Derivation of Colpitts feedback capacitor values

Derivation of Colpitts feedback capacitor values.
Vin
Iin
Zin
C1
C2
RL
Sheet
1 of 2
Simplified schematic diagram showing the reflection amplifier part of the Colpitts
oscillator. The capacitor network of C1 and C2 would provide positive feedback from
the emitter to base.
V
0 = - I
rearrange to give Ib and sub into equation (1)
V
V
V
in
in
in
in
=
I
(XC
= I
= I
(XC
1
in
in
(XC
(XC ) + I (XC
in
in
(XC
.XC
1
1
1
1
1
+ XC
+ XC
+ I
in
b
.XC
) - I
+ hie)
= ( XC
(XC
+ hie)
Iin.XC1
) -
(XC
(XC + hie)
+ hie)
I
Iin.XC1
+ hie).
.XC
(XC + hie)
1
2
2
2
1
1
b
1
1
- βXC
Iin.XC1
-
.XC
(XC + hie)
1
1
in
.XC
1
2
+ ( XC
)
-
-
- βXC
+ ( XC
1
1
1
(1)
(2)
)
ie
Iin.XC1
+
. βXC
(XC + hie)
1
2
1
+ hie)
I
Iin.XC1
=
(XC + hie)
and multiply out
.XC
Iin.XC1
+ hie)
. βXC
(XC + hie)
1
I
in
b
2
-
1
2
2
x both sides by XC
and multiply
1
+ hie
out again.

Sheet
2 of 2
( ) ( )
[ ]
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( )
[ ]
2
1
2
1
2
1
in
in
in
2
2
1
2
1
in
in
in
2
1
2
1
in
in
in
2
1
2
1
2
1
2
1
in
in
in
1
1
2
1
2
1
in
in
in
in
2
1
2
1
in
1
in
in
2
in
1
2
in
2
in
1
1
in
1
in
2
in
1
2
1
in
2
in
2
in
1
1
in
2
1
in
1
in
C
C
C
.
C
.
1
C
.
C
.
1
gm.
Z
I
V
then
1
-
j
as
C
.
1
C
.
1
C
.
1
.
C
.
1
gm.
Z
I
V
-
:
reactances
expand
1
gm
Let
1
.XC
XC
XC
XC
Z
I
V
1
.XC
XC
XC
XC
.
1
.XC
XC
XC
XC
.
Z
I
V
-
:
then
XC
that
assume
we
If
XC
1
.XC
XC
XC
XC
.
Z
I
V
1
I
.
.XC
XC
XC
XC
.
I
)
(XC
V
I
for
Rearrange
.
.XC
I
.
XC
.XC
I
.
.XC
I
.
XC
.XC
I
.
)
(XC
V
.
.XC
I
.
XC
XC
I
-
.XC
I
.
.XC
I
.
XC
.XC
I
.
XC
I
)
(XC
V
+
+
−
=
=
=
+
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
=
+
=
+
+
+
=
=
+
+
+
⇒
+
+
+
=
=