Sinusoidal oscillators

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2. Objectives
 Describe the basic concept of an oscillator
 Discuss the basic principles of operation of an
oscillator
 Analyze the operation of RC oscillators

3. Introduction
 Oscillator is an electronic circuit that generates a
periodic waveform on its output without an external
signal source. It is used to convert dc to ac.
 Oscillators are circuits that produce a continuous
signal of some type without the need of an input.
 These signals serve a variety of purposes.
 Communications systems, digital systems
(including computers), and test equipment make use
of oscillators

4. Introduction
 An oscillator is a circuit that produces a repetitive signal from
a dc voltage.
 The feedback oscillator relies on a positive feedback of the
output to maintain the oscillations.
 The relaxation oscillator makes use of an RC timing circuit to
generate a nonsinusoidal signal such as square wave
Sine wave
Square wave
Sawtooth wave

5. Types of oscillators
1.RC oscillators
Wien Bridge
Phase-Shift
1.LC oscillators
Hartley
Colpitts
Crystal
1.Unijunction / relaxation oscillators

6. Feedback Oscillator Principles
 When switch at the amplifier input is open, no oscillation occurs.
 Consider V
i,
, results in V
o
=AV
i
(after amplifier stage) and V
f
= b(AV
i
)
(after feedback stage)
 Feedback voltage V
f
= b(AV
i
) where bA is called loop gain.
 In order to maintain V
f
= V
i
, bA must be in the correct magnitude and
phase.
 When the switch is closed and V
i
is removed, the circuit will continue
operating since the feedback voltage is sufficient to drive the amplifier and
feedback circuit, resulting in proper input voltage to sustain the loop
operation.

Feedback circuit used as an oscillator

7. Basic principles for oscillation
 An oscillator is an amplifier with positive feedback.
A
b
V
e
V
f
V
s
V
o
+
(1)
fse VVV +=
(2)
ofβVV=
( ) ( )(3)
osfseo βVVAVVAAVV +=+==

8. Basic principles for oscillation
 The closed loop gain is:
( ) ( )
osfs
eo
βVVAVVA
AVV
+=+=
=

oso VAAVV b+=
( )
soAVVA =-b1
( )Aβ
A
V
V
A
s
o
f
-
=º
1

9. Basic principles for oscillation
 In general A and b are functions of frequency and
thus may be written as;
is known as loop gain
() ()
()
()()sβsA1
sA
s
V
V
sA
s
o
f
-
==
()()sβsA

10. Basic principles for oscillation
 Writing the loop gain becomes;
 Replacing s with jw
 and
() ()()ssβAsT=
()
()
()sT1
sA
sA
f
-
=
( )
( )
( )jωT1
jωA
jωA
f
-
=
( ) ( )( )jωβjωAjωT =

11. Basic principles for oscillation
 At a specific frequency f
0


At this frequency, the closed loop gain;
will be infinite, i.e. the circuit will have finite output
for zero input signal - oscillation
( ) ( )( )1
000
== jωβjωAjωT
( )
( )
( )( )
00
0
0
jωβjωA1
jωA
jωA
f
-
=

12. Basic principles for oscillation
 Thus, the condition for sinusoidal oscillation of
frequency f
0
is;
 This is known as Barkhausen criterion.
 The frequency of oscillation is solely determined by
the phase characteristic of the feedback loop – the
loop oscillates at the frequency for which the phase
is zero.
( )( )1
00
=jωβjωA

13. Basic principles for oscillation
 The feedback oscillator is widely used for
generation of sine wave signals.
The positive (in phase) feedback arrangement
maintains the oscillations.
The feedback gain must be kept to unity to keep the
output from distorting.

14. Basic principles for oscillation
In phase
Noninverting
amplifier
V
f V
o
A
v
Feedback
circuit

15. Design Criteria for Oscillators
1.The magnitude of the loop gain must be unity or
slightly larger
– Barkhaussen criterion
2.Total phase shift,f of loop gain must be 0 ° or 360°
1=Aβ

16. RC Oscillators
 RC feedback oscillators are generally limited to
frequencies of 1 MHz or less.
 The types of RC oscillators that we will discuss are
the Wien-bridge and the phase-shift

17. Wien-bridge Oscillator
 It is a low frequency oscillator which ranges from a
few kHz to 1 MHz.
The Wien-bridge oscillator schematic drawn in two different but equivalent ways

