the source of MPP2 may be coupled to the voltage supply  7041  via a resistor R d . In one embodiment, R d =R c /m. The drain of MPP2 may be coupled to an ICO  7043 . 
     The control voltage V ctrl     —     A  may control the current I A  flowing through MNA, and the control voltage V ctrl     —     B  may control the current I B  flowing through MNB. The reference current of the current mirror  7044  is I C , which is the sum of I A  and I B . The output current I C ′ of the current mirror  7044  may be provided to the ICO  7043  as its control current, wherein I C ′=I C *m=(I A +I B )*m. 
       FIG. 8  illustrates a transient simulation result for V ctrl     —     A  and V ctrl     —     B  in  FIG. 3  when the PLL  300  is acquiring a lock. As shown, instead of a DC voltage in  FIG. 2 , the control voltage V ctrl     —     B  in the PLL  300  in  FIG. 3  may change with the control voltage V ctrl     —     A  slowly, and it may take a few microseconds for V ctrl     —     B  to catch up with the control voltage V ctrl     —     A . Taking the variation of V ctrl     —     B  into consideration, the frequency tuning gain of the PLL shown in  FIG. 3  may be: 
     
       
         
           
             
               
                 
                   
                     K 
                     
                       VCO 
                       ′ 
                     
                   
                   = 
                   
                     
                       lim 
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           V 
                         
                         → 
                         0 
                       
                     
                     ⁢ 
                     
                       
                         
                           f 
                           ⁡ 
                           
                             ( 
                             
                               
                                 V 
                                 ctrl_A 
                               
                               = 
                               
                                 
                                   V 
                                   ctrl_B 
                                 
                                 + 
                                 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   V 
                                 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           f 
                           ⁡ 
                           
                             ( 
                             
                               
                                 V 
                                 ctrl_A 
                               
                               = 
                               
                                 
                                   V 
                                   ctrl_B 
                                 
                                 - 
                                 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   V 
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         2 
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         V 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
       FIGS. 9A ,  9 B,  9 C and  9 D illustrate exemplary relationships between K VCO ′ and the difference between V ctrl     —     A  and V ctrl     —     B , V ctrl     —     A −V ctrl     —     B , in the PLL  300  of  FIG. 3 . As shown, if V ctrl     —     B  is set to a absolutely DC value and the output frequency is scanned by increasing V ctrl     —     A  only, K VCO ′ may vary significantly with (V ctrl     —     A −V ctrl     —     B ), changing from 220 MHz/V to 450 MHz/V. This may make the bandwidth of the PLL loop vary and deteriorate the loop stability. However, it is found that, when V ctrl     —     A −V ctrl     —     B =0, Kvco′ almost does not vary with the absolute value of V ctrl     —     B . Thus, in one embodiment, V ctrl     —     B  may be biased with a relative DC voltage, which may follow V ctrl     —     A  in a long period of time. One way to implement this is shown in  FIG. 3 , in which V ctrl     —     B  is taken from the junction of R1 and C1. Since C1&gt;&gt;C2, the difference between V ctrl     —     A , and V ctrl     —     B  may be a DC bias point. Furthermore, since V ctrl     —     B  is connected to V ctrl     —     A  via R1, V ctrl     —     B  may follow V ctrl     —     A  at last. 
     Some properties of the PLL  300  are as follows: 
     
       
         
           
             
               
                 
                   
                     V 
                     ctrl_B 
                   
                   = 
                   
                     
                       I 
                       CP 
                     
                     * 
                     
                       ( 
                       
                         
                           1 
                           
                             sC 
                             2 
                           
                         
                         // 
                         
                           ( 
                           
                             
                               R 
                               1 
                             
                             + 
                             
                               1 
                               
                                 sC 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     wherein “∥” is a functor which means: 
     
       
         
           
             
               A 
               // 
               B 
             
             = 
             
               
                 
                   A 
                   * 
                   B 
                 
                 
                   A 
                   + 
                   B 
                 
               
               . 
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     V 
                     ctrl_B 
                   
