Microsoft Word - Thesis Lai_vfOscillator design in III-V technologies
Szhau Lai
Chalmers University of Technology
Szhau Lai © Szhau Lai, 2014 ISBN 978-91-7597-065-3 Doktorsavhandlingar vid Chalmers tekniska högskola
Ny serie nr 3746 ISSN 0346-718X Technical report MC2-284 ISSN 1652-0769 Microwave Electronics Laboratory Department of Microtechnology and Nanoscience -MC2 Chalmers University of Technology SE-412 96 Göteborg, Sweden Phone: +46 (0) 31 772 1000 Printed by Chalmers Reproservice Gothenburg, Sweden 2014
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Abstract The thesis treats the design of low phase noise oscillators/VCOs in GaAs/InGaP HBT and GaN HEMT technologies. The covered topics are: active device modeling, noise characterization, passive structures, phase noise models, simulation/measurement tools, circuit topologies, and design techniques.
The mature InGaP HBT technology is known to be good for design of low phase noise oscillators, thanks to its low flicker noise and high breakdown voltage. The emerging GaN HEMT technology with its higher breakdown voltage also indicates the potential of low phase noise oscillator design. Large signal models for both devices were extracted based on Chalmers’ in-house models. Investigation of devices’ low frequency noise (LFN) characteristics showed that GaN HEMTs have noise levels close to InGaP HBT at low power, nevertheless, noise levels are high at high bias power. The bias dependency of LFN was utilized to develop low phase noise oscillators in GaN HEMT technology.
An accurate method for phase noise calculation is implemented. The method is integrated with a commercial simulator and the phase noise is calculated posteriorly based on LFN and Hajimiri’s impulse sensitivity function. The method is validated for three different InGaP HBT VCOs designed at 7 GHz using well-known topologies, i.e. balanced Colpitts, cross- coupled, and Gm-boosted balanced Colpitts. The method is also applied to a 9 GHz GaN HEMT balanced Colpitts oscillator and proven to predict phase noise accurately over a wide range of bias points.
The selection of oscillator topology is a key step in VCO design. A number of different topologies, suitable for various applications have been investigated and analyzed in this work. A 9 GHz double-tuned balanced Colpitts VCO demonstrates a 14.4% tuning range and phase noise better than -102dBc/Hz@100kHz over the tuning range. A wide tuning range of 34.4% is demonstrated in a 14.3 GHz mixer-based VCO. It is shown how a pair of modified Darlington HBTs is used in a gm boosted balanced Colpitts VCO to lower phase noise. In GaN HEMT oscillator designs, biasing strategies to avoid LFN are proposed. Two designed oscillators demonstrate the-state-of- the-art low phase noise in GaN HEMT technology. A MMIC balanced Colpitts oscillator at 10 GHz demonstrates a minimum phase noise of -136dBc/Hz at 1 MHz. A hybrid negative resistance oscillator with switch mode waveform designed at 2 GHz demonstrates a phase noise of -149dBc/Hz at 1 MHz.
Keywords: MMIC, InGaP HBT, GaN HEMT, device model, varactor, VCO, phase noise, low frequency noise, simulation, CAD, Colpitts oscillator, mixer, Darlington pair, switch mode.
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This thesis is based on the following papers.
[A] Dan Kuylenstierna, Szhau Lai, Mingquan Bao, Herbert Zirath, "Design of Low Phase noise Oscillators and Wideband VCOs in InGaP HBT Technology," IEEE Transactions on Microwave Theory and Techniques, vol. 60, no. 11, pp. 3420-3430, Nov. 2012.
[B] Szhau Lai, Dan Kuylenstierna, Mikael Hörberg, Niklas Rorsman, Iltcho Angelov, Kristoffer Andersson, Herbert Zirath, "Accurate Phase noise Prediction for a balanced Colpitts GaN HEMT MMIC Oscillator, " IEEE Transactions on Microwave Theory and Techniques, vol. 61, no. 11, pp. 3916-3926, Nov. 2013.
[C] Szhau Lai, Mingquan Bao, Dan Kuylenstierna, Herbert Zirath, "Integrated wideband and low phase noise signal source using two voltage-controlled oscillators and a mixer," IET Microwaves, Antennas & Propagation, vol. 7, no. 2, pp. 123-130, Jan. 2013.
[D] Szhau Lai, Mingquan Bao, Dan Kuylenstierna, Herbert Zirath, "A method to lower VCO phase noise by using HBT Darlington pair," in 2012 IEEE MTT-S International Microwave Symposium Digest, Montreal, Canada, pp. 1-3, 17-22 Jun. 2012.
[E] Szhau Lai, Dan Kuylenstierna, Rumen Kozhuharov, Bertil Hansson, Herbert Zirath, "An LC VCO for High Power Millimeter-Wave signal generation," in 2013 IEEE Compound Semiconductor Integrated Circuit Symposium (CSICS), Monterey, California, USA, 13-16 Oct. 2013.
[F] Szhau Lai, Dan Kuylenstierna, Mustafa Özen, Mikael Hörberg, Niklas Rorsman, Iltcho Angelov, Herbert Zirath, "Low phase noise GaN HEMT oscillators with excellent figure of merit" IEEE Microwave and Wireless Components Letters, vol. 24, no. 6, pp. 412-414, June 2014.
[G] Szhau Lai, Christian Fager, Dan Kuylenstierna, Iltcho Angelov, "LDMOS Modeling," IEEE Microwave Magazine, vol. 14, no. 1, pp. 108-116, Jan.-Feb. 2013.
[H] Szhau Lai, Mingquan Bao, Dan Kuylenstierna, Herbert Zirath, "A 20 GHz Low Phase noise Signal Source Using VCO and Mixer in InGaP/GaAs HBT," 2012 IEEE Compound Semiconductor Integrated Circuit Symposium (CSICS), La Jolla, California, USA, 14-17 Oct. 2012.
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[I] Szhau Lai, Dan Kuylenstierna, Iltcho Angelov, Klas Eriksson, Vessen Vassilev, Rumen Kozhuharov, Herbert Zirath, "A varactor model including avalanche noise source for VCOs phase noise simulation," in 41st European Microwave Conference Proceedings, 10-13 Oct. 2011, Manchester, UK, 2011, pp. 591-594.
Other publications
[a] Herbert Zirath, Szhau Lai, Dan Kuylenstierna; Jonathan Felbinger, Kristoffer Andersson, Niklas Rorsman, "An X-Band Low Phase noise AlGaN-GaN-HEMT MMIC Push-Push Oscillator," 2011 IEEE Compound Semiconductor Integrated Circuit Symposium (CSICS), La Jolla, California, USA, 16-19 Oct. 2011.
[b] Szhau Lai, Dan Kuylenstierna, Iltcho Angelov, Bertil Hansson, Rumen Kozhuharov, Herbert Zirath, "Gm-boosted balanced Colpitts compared to conventional balanced Colpitts and cross-coupled VCOs in InGaP HBT technology," in 2010 Asia-Pacific Microwave Conference (APMC) Proceedings, pp. 386-389, 7-10 Dec. 2010.
[c] Thanh Ngoc Thi Do, Mikael Hörberg, Szhau Lai, Dan Kuylenstierna, " Low Frequency Noise Measurements - A Technology Benchmark with Target on Oscillator Applications," to be presented in 2014 44st European Microwave Conference (EuMC), Oct. 2014.
[d] Vessen Vassilev, Herbert Zirath, Rumen Kozhuharov, Szhau Lai, "140–220-GHz DHBT Detectors," IEEE Transactions on Microwave Theory and Techniques, vol. 61, no. 6, pp. 2353-2360, Jun. 2013.
[e] Jian Zhang, Mingquan Bao, Dan Kuylenstierna, Szhau Lai, Herbert Zirath, "Broadband Gm-Boosted Differential HBT Doublers With Transformer Balun," IEEE Transactions on Microwave Theory and Techniques, vol. 59, no. 11, pp. 2953-2960, Nov. 2011.
[f] Jian Zhang, Mingquan Bao, Dan Kuylenstierna, Szhau Lai, Herbert Zirath, "Transformer-Based Broadband High-Linearity HBT -Boosted Transconductance Mixers," IEEE Transactions on Microwave Theory and Techniques, vol. 62, no. 1, pp. 92- 99, Jan. 2014.
