research article comparative analyses of phase noise in...

Research ArticleComparative Analyses of Phase Noise in 28 nmCMOS LC Oscillator Circuit Topologies: Hartley, Colpitts,and Common-Source Cross-Coupled Differential Pair

Ilias Chlis,1,2 Domenico Pepe,1 and Domenico Zito1,2

1 Tyndall National Institute, “Lee Maltings”, Dyke Parade, Cork, Ireland2Department of Electrical and Electronic Engineering, University College Cork, Cork, Ireland

Correspondence should be addressed to Domenico Zito; [email protected]

Received 30 August 2013; Accepted 19 November 2013; Published 6 February 2014

Academic Editors: W. Y. Choi and Y. C. Lim

Copyright © 2014 Ilias Chlis et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper reports comparative analyses of phase noise in Hartley, Colpitts, and common-source cross-coupled differential pairLC oscillator topologies in 28 nm CMOS technology. The impulse sensitivity function is used to carry out both qualitative andquantitative analyses of the phase noise exhibited by each circuit component in each circuit topology with oscillation frequencyranging from 1 to 100GHz. The comparative analyses show the existence of four distinct frequency regions in which the threeoscillator topologies rank unevenly in terms of best phase noise performance, due to the combined effects of device noise andcircuit node sensitivity.

1. Introduction

Oscillator phase noise (PN) is one of themain bottlenecks forthe information capacity of communication systems, leadingto severe challenges in the design of local oscillators in silicontechnologies, especially at very high frequency [1–5]. In par-ticular, the main difficulties are to achieve a high quality fac-tor LC tank [6–11] and consume reasonable power [12, 13].

Oscillator phase noise has been studied extensively overthe last decades [14–17]. Most of these studies are based onlinear time-invariant (LTI) oscillator models, which provideimportant qualitative design insights, but are limited in thequantitative prediction of the power spectral density lev-els [18], in some cases addressed by adopting nonlinearapproaches [19].

The linear time-variant (LTV) oscillator model allows aquantitative understanding of oscillator phase noise throughthe impulse sensitivity function (ISF), represented as Γ(𝑥)[18]. Since the oscillator is assumed as a linear time-varyingcircuit, the phase sensitivity to noise perturbations can bedescribed in terms of its (time-varying) impulse response.

The evaluation of the ISF involves a significant amountof transient simulations and data extractions, resulting intime consuming calculations, potentially prone to inaccuracy.Recently, new efficient frequency-domainmethods operatingdirectly in the steady-state were proposed [20, 21], allowing aconsistent reduction of the simulation workload. Regardlessof the methods, the analysis of the phase sensitivity cancontribute significantly to a better understanding of theimpact of noise sources to the oscillator phase noise in themost widespread circuit topologies.

A comparative analysis of common-source cross-coupleddifferential pair and differential Colpitts LC oscillators in0.35 𝜇mCMOS technology at 2.9GHzwas carried out in [22],showing the superior performance of the cross-coupled dif-ferential topology. In this perspective, it could be interestingto extend the comparison also to other topologies, technologynodes, and oscillation frequencies. In this regard, in [23],we reported the results of a comparative analysis on Hartley,Colpitts, and common-source cross-coupled differential paircircuit topologies in 28 nm CMOS technology operating at10GHz, confirming the results in [22] and showing that

Hindawi Publishing Corporatione Scientific World JournalVolume 2014, Article ID 421321, 13 pageshttp://dx.doi.org/10.1155/2014/421321

2 The Scientific World Journal

Table 1: Device sizing.

Transistor width (𝜇m) Capacitor value (pF) Inductor value (pH)𝑀1

𝑀2 𝑀

3𝑀4

𝐶1

𝐶2

𝐶3 𝐶

4𝐶5

𝐿1

𝐿2

30 30 30 15 0.97 105 0.495 0.8 0.229 500 250

the Colpitts topology provides superior phase noise perfor-mance with respect to the Hartley topology.

This paper reports an expansion and an extension of ourpreliminary comparative study of PN for the three oscillatortopologies: Hartley, Colpitts, and common-source cross-coupled differential pair circuit topologies in 28 nm CMOStechnology. In particular, we recap the main results andreport additional aspects of the preliminary investigations;then we move forward to wider evaluations on PN contri-butions in relation to the operating frequency. The results ofthe analyses show interesting aspects not addressed by theliterature. In detail, all the steps for an accurate derivation ofthe ISF are summarized and the PN predictions for a wide setof amplitudes of the injected current pulse are compared withthe results obtained by the direct plots obtained by means ofSpectreRF-Cadence Periodic Steady State (PSS) analysis. Thecontributions from each noise source to the overall PN areevaluated qualitatively and quantitatively through the ISF foreach topology operating in a discrete set of frequencies from1 to 100GHz.

The paper is organized as follows. Section 2 reports thedesign of the three oscillator topologies in 28 nmCMOS tech-nology. Section 3 summarizes the key analytical expressionsfor PN predictions through the ISF, the key steps, and settingsfor accurate evaluations and finally reports the results forthe oscillation frequency of 10GHz. In Section 4, qualitativeand qualitative analyses of the PN contributed by each circuitcomponent are carried out for each topology for a discreteset of oscillation frequencies ranging from 1 to 100GHz.Section 5 reports the results that reveal the existence of fourdifferent frequency regions inwhich the best PNperformanceis given case by case by a different topology. In Section 6, theconclusions are drawn.

