Download Chapter 5 Low-Noise Design Methodology

Chapter 7
Oscillators
1
Instabilities, Oscillations and Oscillators
• If positive feedback is applied to an amplifier, the feedback signal is in
phase with the input, a regenerative situation exists.
• If the magnitude of the feedback is large enough, an unstable circuit is
obtained.
• To achieve the oscillator circuit function, we must ensure an unstable
situation. In addition we need to develop the oscillatory power at a
desired frequency, with a given amplitude and with excellent
constancy of envelope amplitude and frequency.
• The design of good oscillators can be quite demanding because the
governing equations of an oscillator are nonlinear, differential
equations. Consequently oscillator analysis and design are not as
advanced as that for linear circuits.
• Typical oscillator analysis involves reasonably simple approximate
analyses of linearized or piecewise-linear-circuit models of the
oscillator together with perturbations and power series techniques.
• There are a few oscillator circuits that can be solved exactly.
2
Ideal Electronic Oscillator
• An ideal harmonic oscillator can be modeled by a lossless L-C circuit.
Due to its lossless nature energy is conserved and alternates between
2
electrical and magnetic forms. The circuit equation is d v   2v  0
1
0
2
2


dt
• where 0 LC
• The solution is v  V1 exp( j 0t )  V2 exp(  j 0t )
•
•
•
•
 V sin  0t
where for the latter expression, the time origin is chosen to produce a zero
phase angle. In the frequency domain, the characteristic equation of the
2
2
circuit is s   0  0
The roots of the equation are s1 , s2   j0 which lie on the imaginary
axis of the complex plane.
A real oscillator will have loss leading to a damped sinusoid. In the
frequency domain, the characteristic equation includes a linear term as
shown s 2  2s   02  0
The natural frequencies, which are the roots, lie in the left-hand plane
3
For  0  

s1 , s2    j  02   2

0.5
   j 0
• If oscillation are to be maintained, energy must continuously be
supplied to the circuit on a time average. This power is usually
supplied by dc bias to the devices that convert the bias power into
signal power in the form of a negative, nonlinear conductance or as
regenerative feedback.
Tunnel-Diode Oscillator
• The model for a tunnel
-diode oscillator is shown
4
• If the diode is biased at VDD the incremental input conductance presented to
the passive resonant circuit is negative and can compensate for the positive
losses of the inductance, capacitance, leads, etc. Such losses are modeled by
a single conductance G=1/R.
'
• Shifting the axis to the Q-point, the new voltage variable is v  V  VDD .
The original diode I-V characteristic is described functionally as I  f1 (V )
• The translated diode-current variable is i '  I  I 0  I  f1 (VDD )
• The circuit equation in the original variables is 1  (VDD  V )dt  GV  C dV  f1 (V )
L
dt
• In terms of the new variables we have
•
'
1 '
dv
'
'
v
dt

Gv

C

f
(
v
)  GVDD  I 0  0
2

L
dt
dv ' dV
where

dt
dt
I 0  f1 (VDD ) and f 2 (v ' )  f1 (V )  f1 (VDD )
5
• Both sides of equation are next differentiated and multiplied by L


d 2v '
d
LC 2  L Gv'  f 2 (v ' )  v '  0
dt
dt
• The combined first-order term in the equation can be viewed as the net
nonlinearity of the oscillator.
• We need to determine if the equilibrium point of the circuit is unstable.
At the Q-point, an incremental analysis is made. For the diode, the
slope of the translated I-V characteristic at v’= 0 is designated -a. The
characteristic equation at the bias point and in the frequency domain is
LCs 2  L(G  a)s  1  0
• The natural frequencies of the linearized circuit about the bias point are
G  a 
s1 , s2   