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Notations and abbreviations
Notations α(t) Normalized periodic function Current gain of bipolar transistor in common emitter configuration C Capacitance Cmax Maximum capacitance of varactor Cmin Minimum capacitance of varactor Cout Output capacitance c0 Mean value of impulse sensitive function f Frequency fluctuation Phase fluctuation V Voltage fluctuation f Frequency offset f1/f Corner frequency between 1/f noise and white noise f Frequency fmax Maximum oscillator frequency fT Current-gain cut-off frequency f0 Oscillation frequency F Noise figure Phase shift gm Transconductance Imax Transistor maximum current capability Ib Base current Id Drain current i Current waveform in Shot noise current i1/f 1/f noise current Ic Collector current k Boltzmann’s constant k0 Free space propagation constant K Demodulator sensitivity l Length of transmission line L Inductance n Colpitts capacitance division ratio ni Ideality factor in diode model Nic Normalized noise waveform q Electron charge qmax Maximum charge stored in the tank
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QL Loaded quality factor R Resistance Rp Parallel resistance of parallel resonator Re Emitter resistance r HBT base input impedance Γ Impulse sensitive function ΓNMF Noise modulation function T Temperature τd Delay time of the delay line VBR Breakdown voltage Vd Drain bias Vc Collector bias VT The thermal voltage v Voltage waveform ω Angular frequency r Relative dielectric constant e Effective dielectric constant Pout Output power or Output port in the figure Ps Power delivered to the resonator Zc Characteristic impedance
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Abbreviation
AC Alternating current AM Amplitude modulation AlGaN Aluminum gallium nitride CAD Computer-aided design CMOS Complementary Metal Oxide Semiconductor DC Direct Current DRO Dielectric resonator oscillator DUT Device under test FET Field effect transistor FFT Fast Fourier transform FM Frequency modulation FOM Figure of merit GaN Gallium Nitride GaAs Gallium Arsenide G-R noise Generation Recombination noise HB Harmonic Balance HBT Heterojunction Bipolar Transistor HEMT High Electron Mobility Transistor IF Intermediate frequency InGaP Indium gallium phosphide InP Indium phosphide ISF Impulse sensitivity function LDMOS laterally diffused metal oxide semiconductor LFN Low frequency noise LO Local oscillator LTI Linear time invariant LTV Linear time variant mHEMT Metamorphic High Electron Mobility Transistor MMIC Monolithic Microwave Integrated Circuit MIC Microwave Integrated Circuit MOS Metal-Oxide-semiconductor NMF Noise modulation function PCB Printed circuit board pHEMT Pseudomorphic High Electron Mobility Transistor RF Radio Frequency RPN Residual phase noise SiN Silicon Nitride SOI Silicon on insulator TITO Tuned input tuned output VCO Voltage-controlled oscillator
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Contents
2.1 Transistor characteristics and models ..................................................... 5 
2.2 Low frequency noise properties ................................................................ 8 
2.2.1 Low frequency noise measurements .............................................. 9 
2.2.2 Residual phase noise measurements ........................................... 13 
2.3 Passive components ................................................................................. 16 
2.3.3 Varactor ........................................................................................... 20 
3.1 Phase noise model .................................................................................... 21 
3.1.1 Linear time invariant theory ......................................................... 21 
3.1.2 Linear Time Variant theory .......................................................... 22 
3.2 Numerical simulations used for oscillator design ................................ 26 
3.3 Phase noise measurement methodologies ............................................. 31 
4 VCO design 35 
4.1.1 Conventional VCO topologies ........................................................ 37 
4.1.2 Double-tuned balanced Colpitts VCO .......................................... 47 
4.1.3 A gm-boosted VCO using a pair of modified Darlington HBTs .. 49 
4.1.4 Wideband VCO design using two VCOs and a mixer................. 51 
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4.1.5 Comparison with the state of the art wideband MMIC VCOs .. 52 
4.2 Oscillator design in GaN HEMT technology ........................................ 53 
4.2.1 An MMIC 10 GHz balanced Colpitts oscillator ........................... 54 
4.2.2 A hybrid 2 GHz switch mode negative resistance oscillator ...... 56 
4.2.3 A 26 GHz push-push MMIC oscillator ......................................... 59 
4.2.4 Comparison with different GaN HEMT oscillator in literature 61 
4.3 High frequency signal source MMIC design ......................................... 62 
4.3.1 Second harmonic signal generation techniques verified in InP DHBT .............................................................................................. 63 
4.3.2 Mixer-based third harmonic signal generation in InGaP HBT . 64 
5 Conclusions 67 
Acknowledgements 73 
Bibliography 75 
Introduction
Microwaves are electromagnetic waves in the frequency range between 300 MHz and 300 GHz. The use of microwave technology has been continuously increasing ever since Marconi developed the first wireless telegraph system. Numerous applications developed over the last century such as the mobile communication, have brought extreme conveniences into our daily lives. In any microwave transmitter, there exist two necessary components: a signal source and an antenna. Take the Marconi’s telegraph transmitter for example; it includes a signal source made of Ruhmkorff coil and an antenna for transmitting the signal [1]. Ruhmkorff coil is less often heard of nowadays, however, the signal source is still essential in any microwave system. Thanks to the development of semiconductor technology, the use of transistor-based oscillators has gradually become the wide spread solution for signal generation.
When selecting a suitable oscillator, the specifications vary in different systems. Oscillation frequency, output power, tuning range, power consumption, and phase-noise are often the parameters considered by the system designers. Among all the specifications, phase-noise may be the most important parameter in determining the system performance. In wireless communication systems, the ideal case would be having a pure signal at a single frequency, i.e. an impulse function in the frequency domain. However, noise from both the environment and the transistor itself modulate the oscillator, resulting in frequency fluctuations. This type of fluctuations can be seen in the spectrum as a skirt shape around the carrier frequency and is usually called phase-noise due to the fact that it in time domain can be represented as a random variation of the phase1.
The requirements on a low phase-noise oscillator in modern communication systems vary based on two different concerns. Firstly, phase-
1 Note that also amplitude noise creates skirts around the carrier. However, amplitude is
of less concern in many applications as its effect is often eliminated due to compression acting as an amplitude limiting mechanism.
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noise determines the distinguishability on the constellation diagram. Secondly, the spectrum regulation has put demands on the signal level for different applications within a limited and condensed spectrum. System designers need to consider the detectable range of the desired signals without being interfered from adjacent signals.
An example of the first concern is point-to-point microwave links in the backhaul of mobile communication systems [2]. In point-to-point systems, interference with other sources is generally of less concern due to directivity of antennas. On the other hand, the requirements on data capacity can be very tough as one link may carry data from several base stations. To handle these high data rates, advanced modulation formats, e.g., high order QAM and sometimes multi-carrier transceiver architecture using MIMO [3] are utilized. Higher order QAM and MIMO both lead to tougher requirements on phase noise [4].
An example where interference between adjacent channels sets the phase noise requirements is in the access network, e.g., communication between the cell phone and the base station [5]. The phase-noise requirement is then determined by an effect called reciprocal mixing [6]. The effect describes the relation between the received signal power and the blocker signal power in the adjacent frequency. When a high power blocker signal and a low power received signal are mixed with the LO, they are both down converted to the IF for demodulation. The unwanted down-converted high power blocker signal may have a phase-noise skirt overlapping the wanted signal and causing the wanted signal to drown inside the phase noise. Some examples of access networks where reciprocal mixing is of concern are: GSM 900, DECT, DCS1800, and WCDMA [5].
In the design of oscillators for above mentioned applications, the question is how to achieve the requirements to a minimum cost, compact size, and low power consumption. Along with the rapid progress of modern communication systems, performance of MMIC/RFIC technologies has improved quickly and mass production of circuits fulfilling most of the demands is feasible. However, the phase-noise requirement is still difficult to reach. Semiconductor technologies and many circuit design techniques have been developed to address the challenge, but the progress is slow.
The challenge of designing low phase noise MMIC oscillators can be approached by choosing a good technology and proposing a good circuit topology. From technology perspective, three major factors affect the phase-noise of MMIC oscillators: low frequency noise (LFN) of active device, power, and Q factor of passive elements. All three factors are covered by Leeson’s equation [7] and discussed in Chapter 2. Without going into details, Fig. 1-1 illustrates how phase noise in different technologies depends on the three factors. FET devices generally have high LFN while bipolar devices usually have less LFN. III-V technologies can usually provide higher Q factor
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and power, compared to silicon technologies. Combining all three factor, the green arrow points in the direction towards better phase noise, see Fig. 1-1. The best choice for low phase noise is a technology with high power, negligible LFN in the active device and high Q passive. The best available option currently known is the GaAs-InGaP HBT. GaN HEMT has potentials for low phase noise thanks to the excellent power capability, provided that the effect of LFN can be limited.
The state of the art phase noise performance of MMIC/RFIC oscillators/VCOs based on SiGe HBT [8]-[10], InGaP HBT [11]-[13], [A, C], CMOS [14]-[24], and GaN HEMT [28]-[35],[F] devices are presented in Fig. 1-2. The results are collected from open literature and datasheets of commercial products [13]. At f=1 MHz, which is benchmarked in Fig. 1-2, the best performance is reached for InGaP HBT. Apart from InGaP HBT, Fig. 1-2, also shows some good GaN HEMT oscillators, among these the best one is a result from work in this Thesis [F].
Two objectives of this work have been to investigate which device technologies that are best suited for low phase noise oscillator design and to investigate different oscillator topologies and/or design techniques that can improve oscillator performance, primarily in the terms of phase noise and tuning range. A number of different topologies and design techniques have been studied in GaAs-InGaP HBT technology and GaN HEMT technology. A particular objective has been to investigate to what extent the high breakdown voltage of GaN HEMT can be transferred to good phase noise.
A third objective in this work has been to develop an accurate phase noise prediction method that can be used as a reliable design tool. This has been a particular challenge in GaN HEMT technology, where up-conversion of LFN
Fig. 1-1 Low frequency noise, Q factor, and Power in different MMIC technologies.
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has to be accurately modeled. To address this issue, Hajimiri’s impulse sensitivity theory [36] has been used to calculate phase noise. The input to the calculation has been measured LFN data and waveforms from large-signal simulation.
The thesis is outlined as follows. The background on transistor modelling, noise characteristics, and passive device properties related to VCO design are introduced in Chapter 2. The phase noise models, the functionality of simulators and how simulators use the developed models are discussed in sections 3.1 and 3.2. Then, section 3.3 presents the phase noise measurement methods used in this work. The oscillators and VCOs designed in this work are described in Chapter 4. Section 4.1 focuses on oscillator designs in InGaP HBT while GaN HEMT oscillators are presented in section 4.2. Section 4.3 highlights two techniques to design high frequency oscillators.
