ESE319 Introduction to Microelectronics
Colpitts Oscillator
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Basic positive feedback oscillator
The Colpitts LC Oscillator circuit
Open-loop analysis
Closed-loop analysis
Root locus
Stability limit
Colpitts design
2008 Kenneth R. Laker updated 10Nov09 KRL
1
ESE319 Introduction to Microelectronics
Basic Positive Feedback Oscillator
closed-loop oscillator
Vi = 0
+
Vf
A(s)
+
pos. fdbk
open-loop: determine loop-gain
Vo
Acl  s=
A s
1− A s
V o= A sV i V f = As0V o ⇒
V o 1− As=0
Vi
+
A(s)
Vo
Vf
V f Vo
K N s
= = As=
Vi Vi
D s
Since: V o ≠ 0 ⇒1−A s=0⇒ A s=1 ⇒ D s−K N s=0
Condition for oscillation at s = jω0: As=1 e j ± 2k  Barkhausen criterion
for k = 0, 1, 2 ..
2008 Kenneth R. Laker updated 10Nov09 KRL
2
ESE319 Introduction to Microelectronics
Colpitts Oscillator Basic Schematic
Assumptions:
1. rπ large (compared to 1/ωC2).
L
R
C1
2. Cµ negligible (compared to C1, C2)
C2
3. Cπ part of C2 (in closed loop)
4. R represents total resistance
in collector circuit, i.e. R∥r o≈ R
Emphasizing feedback
L
Vπ
R
C1
Vf
C2
Emphasizing loop gain A s
2008 Kenneth R. Laker updated 10Nov09 KRL
C
B
V  r
C
C
gmV 
ro
E
E
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ESE319 Introduction to Microelectronics
Loop-Gain Analysis
Vπ
Cµ
L
Vc
gmVπ
R
C1
Vf
Vf
C2
Vf
loop-gain: A s=
V
Node equation at Vc:
∣ j C ∣≪ g m
1 1
1
 s C 1  V c − V f  g m −s C  V  =0
sL R
sL
1 1
1
 s C 1  V c − V f g m V =0
sL R
sL
−1
1
at Vf:
V c  sC 2  V f =0
sL
sL
2008 Kenneth R. Laker updated 10Nov09 KRL
(1)
Note that
Vf = Vf (s)
Vc = Vc(s)
Vπ = Vπ(s)
(2)
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ESE319 Introduction to Microelectronics
Open Loop Analysis - cont.
from previous slide
Rearranging (1) and (2):

sC 1 
−

(1)
1 1
1
 V c − V f =−g m V 
sL R
sL


1
1
V c  s C 2
V f =0
sL
sL
(3)
(2)
Eq. (1) – gm Vπ
(4)
Eq. (2)
Further rearrangement of (3) and (4):


s 2 LC 1 1 1
1
 V c − V f =−g m V 
sL
R
sL


s 2 LC 2 1
1
− V c
V f =0
sL
sL
2008 Kenneth R. Laker updated 10Nov09 KRL
(5)
(6)
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ESE319 Introduction to Microelectronics
Open Loop Analysis - cont.
from previous slide

Prepare to add the two equations:


  
 
Adding (to eliminate Vc terms):
  
2
2
s
L C 11 1
1
−


sL
sL
R
2008 Kenneth R. Laker updated 10Nov09 KRL




2

s L C 21
1
− V c
V f =0
sL
sL
2
2
−g m
1 s L C 11 1
1
 V c−
V f=
V
sL
sL
R
sL
sL
2
s 2 LC 1 1 1
1 s L C 1 1 1
−
 V c

sL
sL
R
sL
R
2
s L C 11 1
1
 V c − V f =−g m V  (5)
sL
R
sL

s 2 LC 2 1
V f =0
sL

s2 L C 2 1
−g m
V f=
V
sL
sL
Eq 5∗
(6)
1
sL

2
s LC 1 1 1
Eq 6∗

sL
R
(7)
6

ESE319 Introduction to Microelectronics
Open Loop Analysis – cont.
From previous slide
2
s2 L C 11 1
1
−


sL
sL
R
  


s2 L C 2 1
−g m
V f=
V
sL
sL
(7)
Multiply (7) by (sL)2:
 


sL 2
−1 s LC 1 1
 s LC 21  V f =−g m sL V 
R
2
Expand and collect terms according to sn:


2
L
C2
L
4
2
2
3
−1s C 1 C 2 L s  LC 1 LC 2 s
s 1 V f =−g m sLV 
R
R
2008 Kenneth R. Laker updated 10Nov09 KRL
(8)
7
ESE319 Introduction to Microelectronics
Open Loop Analysis - cont.
From previous slide


3 2
s
L C2 s L
4
2
2
−1s C 1 C 2 L s  LC 1 LC 2 
 1 V f =−g m sLV 
R
R
(8)
Canceling (-1 by 1) in (8) and dividing by sL:


2
s
L C2 1
3
s C 1 C 2 Ls  C 1C 2  
 V f =−g m V 
R
R
Multiply by R:
s
3
R C 1 C 2 Ls R  C 1 C 2 s LC 21  V f =−g m R V 
2008 Kenneth R. Laker updated 10Nov09 KRL
2
(9)
8
ESE319 Introduction to Microelectronics
Open Loop Analysis - cont.
From previous slide
 s3 R C 1 C 2 Ls R  C 1C 2 s 2 LC 21  V f =−g m R V 
(9)
Divide (9) by RC1C2L:

