IEEE Solid-States Circuits Magazine - Fall 2020 - 93

layout. The artificial transmission
line can, in turn, be used as a build-
ing block for lumped-element imple-
mentations of distributed circuits.
Figure 10(b) illustrates, for example,
a lumped-element X-band Wilkin-
son divider that uses this concept
[23]. Another distributed circuit that
lends itself to a lumped-element
implementation is the quadrature
hybrid directional coupler [24], [25],
which is used for the generation of
quadrature signals as well as for
power splitting/combining in bal-
anced and Doherty amplifiers. The
bifilar transformer concept is used
in a very effective way in the design
of a differential quadrature hybrid
reported in [25] to reduce the over-
all silicon footprint of the device.
Magnetic transformers can be
used to combine the output power
of a set of identical amplifiers [26].
One possibility is to connect the
secondary windings of N S identical
transformers in series, as displayed
in Figure 11(a). The load voltage is
shared among the amplifiers, which
decreases the voltage stress on
the amplifiers' devices. Moreover,
the combiner features an intrinsic

R′

R1

k

R2

step-down load impedance transfor-
mation. Since I L = - I 2 and VL = N S V2 ,
the equivalent load resistance seen
by each amplifier is
	

Rl =

transformer parasitics (i.e., the mag-
netizing and leakage inductance)
into the combiner network. This can
be done, for example, by absorbing
the transformer parasitics in a lad-
der structure, as discussed in detail
in the balun example.

R L .(15)
NS m2

On top of the impedance transforma-
tion due to the combiner operation,
there is the intrinsic transformation
due to the use of magnetic trans-
formers, which is taken into account
in (15) by the parameter m. Assum-
ing (for simplicity) that there is
no flux leakage, m is equal to the
transformer's turn ratio, n. Parallel
power combining is also possible,
as demonstrated in Figure 11(b). In
this case, the load current is shared
among N P identical transform-
ers (I L = - N P I 2 and VL = V2), which
results in an intrinsic step-up trans-
formation of the load impedance:

Matching Networks
Magnetic transformers are broadly
used in the implementation of input,
interstage, and output matching
networks because of their inherent
impedance transformation feature.
A key point in the design of such
networks is that the transformer's
parasitics are to be embedded in
the design leveraging, e.g., ladder
or doubly tuned networks. We have
already discussed the use of equiv-
alent ladder networks to absorb the
intrinsic parasitic elements of the
coupled inductors, so now we will
focus on doubly tuned networks.
A schematic of a doubly tuned
network is sketched in Figure 12.
Such a network arises every time we
cannot neglect the C 1 and C 2 capaci-
tances shunting the primary and sec-
ondary windings of the transformer.
These capacitors can be explicit or

Rl = N P R2 L .(16)
m

	

Hybrid series/parallel power com-
biners can also be realized, as
reported in [26]. In any case, similar
to the previous discussion about the
balun, there is the need to embed the

R′

I2

R1

k

R2

+
L1

R′

L2

R1

k

R2

+
L1

V2
-
IL

I2
+

L1

R′

L2

R1

k

R2

V2
-

R′

L2

R1

k

L2

-

R1

k

(a)

V2
-

+

V2
-

VL
RL

-

I2

R2

+
L2

L1

IL

I2
+

L1

VL
RL

V2
-

R2

+

R′

I2

I2

+
L2

L1

V2
-

(b)

FIGURE 11: The (a) transformer-based series power combiner and (b) transformer-based parallel power combiner.

	 IEEE SOLID-STATE CIRCUITS MAGAZINE	

FA L L 2 0 2 0	

93
IEEE Solid-States Circuits Magazine - Fall 2020

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