IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 54, NO. 4, APRIL 2007 293
Transformer-Based Dual-Mode Voltage-Controlled Oscillators
Andrea Bevilacqua, Member, IEEE, Federico P. Pavan, Christoph Sandner, Member, IEEE, Andrea Gerosa, Member, IEEE, and Andrea Neviani, Member, IEEE
Abstract—In this brief, we propose to use a transformer-based resonator to build a dual-mode oscillator, e.g., a system capable of oscillating at two different frequencies without recurring to switched inductors, switched capacitors, or varactors. The be- havior of the resonator configured as a one-port and a two-port network is studied analytically, and the dependence of the quality factor on the design parameters is thoroughly explored. These results, combined with the use of traditional frequency tuning tech- niques, are applied to the design of a wide-band voltage-controlled oscillator (VCO) that covers the frequency range 3.6–7.8 GHz. The performance of the designed VCO, implemented in a digital 0.13- m CMOS technology, has been studied by transistor-level and 2.5D electromagnetic simulation (Agilent Momentum). A typical phase noise performance at 1-MHz offset of dBc Hz has been predicted, while the power consumption ranges from 1 to 8 mW, depending on the VCO configuration.
Index Terms—Feedback, microwave oscillators, reconfigurable architectures, transformers, voltage-controlled oscillators (VCOs).
I. INTRODUCTION
FULLY integrated voltage-controlled oscillators (VCOs) capable of operating at several frequency bands while showing low phase noise are key building blocks in both emerging multiband, multistandard, and broad-band radios, as well as in wire-based communication systems [1]–[11]. LC VCOs have been preferred over ring and relaxation oscillators because of their better phase noise performance. In order to achieve a wide tuning range, several techniques can be used. Inversion- or accumulation-type MOS varactors support a large capacitance variation, at the price of higher tuning sensitivity, and, in the end, worse phase noise performance. Switched ca- pacitor banks suffer from the resistive and capacitive parasitics associated with the switches. A variable inductor is beneficial, since it results in both a smaller capacitance and a smaller capacitance variation requirements, improving tuning range, phase noise and power consumption simultaneously. Inductor switching has been proposed [8], [9], although this technique is severely limited by the switches.
In this brief, we propose a different technique to extend the tuning range of LC VCOs. An oscillator built around a trans- former-based resonator is shown to feature two modes of os- cillation. By selecting the proper feedback mechanism either of
Manuscript received April 4, 2006; revised November 15, 2006. This paper was recommended by Associate Editor P. P. Sotiriadis.
A. Bevilacqua, F. P. Pavan, A. Gerosa, and A. Neviani are with Diparti- mento di Ingegneria dell’Informazione, Università di Padova, 35131 Padova Italy (e-mail: andrea.bevilacqua@dei.unipd.it).
C. Sandner is with Infineon Technologies Austria AG, A-9500 Villach, Austria.
Digital Object Identifier 10.1109/TCSII.2006.889734
the two modes of oscillation can be selected. This results in two bands of operation. Compared to the inductor switching tech- nique, the proposed approach does not require any switch con- nected to the LC tank that would degrade the resonator . By combining the dual-mode operation with capacitance variation a very wide tuning range is achieved.
The brief is organized as follows. In Section II, the trans- formed-based resonator employed in this work is described. Sections III and IV discuss how a dual-mode oscillator can be built by using the resonator as a one-port or a two-port network, respectively. In Section V, the quality factor of the transformer-based resonator is analyzed. The design of a VCO which demonstrates the proposed technique is presented in Section VI, along with some simulation results. Finally, the work is wrapped up in Section VII.
II. TRANSFORMER-BASED RESONATOR
A transformer, made of two magnetically coupled coils with inductance and , respectively, and coupling coefficient is turned into a resonator by loading its primary and secondary ports with capacitors and , as shown in Fig. 1(a). The transformer is described by means of the impedence matrix
(1)
where the losses are simply modelled by means of the resis- tances and . The primary and secondary coil quality fac- tors are and , respectively.
The resonator can be described as a two-port network by means of the impedance matrix
(2)
The use of a transformer as a resonator allows the possi- bility of having two modes of oscillation, as explained in the following.
