1. Communication Circuits Prof. M. Kamarei
Oscillators
Communication Circuits
Prof. M. Kamarei

2. Contents Introduction General Considerations Ring Oscillators
LC Oscillators
Other Oscillators
Voltage-Controlled Oscillators
M. Kamarei, Spring 2009
Oscillators

3. Introduction
An oscillator is an autonomous circuit that converts DC power into a periodic waveform.
Oscillators are extensively used in both receive and transmit paths. They are used to provide the local oscillation for the mixers for up and down conversion.
M. Kamarei, Spring 2009
Oscillators

4. Contents Introduction General Considerations Ring Oscillators
Oscillator As Feedback System
Barkhausen Criteria
One-Port View of Resonator-Based Oscillators
Ring Oscillators
LC Oscillators
Other Oscillators
Voltage-Controlled Oscillators
M. Kamarei, Spring 2009
Oscillators

5. Oscillator As Feedback System
General feedback system
We assume s=jω
If there is a frequency in which H(jω0)=1180º then
M. Kamarei, Spring 2009
Oscillators

6. How a circuit can oscillate?
Input noise component at ω0 experiences unity gain and 180º phase shift in H(s). Then it is subtracted from input and gives larger difference.
The circuit continues regenerating and allows the ω0 to grow.
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Oscillators

7. How a circuit can oscillate?
Following the signal around the loop over several cycles:
|H(jω0)|< |H(jω0)|>1
VX : Diverged
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Oscillators

8. Barkhausen Criteria
If a negative feedback circuit satisfies these two conditions, the circuit may oscillate at ω0 .
First criterion
We typically choose the loop gain to be at least two or three times more than the required value to ensure oscillation in presence temperature and process variations.
M. Kamarei, Spring 2009
Oscillators

9. Barkhausen Criteria Second criterion
180º phase shift criterion is equivalent to totally 360º loop phase shift.
(a): 180º phase shift is provided by negative feedback and another 180º by amplifier.
(b): 360º phase shift is totally provided by amplifier.
(c): 2nπ phase shift function same as (b)
M. Kamarei, Spring 2009
Oscillators

10. One-Port View of Resonator-Based Oscillators
Convenient for intuitive analysis
Here we seek to cancel out loss in tank with a negative resistance element
To achieve sustained oscillation, we must have:
M. Kamarei, Spring 2009
Oscillators

11. One-Port View of Resonator-Based Oscillators
Simple tank stimulated with current impulse
Decaying oscillation
In each cycle energy is lost in the resistor.
Canceling out RP with a negative resistance –RP
Stable oscillation
M. Kamarei, Spring 2009
Oscillators

12. Contents Introduction General Considerations Ring Oscillators
Three-Stages Ring Oscillator
Natural Response
Some Other Implementations
LC Oscillators
Other Oscillators
Voltage-Controlled Oscillators
M. Kamarei, Spring 2009
Oscillators

13. Simple Ring Oscillator
Ring oscillator consists of some gain stages in a loop.
Dose a single common-source stage oscillate in loop?
Max. Phase Shift = 180º + 90º < 360º
Common-source negative gain
Max. Phase shift of the output single pole
 No Oscillation
M. Kamarei, Spring 2009
Oscillators

14. Three-Stages Ring Oscillator
Negative feedback provides 180º phase shift.
3 single pole gain stages can provide 0 to 270º phase shift
Phase criterion → Oscillation frequency
Negative feedback
Each stage phase delay
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Oscillators

15. Three-Stages Ring Oscillator
Gain criterion → The minimum gain of stages
Phase delay between 2 outputs:
180º-60º=120º
The ability to generate
multiple phases
M. Kamarei, Spring 2009
Oscillators

16. Three-Stages Ring Oscillator
Amplitude limiting
Poles:
Neglecting s1
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Oscillators

17. Three-Stages Ring Oscillator (Root Locus)
Pure Complex poles → Stable oscillation
A0<2
Circuit is stable (LHP poles) → Damped oscillation
A0>2
Circuit is unstable (RHP poles) → Oscillation amplitude grows
M. Kamarei, Spring 2009
Oscillators

18. Natural Response Given a transfer function
The total response of the system can be partitioned into the natural response and the forced response
where f2(si(t)) is the forced response whereas the first term f1() is the natural response of the system, even in the absence of the input signal. The natural response is determined by the initial conditions of the system.
M. Kamarei, Spring 2009
Oscillators

