Improved Biquad Structures Using Double-Output Transconductance Blocks

for Tunable Continuous-Time Filters

VLADIMIR I. PRODANOV AND MICHAEL M. GREEN

Department of Electrical Engineering, University at Stony Brook, Stony Brook, NY 11794-2350

Abstract. A family of gm −C biquad structures is derived. These biquads require only a pair of grounded capacitors
and three transconductors. It is shown that a pair of complex zeros can be realized simply by replicating the output
stage of the transconductance block, thereby constructing a second output current that is proportional to the original
output current. Although these biquad structures are very compact, they allow independent programming of the
filter’s center frequency and Q. IC simulations and measurements are presented using a fifth-order tunable filter as
an example.

Key Words: active filters, biquadratic filters, continous-time filters, tunable filters, analog integrated circuits

1. Introduction In the next section we present two biquad struc­

tures which employ only three transconductors and two
In general there are two analog filter synthesis tech- grounded capacitors, hence conserving both power dis­
niques: the passive prototype-based technique and the sipation and chip area. In Section 3 we discuss the ef­
biquad-based technique. Biquads are preferred when fect of the transconductance block nonidealities on the
the desired filter transfer function has finite zeros. In performance. In Section 4 we give measured results of
the last decade many different gm − C biquad struc- a fifth-order filter that was designed using these biquad
tures for monolithic filter implementation have been structures.
reported [2], [5], [1]. These biquads are all based on
realizing a pair of poles using a set of integrators con­
2. Derivation of the Biquad Structures
nected with feedback; the transmission zeros are then
obtained by injecting weighted signals into the loop. Consider the simple structure shown in Fig. 1. The
As discussed in [5], there are two ways of achieving transfer functions V1 /Vin and V2 /Vin are given by:
these zeros without disturbing the location of the two
poles: V1 1 + s gCm22
1. By coupling the input voltage to an appropriate = (1)
Vin 1 + s gCm11 + s 2 gCm11 Cgm2
2

node (or set of nodes). In many cases this pro­

cedure results in a structure that contains floating V2 1

= (2)
capacitors and hence requires input buffering. Vin 1 + s gm1 + s 2 gCm11 Cgm2
C 1 2

2. By feeding additional current (generated using an

additional transconductor) into a node with some

The poles of this circuit are thus given by:
fixed impedance connected to ground. Using this � � �
approach one can design highly flexible biquads gm2 gm1 C2
([1], [4]) that use only grounded capacitors and s= −1 ± 1 − 4 (3)
2C2 gm2 C1
hence do not require input buffering. Unfortu­
nately, such biquads employ many (7–8) transcon- A pair of zeros can be realized by adding an ex­
ductance blocks, which can dissipate excessive tra feedforward path; this can be done conveniently
power and require large chip area. by simply adding an extra proportional output to each
transconductance block, as shown in Fig. 2.
 Incorporating (1) and (2) into (5), we have the following
filter transfer function:
� �
Vout 1 + s 1 C1
k1 gm3
+ 1 C2
k2 gm3
+ s 2 k1Cgm3
1 C2
gm2
= (6)
Vin 1+sg +s g g
C1
m1
2 C1 C2
m1 m2

�
where ki 1 ≡ gmi /gmi , i = 1, 2.
The double-output transconductance blocks shown
in Fig. 2 are realized by an additional output stage
Fig. 1. gm − C lowpass structure. as illustrated in Fig. 4. Fig. 4(a) shows a conven­
tional single-output transconductor. Fig. 4(b) shows
a double-output transconductor with gm and gm� of
(W/L)
the same sign; notice that the ratio gm /gm� = (W/L) �.

Fig. 4(c) shows a double-output transconductor with

gm and gm� of opposite sign which is a result of chang­
ing the diode connection from transistor M1 in Fig. 4(b)
�
to transistor M2 and moving output current i out to the
drain of M1 .
From the transfer function given in (6) we can derive
the ω0 and Q of the quadratic function in the numerator
Fig. 2. gm − C biquad structure that realizes a pair of transmission and the denominator:
zeros.
For the denominator,

