rf module design - [chapter 7] voltage-controlled oscillator

RF Transceiver Module DesignChapter 7

Voltage-Controlled Oscillator李健榮助理教授

Department of Electronic EngineeringNational Taipei University of Technology

Outline

• Resonator

• Feedback Loop Analysis

• Amplifier Configurations

• Capacitor Ration with Copitts Oscillators

• Phase Noise and Lesson’s Model

• Summary

Department of Electronic Engineering, NTUT2/43

Introduction

• An oscillator is a circuit that generates a periodic waveform.

• Oscillators are used with applications in which a referencetone is required. In most RF applications, sinusoidal referenceswith a high degree of spectral purity (lowphase noise) arerequired.

90

( )I t

cos ctω

( )Q t

Low Noise Amplifier(LNA)

Bas

eban

d

Pro

cess

orLPF

LPF

90

( )I t

cos ctω

( )Q t

( )ms t

Power Amplifier(PA)

Antenna

Bas

eb

and

P

roce

sso

r


LC Resonator

• An LC resonator determines the oscillation frequency andoften forms part of the feedback mechanism.

If i (t) = Ipulsed (is applied to the parallelresonator, the system time response:

( )2

2 2

2 1 1cos

4

t

RCpulse

out

I ev t t

C LC R C

−

= − ⋅

2 2

1 1

4osc LC R Cω = − 1

osc LCω =

L C R

( )outv t

( )i t

Time

Am

plit

ud

e

R→ ∞

( ) ( )pulsedi t I t=


Adding Negative Resistance Through Feedback

• In any practical circuit, oscillations will die away unlessfeedback is added to generate a negative resistance in order tosustain the oscillation.

L C pRnR−

L C sr

nr−

feedback active device

Parallel RLC Resonator Series RLC Resonator

feedback active device


Feedback System

• The oscillator can be seen as a linear feedback system.• The gain of the system:

• Barkhausen’s criterion:For sustained oscillation at constant amplitude, the poles must be on the jω axis

which states that theopen-loop gain around the loop is 1 and the phase around theloop is 0 or some multiple of 2π.

• To find the poles of the closed-loop system, one can equate thisexpression to zero, as in .

( )( )

( )( ) ( )1

out

in

V s G s

V s G s H s=

−

( ) ( )1 0G s H s− =

( )G s

( )H s

( )outV s( )inV s +

+

( ) ( ) 1G j H jω ω = ( ) ( ) 1G j H jω ω = ( ) ( ) 2G j H j nω ω π∠ =and


Current Limiting

• If the overall resistance is negative, then the oscillationamplitude will continue to growindefinitely. In a practicalcircuit, this is, of course, not possible.

• Current limiting (power rails, or nonlinearity) eventually limitsthe oscillating magnitude to some finite value effect of thenegative resistance in the circuit until the losses are justcanceled, which is equivalent to reducing the loop gain to 1.

v

growth

t

limited


Implementations of Feedback

• Feedback (or−Gm ) is usually provided in one of three ways:

Colpitts oscillator:Using a tapped capacitor and amplifier to form a feedback loop

Hartley oscillator: Using a tapped inductor and amplifier to form a feedback loop

−−−−Gm oscillator: Using two amplifiers in a positive feedback configuration

G

amplifier

G

amplifier

amplifier

G

L

buffer


Amplifier Configuration (Colpitts or –Gm)

• The−Gm oscillator has either

A CC amplifier made up ofQ2 , andQ1 forms feedback

A CB amplifier consisting ofQ1, andQ2 forms feedback

• Colpitts and Hartley oscillators can be made either CB or CC.

C L

1Q

2Q

1C

2C

L

1Q1C

2C

1Q

L

CB Colpitts CC Colpitts−Gm Oscillator

CC

CB


Loop Analysis (I)

• Loop analysis gives information about the oscillator :

(1) Determine the frequency of oscillation

(2) The amount of gain required to start the oscillation

1C

2C

L

1Q

Common base

2C

1C

LpR

er

evc m ei g v=

At the collector, ( )1 1

1 10c e m

p

v sC v sC gR sL

+ + − + =

At the emitter, 1 2 1

10e c

e

v sC sC v sCr

+ + − =

1 1

1 1 2

1 1

0

01

mp c

e

e

sC sC gR sL v

vsC sC sC

r

+ + − − = − + +


Loop Analysis (II)

• The conditions for oscillation:

whereωconr is the corner frequency of the HPF formedby the capacitive feedback divider.

