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					Phase-Locked Loop




                    1
  Phase-Locked Loop in RF Receiver
Antenna

        BPF1   LNA     BPF2    Mixer BPF3 IF Amp



       RF front end                                Demodulator

                              LO
                           VCO
Ref.              Loop
          PD      Filter
                                   Phase-
                 1/N
                                   Locked
                                   Loop
                                                             2
        Functional Blocks in PLL
                              VCO
     Ref             Loop           LO
             PD      Filter
                                    Phase-
                    1/N             Locked
                                    Loop
• Phase detector (PD): find difference between
  phases of two signals
• Loop filter: provide appropriate control voltage
  for the voltage-controlled oscillator (VCO)
• VCO: generate signals with phase determined
  by the control voltage
• Divide-by-N: LO phase changes N times faster
  than Ref phase
                                                     3
               Design Issues
• Tracking behavior
• Noise performance
• Jitter characteristics
  – Jitter tolerance
  – Jitter transfer
  – Jitter generation
• Power consumption



                               4
              System Modeling
                                    VCO
    vRef           vd          vC
              PD        F(s)              vLO



•   vRef: input reference signal
•   vLO: local oscillator (LO) output signal
•   vd: detector output
•   F(s): transfer function of loop filter
•   vC: control voltage for VCO

                                                5
                    System Modeling
             qRef        Kdqe          VCO
    vRef                                     vLO
                    PD          F(s)
           qLO


•   Phase signals contain information
•   qRef: phase of reference signal
•   qLO: phase of local oscillator (LO) signal
•   qe: phase difference between qRef and qLO


                                                   6
Jump in Phase




dq LO
       0  5(q REF  q LO )   7
 dt
Ramp in Phase




dq LO
       0  5(q REF  q LO )   8
 dt
       Ramp in Phase




dq LO                            t
       0  5(q REF  q LO )  5 (q REF  q LO )d
 dt                              0                     9
            Phase Detector
           qREF   +       qe        vd
                               Kd
                      

                      qLO


• Vd=Kdqe=Kd(qREF – qLO)
• Kd: gain of phase detector




                                         10
                Loop Filter
               vd             vC
                       F(s)



• VC(s) = F(s) Vd(s)
• Low-pass filter
  – Extract phase error
  – Remove high frequency noises
• Passive filter for integrated PLL
• Active filter for discrete component PLL
                                             11
                  Passive Lag Filter

      R1                                          1
                                 1  s 2
+            +     F ( s) 
                              1  s( 1   2 )
        R2
vd           vC     1  R1C                                            2
                                                                      1   2
        C           2  R2C
–            –                                           1       1
                                                      1   2   2

    • Lag filter: pole magnitude smaller than zero
    • Passive components: high linearity, gain < 1


                                                                                 12
                     Active Lag Filter
      R1   C1   R2    C2
                                                                     C1
                                              1  s 2        Ka 
+                          +    F (s)  K a                          C2
                                              1  s 1
                –              1  R1C1
vd                         vC   R C
                +               2     2 2
                                                                     2 R2
                                K a  C1 / C2                 Ka      
–                          –                                         1 R1
                                                         1                   1
                                                         1                  2

     • Can adjust pole and zero locations
     • Can have gain
     • Op amp limitations

                                                                                  13
Active Proportional-Integral (PI) Filter
      R1   R2   C
+                   +                 1  s 2
                         F ( s)  
                                        s 1
           –
vd                  vC    1  R1C                    R2
           +                                          R1
                          2  R2C
–                   –                            1
                                                 2
    • Large open loop gain at low frequency
    • Op amp limitations
      – Linearity
      – Noise
      – Open loop gain
                                                           14
    Voltage-Controlled Oscillator
        vC           +              qLO
             KVCO             1/s
                         

                         0

  
 qLO  0  KVCOvC
• KVCO: gain of VCO




                                          15
               Transfer Function of PLL
                                                       0
qREF   +       qe        vd          vC            +
                                                       +
                                                                  qLO
                    Kd        F(s)        KVCO              1/s
           

           qLO


   • Open-loop transfer function from qe to qLO
                                 K d KVCO F ( s)
                         A( s) 
                                       s