� �
gm1 gm2 gm1 C2
ω0d = Qd = (7)
C1 C2 gm2 C1

For the numerator,

Fig. 3. Using a single-output transconductor as a summer: the circuit
and its small signal equivalent. �
� k1 ggm3 C2
gm3 gm2 m2 C 1
ω0n = k1 Qn = (8)
C1 C2 1 + kk12 CC21
The core of the Fig. 2 biquad, shown within the dot­
ted lines, is simply the lowpass structure given in Fig. 1.
� � Note that if k2 is controlled independently from the
The second outputs (labeled gm1 and gm2 —these will be �
other transconductances (by varying gm2 ), Q n can be
explained shortly), along with transconductance block
varied while keeping ω0n fixed.
gm3 , are used to realize the zeros by summing currents
By interchanging the identity of the input and output
i 1� and i 2� , converting them into a voltage and adding
terminals of the Fig. 2 biquad, as shown in Fig. 5, it
this voltage to V2 . These operations are illustrated in
is straightforward to show that (1), (2) and (5) will
Fig. 3.
still hold, but with Vin and Vout interchanged. Thus
To derive the transfer function of this biquad we can
the transfer function of the Fig. 5 circuit will be the
write, using Fig. 3,
reciprocal of the transfer function given in (6). In this
1 � filter, the center frequency and filter Q (determined by
Vout = V2 + (i + i 2� ) (4) the poles) will be given by (8).
gm3 1
Since in the Fig. 5 structure the core two-integrator
which gives: loop formed by gm1 , gm2 , C1 and C2 determines the ze­
ros of its transfer function, it is convenient to break this
1 � � feedback in the various ways shown in Fig. 6 to realize
Vout = V2 + [g (Vin − V2 ) + gm2 (V1 − V2 )] (5)
gm3 m1 the following different types of transfer functions:
Fig. 4. (a) Single-output transconductor; (b) and (c) Double-output transconductors.
 Fig. 5. gm − C biquad structure reciprocal to the one in Fig. 2.

Fig. 6(a):
� �
Vout s gCm11 1 + s gCm22
= � � (9)
Vin 1 + s k11 gCm31 + k12 gCm32 + s 2 k1Cgm2
1 C2
gm3

Fig. 6(b):

Vout 1 + s 2 gCm11 Cgm2

2

Notch : = (10)
Vin 1 + s k2Cg2m3 + s 2 k1Cgm2
1 C2
gm3

V1 s gCm22
BP : = (11)
Vin 1 + s k2Cg2m3 + s 2 k1Cgm2
1 C2
gm3

V2 1
LP : = (12)
Vin 1 + s k2 gm3 + s 2 k1Cgm2
C 2 1 C2
gm3

Fig. 6(c):

Vout s 2 gCm11 Cgm2

HP : = (13)
Vin 1 + s k2Cg2m3 + s 2 k1Cgm2
1 C2
gm3

V1 s gCm22 Fig. 6. Additional structures derived by breaking a feedback loop in

BP : = (14) Fig. 5 circuit.
Vin 1 + s k2Cg2m3 + s 2 k1Cgm2
1 C2
gm3

V2 1
LP : = (15)
Vin 1 + s k2 gm3 + s 2 k1Cgm2
C 2 1 C2
gm3

Notice that the structure in Fig. 6(c) realizes a high pass

filter whose loss at low frequencies neither depends on
subtraction of signal currents nor requires the use of a without the need for extra current mirrors, thereby
floating capacitor. Another gm − C topology with this making transistors M1 and M2 in the double-output
same desirable property was first reported in [5]. structure (Fig. 4(b) and 4(c)) unnecessary. Hence, we
A further simplification of the biquads presented conclude that only two transistors must be added to a
here is possible. Due to the way the two basic struc­ single-output transconductor to create a double-output
tures are implemented (see equation (4)), the currents transconductor suitable for implementation of the pro­
�
I1+ �
, I1� − , I2+ �
and I2− can be directly injected into the posed biquad structures, thus further simplifying the
output stage of the third transconductor (e.g. Fig. 7) circuitry and reducing the output noise.
Fig. 7. Avoiding the use of additional current mirrors in the double-output transconductors.

Fig. 8. Block diagram of inverse Gaussian filter.

Fig. 9. Ideal and measured magnitude frequency response of filter.



Table 1. Transfer function magnitudes at zero frequency.

