1
OSCILLATORS
An oscillator is a circuit, which produces a periodic signal withoutany input signal. It converts DC power (from the supply) to a periodic signal.
Oscillators are extensively used in both receive and transmit paths.They are used to provide the local oscillation for the mixers for upand down conversion.
On this topic, we cover:
1. Oscillator basics2. Oscillator topologies3. Phase noise issues4. Oscillator implementations and design
ECEN 665
2
Oscillator types:
1. Crystal oscillators2. Active-RC and Gm-C oscillators3. Ring oscillators4. LC timed oscillators5. Relaxation oscillators.
… etc.
Voltage Controlled Oscillators:
VCO’s are used in RF applications, usually, in a Phase locked loop to provide Different LO frequencies needed for channel selection. Besides frequency synthesis it is used for other applications.
3
OSCILLATOR BASICS
Most RF oscillators can be viewed as + ve feedback system.
+ A(s)
β(s)
+
+vovi
)()()(1
)()( svssA
sAsv io β−=
The Barkhausen Criterion states that sustained oscillation can be achieved.
if Loop gain equal to unity.
and
( ) ( ) 1=oo ssA β
( ) ( ) ooo ssA 0=β
For zero vi the output will be finite at a given frequency (ωo). For loop gain >1 oscillation will grow.
4
Two-Port vs One-Port ModelsThe previously shown model is known as the two-port model, because the A(s) and β(s) Networks are both two-port nets in a closed loop.
The one-port model of oscillators is shown below:
-Rp Rp L C
Active Network Resonator
The resonator (LC tank for example) has parasitic resistances (RP) which prevents the resonator from oscillating because the stored energy will leak through the resistance. To compensate for this loss a positive feedback negative parallel resistances (-Rp) will be added to the resonator so that the energy loss in Rp is replenished by the negative resistance. The negative resistance’s implementation is typically an active network.
5
What sets the Frequency and Amplitude of an Oscillator?
The frequency is usually set by Barkhausen Criterion. It is the frequency at which the loop gain is greater or equal to 1 and the phase is zero.
In many cases, as in LC tuned oscillators, the ideal oscillation frequency is determined by the LC tank.
LCfo π2
1=
The amplitude is, however, more a complicated parameter to set.
6
More on Oscillator Theory:Ideal Oscillator:
iL iC
CL
ωj
LCo1
=ω
σ
Oω−
Once the tank is excited (by a current pulse for example) a certain amount of energywill be conserved. This energy will alternate between magnetic and electric forms.
In the time domain:
LCv
dtd
dtdLCv
dtdvCi
iidtdLv
oov
v
c
cLiL
1 ere wh 0or
and
but
22
2
2
2
==+
−=⇒
=
−==
ωω
7
The Differential Equation of the Ideal Oscillator
022
2=+ v
dtd
ov ω
When solved yields:tojtoj eVeVtv
ωω −+= 21)(
Choosing the case where the phase angle is zero at t=0
tVtv oωsin)( =
In the frequency domain:
( )
22
2 )()(
)( )()( but )()()(
o
Lo
c
LcLL
soisLsV
svsCIsisIoisisLsV
ωω
+=⇒
⋅=−=+=
Both poles are at as expected !! ojω±
8
Real Oscillator:
Initial excitation well start some oscillation.This will die out after a while because theEnergy leaks through the parasitic resistances.
L
RL
C
RC
σ−a
ωj
2411Qo −ω
2411Qo −− ω
oω
oωCharacteristic equation
( )
C
L
C
CL
L
CLC
LCRR
LCRCRRLss
RCRRLsRLCs
++
+
+++
2
2
or
C
CL
CCo
LCRCRRL
RLRL
2
2
+=
=
τ
ω
9
If oscillation is to be maintained. The energy loss (due to parasitic resistance) is to becompensated for. This energy is usually provided from the power supply via a negativeresistance, a nonlinear conductance or regenerative feedback, which convert dc power to signal power.
CoC
L
o
s
cL
ccLL
CRQ
RL
CLR
RRRQRQ
G
ω
ω
1
Q where
ly.respective , and theof parasitic series
theare and //1
L
22
=
=
=
LLiC
ci ri
Ggmi
v
q p
flatter
steeper rgm ii +
slope = - a + G
ri
Slope = G
gmi
Slope = - a
v
v
v
NegativeResistor
10
The following model equation can be written
vr
c
L
gmrcL
GidtdvCi
vdTL
i
iiii
=
=
∫=
=+++
1
0
Differentiating both sides of the above equation:
012
2=+++
dtdi
dtdvG
dtvdCv
Lgm
[ ] 02
2=+++ viG
dtdL
dtvdLC gmv
The above DE describes the behaviour of many oscillator implementations.