19. Real LHP Poles
Stable systems have all poles in the left-half plane (LHP).
Consider the natural response when the pole is on the negative real axis, such as s1 for our examples.
The response is a decaying exponential that dies away with a time-constant determined by the pole magnitude.
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Oscillators

20. Complex Conjugate LHP Poles
Since s2,3 are a complex
conjugate pair
We can group these
responses since a3 = a*2
into a single term
When the real part of the complex conjugate pair σ is negative, the response also decays exponentially.
M. Kamarei, Spring 2009
Oscillators

21. Complex Conjugate Poles (RHP)
When σ is positive (RHP), the response is an exponential growing oscillation at a frequency determined by the imaginary part ω0.
Thus we see for any amplifier with three identical poles, if feedback is applied with loop gain A03 > 23 = 8, the amplifier will oscillate.
M. Kamarei, Spring 2009
Oscillators

22. Frequency Domain Perspective
In the frequency
domain perspective,
we see that a
feedback amplifier
has a transfer
function
If the loop gain a0 f = 8, then we have with purely imaginary poles at a frequency ωosc = √3×ω0 where the transfer function a(jωosc)f = -1 blows up. Apparently, the feedback amplifier has infinite gain at this frequency.
M. Kamarei, Spring 2009
Oscillators

23. Oscillation Build Up
In a real oscillator, as the oscillation amplitude increases, the stages in the signal path experience nonlinearity and eventually saturation, limiting the maximum amplitude.
We may say the poles being in the RHP and finally move to imaginary axis to stop
the growth.
The circuit must spend
enough time in saturation
so that the average loop
gain is still equal to unity.
M. Kamarei, Spring 2009
Oscillators

24. Some Other Implementations
Differential implementation
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Oscillators

25. Some Other Implementations
CMOS inverter ring oscillator
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Oscillators

26. Some Other Implementations
Five-stages single-ended ring oscillator
Four-stages differential ring oscillator
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Oscillators

27. Contents Introduction General Considerations Ring Oscillators
LC Oscillators
RLC Circuits
A Simple Negative Resistance Oscillator
Differential Negative Resistance Oscilator
Colpitts Oscillator
Clapp Oscillator
Hartley Oscillator
CE and CC Oscillators
Pierce Oscillator
Other Oscillators
Voltage-Controlled Oscillators
M. Kamarei, Spring 2009
Oscillators

28. RLC Circuits Resonance of LC tank
The impedance of the circuit goes to infinity at resonance frequency
Q of the tank in infinite.
(Q is the quality factor of the tank and defined as energy stored, divided by energy dissipated per cycle)
Ideal Tank
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Oscillators

29. RLC Circuits Realistic inductors suffer from resistive components.
Impedance peaks at resonance frequency.
Q in finite.
Realistic Tank
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Oscillators

30. Series RL to Parallel RL
In narrow frequency range it is possible to convert circuit to parallel configuration.
For the two impedances to be equivalent:
L1ω/Rs=Q
typically more than 3
M. Kamarei, Spring 2009
Oscillators

31. A Simple RLC Model RP is typically higher than RS CP is equal to CS
In resonance
frequency the tank reduces
to a simple resistor
Phase difference between
voltage and current is zero
For ω<ω1
Inductive behavior, Positive phase
For ω>ω1
Capacitive behavior, Negative phase
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Oscillators

32. A Simple Negative Resistance OSC
How can a circuit provide negative resistance?
Positive feedback circuits with sufficient gain can provide it
Small signal model
If gm1=gm2=gm
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Oscillators

33. A Simple Negative Resistance Osc.
Adding a tank to the negative resistor,
we make an oscillator
For oscillation build-up
RP-2/gm  0
Redrawing the oscillator
Converting to differential negative resistance oscillator
Differential
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Oscillators

34. Differential Negative Resistance Osc.
This type of oscillator structure is quite popular in current CMOS implementations
Advantages
Simple topology
Differential implementation (good for feeding differential circuits)
Good phase noise performance can be achieved
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Oscillators

35. Analysis of Negative Resistance Osc. (Step 1)
Derive a parallel RLC network that includes the loss of the tank inductor and capacitor
Typically, such loss is dominated by series resistance in the inductor
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Oscillators

