enhancing the phase-noise-figure-of-merit of a resonator using … · 2020-01-21 · enhancing the...

Enhancing the Phase-Noise-Figure-of-Merit of aResonator using Frequency Transformations

Sreeni Poolakkal and Nagarjuna Nallam

Department of EEE,Indian Institute of Technology Guwahati,

Assam - 781039, India.

Email: [email protected]

Enhancing the PNFOM of a Resonator using Freq. Trans. VLSID 2020

Outline

Motivation

Phase Noise in Feedback OscillatorsReview of Two-Port OscillatorsSingle-Port LC -Oscillators

Enhancing the PNFOM of ResonatorsFrequency TransformationsEffect on the Tank Performance

Oscillators with Fourth-Order TanksDesign ExampleSimulated Results

Summary


Motivation

L1

C1

L2

C2

L3

C3 L4

L5

C5 L6

C6

Which of these tanks offer lowest phase noise when used in thedesign of an oscillator at ω0?


Phase Noise in Feedback Oscillators


Leeson’s Phase Noise Model [1]

Amp.

BPF|S21| = L

PoutPin

(a) (b)

∆ω

Sφ(∆ω)

vout = A cos(ω0t + φ(t))ω1/f

L(∆ω) ≈[

2LFkT

Pout

(1 +

ω20

4Q2∆ω2+

ω20ω1/f

4Q2∆ω3

)],

[1]. D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” in Proceedings of the IEEE, vol. 54, no.2, pp. 329-330, Feb. 1966.


Open-Loop Quality Factor (Q)

Q =ω0

2| ddω

ln(Ψ)|,

where Ψ is an immittance function that depends on the activedevice model (CCCS/VCCS) and the oscillator topology [2].

A simplified expression for Q is

Q ≈ ω0

2|τd |,

where τd is the group delay of the open-loop system.

[2] T. Ohira, “Rigorous Q-factor formulation for one- and two-port passive linear networks from an oscillator noisespectrum viewpoint,” in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 52, no. 12, pp. 846-850,Dec. 2005.


Phase-Noise-Figure-Of-Merit (PNFOM) of BPF

PN expression in terms of group delay and loss is

L(∆ω) ≈[

2LFkT

Pout

(1 +

1

τ2d∆ω2

+ω1/f

τ2d∆ω3

)]PNFOM of a BPF is given below [3].

(PNFOM)bpf = 10log( L

τ2d

)

A BPF with low (PNFOM)bpf is preferred in oscillators.

[3] J. Choi and A. Mortazawi, ”A New X -Band Low Phase-Noise Multiple-Device Oscillator Based on theExtended-Resonance Technique,” in IEEE Transactions on Microwave Theory and Techniques, vol. 55, no. 8,pp. 1642-1648, Aug. 2007.


A Single-Port LC -Oscillator

Lp

Cp −Ra

Activet

Vp

Z (jω0) = Rp = QLω0Lp

X

VX

Rs

Assumptions:

1. Voltage-Limited [4]

2. Inductor is the only

lossy element

L(∆ω) ≈[

2FkT (2Rp)

V 2p

(1 +

1

τ2d∆ω2

+ω1/f

τ2d∆ω3

)]

[4] C. Samori, “Understanding Phase Noise in LC VCOs: A Key Problem in RF Integrated Circuits,” in IEEESolid-State Circuits Magazine, vol. 8, no. 4, pp. 81-91, Fall 2016.


PNFOM of a Resonator

(PNFOM)res = 10log

(Rp

τ2d

),

where Rp is the equivalent resistance across the tank at ω0 and τdis the group delay at ω0.

τd =d∠Zdω

∣∣∣ω=ω0

, where ∠Z is the phase of the tank impedance.

For a second-order tank:

τd ≈ − 2QL

ω0or − 2Lp

Rs; (PNFOM)2res = 10log

(ω2

0Rs

4

),

where QL is the quality factor of the inductor Lp.


Lowest Achievable (PNFOM)2res in CMOS 65 nm

0 1 2 3 4 5 66

8

10

12

14

16

18

200

202

204

206

208

210

212

Inductance (nH)

Qmax

PN

FO

M(d

B)

@ 3.6 GHz

An Estimate of PNFOM from QL (only).


Enhancing the PNFOM of Resonators


Frequency Transformations

I Mathematically: Simple variable substitution. One-to-one andone-to-many mappings are possible.

I Physically: Not all transformations are realizable.

I Historically: Used in the design of bandpass and bandstopfilters.