Fig. 1-2 Reported phase noise performance at 1 MHz offset for oscillators/VCOs using different active devices.
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Material and device technologies for oscillator design
In this work, most of the VCOs are designed based on MMIC technologies, specifically: InGaP HBT, GaN HEMT, and InP HBT. One hybrid GaN HEMT oscillator based on a bare-die transistor is also presented. The first section of this chapter introduces the transistor models used in this work and a set of active device parameters that are related to oscillator design. Section 2.2 presents the LFN characterization of InGaP HBT and GaN HEMT devices used in this work. In section 2.3, properties of passive elements in III-V technologies are discussed. Characterization methods are presented, and the achievable Q factors are covered.
2.1 Transistor characteristics and models
For oscillator design, the oscillation condition is usually related to the transconductance (gm). Hence, accurately modeling transistors’ gm is of high importance. The shape of gm typically resembles a bell for both bipolar and FET devices, see Fig. 2-2. Depending on the technologies and transistor sizes, transconductances present different peak values and asymmetric bell shapes. For the devices that have been characterized in this work, bipolar devices have higher peak gm than FET devices have. Table 2-1 compares an InGaP HBT and a GaN HEMT with sizes in the same order. The peak gm in the GaN HEMT is less than 2% of that in the InGaP HBT.
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(a) (b)
Fig. 2-1 Illustration of (a) knee voltage and breakdown voltage in Id vs. Vds, and (b) different operation regions in Id vs. Vgs.
Fig. 2-2 Measured (red) and simulated (blue) transconductance (gm) of an LDMOS transistor. Drain bias swept from 0 to 20V. The operation regions are indicated in Fig. 2-1(b).
Table 2-1 Characteristics of 0.25um GaN HEMT benchmarked versus 1um InGaP HBT Technology 1um InGaP HBT
4(fingers) x 20um (emitter length) Occupied area (40um x 40um)
0.25um GaN HEMT 2(fingers) x 30um (gate width) Occupied area (30um x 110um)
VBR 9V 30V fT / fmax 54 / 40>100 GHz @ VCE= 3.6V 29/68 GHz @VDS=15V Peak gm >1 S @ VCE=1.4V 0.015 S @ VDS=15V
For phase noise consideration, the available power provided by transistors is an important characteristic which is strongly dependent on knee voltage and breakdown voltage, illustrated in Fig. 2-1(a). The knee region is important for general RF circuit designs and it is normally well modelled.
On the other hand, breakdown effect is often not well modelled. From a
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designer’s perspective, as long as the operation is within the indicated safe operation region, modeling breakdown effect is not necessary. An important characteristic to be aware of is the self-heating effect in HBTs, which presents a positive thermal feedback constant. It means that when the device temperature rises due to a high bias power, the current increases. To avoid the run-away breakdown current from burning the transistor, designers need to consider the heat dissipation and keep a large margin below the breakdown voltage. On the contrary, HEMT devices have a negative thermal feedback constant, and less problems with the run-away breakdown current. Table 2-1 gives examples on the breakdown voltages of InGaP HBT and GaN HEMT in common emitter/source configurations. There exists over 20V difference between the two technologies, GaN HEMTs can support considerably higher voltage swing compared to InGaP HBTs.
The capacitance models are important, e.g. they influence fT / fmax and are often the cause of gain reduction in high frequency design. In addition, these capacitances are bias dependent. Through the dynamic voltage swing, these time-varying capacitances bring uncertainty in determining the average capacitance, resulting in inaccurate prediction of oscillation frequency. To model the bias dependent capacitances, suitable fitting equations and sometimes many fitting parameters are required.
By combining all the parasitics, both bias dependent and independent, a large signal model is formed. An accurate large signal model is essential to simulate the transistor load line and waveform accurately, and it is the bases to perform an accurate phase noise prediction.
In this work, the InGaP HBT and the GaN HEMT models are extracted, all models are based on in-house developed equations. The extracted InGaP HBT model is based on the empirical HBT model developed by Iltcho Angelov and Mitsubishi in 2002 [1]. In this model, the DC model equations are similar to Chalmers HEMT model [38][39], which defines the equation at a referred bias point where the device is typically used. The relevant currents, voltages and derivatives have the best accuracy at the reference point. In the capacitance model, the physical expression of the depletion capacitance is modified to prevent the pole when biasing voltage equal to the built-in potential. This modification helps simulations to converge better when a voltage swing is large.
About the GaN HEMT model, two different model topologies, i.e. Angelov’s GaN HEMT model and Fager’s model are used. Angelov’s GaN HEMT model is originated from Chalmers HEMT model [38][39]. In Angelov’s GaN HEMT model, GaN HEMT effects, such as knee walk out and the breakdown effect are added to the basic HEMT model. However, more components could cause longer simulation time or convergence problems. To be efficient in simulation, designer should know the application and use the least complex model that includes the wanted effects.
In Fager’s model, a set of standard equations to model DC characteristic is
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described in [40]. The bias dependent capacitances require different fitting equations, depending on their shapes [41][42]. The model was designed to model LDMOS originally [40][paper G], but it can also be used as a general HEMT model.
Compared to Angelov’s GaN HEMT model, the equations in Fager’s model can flexibly fit the gm peak, even if the gm peak is rather flat or asymmetric, shown in Fig. 2-2. In Angelov HEMT model, the gm peak presents a bell shape and is symmetric around the peak. It requires adding higher order power series (order up to three is used in the standard equations) or arranging a second gm peak to model the asymmetry. In most HEMT devices, gm usually has a symmetric shape around the peak. Then, the standard equation in Angelov model can fit most of them.
2.2 Low frequency noise properties
The noise characteristic of a transistor is important for the prediction of oscillator phase noise. In this work, oscillator’s near carrier phase noise at 100 kHz and 1 MHz attracts more attention. Within this offset frequency range, the noise behaviors of most HEMTs are dominated by 1/f noise and generation-recombination (G-R) noise while some low noise HBTs are dominated by shot noise.
LFN can be measured under a static bias condition or an RF-pumped condition. In most of this work, LFN is measured in the static bias condition, presented in section 2.2.1. The measurement under the RF-pumped condition is not studied in this work. It is introduced recently to characterize LFN under the large signal excitation [48]. Particularly, the method of LFN modelling is believed to be closer to the oscillator’s large signal condition and has shown a good agreement between the VCO phase noise measurement and the simulation result [48].
Residual phase noise (RPN) measurement is another method to characterize 1/f noise even closer to the real operation of an oscillator. The method injects a carrier signal into the DUT, the same procedure as in the RF-pumped LFN measurement. However, the method measures the additive phase noise at the carrier signal sideband instead of measure noise directly at baseband. The measurement accounts the up-conversion effect when LFN is converted into phase noise.
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2.2.1 Low frequency noise measurements
LFN measurement is an essential step in device characterization for VCO design. Depending on different technologies, the corner frequency between 1/f noise and white noise can vary from a few kHz to a few MHz. In general, HBTs have lower corner frequency compared to FETs. Hence, HBTs are often preferred for low phase noise VCO design. For example, InGaP/GaAs HBT has corner frequency as low as 20 kHz. A detailed comparison of the noise level at 100 kHz between InGaP HBT, GaN HEMT and GaAs pHEMT is presented in [43]. A short summary is listed in Table 2-2. Among the three technologies, noise in GaAs pHEMT is higher than that in the two other technologies. When LFN is normalized to DC power, it is found that for low current bias, GaN HEMTs at 100 kHz have levels comparable to InGaP HBTs.
To exemplify how the LFN varies with bias conditions, Fig. 2-3 shows the noise current density measured at 100 kHz for a GaN HEMT device. In the low voltage/ high current and high voltage/ low current regions, the noise is relatively lower than their vicinity, low-noise regions are marked by elliptic shapes in Fig. 2-3. At a detailed look, the noise performance in the low voltage/high current region is lower than that in the high voltage/ low current region, shown in Fig. 2-4. The bias dependency of the LFN is taken into account in the oscillator design, presented in section 4.2.
Table 2-2 Comparison of LFN in different technologies
Device Size (μm)
(A2/Hz)
(A/Hz.V)
InGaP HBT 1  4x20  1000  1.2  9  3  5.30×1019  1.96×1017  1.10×1019  4.07×1018
GaN HEMT 6  4x50   250  1.5  4.7  10  1.71×1017  3.62×1016  1.43×1019  3.03×1018
GaN HEMT 4  4x50  250  1.3  39  10  6.66×1017  1.71×1016  1.75×1018  4.50×1018 
GaAs pHEMT 6  4x40  100  1  25  3  3.88×1017  5.19×1016  4.71×1018  6.30×1017 
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Fig. 2-3 GaN HEMT LFN at 100 kHz measured versus drain voltage and current. The measured transistor is from Triquint’s 0.25 m GaN HEMT process with a size of 4x100um. Similar noise distribution but different noise levels are seen in transistors of different sizes and from different processes.
Fig. 2-4 Measurements of GaN HEMT LFN at 100kHz versus drain current (Vd=2V) and drain voltage (Id=12mA). The test GaN HEMT has a size of 4x50um.
