Oscillators

Jayanta Mukherjee 1
Oscillator Models
• Feedback oscillator system (a)
• Negative resistance oscillator (b)
• Equivalence of representation using S-parameters (c)

noise
ain
an
aL ain

L in

aL
Z in

Note: the impedances shown are for the first harmonic (fundamental).
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Condition for Starting the Oscillations: Nyquist Test
Consider the transfer functions:

Vout A a Γ in
 or L  90
Vin 1  AH(s) an 1  Γ in Γ L Γin Γ L 
1+j0
s  3 s  6
The number of encirclements of the point (1  j0)
by Γ in Γ L curve should be non-zero. For example:

10  s  3
1+j0
Γin ΓL 
 s  20  s  2

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Operation Point: Barkhausen's Criteria
• The oscillation needs to sustain themselves in the absence of noise. This occurs
for:
A0 H ( s  j0 , A0 )  in (0 , A)  L (0 )  1
where A 0 is the amplitude of oscillation and 0 the oscillation frequency in
steady state.
1
in ( A0 , 0 ) 
 L (0 )
 Z in  A0 , 0    Z L 0 
 Z in  A0 , 0   Z L 0   0
-Zin(A)

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Stability

An oscillator is stable if at the operating pt,

Rin ( A0 , 0 ) X L (0 ) X in ( A0 , 0 ) RL (0 )
 0
A  A 

Kurokawa condition of stability

 It can be shown that this translates to,

      0 (prove it)

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Performance Parameters
1. Frequency Range
• e.g. GSM 935 – 960 MHz tuning range.
• Allowance for temperature variation and modeling inaccuracies.
• Quadrature topology or Injection pulling can necessitate higher
frequency generation.

2. O/P voltage swing

• Higher swing ensures sharp
switching.
• Lower swing can amplify
internal oscillator noise.
• For 1 V supply swing should
be between 0.6 to 0.8 Vpp.
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Performance Parameters
3. Load Pulling
• May need to supply to more than
one load like frequency divider and mixer.
• Problem amplified in Tx where PA i/p
capacitances can cause PA pulling.
• Usually a buffer stage is used to
decouple oscillator from o/p loading.
• Buffers can also amplify o/p of
oscillator thereby increasing voltage
swing.

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Performance Parameters
4. Power dissipation – Often in conflict with phase noise and tuning
range. Since tuning range depends on range of control voltage. For increasing
control voltage supply voltage and hence power consumption needs to be
increased.
Phase Noise is related to voltage swing → lower noise requires higher voltage
swing and hence higher power consumption.

5. Supply Sensitivity – Supply variation is equivalent to flicker noise which affects

oscillator frequency. Flicker noise cannot be easily removed by bypass
capacitors.

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Performance Parameters
6. Output Waveform
• Abrupt LO transitions reduce noise and increase conversion gain.
and Improve frequency divider performance.
• Differential signals with 50% duty cycle reduce
second order non linear effects and also dc
Both M1 and M2 on. Only common mode gain provided.
feedthrough. Also both M1 and M2 contribute to o/p noise.
• Pure square waves difficult to achieve due to
harmonic suppression due to narrowband nature
of oscillator tank and the o/p parasitics of
buffer.
• Hence the next best approach is to increase O/P
voltage swing (create large ID changes
in M1 and M2) by increasing VLO swing or
increasing size of M1 and M2.

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Performance Parameters
7. Phase Noise
 
 
Say , xLO(t)  ALO cos ωc t  φn(t) 
  
 change is random ,
causes random 
 change in frequency 

Ln ( f 0  f ) dBc / Hz  Pnoise dBm  B dBHz  PLO dB

Random change in frequency

Oscillator Output spectrum

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Effect of LO Phase Noise On Receiver
Pnoise proportional to Pint BPLO Ln
• Degradation (decrease) of SNR → increase of BER ( f noise  fint  f 0  f sig )
PIF proportional to Psig  PLO
• Degradation of selectivity (in addition to IMD)
PIF Psig  PLO Psig
We must have at worst : SNRmin   
• For small bandwidth B the noise power is: Pnoise Pint BPLO Ln BPint Ln (f )
Giving the maximum allowable phase noise at f  f as,
Ln (f ) dBc  Psig  Pint dBm
 B dBHz  SNRmin dB
dBm

PLO
B|dBHz=10*log10(B)

Pnoise

fnoise

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GSM Example

Frequency Offset Interfering Signal Ln(Δf )

Δf (MHz) Level (dBm) dBc/Hz
3.0 -23 -138
1.6 -33 -128
0.6 -43 -118

The channel Bandwidth is 200 kHz. The carrier signal is -99 dBm. The required SNRmin is 9 dB.