2
C


 1  G  a 
j


LC
2
C



1/ 2
   j
6
• If the magnitude of the slope of the diode characteristic at the bias
point a, is greater than G, the loss conductance of the oscillator, the
natural frequencies lie in the right-half-plane. Therefore, given an
excitation, the oscillatory response grows exponentially. If the natural
frequencies are complex, the response is an exponentially growing
sinusoid v ' (t )  Ae  t cos t
• This growth cannot continue indefinitely. As the diode voltage
excursion gets large, the end points, p and q, of the negative slope
region of the diode I-V characteristic are encountered the diode
introduces more loss into the system. The voltage response can be
illustrated as shown below
7
• For very small excursions of the bias point, an exponential growth is
produced. As the oscillation becomes larger, the total loss introduce a
limiting condition, and a steady-state oscillation is produced. The
output cannot be pure sinusoidal, the tips of the voltage output must be
compressed, and the output waveform must contain harmonics. For
the tunnel-diode oscillator, the output voltage will be quite
nonsinusoidal in the best case since the non-linear I-V is not
antisymmetrical about the bias point in the negative-conductance
region
• It appears that by setting |a| = G, we may exactly cancel G and result in
an ideal oscillator. However, the I-V is not linear at the bias point, or
anywhere, thus the equality condition is not satisfied over the
excursion of the variable. Moreover the diode characteristics are
temperature and age dependent and therefore cancellation is only
temporary
8
van der Pol Approximation
• About the bias point of the total I-V characteristic, he proposed a cubic
polynomial approximation. The slope of the bias point is equal in
magnitude to the total conductance. The cubic term of the
approximation produces the essential limiting action
• For the tunnel-diode oscillator the nonlinear differential equation is
repeated for easy reference
d 2v
d
LC 2  L [Gv  f (v)]  v  0
dt
dt
• in which the primes are dropped. The 1st order term combines the
effects of the passive conductance and of the diode’s I-V characteristic.
It is convenient next to introduce a time scaling normalization. The
dimensionless independent variable is T  t
0.5
(LC
)
• Thus the differential equation becomes
d 2v  L 
 
2
dT
C 
0.5
d
[ F (v)]  v  0
dT
9
• where the total device and conductance function is defined as F (v)  Gv  f (v)
• A plot of this total function is the addition of f(v) and Gv. The slope of this total
function at the origin is -(a-G) where -a is the slope of the diode characteristic
alone at the bias point. The van der Pol approximation for F(v) is labeled Fv(v)
Fv (v)  a1v  b1v 3
• for v=0, the slope of the approximation is set equal to the slope of the actual
total characteristic a1  (a  G)
• To determine an appropriate value for b1 for the van der Pol approximation are
1/ 2
a 
Vx    1 
 b1 
a1
b

• This leads to 1 V 2 where  Vx are the crossover points on the voltage axis
x
• The oscillator equation containing the cubic approximation is transformed into a
standard form, from which an approximate closed-form solution for the
equation is obtained and the output is nearly sinusoidal
•
10
11
• To obtain th evan der Pol Equation we introduce the parameter 
which is called the van der Pol parameter. This is the negative of the
slope of the normalized total nonlinear function at the origin,
1/ 2
1/ 2
1/ 2
multiplied by ( L / C ) . Thus
L
L
 
 a1    (a  G )
C
 
C 
• Next, a scaling of the voltage variable is introduced
v  hu where h 
2

L
3b1  
C 
0.5
• using these parameters, we obtain the standard form of the van der Pol
equation
•
d 2u
2 du


(
1

u
)
u  0
2
dT
dT
About the equilibrium (bias) point the differential equation of the
system becomes p 2  p  1  0 where p is the normalized complex
frequency variable, p=(LC)0.5s. The natural frequencies of the
linearized, normalized system are
12
   2 

p1 , p2   j 1    
2   2  
0.5
• For positive  , the natural frequencies are in the RHP and complex if
 < 2.
For

> 0 but very small, the buildup of oscillation has the
 T 
form u (T )  A exp 
 cos T
 2 
• We expect that after a steady state has been reached, the form of the
response should be u (T )  Const  cos T van der Pol proposed a
solution which after making several reasonable assumptions and
simplifications, which are valid for near-sinusoidal oscillation, the
solution in terms of the original variables becomes
 4a 
v(t )   1 
 3b1 
0.5
 1

cos
t


0
 ( LC ) 0.5
0.5



  (t  t0 ) 
1

exp


0.5 
(
LC
)



1
13
• In this equation, a phase angle 0 is introduced to provide the proper
phase for the cosine term in relation to the choice of the constant t0.
For small values of time, the envelope of the oscillation grows
exponentially. After a long period of time, the zero-to-peak amplitude
0.5
reaches a maximum value
 4a 
Vmax   1 
 3b1 
•
Since V   a1 
x
b 
 2
0.5
 a G 

 
b
 1 
0.5
0.5
Vmax
4
   Vx  1.15Vx
3
• Since the value of the voltage at the negative peak of the total
characteristic is
0 .5