Measurement setup In this work, a LFN measurement system introduced by Agilent [44] is
used. The method uses a current-to-voltage transimpedance preamplifier with an internal voltage supply from Stanford Research (SR570) for noise amplification. The detectable range of the dynamic signal analyzer (Agilent 35670A) can go much lower than its noise floor depending on the amplification of the preamplifier. In the measurement setup, the detectable noise current is as low as 10-20 (A2/Hz). The biasing setup uses a parameter analyzer to control
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gate voltage or base current. A low pass filter (1 Hz) is connected to the base of the device under test to avoid noise leakage from the low impedance base. Further, the noise from the bias supply is blocked to ensure a noise free environment for measurement. The measurement setup is shown in Fig. 2-5. However, the setup using internal bias from SR570 has a supply limitation (4V/5mA). The limitation can be overcome by building an external battery power supply or using the parameter analyzer with a low frequency bias T [B]. In this work, the latter method is adopted. The low frequency bias T is composed of a high inductance (150mH) and a large capacitance (22mF). To find a capacitor with such a high value, electrolyte capacitor is the only option commercially available. The electrolyte capacitor has a high leakage current, which disturbs the measurement. In addition, charging and de-charging a big capacitance is a time consuming procedure, which is unfavorable for the already time-consuming LFN measurement.
Another type of setup using a voltage-to-voltage preamplifier (SR560) can overcome the power limitation and avoid the problem caused by the huge bias T. The setup requires an additional bias resistance to connect the output to avoid noise being grounded. The current noise is then transformed to voltage noise by the bias resistance in parallel with output resistance Rds of a transistor. The input resistance of SR560 is around 1 Mohm, which does not affect the measurement result. A drawback in the setup is that the real bias voltage on the transistor should be pre-calculated by deducting the voltage drop on the bias resistance.
Fig. 2-6 shows LFN measurement and extracted model for a 4x20um2 InGaP HBT measured using the transimpedance amplifier setup. In the system, the measurable spectrum is up to 100 kHz, which is limited by the dynamic signal analyser. This frequency range is sufficient to extract 1/f noise parameters, however, not enough to see the corner frequency and white noise level in most FET devices.
Fig. 2-5 The LFN measurement setup using a transimpedance amplifier.
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Fig. 2-6 The LFN measurements (blue curve) of an InGaP HBT compared to simulations using the extracted model (red curve).
In the development of semiconductor technologies, the physical based Hooge’s model [45] is a common mean to model LFN and monitor the process related parameters. However, with the increased complexity of modern transistors, the Hooge model cannot fit measured LFN data without adding additional fitting parameters. Instead, LFN models used today are more empirical than physical, although, the basic characteristic of 1/f slope and the proportionality to current are still maintained. The adopted model in this work has a basic format similar to Hooge’s model [46],
,2 Ffe
Af c
I Kfi ( 2-1 )
where Ic is the collector current, Af is the current dependency factor, and Kf is for the level fitting. Ffe is the frequency dependent power factor, which is close to 1 and from where LFN gets the name of 1/f noise.
On top of 1/f noise, G-R noise appears in some devices, especially in HEMTs. The G-R noise is one part of LFN and it presents a Lorentzian shape. In history, McWhorter developed a LFN model [47] with a 1/f shape based on multiple Lorentzian curves. In this work, the detailed G-R noise is not modelled. In the procedure of LFN extraction, the factor Ffe could differ from 1 when additional G-R noise presents.
Beside 1/f and G-R noise, white noise also has influence on phase noise, especially on the -20dB/decade region. The white noise is dominated by the shot noise ( 2-2 ) in bipolar device, while in FET device, white noise is determined by thermal noise ( 2-3 ).
10 0
10 1
10 2
10 3
10 4
10 5
10 6
10 -21
10 -19
10 -17
10 -15
10 -13
Frequency (Hz)
N o
is e
c u
rr e
n t (
Shot noise is expressed as
fqIi c 22 (A2), ( 2-2 )
where q is the electron charge 1.6×10-19 Coulomb, Ic is the bias collector current, and f is the noise measurement bandwidth.
Thermal noise is expressed as
RfkTi /42 (A2), ( 2-3 )
where k is the Boltzmann constant, T is the temperature and R is the equivalent noise resistance between drain and source. In this work, the white noise floor is not reached due to finite bandwidth of the signal source analyser.
2.2.2 Residual phase noise measurements
A residual phase noise (RPN) measurement system with state of the art noise floor [49] is used, see Fig. 2-7. The injection signal, generated by a 3.5 GHz low noise dielectric resonator oscillator (DRO), is split into two paths. One path goes through the device under test before being mixed with a direct path. Therefore, in the two paths, the phase noise of signals from the DRO is correlated and canceled, the only thing left at the output is the phase noise induced by the device under test. There is a cross correlation arrangement in the setup using two splitters and two mixers, which are intended to cancel noise from the mixer. At the mixer output, the phase noise at carrier frequency is down converted to the baseband frequency, and the noise can be amplified by the operational amplifier and detected by the FFT analyzer.
8kΩ DC
14
It is important to mention that the measured noise is not shown in absolute value as in LFN measurement. The result is expressed as the noise to carrier ratio dBrad/Hz, which considers the LFN up-conversion effect. The unit dBrad/Hz is the phase fluctuations in double sideband. Compared to the unit of single sideband noise dBc/Hz, dBrad/Hz is 3 dB higher.
A special arrangement is made in the base-biasing network. The base is biased by a voltage source with an 8k resistance, which can be seen as a current source with a high impedance loading. A bypath 20uF capacitor is used as a low impedance load to direct LFN to ground.
In the measurement, two different HBT operation conditions are considered on the same device under DC bias, Ib=122uA and Vc=2V. Firstly, the linear mode operation is measured using an injection signal of 0dBm, which is shown as black load line in Fig. 2-8. While the second condition is measured using an injection signal of 5.5 dBm to reach the saturation mode operation, shown as blue load line in Fig. 2-8. In the measurement, the results of RPN measurements from linear mode operation and saturation mode operation are compared. Further investigation of load line running into the deep saturation region is not included in this work.
In each measurement, the data is collected under two different base terminations, i.e. the high impedance 8k from the bias resistor and the low impedance of the 20uF bypass capacitor. When the high impedance is used, the noise mainly goes out from output collector port, where a higher RPN can be seen compared to the low impedance cases in Fig. 2-9 and Fig. 2-10. When the bypass capacitor is used, part of the device noise is grounded, therefore lowering the RPN. The largest available capacitor without leakage current is only 20uF, which limited the bandwidth. Therefore, the noise at low frequency cannot be perfectly shorted. The effect causes the RPN slope deviating from the 1/f noise slope (-10dB/decade), shown in Fig. 2-9. On the other hand, the saturation mode operation, shown in Fig. 2-10, shows that the RPN is so low that the base termination has little influence on the result.
To summarize in brief, the measurements of InGaP HBT devices reveals that in saturation operation, the induced phase noise is lower than in linear mode operation. The result is in contradiction with the intuitive feeling that the nonlinearity of saturation mode operation causes noise generation and degrades phase noise. A proposed explanation is that there is an optimum point in the saturation mode, where the capacitance has the minimum sensitivity to noise and can suppress AM-FM noise conversion to the maximum, similar phenomenon is found for MOSFETs [50].
15
Fig. 2-8 Two Dynamic load lines superposed on the transistor IV curves indicate a linear mode (black line) and a saturation mode (blue line) operation.
Fig. 2-9 The RPN measured under the linear mode operation.
Fig. 2-10 The RPN measured under the saturation mode operation.
0.5 1.0 1.5 2.0 2.5 3.00.0 3.5
0.00 0.01 0.02 0.03 0.04 0.05
-0.01
0.06
Vc
Ic
2.3 Passive components
The resonator is the passive part in an oscillator. In MMIC design, it can be designed using lumped LC components or distributed lines. To enable tuning of the oscillation frequency, the resonator in VCOs generally incorporates a varactor, i.e. a variable capacitance. The Q factor of the resonator plays an important role for the phase noise performance. This section addresses how to characterize and optimize Q factors of passive elements.
2.3.1 Characterization methods for extraction of quality factor
Q factor of passive components can be extracted by two different methods: reflection-type and resonant-type measurements. In reflection-type characterization one performs a one-port S-parameter simulation or measurement. By simply transforming the one port S-parameter to Y- or Z-parameters, the Q factor of the device under test (DUT), i.e., a capacitor or an inductor, can be extracted by taking the ratio between the imaginary and the real parts of Y or Z. The reflection type method can be used below the self-resonant frequency of the component, but the accuracy of the measured Q factor depends strongly on the calibration accuracy [51]. A oneport test structure for reflection-type measurements of an InGaP HBT varactor is shown in Fig. 2.11(a)
Resonant-type measurements are much more robust to calibration inaccuracies [51]. The principle is that the device under test (DUT) is made series resonant with another reactive element, e.g., an inductor if the DUT is a capacitor and vice versa if the device under test is an inductor. Then the Q factor can be calculated from the 3dB bandwidth of the resonator. Under the assumption that the Q factor of the element setting the DUT in resonance is much higher than the Q of the DUT, this method would give the DUT’s Q directly. If the Q factors of the two reactive elements are in the same order of magnitude, the DUT’s Q factor can be calculated from
rLc QQQ
111 ( 2-4 )
where Qc is capacitance Q factor, QL is inductance Q factor and Qr is the total Q for the resonator, i.e., what is calculated from the 3dB bandwidth. A particular type of resonant measurement method is the Deloach method where a series resonator center-taps a transmission line [52]. A Deloach test
17
structure for an InGaP HBT varactor is shown in Fig. 2-11(b). Given QL is known, e.g., EM-simulation; Qc factor can be calculated from ( 2-4 ). An example of the extracted result is shown in Fig. 2-12(b).