( )1 1 2 1 1

1 1 10m

p e

sC sC sC sC sC gR sL r

+ + + + − + =

( )1 23 2 11 2 1 1 2

10m

p e p e e

L C C LC Ls LC C s LC g s C C

R r R r r

++ + − + + + + =

1 2

1 2 1 2

1 1

e p

C C

C C L r R C C Lω

+= +

pL

RQ

Lω=

( ) ( )2

2 00 0 0

1 2 1 2

1 11 1

e p p e L conr

L L

r R C C R r C C Q

ω ωω ω ω ωω ω ω

= + = + = ++ +

( )1 2m

L

C Cg

Q

ω +=

Tells us what value ofgm

(and corresponding valueof re) will result insustained oscillation. Fora real oscillatorgm wouldhave to be made largerthan this value toovercome any additionallosses not properlymodeled.

Department of Electronic Engineering, NTUT

, , and

11/43

Capacitor Ratios with Colpitts (I)

• The capacitive divider (C1 ,C2 , andre) affects oscillationfrequency and feedback gain,which acts like a HPF.

( )1 1

1 2 1 21 1

e e cor

c e

cor

jv j r C C

v j r C C C C j

ωω ω

ωωω

′

= = + + + +

L pR

1C

2Cer

ev′

cv

Frequency

1

1 2

C

C C+

Gain

0A

corω

10

1 2

CA

C C=

+ ( )1 2

1cor

er C Cω =

+

1tan2 c

π ωφω

− = −

If the frequency of operation is wellabove the corner frequencyωcor , thegain is given by the capacitor ratioand the phase shift is zero.

90

0

Phase

Frequencycorω


Capacitor Ratios with Colpitts (II)

• re is transformed to a higher value through the capacitordivider, which effectively prevents this lowimpedance fromreducing theQ of theLC resonator.

• The resulting transformed circuit as seen by the tank

2

2,tank

1

1e e

Cr r

C

= +

L pR1C

2C,tanker

cv

1 2

1 2

1

T

C C

LC CLCω += =

(makeC2 large andC1 small to get the maximumeffect of the impedance transformation)


Negative Resistance

• Negative resistance of CB Colpitts oscillator

• Input impedance:

• A necessary condition for oscillation:

This is just a negative resistor in series with the two capacitors.

where rs is the equivalent series resistance on the resonator.

2i me

vi g v j C v

rπ

π πω′′ ′+ = +

1m

e

gr≃

2

iivj Cπ ω

′ =1

i mce

i g vv

j Cπ

ω′+=

1 2

1 m ice i

g iv i

j C j Cω ω

= +

21 2 1 2

1 1i ce mi

i i

v v v gZ

i i j C j C C Cπ

ω ω ω′ += = = + −

21 2

ms

gr

C Cω<

ii

iv

cev

vπ′er

1C

2C

mg vπ′

+

−

+

−

−

+


where , , and

14/43

Negative Resistance for Series/Parallel Circuits

• Since the resonance is actually a parallel one, the seriescomponents need to be converted back to parallel ones.

• However, if the equivalentQ of the RC circuit is high, theparallel capacitorCp will be approximately equal to the seriescapacitorCs , and the above analysis is valid. Even for lowQ,these simple equations are useful for quick calculations.

2C

1C

LpR

er

xvm xg v

cvsr negR

LTC


Example

• AssumeL = 10 nH, Rp = 300Ω, C1 = 2.5 pF,C2 = 10 pF, andthe transistor is operating at 1 mA, orre = 25Ω andgm = 0.04.Using negative resistance, determine the oscillator resonantfrequency and apparent frequency shift.

( |negative resistance| > original resistance,the oscillator should start up successfully)

This is a frequency of 1.2353 GHz, which is closeto a 10% change in frequency. Further refinementshould come from a simulator.

