IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 38, NO. 6, JUNE 2003
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Varactor Characteristics, Oscillator Tuning Curves, and AM–FM Conversion
Emad Hegazi, Member, IEEE, and Asad A. Abidi, Fellow, IEEE
Abstract—A simple analysis relates the small-signal specification of a varactor’s capacitance to an oscillator’s tuning curve. The
notion of an effective capacitance across the amplitude of oscillation is introduced. The analysis also explains how the varactor converts AM noise on the oscillation into FM, which is phase noise. The
analysis is experimentally validated.
Index Terms—AM-FM conversion, differential LC oscillator, oscillator, phase noise, tuning curve, varactor, voltage-controlled oscillator (VCO) sensitivity.
I. INTRODUCTION
D
ESPITE the many publications on LC oscillators
prompted by their widespread use in RF circuits, many
fundamental questions remain unanswered. Some of these
questions are deceptively simple.
In this paper, we answer this basic question: If a varactor with
a certain – characteristic is attached across the LC tank, what
is the resulting shape of the oscillator’s frequency-tuning curve?
The answer requires a careful large-signal analysis. We present
the analysis here in the simplest possible terms, and we physically interpret the results and validate them with measurements
on two fabricated oscillators as well as with simulations. The
analysis also sheds light on the varactor’s susceptibility to convert AM into FM, which has repercussions on oscillator phase
noise.
II. LARGE SIGNAL ANALYSIS WITH VARACTOR
A. Quasi-Sinusoidal Approximation
We will use as an example the MOS differential LC oscillator
(Fig. 1), whose operation is better understood than other oscillator circuits [1], [2]. In autonomous steady state, the oscillating
voltage across the LC tank is large enough to fully commutate
the tail current through the differential pair. The resulting square
wave of current sustains oscillation by replenishing losses in the
tank.
A quasi-sinusoidal analysis suffices to explain almost all
properties. What this means is that although the sustaining
square wave current is rich in harmonics, the harmonics do not
play an important role within the tank (Fig. 2). This is because
only the fundamental component of the square wave current,
whose frequency (almost) coincides with the resonant frequency, will flow through the shunt loss conductor, whereas the
L and C in parallel present an infinite reactance at resonance.
Although harmonics are present, the fundamental component
alone defines the steady-state amplitude of oscillation.
Fig. 1.
Differential CMOS LC oscillator.
Fig. 2. Steady-state current flow paths in oscillator. Controlled current source
represents the action of the differential pair.
The harmonics in the square wave take the path of least
impedance, which is through the capacitor. The resulting
harmonic voltage amplitudes across the tank are related as
follows:
Fundamental voltage
th Harmonic voltage
Fundamental
th Harmonic
(1)
It can be seen that in a tank with a reasonably high quality
factor the amplitude of the third and higher harmonics of the
square wave can be neglected compared to the fundamental.
Operation of the differential LC oscillator is most easily understood in the current-limited regime, when the amplitude is
. The differential osnot yet limited by the supply voltage
cillation amplitude is controlled by the tail current
(2)
B. Varactors in Oscillators
Manuscript received May 30, 2002; revised December 15, 2002.
The authors are with the Department of Electrical Engineering, University of
California, Los Angeles, CA 90095 USA (e-mail: abidi@icsl.ucla.edu).
Digital Object Identifier 10.1109/JSSC.2003.811968
A varactor (variable reactor) is usually a voltage-dependent
capacitor whose capacitance depends on a control voltage
. In a two-terminal varactor such as a reverse-biased p-n
0018-9200/03$17.00 © 2003 IEEE
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IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 38, NO. 6, JUNE 2003
captures the effect of the third terminal, the substrate, which is
fixed to ground. Now we must determine that if the varactor’s
small-signal capacitance changes periodically with the oscillation, then what effective capacitance in parallel with the tank
circuit determines the final oscillation frequency?
At first sight, it might seem that the effective capacitance is
simply the time average of the incremental capacitance over
one oscillation cycle. However, this gross simplification gives a
large error in the slope of the frequency-tuning curve and is not
correct. This is because the time average capacitance does not
properly account for the balance of current and voltage required
at every instant of time between the inductor and capacitor to
satisfy the condition for autonomous oscillation.