(a) (b)
Fig. 2-11 (a) One port structure and (b) two port Deloach structure of a varactor with a size of 60um(finger length) x 12um(finger width) x 8(#fingers)=5760 um2.
(a) (b)
Fig. 2-12 Measurement and extracted model of the varactor with a size of 60um(finger length) x 12um(finger width) x 8(#fingers)=5760 um2, shown in Fig. 2-11. (a) The transmission of resonance measurement (blue) and simulation (red). (b) The measurement (mark) and the extracted model (line) of Q factor and CV curve.
4 8 12 160 20
-30
-20
-10
0
-40
10
20
40
60
80
100
Q
Qsim
Qmeas
2
4
6
8
10
2.3.2 Quality factors of lumped and distributed elements
In the design of III-V MMIC passive components, the dominating loss mechanism is conductor loss. The conductor loss plays an important role in the series resistance of inductors and capacitors. In MMIC technology, MIM capacitors are the most commonly used capacitors, which consists of a dielectric layer, e.g. a SiN, sandwiched between two metal layers. The dielectric loss is usually negligible compared to loss in the metal. Therefore, the capacitor model is usually described as a resistance in series with a capacitance. This description indicates that the Q factor decreases as the frequency is increased,
RC QC
1 . ( 2-5 )
The capacitor’s Q factor is strongly dependent on its geometry, and the capacitance value is proportional to its occupied area. Fig. 2-13 demonstrates an example of the Q factor versus frequency for a 2.5pF MIM capacitor simulated using a momentum method. The MIM capacitor is formed based on a layer profile from a III-V process, shown in Fig. 2-14. The MIM capacitor is formed by a 0.27um height SiN layer, sandwiched between a 7 um height top metal and a 2um height bottom metal. It is shown that the Q factor can be larger than 50 at 12 GHz.
Fig. 2-13 The extracted Q factors of a 2.5pF capacitor and a 0.3nH inductor on a III-V substrate, see Fig. 2-14. The capacitor and the inductor are simulated by a momentum method. The capacitor has a size of 100um (width) x 100um (length), and the inductor has a size of 50um (width) x 500um (length).
2 4 6 8 10 12 14 16 180 20
30
60
90
120
0
150
19
Fig. 2-14 An example III-V process uses a 100um SiC (r=10.2) substrate. A MIM capacitor can be formed by a 0.27um height SiN (r=7.77) layer, sandwiched between a 7 um height top metal and a 2 um height bottom metal.
MMIC inductors are usually realized by a distributed line. In that sense, the transmission line model can be used to model inductors. At frequency far below the self-resonant frequency, the inductor model can be simplified as an inductance in series with a resistance, and the Q factor becomes proportional to the operation frequency, i.e.,
R
. ( 2-6 )
A high Q inductor is often realized with a wide line to reduce the conductor loss. Further, a high inductance may be realized by a coil shape line to enhance the inductance due to positive mutual inductance. When combining these two purposes, the inductor could occupy an area much larger than any other part of the circuit and further the wide strips will reduce the self-resonant frequency of the inductor. These large inductors can be easily seen from the chip photos of many VCOs in open literature.
An example inductor using the substrate profile from Fig. 2-14 is simulated by a momentum method. The inductor Q factor versus frequency is also presented in Fig. 2-13. The example inductor with a reasonable width of 50 um and a length of 500 um is formed by a 2um metal on top of a 100um SiC substrate. It can be seen that the Q factor of the inductor is lower than that in the presented MIM capacitor when the operation frequency is below the line where Qc=QL, see Fig. 2-13. According to ( 2-4 ), the Q factor of an LC resonator is dominated by the element with the lower Q factor, i.e. an inductor decides the Q factor of a resonator at low frequency. In this work, LC resonators with a Q factor above 20 are adopted in many designs.
20
2.3.3 Varactor
The Q factor and the tuning range of varactors are among the most important parameters in VCO design. A varactor’s tuning range can be expressed as the ratio between the maximum and minimum capacitance, i.e., Cmax/Cmin. In MMIC designs, varactors are generally implemented using diode structures available from the transistor process.
In HBT processes, the base-collector junction is most commonly used. The parameters in the base-collector junction are less sensitive to the characteristic of the bipolar device and its doping profile can be customized for varactor functionality without seriously degrading the transistor’s performance. In addition, the lightly doped collector has a longer diffusion distance, in which the junction has a larger reverse breakdown voltage to support a high voltage swing and the capacitance can be tuned with a lower voltage sensitivity. A standard abrupt junction and a customized hyper- abrupt junction are used in this work. An example of an abrupt varactor is shown in Fig. 2-12, achieving a capacitance ratio of 2.5 and a Q factor of 40.
In HEMT processes, hyper-abrupt varactors cannot be realized without adding additional process steps, the HEMT epitaxy has to be optimized for the transistor functionality. Varactors are generally realized by connecting the drain and source of a HEMT. Then the combination of capacitances Cgs and Cgd will form a varactor. Through the optimization of the varactor geometry, e.g. number of fingers and finger width, this type of varactor can reach a medium Q factor. Q factors above 20 and capacitance ratios in the range 2 to 6 have been reported for varactors in III-V HEMT technologies [53]-[55]. Other types of varactors that require fabrication processes in addition to HEMT process are not included in the discussion.
A less commonly studied characteristic of varactors related to VCO design is the noise property. Thermal noise that originates from the resistance is related to the Q factor and is included in all varactor models. Avalanche breakdown noise is often not included. Operation in this region is avoided by keeping the voltage swing below the breakdown voltage. A research activity included in this work has investigated the influence of varactor avalanche noise on VCO phase noise degradation [I]. The report is the first in open literature to relate the VCO phase noise simulation with an avalanche noise model.
21
Oscillator simulation, phase noise modeling and measurement
This chapter deals with oscillator phase noise. Phase noise models are introduced first to show the parameters that influence phase noise performance. These models are often used as a brief guide to optimize the phase noise in oscillator design. A simple model based on a linear time invariant (LTI) theory and a more complicated model based on a linear time variant (LTV) theory are introduced in section 3.1. Section 3.2 presents oscillator simulation methods commonly used in CAD tools and explains these methods’ insufficiencies in phase noise prediction. A proposed auxiliary method that integrates the harmonic balance simulation and the LTV phase noise model to predict phase noise accurately [B] is also covered in Section 3.2. In section 3.3, some commonly used phase noise measurement methods are presented.
3.1 Phase noise model
3.1.1 Linear time invariant theory
An early phase noise model was proposed by Leeson in 1966 [7], this model and its extension are often referred as linear time invariant (LTI) model. Originally, the proposed equation was intuitively derived. The concept of phase noise is related to the resonator bandwidth (ω/2Q) and the noise sources that consist of white noise and transistor 1/f noise. When the two terms are multiplied together, the single sideband phase noise is similar to the following form
22
S
( 3-1 )
where PS is the RF power dissipated in the resonator, Q is the loaded quality factor, f0 is the oscillation frequency, f is the offset frequency, k is the Boltzmann constant, T is the operating temperature, and F is the effective noise figure of the feedback oscillator[7].
It is worth to mention that, in the literature there are discrepancies in the place of the factor 2 below F in (3-1). In (3-1) the factor 2 is placed in the denominator, in accordance with in Everard’s [56] and Rohde’s [6] works, while in Leeson’s original paper it is placed in the numerator. The reason for the factor of four difference can be explained by the fact that Leeson refers to double sideband phase noise and does not separate the amplitude noise and phase noise. A confirmation that (3-1) is correct was presented in [57] where a noise floor -177dBc/Hz was experimentally demonstrated in a residual phase noise measurement.
It is also at its place to discuss the meaning of “effective noise figure” F. In brief it represents the shortcomings of LTI theories and is the reason why these theories never can predict phase noise a priori but only fit a measured spectra a posteriori. Under nonlinear operation as is the case in any practical oscillator, noise will be mixed from harmonic frequencies, which makes the noise figure to appear higher than what is the case for a transistor under linear operation. Linear time varying (LTV) theories can predict how the noise figure is increased due to nonlinear mixing [69].
3.1.2 Linear Time Variant theory
In LTI theory, the phase noise is calculated directly from the frequency domain, it does not consider the effect of the noise injection at different time instants. In LTV theory, the phase noise is derived from the time domain, in which the noise injections at different time instants in the oscillator orbit result in different phase shifts. Then, the phase noise is the accumulation of those phase shifts.
Many phase noise models were derived based on time variant theory. Some are used in phase noise simulation in CAD tools. Among them, Hajimiri’s theory [58] is the most intuitive and the model parameters are directly related to the circuit design parameters. In addition, the phase noise can be organized into a compact expression.
Hajimiri introduces the impulse sensitive function (ISF), which is a function to describe the phase shift caused by the noise injection at different time instants. To use the theory for phase noise prediction, calculating the ISF is a necessary step. The most accurate method to calculate the ISF is using
23
transient simulation. However, a drawback of transient simulation is that the convergence property degrades drastically with complexity of the transistor models. The following paragraph introduces an approximate method for the ISF calculation described in [36] and the procedure of phase noise calculation that is used in oscillator design.
ISF calculation In a well-defined LC resonator, where the inductance (L) and the
capacitance (C) can be identified between two nodes. The cross tank voltage waveform v(t) and tank current waveform i(t) are defined as
),()( 0 tfVtv ( 3-2 )
),()( 0 tg L
( 3-3 )
where V0 is the amplitude, f(t) and g(t) are the unity function of voltage and current waveforms, respectively. The relation between voltage and current is
scaled by the tank impedance CLZc .
