IEEE Solid-States Circuits Magazine - Fall 2020 - 98

is limited by the magnetic compo-
nents, its value increases with fre-
quency. Hence, in applications where
we want to use both modes of oscil-
lation, it is still possible to achieve
comparable quality factors at both
~ L and ~ H .
The mutual inductance increases
the quality factor of the transformerbased resonator at ~ L. However, the
same happens to any other inductorbased tank because of the coupling
among the turns of the coil. This can
be better understood if we think of
wiring the primary and secondary

windings of the transformer as a
single inductor, as displayed in Fig-
ure 17. It can be shown [28] that all of
the resonators in Figure 17 have the
same quality factor, such that there
is no intrinsic improvement of the
quality factor due to the sheer use of
a transformer.
Neglecting the 1/f noise, the phase
noise of the oscillators in Figure 14
can be expressed as [28]
L (T~) =
	

R eq, L, H ~ L, H 2 
`
j G, (28)
10 log 10 = k B TF
T~
V 21 Q L2, H

where Q L, H is the resonator's quality
factor evaluated at ~ L, H , V1 is the peak
amplitude at the primary port, and
1+c
one-port oscillator
c
F=*
,
	
1+
two-port oscillator
A v21, L, H
(29)
with c being the transistor excess
noise factor. The use of a transformerbased resonator does not increase the
quality factor per se, as discussed,
but it helps achieve a smaller R eq
for a given value of the quality fac-
tor [27], [28]. This results in a lower

Root Locus
50

+
Gm

Ceq

Leq

V1
-

+

Req

V2
-

ℑm (Grad/s)

25

1:Av21

0

Gm > 0

Gm < 0

ω 0 = ωL

ω 0 = ωH

-25
-50
-15

-10

-5
0
ℜe (Grad/s)

5

10

FIGURE 16: The small-signal equivalent circuit of a two-port oscillator.

Parallel Inductors Tank at ω 0
l1
+

l1
+
V1

Transformer Resonator at ωL = ω 0
R2
R1
k
C1

L1

L2

C2

-

V1

C

L2

L1

V2
Series Inductors Tank at ω 0
l1
+
V1

R1
C

k

(b)
FIGURE 17: A comparison between (a) a doubly tuned resonator and (b) a standard LC tank.

FA L L 2 0 2 0	

IEEE SOLID-STATE CIRCUITS MAGAZINE	

R2

L2

L1

-

98	

R2

Same Q

-
(a)

k

-

I2
+

R1



IEEE Solid-States Circuits Magazine - Fall 2020

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

Contents
IEEE Solid-States Circuits Magazine - Fall 2020 - Cover1
IEEE Solid-States Circuits Magazine - Fall 2020 - Cover2
IEEE Solid-States Circuits Magazine - Fall 2020 - Contents
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