04/03/2019 1 1 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 j A j f R L ) 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 (Z i is the input impedance of block A): • The complex condition corresponds to 2 real conditions: or R L ) j ( A ) j ( f Z i Z i V i i i V j A j f V ) ( ) ( 0 )] ( ) ( Im[ 1 )] ( ) ( Re[ j A j f j A j f 2 )] ( ) ( [ 1 ) ( ) ( k j A j f Arg j A j f
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04/03/2019
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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
Resonant networks - 2
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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Resonant networks - 3
)
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
Resonant networks - 4
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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Resonant networks - 5
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
Resonant networks - 6
R: 10.1
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Resonant networks - 7
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
Resonant networks - 8
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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Resonant networks - 9
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
Resonant networks - 10
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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Resonant networks - 11
)
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
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Resonant networks - 12
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
Passive component models - 2
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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Passive component models - 3
• 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
18
Passive component models - 4
• 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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Passive component models - 5
• 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)
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Passive component models - 6
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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Passive component models - 7
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
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Passive component models - 8
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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Passive component models - 9
)
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
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Passive component models - 10
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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Passive component models - 11
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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Colpitts network - 2
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
Colpitts network - 3
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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Colpitts network - 4
The real part of complex poles is derived by considering the sum