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Effect of LO Phase Noise On Transmitter
SNR at f2 reduced due to phase noise
of user 1

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Effect of LO Phase Noise On Modulation and
Demodulation Overall phase is random

𝜋
𝑥𝑄𝑃𝑆𝐾 𝑡 = 𝐴 cos 𝜔𝑐𝑡 + 2𝑘 + 1 + 𝜑𝑛(𝑡) 𝑘 = 0, … . , 3
4
From Phase Noise
Amplitude remains same
but phase is random.

If noise is very high then symbol

will be detected falsely.

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Phase Noise Model

Transfer function of a parallel RLC resonator

V 1 1 V1 VV2
H ω  2  
V1  ω ω 0  1  jQ 2 Δω
1 j Q   
ω
 0 ω  ω0
 0L
For calculating the output oscillator PSD we need, Q ,
R
2
Sout      
2 2
1  ω0   h  0
  1    1    
Sin  ω  1  H  ω   2QΔ   
ω H()
where, ωh  0
2Q
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Leeson's Model
dBm/Hz

Δω2

ωLO ωh Δω(log scale)

  ω   2

Sout  Sin 1/ (1  H ( ))  Sin 1  

2 0
   o/p PSD
  2QΔ  
Sin  kT0 F  i/p PSD, where F is an empirical noise factor
 Sout   FkT   2  
L( )  10log    10 log  1  
0 0
 
 PLO    2Q   
 PLO
where PLO is the oscillator output power.
Better phase noise characteristics are achieved with a high Q resonator.
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Problems with Leeson's Model
• Empirical, often fitting factors are needed to match with experiments.
• No device parameters are present in the model.
• Does not take into consideration up conversion of flicker noise since
at very low , the PSD varies inversely with 3 instead of 2.

Advantages of Leeson's Model

• Simple linear model.
• Predicts correctly that for large , the PSD varies inversely with 2.
• Predicts correctly that phase noise reduces with an increase in PLO.

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Basic LC Oscillator Topologies (Colpitt’s and Clapp’s)

At operating point,1
gm
R S  -R IN  1
C1C2 2 id  g m vgs , vgs   I X 
sC1
j  1 1  gm I X
X L  j0 L1   X IN     I C1  I X , I C 2  I X  id  I X 
 0  C1 C2  sC1
1
 0   g  1 1
CC VX  I X  1  m    IX 
L1 1 2  sC1  sC2 sC 1
C1  C2
VX ( j ) 1 1 gm
Condition of startup,   
I X ( j ) jC1 jC2 C1C2 2
| RIN |  RS     
X IN RIN
gm
  RS (which is usually satisfied at dc)
C1C2 ω2 Jayanta Mukherjee 18
Variants of 3 point oscillator
RP
AC open
L1 contains RS
AC short
RS
LS
LS
Vout

I BIAS
C2

C1

For Colpitt's oscillator, 2

Vtank  IBIAS Vout  I BIAS R p

 RP
where, R P   Ls 0  2
/ RS , is the equivalent shunt resistance of the inductor at  0 .
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Differential Negative Gm Oscillators

Max phase shift of

270 degree,
Not sufficient
to meet
Bark hausen
criteria

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Differential Negative Gm Oscillators
Each stage provides 180 degree phase shift

Voltage swing VXY  (4/π) ISS Rp

(See e.g. 8.11 in razavi new edition) Less affected

by VT variation
when ISS is present
(Gray & Meyer 4th Ed,
Section 3.5.3)

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 I X   g m1V1  g m 2V2
VYX V1  V2  I X / g m1  I X / g m 2
 
IX IX IX
 1 1  2
     
g
 m1 g m2  gm
At steady state,
2 1
2Rp   Rp 
gm gm
VYX  Rp gm  1

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Phase Noise of differential oscillator

= -2/gm

C1/2

2Rp
Noise Say net conductance G=1/2Rp- gm/2 = 0
2L1 sources

Hence net resistance R = 1/G = 

With MOSFET noise sources

23
 LO power
I n 2  I n1 2 2kT  g m (for homodyne - DSB NF)
IX  , I n1 =  4 
2
2  kT  g m (for heterodyne - SSB NF)    I SS R p  / 2
 
