IEEE Solid-States Circuits Magazine - Fall 2020 - 95

resistance Rl is (approximately) pro-
portional to p 2.
The equivalent resistance R eq in
Figure 12 models the losses in the res-
onator and can be used to assess the
efficiency of the doubly tuned match-
ing network. Its analytic expression
is quite involved but can be approxi-
mated as follows [27]. At ~ L,

In the neighborhood of this reso-
nance, the circuit behaves as an
impedance inverter, as illustrated
in Figure 12; that is, it behaves as
a quarter-wavelength transmission
line with an equivalent characteris-
tic impedance:
	

Z0 =

~ S L 2 (1 - k 2)
.(25)
n·k

Z
2
if p % 1
] . R1 Q1
In this case, it is convenient to
2
]
2 ^1 + k h
model the resonator losses as an
if p = 1 ,		
	 R eq,L = ] R 1 Q 1 $
Q1
[
equivalent resistance in series
1+
			
Q2
]
with R L:
			
]
Q
] . R 1 Q 21 $ 2 k 2 if p & 1
Q2
Q
k2 +
1
	
\
Q1
(22)
	
R eq,S = R 2
.(26)
k2
where Q 1 = ~L 1 /R 1 and Q 2 = ~L 2 /R 2,
At resonance, the load resistance
while at ~ H ,
referred to the primary port becomes
Rl = Z 20 / (R L + R eq,S ) . Z 20 /R L.
R eq,H =
At this point, it is quite clear that
Z
2 2
2 2
]
2 k p ^1 - k h
a doubly tuned network offers more
if p % 1
] . R1 Q1 $
Q1
k2 +
]
degrees of freedom, but also higher
Q2
]
2
complexity, than a second-order
]]
^
h
1
k
2
	 R1 Q1 $
if p = 1 .
[
tank. In the design of a matching
Q1
1+
 ]
Q2
network, the typical constraints are
]
2 2
] . R Q 2 $ ^1 - k h
a desired input impedance, Rl, and
if
1
&
p
1
1
]
2 Q1
a given operating frequency, ~ 0.
1+k
]
Q2
\
At the same time, we want to mini-
(23)
mize the power loss in the doubly
From (22) and (23), we learn that a
tuned network to maximize network
higher magnetic coupling increases
efficiency. This means maximizing
the equivalent resistance at ~ L but,
the equivalent loss resistance in the
on the contrary, reduces R eq at ~ H .
high-Q S case, as it is in parallel with
Hence, a larger ; k ; improves the effi-
Rl (see Figure 12), or, on the contrary,
ciency of the network if it is operated
minimizing R eq in the low-Q S case, as
in this case R eq is in series with the
at the lower resonance frequency,
load (see Figure 12).
while it decreases the efficiency of
A few design examples help put
the network if it is operated at the
the foregoing discussion in perspec-
higher resonance frequency. More-
tive. First, we consider the case of a
over, (23) shows that R eq is, like-
step-down matching network, like
wise, the transformed resistance
Rl , approximately proportional to
the one depicted in Figure 13(a). This
p 2 at ~ H .
is the typical scenario that emerges
In the low-Q S case, the doubly
in the design of a power amplifier,
tuned network behaves quite dif-
especially if a technology with a low
ferently [27]. From (17), we see that
supply voltage is used. We design
the network to operate in the highhaving a small value of Q S means
C 2 is negligible, as its impedance is
Q S regime, such that its efficiency
shunted by a much lower impedance:
is maximized when, as discussed,
R L. The circuit hence degenerates
R eq is maximized. Comparing (22)
and shows a single parallel reso-
to (23), we see that it is more con-
nance at frequency
venient to use the network at its
lower frequency resonance, that is,
to set ~ L = ~ 0. As discussed, a large
~1
	
~S =
.(24)
magnetic coupling is beneficial for
1 - k2

	

operation at ~ L. Moreover, from
(22), we see that larger inductances
are also beneficial to maximize net-
work efficiency (as long as Q S 2 1).
It can be proven [27] that the value
of p that maximizes R eq,L is roughly
unity. With this design choice, the
impedance transformation w ill
be set only by the turn ratio n, as
shown by (20).
A different scenario is illus-
trated in Figure 13(b), in which a
step-up impedance transformation
is desired. This could be the case
of an interstage matching network
for, say, a cascode amplifier. In this
case, we would like to use large
inductors to improve the network
efficiency, but this choice, com-
bined with a small load resistance,
inevitably leads the network to
operate in the low-Q S regime. Thus,
the specification on the trans-
formed impedance sets a constraint
on the equivalent characteristic
impedance, Z 0. If we combine this
constraint with (25) and (26), we can
recast the latter as

	 R eq,S =

k 2 + ^Q 2 /Q 1h
Rl R L
.(27)
n$
Q2
k (1 - k 2)

To maximize network efficiency, we
thus need to minimize n. Moreover,
(27) shows a remarkable result:
there is an optimal value of k that
minimizes R eq,S , hence maximizing
network efficiency.
One interesting feature of the dou-
bly tuned network is that it can be
used in the high-Q S regime to achieve
broadband operation [27], [30]. This
scenario is illustrated in Figure 13(c).
To achieve a flat passband, we need
Rl to be equal at ~ L and ~ H . This
means we select, as a start, p = 1.
Moreover, to minimize the ripple
between ~ L and ~ H , we need to set
; k ; Q S = 1, as explained in [27]. With
these design choices, the bandwidth
is set only by the value of the mag-
netic coupling. At this point, we need
to take into account the losses of the
coupled coils. From (22) and (23), we
see that R eq, L 2 R eq, H , such that the
losses are larger at the higher edge

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