Let us now carefully account for this balance. Kirchhoff’s
laws applied to the reactive components dictate that
(a)
(4)
periodically sweeps the
As the oscillating control voltage
small-signal capacitance of the varactor across the part of its
characteristic covered by the amplitude , the oscillating capacitance itself can be represented as a Fourier series:
(5)
(b)
Fig. 3. Small-signal capacitance. (a) Reverse-biased p-n junction varactor.
(b) MOS varactor. Typical oscillation waveform shows that the small-signal
capacitance changes greatly over a cycle.
junction, the instantaneous voltage across the terminals is also
the control [Fig. 3(a)]. A three-terminal varactor such as a
MOSFET can decouple the signal and the control, in that the
control voltage might be the bias across, say, the substrate
and shorted source–drain, while the oscillation appears across
the gate and source–drain [Fig. 3(b)]. The large oscillation
amplitude causes the bias itself to vary periodically at the
oscillation frequency.
The standalone varactor is specified by its small-signal, or
versus . This is defined in terms
incremental, capacitance
of the instantaneous charge and voltage across the varactor
as follows:
The coefficients of the Fourier series may be obtained by
analysis or simulation of the varactor’s nonlinear – characsubject to the oscillating voltage
teristic around the bias point
. For two identical MOSFETs or p-n junctions connected back-to-back in series driven by a symmetric oscillation,
the differential varactor characteristic is even-symmetric.
The nonlinear varactor current must itself contain harmonics,
which are all reactive. Continuity of reactive current requires
these harmonics to flow through the inductor. A linear inductor
can only carry current harmonics if the voltage across it also
contains harmonics. Therefore, we must express the tank
voltage as a Fourier series:
(6)
Simulation shows that even with the extremely nonlinear varactor characteristics of an abrupt step in capacitance, the current
and voltage remain quasi-sinusoidal (Fig. 4). Still the analysis
must include harmonics because, as we will show, they contribute through nonlinear action to the fundamental component.
The complete inductor current is
(7)
and from (3) the capacitor current is
(3)
versus
are shown for the reverse-biTypical curves for
ased p-n junction [3] and for a MOS capacitor [Fig. 3(a) and
(b)]. In the case of the MOS capacitor, the family of curves
(8)
IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 38, NO. 6, JUNE 2003
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Fig. 4. Simulated currents and voltages in a differential LC oscillator.
Capacitor absorbs high frequencies contained in current commutation step.
Now from (4), the coefficients at every frequency must balance between (7) and (8). At the fundamental frequency, balance
requires
(9)
Owing to the quasi-sinusoidal approximation, (9) neglects the
) of the caeffect of mixing higher harmonics (at
pacitance waveform with higher order derivatives of the voltage
). These terms will also contribute a
waveform (at
small capacitive current at the fundamental frequency, which we
, which
neglect. Thus we arrive at the effective capacitance
sets the frequency of oscillation
(10)
is the time-average capacitance, which inThe first term
cludes any fixed capacitance in parallel with the varactor. The
is the second-order Fourier coefficient of the
second term
nonlinear varactor driven by the oscillation.
This leads to the frequency-tuning characteristic of the oscildepends on the average voltage
present across the
lator.
varactor, which centers the oscillation on different parts of the
characteristic. It also depends on , the amplitude of oscillation.
This is set by (resistive) limiting in the oscillator circuit and it
is reasonable to assume that remains unchanged across the
tuning range. However, as we will show later, fluctuations in
due to noise can cause fluctuations in
and thus in frequency,
a process called AM-to-FM conversion.
which is
We have found a graphical way to calculate
sometimes more convenient than calculating the Fourier coefficients. This is based on the area enclosed by the – locus of
a capacitor subject to periodic stimulus. The relationship is
(11)
For a linear capacitor , the – locus is an ellipse with
and
, which encloses an area
, leading, as expected, to
. As shown in the
Fig. 5. I –V locus of a MOS varactor subject to a large sinewave voltage, for
several values of bias. The locus consists of two different ellipses joined at the
transition voltage.
Appendix, this relationship holds for a nonlinear capacitor as
well, and yields an equivalent result to (10).
