IEEE Solid-States Circuits Magazine - Spring 2020 - 8

to that in Figure 2(b). We analyze this
topology subsequently, but should
remark that G m1 G m2 must be negative. Also, the loop consisting of G m1
and G m2 is called a gyrator, as it rotates the impedance of C 1 (by 90°),
transforming it to inductance.
In the spirit of Christensen's topology, we can construct the circuit
shown in Figure 4, where C 1 creates
low-pass action at the drain of M 1.
We can also view M 1 as a transconductance stage, G m1, and R 1 as
a "poor man's" approximation of a
feedback transconductance stage,

H (s )

G m2, concluding that the circuit gyrates C 1 to an inductor.
There is yet another perspective
for creating active inductance. Let us
suppose the feedback in Figure 2(c) is
positive and write
	

Z2
. (3)
1 - KH (s)

Z out =

We surmise that, in this case, a highpass KH(s) causes Z out to rise with
frequency. The high-pass KH(s) also
avoids latch-up (which can occur in
the presence of positive feedback) by
suppressing the loop gain at low frequencies. As illustrated in Figure 5,
this topology can be realized by a
source follower. Interestingly, here
H(s) is not a high-pass function, but

Gm 1
C1

Z1

Zin

K is. Of course, if M 1, R 1, and C 1 are
viewed as forming a two-terminal
impedance, the two circuits in Figures 2(b) and 5 are equivalent. We
reexamine both later in this article.

Properties of Active Inductors
As our first step toward quantifying
the impedance of active inductors,
we return to Figure 2(b) and, from
the open-loop circuit shown in Figure 6, write
	

where channel-length modulation
is neglected. Also, since the smallsignal drain current of M 1 entirely
flows through C 1, the loop gain (loop
transmission) is given by
	

R1

Z 1 = R 1 + 1 , (4)
C1 s

KH (s) = - VF (5)
Vt

	
K

Gm 2

Z1

FIGURE 3: An active inductor incorporating
a gyrator.

M3

VF

+

C1

Vt

-

FIGURE 6: The open-loop circuit of an active inductor.

R1

Zin
C1

M1

Zin

R1
1
gm

Gm 1

Leq

M1
Vout

Zout

FIGURE 5: A source follower providing an
inductive output impedance.

8	

S P R I N G 2 0 2 0	

ω

H (s )

R1
C1

C1

FIGURE 7: The frequency response of an
active inductor.

VDD

K

gm

1
R1C1

FIGURE 4: An MOS implementation of
Christensen's circuit.

Zin

R1 -

1
gm

1
gm

FIGURE 8: The equivalent circuit of an active inductor.

IEEE SOLID-STATE CIRCUITS MAGAZINE	

gm
.(6)
C1 s

It follows from (1) that the closedloop input impedance is equal to

	
Gm 2

=

	

Z in =

R1 C1 s + 1
C1 s
(7)
gm
1+
C1 s

= R 1 C 1 s + 1 .(8)
C1 s + gm

This result agrees with our intuition
in Figure 2(b). At low frequencies,
C 1 acts as an open circuit and M 1
as a diode-connected device, yielding Z in . 1/g m . At high frequencies,
the gate of M 1 is at ac ground, and
Z in . R 1. Sketched in Figure 7, Z in
rises with frequency if 1/g m 1 R 1,
i.e., if the zero of Z in, ~ z, has a lower magnitude than its pole, ~ p. The
circuit thus behaves inductively between ~ z and ~ p.
We wish to formulate the equivalent inductance of this topology. As an
approximation, we can find the slope
of Z in in Figure 7 between ~ z and
~ p. But an exact expression is also
possible. Since Z in reduces to 1/g m
at low frequencies and to R 1 at high
frequencies, we envision the arrangement shown in Figure 8. Let us first
subtract the series element, 1/g m,
from Z in:
IEEE Solid-States Circuits Magazine - Spring 2020

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