11
At the quiescent bias point (which is the equilibrium point)
( ) aidtd
vgm −==0
( ) 02
=+−+ vaGLdt
vdLC
In the frequency domain, the characteristic equation at the equilibrium point is given by:
( ) 012 =+−+ saGLLCsSolving for the poles of the system:
βα jC
aGLC
jC
aGs
±=
⎟⎠⎞
⎜⎝⎛ −
−±⎟⎠⎞
⎜⎝⎛ −
−=
2
12
2
2,1
In the time domain:
teAtv t βα cos)( =
Oscillationenvelope
Oscillation
12
CGa
teAtv t
2 where
cos)(−
=
=
α
βα
From the above equ., the following observations can be made:
1. The oscillator frequency is determined by
2. if
3. if
LC1
≅β
( )aGo <>α
Decaying oscillation
G < a G > a
RHP LHP
t
t
( )aGo >< α
growing oscillation.
13
In practical oscillators, the slope of the negative resistance or nonlinear element (a) is made
greater than G . This results in growing oscillations as long as the swing is limited
between points p and q.
( )Ga 3≅
p q
( )Ga −−
If the oscillation swing grows beyond p or q, the
slope (a-G) or α become negative and the poles
move the LHP causing the oscillation to decay
tempe. Until the swing drops within below p
and/or q. This is how a sustained oscillation is
produced and a steady sate can be reached.
Note that the oscillations will not be pure sinusoid. The harmonics are
Usually far out and can be easily filtered.
Also note that it is hard to determine the oscillation swing exactly!
14
The van der Pol Approximation
In the 1920’s , Van der Pol proposed to model the total I-V Chcis the simplest possible manner. He assumed a cubic approximation.
( )[ ]vvsii rgm +
( ) 311 vbvavi +−=
where( )vi
xv−
xv v21
1
xvGab
Gaa−
=
−=
Van der Pol’s analysis leads to the following time domain expression for the Oscillation voltage.
( )
CL where
cos1 34)(
1
21
1
1
a
LCte
batv o
LCott
=
⎟⎠⎞
⎜⎝⎛ +⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛+=
−−−
ε
φε
Φο is introduced to provide the proper phase in relation to the constant to
15
At steady state, the zero-to-peak voltage amplitude reaches a maximumvalue of
xvbav 15.1
34
1
1max ==
16
Basic LC Oscillator Topologies(a) Feedback model:
ii
Σ
-Gm+-
vGi mf =
if+ii vL C R
( ) ( )
( ) ( ) 1/
11 when
1
)1(
222
22
+−+=
+−+=
++=
++=
−=
=−
+=+=
mi
omi
o
mi
im
miif
GGsLLCssLi
cGGsscsiv
CGss
cssL
sCGZ
ZGZiv
ZiZGv
ZvGiZiiv
ω
ω
( ) 01s
:sticcharacteri The2 =+−+ sGGLLC m NOTE: Similarly to that on pp 11
17
(b) Negative Resistance Model:
L C R
vGm−
ri
+
-mG
vgm
v
negative resistance
Negative Resistance Realization:
v
vα
( )vgi m α−=
( ) 1 ,11
<−
−=−
= αα
αα mm g
viG
A MOS device can also be used.
v
vgmmg1
−
v
18
See the sustained on the next page. The frequency LC1
ggain=-100.0uv
v
Tvpairs=3Periodic:FALSE
Delay:0V2
G0C15C:1e-12 oL12
I:1e-9
VOR22R=10000
gnd
Rgm
1=
A small narrow pulse to start simulations.The pulse (to produce freq. at µi. o
p fT 1
<
19
20
21
22
23
Full Oscillator Realization:
To avoid loading the LC tank (to prevent reducing its Q), an impedance transformerto up-convert the impedance . ( )12 ZZ >
Different realization for the impedance transformation are shown below:
LCTank
ImpedanceTransformer
CCV
v2Z
1Zvα
CCV1:n
CCVCCV
1C
2CL 1L
2LC
12
2 ZnZ = ( ) 12
212 1 ZCCZ += ( ) LZLLZ 2122 1+=
COLPITTS OSC. HARTLEY OSC.