                                                                    16
                Transfer Function of PLL
                                                      0
qREF   +        qe        vd          vC          +
                                                      +
                                                                  qLO
                     Kd        F(s)        KVCO            1/s
            

            qLO


   • Closed-loop transfer function from qREF
     to qLO
                     LO ( s )    A( s )     K d KVCO F ( s )
           H ( s)                       
                     REF ( s ) 1  A( s ) s  K d KVCO F ( s )
                                                                    17
        Transfer Function from qREF to qe
                                                             0
qREF     +        qe        vd          vC              +
                                                             +
                                                                             qLO
                       Kd        F(s)         KVCO                   1/s
              

              qLO


   • Closed-loop transfer function
        e ( s )   REF ( s )   LO ( s )   REF ( s )  H ( s ) REF ( s )
                  e (s)                             s
       H e (s)               1  H ( s) 
                  REF ( s )                s  K d KVCO F ( s )
                                                                                 18
                    Other TF of Interest
                                            vCn
qREF   +       qe        vd          vC +   +
                                                                        qLO
                    Kd        F(s)                  KVCO          1/s
           

           qLO


   • Noise in control voltage
           [VCn ( s )  K d F ( s ) LO ( s )]KVCO / s   LO ( s )
            LO ( s )      KVCO
                      
           VCn ( s ) s  K d KVCO F ( s )
                                                                          19
                    Other TF of Interest
                                                                     qn
qREF   +       qe        vd          vC                          +
                                                                     +
                                                                          qLO
                    Kd        F(s)        KVCO          1/s
           

           qLO


   • Phase noise of VCO
            n ( s )  K d KVCO F ( s ) LO ( s ) / s   LO ( s )
            LO ( s )         s
                      
            n ( s ) s  K d KVCO F ( s )
                                                                            20
  Transfer Functions for Different Loop
                 Filters
• Passive lag filter                            K d KVCO
                                                          (1  s 2 )
               1  s 2                         1   2
                                H (s) 
   F ( s)                                   1  K d KVCO 2      K K
            1  s( 1   2 )           s2                   s  d VCO
                                                 1   2          1   2
• Active lag filter
                                                    K d KVCO K a
                                                                      (1  s 2 )
                1  s 2                                  1
   F ( s)  K a                 H (s) 
                1  s 1                        1  K d KVCO K a 2           K d KVCO K a
                                          s 
                                          2
                                                                        s
• Active PI filter                                        1                        1
           1  s 2                              K d KVCO
                                                               (1  s 2 )
   F (s)                                           1
             s 1               H (s) 
                                                K d KVCO 2           K d KVCO
                                          s 
                                           2
                                                                 s
                                                     1                  1
                                                                                         21
      Normalizing Transfer Function
 • Normalized denominator
D(s)  s 2  2n s  n ,
                       2
                             n : natural frequency;  : damping ratio
 • Passive lag filter
            K d KVCO               n       1
    n                        ( 2           )
            1   2            2       K d KVCO
 • Active lag filter
            K d KVCO K a           n                 1
    n                               ( 2                 )
                 1                2             K d KVCO K a
 • Active PI Filter
            K d KVCO               n
    n                               2
               1                  2
                                                                    22
    Normalized Transfer Function
• Passive lag filter
                            n
                             2
                (2 n           ) s  n
                                         2

                          K d KVCO
     H (s) 
                    s 2  2 n s  n
                                     2



• Active lag filter
                             n
                              2
                (2 n              ) s  n
                                            2

                         K d KVCO K a
     H ( s) 
                     s 2  2 n s  n
                                      2



• Active PI Filter
               2 n s  n
                          2
     H (s)  2
            s  2 n s  n2


                                                23
    Normalized Transfer Function
          2 n s  n
                     2
                                                  s2
H (s)  2                      H e (s) 
       s  2 n s  n2
                                           s 2  2 n s  n
                                                            2



• Passive lag filter
                               1
                 K d KVCO 
                               2
• Active lag filter
                                    1
                 K d KVCO K a 
                                    2

                                                                24
Frequency Response of H(s)




           2 n s  n
                      2
 H (s)  2
        s  2 n s  n2




                             25
Frequency Response of He(s)




                              s2
           H e (s) 
                       s 2  2 n s  n
                                        2




                                            26
          Step Response of PLL
• Phase step
      qREF (t )  q  u(t )  REF (s)  q / s

• Phase Error
                                          q  s
      e ( s)  H e ( s)q / s 
                                   s 2  2 n s  n
                                                    2


                                              
                 q [cos( 1   2 nt )            sin( 1   2 nt )]ent ,   1;
                                             1  2
                
      qe (t )  q (1  nt ) exp(  nt ),   1;
                                                
                q [cosh(  2  1nt )               sinh(  2  1nt )]ent ,   1.
                