                   = 
                   
                     
                       I 
                       CP 
                     
                     * 
                     
                       ( 
                       
                         
                           1 
                           
                             sC 
                             2 
                           
                         
                         // 
                         
                           ( 
                           
                             
                               R 
                               1 
                             
                             + 
                             
                               1 
                               
                                 sC 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         1 
                         
                           sC 
                           1 
                         
                       
                       
                         
                           R 
                           1 
                         
                         + 
                         
                           1 
                           
                             sC 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     When I A =V ctrl     —     A *g m     —     A , I B =V ctrl     —     B *g m     —     B , and 
     
       
         
           
             
               
                 g 
                 m_A 
               
               = 
               
                 
                   g 
                   m 
                 
                 
                   β 
                   + 
                   1 
                 
               
             
             , 
             
                 
             
             ⁢ 
             
               
                 g 
                 m_B 
               
               = 
               
                 
                   
                     
                       g 
                       m 
                     
                     * 
                     β 
                   
                   
                     β 
                     + 
                     1 
                   
                 
                 ⁢ 
                 
                   : 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           I 
                           C 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               I 
                               A 
                             
                             + 
                             
                               I 
                               B 
                             
                           
                           = 
                           
                             
                               
                                 V 
                                 ctrol_A 
                               
                               * 
                               
                                 g 
                                 m_A 
                               
                             
                             + 
                             
                               
                                 V 
                                 ctrl_B 
                               
                               * 
                               
                                 g 
                                 m_B 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             I 
                             CP 
                           
                           * 
                           
                             ( 
                             
                               
                                 1 
                                 
                                   sC 
                                   2 
                                 
                               
                               // 
                               
                                 ( 
                                 
                                   
                                     R 
                                     1 
                                   
                                   + 
                                   
                                     1 
                                     
                                       sC 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                           * 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 1 
                                 
                                   β 
                                   + 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   β 
                                   
                                     β 
                                     + 
                                     1 
                                   
                                 
                                 * 
                                 
                                   
                                     1 
                                     
                                       sC 
                                       1 
                                     
                                   
                                   
                                     
                                       R 
                                       1 
                                     
                                     + 
                                     
                                       1 
                                       
                                         sC 
                                         1 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           * 
                           
                             g 
                             m 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     So the open loop transfer function may be: 
     
       
         
           
             
               
                 
                   
                     
                       H 
                       0 
                     
                     ⁡ 
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         I 
                         CP 
                       
                       
                         2 
                         * 
                         π 
                       
                     
                     * 
                     
                       ( 
                       
                         
                           1 
                           
                             sC 
                             2 
                           
                         
                         // 
                         
                           ( 
                           
                             
                               R 
                               1 
                             
                             + 
                             
                               1 
                               
                                 sC 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     * 
                     
                       ( 
                       
                         
                           1 
                           
                             β 
                             + 
                             1 
                           
                         
                         + 
                         
                           
                             β 
                             
                               β 
                               + 
                               1 
                             
                           
                           * 
                           
                             
                               1 
                               
                                 sC 
                                 1 
                               
                             
                             
                               
                                 R 
                                 1 
                               
                               + 
                               
                                 1 
                                 
                                   sC 
                                   1 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     * 
                     
                       g 
                       m 
                     
                     * 
                     
                       
                         K 
                         ICO 
                       
                       s 
                     
                     * 
                     
                       1 
                       N 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     From equation (15), the pole and zero position may be: 
     
       
         
           
             
               
                 
                   
                     ω 
                     z 
                   
                   = 
                   
                     
                       ( 
                       
                         1 
                         + 
                         β 
                       
                       ) 
                     
                     
                       
                         R 
                         1 
                       
                       ⁢ 
                       
                         C 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                     ω 
                     p 
                   
                   = 
                   
                     
                       1 
                       
                         
                           R 
                           1 
                         
                         ⁢ 
                         
                           C 
                           2 
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             C 
                             2 
                           
                           
                             C 
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ω 
                       p 
                     
                     
                       ω 
                       z 
                     
                   
                   = 
                   
                     
                       1 
                       
                         ( 
                         
                           1 
                           + 
                           β 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         
                           C 
                           1 
                         
