Tuning Methods and Tuning Figures of Merit for LC Oscillators Johan van der Tang and Arthur van Roermund Eindhoven University of Technology (TU/e), Mixed-signal Microelectronics (MsM) Group, EH 5.10 P.O. Box 513, 5600 MB Eindhoven, The Netherlands phone: +31 40 247 2664, fax: +31 40 245 5674 email:
[email protected] Abstract— LC oscillators are used most widely in wireless applications as their resonator can store energy, making its phase noise to carrier ratio given a certain power budget superior to ring oscillators. The tuning range of LC oscillators, especially at high frequencies where parasitic capacitance starts becoming significant, is another important design issue. This paper investigates the influence of tuning methods on the oscillator phase noise. Five tuning methods are compared: varactor tuning, band switching, oscillator switching, active tuning, and phase shift tuning . In addition, for the tuning methods active tuning and phase shift tuning, effective quality factors are introduced as a figure of merit to assess the phase noise performance. Finally, a figure of merit is introduced that makes a comparison between tuning methods possible by normalizing the quality factor of tuning methods for a given tuning range. Keywords— LC oscillators, tuning methods, passive tuning, active tuning, phase noise, figures of merit
I. INTRODUCTION Wireless standards like GSM, UMTS, DECT and Bluetooth, have relative tuning ranges all below 4% [1]. However, process spread makes it even for these standards non trivial to meet the tuning range with a single varactor. For standards like FM radio and Satellite TV, with 21% and 77% nominal tuning range, use of one LC oscillator with only one varactor is out of the question, and techniques like band-switching or oscillator switching must be used. For any standard, the oscillator designer has to optimize both tuning range and phase noise to carrier ratio fm , which are conflicting oscillator properties. This paper, discusses five tuning methods and their influence on f m . Table I presents a qualitative comparison. The tuning range of integrated LC oscillators with only varactor-tuning (e.g a PN-type varactor or MOS varactor [2]) is always problematic (when taking process spread into account). On the other hand, its passive nature makes it a low noise method, and the tuning method is simple. Especially at high frequencies, parasitics make it difficult to achieve more than 20% tuning range. Band-
270
Tuning method
Tuning
Phase
Complexity
noise Varactor Band-switching Osc. switching Active tuning Phase shift tuning
+ + + +/-
+ + + -/+
+ +/+/-
TABLE I A
QUALITATIVE COMPARISON OF TUNING METHODS .
switching, division of the total tuning range into several bands for example by switching capacitors, yields a larger tuning range at the cost of an increased complexity [3]. Because the switched capacitors and the varactor (used for tuning within a frequency band) are passive, the phase noise performance of band-switched LC oscillators can be good. In the case of oscillator switching, the tuning range is also divided into frequency bands, but each band is covered by one oscillator [4]. Two disadvantages of this method are the large chip area (multiple inductors) and the need of a multiplexer-function that selects one of the oscillator outputs. An example of active tuning is the use of impedance converters [5], [6]. Large tuning ranges can be achieved, but the active devices that realize the variable capacitance (or inductance) for this tuning method, in general, cause unacceptable phase noise performance degradation. The last tuning method in Table I, phase shift tuning, uses a phase shifter to change the resonator phase shift and hence the oscillation frequency [7], [8]. As we will see below, this method degrades f m for large nonzero resonator phase shifts. In the following sections each of the tuning methods in Table I, is discussed more in depth. The influence of each tuning method on oscillator phase noise will be assessed by assessing the figure of merit quality factor. For active tuning methods and phase shift tuning, effective quality factors will be defined. In section VIII a novel quality factor will defined that normalizes the quality factor, given
a certain required tuning range. This figure of merit allows fair comparison of each tuning method. Finally in section IX, conclusions are discussed.
