IEEE Solid-States Circuits Magazine - Fall 2020 - 96

Magnetic transformers are key in amplifier
design, as they provide galvanic isolation,
single-ended to differential conversion,
power/signal combining, and impedance
transformation.
of the passband. This will affect the
flatness of the frequency response
of the network. However, since, as
discussed, R eq,H is proportional to
p 2, we can increase p to equalize the
losses in the passband. The increase
of p will expand the bandwidth. If
this behavior is not desired, it can be
compensated for by decreasing k.
To summarize, we have seen that
magnetic transformers are key in
amplifier design, as they provide
galvanic isolation, single-ended to
differential conversion, power/signal
combining, and impedance trans-
formation. Coupling inductors also
enables area savings and different
circuit behaviors in differential and
common mode operation. In addition,
transformers can be conveniently
used in lumped-element implementa-
tions of distributed components. The
key issue we face in integrated imple-
mentations is that the magnetizing

R ′ < RL
C1

R1

The startup conditions for the
one-port and two-port oscillators are
slightly different and can be studied
using the resonator equivalent cir-
cuit introduced in Figure 12 (since
here R L " 3, the network is clearly
in the high-Q S regime) [28], [29],
[32]. The small-signal equivalent cir-
cuit of the one-port oscillator is dis-
played in Figure 15, where - G m is
the negative conductance offered by
the cross-coupled pair in Figure 14.
The oscillations start if G m R eq 2 1,
where R eq is given by (22) and (23) at
~ L or ~ H , respectively. The question
is whether the oscillations start at ~ L
or ~ H . The answer is wherever the
equivalent resistance is larger. Since,
as discussed, R eq is proportional to
p 2 at ~ H , while, at ~ L, it is weakly
dependent on p, the oscillations start
at ~ H if p is large enough; otherwise,
they start at ~ L. The minimum value
of p that triggers oscillations at ~ H
depends on k and Q 1 /Q 2, as illus-
trated in Figure 15 [28], [32].
One interesting feature of the oneport oscillator is that we can select
the oscillation mode by choosing
the port where the negative conduc-
tance is connected. Because of the
symmetry of the circuit, its behav-
ior when the negative conductance

and leakage inductances cannot be
made negligible. Hence, it is impera-
tive to embed them into the design,
e.g., making them part of ladder or
doubly tuned networks.

Transformer-Based Oscillators
An unloaded doubly tuned network
can be used as a resonator to build
a harmonic oscillator. As discussed
earlier and presented in Figure 14,
the resonator features two paral-
lel resonance frequencies, ~ L and
~ H , and, potentially, it allows for two
modes of oscillations. The resonator
can be turned into an oscillator by
connecting at one of its ports a nega-
tive conductance cell, realizing a oneport oscillator (see Figure 14). Another
possibility is to close a feedback loop
around the resonator, achieving a twoport oscillator (see Figure 14). Topolo-
gies that combine both approaches
are also possible [5], [28], [29], [31].

R2

k

C2

L2

L1

RL
R ′ > RL

R1

(a)
C1

R′

R2

k

L2

L1

C2

RL

(b)

C1

L1

k

R2
L2

C2

RL

Gain

R1

ωL
(c)

ωH
Frequency

FIGURE 13: Examples of matching networks: (a) a step-down matching network, (b) a step-up matching network, and (c) a broadband
matching network.

96	

FA L L 2 0 2 0	

IEEE SOLID-STATE CIRCUITS MAGAZINE
IEEE Solid-States Circuits Magazine - Fall 2020

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