18. Oscillator Output Gain and Phase Shift

19. Wien-bridge Oscillator
 The loop gain for the oscillator is;
 where;
 and;
() ()()
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
+==
sp
p
ZZ
Z
R
R
sβsAsT
1
2
1
sRC
R
Z
p
+
=
1
sC
sRC
Z
s
+
=
1

20. Wien-bridge Oscillator
 Hence;
 Substituting for s;
 For oscillation frequency f
0
;
( )
( )
ú
û
ù
ê
ë
é
++
÷
÷
ø
ö
ç
ç
è
æ
+=
RC/jRCjR
R
jT
001
2
0
13
1
1
ww
w
()
( )
ú
û
ù
ê
ë
é
++
÷
÷
ø
ö
ç
ç
è
æ
+=
/sRCsRCR
R
sT
13
1
1
1
2
( )
( )
ú
û
ù
ê
ë
é
++
÷
÷
ø
ö
ç
ç
è
æ
+=
RC/jRCjR
R
jT
ww
w
13
1
1
1
2

21. Wien-bridge Oscillator
 Since at the frequency of oscillation, T(jw) must be
real (for zero phase condition), the imaginary
component must be zero;
 Which gives us;
0
1
0
0 =+
RCj
RCj
w
w
RC
1
0=w

22. Wien-bridge Oscillator
 From the previous equation;
 the magnitude condition is;
or÷
ø
ö
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
+=
3
1
11
1
2
R
R
( )
( )
ú
û
ù
ê
ë
é
++
÷
÷
ø
ö
ç
ç
è
æ
+=
RC/jRCjR
R
jT
001
2
0
13
1
1
ww
w
2
1
2
=
R
R
To ensure oscillation, the ratio R
2
/R
1
must be
slightly greater than 2.

23. Wien-bridge Oscillator
 With the ratio;
 then;
K = 3 ensures the loop gain of unity – oscillation
K > 3 : growing oscillations
K < 3 : decreasing oscillations
2
1
2
=
R
R
31
1
2
=+º
R
R
K

24. T i me
0 s 0 . 2 ms 0 . 4 ms 0 . 6 ms 0 . 8 ms 1 . 0 ms
V ( R5 : 2 )
- 4 . 0 V
0 V
4 . 0 V
G = 3
T i me
0 s 0 . 2 ms 0 . 4 ms 0 . 6 ms 0 . 8 ms 1 . 0 ms
V ( R5 : 2 )
- 4 . 0 V
0 V
4 . 0 V
G = 2.9
T i me
0 s 1 0 0 u s 2 0 0 u s 3 0 0 u s 4 0 0 u s 5 0 0 u s6 0 0 u s
V ( R5 : 2 )
- 2 0 V
0 V
2 0 V
G = 3.05

25. Ideal vs. Non-Ideal Op-Amp
Red is the ideal op-amp.
Green is the 741 op-amp.
T i me
0 s 0 . 2 ms 0 . 4 ms 0 . 6 ms 0 . 8 ms 1 . 0 ms
V ( R1 : 2 )V( R5 : 2 )
- 4 . 0 V
0 V
4 . 0 V

26. Start-Up Conditions
-Initially, the closed-loop gain of the amplifier itself must be
more than 3 until the output signal builds up to a desired
level.
-Ideally, the gain of the amplifier must then decrease to 3 so
that the total gain around the loop is 1 and the output signal
stays at the desired level, thus sustaining oscillation.
- This is illustrated in Figure on next slide.

27. In order to keep the oscillations constant, Hewlett Packard put a positive
temperature co-effient lamp in the circuit at grounding resistor.
The resistance of the lamp is strongly dependent on the temperature of
the filament of the bulb. If the amplitude is too high, the current becomes
large and the resistance of the lamp increases, thereby reducing the gain.
If the amplitude is low, the lamp cools, the resistance decreases, and the
loop gain increases.

28. The feedback fraction at f
R
in this circuit is one-third:
A must be > 3 for oscillations to start. After that, A must
be reduced to avoid driving the op amp to V
SAT
.
in
out
B =
in
out
=
1
3
R
2
@ 2R
1
R
1
A = 1 +
R
2
R
1
One solution is a positive
temperature coefficient
device here to decrease gain.