                         + 
                         
                           C 
                           2 
                         
                       
                       
                         C 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     The gain bandwidth may be: 
     
       
         
           
             
               
                 
                   
                     ω 
                     c 
                   
                   ≈ 
                   
                     
                       
                         I 
                         cp 
                       
                       
                         2 
                         * 
                         π 
                       
                     
                     * 
                     
                       R 
                       1 
                     
                     * 
                     
                       
                         g 
                         m 
                       
                       
                         1 
                         + 
                         β 
                       
                     
                     * 
                     
                       K 
                       ICO 
                     
                     * 
                     
                       1 
                       N 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     As the impedance of C1 is far smaller than that of C2, the thermal noise contributed by R1 at the junction between R1 and C1 is negligible compared to that at the junction R1 and C2. As a result, the output phase noise contributed by R1 at the output frequency ω out  of the PLL  300  may be: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           φ 
                           n 
                           2 
                         
                         = 
                           
                         ⁢ 
                         
                           4 
                           * 
                           K 
                           * 
                           T 
                           * 
                           
                             R 
                             1 
                           
                           * 
                           
                             
                               ( 
                               
                                 
                                   g 
                                   m_A 
                                 
                                 * 
                                 
                                   
                                     K 
                                     ICO 
                                   
                                   s 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             4 
                             * 
                             K 
                             * 
                             T 
                             * 
                             
                               R 
                               1 
                             
                             * 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       g 
                                       m 
                                     
                                     
                                       1 
                                       + 
                                       β 
                                     
                                   
                                   * 
                                   
                                     
                                       K 
                                       ICO 
                                     
                                     s 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   
                                     1 
                                     + 
                                     β 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             * 
                             
                               ϕ 
                               n 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     where φ n   2  is the output phase noise contributed by R1 at the output frequency ω out  of the PLL  100 . 
     Compared to equation (3), equation (18) indicates that the pole over zero ratio of the PLL  300  is 1/(1+β) times of that of the known PLL  100 . This will make the PLL loop unstable. In addition, equation 20 indicates that, due to the frequency tuning gain K VCO  is reduced to 1/(1+β) times, the phase noise contribute by R1 is reduced to 1/(1+β) 2  times. 
     One way to improve the loop stability of the PLL  300  may be to increase the value of C1 to its (1+β) times. However, this may make the die size consumption unacceptable, since C1 is a big capacitor. 
     Another way to improve the loop stability of the PLL  300  may be to decrease the value of C2 to its 1/(1+β) and increase the value of R1 to its (1+β) times. Since the resistor may become R1*(1+β), which will contribute 4*K*T*R1*(1+β) to the noise, the noise reduction benefit depicted by equation 20 may be substituted with equation 23. Equation 23 also indicates that, in the PLL  300 , the phase noise contributed by the filter resistor R1 may be improved to 1/(1+β) times of that in the PLL  100 . Properties of such a PLL may be: 
     
       
         
           
             
               
                 
                   
                     ω 
                     Z 
                     ′ 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           + 
                           β 
                         
                         ) 
                       
                       
                         
                           R 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           C 
                           1 
                         
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             1 
                             + 
                             β 
                           
                           ) 
                         
                         
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   β 
                                 
                                 ) 
                               
                               * 
                               
                                 R 
                                 1 
                               
                             
                             ) 
                           
                           * 
                           
                             C 
                             1 
                           
                         
                       
                       = 
                       
                         1 
                         
                           
                             R 
                             1 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           ω 
                           p 
                           ′ 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               1 
                               
                                 
                                   R 
                                   1 
                                   ′ 
                                 
                                 ⁢ 
                                 
                                   C 
                                   2 
                                   ′ 
                                 
                               
                             
                             * 
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     C 
                                     2 
                                     ′ 
                                   
                                   
                                     C 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               1 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     β 
                                   
                                   ) 
                                 
                                 * 
                                 
                                   R 
                                   1 
                                 
                                 * 
                                 
                                   1 
                                   
                                     1 
                                     + 
                                     β 
                                   
                                 
                                 * 
                                 
                                   C 
                                   2 
                                 
                               
                             