I Recently: Proved to be useful in the design of concurrentmulti-band filters, matching networks, and amplifiers [5].

LP-BP Transformation:

ω = f (ωt) =ω2t − ω2

c

ωt,

where ω is the original frequency, ωt is the transformed frequency,and ωc is a mathematical construct.

[5] N. Nallam and S. Chatterjee, Multi-Band Frequency Transformations, Matching Networks and Amplifiers, IEEETrans. Circuits Syst. I, vol. 60, no. 6, pp. 1635-1647, June 2013.


Applying an LP-BP Transform on an LC -Tank

Lp

Cp −R

Active

Z (jω0) = Rp = Q0ω0Lp

X

Rs

Lp

Ct

−R

Active

Z (jω1) = Z (jω2) = Rp = Q0ω0Lp

X

Rs

CpLt

ω1 ω2ω0Re

(Z)

Rp

Freq. (rad/s)ω0

Re

(Z)

Rp

(a) A second-order tank and its impedance variation

Freq. (rad/s)

(b) Impedance variation of the (fourth-order) tank after the LP-BP transformation

LP

-BP

Tra

nsf

orm

atio

n

Eff

ect

ofL

P-B

PT

ran

sfor

mat

ion


Effect on the Group Delay

Before the transformation: (τd)0 = d∠Zdω

∣∣∣ω=ω0

After the transformation:(τd)i = d∠Z

dω ×dωdωt

∣∣∣ωt=ωi

= (τd)0 × f ′(ωt)∣∣∣ωt=ωi

, i = 1, 2

LP-BP Transformation:

(τd)1 = (τd)0 × (1 +ω2

ω1) and

(τd)2 = (τd)0 × (1 +ω1

ω2),

Assumption:The other inductor (Lt) is lossless.


A Comparison of Tank Parameters

ω1 ω2ω0

ω20L

2p

Rs

ω21L

2p

Rs

ω22L

2p

Rs

Freq. (rad/s)

Re(Z )

ω1 ω2ω0Freq. (rad/s)

2nd order tank

4th order tank (LP-BP)|τd |

|(τd)0|

|(τd)0|(1 + ω1ω2

)

|(τd)0|(1 + ω2ω1

)

Before Trans.

Impedance Group Delay

At ωi , i= 1,2

Rp τd PNFOM

2ndω2i L

2p

Rs

2LPRs

10log(ω2i Rs

4)

4thω2

0L2p

Rs

2LPRs

(ω2+ω1ωi

) 10log(ω2i Rs

4)− 20log( ω2+ω1

ω2−ω1)

20log( ω2+ω1ω2−ω1

) is the improvement in (PNFOM)4res.


Oscillators with Fourth-Order Tanks


One- and Two-Port Oscillators with Fourth-Order Tanks

Vdd Vdd

2LtCp/2

2LpCt Ct

2LtCp/2

Ct 2Lp Ct

Vb

One-port oscillator Two-port oscillator with voltage biasing


A Design Example in CMOS 65 nm

0.8V

1.33nH4.4pF

11.73pF 1nH 11.73pF

0.6V

96µm0.065µm

96µm0.065µm

1nH

4.4pF1.33nH

4.4pF

11.73pF 1nH 11.73pF

LP-BP

1©: Tank at ω0=2.4 GHz, small L/Q 2©: Transformed Tank, ω2 = 3.6 GHz

3©: Add transistors, Vb < Vdd4©: Layout

ωc = 2.08 GHz


Simulated Tank Performance

Frequency (GHz)

Rp

(Ω)

1 1.5 2 2.5 3 3.5 40

50

100

150

200

250

300

1 1.5 2 2.5 3 3.5 4-2

-1.5

-1

-0.5

0

0.5

1

1.5

Frequency (GHz)

τ d(n

s)

2nd order tank at ω2


4th order tank



4th order tank

At 3.6 GHz:

(PNFOM)2res ≈ 204.3 dB

(PNFOM)4res ≈ 199.5 dB XTheoretically predicted improvement: 20log( 3.6+1.2

3.6−1.2 ) ≈ 6 dB


Simulated Phase Noise and FOM

3.5 3.6 3.7 3.8194

195

196

197

198

199

200100k1M3M

FO

M

Frequency (GHz)

-110.7

-132.5-142.5

Frequency offset (Hz)

