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					                        Monolithic RFIC Design


                       Chapter 08
    Oscillator Design & Phase Noise Issues
             (B.Razavi, RF Microelectronics, Ch.7)


1. Introduction : General models, basic topologies, VCO
2. General Considerations : oscillation frequency, pushing / pulling,
                               turning range, VCO gain/sensitivity
                               output power, and phase noise.
3. Performance Evaluation Parameters : Power consumption,
                                          tuning range, VCO gain (Kvco)
                                          phase noise (PN),
                                          pushing/pulling figure.
4. Introduction to “Quadrature Signal Generation”.
                           Introduction

General Considerations:
  •Oscillator     generates a periodic output.
                  needs self-sustaining oscillation mechanism;

         Transfer Function of a Linear Feedback System :
         Y(s)     H(s)
              
         X(s) 1  H(s)               If H(s) =+1, oscillating




                  Fig.1 Feedback oscillator system.
•A real oscillator in the viewpoint of feedback:


  two-port model:




  Barkhausen' s criteria :
  (1) the loop gain :
         H(j0 )  1,
  (2) the totalphase shift around the loop :
         H(j0 )  0 or 360.
  * if the loop in DC is negative, then H(j0 ) has to be 180
  (that is, a negative feedback)
                                                  (analysis)
One-port model :

                   One-port
                   circuit for
                   the LC tank.



                                                            
                                                              r L2
•R1 : negative                                         Rp
                                                                Rs
                                  (in real case, positive)
A concept of oscillation’s start-up :


       The small-signal loop gain must be somewhat
          greater than one.
          oscillation grows up by noise;


        but to achieve a stable amplitude,
          the average loop gain must return to unity.
          Self-sustaining oscillation.
          power constrained / saturation limitation
Automatic Level Control (ALC) Topology:

    •Pros :    enhance the “linearity” of the oscillator;
               thus less AM-to-PM modulation property

    •Corns :  introduce noise into the core.




                                                  (a rectifier)
                            (noise)
Reference Paper : (IEEE Int. Symp. ISCAS2005, pp.2691-2694)
Ideal Waveform -- Square Wave or Sinusoidal Wave?
  •Discontinuous due to the threshold voltage (VT);
  •Square wave is better for the switching purpose;
  •Sinusoidal wave is cleaner in the spectrum
  •Non-linearity of MOSFET  self-mixing  noise;
  •Need trade-off between using Square wave and Sinusoidal Wave.
Basic LC Oscillator Topologies:
    •Discrete RF Oscillator Design:
      -- to minimize the noise (∵high-Q available)
      -- to lower the cost


         •Why usually use “only one active device” in discrete
          VCO design?

         -- Low noise, (a matched device pair is necessary for
            differential topology.)
         -- Low cost, (BOM – Bill of Materials)
         -- Smaller package size.
Positive Feedback for Oscillation:



                                      •Low (load) impedance
                                      •Drop the loop gain
                                      •Tank’s Q degrades.
   feedback
   path
                                      feedback
                                      path

                              1/ gm
Implementation approach:              •Transformer feedback
                                       •Increase (load) impedance
                                       •Enhance Q –factor.




                                                              n2
                                                              gm



                            (with Cap.,
                             Colpitts osc.)
Implementation approach: (cont.)
                                            [更正, 12/18]

                 C                  L
             (1  1 ) 2 / g m   (1  2 ) 2 / g m
                 C2                 L1




                                                   More components,
                                                   side-effects,
                                                   seldom use.
Enlarge L or C of a tank?