noise dominates in f m of an LC oscillator, L should f m [1], [15]. be maximized for optimum III. TUNING WITH VARACTORS
II. PHASE NOISE MODELING
In order to assess the relation between a tuning method and fm , a phase noise model is required. The 6dB per octave region of f m of any LC oscillator can be modeled by [1], [9], [10]:
LC
f m
1 1 2 4Q2
fosc fm
2
i2n 2 icarrier
(1)
in which i2carrier is the squared rms carrier current, i 2n the total noise current across the resonator, and Q the (loaded) resonator quality factor. The influence of each tuning f m will be investigated by assessing its efmethod on fect on the resonator Q. The oscillation frequency f osc , offset frequency f m and 2 icarrier are relatively easy to obtain for the oscillator designer. Two approaches are common to estimate the total equivalent noise current i2n . In case of LTI modeling, i2n is calculated in a linear fashion1 (referred to as i2n LT I ), without taking time variant effects of the nonlinear oscillator into account. Equation (1) can also be used in case of LTV modeling but i2n in (1) must then be calculated making use of the impulse sensitivity function (ISF) [11]. This is more accurate than linear modeling as it takes into account the effect of varying oscillator currents during one period and frequency conversion effects [11], [12], [10]:
i2nLTV
2 i2nLT I
1
2π
2π 0
Γ2eff
x dx
The most common way to tune a LC oscillator is with a varactor. Its quality factor Qvar 1 ω Cvar Rvar needs to be maximized for maximum f m , while having sufficient capacitance ratio to meet the tuning range specification. Especially when using PCB or SMD inductors, bond-wires, or a combination to implement L, the varactor Q will dominate the resonator Q. By adding fixed series capacitance (Cseries ) or fixed parallel capacitance (C par ) with a high Q (for example MIM-capacitors), the varactor Q can be improved:
Qvar new Qvar
Cseries Cseries Cvar C par 2Cseries Cvar 2 Cseries
(3)
However, the improvement in quality factor comes at the cost of a reduction of capacitance ratio: a clear example how phase noise performance and tuning range can be exchanged.
(a)
(d)
(c)
(b)
Fig. 1. Four varactor implementations.
2 i2nLT I Γ2rms
(2)
in which Γeff is the effective ISF, taking into account cyclo-stationary effects. For an oscillator with an ideal sinusoidal waveform and no cyclo-stationary effects Γ 2rms equals 1 2. Note that in this case i2nLTV is identical to i2nLT I . Both with LTI and LTV modeling it can be shown that the inductor (L) of a LC oscillator should be minimized, if the noise of the active oscillator part is dominant and the inductor technology is such that the Q improves for smaller L values [13]. This is beneficial for the tuning range as, in the case of varactor tuning, the varactor can be made larger and becomes more dominant compared to fixed parasitic capacitance. However, the oscillator should startup, and the active part should switch to maximize phase noise performance [1], [14]. In other words, the equivalent parallel resistance of the resonator, that decreases when L decreases, should stay sufficiently large. If the resonator 1 Calculating all noise sources to the output of the oscillator and adding them power wise.
271
Fig. 1 shows four possible varactor implementations; two constructed with bipolar transistors and two with MOS transistors. Table II shows the measured performance of two PN-junction varactors and two MOS accumulation mode varactors. MOS varactors have in prin-
No./Ref.
Type
Cmax Cmin
Qvar
1/[2]
PN-junction
1.83
69@1 GHz
2/[16]
PN-junction
1.7
26@1 GHz
3/[2]
MOS, Acc.
1.76
95@1 GHz
4/[17]
MOS, Acc.
3
23@1 GHz
TABLE II E XAMPLES OF
MEASURED VARACTOR PERFORMANCE .
ciple the advantage that the capacitance variation can be
achieved within a small control voltage swing: an important feature for low-voltage designs. However, this only true for a small voltage swing across the MOS varactor, because a large voltage swing has an averaging effect on the capacitance ratio Cmax Cmin and effectively reduces the oscillator tuning constant Kvco [18].
maximizing W . However, since the parasitic capacitances of transistor Qswitch are proportional to W , the desire to minimize CQ places an upper bound on W . A design procedure for a switched capacitor array, optimizing for quality factor and tuning range, while taking into account the discussed constraints, is described in [20].
IV. BAND SWITCHING
V. OSCILLATOR SWITCHING
Integrated varactors may not provide sufficient tuning range for an LC oscillator in an application. Band switching can then be used to meet the tuning range requirements [4], [3]. In case of a band-switched resonator, a total of Nb switched capacitors divide the tuning range into Nb 1 bands. If the switched capacitors are binary weighted, which is more efficient in terms of the number of needed control signals, the number of bands is equal to 2Nb 1. A varactor realizes continuous tuning within a band. Provided that the frequency bands overlap, all frequencies can be reached. Band switching reduces the requirements of the continuous varactor. Furthermore, it reduces the oscillator’s tuning constant Kvco , making it less sensitive to noise on its tuning input, which is beneficial fm [19]. Therefore, even if a varactor capacitance to ratio is sufficient to meet the tuning range requirements, f m , by reband switching can be needed to improve ducing the tuning slope.