III. ONE-PORT OSCILLATOR
The simplest oscillator configuration is obtained by loading either port of the transformer-based resonator by a negative con- ductance , as sketched in Fig. 1(b) for the primary port. The start-up condition is
(3a) (3b)
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294 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 54, NO. 4, APRIL 2007
Fig. 2. Contour plot of the tuning range as a function of and the magnetic coupling . The shaded area is the region where , corresponding to an oscillation that builds up at .
(4)
Fig. 1. (a) Transformer-based resonator. (b) One-port oscillator. (c) Two-port oscillator.
Equation (3b) yields two possible modes of oscillation at fre- quencies and , which, in the high- case (i.e., ,
, 2), can be expressed as
where , , and . Now, we consider (3a) to determine which mode of oscilla-
tion builds up. If
(5)
is greater than one the system will oscillate in the higher fre- quency mode, while it will oscillate in the lower frequency mode if . To gain insight in the oscillator operation, note that
Fig. 3. Root locus for a one-port oscillator designed to operate at the lower frequency mode when primary is terminated on ( , , nH, nH,, fF, fF).
In any case, (3a) shows that for a sufficiently high value of both modes of oscillation can be excited simultaneously. The resulting multi-oscillation behavior is not desired and must be avoided [12]. The root locus analysis is employed to properly choose the value of . Fig. 3 shows the root locus plot of an oscillator designed to operate in the lower frequency mode when the primary of the transformer-based resonator is terminated on . Thus, , and, for increasing values of , the lower frequency conjugate poles pair first enters the right-hand plane. It crosses the imaginary axis for a value of transconductance equal to . For , (3a) is also satisfied for , as the higher frequency conjugate poles pair also enters the right-hand plane. In order to avoid the multi-oscil- lation behavior, the negative conductance must be chosen such that . In the example illustrated in Fig. 3, we have mS and mS. This shows that a
reliable oscillation without spurious modes can be guaranteed.
IV. TWO-PORT OSCILLATOR
The transformer-based resonator can be also treated as a two- port network to build an oscillator, as shown in Fig. 1(c). At each cycle of oscillation, the energy dissipated in the two-port is restored by a transconductor placed in a feedback loop. The start-up condition is expressed by
(7a) (7b)
both and
the magnetic coupling
are only function of the parameter and , since
(6)
in the high- approximation. In Fig. 2, the contour lines of are plotted as a function of and . The shaded area is the region where . For a given one can choose among several ( , ) pairs. If the selected pair is in the shaded area, the resulting system will oscillate at , otherwise
it will oscillate at .
The foregoing discussion is referred to the case is con-
nected to the primary, but it holds due to symmetry if
is connected to the secondary. The pair to be considered in the plot of Fig. 2 becomes . This means that one can design a dual-mode oscillator. In fact consider, for example,
. Suppose to select and . In Fig. 2 we see that this pair is in the shaded area. On the other hand, the pair is not in the shaded area. The primary/secondary terminations can be easily commutated by switching one on while the other is off, or vice versa, thus
changing the mode of oscillation.
BEVILACQUA et al.: TRANSFORMER-BASED DUAL-MODE VCOs
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Fig. 4. Root locus plots for a two-port oscillator with either positive feedback (solid line), or negative feedback (dashed line) ( , , nH, nH,, fF, fF).
Equation (7b) is satisfied at two frequencies, which, in the high- approximation, are the same as those given by (4). As a consequence, the modes of oscillation are the same for the one-port and the two-port oscillators. This result makes intuitive sense, as the two oscillation frequencies correspond to the two parallel resonances of the tank.
The value of transconductance required to make the system oscillating is derived from (7a) as
(8)
It is interesting to note that, since it can be shown that
, (8) always yields a negative value of transcon- ductance for and it always yields a positive value of for .1 This fact can be interpreted as follows. If the transconductor is connected to the resonator to make up a positive feedback loop, the system will oscillate in the lower frequency mode. Conversely, if the loop is a negative feedback loop, oscillations will build up at . The foregoing discussion is confirmed by the root locus analysis, shown in Fig. 4. The root locus in the case of positive feedback is plotted in solid line. For increasing absolute values of the lower frequency conjugate pole pair is pushed into the right-hand plane, while the higher frequency poles are drawn farther away from the imaginary axis. The minimum conductance (in absolute value) required to start up the oscillations is mS. In Fig. 4 the root locus for the negative feedback loop is also plotted in dashed line. In this case, it is the higher frequency conjugate poles pair the only one that is pushed in the right-hand plane. In the example
of Fig. 4 this occurs for mS.