                           Rp   
                                  r L 2
                                     Rs
                           Assumed L and R s scale proportionally :

                            R p,final    
                                            r mL2    mR p   (Q is enhanced)
                                              mRs




 •Enlarge “L” for better signal swing (S/N ↗)
 •Trade-offs: self-resonant frequency and tuning range decrease
  (for a fixed osc. frequency).
Noise Consideration: (in Colpitts or Hartley oscillators)

    •BJT : base resistance, collector shot noise.
    •MOS : gate noises, drain 1/f noise, channel resistance.
    • Voltage Swing should avoid to “saturate” the active devices.
Negative Resistance in One-Port View:                                     MOS model:
                       Vgs                                 Vx  Vgs +
   I x   I gs              and I d  g m Vgs ; I C2                         g m v gs  id
                    1 / sC1                                 1 / sC 2
                                                                       -
   I x  I d  I C2                          (negative resistance)

  Vx    g     1   1       g     1   1
      2 m           2 m     
  I x s C1C2 sC1 sC 2    C1C2 sC1 sC2
                                                           (imaginary part)
 Voltage Controlled Oscillator Topology:
   •Bias circuit not shown;
   •In (a), a discrete oscillator;
   •In (b), an on-chip, Colpitts oscillator;
   •In (c), the bias ckt (not shown) will degrades tank’s Q  less use.




(上頁colpitts osc
 有一端接地)
Voltage Controlled Oscillator Topology : (cont.)
     •Many communication systems use “channels” concept for
      data exchange.
     •Frequency-tunable oscillator is needed  VCO.
Voltage Controlled Oscillator: (cont.)

         Mathematical Model :
         oscillator output : y(t)  A cos(RF t   (t ))
                                  t
         where,  (t )  KVCO  Vcont (t )dt
                                  
         if only Vcont (t )  constant V0 ,                 Note: in a VCO,
                                                            the phase error
         then                                               is accumulative.)
                       
          y(t)  A cos(( RF  KVCOV0 ) t  0 )


    •Concept: VCO’s function is a “phase modulator” rather than
              a “frequency generator”.
Phase Noise:

    •Noise injection causes (a) frequency variation; and
                            (b) amplitude variation.
    and vice versa.
    •Analog  Phase noise
    •Digital  Jitters
Noise Skirt:
                     [Noise Shaping]




                Ideal Oscillator   Actual Oscillator




                       c                 c

                                           △ω
Phase Noise Calculation – an example:

  PN (dBc/Hz @ △ω offset)
     = △ (in dB) – 10 log (BW)
     = Output S0 (in dB) – Noise level (in dB) @ △ω – 10 log (BW)


                 Ideal Oscillator   Actual Oscillator


                                                 △

                        c                 c

                                            △ω
Leeson’s Equation of Phase Noise:


                1 FkT     f c    fo  2
       L(fm )         f 1  ( 2Q f ) 
                      1              
                2 Po       m      L m 
       where
       L( ) : phase noise;
       f m : offset frequency;
       F : fitting parameter;
       Po : output power;
       f c : corner frequency(1/f effect on VCO ckt)
       f o : output frequency;
       Q L : Q  factor of the resonator.
  Possible PN curves:



                                             1/f3




                                                           PN (dBc/Hz)
              1/f3

                             PN (dBc/Hz)
PN (dBc/Hz)




                     1/f2
                                                                          1/f




              Offset freq.                  Offset freq.                 Offset freq.

              Low-QL                       Medium-QL                     High-QL
Phase Noise Mechanisms:
Discrete Frequency Noise:

     •Noise generated by a certain kind of periodic noise source。


                              Ex. PLL’s spur due to the reference
                                  frequency fREF.
Power-Law (階次方) Noise:

    •Random noise with a probability distribution:
Power-Law Noises:
Effect of Phase Noise on RF Transceivers:

    •For Rx chain:

 (blocking effect)                     “reciprocal mixing”:
                                       Dirty LO signal makes
                                       S/N ratio degrades.



    •For Tx Chain:



  (S/N degradation)
Effect of “Pure” Phase Noise:

           •Diverse in a constellation circle.
           •BER (bit error rate) increases.



 The constellation plot:
    [Measurement]
Definition of Q-factor:

                Pstored per cycle         0 d
      Q  2                          
               Pdissipated per cycle      2 d
      Q  how much the energy is lost as it is transferr ed
          from the capacitor to the inductor and vice versa.