29. After the
oscillations
start, the
lamp heats
to reduce
gain and
clipping.
R
V
out
C
R
L
2R
1
Tungsten
lamp
C R
R
1
V
out
time

30. Making the Oscillations Steady
Add a diode
network to keep
circuit around G =
3
If G = 3, diodes are
off

31. Making the Oscillations Steady
When output
voltage is positive,
D1 turns on and R9
is switched in
parallel causing G
to drop

32. Making the Oscillations Steady
When output
voltage is negative,
D2 turns on and R9
is switched in
parallel causing G
to drop

33. Phase-Shift Oscillator
Phase-shift oscillator
 The phase shift oscillator utilizes three RC circuits to provide
180º phase shift that when coupled with the 180º of the op-amp
itself provides the necessary feedback to sustain oscillations.

35. Phase-Shift Oscillator
vi
v1
v1
v2
v2v3
vo
C
C
C
R
R
R
R2
i
v
sRC
sRC
v ÷
ø
ö
ç
è
æ
+
=
1
1
iv
sRC
sRC
v
2
2
1
÷
ø
ö
ç
è
æ
+
=
iv
sRC
sRC
v
3
3
1
÷
ø
ö
ç
è
æ
+
=
3
3
1
)( ÷
ø
ö
ç
è
æ
+
==
sRC
sRC
s
v
v
i
b
R
R
v
v
sA
o 2
3
)( ==

36. Phase-Shift Oscillator
 Loop gain, T(s):
 Set s=jw
3
2
1
)()()( ÷
ø
ö
ç
è
æ
+
÷
ø
ö
ç
è
æ
==
sRC
sRC
R
R
ssAsT b
[ ] [ ]
222222
2
2
3
2
331
))((
)(
1
)(
CRRCjCR
RCRCj
R
R
jT
RCj
RCj
R
R
jT
www
ww
w
w
w
w
-+-
÷
ø
ö
ç
è
æ
-=
÷
÷
ø
ö
ç
ç
è
æ
+
÷
ø
ö
ç
è
æ
=

37. Phase-Shift Oscillator
 To satisfy condition T(jw
o
)=1, real component must
be zero since the numerator is purely imaginary.
 the oscillation frequency:


Apply w
o
in equation:
To satisfy condition T(jw
o
)=1
031
222
=- CRw
RC3
1
0
=w
[ ]
÷
ø
ö
ç
è
æ
÷
ø
ö
ç
è
æ
-=
-+
÷
ø
ö
ç
è
æ
-=
8
1
)3/1(3)3/(0
)3/1)(3/(
)(
22
R
R
j
j
R
R
jT
ow
8
2
=
R
R
The gain greater than 8, the circuit will
spontaneously begin oscillating & sustain
oscillations

38. 62
1
RC
f
p
=
where b = 1/29 and the phase-shift is 180
o
 For the loop gain bA to be greater than unity, the gain of the amplifier
stage must be greater than 29.
 If we measure the phase-shift per RC section, each section would not
provide the same phase shift (although the overall phase shift is 180
o
).
 In order to obtain exactly 60
o
phase shift for each of three stages,
voltage follower stages would be needed for each RC section.
when voltage follower is not used b/w RC stages

39. RC
f
o
62
1
p
= 29
2
=
R
R
The gain must be at least
29 to maintain the
oscillations

40. LC Oscillators
 Use transistors and LC tuned circuits or crystals in
their feedback network.
 For hundreds of kHz to hundreds of MHz frequency
range.
 Examine Hartley, Colpitts and crystal oscillator.

41. Hartley oscillator
Hartley oscillator was invented in 1915 by the American
engineer Ralph Hartley while he was working for the Western
Electric company. The original design was tube based and he
got a patent for it in the year 1920.
In Hartley oscillator the oscillation frequency is determined
by a tank circuit comprising of two inductors and one
capacitor. The inductors are connected in series and the
capacitor is connected across them in parallel.
Hartley oscillators are commonly used in radio frequency (RF)
oscillator applications and the recommended frequency
range is from 20KHz to 30MHz.

42. In the circuit diagram resistors
R1 and R2 give a potential
divider bias for the transistor
Q1.
Ce is the emitter by pass
capacitor, which by-passes the
amplified AC signals. If the
emitter by-pass capacitor not
there, the amplified ac voltages
will drop across Re and it will get
added on to the base-emitter
voltage of Q1 and will disrupt
the biasing conditions.

