IEEE Solid-States Circuits Magazine - Spring 2020 - 9

	

Z in - 1 =
gm

c R 1 - 1 mC1 s
gm
.(9)
C1 s + gm

Inverting this impedance and decomposing the result into a sum,
-1
c Z in - 1 m =
gm

	

1
C
1
c R1 m 1s
gm gm

1
(10)
,
+
R1 - 1
gm

we identify two elements in parallel.
The first term on the right represents the admittance of an inductor
having a value of
	

L eq = c R 1 - 1 m C 1 , (11)
gm gm

and the second term corresponds to
the inverse of a resistance equal to
R 1 - 1/g m, both of which confirm the
topology predicted in Figure 8. As
expected, we must have R 1 2 1/g m
to obtain an inductance. In practice,
we choose R 1 & 1/g m, obtaining
L eq = R 1 C 1 .(12)
gm

	

The circuits of Figures 4 and 5 follow these results as well. We can
thus select R 1, C 1, and g m to obtain
a desired inductance, and we can
make them programmable so as to
tune L eq.
The two resistive components in
Figure 8 imply a finite quality factor
(Q ) for the active inductor. Returning to Z in in (8), we can define the Q
as the desirable (imaginary) part of
Z in divided by the undesirable (real)
part. It follows that
	

Q=

	

=

I m {Z in}
(13)
Re {Z in}

is considered, and the second term
represents the Q only if the series
resistance, 1/g m, is taken into account. The maximum Q occurs if
g m / (C 1 ~) = R 1 C 1 ~ and is equal to
g m / (2C 1 ~). However, the Q is also
limited by the output resistance of M 1.
For the gyrator-based circuit of
Figure 3, one can readily show that
	

Z in = -

C1
s.(17)
G m1 G m2

(Recall that G m1 G m2 must be negative.) The equivalent inductance is
therefore equal to - C 1 / (G m1 G m2) and
can be set to relatively large values
(e.g., in the microhenry range) by
selecting a small G m1 G m2 product.
Since the G m blocks can incorporate
multiple stages, this structure can
achieve higher inductance values
than that in Figure 2(b).
In contrast to the topology of Figure 2(b), the gyrator-based inductor
appears to exhibit an infinite Q as
(17) does not contain a real part. But
the output resistances of G m1 and
G m2 must be considered. Referring
to Figure 9, we write I F as
	

I F = - G m1

rO1
G V , (18)
rO1 C 1 s + 1 m2 X

and hence
1
	 VX = - C 1 s .(19)
IF
G m1 G m2
G m1 G m2 rO1
The equivalent inductor thus suffers from both a series resistance
equal to - 1/ (G m1 G m2 rO1) and a parallel resistance equal to rO2.

	

The speed improvement afforded
by M 1 is limited by its gate-drain
and source-bulk capacitances. We
can quantify the effect of the former
by redrawing the active inductor as
in Figure 10(b), where C 2 denotes
the gate-drain capacitance. Opening
the feedback loop, noting that the
loop gain is still equal to g m / (C 1 s),
and finding the overall impedance,
we have
	 Z in =


Active inductors have found a multitude of applications in circuit de-

R 1 (C 1 + C 2) s + 1
1
.
$
C1 s + gm
R1 C2 s + 1
(20)

Compared to (8), this impedance exhibits a lower zero frequency and

^g m R 1 - 1h C 1 ~
.(14)
g m + R 1 C 21 ~ 2

Gm1

gm
R1 C1 ~
C
1~
Q=
(15)
gm
+ R1 C1 ~
C1 ~
gm
m ^R 1 C 1 ~h. (16)
=c
C1 ~

We note that the first term on the right
corresponds to the Q of L eq if only the
parallel resistance, R 1 - 1/g m . R 1,

	

In a manner
similar to
their passive
counterparts,
active inductors
can create shunt
peaking, thus
widening the
bandwidth of
amplifiers.

Applications and Design Issues

If R 1 & 1/g m,
	

sign. In this section, we study a few
and describe their tradeoffs.
In a manner similar to their passive counterparts, active inductors
can create shunt peaking, thus widening the bandwidth of amplifiers.
Illustrated in Figure 10(a) is an example [2] where M 1 along with R 1
and the gate-source capacitance,
C GS = C 1, acts as an inductance in
series with R D. This branch exhibits
a higher impedance as the frequency increases, partially counteracting the effect of C L.

VX

+
-

-C1
Gm1Gm2

C1
IF
rO 2

Zin
Gm2

rO 2

−1
Gm1Gm2rO 1

rO 1

FIGURE 9: A gyrator circuit with finite output impedances.

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IEEE Solid-States Circuits Magazine - Spring 2020

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