a
V    1  ; Vmax  2V 
 3b1 
• The maximum amplitude is twice the value of the voltage at which the
negative maximum occurs as shown
•
14
Simple Transistor Oscillator Circuits
• Simply put the condition for oscillation is that the total phase shift
around the loop must be 360° at the frequency of oscillation and the
magnitude of the open-loop gain must be unity at that frequency.
• If a single-stage common-emitter (or common source) amplifier is used
with feedback from the collector to base as shown, then the feedback
network must supply 180° phase shift between the base and the
collector signals
15
• If a common-bse (or common gate) amplifier is used there is no phase
shift between the emitter and collector signals, then a necessary
condition for oscillation is that there be no phase shift between the
input and output of the feedback network.
• If a small phase shift occurs in the forward loop, this must be
compensated for by an equal and opposite phase in the feedback
network.
16
• The primary purpose of the feedback network is to control the
frequency of oscillation, the network is designed so that the Nyquist
criteria are satisfied at only a single frequency.
• The following circuit is analyzed to determine the conditions for
oscillation.
• The equivalent circuit is shown in the next diagram, in which the
transistor output resistance ro is ignored, as is the large biasing resistor
RB. Also, the capacitor connected to the base is assumed to be so large
that the base is at ground potential for the small-signal analysis. Note
that the transistor is connected in the common-base configuration and
has no voltage phase inversion.
17
• The conditions for oscillation are G( jo ) H ( jo )  1 and because the
amplifier is noninverting, the phase shift of the network must also be
arg G ( j o ) H ( j o )  0 .
18
• The loop gain is calculated by opening the feedback loop, applying a
signal, and measuring the return difference. It is necessary that when
the loop is opened, the impedance seen at any point be the same as it is
with the loop closed. In this case it is convenient to open the loop at
the transistor emitter. The impedance shunting the capacitor C2 is the
resistor RE in parallel with the input impedance ri of the common-base
r
amplifier. For common-base amplifier the input resistance is
ri  
so the equivalent circuit of Fig. 7-4a is redrawn in Fig. 7-4b.

• The circuit analysis can be simplified by assuming that
 rR
1
[ (C2  C1 )]2   i E
 ri  RE





2
• and that the Q of the load impedance is high. In this case the circuit
reduces to Fig. 7-5
19
which is the small signal equivalent of the open-loop circuit with the
loop or feedback network opened at the emitter and the circuit
terminated in the correct impedance. The network could be opened at
other points, but opening it at the emitter is very convenient because it
is relatively easy to determine the terminating impedance at this point.
The feedback voltage is given by V  VoC1
and the equivalent
resistance reflected across the coil is C1  C2
rR
Req  i E
ri  RE
 C1  C2 


 C1 
2
Vo
 G ( j )  g m Z L
The forward loop gain is V
where
YL  Z L1  ( jL) 1  Req1  RL1  jC and C 
V
C1
 H ( j ) 
The feedback ratio is Vo
C1  C2
C1C2
C1  C2
•

• A necessary condition for oscillation is that arg G( j ) H ( j )  0
• Such H does not depend on frequency in this example, if arg(GH) is to
be zero, the phase shift of the load impedance ZL must be zero. This
occurs only at the resonant frequency of the circuit where
20
1/ 2 1
 C C 
 o   L 1 2 
 C1  C2 
• At this frequency



ZL 
Req RL
Req  RL
and G ( j ) H ( j )  g m
Req RL
C1
Req  RL C1  C2
• The other condition for oscillation is the magnitude constraint that
Req RL
C1
G ( j ) H ( j )  g m
1
Req  RL C1  C2
• Example
• Design a common-base sinusoidal oscillator using a transistor with a
minimum  of 100.
• Solution: Oscillator design consists of a trial-and-error procedure
using the above equations. We assume a bias current IC=1mA. The
common base input resistance
ri 
r


r
V
 g m1  T  26
g m r
IC
21
• Since this is so small, we can safely assume that the emitter bias
2
resistor RE is much larger than ri so that
 C1  C2 


• Also, the above equations were based Req  ri  C
1


2
2 1
on the assumption that [ (C2  C1 )]   ri
1
1
If we choose C2   8 , then C2   3 and the above
inequality is satisfied so the assumption was justified.
• Practically, a factor of 10 difference will usually satisfy the the “<<”
condition. In addition, for oscillations to occur, the loop gain must be
at least 1. In oscillator design the loop gain is typically selected to
about 3, which allows for some error in the approximation. With a
loop gain > 1, the system is unstable and the oscillations increase in
amplitude until the transistor current begins to saturate. When this
occurs, the  of the transistor is reduced, and thus gm is also reduced.
This reduces the loop gain and stabilizes the amplitude of oscillation.
In this example the loop gain is
GH  g m ri
C1  C2 C1  C2