43. Cin is the input DC decoupling
capacitor while Cout is the output DC
decoupling capacitor. The task of a DC
decoupling capacitor is to prevent DC
voltages from reaching the succeeding
stage. Inductor L1, L2 and capacitor C1
forms the tank circuit.
When the power supply is switched ON
the transistor starts conducting and
the collector current increases. As a
result the capcitor C1 starts charging
and when the capacitor C1 is fully
charged it starts discharging through
coil L1. This charging and discharging
creates a series of damped oscillations
in the tank circuit and it is the key.

44. The oscillations produced in the tank circuit
is coupled (fed back) to the base of Q1 and it
appears in the amplified form across the
collector and emitter of the transistor. The
output voltage of the transistor (voltage
across collector and emitter) will be in phase
with the voltage across inductor L1. Since
the junction of two inductors is grounded,
the voltage across L2 will be 180° out of
phase to that of the voltage across L1.
The voltage across L2 is actually fed back to
the base of Q1. the feed back voltage is 180°
out of phase with the transistor and also the
transistor itself will create another 180°
phase difference. So the total phase
difference between input and output is 360°
and it is very important condition for
creating sustained oscillations.

45. Frequency of the Hartley oscillator.
The frequency “F” of a Hartley oscillator can be expressed using the equation;
C is the capacitance of the capacitor C1 in the tank circuit.
L = L1+L2, the effective series inductance of the inductors L1 and L2 in
the tank circuit.
Here the coils L1 and L2 are assumed to be winded on different cores. If
they are winded on a single core then L=L1+L2+2M where M is the
mutual inductance between the two coils.

46. Colpitts Oscillator
Colpitts oscillator was invented by American
scientist Edwin Colpitts in 1918. It is another type of
sinusoidal LC oscillator which has a lot of
applications. The Colpitts oscillator can be realized
using transistors, FETs or op-amp.
In Colpitts oscillator the tank circuit consists of two
capacitors in series and an inductor connected in
parallel to the serial combination. The frequency of
the oscillations are determined by the value of the
capacitors and inductor in the tank circuit.

47. Collpitts oscillator is generally used in RF
applications and the typical operating range is
20KHz to 300MHz.
In Colpitts oscillator, the capacitive voltage divider
setup in the tank circuit works as the feed back
source and this arrangement gives better frequency
stability.

48. In the circuit diagram
resistors R1 and R2 gives a
voltage divider biasing to the
transistor. Resistor R4 limits
the collector current of the
transistor.
Cin is the input DC
decoupling capacitor while
Cout is the output decoupling
capacitor. Ce is the emitter
by-pass capacitor. Job of the
emitter by-pass capacitor is
to by-pass the amplified AC
signals from dropping across
Re.

49. If the emitter by-pass
capacitor is not there, the
amplified AC signal would
have dropped across Re
and it may have altered
the DC biasing conditions
of the transistor and the
result will be reduced gain.
Capacitors C1, C2 and
inductor L1 forms the tank
circuit. Feedback to the
base of transistor is taken
from the junction of
Capacitor C2 and inductor
L1 in the tank circuit.

50. When power supply is switched
ON, capacitors C1 and C2 starts
charging. When they are fully
charged they starts discharging
through the inductor L1. When
the capacitors are fully
discharged, the electrostatic
energy stored in the capacitors
gets transferred to the inductor
as magnetic flux.
The inductor starts discharging
and capacitors gets charged
again. This transfer of energy
back and forth between
capacitors and inductor is the
basis of oscillation.

51. Voltage across C2 is phase
opposite to that of the voltage
across the C1 and it is the voltage
across C2 that is fed back to the
transistor.
The feedback signal at the base of
transistor appears in the
amplified form across the
collector and emitter of the
transistor.
The energy lost in the tank circuit
is compensated by the transistor
and the oscillations are sustained.

52. The tank circuit produces 180°
phase shift and the transistor itself
produces another 180° phase shift.
That means the input and output
are in phase and it is a necessary
condition of positive feedback for
maintaining sustained oscillations.
The frequency of oscillations of the
Colpitts oscillator can be
determined using the equation.