2. Circuit Topologies

Three LC oscillator topologies have been analysed: single-ended Colpitts, single-ended Hartley, and top-biased com-mon-source cross-coupled differential pair oscillator topolo-gies, as shown in Figure 1. The three oscillator circuit topolo-gies have been implemented in 28 nmbulkCMOS technologyby ST-Microelectronics by adopting the same criteria for afair comparison as follows. The frequency of operation is10GHz. The sizes of the transistors and the values of theinductors and capacitors used are reported in Table 1. Despitethe fact that this work is addressed to the investigations of thecircuit topologies as such, rather than the circuit design andimplementation, that is, regardless of the effects of parasiticcomponents, we considered a reasonable quality factor forthe LC tank in order to carry out the comparative studyof the properties of each circuit topology under the sametypical conditions. Thereby, a quality factor (𝑄) equal to 10

Table 2: Flicker noise modeling.

Transistor 𝑘𝑓(V2 F) 𝑓

1/𝑓(MHz) c

𝑀1

1.09 × 10−22 1390 0.9

𝑀2

2.26 × 10−23 870 0.9

𝑀3

1.88 × 10−23 1100 0.93

𝑀4

1.25 × 10−22 1430 0.9

has been assumed for the inductors, considering a parasiticresistance in series with the inductor, whereas the capacitorshave been considered as ideal devices. In all cases, the powerconsumption is 6.3mW.

A small signal noise analysis by SpectreRF was used forthe derivation of the flicker noise corner of each transistor.Assuming that the power spectral density (PSD) of thethermal and flicker noise currents generated by the transistorin the saturation region is given by (1) and (2), respectively;the flicker noise corner is given by (3) [24].

𝑆𝑖𝑤= 4𝑘𝑇𝛾𝑔

𝑚, (1)

𝑆𝑖𝑓= 𝑔2

𝑚

𝑘𝑓

𝑊𝐿𝐶ox

1

𝑓𝑐, (2)

𝑓1/𝑓

=𝑐

√𝑘𝑓

𝑊𝐿𝐶ox

𝑔𝑚

4𝑘𝑇𝛾, (3)

where 𝑘𝑓is a bias-dependent constant, 𝑐 is a constant with

typical values ranging from 0.7 to 1.2, 𝐶ox is the oxidecapacitance per unit area, and 𝛾 is the excess noise coefficient.For the 28 nm bulk CMOS technology adopted, the thicknessof the oxide 𝑡ox is about 1.4 nm for the n-MOSFET and 1.7 nmfor the p-MOSFET, from which we can derive that 𝐶ox isabout 0.026 and 0.02 F/m2, respectively. The values of 𝑘

𝑓,

𝑓1/𝑓

, and 𝑐 have been derived by means of noise simulationof each single stand-alone transistor of Table 1. They arereported in Table 2.

3. Impulse Sensitivity Function

In order to get an insight of the noise contribution of eachcircuit component in each circuit topology, hereinafter wemake use of the ISF as a predictive tool for quantitative andqualitative PN evaluations.

A detailed procedure for computation of the ISF andPN prediction in a linear time-varying system in the caseof a source-coupled CMOS multivibrator with operating fre-quency up to 2MHz was presented in [25]. All the resultswere achieved only for a single amplitude value of the injectedpulse. However, the time-domain evaluation of the ISFinvolves a number of transient simulations, potentially prone

The Scientific World Journal 3

VDD

M1

M2

L1

C1

C1

C2

VB1

VB2

VOUT

in,tank

in,M2

in,M1

(a)

VDD

M1

M2

L2

L2

C2

C3

C4

VB1

VB2

VOUT

in,tank

in,M2

in,M1

(b)

VDD

M3

M4M4

L1L1

C5

VB3

in,tank

in,M4 in,M4

in,M3

VOUT+VOUT−

(c)

Figure 1: Schematic of the oscillator circuit topologies: (a) single-ended Colpitts, (b) single-ended Hartley, and (c) top-biased common-source cross-coupled differential pair. 𝑉

𝐵1, 𝑉𝐵2, and 𝑉

𝐵3are DC bias voltages. In Colpitts and Hartley topologies, the output voltage is taken

after a 100 nF capacitor in order to remove the DC component.

to inaccuracy. Thereby, it is worth consolidating all the stepsin order to achieve accurate results.

The impulse response from each current noise source tothe oscillator output phase can be written as [18]

ℎ𝜙(𝑡, 𝜏) =

Γ (𝜔0𝜏)

𝑞max𝑢 (𝑡 − 𝜏) , (4)

where 𝑞max is the charge injected into a specific circuit nodeof the oscillator at time 𝑡 = 𝜏, 𝑢(𝑡) is the unity step function,and Γ(𝜔

0𝜏) is a dimensionless periodic function that can be

expressed as a Fourier series [18]:

Γ (𝜔0𝜏) =

𝑐0

2+

∞

∑

𝑛=1

𝑐𝑛cos (𝑛𝜔

0𝜏 + 𝜃𝑛) . (5)

The DC and root mean square (rms) values of Γ(𝜔0𝜏) are

given by the following two equations [18]:

ΓDC =𝑐0

2,

Γrms = √1

2

∞

∑

𝑛=0

𝑐2𝑛.

(6)

The PN of any oscillator is traditionally indicated withL.The thermal noise contribution to the PN spectrum, fromeach given noise source with a white power spectral density,can be expressed as [18]

L {Δ𝜔}|𝑑𝐵 = 10 log[

[

Γ2

rms𝑞2max

(𝑖2𝑛/Δ𝑓)

2Δ𝜔2]

]

, (7)

where 𝑞max is the charge injected into a circuit node by thenoise source 𝑖

𝑛insisting in that node andΔ𝜔 is the offset from

the oscillation angular frequency. Therein [18], it is tacitlyassumed that 𝑐 in (2) is equal to 1, regardless of the technologynode.This assumption leads to the relatively rough but simpleequation (7).