36. Analysis of Negative Resistance Osc. (Step 2)
Split oscillator circuit into half circuits to simplify analysis
Leverages the fact that we can approximate Vs as being incremental ground (this is not quite true, but close enough)
Recognize that we have a diode connected device with a negative transconductance value
Replace with negative resistor
Note: Gm is large signal transconductance value
M. Kamarei, Spring 2009
Oscillators

37. Analysis of Negative Resistance Osc. (Step 3)
Design tank components to achieve high Q
Resulting Rp value is as large as possible
Choose bias current (Ibias) for large swing (without going far into Gm saturation)
We’ll estimate swing as a function of Ibias shortly
Choose transistor size to achieve adequately large gm1
Usually twice as large as 1/Rp1 to guarantee startup
M. Kamarei, Spring 2009
Oscillators

38. Calculation of Oscillator Swing: Max. Sinusoidal Oscillation
If we assume the amplitude is large, Ibias is fully steered to one side at the peak and the bottom of the sinusoid:
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Oscillators

39. Calculation of Oscillator Swing: Square wave Oscillation
If amplitude is very large, we can assume I1(t) is a square wave
We are interested in determining fundamental component
(DC and harmonics filtered by tank)
Fundamental component is
Resulting oscillator amplitude
M. Kamarei, Spring 2009
Oscillators

40. Variations on a Theme Biasing can come from top or bottom
Can use either NMOS, PMOS, or both for transconductor
Use of both NMOS and PMOS for coupled pair would appear to achieve better phase noise at a given power dissipation
M. Kamarei, Spring 2009
Oscillators

41. Colpitts Oscillator
Carryover from discrete designs in which single-ended approaches were preferred for simplicity
Achieves negative resistance with only one transistor
Differential structure can also be implemented, though
Good phase noise can be achieved, but not apparent there is an advantage of this design over negative resistance design for CMOS applications
M. Kamarei, Spring 2009
Oscillators

42. Analysis of Cap Transformer used in Colpitts
Voltage drop across RL is reduced by capacitive voltage divider
Assume that impedances of caps are less than RL at resonant frequency of tank (simplifies analysis)
Ratio of V1 to Vout set by caps and not RL
Power conservation leads to transformer relationship
M. Kamarei, Spring 2009
Oscillators

43. Simplified Model of Colpitts
Purpose of cap transformer
Reduces loading on tank
Reduces swing at source node (important for bipolar version)
Transformer ratio set to achieve best noise performance
M. Kamarei, Spring 2009
Oscillators

44. Design of Colpitts Oscillator
Design tank for high Q
Choose bias current (Ibias) for large swing (without going far into Gm saturation)
Choose transformer ratio for best noise
Rule of thumb: choose N = 1/5 according to Tom Lee
Choose transistor size to achieve adequately large gm1
M. Kamarei, Spring 2009
Oscillators

45. Calculation of Oscillator Swing as a Function of Ibias
I1(t) consists of pulses whose shape and width are a function of the transistor behavior and transformer ratio
Approximate as narrow square wave pulses with width W
Fundamental component is
Resulting oscillator amplitude
M. Kamarei, Spring 2009
Oscillators

46. Clapp Oscillator
Same as Colpitts except that inductor portion of tank is isolated from the drain of the device
Allows inductor voltage to achieve a larger amplitude without exceeded the max allowable voltage at the drain
Good for achieving lower phase noise
M. Kamarei, Spring 2009
Oscillators

47. Simplified Model of Clapp Oscillator
Looks similar to Colpitts model
Be careful of parasitic resonances!
M. Kamarei, Spring 2009
Oscillators

48. Hartley Oscillator
Same as Colpitts, but uses a tapped inductor rather than series capacitors to implement the transformer portion of the circuit
Not popular for IC implementations due to the fact that capacitors are easier to realize than inductors
M. Kamarei, Spring 2009
Oscillators

49. Simplified Model of Hartley Oscillator
Similar to Colpitts, again be wary of parasitic resonances
M. Kamarei, Spring 2009
Oscillators

50. CE and CC Oscillators
If we ground the emitter, we have a new oscillator topology, called the Pierce Oscillator. Note that the amplifier is in CE mode, but we don’t need a transformer!
Likewise, if we ground the collector, we have an emitter follower oscillator.
A fraction of the tank resonant current flows through C1,2. In fact, we can use C1,2 as the tank capacitance. C1 C2 C1 C2
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Oscillators