3
C1
C1
22
• So C1=C2/2=500pF. The value of the inductance is found to be L=0.19
H. also, Req  234 . A load resistor RL can be shunted across the
inductor without affecting the calculations if it is much larger than Req.
In this case a resistance of 1500 could be safely added. The
complete design would require selecting a supply voltage and bias
resistors so that the Quiescent collector current is 1 mA. A completed
circuit diagram is given below:
23
Practical Consideration
• The condition T  1 may not be met due to the aging of the transistors
and the circuit. It is therefore necessary to have |T| somewhat larger
than unity (say 5%) ensure |T| does not fall below unity in case of
incidental changes of circuit parameters. In the case where T  1 the
magnitude of oscillation is limited by the onset of nonlinearity.
Phase-shift Oscillator
• A typical phase-shift oscillator is shown below. Here a discretecomponent JFET amplifier is followed by three cascaded arrangements
of a capacitor C and a resistor R, the output of the last RC combination
being returned to the gate. If the loading of the phase-shift network on
the amplifier can be neglected that is R>> RL, the amplifier shifts by
180 the phase of any voltage which appears on the gate, and the
network of resistors and capacitors shifts by an additional amount. At
some frequency the phase shift introduced by the RC network will be
precisely 180 , and at this frequency the total phase shift from the gate
around the circuit and back to the gate will be exactly zero.
24
25
• This particular frequency will be the one at which the circuit oscillates,
provided that the magnitude of the amplification is sufficiently large.
jgm RL N3
T ( j ) 
(1  6 N2 )  j N (5   N2 )
• where  N  RC
. For the circuit to oscillate T should be real and
2
jgm RL / 6 6
•  N  1 / 6 . At  2  1 / 6
T ( j N ) 
N
j (5  1 / 6) 6
• and g m RL  29 to sustain oscillation.
• It is possible to use an Op-Amp in place of the transistor as shown. In
this case R2/R1=29.
26
Wien Bridge Oscillator
• An oscillator circuit in which a balanced bridge is used as the feedback
network is the Wien bridge oscillator as shown
27
• The four arms of the bridge are Z1, Z2, R1, R2. The input to the bridge
is the output Vo of the Op-Amp, and the output of the bridge between
nodes 1 and 2 supplies the differential input to the Op-Amp. there are
two feedback paths in the figure: positive feedback through Z1 and Z2,
whose components determine the frequency of oscillation, and
negative feedback through R1 and R2, whose elements affect the
amplitude of oscillation and set the gain of the Op-Amp. The loop
gain is given by
 R2  Z 2
T ( s)  1  
 R1  Z1  Z 2
•
For Z1  ( RCs  1) / Cs and Z2  R /( RCs  1)
• The equivalent circuit is as shown
28
General Form of an Oscillator
• Many oscillator fall into the general form as shown.
• In this case the return ratio is
V
T   13
Vˆ13
T
AvVˆ13Z L
Vo 
Z L  Ro
V13 
Z2
Vo
Z1  Z 2
Av Z1Z 2
Ro ( Z1  Z 2  Z 3 )  Z 2 ( Z1  Z 3 )
29
LC-Tunable Oscillators
• The oscillators described so far are RC tunable circuits which are often
limited to operations in the range of a few hundred kHz. For higher
frequencies LC tunable circuits are needed. Two types of LC-tunable
oscillators are commonly used.
30
•
From T 
Av X 1 X 2
jRo ( X 1  X 2  X 3 )  X 2 ( X 1  X 3 )
Z1  jX1
Z 2  jX 2
Z 3  jX 3
• where
X  L for inductance and - 1
•
C for capacitors
Crystal Oscillators
• If a piezoelectric crystal, usually quartz, has electrodes plated on
opposite faces and a potential is applied between these electrodes,
forces will be exerted on the bound charges within the crystal. If this
device is properly mounted, deformations takes place within the
crystal, and an electromechanical system is formed which will vibrate
when properly excited. Frequencies ranging from a few kHz to a few
hundred MHz. Q values range from several thousand to several
hundred thousand. The equivalent circuit of a crystal is shown below
L,C and R are analogs of mass, compliance (reciprocal of spring
constant) and viscous-damping factor of the mechanical system. If we
neglect R the impedance of the crystal is
31
j  2   s2
jX  
C '  2   2p
2
• where  s  1 / LC
•
 2p  1/ L(1 / C  1/ C ' )
32
• A variety of crystal-oscillator circuits is possible. If a crystal is used
for Z1 in the basic configuration shown before, a tuned LC
combination for Z2 and capacitance Cgd between the gate and drain for
Z3, the resulting circuit is as indicated
33