Hence total noise current psd in one branch,
4kT
= I 2X  Hence, Phase noise
2 Rp
 I 2n1 2kT  1
 I2n1 2kT     2 2
Vn2,out  2 R 
p  C1 
Total noise voltage psd, V 2
   | Z tank |2 L( )   
n , out
 2 R p  LO power 2
 4 
   I SS R p  / 2
 I 2n1 2kT  4 L12 2
   (Taking R P =  )
 2 R p  1  L C  
 2 2  2  I 2n1 2kT  1
     2 2 2 2,
8  2 R p  I SS R p C1 
1 1

 I2n1 2kT  4 L2 2  I 2n1 2kT  1

=   1
   2 2  2  I 2n1
2kT  02  C /2
 2 R 
p   
2
2  2 R 
p  C1 
=    2 2 2 , Q  0 1
  8  2 Rp  I SS Q  1/ 2 RP
1  2  
 0 

where,   0   and 0 >>

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• 𝐿(𝛥𝜔) is inversely proportional to I SS2 (which is proportional to voltage
swing).
• 𝐿(𝛥𝜔) is inversely proportional to  2 (the offset frequency).
• 𝐿(𝛥𝜔) is inversely proportional to Q2 .

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Voltage controlled oscillator •
•
•
Range of Cvar depends on channel length.
Lesser the value of L higher is the relative variation of Cvar .
However overlap capacitance(Cox) dominates Cvar when L is
too less.
• Need to increase L for optimum Cvar variation.
• Increasing L decreases the Q of the varactor.

Cvar Cvar

1
0 
L1 C1  Cvar 
constant

out  0  KVCO Vcont

Ideal characteristics
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Advantages of Differential topology
• Less affected by power supply variations due to symmetry.
• Higher voltage swing
• Less affected by VT and temperature variations

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Quadrature LC oscillator

Z L Z in
Gm 2VX  VY (1)
Z L  Z in
Z L Z in
Gm1VY  VX (2)
Z L  Z in

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Equating (1) and (2) we get,
Z L Z in
G V  G V  Z  Z  0
2
m2 X m1 Y
2

L in

Since ZL ||Zin cannot be zero,

Gm 2VX2  Gm1VY2

If Gm1  Gm 2
 VX =  VY i.e. a oscillator with 180 phase shift .
If Gm1   Gm 2
 VX =  j VY i.e. a quadrature oscillator

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In Phase coupling (VA=-VC)
ZA and ZB are resonators

IZA VA = -IZA ZA

ID3
ID1 ID1

ID3
Injected currents IZA VA

I ZA  I ZA exp( j ), Z A  Z A exp(0)
Hence, VA = -I ZA ZA  I ZA Z A ,
Thus VA can have phase = 0,

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Cross Coupling (VC = j VA)
ZA and ZB are resonators, ID1 and ID3 follow the phases of VB and VD

IZA  -/2 Phasor diagram of

 +/2
voltages and currents
ID3
ID1

Injected currents
 + 
VA

-
 Determines relative magnitude of ID3 and ID1
VA = -IZA ZA I ZA  I ZA exp( j    ), Z A  Z A exp( j )
Hence, VA = -I ZA ZA  I ZA Z A ,
Thus VA can have phase = 0, even though I D1
and ID3 are normal to each other. Jayanta Mukherjee 31
Injection based frequency division

= ISS + Irf

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Monolithic Inductors
Am5/3
 Empirical formula for spiral inductor L  1.3  10 -7

W 1.75 W  G 
1/6 1/4
Atot
where A m is the metal area, A tot  S2 is the inductor area, W the line width,
G the line spacing. Formula is accurate from 5 to 50 nH.
 Image currents in a lossy substrate decrease the inductor Q (eddy currents).

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Shielded and Not Shielded Inductors

• The shield suppresses the substrate loss and noise coupling. A patterned shield
reduces the image current.
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Electrical Model for Non-Shielded Inductors

IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 33, NO. 5, MAY 1998, "On-Chip Spiral Inductors with Patterned Ground Shields for Si-Based RF IC’s" Patrick Yue, and S. Simon Wong
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Effective Inductance

Leff / Ls

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