When a charge impulse Δq is injected into the tank at t=t0, after n periods of self-restoring mechanism, the voltage fluctuation Δq/C will result in a phase shift 2n+ and amplitude change ΔV. The phase shift comes into phase noise, and ΔV results in amplitude noise. Usually, amplitude noise is much smaller than phase noise after a few cycles of self-restoring mechanism and can be ignored. The final waveform can be expressed as
),'()()'( 0 tfVVtv ( 3-4 )
),'()()'( 0 tg L
( 3-5 )
where t’ is the time after n periods, here the time of restoring cycles nT is ignored due to its periodicity v(nT+t)=v(t). To calculate the phase shift due to a charge impulse, the final voltage and current waveforms are made equal to the initial condition
,)()'()( 000 C
q tfVtfVV
( 3-7 )
Since the voltage fluctuation is small, a Taylor’s expansion can be used to approximate f(t’) and g(t’). In the calculation, only the first order linear term is included in the expansion,
24
).()')((')()( 000000 tgVtttgtgVV ( 3-9 )
, ))(')()()('(
( 3-10 )
.
)(')()()('
( 3-11 )
The required time domain waveform to calculate the ISF can be found from the large signal simulation methods, introduced in section 3.2.
Phase noise calculation
When the ISF is known, the phase noise can be calculated by the following equation [36],
). 2
( 3-12 )
The calculation of phase noise can be treated in two parts. The phase noise with -20dB/decade slope can be calculated by ( 3-12 ). The phase noise with -30dB/decade slope can be further simplified as
), 2
c L f ( 3-13 )
where c0 is the mean value of the ISF. For a transistor-based oscillator, the two noise sources that have the
largest contribution to phase noise are included in this calculation. These noise sources are bias dependent noise, i.e. 1/f noise and shot noise, expressed as
,)(22 tIqfin ( 3-14 )
where 2 ni and 2
/1 fi is the shot noise and the 1/f noise, respectively. Depending
on the devices’ characteristic, the shot noise is not so significant in FET device and will be excluded in the phase noise calculation of HEMT oscillators. These noise sources are excited by the transistor intrinsic current I(t). It is important to strengthen that the noise sources are time variant under oscillator operation with the same period as the oscillation. When the periodic current modulates the noise, the noise sources become cyclo-stationary and can be expressed as stationary noise in0(t) multiplied by the normalized periodic function (t) with unity amplitude,
).()()( 0 ttiti nn ( 3-16 )
With the new definition of the time variant noise source, the ISF in ( 3-11 ) needs to be modified according to (t),
).()()( tttNMF ( 3-17 )
The new ISF is called noise modulation function (NMF) and is used in ( 3-12 ) and ( 3-13 ) for a more accurate prediction of phase noise. An example to demonstrate the calculation will be shown in 3.2.
26
3.2 Numerical simulations used for oscillator design
Usually the oscillator design is carried out in several steps. It generally starts from a small signal simulation to check the oscillation condition, followed by a large signal simulation for stable oscillation condition and finally a noise simulation for phase noise prediction. Depending on the specifications, the simulation effort varies.
To ensure oscillation condition, small-signal S-parameters are normally used. To predict output power and large-signal characteristics, a large signal model and a simulation method are required. Different large-signal simulation methods are available, e.g., harmonic balance, transient simulation, and periodic steady state.
),()()( ))((
( 3-18 )
where i(v(t)) and q(v(t)) are current, and charge respectively. They can be linear or nonlinear depending on the circuit composition. The third term y(t) is the admittance (impulse response), which is determined by the passive circuit and is usually linear. The right hand side iext is the excitation current source.
In transient simulation, the equation is solved in the time domain using discretized samples. A drawback in transient simulation is that the time step between each sample and the time length to be simulated are difficult to determine in the simulator setup. A small time step is required to improve accuracy; on the other hand, a long length of time is required to simulate the low frequency effect. Although transient simulation can cover the effect over a wide frequency range, this method requires computation resources proportional to the number of time samples. In oscillator design, this method is only used to simulate oscillation startup. If the design interest concerns only the steady state oscillation period, two efficient methods can be used, i.e. harmonic balance and periodic steady state.
In the harmonic balance method, the steady state solution is solved in frequency domain, using an algorithm that assumes the input frequency to be known. [61][62][63]. The method is used in commercial softwares like ADS and Microwave office. An autonomous circuit, like an oscillator lacks input frequency and thus requires a reasonable guess as starting condition. A good guess on the initial frequency could be the frequency that fulfills the small signal oscillation condition, found from S-parameter simulation. The large signal oscillation condition can be different from its small signal condition and
27
may cause a frequency shift in the final oscillation frequency. To find the steady state oscillation, an oscillation probe is inserted into the oscillator feedback loop to check the large signal oscillation condition. The oscillation probe is a voltage source with a voltage amplitude and a frequency as the control parameters. These two parameters are swept in the simulation, and a harmonic balance simulation is carried out once at every swept point. Then, the function of the probe is to detect the condition when the large signal loop gain equals to one and phase is zero.
The periodic steady state method is based on an algorithm in time domain
[60]. The method is used in Cadence. Compared to the transient simulation, periodic steady state has different boundary conditions. This boundary condition comes from a reasonable assumption that at the steady state, the signal is repeating itself in each period,
).()( Ttxtx ( 3-19 )
In the sense, the time length required in the simulation is only one period T. The method is similar to the harmonic balance method, which requires a guess on initial frequency and a probe to detect the large signal oscillation condition. By inserting a voltage probe, the large signal oscillation can be found from simulations at several voltages and frequencies. Compared with harmonic balance simulation where the waveform is limited to the combination of harmonic signals, the time domain method can solve the waveform in an arbitrary shape. This characteristic helps improve the convergence for the simulation.
The final step is the phase noise simulation. Phase noise simulation requires the result from a large signal simulation. Since the noise can be regarded as a small signal quantity, when it applies to the oscillator trajectory, the circuit equations can be linearized around the large signal solution in order to calculate the small signal response. In harmonic balance simulation, the result is expressed in the frequency domain. Therefore, the linearization is also performed in the frequency domain. Through the linearization, the Fourier coefficients from the harmonic balance solution can be rearranged into a conversion matrix. The matrix can convert the noise sources around harmonics frequencies into phase noise around the oscillation frequency[61][62]. In periodic steady state simulation, the same linearization is performed in a similar manner but in the time domain. Then a sensitivity matrix works in a similar way to Hajimiri’s sensitivity function in phase noise calculation.
The algorithms behind each software are not the focus of this work, but it is important to know what effects are genuinely simulated. In this work, the commercial CAD tool ADS is used as the main design platform, in which the noise source are defined by DC currents [64]. However, in oscillator operation,
28
the noise sources are strongly excited by AC currents. The AC-modulated noise sources are often referred to as cyclo-stationary noise. If the bias dependent noise sources are not cyclo-stationary, the phase noise calculation will not give an accurate prediction, especially not in the -30dB/decade region. The correct procedure is that the bias dependent noise sources should be modulated by the AC currents to perform the first frequency conversion. In that sense, the bias dependent noise sources are converted to noise sources at the harmonic frequencies of the oscillator, keeping the same shape as the original noise sources. Then, the phase noise calculation is performed using conversion matrix for the second frequency conversion. If the bias dependent noise sources are not modulated by AC currents, the noise conversion to signal sideband noise could be seriously underestimated, especially in the case of 1/f noise. A detailed explanation is provided by Rudolph in [68].
Several methods to implement the cyclo-stationary noise in ADS are proposed [65][66][67], which requires knowledge on writing a script model in verilog-A or a complex arrangement using a mixer based SDD model. A clear improvement of phase noise simulation accuracy is achieved when cyclo-stationary noise representation is used [66][67]. Paper [B] reports on improved accuracy in phase noise prediction when using Hajimiri’s LTV theory instead of ADS’s inherent algorithm for phase noise simulation. The proposed method is demonstrated on a GaN HEMT balanced Colpitts oscillator. The circuit schematic is shown in Fig. 3-1(a).
29
To present the phase noise calculation procedure, the detailed steps are
summarized from section 3.1 and 3.2 and listed as follows: 1. Develop a large signal transistor model that enables access to the
intrinsic current. 2. Extract the LFN parameters from measurements using ( 2-1 ). 3. Perform a harmonic balance simulation to derive the tank voltage
waveform, i.e. voltage waveform between Vt+ and Vt- in Fig. 3-1(a), and tank current waveform, labeled as Itank in Fig. 3-1(a). In addition, the intrinsic current waveform is also required for the following calculation.
4. Calculate ISF waveform using ( 3-11 ). The required information is the tank voltage and current waveforms derived from step 3.
5. Calculate the cyclo-stationary LFN using ( 3-15 ). The required low frequency parameters are derived from step 2, and the intrinsic current is from step 3. An example of cyclo-stationary LFN at 100 kHz is shown in Fig. 3-2.
6. Calculate NMF waveform using ( 3-17 ). The normalized noise waveform (t) can be extracted from step 5, and the ISF was calculated in step 4. Examples of ISF and NMF waveforms are presented in Fig. 3-2.
7. Phase noise in the -30dB/decade region can be calculated using ( 3-13 ).
The phase noise comparison is shown in Fig. 3-1(b). There is a good correspondence in the simulation and the measurement results for frequencies around 10 MHz, where 1/f noise is less significant. At the offset frequency from 1 kHz to 1 MHz, the simulation result shows more than a 10dB difference compared to the measurement result. In contrast to simulation, the proposed method without fitting parameters shows a good agreement to the measurement result.
