Proceedings of the World Congress on Engineering 2013 Vol II,
WCE 2013, July 3 - 5, 2013, London, U.K.
Comparative Analysis of Tuning Range of
Regulated Cascode Cross Coupled
CMOS Oscillators
Kittipong Tripetch

II. ANALYSIS AND SIMULATION OF SINGLE ENDED COLPITTS




o
s a
C 2s
C2
a1
C1
gm
2
s
LP P
s a
2
R
LP LP
2
2C
R Ps
3 C
P
P
1
a 3C PC 1 R m R m
R
g g









(1)

S

H
VG

n
VT
ID
2
ID
WL
x
Co
2
gm


where




is small
signal transconductance of the MOS transistor.
D
VD
t
u
Vo
C2
b
a
 
RP
LP
Ib
C1
C1
n
Ii
M1
Vb
C2
V1
gm
V1
t
u
Vo
RP
LP
The conventional method of analysis and design of
Colpitts based CMOS Oscillator is not following
Barkhausen criterion [1]. By substituting low frequency
small signal equivalent circuit of CMOS into single ended
Colpitts based CMOS oscillator and injecting input current
source into the circuit, we can derive the transimpedance
gain of the circuit. Then, we can equate the real part and
imaginary part of the function to zero. After that, we can
obtain two equations of oscillating frequency as a function
of passive components. Then, if we equate both of the
equations, substitute the capacitance ratio. We can derive
minimum required gain for the Colpitts based circuits.
In contrast to the single ended Colpitts based CMOS
oscillator. The cross coupled differential Colpitts based
CMOS oscillator can not be designed by the same way as
single ended Colpitts based CMOS oscillator but it follows
Barkhausen criterion [2]. It is different because cross couple
differential Colpitts based CMOS oscillator can be seen as a
2 stage amplifier in cascade while output of the second stage
amplifier is fed back into input of the first stage amplifier
Section II describes how to design tuning range of single
ended Colpitts based CMOS oscillator by switching the bias
resistor. Section III describes how to design tuning range of
cross coupled differential Colpitts based CMOS oscillator.
Section IV describes how to design cross coupled
differential Regulated Cascode based CMOS oscillator.
designed in [1]. It was given by
n
I. INTRODUCTION
The conventional Colpitts based CMOS oscillator was
t
u
Index Terms— Tuning Range, CMOS oscillators, Regulated
Cascode cross coupled VCO
BASED CMOS OSCILLATOR
V oI ia 3a 2 a 1 a o
Abstract— Tuning Range of Oscillator is one of the
important specification which is used in phase locked loop. The
more wider tuning range, the better the phase lock loop. This
paper proposed regulated cascode cross coupled based CMOS
oscillator. The analysis and simulation results of tuning range
of this type of oscillator are compared with conventional
Colpitts based CMOS oscillator as a function of bias current
consumption.
 
Fig. 1 (a) Single ended Colpitts based CMOS oscillator
(b) Small signal equivalent circuit of (a)
Manuscript received March 12, 2013
Kittipong Tripetch, Division of Electronic and Telecommunication
Engineering, Faculty of Engineering and Architecture, Rajamangala
University of Technology Suvarnabhumi, Nonthaburi Campus
7/1 Nonthaburi1 Road, Suan-Yai, Muang, Nonthaburi Province,
Thailand, 11000, k3101600678282@windowslive.com
ISBN: 978-988-19252-8-2
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCE 2013
Proceedings of the World Congress on Engineering 2013 Vol II,
WCE 2013, July 3 - 5, 2013, London, U.K.
(4)
Then substitute value of inductance back into equation (2)
to see if oscillating frequency is changed by real part
condition on the rightmost of equation (2)
Oscillating frequency was derived as

LP
R PC 2
gm
z
H
G
2
9
Rp

k
5 AV
7
m
.
0
1
7
.
5A 5
.
1
m
D7
.
3 2I
2 0




1
gm
A,V
1


2

RP

,


k
gm

m
5
.
5
1
8
.
5A 2
.
1
m I D7
.
3 12 0

1
RP
z
H
G
1
.
2
(6)
H
n


voltage. Assume that if we use supply voltage equal to 3
volt. We can design current to flow 1mA and 2mA, as a
result, its oscillating frequency are changed which may be
called “tuning range”of Colpitts oscillator. By setting dc
output voltage to be half of the supply voltage, 1.5 volt, we
can determine resistive load and small signal
transconductance to be
1
1

