sinusoidal oscillators /1

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Sinusoidal Oscillators /1

• A sinusoidal oscillator is a system described by two conjugate

imaginary poles

• A linear amplifier can be closed within a feedback loop in order to

build an oscillator, if Barkhausen condition is satisfied, i.e. the

system shows a pair of conjugate imaginary poles

0)()(1 jAjf

RL

)j(A

)j(f

2

Sinusoidal Oscillators /2• Barkhausen condition means that system oscillation is self-sustained, i.e.

the loop gain is equal to 1 (Zi is the input impedance of block A):

• The complex condition corresponds to 2 real conditions:

or

RL

)j(A

)j(f

Zi

Zi

Vi

ii VjAjfV )()(

0)]()(Im[

1)]()(Re[

jAjf

jAjf

2)]()([

1)()(

kjAjfArg

jAjf

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Resonant networks - 1

• The transfer function of a resonant network contains a pair of

conjugate complex poles (at least a L-C pair is needed)

• The resonance frequency is defined as the frequency where

the transfer function is real (and also maximum)

• Please, let’s consider a series resonant network as the

following one.

I

4


If we evaluate the current flowing in the loop: I(j)= V (j)Y(j)

) ) ) CLs1sCR

sC

sLsC

1R

1

sV

sIsY

2

The network poles of the transfer function Y(s) are:

)LC

LCRCRCS

2

42

2,1

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5


)

jCR

L

L

R

LCL

R

L

R

LC

LCRCRCS

2

22

2,1

411

2

1

222

4

The cutoff radian frequency 0 is evaluated as the modulus of

the conjugate complex poles:

LC

1220

6


C

L

R

1

LC

RL

R

L

RI2/1

LI2/1

P

EQ 0

2

20

diss

imm0

The quality factor Q of the resonant network is defined as the ratio:

where Eimm is the average energy stored in the network at radian

frequency 0, and Pdiss is the average dissipated power in network

resistors (i.e. the average power loss)

Poles can be expressed as a function of Q and 0:

LC

1220

20

2

02,14

11

241

2 Qj

L

R

L

CRj

L

RS

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Phase (degrees) and Modulus (dB) of the transfer function by

considering R (losses) as a parameter: a steeper phase and a

higher modulus is found around 0

R: 101

8


R: 10.1

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The -3dB bandwidth of the resonant network transfer function

can be related in a simple way to the quality factor Q

For instance, if we consider the series resonant network

previously presented, we have at resonant frequency:

that is just the maximum value for the transfer function

Let’s evaluate the frequency at which 3dB power attenuation

is obtained

)R

1Y 0

10


The modulus of the transfer function can be expressed as:

)

)

2

2

2o222

2

2222

22

1LR

1

LC

11LR

1

C

1LR

1jY

LjCj

1R

1jY

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11


L22

L

LLR

0

0

odB3

dB3

odB3

dB3

2o

2dB3

The fractional bandwidth (FBW) is defined as the ratio of the

bilateral -3dB bandwidth (BW = 2) on the resonance radian

frequency. We find:

Q

1

L2

R2BWFBW

00

12


As the Q factor increases, a steeper phase transition as well as a

lower -3dB bandwidth (i.e. more frequency selectivity) are

obtained

Both the above mentioned property can be exploited in order to

design a frequency-stable oscillator:

1) Frequency selectivity allows decreasing of the effect of

amplifier additive noise: lower phase noise is obtained, we’ll see

later…

2) Steep phase transition of the resonant network makes the

oscillation frequency less dependent on amplifier characteristics

variation

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)

0d

dS 0F

The frequency stability coefficient SF gives us an idea of the

variation speed of the phase of the transfer function around

the resonance frequency:

An approximate estimation can be obtained for a resonant

network showing quality factor Q:

Q2SF

14


From expressions above, we can see that resistive loss has

to be reduced in order to increase the Q factor

Resistive loss is due to:

1) Loss in passive components.