30
(a) (b)
Fig. 3-1 (a) The schematic of the balanced Colpitts GaN HEMT oscillator. (b) The comparison of phase noise between calculation method, ADS simulation and measurement result at frequency offsets from 1 kHz to 10MHz.
Fig. 3-2 (a) Tank voltage (between the nodes Vt+ and Vt- in Fig. 3-1) and tank current waveforms (labeled as Itank in Fig. 3-1) extracted from harmonic balance simulation. (b) LFN at 100 kHz being modulated by the transistor intrinsic current. (c) Calculated ISF and NMF using ( 3-11 ) and ( 3-17 ), respectively .
10 3
10 4
10 5
10 6
10 7
0
2
-0.5
0
0.5
1
3.3 Phase noise measurement methodologies
Phase noise measurement can roughly be categorized into two different techniques. One technique called direct spectrum measurement, measures the frequency offset directly from a spectrum analyzer and normalized it to the signal power. Another technique demodulates the signal into base-band and then measures the time domain fluctuation, similar to the setup in Fig. 2-7. Two examples using this technique are delay line method and PLL method. It is important to know the restriction of each measurement method and choose the right one for reliable measurements.
Direct spectrum measurement
Direct spectrum measurement is suitable for an oscillator with amplitude noise significantly less than the phase noise [71]. This is usually the case since the amplitude fluctuation is limited by amplitude restoring mechanism. This type of measurement can be carried out using an ordinary spectrum analyzer. Before measuring the phase noise at certain offset frequencies, one should consider the noise floor of the spectrum analyzer and the resolution bandwidth. The noise of the measurement system consists of the thermal noise, and phase noise of the local oscillator. If an amplifier is used to measure the phase noise of a low power signal, the noise figure of the amplifier also contributes to the total noise. The noise floor presented on the spectrum analyzer is the integration of the total system noise in the resolution bandwidth. It is important to be aware that the phase noise of the measured oscillator must be higher than the noise floor. Once the noise floor is guaranteed, the phase noise can be calculated as follows
),log(10)()( RBWPfPfL sig ( 3-20 )
where P(f) is the power at f, Psig is the carrier signal power, and RBW is the resolution bandwidth.
For measurements of high frequency signals (>100GHz), most of the results found in open literature are measured using this method. In this work, the method is used in [E].
32
Demodulation measurement The key concept of demodulation measurement is to use a phase detector
to measure only the phase fluctuation [70]. Therefore, the amplitude fluctuation is excluded in the measurement result. The technique can be implemented by measuring a single source (DUT) or two sources (a DUT and a reference oscillator). In the single source measurement, the DUT signal is split into two paths, one path goes through a delay line, which converts frequency fluctuations into phase fluctuations, and another path goes to a phase shifter to control signals in the two paths with a 90-degree phase difference, a system setup is shown in Fig. 3-3. Then the phase detector gives an output of
, )sin(
( 3-21 )
where K is the sensitivity of the demodulator,ffis the frequency fluctuation at f frequency offset, and τd is the time delay of the delay line.
The measurement requires calibration of the transfer function ( 3-21 ) to calculate the phase noise. When measuring phase noise at frequency offset close to 1/τd , the transfer function is zero, and measurement is not possible. Changing the delay line manually is then required in order to measure the phase noise at the particular offset frequency.
Fig. 3-3 Phase noise measurement using the delay line method (single source measurement).
An additional problem in the setup is that the measurement is not
automatic. It requires manual tuning on the passive phase shifter. Thus, this method is not efficient to perform phase noise measurement at multiple bias points. In this work, the method is used in the early stage in the project. The result in [A, I] are measured using this method.
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The latest commercial setups for phase noise measurements are automatic and simple to use. Most of them are based on the two sources measurement using an internal reference oscillator and a PLL. The PLL can maintain a 90-degree difference between the reference oscillator and the DUT automatically [72], shown in Fig. 3-4. The range of measurable phase noise is then limited by the phase noise of the reference oscillator. In this work, most phase noise measurements are carried out using this method.
Fig. 3-4 Phase noise measurement using the PLL method (two sources measurement).
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Chapter 4
VCO design
The VCO design serves different purposes and the specifications vary respectively. The different varieties of an oscillator can be roughly categorized as low phase noise oscillator, wide tuning oscillator, and high-frequency/high- power signal source. On top of these performances, power consumption is a general consideration in all type of designs. It is interesting to see that, these specifications mentioned above are directly or indirectly included in the Leeson’s phase noise model ( 3-1 ). Designing oscillator with multiple functionalities, e.g. low phase noise and wide tuning range, brings trade off to the design. In this chapter, the techniques to improve oscillator performance presented in the appended paper are summarized. In this work, InGaP HBT technology and GaN HEMT technology are used to design several oscillators/VCOs. The device characteristics introduced in Chapter 2, give designers indications what performance that can be expected.
In section 4.1, the work investigates the design of low phase noise oscillators/VCOs in InGaP HBT. Conventional topologies are considered as well as some modifications with purpose to improve phase noise and/or tuning range. In section 4.2, GaN HEMT oscillator design techniques are presented. Compared to InGaP HBT, the LFN in GaN HEMT has a large variation depending on bias conditions. The design concept focuses on biasing the transistor in the low noise region. The result is compared, related to their device characteristics, and bench marked versus the state of the art. In section 4.3, methods for design of high frequency signal sources are presented. A method proposed for designing oscillator above 100 GHz is implemented in InP HBT technology. In addition, a technique for 3rd harmonic signal generation is implemented in InGaP HBT technology. Some other techniques to design high frequency oscillators are also addressed and compared.
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4.1 VCO design in InGaP HBT technology
In the design of low phase noise oscillators, Leeson’s equation ( 3-1 ) is the most common and intuitive guide line to optimize the phase noise performance. When designing an MMIC oscillator, attention is focused on optimizing the Q factor and the power PS coupled to the resonator. Both factors are limited by technologies.
In section 4.1.1, three conventional VCO topologies: Balanced Colpitts, cross-coupled and gm-boosted oscillators, are studied. The small signal oscillation condition is derived for each topology and the design tool to predict phase noise, see section 3.2, are verified in the three VCOs. The oscillators are designed as balanced, which has the advantages of increasing the tank voltage swing in order to lower phase noise according to ( 3-1 ). In addition, the structure decouples the loading effect of bias supply by connecting DC bias at a virtual ground. Finally, the balanced VCO provides differential output, which is suitable to drive a mixer having a balanced topology. Three VCOs using the mentioned topologies were designed for the same frequency band (6 to 8 GHz). The three VCOs use the same inductance and varactor in the layout for a fair comparison of phase noise.
Further modifications of the conventional topologies are discussed in sections 4.1.2 and 4.1.3. To pursue higher tuning range as well as low phase noise, the two design goals are in contradiction to each other. Often, VCOs with large tuning range, exhibit higher noise. The most intuitive explanation is to look at the influence of tuning sensitivity in the unit of Hz/V. Any voltage fluctuation is translated to a frequency fluctuation that is proportional to the tuning sensitivity. The tuning sensitivity can be included as a parameter in the extension of Leeson’s equation, several researchers have derived different expressions [6], [73]. The expression arranged by Rohde [6] is an example
, 2
L ( 4-1 )
where R is the equivalent noise resistance of the tuning diode, and K0 is the tuning sensitivity (sometimes called oscillator voltage gain) with unit of Hz/V.
In applications where a good phase noise and a reasonable tuning range are required, a technique to design double-tuned balanced Colpitts VCO [A] is introduced in section 4.1.2. The proposed design scheme can provide a fair tuning bandwidth and maintain a good phase noise over the tuning range. Another technique proposed a method to improve the performances of a gm-boosted Colpitts using a pair of Darlington transistors [D], introduced in section 4.1.3. The novel implementation has proven to improve both the phase
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noise and the tuning range. The tradeoff between phase noise and tuning range can be overcome by
separating the complete tuning bandwidth into small fractions, i.e. a VCO with two tuning mechanisms: a coarse-tuning to switch between different frequency bands and a fine-tuning to tune the in-band signal. A similar technique using two VCOs and a mixer [C] is introduced in section 4.1.4.
4.1.1 Conventional VCO topologies
Balanced Colpitts VCO
In the design of oscillators, the phase noise performance is optimized by maximizing the resonator’s Q factor and the RF power delivered to the resonator. In an oscillator, the RF power is generated by a transistor. The transistor, having different impedances at its three terminals, requires a feedback loop that couples the optimum power to the resonator and at the same time maintains a high Q factor. The feedback loop can be designed by impedance transformations. Two commonly used methods are capacitive and inductive voltage dividers as used in Colpitts and Hartley topologies, respectively. In MMIC VCO design, the inductor in Hartley topology requires a long distance in terms of wavelength, and is suitable for high frequency design. In the frequency band 6 to 8 GHz, the Colpitts topology is more suitable for a compact and small-size layout. The design schematic, layout and detailed parameters are shown in Fig. 4-1 for an InGaP HBT balanced Colpitts VCO.
C1=1.6pF C2=2.8pF Re=100
(a) (b) Fig. 4-1 (a) Schematic and (b) layout of the balanced Colpitts VCO using collector to emitter feedback.
