IEEE Circuits and Systems Magazine - Q4 2020 - 12

In the past, the Colpitts oscillator was chosen due to its stability and simple design.
Today, it is chosen because of its compatibility with modern
technologies such as ASIC and MEMS.
part of many electronic systems. Due to limitations and
scope of the paper we will not give an in-depth analysis of the various topologies presented but refer to the
relevant publications. As the reader might notice when
reading this paper, FBAR is used to represent all types
of the resonating components such as SAW, BAW and
crystals with similar behaviour in the figures.

to cover various oscillator structures and parameters
[50], [51]. As reported in [47], [52]-[54], the phase noise
in an oscillator is inverse proportional of the voltage
swing, current and proportional to the temperature and
can be calculated as:

II. Phase Noise in Colpitts Oscillator
One of the reasons for the continues popularity of Colpitts oscillators even today, is their good cyclostationary noise properties compared to other topologies
such as ring oscillators and cross-coupled [42], [47].
In general, the main sources of noise in oscillators are
the active components providing the gain such as FETs
and BJTs and the resistive real parts of the frequency
selective components, i.e. inductors or resonators. The
total phase noise (PN) in oscillators L {T~} tot,(dBc/Hz) is
caused by the presence of the flicker and the thermal
noise (white noise) in the circuit and can be calculated
as follows [48]:

where I DD is the DC current, VSwing is the maximum voltage swing and Q is the quality factor of the oscillator.
From equation above, it is clear that to decrease the
power of the noise, one should decrease the current and
increase the voltage swing.
Also, the configuration of the amplifier and type of
the active component will play an important role for
the noise properties of the oscillator. For example, in
common drain (collector) Colpitts, the active device is
turned on for shorter time of the total period of the output signal, which limits the cyclostationary noise.
The active components may in addition modulate the
low frequency flicker noise to the oscillation frequency,
which can be observed in the phase noise spectrum
as decreasing noise in the side band. By using PMOS
instead of NMOS as the active components, the flicker
noise can be reduced [55].
The thermal noise is caused by both the passive and
the active components in the circuit. In [56] a strategy
for optimizing the inductor in order to reduce the noise
in oscillators is proposed. [48], [57] show that by optimization of current, inductive degeneration and noise
filtering, the phase noise can be further reduced, up
to 15 dB. In [58], the size and ratio between the capacitances C1 and C2 in the feedback are analyzed and they
show that one can optimize the far-out and close-in phase
noises by tuning these capacitance values.
Comparing the phase noise performance of Colpitts
with other oscillators topologies is not straight forward
as they are different in the way they are implemented.
However, some comparison studies can be found in the
literature. In [59] a differential Colpitts is compared to
three other differential configurations i.e. Hartley, Armstrong and a common-source cross-coupled differential
pair, all realized in 25 nm process. In this study, the Colpitts exhibits the worse PN, however, the difference with
the PN of the Hartley is only of 3-4 dBc/Hz. In [60] a
differential version of the common gate is compared to
LC cross-coupled. This comparison shows a better PN in

	

10 log 10 (L {T~} flicker + L {T~} thermal) (2)

The total phase noise in the circuit will result in random variations in both amplitude and the phase of the
signal given by:
	

Vout (t) = A (t) f [~ 0 t + z (t)](3)

The amplitude noise can however be limited or minimized by limiting the amplitude of the signal. The phase
noise in frequency domain at an offset frequency at one
of the side bands is given by:
	

L {T~} (dBc/Hz) = 10 log 10 ;

PSideband (~ 0 + T~)
E (4)
PCarrier

The phase Noise level at an offset frequency,
L (T~) (dBc/Hz) was first modeled by Leeson [46] as:
	

10 log 10 ;1 + `

~0

2Q L T ~

2
(~ c)
E (5)
j E FKT ;1 +
PCarrier
T~

where ~ 0 is the center frequency, F is noise factor, Q L is
loaded quality factor, T is absolute temperature, Pcarrier is
the carrier power in dBm, and ~ c is corner frequency for
flicker noise in Hz. This model is a simple model suited
for general feedback circuits and needs to be modified
12â	

	

IEEE CIRCUITS AND SYSTEMS MAGAZINE 		

L {T~} ? `

~0

T~

j

2

kT
(6)
V 2Swing I DD Q

FOURTH QUARTER 2020
IEEE Circuits and Systems Magazine - Q4 2020

Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q4 2020