                             * 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   
                                     1 
                                     
                                       1 
                                       + 
                                       β 
                                     
                                   
                                   * 
                                   
                                     C 
                                     2 
                                   
                                 
                                 
                                   C 
                                   1 
                                 
                               
                             
                             ) 
                           
                           ≈ 
                           
                             1 
                             
                               
                                 R 
                                 1 
                               
                               ⁢ 
                               
                                 C 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           φ 
                           n 
                           2 
                         
                         = 
                           
                         ⁢ 
                         
                           4 
                           * 
                           K 
                           * 
                           T 
                           * 
                           
                             R 
                             1 
                           
                           * 
                           
                             
                               ( 
                               
                                 
                                   g 
                                   m_A 
                                 
                                 * 
                                 
                                   
                                     K 
                                     ICO 
                                   
                                   s 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             4 
                             * 
                             K 
                             * 
                             T 
                             * 
                             
                               ( 
                               
                                 1 
                                 + 
                                 β 
                               
                               ) 
                             
                             * 
                             
                               R 
                               1 
                             
                             * 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       g 
                                       m 
                                     
                                     
                                       1 
                                       + 
                                       β 
                                     
                                   
                                   * 
                                   
                                     
                                       K 
                                       ICO 
                                     
                                     s 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           = 
                           
                             
                               1 
                               
                                 1 
                                 + 
                                 β 
                               
                             
                             * 
                             
                               ϕ 
                               n 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     One way to both improve the loop stability and further reduce the output phase noise of the PLL  300  is to use capacitor multiplication.  FIG. 10  is a circuit schematic depicting a PLL corresponding to the PLL  300  of  FIG. 3 , with the VCO  304  corresponding to VCO  204  in  FIG. 2 , but with a passive loop filter with capacitor multiplication according to one embodiment. A replica charge pump  1002  may be added to construct an equivalent capacitor multiplication for C1. The replica charge pump  1002  may have a current source  10021 , switches  10022  and  10023 , and a current source  10024  coupled in series between the fixed voltage AVDD and the fixed voltage P, in parallel with the charge pump  302 . The switch  10022  may be controlled by the switching signal PD from the PFD  301 , and be turned on and off approximately simultaneously with the switch  3022 , but in the opposite direction. The switch  10023  may be controlled by the switching signal PU from the PFD  301 , and be turned on and off approximately simultaneously with the switch  3023 , but in the opposite direction. The junction of R1 and C1 in the loop filter  303 , from where the control voltage V ctrl     —     B  of the voltage to current converter  3042 B is taken, may be coupled to the junction of switches  10022  and  10023 . The current provided by the current source  10021  may be a times the current provided by the current source  3021  in the charge pump  302 , wherein α&lt;1. 
     When ω fb  is lower than the reference frequency ω ref , the PFD  301  may output switching signals PU and PD to the charge pump  302  and the replica charge pump  1002 , closing switches  3022  and  10023  and keeping switches  3023  and  10022  open. When f&lt;1/(C1R1), since 1/(sC2)&gt;&gt;(R1+1/(sC1), most of the current I CP  coming from the current source  3021  in the charge pump  302  may go through C1. At the same time, since 1/(sC1)&lt;&lt;(R1+1/sC2)), most of the current from the current source  10024  in the replica charge pump  1002 , which is in the opposite direction of the current I CP  coming from the current source  3021  and is αI CP , may go through C1 as well. Thus, the actual current going through C1 is I CP *(1−α), and the actual voltage drop over C1 is about I CP *(1−α)/(s*C1), which means that the equivalent value of C1 may be amplified by 1/(1−α). 
     When α=⅞, for example, for each charge unit charged to C1 from the current source  3021  via the switch  3022  and R1, ⅞ of the charge unit may be drawn from C1 by the current source  10024  via R1 and the switch  10023 . Thus, the value of C1 may be amplified by 8 times, without changing the size of C1 and reducing chip area efficiency, and the variation of V ctrl     —     B  in the PLL  1000  may reduce ⅞. 
     When ω fb  is higher than the reference frequency ω ref , the PFD  301  may send switching signals PU and PD to the charge pump  302  and  1002 , keeping switches  3022  and  10023  open while closing switches  3023  and  10022 . Consequently, C1 may discharge via R1 and the switch  3023  and the discharge current is I CP . At the same time, the current source  10022  may charge C1 via the switch  10022  and R1, and the charging current may be α*I CP . 
     When ω fb  equals ω ref , the PFD  301  may keep the switches  3022  and  3023  in the charge pump  302  and switches  10022  and  10023  in the replica charge pump closed to maintain the relationship. 
     The frequency tuning gain of the PLL  1000  may be similar to that of PLL  300 . However, since the variation of V ctrl     —     B  is significantly reduced, the loop stability of PLL  1000  may be much better than that of PLL  300 . 
     Properties of the PLL  1000  may be: 
     
