Oscillators
Electronic Engineering
© University of Wales Newport 2009 This work is licensed under a Creative Commons Attribution 2.0 License.
The following presentation is a part of the level 5 module -- Electronic Engineering. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1 st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
Contents Oscillator Circuits Wien Bridge Oscillator R. C. Phase Shift Oscillator. Tuned Collector L C Oscillator. Generalised Oscillator Circuit Colpitts Oscillator Hartley Oscillator Crystal Oscillator Crystal Modelling Credits
In addition to the resource below, there are supporting documents which should be used in combination with this resource. Please see:Clayton G, 2000, Operational Amplifiers 4th Ed, Newnes James M, 2004, Higher Electronics, Newnes
Oscillators
Oscillator CircuitsAn amplifier will become unstable if it has positive
feedback applied to itSee below
VoutA
B
A is the gain of the amplifier and B is the proportion of the output fed back to the input.
Oscillators
A will amplify a signal on the input and then a proportion of the amplified signal will be fed back. If this is large enough to replace the original signal the system keeps generating an output. There are a few rules that determine the nature of the output.
These are stated in the Barkhausen Criterion.It states:
For Sinusoidal Oscillation to occur at a single frequency we must have:
1. A loop gain of unity (A x B = 1)2. A loop phase shift of zero (A + B = 0 or 360)3. Condition 1 or 2 to be true only at a single
frequency.
We will examine a range of sinusoidal oscillators.Oscillators
Wien Bridge Oscillator
This uses a feedback network of the following form:Note the resistors have the same value as do the capacitors.
R
R
C
C
ZS
ZP
Vout
Vin
As the input frequency is varied the output will have a different gain and phase relationship with the input. At one frequency the input and output will be in phase.
Cj
CRj
CjRZS
11
11
1
CRj
R
CjR
CjR
ZP
Cj
CRj
CRj
RCRj
R
VinZZ
ZVinVout
SP
P
1
1
1
11
11
1
1
CRj
CRj
CRjCRj
CRj
Cj
CRj
CRj
RCRj
R
Vin
VoutGain
22 311 )()( CRCRj
CRj
CRjCRj
CRjGain
Oscillators
CRjCR
CRj
CRCRj
CRjGain
3131 22
)()(
222
23
222
2
31
3
31
31
)())((
)()(
)())((
))((
CRCR
CRCRjCRj
CRCR
CRjCRCRjGain
222
2
222
2
31
1
31
3
)())((
))((
)())((
)(
CRCR
CRCRj
CRCR
CRGain
222
2222
31
13
)())((
)))((())((
CRCR
CRCRCRGain
2
21
3
1
)(
))((
CR
CRCRTanPhase
Oscillators
Example R = 10k and C = 10nF
Frequency Gain Phase
900 0.3093889 2.34789
1000 0.3173753 2.864766
1100 0.3232355 3.289809
1200 0.3274285 3.527987
1300 0.3303008 3.459755
1400 0.3321152 2.938197
1500 0.3330734 1.785935
1600 0.3333313 -0.20661
1700 0.3330114 -3.27647
1800 0.3322112 -7.6651
1900 0.331009 -13.5567
2000 0.3294684 -20.9608
Wien Bridge
0.305
0.31
0.315
0.32
0.325
0.33
0.335
900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
Frequency
-25
-20
-15
-10
-5
0
5
Gain curve
Phase curve
1.59kHz
0.333
Oscillators
From the graph:
Zero degrees phase shift occurs at 1.59kHz
The gain at this point equals approximately 0.333
Can we determine this from the equations?
For this to equal 0 then:
2
21
3
1
)(
))((
CR
CRCRTanPhase
01 2 )( CR12 )( CR 1CR
CR
1
kHznFkCR
f 591110102
1
2
1.
Gain at this frequency:
But –
Which leaves us with:
To use this to produce an oscillator we need an amplifier with the following characteristics:
1. Gain = 3 (to give loop gain of unity)2. Phase shift of 0 (to give a loop phase shift of
zero)
222
2222
31
13
)())((
)))((())((
CRCR
CRCRCRGain
01 2 )( CR
3
1
9
3
3
3
3
32
2
2
22
)()(
)())((
CR
CR
CR
CRGain
Oscillators
R2
R1
R
RC
C
Vout
Feedback
Amplifier
32
11
R
RGain 221 RR
If the gain is too large the sine wave will clip at the supply rails and if it is too small it will not oscillate at all.