1 17.07 Grad/s

10 nH 2 pFTLCω = = =

×2 pFTC =

( )221 2

0.0432

7.07107 Grad/s 2 pF 10 pFm

s

gr

C Cω−= − = = − Ω

⋅ ⋅

1 12.2097

7.07107 Grad/s 32 2 pFs T

Qr Cω

−= − = = −⋅ ⋅

( ) ( )2 2par 1 32 1 2.2097 188 sr r Q= + = − + = − Ω

0 1.1254 GHzf =

( )par 2

2

2 pF1.66 pF

1 1 1 2.20971

sCC

Q

= = =++

par

1 17.7615 Grad/s

10 nH 1.66 pFLCω = = =

⋅


Negative Resistance of −Gm Oscillator

• Assume that both transistors are biased identically, thengm1 =gm2 , re1 = re2 , vπ1 = vπ2 , and solve forZi = vi /i i .

• Input impedance:

• Necessary condition for oscillation:

where Rp is the equivalent parallel resistance of the resonator.

1 1 2 21 2

ii m m

e e

vi g v g v

r r π π= − −+

2i

m

Zg

−=

2m

p

gR

>

ii

iv1 1mg vπ

+

−1vπ

2vπ

1er

2er

2 2mg vπ

+

−

+

−


Minimum Current for Oscillation (I)

• Using a 5-nHinductor with Q = 5 and assuming no otherloading on the resonator, determine the minimumcurrentrequired to start the oscillations of 3 GHz if a Colpittsoscillator is used or if a –Gm oscillator is used.

To find the minimum current, we find the maximum rneg by taking thederivative with respect to C1.

The maximum obtainable negative resistance is achieved when the two capacitorsare equal in value, C1 = C2 = 1.1258 pF, and twice the Ctot.

( )22

1 1562.9 fF

2 3 GHz 5 nHtot

osc

CLω π

= = =⋅ ⋅

1 2

1 2tot

C CC

C C=

+1

21

tot

tot

C CC

C C=

−

neg 2 2 2 21 2 1 1

m m m

tot

g g gr

C C C C Cω ω ω= = −

neg

2 2 2 31 1 1

20m m

tot

dr g g

dC C C Cω ω−= + = 1 2 totC C=


Minimum Current for Oscillation (II)

Now the loss in the resonator at 3 GHz is due to the finite Q of the inductor. The series resistance of the inductor is

Therefore, rneg= r s = 18.85 Ω. Noting that gm = I c /vT ,

In −Gm oscillator, there is no capacitor ratio to consider. The parallelresistance of the inductor is

A −Gm oscillator can start with half as much collector current in each transistor as a Colpittsoscillator under the same loading conditions.

( )2 3 GHz 5 nH18.85

5s

Lr

Q

πω ⋅ ⋅= = = Ω

( ) ( )2 221 2 neg 2 3 GHz 1.1258 pF 25 mV 18.85 212.2 AC TI C C v rω π µ= = ⋅ ⋅ ⋅ ⋅ Ω =

( )2 3 GHz 5 nH 5=471.2 pR LQω π= = ⋅ ⋅ ⋅ Ω

m C Tg I v= 2 2 25 mV 471.2 106.1 AC T pI v R µ= = ⋅ Ω =


Basic Differential Topologies

• Take two single-ended oscillators and place themback to back.

1C1C

CCV

1Q 2Q

2 2C

biasV

biasI biasI

L

CCVCCV

1Q 2Q

L

1C 1C

2 2C

biasIbiasI

CCV

L

C

1Q2Q

biasI

Copitts CB Copitts CC −Gm


Modified CC Colpitts with Buffering

• Oscillators are usually buffered (use emitter follower) in orderto drive a low impedance. Any load that is a significantfraction of the Rp of the oscillator would lower the outputswing and increase the phase noise.

• CC oscillator is modified slightly byplacing resistors in the collector.The output is then taken fromthecollector. Since this is a high-impedance node, the resonator isisolated fromthe load. However, theaddition of these resistors will alsoreduce the headroomavailable tothe oscillator.