The flicker noise contribution to the PN spectrum for anyoscillator, from each given noise source with a 1/𝑓 spectrum,can be expressed as follows [18], where𝜔

1/𝑓is the flicker noise

corner of the device:

L {Δ𝜔}|𝑑𝐵= 10 log[

[

𝑐2

0

𝑞2max

(𝑖2𝑛/Δ𝑓)

8Δ𝜔2

𝜔1/𝑓

Δ𝜔

]

]

. (8)

3.1. Simulation Steps and Settings. All the simulations havebeen carried out by using the SpectreRF simulator in theCadence design environment.The ISF of the oscillator topol-ogies has been evaluated for an oscillation frequency of10GHz, which will be considered hereinafter as a referencefor all the other cases. First we run a transient simulation inorder to observe and record when the amplitude of the oscil-lation waveform reaches the steady state regime. In our case,this occurs with large margins after 5 ns. Afterwards, we per-form other transient simulations applying current impulsivesources acting in parallel with the actual inherent currentnoise sources of the LC tank and transistors, by activatingonly one noise source at one time. The current impulses areset to occur in the steady state regime starting from a giventime reference for the unperturbed solution. The pulse widthof each current source has been chosen equal to 1 ps (i.e., onehundredth of the oscillation period) with 0.1 ps rise and falltime, as shown in Figure 2.

The simulation has been repeated for amplitudes of theinjected current of 1, 10, and 100𝜇A and 1 and 10mA. Eachtransient analysis is performed using the conservative modeand a maximum time step of 10 fs (i.e., one 10-thousandth of


t

0.1ps 0.1ps

1psIpulse

Figure 2: The injected current pulse.

Δt

t

Vout

𝜏

Figure 3: Time shift Δ𝑡 caused by the impulse injection occurringat the time 𝜏.

the oscillation period), in order to have good accuracy evenin the case of the smallest injected current pulse (i.e., 1 𝜇A).The charge 𝑞max injected in each node corresponds to the areaunder each pulse, that is, the area of the trapezoid of Figure 2:

𝑞max = 𝐼Pulse × 1.1 × 10−12 Coulombs, (9)

where 𝐼pulse is the amplitude value of each source pulse.This isrepeated for all the𝑁 noise sources connected in parallel andfor all the𝑀 instants of time over one period of oscillation,where𝑁 = 3 and𝑀 = 40, in our case.The time instants havebeen chosen to be equally spaced in an oscillation period.Thetime shift caused by the impulse injection can be extracted bycomparing the perturbed and unperturbed waveforms. Thismeans that when the oscillation has reached the steady stateregime, the time shift Δ𝑡

𝑖of the zero-crossing instant of the

perturbed oscillation with respect to the unperturbed one,that is, when no impulse is applied, is calculated as shown inFigure 3.

Then, these time shifts are converted into phase shifts byusing the following relation:

Γ (𝑥 = 𝜔0𝑡) = 2𝜋

Δ𝑡𝑖(𝑡)

𝑇. (10)

In order to take into account the cyclostationary natureof the active device noise sources, Γ(𝑥) is multiplied with

𝛼(𝑥), where 𝛼(𝑥) is the absolute value of the unperturbedcurrent flowing in the respective node in which the impulsesare injected, normalized to its maximum value in the period.Then, the DC and root mean square (rms) components of theproduct Γ(𝑥) × 𝛼(𝑥) can be calculated as follows:

ΓDC =∑40

𝑖=1[Γ (𝑥) 𝛼 (𝑥)]

40,

Γrms =√∑40

𝑖=1{[Γ (𝑥) 𝛼 (𝑥)]

2}

40.

(11)

Finally, the total PN of the oscillator is computed byadding the contributions from all the noise sources actingin the circuit, according to (7) and (8). In particular, theactive devices inject noise to the terms responsible for bothflicker and thermal noise contributions to the oscillator PN,whereas the LC tank participates only to the thermal noisecontribution to PN. Equation (12) gives the total PN for eachof the three oscillators, where 𝑚 is the number of transistorsof the oscillator circuit.The first sum in (12) describes the PNcomponent contributed by the thermal noise. As a result, itcontains an additional term (𝑚 + 1), due to thermal noisecoming from the LC tank:

L {Δ𝜔}|𝑑𝐵 = 10 log{

{

{

𝑚+1

∑

𝑖=1

[

[

Γ2

rms𝑞2max

(𝑖2𝑛/Δ𝑓)𝑖

2Δ𝜔2]

]

+

𝑚

∑

𝑖=1

[

[

Γ2

DC𝑞2max

(𝑖2𝑛/Δ𝑓)𝑖

8Δ𝜔2

(𝜔1/𝑓)𝑖

Δ𝜔

]

]

}

}

}

.

(12)

3.2. Results. Figure 4 reports Γ(𝑥)𝛼(𝑥) for an injected currentpulse amplitude of 1 𝜇Aversus the phase for the injected noisesources, during one oscillation period, for the three oscillatorcircuit topologies.

Figure 5 reports the comparison between the PNobtained through the ISF and the PN obtained by directplots from PSS and periodic noise simulations, for the threeoscillator circuit topologies. Note that the PN predicted bythe ISF is very close to the values obtained by means ofSpectreRF simulations. Table 3 provides the PN results for allthe current impulse amplitude values, for a 1MHz frequencyoffset from the carrier.