2) Real parts of active device (BJT or MOS) impedances

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Passive component models - 1

Capacitor, inductors, and transformers are affected by both

loss and self-resonance:

• Loss is accounted for in the model by means of a resistor

• In order to account for self-resonance, a dual component has

to be inserted in the model (L C)

16


Ideal capacitor

Lossy capacitor

Lossy capacitor with self-resonance

A RF capacitor shows a set of harmonic self-resonance frequencies:

It can be used only below the lowest resonance frequency.

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• Loss is caused by finite resistance of both

dielectrics and conductors

• In capacitors, the capacitance is increased by

increasing area and / or by lowering plates mutual

distance: in both cases, for a given dielectric

material, resistance is lowered

• Moreover, in capacitors also radiation loss has to

be considered, as the e.m. field is not confined

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• The presence of an inductor in the capacitor model

allows modeling of both magnetic loss of access

metallic pins, and cavity resonance modes

• Inductor loss is due to conductor electric loss and,

at higher frequencies, to the so-called skin effect that

produces series resistance increase

• The presence of a capacitor in the inductor model

allows modeling of mutual parasitic capacitance

between inductor coils

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• As a consequence of loss effect, the quality factor of real

networks is lowered and therefore frequency selectivity (-3dB

bandwidth is increased)

• A quality factor Q is considered also for the single passive

component (L and C)

20


If only resistive loss is considered,

the following models are obtained:

• Inductor with series resistor

(conductor loss)

• Capacitor with parallel resistor

(dielectric loss)

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PPC

PCC

SLL

CsR1

R)s(Z

sLR)s(Z

The Q factor of a component (an inductor, for

instance) with series model can be defined as the

ratio of reactance on resistance sLsL

R

LQ

pCpC CRQ The Q factor of a component (a capacitor, for

instance) with parallel model can be defined as

the ratio of susceptance on conductance

22


It is possible to consider a parallel

model for the inductor

and a series model for the capacitor

LjRLjR

LjRZ sL

ppL

ppLpL

At a fixed frequency we can write:

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)

2

2

pL

2

p

2

pL

2

P

2

2

pL

2

pLppL

2

P

2

2

pL

2

P

2

2

pL

2

pLppL

2

P

2

2

P

22

pL

2

pLppL

2

P

2

2

ppL

ppLppL

ppL

ppL

pL

Q

11Q

R

Q

11

Lj

R

L1R

RLjRL

R

L1R

RLjRL

LR

RLjRL

LjR

LjRLjR

LjR

LjRZ

24


2

pL

2

2

pLS

p

2

pS

2

2

pL

2

ppL

Q

R

Q

11Q

RR

L

Q

11

LL

Q

11Q

R

Q

11

LjZ

The quality factor is dependent

on frequency: it is lowered as

frequency increases: see RF

components datasheets

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L - R MODEL C - R MODEL

QS = LS / RS QS = 1 / ( RS CS)

QP = RP / LP QP = RP CP

RP = RS (1 + QS2) RS = RP / (1 + QP

2)

LP = LS (1 + 1 / QS2) CS = CP (1 + 1 / QP

2)

TRANSFORMATIONS TABEL

26

Colpitts network - 1

R1 and R2 contain loading of the BJT used to close the

oscillator loop (we’ll see later), and the quality factor model

of capacitors C1 and C2

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The Vout/Iin transfer function is evaluated, as the BJT can be

considered as a voltage-controlled current source

) ) ) ) 21212211In

Out

CCsGGsCGsCGsL

1s

I

V

The poles of the network and the resonance frequency are

evaluated from coefficients of transfer function denominator

) ) ) )

0GG

CCGLGsGCGCLsCLCs

CCsGGsCGsCGsLsD

12

212112212

213

21212211

28


The resonance frequency is evaluated by putting to 0 the imaginary

part of denominator:

The real part of complex poles is needed in order to find the Q factor.