24
Note that a resonance, the phase shift of the loop is supposed to be zero (according toThe Barkhausan Criterion). That is why the output signal is fed back to the emitter. TheZero-phase condition may be satisfied if the output signal is inverted and then fed back to the base as shown below:
This topology is widely used and is known as the –v Gm oscillator
inverter
Load
CCV
LC
2Q 1Q
C
Lv v−
2Q 1Q
25
It is interesting to note that the cross-coupled pair (BJT an MOS) presents a negativeresistance to the tank. That is why it can be classified as a negative resistance oscillator.
Since it is a fully differential circuit we may model the cross-coupled pair as follows:
a’av
v−vgm−
vv−
vgm−
The resistance between a and a’ can be expressed as:
( )mm gvg
vvR 2−=
−−−
=
26
1 – BJT Realization:CCV
L LvpC vpC
C′ C′
BBV
The capacitor divider is used to reduce the voltage swing at the base compared tothat at the collector to avoid saturating the BJTs.
vpvp
collectorbase CCCC
Cvv >′
′+×=
The AC coupling resistors add noise.
27
2 – NMOS (or PMOS) Realization
It is similar to the BJT except that it does not need capacitive division.
3 – CMOS Realization:
L
C
CCV
1P 2p
1v 2v
1N 2N
oI
28
Current Limited Oscillation
In this case the current of the tail current sources is fully switched from one side of thepair to the other at a frequency Assuming that the BJT or MOS switches
are fast enough, the current waveform in each branch is a square wave with 50% dutycycle. Assuming that the tail current is Io , the current wave form in one side of thepair has the following shape.
LCo1
=ω
oI
00 T 2T 3T
This can be described by its power series as
( ) ⎟⎠⎞
⎜⎝⎛ +++= LttIi ooof ω
πω
π3sin
34sin41
21
NOTE: If the current wave form deviate from a square wave which is typically the
π4
case the factor will change to some other number depending on the
on the current waveform shape!!
29
Consider the NMOS oscillator:
LCo1
=ω
2LR 2LR2L2LCCV
C2 C21v
2v
oIoI
2LC2
1v
222 RRQ LL =
L
oL R
LQ ω=
2//
2QR 2
sinsin2
2sin2)(
22L2112
1
CC
Loppp
oo
oooooo
RQRRIvvv
tRItRIZtItv
=⋅=−=
==×⎟⎠⎞
⎜⎝⎛= ==
π
ωπ
ωπ
ωπ ωωωω
NOTE: The current components at 3ωο, 5ωο, 7ωο … will produce voltage componentswith small power because the impedance at these frequencies is small.
30
C
GHzLC
LCC
LL
o
2092R
2.314 25
3.628 50
56.121
2
=
==
==
==ω
Switching of square wave current (pulses)
31
32
Now consider the CMOS oscillator with a similar tank as that used for the NMOSOscillator (pp 24).
In one half cycle, the devices P1 and N2 are on while N1 and P2 are off, so the oscillator can be modeled as follows:
On the second half cycle, the current through R is still Io but flows in the oppositedirection. So the current waveform (assuming that the PMOS and NMOS devicesswitch enough) looks as follows:
RIo
Io
Io
L
v1v1
C
+Io
-Io
33
This is different from the wave form previously shown, because it is bipolar. TheFourier expansion for the CMOS oscillator is:
⎟⎠⎞
⎜⎝⎛ ++−= LtttIti oooo ω
πω
πω
π5sin
543sin
34sin4)(
The differential voltage of the CMOS oscillator is hence, given by:
RIv
tRItv
op
oo
o
π
ωπωω
4
sin4)(
12
12
=
⋅==
• Note that swing of the CMOS oscillator is double that of the NMOS (or PMOS)oscillator as seen before.
34
Some Important Points:
1. In the oscillators cases we considered so far we have assumed that the swing issmall enough so that the swing is determined by the current and the tank parallel resistance. If this is the case, we consider the oscillator to be current limited. IfThe swing grows too much, the swing will be limited by the supply voltage and in the oscillator is known to be voltage limited.
2. In current limited cases, the swing increases by increasing the current (tail current)or by using high Q tank (which increases the resistance.).
3. The switching speed of the core devices will determine the current waveform shapewhich in turn determines magnitude of I at ωo (fundamental component).
35
Voltage Controlled Oscillators
This is one of the most critical blocks in any RF frequency synthesizers. Adjustableoscillators are used for tuning to desired frequencies. A VCO is used as part of aPLL to obtain precise frequencies.
The frequency tuning is done by changing the LC tank capacitance value. This is morepractical than varying the inductance. Different types of variations have been used toobtain a voltage controlled capacitance. Among them are pm junction varactors andMOS varactors and MOS varactors.
In a pn junction varactor the oscillator output node is connected to either the p or n sideand the controlling voltage is applied to the opposite terminal
orVosc.VC VC
Vosc.