V. QUALITY-FACTOR ANALYSIS
It is well known that the quality factor of the tank plays a key role in the frequency stability of LC oscillators. The quality factor of the transformer-based resonator can be written as
Fig. 5. Contour plot of in the one-port case. is evaluated for the lower frequency mode in the shaded area, where , and for the higher frequency mode elsewhere ( ).
Fig. 6. Contour plot of in the two-port case for the lower fre- quency mode (solid line) and for the higher frequency mode (dotted line) ( ).
where
intheone portcase inthetwo portcase
(10)
(9)
In the one-port case, it can be shown that the factor is uniquely determined by , , and for both modes of operation. For a given design (i.e., for a fixed ) is, to a first order approximation, a purely technology-related param- eter. Moreover, it can be verified that the higher , the higher the resonator , as one may intuitively expect. In Fig. 5 the resonator is plotted for both modes of operation in the one-port case. In this example, . The shaded region is where the oscillator operates in the higher frequency mode. In this mode of operation the factor increases with increasing
and it increases with decreasing . On the other hand, in the lower frequency mode is maximum for and it increases for increasing values of . In summary, the optimization of the resonator for one mode of operation leads to its degradation for the other mode. Moreover, Fig. 5 shows that in the lower fre- quency mode the best is obtained when that mode is the only attainable with the one-port configuration.
In the two-port oscillator, the quality factor only depends on , , and the technology-related parameters , and for both modes of operation. Note that it is reasonable to assume . The contour plot of in the two-port case is shown in Fig. 6 for both the lower frequency
1Without lack of generality, we assume .
296 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 54, NO. 4, APRIL 2007
Fig. 7. Simplified schematic of the designed dual-mode wide-band VCO.
mode (solid-line) and the higher frequency mode (dotted line). Similarities to the one-port case can be observed. If the two coils are loosely coupled, the resonator is enhanced for operation at
, while it is degradated for operation at . On the contrary, if one selects and tightly coupled primary and secondary in- ductors, the in the lower frequency mode is optimized, while the frequency stability is impaired in the higher frequency mode of operation.
As a final remark, it is worth to notice that a higher
is associated with a higher imbalance between the two modes of operation, as illustrated by Figs. 2, 5, and 6.
VI. VCO DESIGN
To put the foregoing discussion in perspective, we apply the theory of dual-mode oscillators to the design of a wide-band VCO. By exploiting the dual-mode behavior only two frequen- cies of oscillation are generated. As a consequence, capacitance variation is required to tune the oscillator over a frequency band for each mode of oscillation. As previously illustrated, the VCO operation is basically determined by parameters and . As a consequence, for a given transformer, i.e., for a given set of
, , and , the frequencies of both modes are scaled wihout changing , nor affecting the VCO behavior, if and are varied while keeping their ratio constant. The result is that, for a target overall tuning range , the re- quired capacitance variation is only , as opposed to as in the conventional tuning approach. This is due to the use of the dual-mode operation that expands the tuning range by a factor . Suppose to implement and as a capacitor bank plus a small varactor. In [5], a thorough analysis of the trade-off between the quality factor of the capacitor array ( ) and the frequency variation due to capacitance switching is carried out. The main point is that a wider capacitance variation results in a lower , which means a higher degradation of the resonator loaded quality factor. As a consequence, the reduced capaci- tance variation requirement of the dual-mode oscillator is both beneficial for the tuning range and for the noise performance of
the VCO.
A design procedure can be sketched as follows.
1) Set .
2) Choose based on the optimization of for both
modes of operation. For a balanced design is chosen close to one, while maximizing . In this case, the higher fre- quency mode is obtained by means of the two-port config- uration, while the lower frequency mode can be obtained with either configuration.
3) and are selected by keeping in mind that higher inductance means lower power consumption and phase noise, but also higher capacitive parasitics and reduced tuning range. Layout constraints also play an important role when choosing the transformer parameters.