                                               2 1

• Steady state error (final value theorem)
                                         q  s 2
      qe ()  lim s e ( s)  lim 2                  0
                s 0            s 0 s  2  s   2
                                             n     n

                                                                                            27
Step Response



                     s2
  H e (s) 
              s 2  2 n s  n
                               2




                                   28
        Ramp Response of PLL
• Phase ramp
      q REF (t )    tu (t )  REF ( s)   / s 2

• Phase Error
                                             
      e ( s )  H e ( s) / s 2 
                                      s 2  2 n s  n
                                                       2


                        1
                              sin( 1   2 nt )e nt ,   1;
                   n    1  2
                 
      q e (t )    t exp(  nt ),   1;
                        1
                               sinh(  2  1nt )e nt ,   1.
                  n  2  1
                 

• Steady state error (final value theorem)
                                              s
      qe ()  lim s e ( s)  lim                       0
                s 0             s 0   s  2 n s  n
                                         2             2


                                                                     29
Ramp Response


                      s2
   H e (s) 
               s 2  2 n s  n
                                2




                                    30
General Steady State Error in Ramp Response
                                     s
    H e ( s)  1  H ( s) 
                            s  K d KVCO F ( s)
                                        / s
   e ( s)  H e ( s) / s 2 
                                  s  K d KVCO F ( s)

 • High loop gain
                                                      
    F (0)    qe ()  lim s e ( s )  lim                      0
                          s 0             s 0 s  K K
                                                     d VCO F ( s )

 • Low loop gain
                                                         
   qe ()  lim s e ( s)  lim                     
             s 0            s0 s  K K
                                      d VCO F ( s )   K d KVCO F (0)

                                                                        31
               Stability of PLL
• Criterion for stability
  – Closed-loop pole at left half plane
  – Sufficient phase margin
• Control of pole location
  – Open loop gain
  – Open loop zero
• Check root locus
                   A( s )    K d KVCO F ( s) / s
        H ( s)           
                 1  A( s) 1  K d KVCO F ( s ) / s

                                                      32
            Root Locus Method
• Closed-loop TF
                   K d KVCO F ( s ) / s       K  n( s )
        H ( s)                         
                 1  K d KVCO F ( s) / s d ( s )  K  n( s )
        where
        K  K d KVCO and F ( s) / s  n( s) / d ( s).
• Closed-loop poles make
        d ( s )  K  n( s )  0
  – K=0, open-loop poles
  – K infinity, open-loop zeros or infinity


                                                                33
Root Locus for Passive Lag Filter

     F ( s ) 1 1  s 2
            
       s      s 1  s ( 1   2 )




                                     34
Root Locus for Active Lag Filter

    F ( s ) 1 1  s 2
           
      s      s 1  s 1




                                   35
Root Locus for Active PI Filter

   F ( s ) 1 1  s 2
          
     s      s s 1




                                  36
Root Locus for 1st-Order LP Filter

     F ( s) 1 1
           
       s     s 1  s 1




                                     37
   Effects of Parasitics

F (s) 1 1              1
     
  s    s 1  s 1 1  0.1s 1




                                38
         Effects of Zero

F ( s ) 1 1 1  2s 2
       
  s      s 1  s 1 1  0.1s 1




                                  39
           Phase Noise and Jitter
• Phase noise
   – Fluctuation in phase
   – Frequency domain
   – Discussed in RF circuits
• Jitter
   – Error in clock edge (period)
   – Time domain
   – Significant in communications circuits
• Two concepts
   – Related to each other
   – Exact relationship not clear


                                              40
                  Jitter Measurements




Agilent, “Understanding Jitter and Wander Measurements and Standards.”   41
              Jitter Tolerance
• Ability of a PLL to operate with jitter
  – Applied to its reference
  – Various magnitudes
  – Different frequencies
• Usually specified using an input jitter mask
  – Jitter magnitude and corner frequencies
  – BER requirement
  – Various for standards



                                                 42
         PLL in Clock and Data Recovery
         0   1       0       0       1       0       1       0       1
Ideal
signal