36
(VC)
MOS varactor come in two types. Let us consider NMOS varactors.
Conventional MOS
p+ p+
p-well
VoscVW
On this device evenat high frequenciesthe capitance willreturn to Cox at some point.
n+ n+p-well
Vosc
VW(VC)
MOS Cap
For highfrequency
VW
37
For MOS Cap:
OXCC
modeon Accumulati0
=
⇒<GWV
dVdQ
C
CCC
VV
depdep
depox
dep
GWT
=
+=
⇒>>
oxCC
modeDepletion 0
M O S
Cox
Qm
Qssm QQV ∆∆∆ &→
depQV ∆∆ →
Cox
Qm
Qdep
Cdep
38
Qm
Qdep
Qinv
0=→ depinv QQV ∆∆∆
VGW > VT
At low frequency:
Since the frequency is low, the minoritycharges have time to be pulled from thebulk to the surface to make up the inv.layer and the depletion layer does notchange any more so
oxinv C
dVdQC ==
At high frequency:
There is no time to attract minority changesto the surface. So the change comes form the dep layer
depox
depox
CCCC
C+
=
39
An ideal VCO has the following law
CVCOoosc VKff +=
Free runningfrequency
VCO gain(Hz/V)
Control
fosc
fo
oVC
40
The spectrum of is up-converted to )(tnφ Cω±
Cω
Ideal Oscillator
Qn
Cω
Real Oscillator
The phase noise is measured by dividing the power in 1Hz bandwidth at an offset ∆fby the power of the carrier (fc).
( ) HzdBffL c⎟⎠
⎞⎜⎝
⎛==
carrier theofpower offset at 1Hzin power 40log noise Phase ∆∆
41
~
Implications of Phase Noise
In the case of receiver:
If wantedsignal
Interfer
Signal
Interfer
RFIF
LOOscillator
LOωc
Overlap of theinterference withthe wanted signalafter down conversion
42
Wantedsignal
ωTX
TXRX
TXω~
In the case of transmitter:
If the phase noise of the LO at the transmitters side is large, the skirtof the RF transmitted signal overlaps nearby wanted signals and hencereduces the SNR.
43
f1 f2f∆
Wantedsignal
Pint)( fS
Interferencewith skirtdue to phase noise
BWch
Assume that, from the blocking specs, that
Pint - Psig = YdB
Psig = the signal power integrated over
channel BW.
Assume also that a given signal to noise ratio (SNR) is to be met.
( )( ) ( )
( )
( )( ) HzdBYSNRBW
HzdBYSNRBW
PP
PP
BWPBWP
PBWP
PP
BWPffPPff
PfS
dffSP
cchn
cchn
sig
sig
n
chch
nchnnon
chnonon
non
nffn
++−=−−−=
×=×===
×=−=∴
∫==
log10log10
11
.&between constant is theassume simplicityfor
power noise The
intintintint
12
21
21
φφ
φ
44
LEESON’S MODEL
Based on a linear time in variant approach for timed LC tank oscillators, one canderive an expression for the phase noise.
In(w) L C GL -GactiveEffective noise source(white noise)
It can be shown that impedance seen by the effective noise source at ωo + (where is given by:
( )
tank theof Q loaded theis tank theof econductanc parasitic parallel theis
21
L
L
L
o
Loo
QG
QGjZ
ω∆ωω∆ω −=+
oω∆
oω∆oωω∆ <<
45
The phase noise (for the case of white noise) can, hence be given by:
( )
⎟⎟⎠
⎞⎜⎜⎝
⎛=
⋅+=
=
ω∆ω
ω
L
o
Lsig
n
sig
no
sig
noise
QGV
i
V
iBWZ
vvBWL
21
ˆ2121
log10
ˆ21
21
log10
log 10)(
2
2
2
22
2
2
2ni can be written as 4 F KT GL this expression for the effective noise curent density is
not physical. (Just a model!) F is difficult to derive and is just a fitting parameter!
Lsigos
L
o
sig
GViP
QPFKTL
ˆ21
22log10)(
2
==
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
ω∆ωω∆
The rms power dissipated in GL.
46
Even though the LTI model is not accurate, it yields important facts for oscillator design:
( )
( )
( ) .oscillator theinside noise reduce 3.
and increasingby increase1 2.
increase1 1. 2
⇒
⇒
⇒
FL
IQPP
L
L
Lss
LL
αω∆
αω∆
αω∆
Leeson’s model is not accurate and fails top account for the large signalBehavior of the oscillator. Yet it gives insight for oscillator design.
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