4) Size and to satisfy the choice and to center the frequency range of interest.
5) Use a bank of switched capacitors and/or varactors to im- plement the capacitance variation, while keeping constant.
As an example of the proposed procedure, we designed a wide-band VCO, whose simplified schematic is shown in Fig. 7. The band of operation is from 3.6 to 7.8 GHz, corresponding to , while the target phase noise is dBc Hz at 1-MHz frequency offset from the carrier. We selected
and . The oscillator is operated in the lower fre- quency mode exploiting the one-port configuration because of a slightly better phase noise performance compared to the two- port scheme.2 The two-port topology is employed for the higher frequency mode. The two modes are commutated by switching the current generators and alternatively on. The in- ductances are selected to be nH and nH. The transformer is implemented by means of coupled octagonal symmetrical coils in a fully differential fashion. The center taps of the coils are connected to the supply voltage, equal to 1 V . The output voltage is taken at the secondary for higher voltage swing by a buffer not shown in Fig. 7 for clarity. The capacitance varia- tion is obtained by combining a 3-bit binary-weighted capacitor array and a MOS varactor, designed following the guidelines reported in [5]. The VCO is implemented in a digital 0.13- m CMOS technology. All the circuit components have been care- fully modelled. In particular, the transformer performance was assessed by means of a 2.5D electromagnetic simulator (Agilent Momentum).
In Fig. 8 the simulated VCO frequency range is shown. is the 3-bit control word of the capacitor bank.
2Simulations showed a difference in the phase noise performance despite the substancial equivalence in the factor reported in Figs. 5 and 6. A complete analysis of the phase noise contributors other than the resonator losses is be- yond the scope of the brief. Note, however, the similarities in both modes of operation between the presented VCO and an oscillator made of a standard LC tank and a cross-coupled pair, namely the sinusoidal resonator voltage and the hard-switching differential pair. Consequently, once the resonator is calcu- lated as in Section V, the phase noise analysis follows the guidelines reported, for example, in [2], [5], [13], or [14], the cyclostationary nature of the phase noise process being captured by Hajimiri’s impulse sensitivity function (ISF) that can be evaluated numerically [13] or analytically [14].
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Fig. 8.
Simulated VCO frequency range.
based resonator as a one-port network and as a two-port net- work. The dependence of the quality factor on the design pa- rameters in the two configurations was shown to be similar, with the resonator enhanced for operation in the higher fre- quency mode and degraded in the lower frequency mode when the coils are loosely coupled, the other way around when the coupling is strong. The application of the theory to the design of a wide-band VCO showed that, in practice, it can be conve- nient to rely on a hybrid approach, where the lower frequency mode exploits the one-port configuration, while the two-port topology is employed for the higher frequency mode. The sim- ulation results are promising and suggest that this approach can be a viable alternative to the switched inductor technique, al- though some additional design effort is likely to be required to equalize the VCO performance over the entire operation range.
is the varactor control voltage. Modes 1 and 2 are the lower frequency and higher frequency modes of operation, respec- tively. The oscillator is capable of continuos tuning from 3.6 to 7.8 GHz, although the capacitance variation alone would only cover a tuning range equal to .
The overall VCO performance is best described by means of the figure of merit (FOM) [13]
(11)
where is the oscillator power consumption, and
is the phase noise at a frequency offset from the carrier. The simulated VCO FOM shows quite some variation, ranging from dB to 4 dB over the entire frequency range. In particular, it is fairly degradated when the varactor capacitance is maximum ( V), suggesting a low varactor factor. Nonetheless, it is comparable with the state-of-the-art (e.g., dB [8], 6.5 dB [9], 7 dB [13], 11 dB [5], 15 dB [6]) confirming the potential of the proposed approach. The phase noise at 1-MHz frequency offset, simulated by means of the periodic noise analysis of SpectreRF, is ap- proximately constant ( MHz dBc Hz) over the tuning range for V, while it is maximally degraded ( MHz dBc Hz) for V at the maximum frequency of operation. The power consumption ranges between 1 and 8 mW over the entire range of operation conditions.
VII. CONCLUSION
The analysis proposed in this work showed that two modes of oscillation can be obtained both configuring the transformer-
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