Note that for this oscillation frequency (10GHz) the PNofthe common-source cross-coupled differential pair topologyis lower to the PN of the Colpitts topology, in agreementwith [22], and that the PN of Colpitts is lower to the PNexhibited by the Hartley topology. Moreover, note that theagreement degrades for higher pulse amplitudes, when thecurrent-to-phase transfer function starts becoming nonlin-ear.The amplitude in which this occurs is slightly different foreach oscillator topology, but for the injected current impulseof 1 𝜇A, the difference between the PN predicted by the ISFmethod and the one given by PSS and periodic noise (Pnoise)analysis is lower than 1% at a 1MHz frequency offset.


Table 3: Summary of the PN results obtained by SpectreRF and ISF.

PN (dBc/Hz) @ 1MHz frequency offset

Topology SpectreRF ISF1 𝜇A 10𝜇A 100 𝜇A 1mA 10mA

Colpitts −96.25 −96.20 −98.33 −98.49 −98.50 −98.45Hartley −92.75 −92.79 −95.18 −94.36 −94.85 −95.29Cross coupled −102.66 −102.69 −102.84 −102.83 −102.84 −102.94

0 1 2 3 4 5 6

0.0

0.1

0.2

0.3

0.4

0.5

0.67

−0.1

−0.2

−0.3

VO

UT

(V)

Colpitts

01

2

3

4

5

6

−1

−2

−3

−4

Γ(x)𝛼(x)

x (rads)

Γ(x)𝛼(x) − M1

Γ(x)𝛼(x) − M2

Γ(x) − LC tankVOUT

×10−6

(a)

0

0 1 2 3 4 5 6

Hartley

0.0

0.1

0.2

0.3

0.4

0.5

0.6

−0.1

−0.2

−0.3

VO

UT

(V)

1

2

3

4

5

6

7

−1

−2

−3

−4

Γ(x)𝛼(x)

x (rads)

Γ(x)𝛼(x) − M1

Γ(x)𝛼(x) − M2

Γ(x) − LC tankVOUT

×10−6

(b)

0 1 2 3 4 5 6

Cross-coupled differential

0.00.51.01.52.02.53.03.5

01

2

3

4

5

6

7

−1

−2

−3

−4

Γ(x)𝛼(x)

x (rads)

Γ(x)𝛼(x) − M3

Γ(x)𝛼(x) − M4

Γ(x) − LC tank

−2.0

−1.0

−1.5

−0.5

VOUT+ − VOUT−

×10−6

VO

UT+−V

OU

T−

(V)

(c)

Figure 4: Γ(𝑥)𝛼(𝑥) of the MOSFETs and Γ(𝑥) of the LC tank versus phase for a 1 𝜇A amplitude current impulse, for the oscillation frequencyof 10GHz: (a) Colpitts topology, (b) Hartley topology, and (c) common-source cross-coupled differential pair topology.


ColpittsPN−ISFPN−SpectreRF

Frequency offset (Hz)10k 100 k 1M 10M

PN (d

Bc/H

z)

−110

−100

−130

−120

−90

−80

−70

−60

−50

−40

−30

−20

−10

Flicker noise contr.−ISFThermal noise contr.−ISF

(a)

Hartley


PN−ISFPN−SpectreRF


PN (d

Bc/H

z)

−110

−100

−130

−120

−90

−80

−70

−60

−50

−40

−30

−20

−10

(b)

Cross-coupled differential


PN−ISFPN−SpectreRF


PN (d

Bc/H

z)

−110

−100

−130

−120

−90

−80

−70

−60

−50

−40

−30

−20

−10

(c)

Figure 5: PN versus frequency offset for the three oscillator circuit topologies, obtained through the ISF for a 1 𝜇A current impulse anddirect plot from PSS and periodic noise (Pnoise) SpectreRF simulations, for the oscillation frequency of 10GHz. The flicker and thermalnoise contributions to the overall PN are also plotted in order to identify the 1/𝑓3 PN frequency corner. (a) Colpitts: the 1/𝑓3 PN corner isat the frequency offset of 3.1MHz. (b) Hartley: the 1/𝑓3 PN corner is at the frequency offset of 5.7MHz. (c) Common-source cross-coupleddifferential pair: the 1/𝑓3 PN corner is at the frequency offset of 7.5MHz.

4. Analyses and Comparison versusOscillation Frequency

The investigations through the ISF can provide a betterunderstanding of the PN in each oscillator topology. In orderto be able to extract further useful considerations aboutthe devices and topologies, the previous analyses have beenreiterated also for other oscillation frequencies. In detail,

the three oscillator topologies have been implemented alsofor 1 and 100GHz operations, by keeping the quality factorof 10 for the LC tank and preserving the same power con-sumption of 6.3mW as in the case of the 10GHz oscillationfrequency. The transistor sizes were also kept the same as inthe previous case. As a consequence of the results reported inthe previous section, we injected noise current impulses withamplitude of 1𝜇A.


Table 4: Device sizing for oscillation frequencies of 1 and 100GHz.

Frequency (GHz) Transistor width (𝜇m) Capacitor value (pF) Inductor value (pH)𝑀1

𝑀2 𝑀

3𝑀4

𝐶1

𝐶2

𝐶3 𝐶

4𝐶5

𝐿1

𝐿2

1 30 30 30 15 10 105 5 10 2.5 5 × 10

32.5 × 10

3

100 30 30 30 15 0.0515 105 0.023 0.1 0.0057 50 25

Table 5: Summary of the PN results obtained by SpectreRF and ISF.

PN (dBc/Hz) @ 1MHz frequency offset

Topology 1GHz 100GHzSpectreRF ISF (1 𝜇A) SpectreRF ISF (1𝜇A)

Colpitts −115.31 −116 −77.06 −77.69Hartley −114.52 −114.85 −81.18 −81.38Cross coupled −123.7 −124.06 −74.78 −75.59

Table 4 reports the values of the individual circuit com-ponents for the topologies of Figure 1, used for the oscillationfrequencies of 1 and 100GHz.