2
1

2
2
LP
0
1
F
p
2
F
1p
9
0
1F2 2 1
p
1
.
2
1
LP
R
 

0
1
1
1
1
/
C1
C2

 




, it can be used to set dc output
designed is resistive load



Supposed that we want to design capacitance, inductance
value at oscillating frequency which is equal to 2.1 GHz.
For typical design, we can set capacitance to be 1pF. Then,
we can use equation (2) to design inductance value as
follow.



 

The other parameter in the circuit which is still not
then, the minimum required gain is 4.
 


(5)
(3)
From equation (3), if we set capacitance ratio


2
1



2
C 2C 1
1
C 1C 2
RP
gm

 


0
1
2
 
(2)
If we equate two oscillating equation in (2) then we can
derived the condition as below.
6
4
1
.
2


4

R




C1



C2
C2
1 C 1C 1
LP
R
 
Bode Diagram
200
150
Magnitude (dB)
100
50
0
-50
-100
135
Phase (deg)
90
45
0
-45
-90
0
5
10
10
10
15
10
10
Frequency (Hz)

S
m

1

,
and circle line is for
5
7
.
0
5
8
.
2
ISBN: 978-988-19252-8-2
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

1

,

5
.
1
Where solid line is for
1
7
.
5
gm
k
Rp
S
m
gm
k
Rp
Fig. 2 Magnitude and phase response of the single ended Colpitts based CMOS oscillator
WCE 2013
Proceedings of the World Congress on Engineering 2013 Vol II,
WCE 2013, July 3 - 5, 2013, London, U.K.
III. ANALYSIS
AND SIMULATION OF CROSS COUPLED
DIFFERENTIAL COLPITTS BASED CMOS OSCILLATOR
 
c0
b 0c 1
s
b1
s
c2
2
b 2s
2
s
c3
3
b 3s
3
s
c4
4
b 4s
4
s
c5
5
b 5s
5
s
c6
6
s




D
VD
D
VD

RP
LP

C4
Ib
C1
C1
2
2P
L
1
LP 3 m
C1 g
1 2
gm
a1
2
a1 a
a3 2
a2 2 a0
a3 a
a3
23
a
2 22 2
C3
M2
C2
M1
Ib

k
a1 a0
ISBN: 978-988-19252-8-2
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
(8)
5
7
.
0
,

5
.
1
1


Because denominator polynomial of this circuit has 6th
order. Thus, we could not manipulate closed form
expression of oscillating frequency anymore.
After substitute the inductance 1 nH, capacitance 1pF,

 and transconductances (the
resistance
transistor has the same value as equation (6)) into the
coefficients of the polynomial. We can plot magnitude and
phase response by bode function in MATLAB as fig 4.
k
2 ma 0
g
LP 1
a
s s



(11)
C1
b
a3 a2




1
RP
LP
RP




4m
g
2P
L
C2
V2
2
gm
V2
t
u
LP
C1
a2
LP 2
C 1s
1
1
g ma 3
gm
3
s
P
2
s
L
C2
1
t n
u
g mR P
V oV i
P
C2 L
s LP
P
C 1C 1R C 2
1
C 2L P
m
C1 g
s
A

2 2
Vo
C2
V1
gm
V1
 




a0 a0
a 2 a 1a
20
a
n
Vi

 



21
a
c 6c 5 c 4 c 3 c 2 c 1c 0
t
u
Vo
RP
LP
n
Vi




(10)
Fig. 3 (a) Cross couple differential Colpitts based CMOS
oscillator (b) Small signal equivalent circuit of (a)




 









1
1
LP 2 m
2m
C 1g Pg P
1L L
g ma 2a 1
a2
LP
L PL P
1
C 1g
2 mC 1C 1a 0
1
1
1
1
2m
g mL Pg mg mg
a 3a 3a 1a 0L P0
1


  

