Oscillators
Communication Circuits
Prof. M. Kamarei
Contents
●
●
●
●
●
●
Introduction
General Considerations
Ring Oscillators
LC Oscillators
Other Oscillators
Voltage-Controlled Oscillators
M. Kamarei, Spring
2009
Oscillators
2
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
3
Contents
● Introduction
● General Considerations
– 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
4
Oscillator As Feedback System
● General feedback system
V out
(s) 
V in
V out
● We assume s=jω
V in
( j ) 
H (s)
1  H (s)
H ( j )
1  H ( j )
● If there is a frequency in which H(jω0)=1180º then
V out
V in
M. Kamarei, Spring
2009
( j 0 ) 
 
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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.
M. Kamarei, Spring
2009
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How a circuit can oscillate?
● Following the signal around the loop over several cycles:
|H(jω0)|<1
|H(jω0)|>1
VX : Diverged
M. Kamarei, Spring
2009
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Barkhausen Criteria
● If a negative feedback circuit satisfies these two
conditions, the circuit may oscillate at ω0 .
 H ( j )  1

  H ( j  )  180 
● 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
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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
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9
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
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10
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
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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
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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
13
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
 H ( j  )  360   180   3  60   tan
Negative
feedback
M. Kamarei, Spring
2009
1
 osc
0
 60    osc 
3 0
Each stage
phase delay
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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
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Three-Stages Ring Oscillator
● Amplitude limiting
Poles:
Neglecting s1
M. Kamarei, Spring
2009
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Three-Stages Ring Oscillator (Root Locus)
● A0=2
– 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
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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
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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.
M. Kamarei, Spring
2009
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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
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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
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Frequency Domain Perspective
● In the frequency
domain perspective,
we see that a
feedback amplifier
has a transfer
function
 osc
● 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
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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
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Some Other Implementations
● Differential implementation
M. Kamarei, Spring
2009
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Some Other Implementations
● CMOS inverter ring oscillator
M. Kamarei, Spring
2009
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Some Other Implementations
● Five-stages single-ended ring oscillator
● Four-stages differential ring oscillator
M. Kamarei, Spring
2009
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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
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RLC Circuits
● Resonance of LC tank
Z  j    jL  ||
Z  j  res    
jL  res 
1
jC  res
1
jC 
0 
 res 
1
LC
Ideal 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)
M. Kamarei, Spring
2009
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RLC Circuits
● Realistic inductors suffer from resistive components.
Realistic Tank
● Impedance peaks at resonance frequency.
● Q in finite.
M. Kamarei, Spring
2009
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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
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2009
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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
M. Kamarei, Spring
2009
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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
M. Kamarei, Spring
2009
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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
M. Kamarei, Spring
2009
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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
M. Kamarei, Spring
2009
Oscillators
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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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2009
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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
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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
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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:
M. Kamarei, Spring
2009
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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
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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
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2009
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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
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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
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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
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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
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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
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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
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Simplified Model of Clapp Oscillator
● Looks similar to Colpitts model
– Be careful of parasitic resonances!
M. Kamarei, Spring
2009
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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
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2009
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Simplified Model of Hartley Oscillator
● Similar to Colpitts, again be wary
of parasitic resonances
M. Kamarei, Spring
2009
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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.
C2
C1
C2
M. Kamarei, Spring
2009
C1
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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
vo  I 1
vi   I  1
M. Kamarei, Spring
2009
1
j C1
vo
vi
1
n
C1
C2
j C 2
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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.
M. Kamarei, Spring
2009
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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
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Wien-Bridge Oscillator
● By breaking the loop at the positive input, the return
signal is calculated as:
R
V RETURN
V OUT

RCs  1
R
RCs  1
R
1

1
3  RCs 
Cs
1

RCs
1
1 

3  j  RC  

RC



.
● 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.
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2009
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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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2009
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Phase-Shift Oscillator
● With one op amp
● Assuming that the phase-shift sections are independent
of each other:


A   A

 RCs  1 
1
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2009
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56
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.
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2009
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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.
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2009
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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 104 - 106
● Electrical model:
contacts and lead wires
mechanical energy storage
loss (RS ≈ k/f0)
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2009
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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.
Modified
– This topology is useful if one terminal of the
crystal must be grounded.
M. Kamarei, Spring
2009
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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.
M. Kamarei, Spring
2009
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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
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2009
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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
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2009
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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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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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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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2009
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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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2009
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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
V cont
M. Kamarei, Spring
2009
K VCO
s
 ex
cos  0 t   ex 
Oscillators
V out
68
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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2009
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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.
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2009
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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
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2009
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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.
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2009
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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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2009
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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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2009
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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
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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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2009
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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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2009
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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.
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2009
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