Table 5 reports the PN values at a 1MHz offset predictedby the ISF along with the values obtained by means ofSpectreRF simulations for the oscillation frequencies of 1 and100GHz.

Figure 6 reports the relative contributions of𝑀1,𝑀2, and

LC tank to the overall PN versus the oscillation frequency forthe Colpitts topology, in both the flicker and thermal noisecontributions to PN.

Figures 7 and 8 report the results for the Hartley andcommon-source cross-coupled differential pair topologies,respectively.

4.1. Comparative Analysis between Devices. The relative con-tributions to the overall current flicker and thermal noisefromMOSFETs and LC tank of the three oscillator topologiesare summarized in Table 6, as well as the values of ΓDC andΓrms calculated for the 1𝜇A injected noise source.

These results stimulate some careful evaluations about thenoise contributions of each device in each oscillator topologyat different oscillation frequencies.

To do this, we could refer again to (7), (8), and (12) andconsider preliminarily that the amount of flicker or thermalnoise of the transistor in a certain region of operation doesnot determine exclusively the flicker or thermal noise contri-bution to the oscillator PN, as reported in [18]. In particular,we can observe that, for a given 𝑞max (9) and a given frequencyoffsetΔ𝜔 = 2𝜋×106, the amount of flicker noise contributionto PN is proportional to the product of the transistor flickernoise and Γ2DC:

L {Δ𝜔} ∝

𝑚

∑

𝑖=1

{Γ2

DC [(𝑖2𝑛

Δ𝑓)

𝜔1/𝑓

Δ𝜔]} , (13)

whereas the amount of thermal noise contribution to PNis proportional to the product of the thermal noise of thetransistor and LC tank and their respective Γ2rms:

L {Δ𝜔} ∝

𝑚+1

∑

𝑖=1

{Γ2

rms(𝑖2𝑛

Δ𝑓)} . (14)

In other words, the flicker noise is weighted by Γ2DC,whereas the thermal noise is weighted by Γ2rms, as mentionedin [18]. On the other hand, ΓDC and Γrms do not depend onthe device noise sources but on the node in which the noisecurrent is injected in a circuit topology.

Considering these aspects, it is worth highlighting thefollowing observations on the above results.

In the Colpitts topology, we observe from Figure 6(a) andTable 6 that, for oscillation frequencies higher than about70GHz, transistor 𝑀

1dominates the flicker noise contri-

bution to PN. However,𝑀2dominates at frequencies lower

than 70GHz, despite the fact that 𝑀2generates a lower

flicker noise than𝑀1. This is due to the fact that, according

to Table 6, the absolute value of ΓDC for 𝑀2is larger than

ΓDC for𝑀1at low frequencies of oscillation. In other terms,

this means that the oscillation waveform at the node (drainnode of 𝑀

2) into which the noise current is injected is

less symmetrical with respect to the rise and fall times [18].Regarding the thermal noise contribution to PN, shown inFigure 6(b), at oscillation frequencies above 20GHz,𝑀

1has

themajor PN contribution.However, below 20GHz,𝑀2has a

higher Γrms than𝑀1, as shown in Table 6. As a result, it takesa larger portion of the thermal noise contribution to PN atoscillation frequencies below 20GHz. This happens despitethe thermal noise contribution of𝑀

2is half that of𝑀

1.

As for the Hartley oscillator topology, we observe fromFigure 7(a) that, at oscillation frequencies between 3 and50GHz, transistor 𝑀

1dominates the flicker noise contri-

bution to PN. Nonetheless, in lower and higher oscillationfrequencies,𝑀

2dominates, despite its lower flicker noisewith

respect to 𝑀1as shown in Table 6, since its contribution

is characterized by a higher absolute value of ΓDC, as againshown in Table 6. In the thermal noise contribution to PNreported in Figure 7(b),𝑀

2presents the major contribution,

because, fromTable 6,𝑀2has a higher Γrms than𝑀1.Thereby,

it takes a larger portion of the thermal noise contribution toPN.This happens regardless of the fact that the thermal noisecontribution of𝑀

2is half that of𝑀

1, according to Table 6.

As for the common-source cross-coupled differentialpair oscillator topology, we see in Figure 8(a) that the pairof n-MOSFETs 𝑀

4, at frequencies lower than 50GHz, is

responsible for most of the flicker noise contributions to PN,


1 10 1000

102030405060708090

100110120130140

Flicker noise contribution to PN

Oscillation frequency (GHz)

Rela

tive c

ontr

ibut

ion

(%)

M1

M2

(a)

1 10 1000

102030405060708090

100110

Thermal noise contribution to PN

LC tank


M1

M2

Rela

tive c

ontr

ibut

ion

(%)

(b)

Figure 6: Relative contributions of𝑀1,𝑀2, and the LC tank for the Colpitts topology versus oscillation frequency @ 1MHz offset. (a) Flicker

noise contribution to PN. (b) Thermal noise contribution to PN.