(9)
The coefficient of the numerator and denominator
polynomial can be written as following
(7)
Because voltage gain of amplifier stage and voltage gain
of feedback network in fig. 3 can be derived as



b 5b 4b 3b 2b 1b 0
s
A
s s
A
s
H
  .
   
 
s
H
In contrast to the derivation of single ended Colpitts
based CMOS oscillator, we can analyze and design cross
coupled differential Colpitts based CMOS oscillator
according to principle of feedback circuit analysis [3]
General transfer function of amplifier with feedback
network can be written as following.
After substitute voltage gain of the amplifier and voltage
gain of feedback network into equation (7), we can derived
It can be seen from fig 4 that magnitude response seems
to have band-pass characteristic. But phase response is not
360 degree phase shifted at resonance frequency. It means
this circuit is not oscillate.
WCE 2013
Proceedings of the World Congress on Engineering 2013 Vol II,
WCE 2013, July 3 - 5, 2013, London, U.K.
Bode Diagram
-10
-15
-20
Magnitude (dB)
-25
-30
-35
-40
-45
-50
-55
90
85
Phase (deg)
80
75
70
65
60
55
50
7
8
10
9
10
10
10
11
10
10
Frequency (Hz)
Fig. 4 Magnitude and phase response of cross coupled differential Colpitts based CMOS oscillator.
S
m
1

1
7
.
5
gm

,
k

and circle line is for
5
7
.
0
Rp
S
m
1

5
8
.
2
,

gm
k

5
.
1
Rp
solid line is for
k
c
a
b
d
e
e
F
VD
k
r
o
w
t
e
N
VD
D
D
2
t
u
Vo
M2
M3
M1
2
n
Vi
a
 
Vo
t
u
RB

CP
LP
2
s
rd

VX
VY
2
s
g mr d

RP
VY
3
VX
3
gm
VX
1
s
rd
n
Vi
1
gm
n
b



(13)
Vi

2
s
1 rd

 

2




gm
RB
3

 





2


gm

 

gm
gx
2
s
rd
 
  
 
2
s
rd
RP
gz


3
s
1 rd





 


2
s
1 rd

 

2
gy

 

s
1 rd
2
gm
1
1
s
1 rd
3
s
1 rd
3
s
1 rd
1 RB 1
1 RB
3
m
1
g mg
gx
2
1 gm
1
1
gm x
2g
2
1
m
gm g
gx


 

CP
RB
M1
  

(12)

LP
CP
t
u
Vo
M2
n
Vi
s

s
A
 
1
gz
LP
g Ys
LP
s
CP
LP
2
s
t
u n
V oV i

M3
With the cross coupled family. The suitable analysis
method should be the same as section III. Voltage gain of
cross coupled differential regulated cascode based CMOS
oscillator in fig. 5 is derived as following

RP
RP
RB
OSCILLATOR
LP
IV. ANALYSIS AND SIMULATION OF CROSS COUPLED
DIFFERENTIAL REGULATED CASCODE BASED CMOS
 
Fig. 5 (a) Cross Couple Differential Regulated Cascode
based CMOS oscillator (b) Small signal of Regulated
Cascode Bandpass Amplifier
ISBN: 978-988-19252-8-2
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCE 2013
Proceedings of the World Congress on Engineering 2013 Vol II,
WCE 2013, July 3 - 5, 2013, London, U.K.
1
2
.
2
3
3
ID
2
2 V
/
A
2
7
.
0
8
.
0
4
9
1
3



M3
RB
A 3
.
2
5
2
5
.
2 3 1
.
3
1
3
k


can be
0
4
.
2
2

A
RB


(19)
V
/
A
m
2
6
.
0
  

A


5
2
.
1
3
2
3
2
V
/
A
4
9
1
2
3
3



ID



x
n

WL
Co
2
3
gm



(20)
k
0
5

A
m
1 2
1
0
.
0
2
1

1 ID

2


s
rd
1
A
m
1
1 ID
s
rd





(21)


M

2
.
3

A

5
2
.
1 1
3
1
0
.
0

3

3
is
1 ID
s
rd

(16)
process
2
V
/
A
4
9
1
the


used. Then, aspect ratio of the transistor can be designed to
be



1
2
.
2
3

2

 
A
1

7
.
0
5
A .
m 1
2 2
2 V
/

 