       
         
           
             
               
                 
                   
                     V 
                     ctrl_A 
                   
                   = 
                   
                     
                       
                         I 
                         CP 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             1 
                             
                               sC 
                               2 
                             
                           
                           // 
                           
                             ( 
                             
                               
                                 R 
                                 1 
                               
                               + 
                               
                                 1 
                                 
                                   sC 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         I 
                         CP 
                       
                       ⁢ 
                       
                         α 
                         ⁡ 
                         
                           ( 
                           
                             
                               1 
                               
                                 sC 
                                 1 
                               
                             
                             // 
                             
                               ( 
                               
                                 
                                   R 
                                   1 
                                 
                                 + 
                                 
                                   1 
                                   
                                     sC 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           1 
                           
                             sC 
                             2 
                           
                         
                         
                           
                             R 
                             1 
                           
                           + 
                           
                             1 
                             
                               sC 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     ctrl_B 
                   
                   = 
                   
                     
                       
                         
                           I 
                           CP 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               1 
                               
                                 sC 
                                 2 
                               
                             
                             // 
                             
                               ( 
                               
                                 
                                   R 
                                   1 
                                 
                                 + 
                                 
                                   1 
                                   
                                     sC 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           1 
                           
                             sC 
                             1 
                           
                         
                         
                           
                             R 
                             1 
                           
                           + 
                           
                             1 
                             
                               sC 
                               1 
                             
                           
                         
                       
                     
                     - 
                     
                       
                         I 
                         CP 
                       
                       ⁢ 
                       
                         α 
                         ⁡ 
                         
                           ( 
                           
                             
                               1 
                               
                                 sC 
                                 1 
                               
                             
                             // 
                             
                               ( 
                               
                                 
                                   R 
                                   1 
                                 
                                 + 
                                 
                                   1 
                                   
                                     sC 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     When I A =V ctrl     —     A *g m     —     A , I B =V ctrl     —     B *g m     —     B , and 
                 g   m_A     =       g   m       β   +   1         ,           ⁢       g   m_B     =         g   m     *   β       β   +   1         ,         
then:
 
     
       
         
           
             
               
                 
                   
                     I 
                     C 
                   
                   = 
                   
                     
                       
                         I 
                         A 
                       
                       + 
                       
                         I 
                         B 
                       
                     
                     = 
                     
                       
                         
                           
                             V 
                             ctrl_A 
                           
                           * 
                           
                             g 
                             m_A 
                           
                         
                         + 
                         
                           
                             V 
                             ctrl_B 
                           
                           * 
                           
                             g 
                             m_B 
                           
                         
                       
                       = 
                       
                         
                           1 
                           
                             1 
                             + 
                             β 
                           
                         
                         * 
                         
                           I 
                           cp 
                         
                         * 
                         
                           
                             
                               s 
                               * 
                               
                                 ( 
                                 
                                   
                                     
                                       R 
                                       1 
                                     
                                     * 
                                     
                                       C 
                                       1 
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       1 
                                     
                                     * 
                                     
                                       C 
                                       2 
                                     
                                     * 
                                     β 
                                     * 
                                     α 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   β 
                                 
                                 ) 
                               
                               * 
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   α 
                                 
                                 ) 
                               
                             
                           
