IEEE Solid-States Circuits Magazine - Fall 2020 - 89

with many turns and large coil sec-
tions are sought, especially at lower
frequencies, to get large inductance
values, while tightly wound coils
help increase the magnetic coupling.
These features help implement a
good magnetic transformer and, at
the same time, explain why the trans-
formers are inherently bulky devices.
The metal wire used to build a coil
has some nonnegligible resistivity,
t -that is, any practical inductor is
lossy. We can model the losses asso-
ciated to the wire by means of a resis-
tor, R , in series to an ideal lossless
inductor. In the case of a solenoidal
coil, we can write
R = t N22rr ,(10)
rd /4

	

where d is the wire diameter. The
question now is whether, given a
generic winding of wire, the result-
ing device is an inductor or a resis-
tor, and, if it is an inductor, how
good that inductor is. To answer this
question, we borrow a concept from
second-order resonators, namely,
the quality factor. For any frequency
of interest, ~ 0 , we imagine hooking
up our inductor to an ideal lossless
capacitor, such that it resonates at
~ 0 , and we apply the definition of
quality factor, i.e., we take the ratio
between the total energy stored in
the resonator and the energy lost
during the time span, 1/~ 0 . Since
only the inductor is lossy, the qual-
ity factor of the resonator will define
the quality factor of the inductor.
For a solenoidal inductor, using (4)
and (10) we have

	

QL =

nrd 2
~0 L
= ~0
Nr.(11)
R
8t,

If we compare (11) to (4), we notice
that Q L is approximately propor-
tional to L , which means that,
at least in the case of a solenoid, a
larger inductance comes along
with a higher inductor quality fac-
tor. Again, this shows why larger,
bulkier inductors and magnetic
transformers are favored (at least at
lower frequencies).

Integrated Inductors and
Transformers
Integrated inductors are imple-
mented as a planar version of the
solenoid geometry [18]. They fea-
ture a spiral shape. A circular spiral
shape maximizes the inductor qual-
ity factor. However, this shape is not
always allowed by the design rules.
Common alternatives are octagonal
or square shapes. Planar inductors
can be asymmetric or symmetric,
and they can have a center tap,
as illustrated in Figure 4. In a pla-
nar inductor, the field distribution
within the coil is not uniform, as in
the solenoid. This makes the com-
putation of the inductance much
more complex. However, some com-
pact approximated expressions are
available, such as [19]
	

L = K1 n0

N 2 d avg
,(12)
1 + K2 td

where the average inductor diam-
eter d avg and fill factor t d are
defined in Figure 4. The parameters
K 1 and K 2 depend on the inductor's
shape; they are reported in Figure 4
for the octagonal and square cases.
We can use (12) to gauge the size of
a planar inductor for typical induc-
tance values, as used in integrated
circuits. Assuming an octagonal
coil with trace width w = 10 nm and
intertrace spacing s = 5 nm (see
Figure 4), we see that a single-turn,
100-pH inductor has outer diameter
d out = 70 nm , while a seven-turn,
10-nH inductor has outer diameter

d ++ddin
out
in
ddavg == dout
avg
22
d --ddin
out
in
ρρd == dout
d d
dout ++ddin
out

in

d out = 300 nm . In both cases, they
are huge devices, much bigger than,
say, a transistor.
Integrated inductors have some
limitations. Magnetic materials are
typically not available in integrated
processes (i.e., n = n 0 ), which results
in a larger device size for a given
inductance value. As previously
noted, the magnetic field is not uni-
form across the planar winding,
such that the inductance is propor-
tional to the coil diameter rather
than the area of its section, as in
the solenoid's case. Because of the
planar spiral shape, there is lim-
ited magnetic coupling between the
turns, as inner turns have smaller
areas and the mutual inductance
depends on only the smaller coil
section. Hence, the inductance is
not proportional to the square of
the number of turns. Again, this
means a larger inductor size for a
target inductance value.
Since the inner turns of the spi-
ral contribute less additional induc-
tance but still provide losses, a
hollow spiral layout is preferable:
it is recommended that the fill fac-
tor t d be limited to less than 0.5.
Since t d increases with the number
of turns N, (12) shows that L is pro-
portional to N p, where, as discussed,
p 1 2. However, the parasitic series
resistance is still approximately
proportional to N . As a result, the
inductor quality factor is propor-
tional to Lb with b . 0.2, 0.3. The
quality factor still increases with

ddout
out
ddin
in
ww
ss

N 22d
avg
LL==KK1µµ0 N davg
1 0
11++KK2 ρρd

2 d

KK1
1
KK2
2

Square
Square
2.34
2.34
2.75
2.75

Octagonal
Octagonal
2.25
2.25
3.55
3.55

FIGURE 4: Examples of planar integrated inductors.

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IEEE Solid-States Circuits Magazine - Fall 2020

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