   •For a resonant circuit:
                        Vx            1
Colpitts Oscillator:         sL //      // R p
                        I in         sCeq
                        where
                                  C1C2
                        Ceq                   and
                                (C1  C2 )
                             (1  C1 / C2 ) 2
                        RP 
                                   gm
                        If C1  C2 s  g m ,
                        the voltage applied to
                        the sourceend of M1 is :
                                       C2
                        Vs  Vx [              ]
                                    (C1  C2 )
                Vs      the drain currentof M1 is :
                                             C2
                        I out   g mVx
                                          (C1  C2 )
Colpitts Oscillator: (cont.)                    1          x3 x5 x7
                                           tan        x  x         
                                                             3   5   7

 (open-loop     I out               C2    sL //  1          
 transfer             (s)   g m                      // R p 
 function)       I in             C1  C2 
                                                 sCeq        
                                                              
                                                L
                ( )         tan 1
                          2              R p (1   2 LCeq )
               
                                    1
               at   0               ,
                                   LCeq
                d                                                   Rp
                          2Ceq R p           Q  0Ceq R p 
                d  0                                            0 L
                                                          (並聯)
Sources of Phase Noises:
Noise in Signal Path:
  from the noise viewpoint :
  Y(s)     H(s)
       
   X(s) 1  H(s)
  where if   0   , and   0
                             dH
  H ( j )  H ( j0 )   
                             d  
                                     0
  Since at   0 , H ( j0 )  1 ... (at resonant)
  
  Y ( j (0   ))   H ( j (0   ))                            H ( j 0 )
                                         
  X ( j (0   )) 1  H ( j (0   ))                                       dH
                                                       1  [ H ( j0 )                ]
                                                                                d  
                                                                                        0
                  1                           1
                                   
                      dH                     dH
      1  [1                ]        
                      d                  d  
                              0                        0
Noise in Signal Path: (cont.)
                2
                     voltage gain   power
            Y                                 2
            X
                                     2
           Y                                     1
           X ( j (0   )) 
                            
                                                           2
                                                      dH
                                              ( ) 2
                                                      d
                                                                  Let is the
                                    j ( )
           Let H( )  H  e                                      phase of the
                                                                  open-loop
               dH     dH       d j
                  (      jH    )e                              transfer function.
                d    d       d
           Since H  constant,
                                   d   d
                     2                 2          2
                dH             2
                         H                          ...H  1 at resonant.
                d                 d   d
Noise in Signal Path: (cont.)


         
                                                                     d
                             2                                 2          2
         Y                           1                  dH
         X ( j (0   ))                       and            
                                         dH
                                                2
                                                          d         d
                                 ( ) 2
                                           d
         at   0   ,
                                 Added intentionally!
         the Noise Gain is
                        0                      1  0 
                   2                                       2
         Y        
                           2
                                    1
         X ( j )                             2     
                     4  0 d 
                             2             2
                                               4Q     
                                   ( )
                                2 d 
                               
         * This noise gain is due to the transferfunction!
Noise in Control Path:
           •Be directly converted as follow:
Phase Noise Measurement:                             L
                                                                                       Note:
                                                                                       due to
                                                                                       the transfer
                                                                 1/ f 3 -30dB/decade   function effect

                                                                              2
                                                                           1/f -20dB/decade
                                                                                                1       
                                                                                          10log  FkT
                                                                                                2 P     
                                                                                                         
                                                                                                        
  Zin                        1         1
                                                                                                    s

            Z (0   )      
                             GL 1  j 2Q                    
                                                                  1/ f 3
                                                                                  o
                                                                                               log
                                                                                  2Q
                                         L
                                           0

   vout (o  w)    H (o  w)       1    o
                                  j                               (after closed-loop)
    I in (o  w) 1  H (o  w)     GL 2QL 