Band switching using switched capacitors has already been discussed. Instead of switching resonator elements on and off, oscillators can be switched on and off [4], [21]. Each oscillator covers one of the overlapping frequency bands, and the varactor or varactors in each oscillator cover the frequencies within one band. The disadvantages of oscillator switching as tuning method are that much more chip area is needed and circuitry is needed to select and switch unused oscillators off, as well as circuitry that connects the selected oscillator to the blocks where its signal is used (a “multiplexer”). An advantage of oscillator switching is that the overall phase noise performance can be superior when using multiple oscillators. f m inEach oscillator can be optimized for maximum dependent of the other oscillators. Activating and deactivating an oscillator can be as simple as switching a tail current on and off, and this can be easier to realize than making high-quality switched capacitors at high frequencies.
V switch
C switch
C switch
C switch
Q switch
CQ
RQ
(b)
(a)
VI. ACTIVE CAPACITIVE TUNING on
(c)
Fig. 2. Simplified model for a switched capacitor.
In Fig. 2, simplified models for the off-state and onstate of a typical band switch circuit implementation are shown. In the off-state, the minimum parasitic capacitance of the switch is an important design parameter. The minimum value of the switched capacitor is equal 1 1 . Therefore C should be minimized to CQ 1 Cswitch Q for a minimum fixed capacitance introduced by the bandswitches, since this reduces the effective tuning range of the varactor in the resonator. In the on-state, the quality factor of a band switch is of primary importance, and in the first order it is equal to
Qb
1
ω Cswitch RQon
Circuit topologies can be used to emulate a varactor function. Large tuning ranges can be achieved at high frequencies using variable impedance converters [5], [6]. However, since active devices are per definition part of active capacitive tuning solutions, it is likely that the additional noise of the active devices will degrade the spectral purity of an oscillator severely, compared to passive tuning solutions. In order to be able to compare active varactors with passive varactor quality factors , it is useful to define an “effective quality factor” QC . This quality factor defactive inition also takes active device noise into account. If we model the active varactor as a parallel circuit of Cactive , a resistance R and a current noise source with current density i2ntot , the quality factor of an active varactor can be defined as:
1
ωC 1 switch µnCoxW L Vswitch VT H
(4)
Resistance RQon needs to be minimized for a high Qb , and this can be achieved by applying maximum Vswitch and
272
QC
active
Qvaractor
i2nR i2ntot
(5)
with Qvaractor ω Cactive R, and i2nR the noise current density of R: 4kT R. The formula for Qvaractor is identical
have i2ntot i2nR , and this can result in a significant decrease of QC . active
Fig. 3 shows the die photo of 240-310 MHz LC oscillator with external coils that was used to investigate the fm of active tuned oscillators [6]. Measured phase noise at the extremes of the tuning range was quite good: measured 10 kHz was better than -75 dBc/Hz at 240 MHz and 310 MHz. However, in the middle of the tuning range, QC calculated to less than unity for this oscilactive lator. The poor performance predicted by QC in the active middle of the tuning, was confirmed by measurements. 10 kHz was worst case more than 20 dB below 75 dBc/Hz. Especially for active tuned LC oscillators, fm should always be evaluated across the whole tuning range.
2
ω LC ω LC−res
to the quality factor definition of a passive varactor with effective parallel resistance R. Since for a passive varactor i2nR i2ntot , (5) reduces to Qvaractor . An active varactor will
Q p =2 1
Q p =5 Q p =10
0-180.0 −90
-120.0
−60
−30
-150.0
30
0.0
60
90
-30.0
φ
res
[
]
Fig. 4. The normalized frequency ωLC ωLC res of a quadrature LC oscillator versus phase shift φres , for Q p is 2, 5 and 10.
VII. PHASE SHIFT TUNING The frequency of an LC oscillator can be varied by varying resonator phase shift φres [25]:
ωLC
tan φres
ω
4 Q2p tan2 φres 2Q p
LC res
(6)
with ωLC res equal to LC 1 , and Q p the quality factor of the LC resonator for φres 0. For single-phase oscillators, phase shift tuning is not a logical choice because there is in general not a phase-shifter in the circuit present. In quadrature LC oscillators the two single-phase oscillators are coupled by circuits that can act as tunable phase shifters. Large tuning ranges can be obtained with phase shift tuning [7], [8]. In Fig. 4, (6) is plotted for three Q p values. The tuning range increases when Q p decreases, since the resonator phase characteristic versus frequency becomes less steep for lower Q p values. An effective quality factor can be defined for phase shift tuned LC oscillators [26]:
variable capacitance topology
active osc. part band gap
Fig. 3. LC oscillator with active capacitive tuning.
As (3) suggests, it is important to note that the quality factor of active varactors can be improved by placing highquality capacitors in series or in parallel. Therefore, when only a very small tuning range is required, active capacitive tuning may be a viable solution. A good example of a design achieving excellent phase noise performance with active varactors, uses an active varactor topology to tune a crystal oscillator for an FM Stereo Decoder [22]. Similar to the quality factor definition of an active capacitance, an effective quality factor can be defined for active inductances2 [1].