Ph

ase

Noi

se(d

Bc/

Hz) 3.46 GHz-3.77 GHz

-180

-160

-140

-120

-100

-80

-60

10k 100k 1M 10M 100M

1/f corner frequency ≈ 150 kHz


Comparison with state-of-the-art oscillators

Reference JSSC’17 JSSC’13 ISSCC’01 JSSC’08 ISSCC’12 ISSCC’12 This Work†

Technology (nm) 28 65 350 130 55 65 65

Supply voltage (V) 0.7 1.25 2.5 1 1.5 1.2 0.8

Frequency (GHz) 3.7 3.7 1.2 5.2 3.35 3.92 3.6

Tuning range (%) 27.2 25 18 14 31.4 10.2 8.6

Power (mW) 6.6 15 9.1 1.4 27 25.2 3

Core Area (mm2) 0.15 0.12 NA < 0.11** 0.49 0.19 0.145

at 100k −108.0 −106.6 −109* −100.0* −103.0* NA −110.7

PN (dBc/Hz) at 1M −131.0* −131.5* −140* −122.0* −130* −130.0* −132.5at 3M −139.7 −142.2 −153.2 −141.2 −142 −141.7 −142.5

at 100k 188 186.2* 185* 192.85* 179.18 NA 197.05FOM‡ (dB) at 1M 192.5 192.2 195 195 189 189.9 199.31

at 3M 192.5 192.2 195 195 189 189.9 199.31

Oscillator Common Class Noise Class Class Clip and Class-C withStructure Mode F Filtering C B/C Restore 4th-Order Tank

† Simulation results only, * Estimated from plots, ** Approximated from the die area provided, NA - Not available‡FOM = |PN|+ 20log10(f0/∆f )− 10log10(PDC/1mW )

Good resonator + Class-C biasing - (Current) Source of flicker noise= Best Figure-of-Merit (FOM) at 100 kHz


Summary

I Proposed a PNFOM to compare the quality of resonators

I Frequency transformations can be used to enhance thePNFOM of resonators

I An LP-BP transform improves the PNFOM by20log(ω2+ω1

ω2−ω1) dB

I Presented the design of a fourth-order tank based two-portoscillator with class-C voltage biasing

I The oscillator achieved an FOM of 197 dB at 100 kHz insimulations


References

[1] D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proceedings of the IEEE, vol. 54, no. 2,pp. 329–330, Feb 1966.[2] T. Ohira, “Rigorous Q-factor formulation for one- and two-port passive linear networks from an oscillator noisespectrum viewpoint,” IEEE Trans. Circuits Syst. II, vol. 52, no. 12, pp. 846–850, Dec 2005.[3] J. Choi, M. Nick, and A. Mortazawi, “Low Phase-Noise Planar Oscillators Employing Elliptic-ResponseBandpass Filters,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 8, pp. 1959–1965, Aug 2009.[4] C. Samori, “Understanding Phase Noise in LC VCOs: A Key Problem in RF Integrated Circuits,” in IEEESolid-State Circuits Magazine, vol. 8, no. 4, pp. 81-91, Fall 2016.[5] N. Nallam and S. Chatterjee, “Multi-Band Frequency Transformations, Matching Networks and Amplifiers,”IEEE Trans. Circuits Syst. I, vol. 60, no. 6, pp. 1635–1647, June 2013.[6] D. Murphy, H. Darabi, and H. Wu, “Implicit Common-Mode Resonance in LC Oscillators,” IEEE J. Solid-StateCircuits, vol. 52, no. 3, pp. 812–821, March 2017.[7] E. Hegazi, H. Sjoland, and A. Abidi, “A filtering technique to lower oscillator phase noise,” IEEE Int.Solid-State Circuits Conf. Dig. Tech. Papers (ISSCC), pp. 364–365, Feb 2001.[8] A. Mazzanti and P. Andreani, “Class-C harmonic CMOS VCOs, with a general result on phase noise,” IEEE J.Solid-State Circuits, vol. 43, no. 12, pp. 2716–2729, Dec 2008.[9] L. Fanori, A. Liscidini, and P. Andreani, “A 6.7-to-9.2GHz 55nm CMOS hybrid Class-B/Class-C cellular TXVCO,” IEEE Int. Solid-State Circuits Conf. Dig. Tech. Papers (ISSCC), pp. 354–356, Feb 2012.[10] A. Goel and H. Hashemi, “Frequency Switching in Dual-Resonance Oscillators,” IEEE J. Solid-State Circuits,vol. 42, no. 3, pp. 571–582, March 2007.


Thank You

Contact: [email protected]


enhancing the phase-noise-figure-of-merit of a resonator using … · 2020-01-21 · enhancing the...

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