1 10 1000

102030405060708090

100110120130



M1M2

Rela

tive c

ontr

ibut

ion

(%)

(a)

1 10 1000

102030405060708090

100110120


LC tank

Carrier frequency (GHz)

M1

M2

Rela

tive c

ontr

ibut

ion

(%)

(b)

Figure 7: Relative contributions of𝑀1,𝑀2, and the LC tank for the Hartley topology versus oscillation frequency @ 1MHz offset. (a) Flicker

noise contribution to PN. (b) Thermal noise contribution to PN.

as it not only generatesmore flicker noise but also has a higherabsolute value of ΓDC than 𝑀

3(see Table 6). After 50GHz,

the contribution of𝑀3increases due to its higher ΓDC value

and surpasses that of𝑀4, even though𝑀

4generates a higher

flicker noise. With respect to the behavior of the thermalnoise contribution to PN seen in Figure 8(b), the relative

contribution from the current source 𝑀3gradually drops

with increasing oscillation frequencies, whereas 𝑀4follows

an opposite trend.Moreover, from Figures 6(b) and 7(b), we note that, in the

Colpitts andHartley topologies, the LC tank occupies at loweroscillation frequencies a small portion of the contribution


1 10 100

0

10

20

30

40

50

60

70

80



M3M4−each

Rela

tive c

ontr

ibut

ion

(%)

(a)

1 10 1000

102030405060708090

100110120130


LC tank


M3M4−each

Rela

tive c

ontr

ibut

ion

(%)

(b)

Figure 8: Relative contributions of𝑀3,𝑀4, and the LC tank for the common-source cross-coupled differential pair topology versus frequency

of oscillation @ 1MHz offset. (a) Flicker noise contribution to PN. (b) Thermal noise contribution to PN.

Table 6: ΓDC, Γrms, and relative noise contributions for a 1 𝜇A injected noise source @ 1MHz offset.

ColpittsContribution tototal flicker noise

(%)

Contribution tototal thermal noise

(%)ΓDC Γrms

Oscillation freq. 1 GHz 10GHz 100GHz 1GHz 10GHz 100GHz𝑀1 76.88 67.18 3.8 × 10

−8−2.0 × 10

−85.4 × 10

−75.9 × 10

−75.9 × 10

−71.1 × 10

−6

𝑀2 23.1 32.27 −2.0 × 10

−7−2.0 × 10

−79.9 × 10

−81.1 × 10

−61.0 × 10

−67.9 × 10

−7

LC tank 0.55 2.6 × 10−6

1.8 × 10−6

6.9 × 10−6

HartleyContribution tototal flicker noise

(%)


(%)ΓDC Γrms

Oscillation freq. 1 GHz 10GHz 100GHz 1GHz 10GHz 100GHz𝑀1 76.88 67.18 −4.3 × 10

−91.9 × 10

−71.5 × 10

−76.1 × 10

−77.0 × 10

−75.4 × 10

−7

𝑀2 23.1 32.27 −3.0 × 10

−7−4.1 × 10

−9−3.0 × 10

−71.1 × 10

−61.2 × 10

−66.4 × 10

−7

LC tank 0.55 2.5 × 10−6

2.7 × 10−6

2.0 × 10−6

Cross coupledContribution tototal flicker noise

(%)


(%)ΓDC Γrms

Oscillation freq. 1 GHz 10GHz 100GHz 1GHz 10GHz 100GHz𝑀3 20.60 26.47 −3.3 × 10

−8−1.5 × 10

−83.3 × 10

−74.5 × 10

−74.1 × 10

−76.7 × 10

−7

𝑀4 39.7-each 30.70-each 4.6 × 10

−8−1.5 × 10

−72.0 × 10

−72.7 × 10

−73.7 × 10

−76.7 × 10

−7

LC tank 12.13 1.8 × 10−6

1.8 × 10−6

2.9 × 10−6

of thermal noise to PN graph, because, as Table 6 indicates,the thermal noise generated by the LC tank is at least oneorder of magnitude below the thermal noise generated bythe transistors in each case. However, both in Colpitts andHartley, the contribution of the LC tank increases at higher

oscillation frequencies, where Table 6 indicates that Γrms ofthe tank is notably larger than Γrms of both devices.

On the other hand, from Figures 6(b), 7(b), and 8(b), wenote that, in all three oscillator topologies, the relative contri-bution of the current sources𝑀

2and𝑀

3to the thermal noise


1 10 100

ColpittsHartleyCross-coupled diferential


1 × 10−31

1 × 10−32∑m i=1[Γ

DC

2(in2/Δ

f) i(

/Δ𝜔)]

(A2/H

z)(𝜔

1/f) i

Figure 9: Sum of Γ2DC[(𝑖2

𝑛/Δ𝑓)(𝜔

1/𝑓/Δ𝜔)] for all flicker noise sources

in each oscillator topology @ 1MHz offset versus oscillation fre-quency for Colpitts, Hartley, and common-source cross-coupleddifferential pair topologies.

contribution to PN drops at higher oscillation frequencies.According to Table 6, this is due to the reduction of the Γrmsfor the current sources relative to the Γrms values for the otheroscillator components.

4.2. Comparative Analysis between Topologies. By using thevalues in Table 6 along with (13) and (14), we can determinethe flicker and thermal noise contributions obtained by theISF for a 1 𝜇A injected current source, as reported in Table 7.

In order to provide them in a more intuitive form, theresults in Table 7 are plotted in Figures 9 and 10.

As in the previous section, by considering a given 𝑞max(9) and a given frequency offset Δ𝜔 = 2𝜋 × 106, (13) and (14)can be used in order to compare the flicker and thermal noisecontributions, respectively, to the PN spectrum of variousoscillator topologies. In this perspective, Figures 9 and 10show the variation of the flicker and thermal noise contribu-tions to PN, respectively, for the three oscillator topologiesunder investigation with respect to changes in the oscillationfrequency, at a frequency offset of 1MHz.