It is evaluated by exploiting the relation between poles and

polynomial coefficients:

)

21

2121

21

21

2121

212120

2121021300

CC

CCL

1

CC

GG

CC

CCL

1

CLC

CCGLG

0CCGLGjCLCjjD

) ) ) )

2021

121

21

12*221

12

212112212

213

21212211

1

CLC

GGs

CLC

GGsss

0GG

CCGLGsGCGCLsCLCs

CCsGGsCGsCGsLsD

( 0)

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The real part of complex poles is derived by considering the sum

of poles:

21

1221*221

CC

GCGCsss

21

12

21

1221

2021

12

21

12212

CC

GG

CC

GCGC1

CLC

GG

CC

GCGCsRe2

) ) )

) )2121

12

222

1

2121

122121211212

CCCC

GCGC

CCCC

GCGCCCCCGCCG

121

12212 s

CC

GCGCsRe2

)

)12

222

1

21210

2

0

GCGC

CCCC

sRe2Q

30

Under the hypothesis:

Qc1 >> Qc2 => Q Qc2

VOut(0) = - Iin(0) / [(C1+C2)(G2/C2 + G1/C1) – (G1 + G2)]

=- Iin(0) / [(C1/C2) G2 + [(C2/C1) G1]

C1 >> C2 => VOut(0) = - Iin(0) / [(C1/C2) G2]

)

) )

) )

)2c21c1

212c1c

12

222

1

21210

210

210

12

222

1

21210

2

0

QCQC

CCQQ

GCGC

CCCC

)GG/(

)GG/(

GCGC

CCCC

sRe2Q


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Yin = [s3 LC1C2 + s2 LG C2 + s (C1+C2) + G] / [s2 L C1 + s L G + 1]

Zin = R [(1 - 02L C1) + j 0 G L] / [1 - 0

2L C2] per 0

Rin = R [1 - 02L C1] / [1 - 0

2L C2] = R (C1/C2)2

Xin = 0 L / [1 - 02L C2] = - (Qc2/Qc1) 0 L

Input impedance

evaluation:

32


• Barkhausen condition is never perfectly fulfilled: indeed, poles can

shift both horizontally (oscillation vanishing or saturating), and

vertically (oscillation frequency variation)

• Stability coefficient SF allows evaluation of oscillation frequency

sensitivity to both noise and interference

• If we suppose that Barkhausen condition is satisfied for a frequency

f0’ different from f0, we can consider a first-order Taylor series

expansion of the phase if transfer function:

• Lower frequency shift is obtained for higher SF value

FS

d

d

1'

)'()(0)'(

00

0000

0

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COLPITTS OSCILLATOR (RLC)

34



• The Colpitts oscillator is composed of an amplifier stage (common

emitter) and a Colpitts network

• The amplifier produces 180° phase shift at all frequencies (under the

hypothesis that the high cut-off frequency is much greater than f0), the

Colpitts network only at the resonance frequency: the Barkhausen

condition is fulfilled at the resonance frequency provided that the loop

gain is just equal to 1 at f0

• The presence of a Radio Frequency Coil (RFC) allows to double the

maximum output dynamic

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• Oscillation condition on loop gain modulus is applied by exploiting

Colpitts network properties. Non-idealities of the amplifier block are

moved into the Colpitts network:

• where R1’= RL // ro // R1 RL, R2’= R3 // R4 // r // R2 (R1 and R2 are used

to model capacitors loss, i.e. Ri = Qci / 0 Ci, i=1, 2)

+

-

G

C1C2 R1'R2'

Iout

36



• Under the hypothesis that Qc1 >> Qc2, the overall loop gain is equal to:

– Vbe = - Iout RL C2/C1

• On the other hand, the output current of the amplifier block is:

– Iout = -gm Vbe

• Barkhausen condition is fulfilled if:

– gm = IQ / VT = C1/ (RL C2)

• In order to increase frequency stability of the oscillator, a quartz is used as

a reactive element in the Colpitts network

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Quartz and quartz model /1

sL1 = 1/(sC1)

s= (L1*C1)-1/2

• Crystalline quartz shows a reciprocal relationship, i.e. the

piezoelectric effect, between mechanical deformation along

one crystal axis and the appearance of an electric potential

along another axis: deforming a crystal will produce a

voltage, an applied voltage will deform the crystal

• The phenomenon appears at given resonance frequencies,

with very high Q (104 – 106)

