IEEE Solid-States Circuits Magazine - Fall 2020 - 94

parasitic, possibly due to devices
connected to the coupled inductors
(e.g., pads, transistors, and so on).
We can think of the doubly tuned
network as being made of two induc-
tor-capacitor (LC) tanks (L 1 - C 1 and
L 2 - C 2, with resonance frequencies
~ 1 = 1/ L 1 C 1 and ~ 2 = 1/ L 2 C 2 ,
respectively) that are magnetically
coupled to make up a fourth-order
resonator. The analysis of this net-
work is nontrivial; it is presented, in
detail, in [27], [28]. In this article, we
focus on only a few important con-
cepts. The doubly tuned network can
be represented in the neighborhood
of the resonances by means of equiv-
alent second-order circuits, which
allow one to more easily evaluate the
transformed load impedance at reso-
nance as well as the equivalent loss
resistance and, hence, the network
efficiency. The behavior of the dou-
bly tuned network is heavily affected
by the parameter [27]
	

QS =

RL
L 2 /C 2

behavior. The network behavior also
strongly depends on the ratio of the
LC products of the coupled tanks [29]:
	

p=

~ 21
= L 2 C 2 .(18)
L1 C1
~ 22

We start the analysis of the dou-
bly tuned network by discussing the
high-Q S case. In this case, the two
parallel resonance frequencies are
[27]-[29]
~ 2L, H =

	


2

2

1 + p ! (1 + p) - 4p (1 - k ) 2
~ 2.
2 (1 - k 2)
(19)

Due to magnetic coupling, pole split-
ting occurs, and ~ L and ~ H are,
respectively, lower and higher than
both ~ 1 and ~ 2. Equation (19) is a bit
cumbersome. However, two impor-
tant cases can be singled out. If p = 1,
that is, if the two LC tanks have the
same LC product, ~ 1 = ~ 2, and (19)
simplifies into ~ L, H = ~ 2 / 1 ! ; k ; .
The ratio between the resonance fre-
quencies is set only by the magnetic
coupling. If, instead, p % 1 (or, sym-
metrically, p & 1), then ~ L is almost
identical to the smaller between
~ 1 and ~ 2, while ~ H is pushed at
a much higher frequency than the
larger between ~ 1 and ~ 2. In any
case, for any value of k, the ratio
~ H /~ L increases as p is moved away
from unity.

1
,(17)
1 - k2

which quantifies the loading effect
of R L (see Figure 12) on the reac-
tive network. If Q S & 1, the network
shows two parallel resonances,
around which it behaves as a trans-
former. If Q S 1 1, the network
de--generates, showing a single reso-
nance and an impedance inverter

Zin
+

R1
C1

k

L1

C2

L2

+
V1

R ′ 1:A
v21
Ceq

-

Leq

Req

Since, in the neighborhood of both
resonances, the transformed load
resistance (referred to the primary
port) is simply Rl = R L /A 2v21, we can
get some design insight from (20)
and (21). First, if p = 1, ; A v21 ; = n re--
gardless of the value of k. This
means that, in this condition, the
doubly tuned network behaves as an
ideal transformer, regardless of the
value of the magnetic coupling. Sec-
ond, the voltage gain A v21 has oppo-
site sign at ~ L and ~ H . Third, (21)
implies that, at ~ H , the transformed

RL

QS < 1
l2

l1

+

+

V2

RL

-

FA L L 2 0 2 0	

. - n/ (kp) if p % 1
if p = 1 .(21)
	 A v21, H = * - n
. - n $ k/p if p & 1

-

FIGURE 12: A doubly tuned network and equivalent circuits at resonance.

94	

while at ~ H , we have

+
V2

QS >> 1
Zin

. n $ k if p % 1
A v21, L = * n
if p = 1 ,(20)
. n/k if p & 1

l2

R2

-

l1

	

Explicit or
Parasitic Caps

l1

V1

In the neighborhood of the parallel
resonances, the doubly tuned match-
ing network can be approximated as
a second-order shunt resonator and
an ideal transformer, as depicted in
Figure 12. The voltage gain of the
ideal transformer depends on the
resonance frequency, the magnetic
coupling, the parameter p, and the
turn ratio of the coupled inductors
[27]. At ~ L, we have

IEEE SOLID-STATE CIRCUITS MAGAZINE	

V1
-

Zin

l2
Ceq

Leq

Z0
±90°

Req

+
V2
--

RL
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