5. Topology Performances versus OscillationFrequency Regions

In the previous section, we reported the results of the effectiveISF for every active device of the three oscillator topologiesaccording to (11). Here we try to explain the different PNbehavior achieved for the three oscillator topologies over thefrequency range from 1 to 100GHz.The results of the previoussection suggest considering additional oscillation frequen-cies. For this reason, the three topologies have been designedalso for the additional oscillation frequencies of 30, 50, and70GHz, according to the same criteria of Sections 2 and 4.

1 10 100

ColpittsHartleyCross-coupled differential


1 × 10−32

1 × 10−33

(in2/Δ

f) i

Γ2 rm

s(A

2/H

z)∑

m+1

i=1[

]

Figure 10: Sum of Γ2rms(𝑖2

𝑛/Δ𝑓) for all thermal noise sources in each

oscillator topology @ 1MHz offset versus oscillation frequency forColpitts, Hartley, and common-source cross-coupled differentialpair topologies.

0 10 20 30 40 50 60 70 80 90 100

ColpittsHartleyCross-coupled differential


−130

−120

−110

−100

−90

−80

−70

−60

PN (d

Bc/H

z)

Figure 11: PN at a 1MHz frequency offset from carrier versusoscillation frequency for Colpitts, Hartley, and common-sourcecross-coupled differential pair topologies by SpectreRF.

Table 8 reports the values of the circuit components foreach topology for the oscillation frequencies of 30, 50, and70GHz.

Figure 11 reports the PN results obtained by SpectreRF for1, 10, 30, 50, 70, and 100GHz at a 1MHz frequency offset fromthe carrier.

These results allow us to identify the following four mainfrequency regions: 1–20, 20–30, 30–80, and 80–100GHz.They offer the opportunity to carry out further comparativeanalyses and derive a number of observations.


Table 7: Noise contributions @ 1M𝐻z frequency offset for a 1 𝜇A injected noise current.

Colpitts Γ2

DC [(𝑖2𝑛

Δ𝑓)

𝜔1/𝑓

Δ𝜔] Γ

2

rms (𝑖2𝑛

Δ𝑓)

Oscillation freq. 1 GHz 10GHz 100GHz 1GHz 10GHz 100GHz𝑀1

6.8 × 10−33

2.0 × 10−33

4.2 × 10−32

2.2 × 10−33

2.3 × 10−33

3.3 × 10−33

𝑀2

6.4 × 10−32

6.7 × 10−32

3.0 × 10−32

3.6 × 10−33

3.2 × 10−33

9.1 × 10−34

LC tank 1.6 × 10−34

1.6 × 10−34

1.3 × 10−33

Total (Σ) 7.1 × 10−32

6.9 × 10−32

7.2 × 10−32

6.0 × 10−33

5.6 × 10−33

5.5 × 10−33

Hartley Γ2

DC [(𝑖2𝑛

Δ𝑓)

𝜔1/𝑓

Δ𝜔] Γ

2

rms (𝑖2𝑛

Δ𝑓)


6.2 × 10−35

1.7 × 10−31

4.4 × 10−33

2.4 × 10−33

3.1 × 10−33

1.1 × 10−33

𝑀2

9.8 × 10−32

2.7 × 10−35

1.3 × 10−32

4.2 × 10−33

4.2 × 10−33

1.2 × 10−33

LC tank 1.3 × 10−34

1.3 × 10−34

2.1 × 10−34

Total (Σ) 9.8 × 10−32

1.7 × 10−31

1.8 × 10−32

6.8 × 10−33

7.4 × 10−33

2.5 × 10−33

Cross coupled Γ2

DC [(𝑖2𝑛

Δ𝑓)

𝜔1/𝑓

Δ𝜔] Γ

2

rms (𝑖2𝑛

Δ𝑓)


3.4 × 10−33

1.3 × 10−34

4.2 × 10−32

7.7 × 10−34

5.1 × 10−34

1.7 × 10−33

𝑀4

6.7 × 10−33

1.3 × 10−32

4.8 × 10−32

1.4 × 10−34

2.0 × 10−34

8.9 × 10−34

LC tank 1.7 × 10−34

6.5 × 10−35

4.3 × 10−34

Total (Σ) 1.0 × 10−32

1.3 × 10−32

9.0 × 10−32

1.1 × 10−33

7.8 × 10−34

3.0 × 10−33

Table 8: Device sizing for oscillation frequencies of 30, 50, and 70GHz.

Frequency (GHz) Transistor width (𝜇m) Capacitor value (pF) Inductor value (pH)𝑀1

𝑀2 𝑀

3𝑀4

𝐶1

𝐶2

𝐶3

𝐶4

𝐶5

𝐿1

𝐿2

30 30 30 30 15 0.286 105 0.1385 0.4 0.0627 166.7 83.35

50 30 30 30 15 0.15 105 0.071 0.25 0.0297 100 50

70 30 30 30 15 0.0928 105 0.0439 0.15 0.0158 71.4 35.7

Comparing the total (i.e., sum) contributions of the flickerand thermal noise sources in Table 7 and Figures 9 and 10, wecan note that the flicker noise contribution dominates at thefrequency offset of 1MHz at the oscillation frequencies of 1,10, and 100GHz.We can also note that term (13) determiningthe flicker noise contribution derived from ISF, as in Figure 9,shows an agreement with the PN derived by SpectreRF-Cadence, as in Figure 11.

This is because, as already mentioned in Section 4, from(7), (8), and (12), it can be concluded that the flicker noisecontribution to PN is defined by the product of the transistorflicker noise with ΓDC

2, as expressed in (13) and quantified inTable 7.