 C 1C 2 

−g m R
1
1
s s
s

V f=
V
C1C 2 L
R C1 R C1C 2 L
RC1 C2 L
3
2
The loop-gain transfer function:
−g m
Vf
C1 C 2 L
K N s
= As=
=
V
D s
 C 1C 2 
1
3
2 1
s s
s

R C1
C1 C 2 L R C1C 2 L
2008 Kenneth R. Laker updated 10Nov09 KRL
(10)
9
ESE319 Introduction to Microelectronics
Closed Loop Analysis - cont.
The closed loop equation:
(10)
A s
K N  s
K N  s
Acl  s=
=
= '
1− A s D s−K N  s
D s
where:
K N s
As=
D s
−g m
K=
C1 C 2 L
N  s=1
 C 1 C 2 
1
1
D s=s s
s

RC1
C1 C2 L RC1 C2 L
3
then:
2
(11)
KN(s)
−g m
 C 1 C 2  1g m R
3
2 1
D s=D s−
=s s
s

C1 C 2 L
RC1
C1 C2 L R C1 C2 L
'
2008 Kenneth R. Laker updated 10Nov09 KRL
(12)
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ESE319 Introduction to Microelectronics
Closed Loop Analysis - cont.
We know that the open loop system A(s) is stable. It has poles
in the left half plane, since it is a passive RLC circuit. We
also know that it has 3 poles. One is negative-real, the other
2 can be negative-real or LHP complex conjugates.
 C 1C 2 
1
1
D s=s s
s

R C1
C1C2 L R C1C 2 L
3
2
(11)
So, let's do a rough sketch of the root locus for a feedback system
with a 3 pole A(s).
2008 Kenneth R. Laker updated 10Nov09 KRL
11
ESE319 Introduction to Microelectronics
Root Locus Characteristic
Open loop
poles somewhere
in here
 F = Fx
j
j x
X
●
Closed loop
pole locus
●

 F = Fx X − j  x
●
The loop will become unstable
for any value of  F  Fx .
Rather than sketch the root
locus in more exacting detail – it
has served its purpose by verifying
that oscillation is possible.
Let's solve for the required  Fx .
 C 1C 2 
1
1
D  s =s  s
s

R C1
C 1 C 2 L RC 1 C 2 L (11)
3
2
 C 1C 2  1 g m R (12)
1
D  s= D s − K N s =s s
s

RC 1
C 1 C 2 L RC 1 C 2 L
'
2008 Kenneth R. Laker updated 10Nov09 KRL
3
2
12
ESE319 Introduction to Microelectronics
Stability Limit Calculation
If the closed loop system is at the stability limit point:
D' s=DsK N s=sa s2 2x 
to oscillate at ω = ωx
Multiplying terms:
=0
=0
'
3
2
2
2
'
2
2
2
3
D s=s a s  x sa  x ⇒ D  j =a  x −a   j x − 
Match term by term with:
2
x
a
2
ax
 C 1C 2  1g m R
1
D s=s s
s

RC 1
C1 C 2 L R C1C 2 L
'
1
a=
R C1
3
2
x
=
2
 C 1 C 2 
C1 C2 L
2008 Kenneth R. Laker updated 10Nov09 KRL
1g m R
2
2
a =
s.t. a∗ x =a  x
R C1C 2 L
2
x
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ESE319 Introduction to Microelectronics
Stability Limit
C 1C 2
=
C1C 2 L
1
a=
R C1
2
x
1g m R
a =
R C1C 2 L
2
x
a∗ 2x =a 2x
2
2
To find the “gain” requirement for oscillation, equate a∗ x =a  x
1g m R
1 C 1C 2 C 1 C 2
=
=
2
R C1 C1C 2 L R C1 C 2 L R C1 C2 L
a
 2x
a 2x
C 1 C 2
1g m R
C 1 C 2
C2
=
⇒1g m R=
=1
2
C1
C1
R C1 C2 L R C1C 2 L
C2
g m R=
C1
2008 Kenneth R. Laker updated 10Nov09 KRL

C 1 C 2
 x=
C1 C2 L
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ESE319 Introduction to Microelectronics
Oscillator Design Summary
The oscillation frequency:
The required feedback gain ΓF = gmR:
To insure the start-up of oscillation:
Comments:
1. Unfortunately we don't control “R”.

C 1 C 2
 x=
C1 C2 L
C2
poles at ± jωx
 Fx = g m x R=
C1
C2
g m x R
C1
2. Set C2 & C1 to convenient values, say C2 = C1 = C, and adjust gm using IC
s.t. gmR > C2/C1; then adjust “L” to maintain ωx at the desired oscillation
frequency.
2008 Kenneth R. Laker updated 10Nov09 KRL
15
ESE319 Introduction to Microelectronics
Practical Colpitts Oscillator Circuit
VCC
RFC = RF choke ->
RB1
RFC
C1
2008 Kenneth R. Laker updated 10Nov09 KRL
RE
low resistance at dc
L
large
RB2
large reactance at ωx
C2
large
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