IEEE Solid-States Circuits Magazine - Fall 2020 - 97

V2

+

-

V2

+

C2
Impedance

L2
VDD

C1

L1

k

V1

+

R2

-

C1
-

V1

+

-

l2
C2

L2

VDD

L1

C1

ωH

R1

l1
+

L2
k

L1

Frequency

V1

C2

k

ωL

-

+
V2

lb

lb

-

One-Port Oscillator

Two-Port Oscillator

FIGURE 14: The doubly tuned resonator and the one-port and two-port oscillators.

1:Av21
+
-Gm

Leq

Ceq

+

Req

V1

V2

-

-

0.8

Q1/Q2 = 1

0.7
0.6
0.5
(k)

is connected to the secondary port
is simply obtained by replacing p
with 1/p in the plot in Figure 15.
Hence, if the oscillations start up at
~ H when the negative conductance
is connected to the primary port (for
example, consider the plot in Figure 15
for, say, k = 0.4 and p = 3), then they
will start at ~ L if we move - G m to
the secondary port (now check the
plot in Figure 15 for k = 0.4 and
p = 1/3).
The startup condition for the
two-port oscillator depends on
the loop gain of the feedback loop
around the doubly tuned resona-
tor (see Figure 16). The oscillations
start if A v21 G m R eq 2 1. As previously
noted, (20) and (21) show that the
sign of A v21 is opposite at ~ L and
~ H . As a consequence, by choosing
the sign of G m (i.e., the connections
of the transconductor in Figure 14),
we can univocally select the oscilla-
tion mode. This is particularly vis-
ible in the root locus plot presented
in Figure 16, where, depending on
the sign of G m, only the conjugate
pole pair associated to one oscil-
lation mode enters the right-hand
half-plane [28], [32].

Region With
Oscillations
at ωL

0.4
0.3

Region With
Oscillations
at ωH

0.2
Boundary Depends
on Q1/Q2

0.1
0.2

0.4 0.6 0.8 1
ξ

2

3

4 5

FIGURE 15: The small-signal equivalent circuit of a one-port oscillator.

The computation of the quality
factor of the doubly tuned resonator
in Figure 14 is nontrivial; the details
can be found in [28]. The mutual
inductance between the coupled coils
increases the energy stored in the
resonator at ~ L while decreasing it

at ~ H . This means that, if we want to
operate at a given frequency ~ 0, it is
convenient to leverage the lower fre-
quency resonance and set ~ L = ~ 0, to
maximize the resonator's quality fac-
tor. However, it must be noted that, as
long as the resonator's quality factor

	 IEEE SOLID-STATE CIRCUITS MAGAZINE	

FA L L 2 0 2 0	

97



IEEE Solid-States Circuits Magazine - Fall 2020

Table of Contents for the Digital Edition of IEEE Solid-States Circuits Magazine - Fall 2020

Contents
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IEEE Solid-States Circuits Magazine - Fall 2020 - Contents
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