CCV

L

1C

1Q 2Q

biasI

1C

biasI2 2C

LR LR

CCV


Several Refinements to the −Gm Topology (I)

• Decouple the base from the collector withcapacitors to get larger swings.

• The bases have to be biased separately.

• Rbiashave to be made large to prevent lossof signal at the base. However, theseresistors can be a substantial source ofnoise.

biasV

L

C

1Q2Q

biasI

biasR

CCV

biasV

biasR

cpC cpC


Several Refinements to the −Gm Topology (II)


1Q 2Q

pL

CCV

CsL

biasV

biasI

• Use a transformer to decouple the collectorsfrom the bases.

• Since the bias can be applied through thecenter tap, no need for the RF blocking.

• A turns ratio of greater than unity is chosen,there is the added advantage that the swingon the base can be much smaller than theswing on the collector to prevent transistorsaturation.

23/43

Several Refinements to the −Gm Topology (III)

• Since the tail resistor is not a highimpedance source, the bias current willvary dynamically over the cycle of theoscillation (highest when voltage peaksand lowest during the zero crossings).

• Since the oscillator is most sensitive tophase noise during the zero crossings, thisoscillator can often give very good phasenoise performance.

biasV

L

C

1Q2Q

tailR

biasR

CCV

biasV

biasR

cpC cpC


Time

Amplitude

( )1ci t ( )2ci t

AVEIdcI

24/43

• Using a noise filter in the tail can lead to avery low-noise bias, thus low-phase-noisedesigns.

• Another advantage is that, before startup,the transistor Q3 can be biased insaturation, because during startup the 2nd

harmonic will cause a dc bias shift at Q3

collector, pulling it out of saturation andinto the active region.

• Since 2nd harmonic cannot pass throughLtail, there is no ‘‘ringing’’ at Q3 collector,further reducing its headroom requirement.

biasV

檔案中找不到關聯識別碼rId7 的圖像部分。

C

1Q2Q

tailL

biasR

CCV

biasV

biasR

cpC cpC

3QtailCbiasV


Several Refinements to the −Gm Topology (IV)

25/43

The Effect of Parasitics on the Frequency

• The first task in designing an oscillator is to set the frequencyof oscillation and hence set the value of the total inductanceand capacitance in the circuit.

• To increase output swing, it is usually desirable to make theinductance as large as possible (this will also make theoscillator less sensitive to parasitic resistance). However, itshould be noted that large monolithic inductors suffer fromlimited Q. In addition, as the capacitors become smaller, theirvalue will be more sensitive to parasitics.


Oscillating Frequency Summary

1 2 1

1 2

1osc

C C C CL C

C C Cπ

µπ

ω ≈ + + + +

1 2 2

1 2

1osc

C C C CL C

C C Cπ

µπ

ω ≈ + + + +

1

22

oscC

L C Cπµ

ω ≈ + +

1C

2C

L

1Q1C

2C

1Q

L


CB CC −−−−Gm

C L

1Q

2Q

27/43

Oscillator Phase Noise

( )V f

f1f

( )V f

f1f

( )v t

t

1

1

f

( )v t

t

1

1

f

Time Domain Frequency Domain


Jitter Phase noise

mf

28/43

Phase Disturbance Due to Thermal Noise (I)

• Modeling the noise with the phasor diagram

nP

sP

sP′

Phase disturbance

Amplitude disturbance

FkTB

avsP

Noise-free amplifier

f

0f 0 mf f+

1 Hz1 Hz 1nRMS

FkTV

R=2nRMS

FkTV

R=

avsavsRMS

PV

R=

The input phase noise in a 1-Hz bandwidth at any frequency from the carrier produces a phase deviation.

