A Simplified Method for

Phase Noise Calculation

Massoud Tohidian,

Ali Fotowat Ahmady*

and Mahmoud Kamarei

University of Tehran, *Sharif University of Technology, Tehran, Iran

Poster: T-18

Outline

• Introduction

• Preliminary Assumptions

– Noise Approximation

– Noise Frequency Translation

– PM and AM Noise Separation

– Output Phase Noise Spectra

• Phase Noise Calculation Method

– The Proposed Method

– Comparison with Hajimiri’s ISF Method

• Experimental Results

• Conclusion

2

Introduction

• Leeson’s Equation [1]

– A heuristic and historical model for phase noise of LC oscillators

• Hajimiri’s ISF Method [2]

– General

– Accurate

– Impulse Sensitivity Function (ISF)

• Transient-time simulation

m

mQV

kTRFL

2

4 0

2

3

Noise Approximation

• A noise signal is approximated by a series of several

sinusoidal signals [3].

4

No

ise

So

urc

e

Sp

ectr

a

ω

. . .

ω

No

ise

Ap

pro

xim

atio

n

. . .

Δω

Random Phase

Equal Power

Noise Frequency Translation

tωωAAα

tωωAAα

tωAαY

nn

nn

nnn

)2cos(3

)cos(2

cos

0

2

03

002

1

3

3

2

21 XαXαXαY

)cos(

)cos( 00

tA

tAX

nn

• Consider a general nonlinear system

– Input: carrier + noise

– Output: translated noise (neglecting the carrier and small terms)

5

“Prominent” Frequencies

• Only “prominent” frequencies translate to around the carrier.

ωn0 ωn1+ωn1- ωn2+ωn2- ωni+ωni-

Carrier

@ ω0

Ou

tpu

t

Sp

ectr

a

No

ise

So

urc

e

Sp

ectr

a

ω

ω

. . .

ωn0 + ω0

2ω0 - ωn1-

ωn1+

3ω0 - ωn2-

ωn2+ - ω0

6

PM and AM Noise Separation

• A single tone noise at an prominent frequency generates

two noise tones around the carrier.

• These two carriers generally modulate both Phase (PM)

and Amplitude (AM) of the carrier.

+ωs

-ωs

V0

AM

PM

ωω0 ω0+ωs

nnA

nnA

V0

ω0-ωs

Output Spectra Phasor Representation

7

PM and AM Noise Separation

• The output noise can be separated into PM and AM noise.

nnnnnnAM

nnnnnnPM

AAAAA

AAAAA

cos22

1

cos22

1

22

22

-ωs+ωs

V0PM

-ωs

+ωs

V0

AM

Pure PM Noise

Pure AM Noise

8

Output Phase Noise Spectra

• Leeson’s equation:

– Phase noise skirt around the carrier signal falls with 20dB/dec for

a white noise source [1].

• General phase noise equation:

22

22

2log10log10sm

nn

sig

noisem

V

iZ

P

PL

carrier

amplitude

total noise

translation gain noise current

frequency at

which Zn is

simulated

phase noise

offset frequency

9

Noise Translation Gain

2

3

2

2

2

1

2

0

2

3

2

3

2

2

2

2

2

1

2

1

2

0

2

222 nnnn

nnnnnnnn

zzzz

zzzzzzzZ

• The translation gains are calculable using simulation

sniniiniPMni

sniniiniPMni

iiVz

iiVz

0,,

0,,

,/

,/

10

ωn0 ωn1+ωn1- ωn2+ωn2- ωni+ωni-

Carrier

@ ω0

Noise @

ω0+ωs

Noise @

ω0-ωs

Ou

tpu

t

Sp

ectr

a

No

ise

So

urc

e

Sp

ectr

a

ω

ω

zn0zn1- zn1+ zn2-

zn2+zni- zni+

. . .

Noise Sources

• Stationary Noise Sources

– Resistors, constant current source, etc.

– Noise source is modeled with a single

tone (ST) source.

• Cyclostationary Noise Sources

– Modulated sources, switching

transistors, etc.

– Noise source is modeled with a single

tone source with amplitude modulation

in @

ωni

iout=f(id,in)

in @

ωni

id

11

Proposed ST Simulation Method

• Calculate the noise translation gain

– Applying the ST noise source at prominent frequencies, simulating

the circuit and measuring the output spectrum around the carrier.