Region 1 (1–20GHz). According to Figure 11, the common-source cross-coupled differential pair topology exhibits thelowest PN with respect to the other two topologies. Thehighest PN is exhibited by the Hartley topology. This is inagreement with the trend reported in Figure 9. Delving intothe separate noise sources as addressed in Section 4 andshown in Figures 6(a), 7(a), and 8(a), the nodes mostly proneto the current noise injection are the drain of 𝑀

2in the

Colpitts topology, the drain of 𝑀2from 1 to 3GHz and

the drain of𝑀1from 3 to 20GHz in theHartley topology, and

the drain of both𝑀3and𝑀

4in the common-source cross-

coupled differential pair topology.

Region 2 (20–30GHz). In this region, we note from Figure 11that the common-source cross-coupled differential pairtopology still maintains the best PN performance, but, unlikethe above case, we can observe an inversion between theHartley and Colpitts topologies. The latter exhibits the worstPN at 30GHz. Figure 9 follows approximately the sameresults. From Figures 6(a), 7(a), and 8(a), we can see that thenodes mostly sensitive to noise injections are the drain of𝑀2in the Colpitts topology, the drain of𝑀

1in the Hartley

topology, and the drain of𝑀4in the common-source cross-

coupled differential pair topology.

Region 3 (30–80GHz). In Figure 11, we register an inversionfor the best PN performance, given now by the Hartleytopology, whereas the Colpitts topology still exhibits theworst PN as in the previous case. A similar behavior isexhibited in Figure 9. In this region the nodesmostly sensitiveto noise injection according to Figures 6(a), 7(a), and 8(a)are: the drain of 𝑀

2in the Colpitts topology, the drain of


𝑀1up to 50GHz and the drain of𝑀

2at higher frequencies

in the Hartley topology, and the drain of both 𝑀3and 𝑀

4

until 50GHz and of𝑀3above 50GHz in the common-source

cross-coupled differential pair topology.

Region 4 (80–100GHz). Figure 11 indicates that Hartley con-tinues to exhibit the lowest PN. However, with respect tothe previous case, here we can observe an inversion of per-formance between the Colpitts topology and the common-source cross-coupled differential pair topology, which nowexhibits the highest PN. We can also derive the same con-clusions from Figure 9.The operation in the triode region forsome parts of the oscillation period is themain reason for thisnoise performance degradation at the highest frequencies inthe common-source cross-coupled differential pair topologyaccording to the notes in [26]. Indeed, our design operatesin the voltage-limited regime, thus causing the active devicesto enter in the triode region at the peaks of the differentialoutput node voltage. We notice from Figures 6(a), 7(a), and8(a) that the most sensitive nodes in this frequency range arethe drain of𝑀

1in the Colpitts topology, the drain of𝑀

2in

the Hartley topology; and the drain of𝑀3in the common-

source cross-coupled differential pair topology.At least up to a 1MHz frequency offset from the carrier,

the flicker noise contribution is dominant according toFigures 5(a)–5(c) and Table 7. Therefore, the proportionalincrease of the flicker noise contribution to PN due to𝑀3at the highest oscillation frequencies in the common-

source cross-coupled differential pair topology, as observedin Table 6 and Figure 8(a), is the main cause of the overall PNincrease. Actually, this is an effect of the losses through thep-MOSFET tail current source that become part of the tankcircuit, thus impairing its 𝑄 [27, 28]. Note that the superiorPN performance of the Hartley topology at high frequenciesnoted in Regions 3 and 4 is in agreement with the observa-tions in [26, 29].

6. Conclusions

PN comparative analyses have been carried out for Colpitts,Hartley, and common-source cross-coupled differential pairLC oscillator topologies in the frequency range from 1 to100GHz. The circuit topologies have been implemented in28 nmbulkCMOS technology for operation at 1, 10, 30, 50, 70,and 100GHz, maintaining equal power consumption, qualityfactor, and transistor sizes for a fair comparison among allthe circuit topologies. All the steps and settings for accurateevaluations of the impulse sensitivity function have beendiscussed and clarified in depth. PN performances have alsobeen evaluated directly through periodic steady-state sim-ulations in the SpectreRF-Cadence environment. These lastresults have been comparedwith the results obtained throughthe ISF for a wide set of amplitudes of injected current pulses.The PN predicted by the ISF is in good agreement with theresults obtained by SpectreRF under the given simulationsettings, especially for the pulse amplitude of 1𝜇A.

Moreover, the investigations on the PN contributionsfrom each component of the investigated oscillator circuittopologies have been reported and discussed in detail.

The results show that, under the adopted design conditions,the three oscillator topologies rank unevenly in terms of thebest PN performance rating scale for oscillation frequenciesfrom 1 to 100GHz. This comes as a result of the frequencydependence of both contributions from each circuit com-ponent and the sensitivity to noise injections in the circuitnodes. Recent studies refer to the common-source cross-coupled differential pair topology as the one with the best PNas a consequence of the circuit designs carried out at lowerfrequencies. Our comparative analyses reported here showthat there is no superior topology in the absolute sense,but that the identification of the best circuit topology withrespect to PN is strictly related to the operating frequencyrange. Nowadays, the most popular topology used is thecommon-source cross-coupled differential mainly due to itsreliable startup. However, the results presented here, suggestthe opportunity to invest additional studies and efforts inexploring the circuit design implementations also of othertopologies. The potential of the latter may have been perhapsunderestimated until today, especially at very high frequen-cies. Nowadays, thanks to the recent advances in the nano-scale technology process,MOSFETswith cut-off andmax fre-quencies in excess of 280 and 350GHz [30], respectively, areavailable. Their potential use involves a number of emergingwireless applications in themillimeter-wave frequency range.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work is supported by the Science Foundation Ireland(SFI) and the Higher Education Authority (HEA).

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