51. Pierce Oscillator
If we assume that the current through C1,2 is larger than the collector current (high Q), then we see that the same current flows through both capacitors. The voltage at the input and output is therefore
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Oscillators

52. Pierce Oscillator Bias
A current source or large resistor can bias the Pierce oscillator.
Since the bias current is fixed, the same large signal oscillator analysis applies.
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Oscillators

53. Contents Introduction General Considerations Ring Oscillators
LC Oscillators
Other Oscillators
Wien-Bridge Oscillator
Phase-Shift Oscillator
Crystal Oscillators
Quadrature Oscillator
Voltage-Controlled Oscillators
M. Kamarei, Spring 2009
Oscillators

54. Wien-Bridge Oscillator
By breaking the loop at the positive input, the return signal is calculated as:
When ω=2πf=1/RC, the feedback is in phase, and the gain is 1/3.
→ Oscillation requires an amplifier with the gain of 3.
M. Kamarei, Spring 2009
Oscillators

55. Wien-Bridge Oscillator
When RF = 2RG, amplifier gain is 3 and oscillation occurs at f = 1/2πRC
Example:
[TI, Analog Applications Journal, 2000]
f = 1.59 kHz
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Oscillators

56. Phase-Shift Oscillator
With one op amp
Assuming that the phase-shift sections are independent of each other:
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Oscillators

57. Phase-Shift Oscillator
When the phase shift of each section is -60º, the loop phase shift is -180º.
This occurs when ω=2πf=1.732/RC (since tan(60º) ≈ 1.732)
The magnitude of β is (1/2)3, so the gain A must be 8.
However in practice, the RC sections load each other and this results in discrepancies from the simple calculated gain and frequency equations.
Besides, now op amps are inexpensive and small. So the single op amp phase-shift oscillator is losing popularity.
M. Kamarei, Spring 2009
Oscillators

58. Phase-Shift Oscillator
Buffered Phase-Shift Oscillator
The buffers prevent the RC sections from loading each other, hence closer performance to the simple calculated gain and frequency.
Note: output is a high impedance node, so a high impedance load is mandated to prevent loading and phase shifting.
M. Kamarei, Spring 2009
Oscillators

59. Crystal Oscillators
The most common non-RLC oscillator is made of quartz.
Quartz is a piezoelectric material and thus it exhibits a reciprocal transduction between mechanical strain and electric charge.
Exceptional electrical and mechanical stability
Crystals with very low temperature coefficients are feasible
Q values in the range of
Electrical model:
mechanical energy storage
contacts and lead wires
loss (RS ≈ k/f0)
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Oscillators

60. Crystal Oscillators
In upper fig. the crystal is used in its series resonant mode to close the feedback loop at the desired frequency as in a Colpitts oscillator.
The inductance across the crystal is frequently needed in practical designs to prevent unwanted off-frequency oscillations due to feedback provided by the crystal’s parallel capacitance (C0). The inductance resonates out C0.
The modified scheme is shown in bottom.
This topology is useful if one terminal of the crystal must be grounded.
Modified
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Oscillators

61. Crystal Oscillators
The MOS Pierce oscillator is a popular crystal oscillator.
The crystal behaves like an inductor in the frequency range above the series resonance of the fundamental tone (below the parallel resonance due to the parasitic capacitance).
The crystal is a mechanical resonators. Other structures, such as poly MEMS combs, disks, or beams can also be used.
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Oscillators

62. Quadrature Oscillator (with feedback loop)
Coupling two oscillators in such a way to make them oscillate with 90° difference in phase
Measuring the phase difference and controlling the bias current leads to precise 90° phase shift
M. Kamarei, Spring 2009
Oscillators

63. Contents Introduction General Considerations Ring Oscillators
LC Oscillators
Other Oscillators
Voltage-Controlled Oscillators
Performance Parameters of VCOs
Mathematical Model of VCOs
PN Junction Varactors
Varactor Tuned LC Oscillator
Clapp Varactor Tuned VCO
Varactor Tuned Differential Oscillator
MOS Varactors
M. Kamarei, Spring 2009
Oscillators

64. Voltage-Controlled Osc.
In most applications we need to tune the oscillator to a particular frequency electronically.
A voltage controlled oscillator (VCO) has a separate “control” input where the frequency of oscillation is a function of the control signal Vc
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Oscillators

