IEEE Circuits and Systems Magazine - Q4 2020 - 22

Another way to achieve a differential push-push behaviour, is by using a transformer to couple the inductors in two single-ended oscillators. An inverter-based
push-push Colpitts with differential output was created by using a transformer [104] and it is shown
in Fig. 44. In [104] the circuit is realized using 180 nm
CMOS with a resonance frequency of 1.806 GHz and
uses an on-chip transformer. They achieve a phase
noise â145,6 dBc/Hz at 3 MHz offset. The authors also
propose a source coupled version, where the source of
the two inverters are connected to each other and to
VDD and GND through inductors.
Fig. 45 shows a differential oscillator made by connecting two single-ended oscillators in a ring using inductors [105].

K12

Mp
Vout1

C1

Mp

L1

Vout2

L2
C1

Mn

Mn
C2

C2

B. Pierce Oscillator
If the inductor in the common source oscillator shown in
Fig. 40 is replaced by a resonator such as a crystal or a
FBAR, the oscillator is referred to as a Pierce oscillator.
Pierce first published his paper on the topic in 1923 [17]
and patented it in 1938 [20]. Fig. 46 shows the most common way of presenting the Pierce oscillator, where the
gain stage is an inverting amplifier. The inverting amplifier can be realized in two ways, either by a digital inverter
as shown in Fig. 47 or common source amplifier as shown
in fig. 48. The Pierce oscillator forces the resonator to operate close to its resonance frequency, where it behaves
almost resistive and has the lowest impedance. The two
grounded capacitors connected at each side of the resonator create the necessary phase shift. However, this may
not be the case in real, non-ideal applications because
the total phase shift in the inverting amplifier normally
deviates from 180 degrees, which forces the resonator
to move its operation point toward the anti-resonance in
the inductive region to compensate this [64].
Already in late 50s and early 60s the first Pierce oscillators based on common emitter bipolar technology were
presented [106]. The CMOS inverter-based pierce oscillators was discussed in detail in [99] in 1971 and [107] from
1980 presents a common source pierce oscillator. In [108]

Figure 44. Differential inverter-based Colpitts oscillator with
transformer coupling.
MP

FBAR

L4
Mp

L3

L2

C1

C1
Mn

Mp

Vout1

L1

Vout

Vout2

C2

C1

C2

MN

C2
Mn
Figure 47. CMOS inverter Pierce oscillator.

Figure 45. A differential inverter-based Colpitts differential
ring inductor oscillator [105].
MP

MP
R1

Vbias

Vout

R2

C1

Figure 46. Schematic of a simple Pierce oscillator.
22â

Vout

MN

MN

C2

FBAR

Figure 48. Common source Pierce oscillator.

IEEE CIRCUITS AND SYSTEMS MAGAZINE

FOURTH QUARTER 2020

IEEE Circuits and Systems Magazine - Q4 2020

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