                           
                             s 
                             * 
                             
                               ( 
                               
                                 
                                   s 
                                   * 
                                   
                                     R 
                                     1 
                                   
                                   * 
                                   
                                     C 
                                     1 
                                   
                                   * 
                                   
                                     C 
                                     2 
                                   
                                 
                                 + 
                                 
                                   C 
                                   1 
                                 
                                 + 
                                 
                                   C 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         * 
                         
                           g 
                           m 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     The open loop transfer function may be: 
     
       
         
           
             
               
                 
                   
                     
                       H 
                       o 
                     
                     ⁡ 
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         I 
                         cp 
                       
                       
                         2 
                         * 
                         π 
                       
                     
                     * 
                     
                       1 
                       
                         1 
                         + 
                         β 
                       
                     
                     * 
                     
                       
                         
                           s 
                           * 
                           
                             ( 
                             
                               
                                 
                                   R 
                                   1 
                                 
                                 * 
                                 
                                   C 
                                   1 
                                 
                               
                               - 
                               
                                 
                                   R 
                                   1 
                                 
                                 * 
                                 
                                   C 
                                   2 
                                 
                                 * 
                                 β 
                                 * 
                                 α 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               1 
                               + 
                               β 
                             
                             ) 
                           
                           * 
                           
                             ( 
                             
                               1 
                               - 
                               α 
                             
                             ) 
                           
                         
                       
                       
                         s 
                         * 
                         
                           ( 
                           
                             
                               s 
                               * 
                               
                                 R 
                                 1 
                               
                               * 
                               
                                 C 
                                 1 
                               
                               * 
                               
                                 C 
                                 2 
                               
                             
                             + 
                             
                               C 
                               1 
                             
                             + 
                             
                               C 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     * 
                     
                       g 
                       m 
                     
                     * 
                     
                       
                         K 
                         ICO 
                       
                       s 
                     
                     * 
                     
                       1 
                       N 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     From equation (27), especially when α=β/(β+1), the pole and zero position may be: 
     
       
         
           
             
               
                 
                   
                     ω 
                     z 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             1 
                             + 
                             β 
                           
                           ) 
                         
                         * 
                         
                           ( 
                           
                             1 
                             - 
                             α 
                           
                           ) 
                         
                       
                       
                         
                           R 
                           1 
                           * 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               C 
                               1 
                             
                             - 
                             
                               
                                 C 
                                 2 
                               
                               * 
                               a 
                             
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           
                             R 
                             1 
                           
                           * 
                           
                             ( 
                             
                               
                                 C 
                                 1 
                               
                               - 
                               
                                 
                                   C 
                                   2 
                                 
                                 * 
                                 α 
                               
                             
                             ) 
                           
                         
                       
                       ≈ 
                       
                         1 
                         
                           
                             R 
                             1 
                           
                           * 
                           
                             C 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
             
               
                 
                   
                     ω 
                     p 
                   
                   = 
                   
                     
                       1 
                       
                         
                           R 
                           1 
                         
                         ⁢ 
                         
                           C 
                           2 
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             C 
                             2 
                           
                           
                             C 
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ω 
                       p 
                     
                     
                       ω 
                       z 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             C 
                             1 
                           
                           - 
                           
                             
                               C 
                               2 
                             
                             * 
                             α 
                           
                         
                         
                           
                             ( 
                             
                               1 
                               + 
                               β 
                             
                             ) 
                           
                           * 
                           
                             ( 
                             
                               1 
                               - 
                               α 
                             
                             ) 
                           
                         
                       
                       * 
                       
                         
                           
                             C 
                             1 
                           
                           + 
                           
                             C 
                             2 
                           
                         
                         
                           
                             C 
                             1 
                           
                           * 
                           
                             C 
                             2 
                           
                         
                       
                     
                     ≈ 
                     
                       
                         
                           C 
                           1 
                         
                         + 
                         
                           C 
                           2 
                         
                       
                       
                         C 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     The output phase noise contributed by R1 may be: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           φ 
                           n 
                           2 
                         
                         = 
                           
                         ⁢ 
                         
                           4 
                           * 
                           K 
                           * 
                           T 
                           * 
                           