• The series resonance (|XL1| = |XC1|) models the piezoelectric

effect

38


• Capacitor C2 models the capacitance between capacitor

terminals, and produces a parallel resonance:

|XC2| = |XL1 + XC1| = |XL1| - |XC1|

1/(pC2) = (pL1)-1/(pC1)

1/(pC2) + 1/(pC1) = pL1

(1/C1)(1+C1/C2) = (p)2L1

(1+C1/C2)/(C1*L1) =

(1+C1/C2)(s)2=(p)

2

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p= s (1+C1/C2)1/2

If C1<<C2, a Taylor-series expansion can be considered:

(1+C1/C2)1/2= 1+(1/2)(C1/C2)

Difference between resonance frequencies p and s is:

= p - s

= (1/2)(C1/C2) s

From quartz reactance diagram above shown, it can be seen

that an inductive impedance is found between s and p

40


WIEN OSCILLATOR (RC)

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• The amplifier block A is a non-inverting feedback amplifier showing

gain: A(j) = 1 + R2 / R1.

• If an ideal operational amplifier is considered, the input impedance is

much greater than the one of the resonant network, and the output

impedance is much lower than the load

• The loss of the resonant is evaluated, and than Barkhausen condition is

applied:

1R

RR

1CRj3CR

CRj)j(A)j(f

1

21

222

42



• The network resonance frequency is found when denominator is purely

imaginary, and therefore we get the following expression:

• From Barkhausen condition on module, the gain of operational

amplifier has to be equal to 3

• A low value of SF (=2/3) is found

CR

10

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Gain control for WIEN OSCILLATOR

• Control of peak output voltage is made by means of a DC feedback

loop:

VR

VOUT

AC/DC

Oscillatore

44



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• Diode and low-pass filter allow reading of the negative peak of the

waveform, so to change the bias point (VGS) of the JFET

• The JFET is biased at IDS = 0 (see the 500uF capacitor), and therefore

its resistance is changed by controlling VGS

• For instance, if waveform peak is rising, VGS is lowered, and JFET

resistance becomes greater so that the overall value of R1 is increased,

so lowering the gain of operational amplifier

46

Pierce Oscillator

quarzo

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DC Analysis

• The amplifier is in CE connection

• The needed output dynamic is 12 Vpp

• The oscillation frequency is 12 MHz

• The power supply is Vcc = 12V, the load R0 = 50

48

Bias point choice

• Component NPN BFG92A/X is used

• IQ = 3 mA is the bias current

• R3 = 0.67 K, R1 = 9.2 K, R2 = 2.8 K

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Small-signal model (Colpitts network)

Qy LjZ

)Cj/1(||R)Cj/1(||r||R||RZ 1Tot121B

)Cj/1(||RZ eq0inC

50

Oscillation conditions /1

beYBC

BCbem V

ZZZ

ZZVg

Ceq1C

eq1

eq1Q

Tot0ineq1

eq1

eq1Q

20

QQ

11

)CC

CC(L

1

RRCC

1

)CC

CC(L

1

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Oscillation conditions /2

1)C/GC/G)(CC()GG(

g

1Toteq0ineq1Tot0in

m

1C/C

Rg

eq1

0inm

52

Matching network use

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Tapped resonant circuits /1

b) Tapped-capacitor circuita) Tapped-inductor circuit

C1

L1RSRL

C2

L2

RL

C L

RS

54


• A first Parallel->Series transformation (resistance is

lowered):

22C

L2

2C

LLS

222C

22C

2S2

Q

R

Q1

1RR

CQ

Q1CC

C1

RLS

C2SL

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55


• A second Series -> Parallel transformation

(resistance is enhanced):

RTOTCL )

) ) ) 2

2C

2C

L22C

2C

L2CLSTOT

S21

S21

2C

2C

S21

S21

Q

QR

Q1

Q1RQ1RR

CC

CC

Q1

Q

CC

CCC

56


• The 2nd transformation produces a parallel resonant circuit, where

input resistance is just the matching resistance at resonance

• I we consider Q expressions we find;

• The matching network produces multiplication of load resistance for a

factor depending on the ratio of capacitances

2

1

2L

2

1

21LTOT

C

C1R

C

CCRR

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• Qtot = 2Q = 2f0 / BW is chosen (in order to account for partition due to

matched load).