4
9
1
WL






5

H
n

2
1

0
1

0
2
1 0
1
4
7
.
1
LP
1
.
2
z
H
G
2
L

(23)
M2
M2
should be design to be 1.5 V. Thus, voltage at
After all numerical values in equation (15) are known.
Then we can substitute it in MATLAB text file and simulate
it. Figure 6 is the result of simulation of equation (15)
should be 2.3 V. Again,
and
ISBN: 978-988-19252-8-2
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
M2
should have the same aspect ratio with
M1
for convenience in design, the regulated transistor
M3
the gate terminal of transistor
 
can be designed to
have the same aspect ratio, thus the gate to source voltage of
transistor
P
1 CP
. For
 
R
V
convenience in design, transistor

8
.
0

7
.
0
5 M2
.
1

1
S
VD
M1
is equal to
(22)
The capacitance in equation (14) is designed to be 1 pF. If
the oscillating frequency is designed to be 2.1 GHz. Then,
inductance is computed to be
(17)
Thus , minimum drain to source voltage of the first
transistor
2

gm
V
/
A
m
1
  

8
9
.
4
A
m


2
2
3
2
V
/
A
4
9
1
2
ID
WL



x



k

x

parameter
n
transconductance
parameters,
Co
process


2





. Bias resistor

7
.
0
5
.
1
WL

x
n
typical
C o2
iD


5
7
.
0
A

2


Co

n
 

can be computed from
The other parameters use in simulation of cross coupled
differential regulated cascode based CMOS oscillator are
computed as following
To design current to flow, we use drain current equation
as following
For



computed to be
2


5m
.
1
2
RP

,
k
5
.
1
A
5m
.
1
1
1
RP




Thus, drain current of transistor
1
 

(15)
To plot magnitude and phase response of equation (15),
we typically design current to flow 1mA and 2mA. We use
3 volt supply and we design output voltage to be half of the
supply voltage which is equal to 1.5 V.
Then

 
(18)
gm

 
 

1

gz
LP
2
s

2y
gyg
2P
LPL
s 2z
g
2P
L
gz
gyCP
2P
L
P
L
2
2
s
2
s
CP
gygz
P
2 PC
L
2P
L
3
s
2
3
s
2P
C
2P
L
4
s
s
H


 

equation (18) to be
(14)
From equation (14), it can be seen that oscillating
frequency of this circuit could not be tune by using bias
current. But the bandwidth of the bandpass filter can be tune
by bias current.
After substitute equation (12) into equation (7), then we
get

WL
P
1 CP
L
R
 
 



j
s
The coefficient in equation (12) are defined in equation (13)
for easy visualization of the whole equation. It can be
understood that equation (12) has bandpass response.
Equation (12) can be used to design oscillator by
substitute   into denominator of equation (12)
Thus, resonance frequency or oscillating frequency can be
derived as
WCE 2013
Proceedings of the World Congress on Engineering 2013 Vol II,
WCE 2013, July 3 - 5, 2013, London, U.K.
Bode Diagram
-5
-10
-15
Magnitude (dB)
-20
-25
-30
-35
-40
-45
-50
-45
-90
Phase (deg)
-135
-180
-225
-270
-315
8
9
10
10
10
11
10
10
Frequency (Hz)
S
m
1

1
7
.
5
,

gm
k

5
7
.
0
and circle line is for
Rp
S
m
1

5
8
.
2
,

gm
k

5
.
1
Rp
solid line is for
Fig.6 Magnitude and phase response of cross coupled
differential regulated cascode based CMOS oscillator
V. CONCLUSION
It is well known that tuning range of cross coupled
common source oscillator is limited by minimum and
maximum capacitance of the varactor. The tuning range of
single ended and differential Colpitts is derived to be
dependent
on
both
passive
capacitance
and
transconductance of the transistor. It means that tuning
range can be changed by switching of the bias current. The
oscillating frequency of cross couple regulated cascode
based was derived to be independent with bias current. But
the bandwidth depends on bias current.
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WCE 2013