We will use this graphical method to obtain a formula for
of a MOS varactor. To simplify analysis, the small-signal
capacitance [Fig. 3(b)] is approximated with a step function; that
steps from a maximum value
to a minimum value
is,
at an effective threshold voltage
:
(12)
and
is the unit step function.
where
The – locus of a discontinuous capacitor driven by a periodic
waveform that spans the discontinuity consists of two ellipses of
different sizes joined with a step transition (Fig. 5). Thus
(13)
where the ellipse is described by
for
and
for
Carrying out the integration in (13) and substituting in (11) we
obtain
(14)
. Outside this range,
This expression holds true when
it is constant at the value of the expression found by setting
or
, as appropriate. This
enables us to trace
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IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 38, NO. 6, JUNE 2003
out the frequency-tuning characteristic versus
, which de. From (14), we see that in spite of the step-like
pends on
characteristic of the small-signal capacitance, the oscillation fredescribes a smooth curve as a funcquency
tion of the control voltage . This is because the large oscillation sweeping across the step-like varactor capacitance smooths
it out by a process of convolution rooted in the current balance
described earlier.
This expression can also be used to predict the incremental
. The relationvoltage-controlled oscillator (VCO) gain
ship is
(15)
Fig. 6. Step-like approximation to the MOS varactor characteristic. The two
sinewaves show the cases when no AM converts into FM.
at an equal offset from the oscillation frequency governed by the
same relation as between the dc tail current and steady-state amplitude [1], [2]
III. AM–FM CONVERSION AND OTHER SOURCES
OF PHASE NOISE
An undesirable side effect associated with a varactor is that
its effective capacitance depends not only on control voltage,
but also on the amplitude of oscillation. It has been shown that
in a varactorless oscillator certain sources of noise only modulate the amplitude of oscillation; for instance, a differential LC
oscillator upconverts low-frequency noise in the tail current into
purely AM sidebands around the oscillation frequency [2]. Usually, AM noise can later be stripped off in a limiter to restore the
close-in spectral purity of the oscillation, and this is why it is
considered benign in most communication applications. However, these amplitude fluctuations also modulate the effective
capacitance of a varactor, which then converts AM noise into
FM noise (see, for example, [4]). Noisy FM sidebands are indistinguishable from phase noise. A limiter cannot suppress either
phase noise or FM noise. This problem must be solved with the
right varactor, which does not strongly convert AM into FM.
Just as we have analyzed the sensitivity of a varactor’s
to control voltage, so also we can analyze its sensitivity to amplitude. In the case of the MOS varactor with a step-like smallsignal capacitance, this sensitivity is
(16)
AM, therefore, converts into FM with the following sensitivity:
(17)
We can use standard formulas to predict the spectral density
at an offset
from the oscillaof these FM sidebands
tion frequency:
(18)
is the root-mean-squared (rms) AM noise voltage
where
is a Bessel
spectral density on the oscillation envelope and
oscillator operfunction of the first kind. For a differential
ating in the current-limited regime, a noise at a particular frequency in the tail current source produces a single AM sideband
(19)
In general, the wider the frequency tuning range, the stronger
the varactor’s proclivity to convert AM into FM. For example,
the tuning range of a MOS varactor grows with the difference
between the maximum and minimum small-signal capacitance
, and we see from (16) and (17) that so does
. This suggests that wide tuning range is at odds with
low phase noise. However, it is possible to decouple tuning
range from phase noise at the price of a somewhat more complex control scheme. This is now described.
An array of fixed capacitors switch selected by a digital word
can tune the oscillator to a set of discrete frequencies [5], [6].
It is sufficient that the varactor’s variable capacitance cover the
largest gap between adjacent discrete frequencies, not the full
tuning range. With more elements in the switched capacitor
array this gap reduces to smaller fractions of the full tuning
range. The main point is that fixed capacitors do not convert
AM into FM, only the varactor does. In this way, the tuning
range may be expanded arbitrarily by adding more elements to
the capacitor array without worsening the sensitivity to AM–FM
conversion; or, for a given tuning range, the sensitivity may be
lowered.
The expression for the MOS varactor [(14) and (17)] shows
another interesting property, namely, that at two biases AM
and
does not convert into FM. These biases are
(Fig. 6). In the latter case, the sensitivity is found
. It is easy to see why this is
by evaluating (17) at
, the oscillation of amplitude
a null. Biased at
never crosses the capacitance step. Therefore, the capacitance
remains constant for small amplitude fluctuations and no FM
? When biased at
results. Why is there a null at
this point, the sinewave oscillation dwells for one half-cycle
and on the other half-cycle on
; so whatever
on
its amplitude, the effective capacitance remains constant at
. Again, random modulation of the oscillation
envelope will not convert into FM.