Wien Bridge Oscillator
Oscillators
R. C. Phase Shift Oscillator.As we are aware, an R C network will alter the amount of
signal it passes as the frequency varies. It also introduces different amounts of phase shift.
R
C
VIN VOUT
The maximum amount of phase shift available from a single R C combination is 90.If we have three such combinations with the same resistors and capacitors we will have up to 270 of phase shift.
Oscillators
VOUTR
C
R
C
R
C
VIN
V1 V2
To derive a relationship between input and output use Nodal Analysis.
@ VOUT R
VCjVV OUT
OUT )( 2
)()(CRj
CRjV
CRjVV OUTOUT
1112
@ V2 CjVVR
VCjVV OUT )()( 2
221
OUTVVCRj
VV 2
21 2
but )(CRj
CRjVV OUT
12
1
12
121 CRj
CRj
CRj
CRjVV OUT
Oscillators
@V1 CjVVR
VCjVVIN )()( 21
11
211 2 VVCRj
VVIN
but
1
12
121 CRj
CRj
CRj
CRjVV OUT
)(CRj
CRjVV OUT
12
CRj
CRj
CRj
CRj
CRj
CRj
CRjCRj
CRj
CRj
CRjVV OUTIN
11
12
12
112
1223
2
2344123 CRj
CRj
CRj
CRj
CRj
CRjVV OUTIN
2
2344123 CRj
CRj
CRj
CRj
CRj
CRjVV OUTIN
3
3232 223441
CRj
CRjCRjCRjCRjCRjCRjVV OUTIN
3
32651
CRj
CRjCRjCRjVV OUTIN
32
3
651 CRjCRjCRj
CRj
V
VGain
IN
OUT
As the frequency is varied the gain magnitude and phase will vary.
Oscillators
32
3
651 CRjCRjCRj
CRj
V
VGain
IN
OUT
32
3
561 CRCRjCR
CRjGain
2322
323
561
561
CRCRCR
CRCRjCRCRjGain
2322
6453
561
56
CRCRCR
CRCRCRjCRjGain
2322
35
2322
46
561
6
561
5
CRCRCR
CRCRj
CRCRCR
CRCRGain
2322
235246
561
65
CRCRCR
CRCRCRCRGain
Oscillators
46
351
5
6
CRCR
CRCRTanPhase
C= 10 nF and R = 10k
Range 100 Hz to 1600 Hz
Frequency Gain Phase
100 0.00024187 -107.824
200 0.00180268 -124.679
300 0.00547793 -139.942
400 0.01144164 -153.412
500 0.01946552 -165.165
600 0.02916192 -175.403
700 0.04013019 -184.346
800 0.05201965 -192.201
900 0.06454692 -199.142
1000 0.07749321 -205.315
1100 0.09069431 -210.839
1200 0.10402935 -215.813
1300 0.11741055 -220.316
1400 0.13077482 -224.417
1500 0.14407717 -228.17
1600 0.15728573 -231.621
RC Phase Shift
0
0.05
0.1
0.15
0.2
0.25
0 200 400 600 800 1000 1200 1400 1600
Frequency
-240
-220
-200
-180
-160
-140
-120
-100
~650Hz
Gain curvePhase
curve
Gain = ~0.035
356 CRCR
46
351
5
6
CRCR
CRCRTanPhase
16 2 CR
From the graph:-180 degrees phase shift occurs at 650HzThe gain at this point equals approximately 0.035Can we determine this from the equations?
For this to equal 0 (-180°)
222
6
1
RC
CR6
1 Hz
nFkRCf 7649
101026
1
26
1.
Oscillators
Now the gain – use: 6
1CR
2322
235246
561
65
CRCRCR
CRCRCRCRGain
22
22
661
65
66
1
66
1636
6365
2161
Gain 0345029
1
2162921629
6629
21629
22
2
.
To use this to produce an oscillator we need an amplifier with the following characteristics:
1. Gain = 29 (to give loop gain of unity)2. Phase shift of 180 (to give a loop phase shift of zero
(360)) Oscillators
R RR
C C C
Rin
Rf
Vout
AmplifierFeedback
R. C. Phase shift Oscillator
The values of Rin and Rf must be selected so that the gain of the amplifier equals twenty nine.