                                         1                       
                                           H ( )  in / f
                                                     2   2
                    V 2                                                                      0 
                                                                                                             2

   L{}  10  log  noise   10  log  2                        10  log[ 2 FkT                ]
                         2 
                     Vsig              
                                         
                                                1
                                                   Vo
                                                       2          
                                                                                 Ps            2Q 
                                               2                 
Leeson’s Equation:

                         2 FkT                 1 / f 3
                                                 2
                                                                 
       L  10  log        1            
                                                      
                                       2Q      1  
                                           0                     
                         Ps   L                               
                                                            
       where
       LΔω the phase noise @ Δω offset
       k     the Boltzmann' s constant
       T     the absolute temperature
       Ps     the signal power
       QL       the loaded Q
       ω1/f 3   the corner frequencyof 1/f 3
       F        a fitting parameter
Noise_Power Tradeoff:              V2
                         P  I 2R      I*  N  I
                                    R
                         Pnoise  N  Pnoise



                                         S/N increases.
Effect of Frequency Division and
                      Multiplication on Phase Noise:
    •Divider contributes the noise much less (can be neglected);
    •Phase noise are also decreased by N factor.
    •From narrow band FM approximation, the phase noise power
     is divided by N2.
    •Frequency multiplication raises the phase noise magnitude by
     the same factor.

     Divide-by-two :

           20  log(2-1 )  6dB (divider, enhancement)
           20  log(21 )  6dB (multiplie r, degradation)
Pushing & Pulling Effects:
                                        [Injection pulling by an
                                         interference]
•Power Supply : pushing effect
•Load / Interference : pulling effect
•Both effects cause
the frequency shift.
•In general, pulling effect
slows down the oscillation.
•For VCO,
the regulator LDO is necessary!
Pulling by Interference in Rx:
Pulling by PA (Load Variation) in Tx:




                                         [to be continued]
Oscillator Types and Features :
                    (a) Negative-Gm / LC-resonant oscillator
                    (b) Interpolative oscillator (two oscillators)
                    (c) Relaxation oscillator / ring oscillator
                    (d) Standing Wave Oscillator

   (a) LC-tuned:
                                                       -R




   Colpitts using active buffer!
Oscillator Types and Features : (cont.)


                                           •Capacitive divider
                                           •Optimized at the current
                                            level where the transistor
                                            noise is lowest; but still
                                            can optimized with the
                                            largest voltage swing!
Oscillator Types and Features : (cont.)
Oscillator Types and Features : (cont.)
    * Push-Push (2x Harmonic) Oscillator




                            2x fosc




                                       PN: -97 dBc/Hz @ 1MHz (reg.)
                                           -102 dBc/Hz @ 1MHz (Micro.)
Oscillator Types and Features : (cont.)

       •Parasitics
       •Q-factor degrades
Oscillator Types and Features : (cont.)

      (b) Interpolative Oscillator:
             •Wide tuning range;
             •Poor phase noise, in general case.

                                      Overall transfer function:
                                      H(s)=α1H1(s) + α2H2(s)
Oscillator Types and Features : (cont.)

      (b) Interpolative Oscillator:
Oscillator Types and Features : (cont.)
     (c) Ring & Relaxation Oscillators:




     •Odd stages (single end)
     •Odd and Even (differential)
Oscillator Types and Features : (cont.)

     (d) Wave-Based Oscillator: Millimeter-Wave Applications
Quadrature Signal Generation:

    Importance : image-rejection; SSB Mixing
    Div-by-2n;
    Div-by-2n+1 ?
    An new example,
        LO generator:

 Let f = 1,350 (MHz),
 
 1,350 x 2 / 3 = 900 MHz
 1,350 x 4 / 3 = 1,800 MHz
Quadrature Signal Generation:
                                 •Havens’ Technique:
    •RC-CR:
Quadrature Signal Generation: (cont.)

     •Frequency Division:
              •Div-by-two, 50 % duty cycle naturally.
              •Easy to implement 2N divider.
Quadrature Signal Generation: (cont.)