For completeness, we should mention that the varactor can
induce FM noise in two other ways. The first is through noise
and the second is through noise on the
on the control voltage
IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 38, NO. 6, JUNE 2003
Fig. 7. Measured frequency tuning curves versus varactor control voltage for
test oscillator circuit shown in inset. Measurements agree very well with analysis
presented in this paper.
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(a)
power supply. Equation (18) with
replaced by
gives the FM resulting from noise on the control voltage. Suppose the control voltage is referred to ground; then noise on the
power supply impresses itself through the inductors on the varactors (Fig. 1) to produce exactly the same effect as noise on
itself, and (18) specifies the resulting FM sidebands.
We should also point out that in our experience, AM–FM conversion in the varactor is seldom the main source of phase noise
in most well-designed oscillators. If it were, the phase noise
would not improve with the quality factor of the resonator.
Additive phase noise arising from sources inside the oscillator
core usually dominates, and the responsible mechanisms are described elsewhere [2].
IV. VALIDATIONS: MEASUREMENT AND SIMULATION
The analytical results and physical insights developed so far
are validated in two different ways: by direct measurement on a
fabricated integrated circuit oscillator and by simulation of this
circuit using SPECTRE-RF.
A first test oscillator is fabricated in 0.35- m CMOS on the
ST Microelectronics BiCMOS6G process. The MOS varactors
consist of a matrix of 64 unit nFETs in parallel, each measuring
5/0.35 m. There is, in addition, a fixed capacitance of 1.1 pF
at each node of the differential oscillator due to the gates and
junctions of the differential pair FETs attached to that node. A
large part of the fixed capacitance is switchable by a 3-b control
word.
We compare the measured frequency-tuning curve of this oscillator with analytical predictions. The oscillator is operated in
the current-limited mode, where the amplitude is proportional
to tail current. A tail current less than 4 mA guarantees that the
supply voltage does not limit amplitude. To verify the dependence of the varactor’s effective capacitance on the oscillation
amplitude, the frequency-tuning curve is measured at two different amplitudes set by tail currents of 2.1 and 4 mA. As ex-
(b)
Fig. 8. (a) AM–FM conversion gain from noise in tail current to FM sidebands
on test oscillator, simulated on SPECTRE-RF and compared with analysis in
this paper. Two nulls are evident. (b) Measured frequency tuning curves at three
tail currents intersect at the same tuning voltages, confirming nulls.
pected, the two curves are different (Fig. 7). The tuning sensitivity lowers with amplitude as the oscillation sweeps across a
greater portion of the varactor characteristic, averaging it more.
We predict the frequency tuning curve with the formulas
given above by approximating the MOS varactor characteristic
with a step. In spite of this simplification, the prediction lies
close to the measurement, and the slopes of the curves match
notably well (Fig. 7).
Next, we search for the predicted nulls in AM–FM conversion. Low-frequency noise in the oscillator’s tail current
first upconverts into AM sidebands [2], and the varactor
converts AM into FM. We study this effect by simulating on
SPECTRE-RF the periodic transfer function from low frequencies in the tail current to phase-noise sidebands in the oscillator
output [Fig. 8(a)]. We compare the simulated transfer function
at
mA with analytical predictions.
versus
1038
Although simulation does not yield perfect nulls, there is a
clear drop in simulated sensitivity at the two anticipated bias
points.
The nulls are also verified by measurement. It is difficult
to directly measure the small-signal response simulated in
SPECTRE-RF, but we can deduce the presence of nulls by
superimposing the frequency-tuning characteristics at three
, and mA [Fig. 8(b)].
different tail currents:
Over this range of currents, the amplitude is proportional to
. Measurement shows that all three tuning curves intersect
of 0 and 1.6 V. This means that at both intersection
at
points the frequency is independent of amplitude, that is, AM
does not cause FM. The tuning voltage at each intersection
point is as predicted, which confirms the analysis.
We also experimentally validate phase noise caused by
AM–FM conversion in an actual oscillator. This is a difficult
experiment because phase noise in a practical circuit results
from many mechanisms at work simultaneously. As we have
shown [2], in a differential oscillator, FM through the varactor
is only one of three different mechanisms responsible for the
noise around the carrier.
upconversion of low-frequency
Therefore, we require a test oscillator with a strong varactor
which emphasizes AM–FM conversion, while
(large
at the same time suppressing the two other mechanisms.