Rin
RfGain
29
RinRf 29Oscillators
Tuned Collector L C Oscillator. This is a development of the four-resistor biased
single stage transistor amplifier:
From equivalent circuit analysis it can be shown that the gain of the amplifier depends (for a limited range) on the value of the collector resistor. If this is replaced by a tuned L C parallel network then we will have a gain which is dependent on frequency.
CL
R
The impedance of this network can be determined in the following way:
CL XXRZ //)(
11
1
2
LCCRj
LjR
CjLjR
CjLjRZ
)(
)(
222
2
21
1
1 CRLC
CRjLCLjR
CRjLC
LjRZ
222
22322
1 CRLC
LCRCLjLjCRjLCRRZ
222
232
222 11 CRLC
CLCRLj
CRLC
RZ
222
232
222 11 CRLC
CLCRLj
CRLC
RZ
222
22322
1 CRLC
CLCRLRZ
R
CLCRLTanPhase
2321
Example
L = 10 mH, R = 10 , C = 10 nF.
Plot the value of Z over the range:
Oscillators
Frequency Impendance Phase12000 1747.19495 88.2391
12500 2049.5822 88.09622
13000 2453.70559 87.89268
13500 3022.7584 87.5925
14000 3885.71647 87.12191
14500 5352.88937 86.30287
15000 8405.51864 84.57067
15500 18571.4156 78.7095
16000 68655.401 -47.2146
16500 13729.3845 -82.6617
17000 7558.08851 -86.202
17500 5253.32655 -87.5099
18000 4049.04035 -88.1861
18500 3308.55693 -88.597
19000 2806.81681 -88.8716
19500 2444.08824 -89.0672
20000 2169.40337 -89.2129
LC Phase Shift
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
12000 13000 14000 15000 16000 17000 18000 19000 20000
Frequency
-90
-70
-50
-30
-10
10
30
50
70
90
15.9 kHzOscillators
Zero phase shift occurs at about 15.9 kHz and this approximately coincides with the peak in the gain (within the limits of our results)Can we determine this from the equations?
R
CLCRLTanPhase
2321 This must equal 0 so:
0232 CLCRL CLCRL 232
CLCRL 222 2
2
2
22 1
L
R
LCCL
CRL
kHzL
R
LCf
L
R
LC9115
1
2
112
2
2
2
.
Impedance at this frequency:
Oscillators
222222
22322
11 CRLC
R
CRLC
CLCRLRZ
Note: 62
210
2
2
10110111
L
Rand
LCL
R
LC
so LC
1
kCR
L
LCRC
RZ 10022
If we use this parallel network instead of the collector resistor we will have an amplifier whose gain is frequency dependent.
Oscillators
Amplifier Characteristic
12000 13000 14000 15000 16000 17000 18000 19000 20000
Frequency
Gain
GMAX
FR
The peak gain GMAX occurs at the resonant frequency FR.We now need to introduce feedback and this is done by converting the inductor of the parallel L C combination into a transformer and applying the feedback signal to the base. An extra capacitor is required to connect the other side of the transformer to ground for ac signals.
Amplifier
Feedback
Oscillators
When the dc bias is set up the new capacitor has no effect and the new inductor acts like a short circuit.When operating a fraction of the output is generated across the new inductor and this is effectively between the base and ground. The amount of feedback is determined by the turns ratio TR of the transformer.If we arrange for the turns ratio of the transformer to have a value given by:
MAXR GT
1
Then we will have a Loop Gain of unity only at the peak gain and therefore this part of the criterion is met only at a single frequency. The Loop Phase Shift can be kept at 0 for all frequencies. Oscillation frequency is given by:
LCFrequency
21
Oscillators
Generalised Oscillator Circuit
Z2 Z3
Z1
-A
The circuit shows an inverting amplifier with three complex components connected to it.
(These components could be inductors or capacitors)
Oscillators
This can be redrawn in the following way –Note Z1 = jX1 etc.