65. Performance Parameters of VCOs
Center Frequency
Tuning Range
The tuning range of the VCO is given by the range in frequency normalized to the average frequency
A typical VCO might have a tuning range of 20% 30% to cover a band of frequencies (over process and temperature)
Tuning Linearity
Higher sensitivity in some regions
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Oscillators

66. Performance Parameters of VCOs
Output Amplitude
Larger oscillation amplitude → Less sensitive to noise
Amplitude trades with power dissipation, supply voltage, and tuning range
Power Dissipation
Trade-offs between speed, power dissipation, and noise
Supply and Common-Mode Rejection
Single-ended forms are more sensitive
Noise may coupled to VCO control line and modulates carrier
Output Signal Purity
Jitter
Phase Noise
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Oscillators

67. Mathematical Model of VCOs
Consider two waveforms
V2(t) accumulate phase faster than V1(t)
The faster the phase of a waveform varies, the higher the frequency of the waveform
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Oscillators

68. Mathematical Model of VCOs
Frequency can be defined as:
Since for a VCO, ωout = ω0 + KVCOVcont we have:
If a VCO is placed in PLL, the only term KVCO ∫Vcont dt is important
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Oscillators

69. An Example of the Model
The Control line of a VCO senses a rectangular signal toggling between V1 and V2 and period Tm.
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Oscillators

70. PN Junction Varactors
The most common way to control the frequency is by using a reverse biased PN junction
The small signal capacitance is given by
The above formula is easily derived by observing that a positive increment in reverse bias voltage requires an increment of growth of the depletion region width. Since charge must flow to the edge of the depletion region, the structure acts like a parallel plate capacitors for small voltage perturbations.
M. Kamarei, Spring 2009
Oscillators

71. PN Junctions Doping
Depending on the doping profile, one can design capacitors with n = ½ (abrupt junction), n = ⅓ (linear grade), and even n = 2.
The n = 2 case is convenient as it leads to a linear relationship between the frequency of oscillation and the control voltage
M. Kamarei, Spring 2009
Oscillators

72. Varactor Tuned LC Oscillator
The DC point of the varactor is isolated from the circuit with a capacitor. For a voltage V < VCC, the capacitor is reverse biased (note the polarity of the varactor). If the varactor is flipped, then a voltage larger than VCC is required to tune the circuit.
M. Kamarei, Spring 2009
Oscillators

73. Clapp Varactor Tuned VCO
Similar to the previous case the varactor is DC isolated
In all varactor tuned VCOs, we must avoid forward biasing the varactor since it will de-Q the tank
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Oscillators

74. Varactor Tuned Differential Osc.
Two varactors are used to tune the circuit. In many processes, though, the n side of the diode is “dirty” due to the large well capacitance and substrate connection resistance.
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Oscillators

75. Varactor Tuned Diff Osc.
By reversing the diode connection, the “dirty” side of the PN junction is connected to a virtual ground. But this requires a control voltage Vc > Vdd. We can avoid this by using capacitors.
The resistors to ground simply provide a DC connection to the varactor n terminals.
M. Kamarei, Spring 2009
Oscillators

76. MOS Varactors
The MOS capacitor can be used as a varactor. The capacitance varies dramatically from accumulation Vgb < Vfb to depletion. In accumulation majority carriers (holes) form the bottom plate of a parallel plate capacitor.
In depletion, the presence of a depletion region with dopant atoms creates a non-linear capacitor that can be modeled as two series capacitors (Cdep and Cox), effectively increasing the plate thickness to tox + tdep
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Oscillators

77. MOS Varactor (Inversion)
For a quasi-static excitation, thermal generation leads to minority carrier generation. Thus the channel will invert for VGB > VT and the capacitance will return to Cox.
The transition around threshold is very rapid. If a MOSFET MOS-C structure is used (with source/drain junctions), then minority carriers are injected from the junctions and the high-frequency capacitance includes the inversion transition.
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Oscillators

78. Accumulation Mode MOS Varactors
The best varactors are accumulation mode electron MOS-C structures. The electrons have higher mobility than holes leading to lower series resistance, and thus a larger Q factor.
Notice that this structure cannot go into inversion at high frequency due to the limited thermal generation of minority carriers.
M. Kamarei, Spring 2009
Oscillators