    •Frequency Division:
        •Div-by-3, important to UWB application.
        •Challenge: to build 50% duty-cycle, quadrature outputs.



   Basic Unit Cell:
   current-switched
   D-type flip-flop.
Important References for div-by-three circuits:
  1. J.-R. Yuan and C. Svensson, “Fast CMOS Nonbinary Divider and Counter”,
     Electronics Letter, vol.29, no.13, pp.1222-1223, June 1993.
  2. R. Magoon and A. Molnar, “RF Local Oscillator Path for GSM Direct
     Conversion Transceiver with True 50% Duty Cycle Divide by Three and Active
     Third Harmonic Cancellation,” in IEEE Radio Frequency Integrated Circuits
      symposium, pp.23-26, June 2002.
  3. http://www.rfdesign.com/ ; “Frequency divider design strategies,”
      by Louis Fan Fei, Broadband Technology, pp.18-26, March 2005.
  4. S.-C. Tseng, Chinchun Meng, and W.-Y Chen, “True 50% Duty-Cycle SSH and
     SHH SiGe BiCMOS Divide-by-3 Prescalers,” IEICE Trans. Electron., vol.E89-C,
     no.6, June 2006.
  5. C.-F. Liang, S.-I. Liu, Y.-H Chen, T.-Y. Yang, and G.-K. Ma, “A 14-band
      Frequency Synthesizer for MB-OFDM UWB Application,” in IEEE Int.
      Solid-State Circuit Conference, 6.7, 2006.
  Patents:
  1. Bo Sun, “Divide-by-three Circuit,” US Patent NO. 6,389,095.
  2. S. J. Clendening and T. D. Adams, “Divide by Three Clock Divider with
      Symmetrical Output,” US Patent NO. 4,366,394.
•Differential type Div-by-3
•Frequency Division: (div-by-3, 50% duty cycle, differential)
•Frequency Division: (div-by-3, 50% duty cycle, quadrature ! )




                                                 Concept:
                                                 Phase alignment
                                                 Is necessary!
Quadrature Divide-by-Three:
                                        Timing Diagram:



                                           66%



                                             66%

         Div-by-3
                    66% Duty Cycle
                                            50%




Delay Generator              AND gate
50% duty-cycle,
 div-by-three circuit:
A new Quadrature 50% duty-cycle,
 Divide-by-Three circuit
 using TSPC technique:
                                    A




         A    B            B    A
                                    B

         A    B            B    A
A New Quadrature Div-by-3 Circuit using TSPC technique: (cont.)


• Experiments:
a. Original [1];
b. Self-link;
c. Cross-link
                                            c.




                                       a.        b.
1. Cross-links between I – IB and Q – QB : 30° phase difference!




                                               Input clock

                                                  Differential pair
                                                  (ref. clock)

                                                  Differential pair
                                                  (lagging 30°)
2. Under Correct Connection: Timing Diagram at each inter-stage node:




                                          Input clock

                                          33.3% Pulse width

                                          66.7% Pulse width

                                          50% Pulse width
3. Under Correct Connection: Timing Diagram at each output node:




                          120.58
                                                      Differential pair
                                                      (ref. clock)
              115.64          135.44

                120.58 – 115.64 (s)
                ________________                      Differential pair
                135.44 – 115.64 (s)                   (lag/lead 90°)
                  ≒ 25%
3. Under Correct Connection: Timing Diagram at each link node:




                                                   Input link Waveform

                                         33.3%      @ I-block

                                                    @ IB-block

                                                     @ Q-block

                                         66.7%       @ QB-block
Single-Side Band (SSB) Generation:




                              cos A  cos B 
                                              1
                                                cos(A  B)  cos(A  B)
                                              2
                              sin A  sin B  cos(A  B)  cos(A  B)
                                             1
                                             2
•Three SSB Mixers;
•3,168 MHz x 2 = 6,336 MHz




                             [the end]

				
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