Distorted
signal
         0   1       0       0       X       1       0       1       0
Ideal
clock

       0         1       0       0       1       0       1       0
Recovered
clock

                                                                         43
Jitter Tolerance Mask




                        44
Jitter Tolerance Measurement




                               45
Jitter Tolerance Measurement




                               46
    Jitter Tolerance Measurement




• Error at corner frequency
  – Insufficient clock recovery bandwidth
  – Incorrect mask used
                                            47
    Jitter Tolerance Measurement


                            Tolerance margin




• Excessive jitter tolerance margin

                                               48
    Jitter Tolerance Measurement




• Occasional fail at specific frequencies
  – Need extra settling time after jitter amplitude change
• Repeating with additional settling time
• Spot measurement
                                                             49
   Jitter Tolerance Measurement




• Limited clock recovery bandwidth
• Eye-width alignment noise
                                     50
    Jitter Tolerance Measurement




• Limited buffer store


                                   51
                  Jitter Transfer
• Jitter transfer or jitter attenuation
• Output jitter vs. input jitter
   – Input jitter with various amplitudes and frequencies
   – Output jitter measured with various bandwidths
• Intrinsic jitter
• Typically specified using a bandwidth plot
   – Amplitude
   – Roll off speed
   – Corner Frequency



                                                            52
Jitter Transfer Mask




                       53
     Jitter Transfer Measurement




• Jitter tolerance mask used to set input jitter level
• Sinusoidal jitter at magnitudes and frequencies
• Narrow-band measurement
                                                         54
    Jitter Transfer Measurement




• Different test masks
• SONET mask: additional amplitude at lower band
                                                   55
    Jitter Transfer Measurement




• Measurement set-up noise
• -40 dB sufficient

                                  56
    Jitter Transfer Measurement




• Low-frequency phase noise
• Power-line crosstalk
• Short measurement time
                                  57
     Jitter Transfer Measurement




• Incorrect filter characteristic
• Excessive peaking

                                    58
                 Jitter Transfer Plot




E. Barari, “Jitter Analysis / Specification,” May 2002.
                                                          59
Measured Jitter Transfer Characteristic




E. Barari, “Jitter Analysis / Specification,” May 2002.
                                                          60
Measured Jitter Transfer Characteristic




E. Barari, “Jitter Analysis / Specification,” May 2002.
                                                          61
Measured Jitter Transfer Characteristic




E. Barari, “Jitter Analysis / Specification,” May 2002.
                                                          62
Measured Jitter Transfer Characteristic




E. Barari, “Jitter Analysis / Specification,” May 2002.
                                                          63
            Jitter Generation
• Intrinsic jitter produced by the PLL
  – Thermal noise
  – Drift in VCO
• Measured at its output
  – Applying a clear reference signal to PLL
  – Measuring its output jitter.
• Usually specified as a peak-to-peak period
  jitter value


                                               64
Jitter Generation Standard




                             65
  Jitter Generation Measurement
• Direct measurement of p-p jitter
• Phase noise measurement
• Eye diagram and histogram




                                     66
Jitter Generation Measurement




                                67
     Measurement Considerations
•   Calibration
•   Measurement range
•   Measurement time
•   Power
•   Frequency offset




                                  68
TF from Noise in VCO Control Voltage

                                      vCn

                                  +
                                      +
                                                           qLO
    -1       Kd         F(s)                  KVCO/s



                 LO ( s)      KVCO              KVCO s
     H C ( s)                              2
                VCn ( s) s  K d KVCO F ( s) s  2 n s  n
                                                            2




 • Can be viewed as low-pass filter

                                                                 69
TF from Noise in VCO Control Voltage




                            s
       H C ( s) 
                    s 2  2 n s  n
                                     2




                                         70
   TF from Phase Noise in VCO

                                                     qn

                                                 +
                                                     +
                                                          qLO
   -1        Kd        F(s)        KVCO/s



                LO ( s )       s                s2
    Hq ( s )                             2
               n ( s) s  K d KVCO F ( s) s  2 n s  n
                                                          2




• High-pass filter
• The same as He(s)

                                                                71
                    Phase Error in VCO
                                         vCn      qn

                                    HC(s)      Hq(s)


qREF   +       qe                               qLO
                    Kd   F(s)   KVCO/s
           

           qLO

   • vCn dominate at low frequencies
   • qn dominate at high frequencies

                                                       72

				
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