Close-in phase noise will also appear through additive mechanisms, but in a differential oscillator this arises from device
noise near the oscillation frequency and its multiples [2]. At
the high frequencies of the test oscillators, this must be white
noise. This makes it easy to discriminate between close-in
phase noise due to additive mechanisms and due to AM–FM
conversion in CMOS; the slope of the former is 20 dB/decade,
whereas it is 30 dB/decade for the latter.
A second test oscillator was fabricated in 0.35- m CMOS on
the ST Microelectronics BiCMOS6M process. Analysis shows
that the differential pair upconverts flicker noise through tail
capacitance [2]. To lower tail capacitance, the tail current source
FET in the test oscillator consists of an array of unit FETs in parallel, each with an annular gate and the drain junction located inside the annulus to lower capacitance. The differential pair FETs
are themselves also of small size, 15 m/0.35 m. Although the
noise rises with small FET area, the upconinput-referred
version gain due to smaller capacitance falls more dramatically
[2].
SPECTRE-RF simulations confirm that varactor nonlinearity
is now the dominant mechanism for flicker-noise upconversion
in this VCO. The tank inductance is 17 nH with a of 8. The
oscillation frequency ranges over 1040–1140 MHz as the bias
on the varactor is changed from 0 to 2.5 V. At 3-MHz offset from
the oscillation frequency, the 20-dB/decade slope indicates that
phase noise arises only from white noise. The expressions in
[2] predict a level of 142 dBc/Hz, which is very close to the
measured value. Extrapolating at 20 dB/decade, we can say that
at 10-kHz offset, white noise will account for a phase noise of
91 dBc/Hz. If the measured phase noise is greater, or if the
slope is 30 dB/decade, then this must be due to upconverted
noise. Fig. 9 shows the measured phase noise at 10-kHz
offset across the full tuning range, versus the prediction of (18).
The measured slope of 30 dB/decade indicates flicker noise.
IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 38, NO. 6, JUNE 2003
Fig. 9. Verification of analysis of AM–FM conversion on an actual oscillator,
designed so that this is the dominant mechanism of upconversion of lowfrequency (1=f ) noise. When this mechanism of upconversion is at a null,
phase noise is limited to some value arising from white noise through additive
mechanisms [2].
Analytical prediction based on (19) assumes flicker noise in the
bias current, obeying the well-known expression for the spectral
density of MOS flicker noise
(20)
1.58 10
W. To improve model accuracy, we
where
caused by changing
. During meainclude variation in
surement, the control voltage is taken from a battery which is
inherently very low noise; thus, tail current fluctuations account
,
for the main source of noise. At a tuning voltage of
which is 1.6 V in this circuit, the measured phase noise falls
close to the floor set by white noise, indicating a null in the conversion of AM into FM.
V. DISCUSSION
The more the – curve of a varactor departs from linearity,
the more it converts low-frequency noise into phase noise. The
– curve may be linearized by connecting a fixed capacitor
either in series or in parallel with the varactor. Series connection [8] achieves linearization at the expense of tuning range,
while parallel connection [5] requires mixed signal control of
the VCO.
There exists some prior literature worthy of summary which
bears on the analysis presented here. Describing functions have
been used before [9] to study the relation between varactor characteristics and tuning range. However, in that work the problem
is posed in such great generality that it is difficult to glean intuition or guidelines for circuit design. The analysis method,
though, bears some similarities to ours.
A recent publication [10], which appeared in print while this
manuscript was in review, also considers the relation between
varactor characteristics, VCO tuning, and conversion of AM
IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 38, NO. 6, JUNE 2003
1039
noise into FM. Whereas we present an analytical solution to
the tank balance equation given by (14), that work [10] settles
for a numerical solution. Our approach results in simple yet
accurate closed-form expressions that capture the nonlinear
behavior of a MOS varactor in an oscillating tank and enable
the prediction of tuning curves. We introduce the notion of
an effective tuning capacitance, which we show is more than
just the time-average small-signal varactor capacitance. Our
analysis predicts how the varactor converts AM noise into
FM. This is an important, although not necessarily dominant,
source of upconversion of low-frequency noise in a differential
oscillator.
APPENDIX
Suppose a nonlinear varactor with small-signal capacitance
is subject to sinusoidal voltage stimulus. Its – curve
will define a closed curve because of the periodic stimulus. The
area enclosed by the – curve is given by
(21)
If
, then
Area
(22)
The integrals define the zero and second-order coefficients of
the periodic function
, that is
(23)
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