53. Where L is the inductance of the inductor in the tank circuit and C is the
effective capacitance of the capacitors in the tank circuit.
If C1 and C2 are the individual capacitance, then the effective capacitance
of the serial combination C= (C1C2)/(C1+C2). By using ganged variable
capacitors in place of C1 and C2, the Colpitts oscillator can be made
variable.
Advantages of Colpitts oscillator.
Main advantage of Colpitts oscillator over Hartley oscillator is the
improved performance in the high frequency region. This is because the
capacitors provide a low reactance path for the high frequency signals and
thus the output signals in the high frequency domain will be more
sinusoidal. Due to the excellent performance in the high frequency region,
the Colpitts oscillator can be even used in microwave applications.

54. Crystal Oscillator

55. Crystal Oscillator
 Most communications and digital applications require the
use of oscillators with extremely stable outputextremely stable output. Crystal
oscillators are invented to overcome the output fluctuationoutput fluctuation
experienced by conventional oscillators.
Crystals used in electronic applications consist of a quartz
wafer held between two metal plates and housed in a
package as shown in Fig. (a) and (b).

56. In crystal oscillators, the usual electrical resonant circuit is
replaced by a mechanically vi­brating crystal. The crystal
(usually quartz) has a high degree of stability in holding con­
stant at whatever frequency the crystal is originally cut to
operate.
The crystal oscillators are, therefore, used whenever great
stability is needed, for example, in communication trans­
mitters, and receivers, digital clocks etc.
A quartz crystal exhibits a very important property known
as piezo-electric effect.

57. Crystal Oscillator
 Piezoelectric Effect
The quartz crystal is made of silicon oxide (SiO
2
) and
exhibits a property called the piezoelectricpiezoelectric
When a alternating voltage is applied across the crystal, it
vibrates at the frequency of the applied voltage.
The thinner the crystal, higher its frequency of vibration.
This phenomenon is called piezoelectric effect.

58. Crystal Oscillator
 Characteristic of Quartz
Crystal
The crystal can have two resonant
frequencies;
One is the series resonance frequency f
1

which occurs when X
L
= X
C
. At this
frequency, crystal offers a very low
impedance to the external circuit where
Z = R.
The other is the parallel resonance (or
anti­resonance) frequency f
2
which
occurs when reactance of the series leg
equals the reactance of C
M
. At this
frequency, crystal offers a very high
impedance to the external circuit
R
L
C
C
M

59. Since, in series resonance,
the crystal impedance is
smallest, it can cause the
crystal to provide the
largest positive feedback.
C
M
R
L
C
C
M

60. Crystal Pierce Oscillator
To excite a crystal for operation in the
series­resonant mode it may be
connected as a series element in a
feedback path, as shown in figure.
In this mode of operation the crystal
impedance is the smallest and the
amount of positive feedback is the
largest.
Resistor R1, R
2
and R
E
provide a
voltage­divider stabilized dc bias
circuit, the capacitor C
E
provides ac
bypass of the emitter resistor Re and
the radio­frequency coil (RFC)
provides for dc bias ­while decoupling
any ac signal on the power lines from
affecting the output signal.

61. The coupling capacitor C
c
has negligible
impedance at the circuit operating frequency
but blocks any dc between collector and base.
The resulting circuit frequency of oscillations
is set by the series resonant frequency of the
crystal.
Variations in supply voltage, transistor
parameters, etc. have no effect on the circuit
operating frequency which is held stabilized by
the crystal.
The circuit frequency stability is set by the
crystal frequency stability, which is good.

62. Colpitts Quartz Crystal Oscillator
The design of a Crystal Oscillator is very
similar to the design of the Colpitts
Oscillator except that the LC tank
circuit that provides the feedback
oscillations has been replaced by a
quartz crystal.
These types of Crystal Oscillators are
designed around the common emitter
amplifier stage of a Colpitts Oscillator.
The input signal to the base of the
transistor is inverted at the transistors
output. The output signal at the
collector is then taken through a 180
o

phase shifting network which includes
the crystal operating in a series
resonant mode.

63. The output is also fed back to the
input which is “in­phase” with the
input providing the necessary
positive feedback.
Resistors, R1 and R2 bias the
resistor in a Class A type
operation while resistor Re is
chosen so that the loop gain is
slightly greater than unity.

64. The circuit diagram of the
Colpitts Crystal Oscillator circuit
shows that capacitors, C1 and C2
shunt the output of the transistor
which reduces the feedback
signal.
The output amplitude should be
kept low in order to avoid
excessive power dissipation in
the crystal otherwise could
destroy itself by excessive
vibration.