2 Example
-60.0
0
-90.0
Qφres
273
(7)
in which N is the number of stages of a multi-phase oscillator3 . Quality factor Qφres is maximal, if φres 0. Equation (7) shows that Qφres stays close to its maximum for small tuning ranges. Phase shift tuning can used to realize a large tuning range (by realizing a large Q φres variation), but at the price of a much reduced effective quality factor, fm . resulting in a degradation of
VIII. NORMALIZING QUALITY FACTOR Five different tuning methods have been described for LC oscillators. Each tuning method can be characterized by an (effective) quality factor and an (effective) 3 The
of active inductances can be found in [23], [24]
N Q p cos φres
formula is also applicable for N 1: a single-phase oscillator
αvar Cmax Cmin . In order to allow a fair comparison between tuning methods that have different combinations of Q and Cmax Cmin , we can normalize the quality factor for the required tuning range. The specified tuning range can be expressed as fmax fmin
Lp
Cmax Cmin
(8)
C series
C fixed
Vtune C tune
Fig. 5. Resonator with fixed parallel and series capacitance.
The normalization of the quality factor is based on Fig. 5. Fig. 5 shows a LC resonator consisting of inductance L p and a compound varactor. The compound varactor consists of a tuning element (in this case a passive varactor, but it also could be an active tuning circuit) with in series capacitance Cseries . In parallel with the series connection of Ctune and Cseries , a capacitor C f ixed is placed. We assume that the quality factor of Cseries and C f ixed is infinite. Normally, fixed capacitors can be realized with a superior quality factor compared to the tuning element. The high quality series and parallel capacitances in Fig. 5 allow improvement of the quality factor of the tuning element (Qvar in this case since we have used a passive varactor) at the expense of a reduced tuning range ( fmax fmin ). In order not to change basic oscillator functionality such as maximum frequency, C f ixed is chosen such that the total capacitance is equal to Ctune . In other words, the total capacitance is such as if C f ixed was zero and Cseries infinite. The total capacitance of the compound varactor in Fig. 5 is identical to Ctune if C f ixed is chosen: C f ixed
2 Ctune Cseries Ctune
(9)
Assuming the required f max fmin is known, the allowable Cseries can be calculated by solving fmax fmin
for Cseries . By calculating the quality factor of the compound varactor with the derived Cseries , for each varactor option, a fair comparison between tuning elements is obtained. This normalized quality factor (when C f ixed is given by (9)) calculates to
2 2 α Cseries αvar Ctune var CseriesCtune 1 αvar Cseries Ctune Cseries Ctune αvar (10)
274
Qnor
C
series
2 Ctune Qvar 2
Cseries
(11)
Fig. 6 (see next page) illustrates how the figure of merit Qnor can be used to compare different passive varactors. As an example Qnor of the varactors numbered 1 to 4 from Table II are plotted versus f max fmin . The plot directly reveals the maximum tuning range for each varactor, as this is the starting point of each curve. Notice that varactor no. 4 outperforms varactor no. 2. The Q normalization shows that the superiour tuning range of varactor no.4 can be used to improve its quality factor, such that it outperforms varactor no. 2. The figure of merit Qnor embodies an important tool for the oscillator designer, which helps him selecting the most optimal tuning solution. IX. CONCLUSIONS In this paper, five tuning methods for LC oscillators have been described and compared, In addition, their inf m is discussed. Passive tuning methfluence on ods such as varactor tuning, band switching and oscillator switching, or combinations, are the best methods to achieve maximum f m in a certain technology, given a certain power budget. Active tuning solutions and phase shift tuning can yield a reasonable f m , but only for small tuning ranges. Effective quality factors have been defined to quantify the effects on f m for these two methods. Finally, a novel figure of merit has been defined that normalizes the quality factor of a tuning method for a given tuning range. This figure of merit, Q nor , helps the oscillator designer with the selection of the most optimal tuning solution.
X. ACKNOWLEDGMENT The authors thank Dieter Kasperkovitz for fruitful and inspiring discussions. R EFERENCES [1]
[2]
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400
3.0
350
2.75 3
Qnor 300
2.5 Cmax Cmin 2.25
250 1
200
2.0
150
1.75
100
1.5 2
50
1.25 4
0 1.0
1.0
1.1
1.1
1.2
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1.4
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1.5
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fmax/fmin
Fig. 6. Normalized quality factor versus f max fmin for the four varactors from Table II. [3]
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