ROUT is the output resistance of the amplifier
jX1
jX2
jX3
ROUT
VinAVin -
+~
Oscillators
The load on the output ZLOAD of the amplifier is given by )//( 213 jXjXjXZLOAD
321
213 )(
jXjXjX
jXjXjXZLOAD
This allows us to determine the gain of the amplifier.
321
213
321
213
)(
)(
jXjXjXjXjXjX
R
jXjXjXjXjXjX
A
ZR
ZAGain
OUTLOADOUT
LOAD
)()(
)(
213321
213jXjXjXjXjXjXR
jXjXjXAGain
OUT
Oscillators
Using j x j = -1
)()(
)(
213321
213XXXXXXjR
XXXAGain
OUT
The feedback ratio for the network involves X1 and X2.
21
2jXjX
jXVoutVin
21
2
XX
X
Vout
VinFeedback
The Loop Gain LG for the circuit is equal to Gain x Feedback
21
2
213321
213)()(
)(
XX
X
XXXXXXjR
XXXALG
OUT
)()( 213321
23XXXXXXjR
XXALG
OUT
For this to be an oscillator the equation must have no j component and therefore:
0321 XXX
This means that for the above equation to be true the three components cannot be the same type (e.g. three capacitors) as this will produce a positive or negative result – there must be a mix e.g. two inductors and one capacitor.
The Loop Gain equation is therefore:
3
2
21
2
213
23)( X
XA
XX
XA
XXX
XXALG
Oscillators
As this value must be positive it means that X2 and X3 must be of the same sign – they must be either capacitors or inductors.
From this general design we can generate two oscillators:
1. X1 = Inductor X2 = Capacitor X3 = Capacitor
2. X1 = Capacitor X2 = Inductor X3 = Inductor
Oscillators
Colpitts Oscillator
032
1 C
j
C
jLj
C2 C3
L1
-A
Oscillation Frequency.0321 XXX
011
321
CjCjLj
32
32
321
2 11
CC
CC
CCL
321
322
CCL
CC
321
32CCL
CC
321
322
1
CCL
CCf
Oscillators
Amplifier Gain
13
2
X
XALG
3
2
2
3
2
31
1
C
C
Cj
Cj
X
XA
Example
100pF470pF
L1
-A
If we need a 25kHz oscillator,
What value of L1 do we require and what should the gain of the amplifier equal?
Oscillators
Hartley Oscillator
0321
LjLjC
j
L2 L3
C1
-A
Oscillation Frequency.
0321 XXX
01
321
LjLjCj
132
2 1)(
CLL
)(
1
321
2
LLC
)(
1
321 LLC
)(2
1
321 LLCf
Oscillators
Amplifier Gain
13
2
X
XALG
2
3
2
3
2
3L
L
Lj
Lj
X
XA
L2 L3
1nF
-10
Example
Determine the values of L2 and L3 if the oscillator is to operate at 10kHz
Oscillators
Crystal OscillatorA crystal oscillator is an electronic circuit that
uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a very precise frequency. This frequency is commonly used to keep track of time (as in quartz wristwatches), to provide a stable clock signal for digital integrated circuits, and to stabilize frequencies for radio transmitters.
Using an amplifier and feedback, it is an especially accurate form of an electronic oscillator. The crystal used therein is sometimes called a "timing crystal". On schematic diagrams a crystal is sometimes labelled with the abbreviation XTAL.
Oscillators
Crystal Modelling *A quartz crystal can be modelled as an electrical
network with a low impedance (series) and a high impedance (parallel) resonance point spaced
closely together.
C0
C1 L1 R1
* The above text is taken from http://en.wikipedia.org/wiki/Crystal_oscillator and is available under the Creative Commons Attribution-ShareAlike License.
Oscillators
Adding additional capacitance across a crystal will cause the parallel resonance to shift downward. This can be used to adjust the frequency that a crystal oscillator oscillates at. Crystal manufacturers normally cut and trim their crystals to have a specified resonant frequency with a known 'load' capacitance added to the crystal. For example, a 6pF 32kHz crystal has a parallel resonance frequency of 32,768 Hz when a 6.0pF capacitor is placed across the crystal. Without this capacitance, the resonance frequency is higher than 32,768.
* The above text is taken from http://en.wikipedia.org/wiki/Crystal_oscillator and is available under the Creative Commons Attribution-ShareAlike License.
Oscillators
Oscillators
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