– PM component:

– Partial translation gain:

– Total noise translation gain of the noise source:

• The total output phase noise contribution of the noise

source

2

3

2

2

2

1

2

0

2

3

2

3

2

2

2

2

2

1

2

1

2

0

2

222 nnnn

nnnnnnnn

zzzz

zzzzzzzZ

nnnnnnPM AAAAA cos22

1 22

22

22

2log10sm

nnm

V

iZL

niiniPMni iVz ,, /

12

Comparison with ISF Method

• Linear Time-Variant Model [2]

• It can be shown that the noise translation gains are

convertible to ISF and vice versa.

max

max

00

2

2

q

cVz

q

cVz

ini

n

ISF Fourier series

coefficients

22

max

22

0

1

00

max

0

log10

cos2

,

m

inm

i

i

i

q

ciL

icc

tuq

th

ISF

max. charge of

the node

13

)(ti

0 t

),( th)(t)(ti

)(t

0 t

ST vs. ISF

• ST Method

– Directly calculates noise contribution in frequency domain.

– The dominant noise sources and noise frequencies are found.

– Separately calculates the Phase and Amplitude Noise.

• Simulation Speed

– Accurate ISF requires several transient time simulations (time

consuming).

– ST method has improved the simulation speed using Harmonic-

Balanced simulation (HB) for just some few prominent frequencies.

• Flicker noise upconversion can be predicted by just one simulation

(considering zn0).

14

Experimental Results

• Case Study:

– QVCO designed for GPS application [4].

VDD VDD

Msw1 Msw2 Msw3 Msw4Mcp1 Mcp2 Mcp3 Mcp4

MtailMbias

Ibias

C1 C2

L1 L2

R1 R2 C3 C4

L3 L4

R3 R4

Ip In Qp Qn

Qp Qn In Ip

Cbp

15

Experimental Results

Die micrograph fabricated in

TSMC 1P6M+ 0.18-µm CMOSMeasured output phase noise spectra

16

• Simulated phase noise frequency contribution @ 1MHz

– The two methods almost predict the same results

• Time-domain ISF for Mtail

– DC + Fourth Harmonic

Phase Noise Frequency Contribution

0 1 2 3 4-160

-150

-140

-130

-120

-110

Noise Source Frequency (carrier hamonic index)

Outp

ut

Phase N

ois

e (

dB

c/H

z)

Msw1

Mcp1

Mtail

Mbias

R1

0 1 2 3 4-160

-150

-140

-130

-120

-110

Noise Source Frequency (carrier hamonic index)

Outp

ut

Phase N

ois

e (

dB

c/H

z)

Msw1

Mcp1

Mtail

Mbias

R1

ST

meth

od

ISF

meth

od0 pi/2 pi 1.5*pi 2*pi

0

0.2

0.4

0.6

0.8

1

Output Phase (rad)

Norm

aliz

ed I

SF

17

• Noise contributions are in good agreement for all

simulations methods and measurement.

Comparison

Noise Source Noise TypeMethod

Single Tone ISF SpectreRF

Msw1-4

Thermal -135.0 -135.0 -135.6

Flicker -141.8 -143.8 -148.2

Mcp1-4

Thermal -144.9 -145.8 -144.4

Flicker -131.8 -131.8 -134.9

Mtail

Thermal -138.8 -137.9 -139.3

Flicker -138.7 -137.6 -138.1

Mbias

Thermal -115.1 -115.3 -116.0

Flicker -114.2 -113.9 -114.7

R1-4 -136.8 -136.4 -137.2

Other Sources neglected neglected -128.2

Total -111.2 -111.1 -111.8

Measurement Result -109.5

Phase Noise @ 1MHz offset

18

Conclusion

• A faster method using nonlinear frequency domain

simulation

• ST directly shows the noise frequency translation

– Facilitates more intuitive design of high purity oscillators by

considering substantial noise frequencies

• Separate prediction of Phase and Amplitude Noise

• Simulation results show good agreement with other

methods and measurements

19

References

[1] D. B. Leeson, “A simple model of feedback oscillator

noise spectrum,” Proc. IEEE, pp. 329–330, Feb. 1966.

[2] A. Hajimiri and T. H. Lee, “A general theory of phase

noise in electrical oscillators,” IEEE J. Solid-State

Circuits, vol. 33, pp. 179-94, 1998.

[3] W.P. Robins, Phase Noise in Signal Sources, Peter

Peregrinus Ltd., London, 1982, pp. 6-8.

[4] F. Behbahani, et al., "A fully integrated low-IF CMOS

GPS radio with on-chip analog image rejection," IEEE J.

Solid-State Circuits, vol.37, no.12, pp. 1721-1727, Dec

2002.

20