IEEE Solid-States Circuits Magazine - Spring 2018 - 12

equal to twice the -3-dB bandwidth
of the filter. This choice represents
a scenario of interest in radio-frequency (RF) receivers where the adja-

integrator inverts and the other does
not. We have for this system

cent channel must be suppressed.
The -3-dB bandwidth is given by
~ 2- 3dB =

~ 2n
1
=2 - 2 +
2
Q

c2 -

1 2+ 4 .
G
m
Q2

Y (s) = k 1 k 2 ,
X
s2 + k1 k2

(6)

X

+

k1
s

-

k2
s

Figure 2: A filter using two lossless integrators in a loop.

α
-
+

A

k1
s

B

Realization of Complex Poles
It can be shown that a passive network consisting of only resistors and
capacitors does not provide complex
poles [4]. We must therefore seek
active implementations that exploit
feedback to create such poles.
Let us begin with the negativefeedback system shown in Figure 2,
where two lossless integrators appear in the loop. Note that Y is negated as it enters the input summer.
This negation can be removed if one

(a)
R2
C1
Vin

R1

-
+

obtaining imaginar y poles if
k 1 k 2 2 0. To stabilize the system,
we must add a term proportional to
s in the denominator. This can be
accomplished by a number of techniques, for example, 1) we can add a
zero to the open-loop transfer function, as practiced in type II phaselocked loops, or 2) we can make
one of the integrators lossy, e.g.,
we can change k 1 /s to k 1 / (s + a) .
The latter is realized if a fraction of
the integrator's output is returned
to its input without phase shift.
Illustrated in Figure 3(a), such an
arrangement yields

If Q = 1 2, the two poles are real
and equal (as in an open-loop cascade of two first-order RC sections),
~ - 3dB = 0.64~ n, and the attenuation
provided by H in (5) at 2~ - 3dB is
0.379. On the other hand, if Q = 1,
~ - 3dB = 1.27~ n a n d t h e at te nu ation at 2~ - 3dB reaches 0.166. That
is, by allowing 1.15 dB of peaking
at the edge of the passband, we
improve the rejection at 2~ - 3dB by
20 log (0.379/0.166) . 7.2 dB.

Y

Vout

(b)
Figure 3: (a) A lossy integrator and (b) its
circuit implementation.

(7)

k1 .
B (s) =
A
s + ak 1

(8)

The circuit implementation is straightforward [Figure 3(b)] and gives
Vout
- R2
(s) =
.
Vin
R 1 (R 2 C 1 s + 1)

(9)

We can now incorporate the lossy
integrator of Figure 3(a) in the architecture of Figure 2 [Figure 4(a)]. In

α
-
X

+

k2
s

k1
s

-

Y

(a)

RF

RF
R2

R2
C1
Vin

R1

-
+

C2
R3

-
+

C1

R5
R4

-
+

R1
Vout

R3
+-
-+

Vin

+
VX
-

+-
-+
R3

R1
(b)

C2

C1

C2

R2
RF
(c)
Figure 4: (a) the use of a lossy integrator in a biquad loop, (b) the tow-thomas biquad, and (c) its differential version.

12

s p r i n g 2 0 18

IEEE SOLID-STATE CIRCUITS MAGAZINE

Vout



Table of Contents for the Digital Edition of IEEE Solid-States Circuits Magazine - Spring 2018

Contents
IEEE Solid-States Circuits Magazine - Spring 2018 - Cover1
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IEEE Solid-States Circuits Magazine - Spring 2018 - Contents
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