• C value is calculated:

• L value is calculated :

• QC2 value is calculated :

Design procedure

S0

tot

R

QC

C

1L

20

1R/R

Q1Q

LS

2tot

2C

58


• C2 value is calculated :

• C2S value is calculated :

• C1 value is calculated :

Design procedure

CC

CCC

S2

S21

L0

2C2

R

QC

22C

22C

2S2Q

Q1CC

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• We need to match a 50 load to a 4K source resistance at 3.0 MHz

resonance frequency, with a loaded Q = 7.5.

• Unloaded Q value is: Qtot = 2·7.5 = 15.

• Overall capacitance is: C = 15 / (0RS) = 200 pF.

• Tuning inductance is: L = 1 / (02 C) = 14 uH.

• If the other expressions previously presented are applied, we find:

– QC2 = 1.34

– C2 = 1.4 nF

– C2S = 2.2 nF

– C1 = 0.22 nF

Design example

60


Design example

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

freq, MHz

-10

-8

-6

-4

-2

0

dB

(S(1

,1))

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Matching network design /1

• The resonance frequency of the network is

chosen slightly lower (9/10) than the one of the

oscillator, so that the network is a capacitor at

the oscillation frequency (Ceq)

• Rin0 is given by the chosen output dinamic and

is equal to 2 K

62

• If we arbitrarily choose for the network Q = 50, we get:

• We choose:

• From network resonance frequency expression:


9.7R

RQQ

0in

0C2C nF2.5CC S22

H1L1

pF217)10/9(L

1C

201

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• At oscillator resonance frequency, the network

admittance is:

pF41C100/19Ceq


pF227CCC

CCC 3

S23

S23

C100/19jGC'

CjGL

1CjGY 00in2

0

20

00in

12

0

00in

64

• From condition on loop gain, we get:

• And the hypothesis: C1 >> Ceq is verified

• Oscillator resonance frequency is fixed by the

quartz, that shows inductive impedance at 0


nF46.9RgCC 0inmeq1

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If we consider the tapped-inductor network, and let’s do a first

Parallel->Series transformation (load resistance RL is lowered)

Again, the transformed component values are found starting

from Q and L2

66


22L

L2

2L

LLS

222L

22L

2S2

Q

R

Q1

1RR

LQ1

QLL

From first transformation:

The 2nd transformation (S P ) produces a parallel

resonant circuit, where input resistance is just the

matching resistance at resonance

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The 2nd transformation (S P ) produces a parallel

resonant circuit, where input resistance is just the

matching resistance at resonance

68


) )

) ) ) 2

2L

2L

L22L

2L

L2LLSTOT

S212L

2L

S21TOT

Q

QR

Q1

Q1RQ1RR

LLQ

Q1LLL

:

In S P (and P S also) transformations, Q is of course unchanged:

We can exploit such property to evaluate the parallel equivalent

resistance RTOT:

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2

2

1L

2

2

21LTOT

L

L1R

L

LLRR

:

If we use Q expressions:

The matching network produces multiplication of load resistance

for a factor depending on the ratio of inductances

.

70

LC Oscillators /1

• The LC oscillator is made of a differential pair (with resistive degeneration if

needed) closed in positive feedback loop, with a tuned load L-C. The

capacitor C is often electronically changed (it is a varactor): the oscillator

shows a resonant radian frequency equal to 1/(L∙C)

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LC Oscillators /2

• The output resistance of the differential pair in positive feedback loop is

quite equal -1/gm of the transistor, as it can be seen from the equivalent

BARTLETT circuit for the differential pair for differential mode:

) mm

2 g/1/11g

1I/VRout

Rpigm Vpi

V2 = -V1

V2V1

II1 I2

I2 = I1

+

V

-

r gmV

72

LC Oscillators /3

• The equivalent negative resistance allows compensation of

tuned load loss: in integrated process (for instance SiGe),

inductors show low Q values (<20)

• Such loss compensation, allows to fulfill Barkhausen

condition on gain module

sinusoidal oscillators /1

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