C3 connected to the output of gm3 will contribute a
parasitic pole at s = −gm3 /C3 . However, it can be
Circuit Low Freq. Transfer Function shown that the desired poles and zeros of the Fig. 2
�
� � �
Fig. 5 Vout
(0) ≈ 1 +
gm2
δ +
gm1
− 1 δ1 circuit are not affected by the presence of this nonzero
Vin gm2 2 gm3 output capacitance. Unfortunately, this property is not
Fig. 6(a) Vout
(0) ≈ δ1 shared by the Fig. 5 and Fig. 6 biquads.
Vin
�
gm2
B. Nonzero Input Admittance:
Fig. 6(b) Notch Vout
Vin
(0) ≈ 1 − δ
gm3 2 All of the biquads discussed in the previous section
BP V1
(0) ≈ δ2 (with the exception of the highpass and notch filters)
Vin
should give a magnitude that approaches zero for suf­
�
gm2
LP V2
Vin
(0) ≈1− δ
gm3 2
ficiently high frequency. However, if the capacitances
that appear across the input terminals of the transcon­
Fig. 6(c) HP Vout
Vin
(0) ≈ δ1 δ2 ductance blocks are taken into account, then a number
BP V1
(0) ≈ δ2 of capacitive loops are formed, giving rise to capacitive
Vin
�
coupling between the outputs and input. This results in
gm2
LP V2
Vin
(0) ≈1− δ
gm3 2
the injection of the input signal into the output nodes at
high frequencies, thus creating additional zeros in the
transfer functions. As a consequence of these zeros,
flattening will be observed at higher frequencies. (As
3. Effect of Nonidealities on the Performance of discussed next, however, the transconductance blocks
the Proposed Structures contribute a set of nondominant poles; hence, this flat­
tening will be observed only over a finite range of fre­
Some of the factors limiting the performance of the pro­ quencies.)
posed structures, and that of any gm −C filter in general,
are the parameters of the individual transconductance C. Nondominant Poles:
blocks, including: (A) nonzero output admittance; (B) All transconductance blocks contribute extra poles due
nonzero input admittance; and (C) phase error due to to internal parasitic capacitances. It is discussed in [3]
nondominant poles. We will give some results con­ and [6] that in gm − C filters, the small phase shift that
cerning the effect of each of these nonidealities. these extra poles contribute can result in a large error
in the overall transfer function. However, since the
A. Nonzero Output Admittance: biquads presented here consist of only three transcon­
The most significant effect of the nonzero output con­ ductance blocks, this error is kept at a minimum.
ductance of the transconductance blocks is on the low
frequency behavior of the biquads, since most of the
transconductance blocks have a capacitor connected to 4. Measured Results
the output. In order to analyze the effect of the nonzero
output conductances on the low frequency behavior, A fifth-order band-limited inverse Gaussian filter using
the substitution sCi → goi can be made in each of the biquad structure in Fig. 2 was designed and fabri­
the transfer functions given in the previous section. A cated using the MOSIS 2 − µ Orbit Analog Process.
listing of the magnitude, in terms of the of each of the The normalized transfer function of the filter, whose
δi ≡ goi /gmi (i.e., 1/δi is the intrinsic gain of the ith block diagram is shown in Fig. 8 is given by:
transconductance block), of these transfer functions at
s = 0 is given in Table 1. We assumed that δi � 1 in 968.38 − 854.03s 2 + 1752.25s 4
order to simplify these expressions. H (s) = (16)
945 + 2268s + 2419.2s 2
We now consider the effect of output capacitance at +1451.5s 3 + 497.7s 4 + 79.6s 5
each transconductance block. Notice that in all circuits
in Fig. 2, Fig. 5 and Fig. 6, there is already a capacitor The poles and zeros are arranged in such a way that
connected at nodes V1 and V2 . Hence any additional in the normalized frequency range ω = 0 − 1.15 its
parasitic capacitance present at these nodes from the amplitude response provides equiripple approximation
transconductance block outputs will only shift the poles of an inverse Gaussian while having linear (Bassel–
and zeros slightly. On the other hand, any capacitance Thompson) phase response. This transfer function can
cipient of the Sigma Xi Prize for Outstanding Graduate the 1994 W. R. G. Baker Award of the IEEE, and a 1994
Science Student at UCLA in 1991, the 1994 Guillemin- National Young Investigator Award from the National
Cauer Award of the IEEE Circuits and Systems Society, Science Foundation.