                             R 
                             1 
                           
                           * 
                           
                             
                               ( 
                               
                                 
                                   g 
                                   m_A 
                                 
                                 * 
                                 
                                   
                                     K 
                                     ICO 
                                   
                                   s 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             4 
                             * 
                             K 
                             * 
                             T 
                             * 
                             
                               R 
                               1 
                             
                             * 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       g 
                                       m 
                                     
                                     
                                       1 
                                       + 
                                       β 
                                     
                                   
                                   * 
                                   
                                     
                                       K 
                                       ICO 
                                     
                                     s 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   
                                     1 
                                     + 
                                     β 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             * 
                             
                               ϕ 
                               n 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
     Equation (30) may indicate that the ratio of pole and zero position of the PLL  1000  in  FIG. 10  is better than that of PLL  300  in  FIG. 3  (expressed by equation (17)), and is similar to that of PLL  100  in  FIG. 1  (expressed by equation (3)). 
     Equation (31) may indicate that the output phase noise contributed by R1 in the PLL  1000  may be 1/(1+β) times of that in the PLL  300  (expressed by equation (19)), and 1/(1+β) 2  times of that in the PLL  100 . 
     In sum, PLL  1000  and PLL  300  may reduce the frequency tuning gain to about 1/(1+β) of that of the known PLL  100 . PLL  1000  may reduce the output noise contributed by R1 to 1/(1+β) times of that in the PLL  300 , and 1/(1+β) 2  times of that in the PLL  100 . In addition, with the capacitor multiplication provided by the replica charge pump, the PLL  1000  may achieve similar loop stability of that of PLL  100  without increasing the size of C1, C2 or R1. Table 1 is an exemplary performance comparison between a known PLL  100  and a PLL  1000  according to one embodiment. 
     
       
         
               
               
               
               
             
               
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
             
               
               
               
               
             
               
               
               
             
           
               
                   
                   
               
               
                   
                   
                 Conventional Structure 
                 This Work 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Kvco 
                 1.5 
                 GHz/V 
                 0.3 
                 GHz/V 
               
               
                   
                 fo 
                 1.3 
                 GHz 
                 1.3 
                 GHz 
               
               
                   
                 f ref   
                 26 
                 MHz 
                 26 
                 MHz 
               
               
                   
                 N 
                 50 
                   
                 50 
                   
               
             
          
           
               
                   
                 α 
                 No data 
                 7/8 
               
             
          
           
               
                   
                 β 
                 No data 
                 4 
                   
               
             
          
           
               
                   
                 Icp 
                 270 
                 uA 
                 340 
                 uA 
               
               
                   
                 R1 
                 2 
                 kΩ 
                 8 
                 kΩ 
               
               
                   
                 C1 
                 120 
                 PF 
                 30 
                 PF 
               
               
                   
                 C2 
                 8 
                 PF 
                 2 
                 PF 
               
               
                   
                 fc 
                 2.6 
                 MHz 
                 2.6 
                 MHz 
               
               
                   
                 fz 
                 0.65 
                 MHz 
                 0.575 
                 MHz 
               
               
                   
                 fp 
                 10.35 
                 MHz 
                 11.04 
                 MHz 
               
               
                   
                 Phase Margin 
                 60 
                 Degree Celsius 
                 63 
                 Degree Celsius 
               
             
          
           
               
                   
                 Phase Noise 
                 −106.3 dBc @ 1.5 MHz 
                 −118 dBc @ 1.5 MHz 
               
               
                   
                 contribute by 
                   
                   
               
             
          
           
               
                   
                 R1 at the out- 
                   
               
               
                   
                 put frequency 
               
               
                   
                   
               
             
          
         
       
     
     Several features and aspects have been illustrated and described in detail with reference to particular embodiments by way of example only, and not by way of limitation. Alternative implementations and various modifications to the disclosed embodiments are within the scope and contemplation of the present disclosure. For example, bipolar junction transistors (BJTs) or junction gate field-effect transistors (JFETs) may be used to replace the FETs in the embodiments. Therefore, it is intended that the invention be considered as limited only by the scope of the appended claims.