0 mf f+


Phasor Diagram

29/43

(noise from )

Phase Disturbance Due to Thermal Noise (II)

• RMS phase deviation

avsavsRMS

nRMSpeak P

FkT

V

V ==∆ 1θ

avsRMS P

FkT

2

11 =∆θ

avsRMS P

FkT

2

12 =∆θ

2 2 1 2RMS total RMS RMS

avs

FkT

Pθ θ θ∆ = ∆ + ∆ =

mω

12 nRMSV

2 avsRMSV

peakθ∆

(total phase deviation)

( ) ( )02 cososc avsRMSv t V t tω θ= + ∆


mf+(noise from )

mf−

30/43

Lesson’s Phase Noise Model (I)

• The spectral density of phase noise :

Due to Thermal Noise

Consider Flicker Noise (modeled)

( ) 2m RMS

avs

FkTBS f

Pθ θ= ∆ =

1)(B dBm/Hz 174 =−=kTB

(due to theoretical noise floor of the amplifier)

1)(B 1)( =

+⋅=

m

c

avsm f

f

P

FkTBfSθ

noise floor flicker noise

( )mS fθ

Noise-free amplifierPhase modulatoravs

FkTB

P

mf

( )mS fθ

cf


Lesson’s Phase Noise Model (II)

• The oscillator may be modeled as an amplifier with feedback

( )

0

1

21

m

L m

LQ

j

ωω

ω

=

+

220 B

QL

=ω

( ) ( )012out m in m

L m

f fj Q

ωθ θω

∆ = + ⋅ ∆

( ) ( )2

0 ,2

11

2out m in mm L

fS f S f

f Qθ θ

= + ⋅

( ), 1 cin m

avs m

FkTB fS f

P fθ

= ⋅ +


Noise-free amplifier

Phase modulator

( ),in mS fθ

Output

Feedback

θ∆

Resonator

Resonator equivalent low-pass

( )in mfθ∆

( )0

2 in mL m

fj Q

ω θω

⋅ ∆

( )out mfθ∆

( )mL ω

32/43

Lesson’s Phase Noise Model (III)

• Lesson’s phase noise model:

( )2

0 2

1 11 ( )

2 2m in mm L

fL f S f

f Q θ

= + ⋅

( ), 1 c

in mavs m

FkTB fS f

P fθ

= ⋅ +

Open-loop


Closed-loop w/ Resonator

( )22

3 2 2

1 11

2 4 2o c o c

mavs m L m l m

FkTB f f f fL f

P f Q f Q f

= + + +

Up-convert 1/f noise

Thermal FM noiseFlicker noise

Thermal noise floor

33/43

Lesson’s Phase Noise Model (IV)


Low-Q oscillator

Phase perturbation

1mf−

3mf−

2mf−

0mf

mf

mf

Resulting phase noise

cf

cf 0 2f Q

High-Q oscillator

Phase perturbation

1mf−

1mf−

0mf

3mf−

mf

mf

Resulting phase noise

cf

cf0 2f Q

34/43

Design Example of Phase Noise Limits (I)

• A 5-GHz receiver including an onchipphase-locked loop(PLL)is argued to be implemented with the VCOrequirements:1.8V supply, <1 mW DC power, and phase noise performance of−105 dBc/Hz at100-kHz offset. It is known that, in the technology to be used, the best inductor Q is15 for a 3-nH device. Assume that capacitors or varactors will have a Q of 50.

Assume a−Gm topology will be used:

2 5 GHz 3 nH 15 1413.7 pr L LQω π= = ⋅ ⋅ ⋅ = ΩParallel resistance due to the inductor:

Required capacitance:( )22

1 1337.7 fF

2 5 GHz 3 nHtot

osc

CLω π

= = =⋅ ⋅

( )50

4712.9 2 5 GHz 337.7 fFp

tot

Qr C

Cω π= = = Ω

⋅ ⋅Parallel resistance due to the capacitor:

Equivalent parallel resistance of the resonator is 1087.5 Ω

Current limit: 1.8-V VCC and PDC< 1 mW: 555.5 µA

Peak voltage swing: ( )tank

2 2555.5 A 1087.5 0.384 Vbias pV I R µ

π π= = ⋅ Ω =


Design Example of Phase Noise Limits (II)

Assume all low-frequency upconverted noise is small and active devicesadd no noise to the circuit (F=1), we can now estimate the phase noise.

( )( )

22tank

0.384 V67.8 W

2 2 1087.5 RFp

VP

Rµ= = =

Ω

337.7 fF1087.5 11.53

3 nHtot

p

CQ R

L= = Ω =

This is−97.5 dBc/Hz at 100-kHz offset, which is 7.5 dB higher than the promisedperformance. Thus, the specifications given to the customer are most likelyverydifficult. This is an example of one of the most important principles in engineering.

