1
Positive-Feedback Oscillators: Illustrations
© Eugene PAPERNO, 2006
I. PHASE-SHIFT OSCILLATOR
)(1)(
1
11
1)( 3
sAA
sA
RC
sRC
s
OL
OLf β
β
−=
=
⎟⎠⎞
⎜⎝⎛ +
=
AOLΣSin
SoSε
β(s)
Im[β (s)]=0
Fig. 1. Abs[β(s)].
Fig. 3. Abs[Af(s)] for AOL=8. [AOLβ(ω1) =1.]
−2 −1.5 −1 −0.5 0 0.5
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
σ
jω
Im[β(s)]=0
0.02
0.125
0.04
0.06
0.10.08
0.2
0.3123
AOL=8AOL=12.5
AOL=25
AOL=3.3AOL=2
Fig. 2. Abs[β(s)].
−1.5 −1 −0.5 0 0.5
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Im[β(s)]=0
AOL=8
σ
jω
Fig. 4. Abs[Af(s)] for AOL=8. [AOLβ(ω1) =1.]
2
Note that AOLβ(ω1) >1 shifts the poles to the right of the jω axis.
−1.5 −1 −0.5 0 0.5
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Im[β(s)]=0
AOL=8
σ
jω
Fig. 5. Abs[Af(s)] for AOL=8. [AOLβ(ω1) =1.]
−1.5 −1 −0.5 0 0.5
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Im[β(s)]=0
AOL=25
σ
jω
Fig. 7. Abs[Af(s)] for AOL=25. [AOLβ(ω1) =3.125.]
−1.5 −1 −0.5 0 0.5
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Im[β(s)]=0
AOL=12.5
σ
jω
Fig. 6. Abs[Af(s)] for AOL=12.5. [AOLβ(ω1) =1.56.]
−1.5 −1 −0.5 0 0.5
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
AOL=3.3
σ
jω
Im[β(s)]=0
Fig. 8. Abs[Af(s)] for AOL=3.3. [AOLβ(ω1) =0.41.]
3
II. WIEN-BRIDGE (HEWLETT) OSCILLATOR
)(1)(
11
1.031
11
1
1
1
)(
sAAsA
CR
sCR
sCR
sCR
sCR
sCR
s
OL
OLf β
β
−=
==
⎟⎠⎞
⎜⎝⎛ −−
+++
+=
AOLΣSin
SoSε
β(s)
Fig. 1. Abs[β(s)].
Fig. 3. Abs[Af(s)] for AOL=8. [AOLβ(ω1) =1.]
−1.5 −1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
Fig. 2. Abs[β(s)].
−1.5 −1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
Fig. 4. Abs[Af(s)] for AOL=10. [AOLβ(ω1) =1.]
4
Note that AOLβ(ω1) >1 shifts the poles to the right of the jω axis.
−1.5 −1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
Fig. 5. Abs[Af(s)] for AOL=10. [AOLβ(ω1) =1.]
−1.5 −1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
Fig. 7. Abs[Af(s)] for AOL=106. [AOLβ(ω1) =105.]
−1.5 −1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
Fig. 6. Abs[Af(s)] for AOL=20. [AOLβ(ω1) =2.]
−1.5 −1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
Fig. 8. Abs[Af(s)] for AOL=5. [AOLβ(ω1) =0.5.]
5
III. HARTLEY-COLPITTS OSCILLATORS
)(1)(
111
1
1
1
1
)(
sAAsA
CLR
sL
sCR
sCR
sL
sCR
sCR
s
OL
OLf β
β
−=
===
++
+=
AOLΣSin
SoSε
β(s)
Fig. 1. Abs[β(s)].
Fig. 3. Abs[Af(s)] for AOL=8. [AOLβ(ω1) =1.]
−1.5 −1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
Fig. 2. Abs[β(s)].
−1.5 −1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
Fig. 4. Abs[Af(s)] for AOL=1. [AOLβ(ω1) =1.]
6
Note that AOLβ(ω1) >1 shifts the poles to the right of the jω axis.
−1.5 −1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
Fig. 5. Abs[Af(s)] for AOL=1. [AOLβ(ω1) =1.]
−1.5 −1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
Fig. 7. Abs[Af(s)] for AOL=2. [AOLβ(ω1) =2.]
−1.5 −1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
Fig. 6. Abs[Af(s)] for AOL=1.4. [AOLβ(ω1) =1.4.]
−1.5 −1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
Fig. 8. Abs[Af(s)] for AOL=0.8. [AOLβ(ω1) =0.8.]
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IV. OSCILLATOR WITH AN UNSTABLE FEEDBACK NETWORK
)(1)(
1
11
1)25(1
25)(
3
sAAsA
RC
sRC
s
OL
OLf β
β
−=
=
⎟⎠⎞
⎜⎝⎛ +
−−
−=
AOLΣSin
SoSε
b(s)
−25 Σ
Phase-shift oscillator
Fig. 1. Abs[β(s)].
Fig. 3. Abs[Af(s)] for AOL=0.85. [AOLβ(ω1) =1.]
−0.1 0 0.1 0.2
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Fig. 2. Abs[β(s)].
−0.1 0 0.1 0.2
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Fig. 4. Abs[Af(s)] for AOL=0.085. [AOLβ(ω1) =1.]
8
Note that AOLβ(ω1) >1 shifts the poles to the left of the jω axis.
Fig. 5. Abs[Af(s)] for AOL=0.085. [AOLβ(ω1) =1.]
Fig. 7. Abs[Af(s)] for AOL=0.01. [AOLβ(ω1) =0.012.]
Fig. 6. Abs[Af(s)] for AOL=0.05. [AOLβ(ω1) =0.59.]
Fig. 8. Abs[Af(s)] for AOL=0.2. [AOLβ(ω1) =2.35.]
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SUMMARY
Hartley-Colpitts Oscillators
AOL2 β( jω1) =1
AOL3 β(s) =1
AOL1 β(s) =1
Im[β (s)]=0
AOL1 < AOL2 < AOL3
β(s)
AOL
Oscillator with an unstable feedback network
AOL2 β( jω1) =1
AOL1 β(s) =1
AOL3 β(s) =1
Im[β (s)]=0
AOL1 > AOL2 > AOL3
β(s)
AOL
Filename: 1._Positive_Feedback_Oscillators_Illustrations Directory: D:\1. Positive-feedback oscillators Template: C:\Documents and Settings\Paperno_E\Application
Data\Microsoft\Templates\Normal.dot Title: Positive-Feedback Oscillators Subject: Author: Paperno_E Keywords: Comments: Creation Date: 12/28/2006 1:06:00 PM Change Number: 147 Last Saved On: 12/30/2006 5:55:00 PM Last Saved By: Paperno_E Total Editing Time: 1,097 Minutes Last Printed On: 12/30/2006 6:56:00 PM As of Last Complete Printing Number of Pages: 9 Number of Words: 399 (approx.) Number of Characters: 2,316 (approx.)
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