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Small-signal analysis
In the analysis of a common-base balanced Colpitts oscillator, the negative conductance between the collectors of HBT1 and HBT2 is calculated as2
. )()/1/)1((
)/1/1(
2
1
21
( 4-2 )
The equation is expressed in the form of the base input resistance r and
the current gain β of a bipolar device in common emitter configuration, it can be simplified to the expression using gm and can be used for other technologies
as well, e.g., FET devices. The relation between gm, r , and β is
mb
c
c
be
b
be
gi
i
i
v
i
c m Vn
I g ( 4-3 )

( 4-5 )
The negative conductance provided by the active component must be large enough to compensate the loss in the tank. At this point, it is essential to introduce the relation between tank parameters for further calculation.
The relation between the Q factor, and the tank resistance, i.e., Rp in a parallel tank is
, c
p
Z
R Q ( 4-6 )
where Zc is the characteristic impedance of the tank. The value can easily be found by
2 Note that ( 4-2 ) is valid for the schematic shown in Fig. 4-1, Colpitts topologies with other
feedback schemes have different expressions.
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, C
L Zc ( 4-7 )
where L and C are the inductance and capacitance of the tank, respectively. As discussed previously in section 2.3.2, the tank’s Q factor in MMIC
technology is typically in the order of 20 around 10 GHz. In this design, a 7 GHz parallel tank with a tank impedance of Zc=15 is used. The equivalent tank conductance 1/Rp =0.0033 S can be calculated from ( 4-6 ). The required start-up condition can be calculated from ( 4-4 ). From the simplified form of negative conductance Re[Y], the required gm is only 0.03 S. Thanks to the high gm of HBT, see Table 2-1, the required gm is easily fulfilled with a low biasing current.
In contrast to a bipolar design, Colpitts topology has critical startup condition using FET devices because of lower gm. For example, Table 2-1 shows that the 2x30um GaN HEMT has peak gm of 0.015 S, which is not enough to start up the oscillation. However, a designer can increase the transistor size to increase gm with the penalty of higher parasitic capacitance. These parasitic capacitances must be taken into account when designing feedback capacitance. If not carefully considered, a considerable frequency shift may be the result.
Phase-noise analysis
With little effort in fulfilling the start-up condition, we have some flexibility in the choice of Re. It decreases the negative conductance in exchange for better Q factor. The purpose of Re is to provide a DC path for transistor biasing current, it cannot be too large due to high power consumption neither be too low as it then will load the tank seriously. To avoid that degradation of the tank’s Q factor, the relation between Re and Rp should follow
,/ 2nRR ep ( 4-8 )
where n is the transformation factor n=C1/(C1+C2). The optimized value of n can be found by systematically sweeping the base bias voltage and n for a fixed Zc, detailed procedure is presented in [A]. In this design, Re/n2 is 2.5 times of Rp, which reduces the Q factor of C2 by 29% and corresponds to 17% degradation of the Q factor for the overall capacitance. The resistor Re is about 100 , the medium voltage drop on the resistor Re will cost acceptable power consumption.
Beside the consideration of Q factor, the transistor noise that is excited by the intrinsic current has a large impact on the phase noise. The impact can be analyzed from the intrinsic current waveform and its conduction angle. In the original Colpitts oscillator, the peak of intrinsic current appears at the minimum voltage of the tank voltage waveform. In the proposed topology, the
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tank has a pair of varactors in parallel to the inductance, which distorts the waveform and shifts the peak current. It is shown that the phase shift of the peak current is proportional to the ratio between varactor capacitance (Cv) added in parallel to L and the total capacitance (Ctotal) in the tank, denoted as nR=Cv/Ctotal [B]. On the other hand, the tuning range will be lower if a low value of varactor is added. Detailed trade-off consideration between the peak current phase shift, oscillation condition, and tuning range are presented in [B].
In the design of InGaP HBT Colpitts VCO, the ratio nR is chosen to be around 0.5. The normalized tank voltage and tank current waveforms are presented in Fig. 4-2. The intrinsic collector current that excites the shot noise and 1/f noise is presented as a normalized unity waveform, shown as NIc in Fig. 4-2. The transistor conducts less than a half cycle and the operation is thus in class C operation. The small shift of current peak from the minimum voltage is due to the parallel varactors. Detailed calculations of ISF and NMF are shown in Fig. 4-3, where the NMF shows both a low amplitude and a small conduction angle. From Hajimiri’s compact phase noise expression ( 3-12 ), the low value of ΓNMF
2 indicates that the phase noise due to shot noise is low. On the other hand, the 1/f noise conversion relates to the DC part of ΓNMF, which is determined by symmetry of ΓNMF waveform and is not obviously seen. The phase noise comparison between calculation result, measured result and ADS simulation is presented in Fig. 4-4(b). The good agreement between calculation result and measurement shows that the calculation method improves the phase noise prediction at the -30dB/decade region.
The load line, shown in Fig. 4-4(a), is expressed as the total collector current (including intrinsic and parasitic currents of HBT) versus the collector to emitter voltage. The figure illustrates the device’s working region under the VCO operation. From the RPN measurement, introduced in section 2.2, it indicated that the transistor should be driven into the saturation mode operation, nevertheless, not overdriven into a deep saturation to avoid the strong distortion. The optimum phase noise can be found around the point where the transistor load line just touches the saturation region.
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Fig. 4-2 The normalized tank voltage and the tank current waveform (top) of the balanced Colpitts VCO, and the normalized intrinsic collector current (bottom).
Fig. 4-3 The calculated ISF and NMF of the balanced Colpitts VCO based on the waveforms in Fig. 4-2. The waveforms are presented for one cycle and the x scale is aligned with the x-axis in Fig. 4-2.
Table 4-1 Design parameters of the balanced Colpitts VCO. Ltank and C are the tank inductance and capacitance. Zc and f can be calculated accordingly. qmax is derived from HB- simulation.
Zc C Ltank f qmax 15.5 1.42pF 0.34nH 7.3GHz 28pC
(a) (b) Fig. 4-4 (a) The balanced Colpitts VCO’s intrinsic (black line) and extrinsic (red dot) load line superimposed on the DC-IV curve (blue line). (b) Comparison of phase noise between the balanced Colpitts VCO’s measurement, calculation and simulation.
0 0.5 1 1.5 2 2.5
x 10 -10
x 10 -10
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Cross-coupled VCO
The cross-coupled oscillator, shown in Fig. 4-5, is one of the most common topologies found in literatures due to the high negative conductance that helps the oscillation start-up.
Small-signal analysis
Similar to balanced Colpitts oscillator, the start-up condition is not the main concern in the cross-coupled oscillator based on InGaP HBT. In addition, the DC current path is also implemented by an emitter resistance Re. The resistance can lower small signal negative conductance and the expression of the small signal admittance between the collectors of HBT1 and HBT2 is
. /1))1((2
1
,
( 4-11 )
If Re=0 and Cc are large, the real part of admittance can be simplified to the familiar term –gm/2.
Cc=0.9 pF Re=60
(a) (b) Fig. 4-5 (a) Schematic and (b) layout of the cross-coupled VCO.
Phase noise analysis
The cross-coupled topology can be seen as a two stage ring oscillator with
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an additional LC resonator to filter harmonic signals. The simulated results are shown in Fig. 4-6. The normalized voltage waveform of the cross-coupled VCO has more harmonic contents than a Colpitts oscillator and presents a waveform similar to a square wave as in a ring oscillator. However, the cross-coupled VCO has better phase noise thanks to the LC tank that stores the energy in the tank. In comparison, the ring oscillator repeatedly charge and discharge its parasitic capacitance in every cycle.
The normalized HBT intrinsic current, labeled as NIc in Fig. 4-6, shows that the transistor operates in class B (50% duty cycle). Therefore, the NMF of the cross-coupled VCO, shown in Fig. 4-7, has a larger ΓNMF
2 than the class C operation of the balanced Colpitts VCO. This indicates that the phase noise of the cross-coupled VCO, shown in Fig. 4-8(b), is worse than the phase noise of the balanced Colpitts VCO. The calculation using the presented theory again shows a good agreement with the measurement results. In Fig. 4-8(a), the dynamic load line has a voltage swing larger than the swing in the balanced Colpitts topology. However, the load line shows only the operation region of the transistor. It does not directly present the voltage swing across the tank. The reasons for a larger voltage swing of the cross-coupled VCO load line comes from the fact that the collector and emitter voltages are out of phase, in contrast, the balanced Colpitts VCO is made in-phase by the feedback capacitance C1.
Fig. 4-6 The normalized tank voltage and the tank current waveform (top) of the cross- coupled VCO, and the normalized intrinsic collector current (bottom).
Fig. 4-7 The calculated ISF and NMF of the cross-coupled VCO based on the waveforms in Fig. 4-6. The waveforms are presented for one cycle and the x scale is aligned with the x-axis in Fig. 4-6.
0 0.5 1 1.5 2
x 10 -10
x 10 -10
x 10 -10
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Table 4-2 Design parameters of the cross-coupled VCO. Ltank and C are the tank inductance and capacitance. Zc and f can be calculated accordingly. qmax is derived from HB-simulation.
Zc C Ltank f qmax 15.1 1.25pF 0.29nH 8.44GHz 16.4pC
(a) (b)
Fig. 4-8 (a) The cross-coupled VCO’s intrinsic (black line) and extrinsic (red dot) load line superimposed on the DC-IV curve (blue line). (b) Comparison of phase noise between the cross-coupled VCO’s measurement, calculation and simulation.
gm-boosted Colpitts VCO
The gm-boosted topology was first proposed by Li [74]. The topology got its name from improving the Colpitts start-up condition by adding the cross-coupled feedback loop. Since then, the technique is frequently utilized and modifications to the initial topology have been proposed to obtain good phase noise and lower power consumption in CMOS technology [75].
Small-signal analysis
The small signal