RF output power:

Oscillator Q:

( ) ( )( )

222 2310

3 2 2

1.12 2 5 GHz1 1 1 1.38 10 J/K 298 K 1 Hz100 kHz 1 1.79 10

2 4 2 2 67.8 W 2 11.53 2 100 kHzo c o c

avs m L m l m

FkTB f f f fL

P f Q f Q f

πµ π

−−

⋅ ⋅ × ⋅ ⋅= + + + = ⋅ = × ⋅ ⋅ ⋅


~ 0 far from carrier

97.5 dBc Hz@100 kHz offset= −

dominant around carrier

36/43

• VCO is an oscillator of which frequency is controlled by atuning voltage.

• VCO is a simple frequency modulator

Voltage Controlled Oscillator (VCO)

vcof

tuneV

tuneV


tuneV

( )oscs t

37/43

Making the Oscillator Tunable

• Varactors in a bipolar process can be realized using either thebase-collector or the base-emitter junctions or else using aMOS varactor in BiCMOS processes.

CCV

CCV

LL

1C 1Q 2Q 1C

R

1BR

2BR

varC varC

CBSubs

CBSubs

Tuning port

CCV

biasIL

varCvarCconR

conV

1Q 2Q

CCV

biasI

L

varCvarC

LR

conV

1Q 2Q

CCV

LR

1C

biasI

1C


• Frequency Range

• Frequency tuning characteristics Tuning sensitivity (Hz/V) :

Linearity

VCO Sensitivity and Tuning Linearity

VK f V= ∆ ∆

vcof

tuneV

,0tV

0f

maxf

minf

,mintV ,maxtV

v∆f∆

Ideal (perfect)

Piecewise good

Piecewise good

Poor


Important Figures

• Output power (50 Ohm)

• Frequency stability: frequency drifting

• Source pushing and load pulling figures

• Harmonics

• Phase noise (or Jitter)


• Phase noise

• Jitter

Cycle jitter

Cycle-to-cycle jitter

Absolute jitter (long-term jitter, accumulated jitter) of N cycles

Phase Noise and Jitter

( ) ( ) ( )1 Hz10log 10log dBc Hz

2noise

carrier

S fPL f

Pϕ ∆

∆ = =

cn nT T T∆ = − ( )2

1

1lim

N

c cnnn

TN

σ→∞

=

= ∆∑

1ccn n nT T T+∆ = − ( )2

1

1lim

N

c ccnnn

TN

σ→∞

=

= ∆∑

( ) ( ) ( )1 1

N N

abs n cnn n

T N T T T= =

∆ = − = ∆∑ ∑ ( )2

1

1lim

N

c ccnnn

TN

σ→∞

=

= ∆∑

for white noise sources ( ) 0

2abs cc

fT t tσ∆ ∆ = ∆ and 2cc cσ σ=


• Relationship between the SSB phase noise and the rms cyclejitter: (Weigandt etal.)

• Relationship between the SSB phase noise and the rms cyclejitter: (Herzel and Razavi)

• Self-referred jitter and phase noise with white noise:(Demir etal.)

Relation of Phase Noise and Jitter

( )( )

3 20

2cf

L ff

σ∆ =∆

( ) ( )( ) ( )

3 20

22 3 40

4

8

cc

cc

Lω π σ

ωω ω π σ

∆ =∆ +

( )( )

20

2 2 4 20

f cL f

f f cπ∆ =

∆ +

( )2 t t cσ ∆ = ∆ ⋅ 20

2 cc

fc σ=


where and

42/43

Summary

• In this chapter, fewkinds of popular oscillator topologies wereintroduced. The CB and CC configurations are good for highfrequency operation while the CE is good for high powerapplication and has good buffering characteristics.

• The active device is configured as feedback loop to provide anegative resistance for resonator.

• For a voltage-controlled frequency application, an oscillator isusually designed with variable capacitors, or varactors, toprovide frequency-tuning capability.

• Lesson’s phase noise model gives an intuitive way tounderstand the behavior of the phase noise generated fromtheoscillator.
