Biquadratic Resonant Filter based on a Fully Differential Multiple
Differences Amplifier
ANTÓNIO J. GANO, NUNO F. ESPECIAL
Electronics. Dept.
INETI – Inst. Nacional de Engenharia e Tecnologia Industrial
Estr. do Paço do Lumiar 22, 1649-038 Lisboa Codex,
PORTUGAL
antonio.gano@mail.ineti.pt ; nuno.especial@mail.ineti.pt
Abstract: - A fully differential biquadratic filter section for data acquisition in smart sensor microsystems is proposed. The
architecture of this biquadratic filter section is built upon a resonant fully differential topology based on a new Fully Differential
Multiple Differences Amplifier cell (FDMDA). This biquadratic low-pass Butterworth filter has two differential inputs that can be
algebraically summed together and voltage tunable cutoff frequency. The internal structure is described and simulation results of
the developed structure are presented and discussed. Finally some advantages of this structure are pointed out, referring possible
applications and the work being carried out.
Key-Words: - Fully Differential, Analogue, Biquadratic, Buterworth, Filter, FDMDA
1 Introduction
The development of smart sensor analogue front-ends
generally require extended flexibility over the more
traditional analogue data acquisition architectures.
These new analogue front-ends offer the possibility to
tune, calibrate or adjust several functional parameters,
like amplifier voltage gains and input ranges, filter
cutoff frequencies, etc. One of the traditional analogue
processing modules for these kinds of applications is the
anti-aliasing filter.
The design process of high performance circuits for
analogue signal processing chains benefit from the use
of fully differential CMOS circuit structures. These
topologies enhance the performance of this kind of
processing chains in mixed-mode analogue-digital
CMOS technology.
A fully programmable analogue CMOS biquadratic low
frequency anti-aliasing filter section was developed,
using a new approach based on a fully differential
topology using multi-input fully differential analogue
CMOS cells.
The possibility to program or adjust the main
characteristics of the filter, namely its order and cutoff
frequency, is an important functional aspect. The
flexibility in tuning the low pass corner frequency and
the minimization of the number active components, are
key specifications in this development process. A
biquadratic section was chosen in this development for
its modularity in building high order filter modules,
giving extra flexibility in the data acquisition interface
circuit design.
The fully differential topology aims better functional
performances in mixed analogue-digital environments,
typical of smart sensor interfaces.
2 Frequency tunable fully differential
biquadratic filter section
The developed biquadratic filter section is based on a
second order resonant fully differential TowThomas[2][3] topology, with two integrator cells (Fig. 1).
-1
Vi
+
ωp
ωp
s+
Q
ωp
s
Vo_LP
Vo_BP
Fig. 1 – Functional block diagram of the 2nd order resonant filter.
This choice was due to the simplicity and flexibility of
the topology, allowing the independent tuning of the
corner frequency ωp=2πfp and the quality factor Q. It
has two independent outputs, one giving a second order
low-pass transfer function (Vo_LP in Fig. 1) and other
implementing a second order band-pass filtering
function (Vo_BP).
The developed filter has a Butterworth transfer function
built on a fully differential topology. The low-pass
cutoff frequency is continuously adjustable by an
analogue controlling voltage in each decade frequency
range, within four decades (10Hz to 10kHz).
To implement this biquadratic section a fully differential
approach was developed using a new Fully Differential
Multiple Differences Amplifier (FDMDA) cell to sum the
input and feedback signals without external components.
developed FDMDA cell has a two-stage topology
including the four input linear V-I ports of Fig. 4.
The Wang and Guggenbühl[11] extended input range
linear transconductors were used, among other
Vdd
Mvc_1a
Mvc_1b
Vdd
Mvc_2a
Mvc_2b
Mvc_3a
Mvc_3b
Mvc_4a
Mvc_4b
Mb1 Mb2
Mb_i0
Mib1
Mb_i3
Mb_i4
Mb_i2
2.1 Fully Differential Multiple Differences
Amplifier cell
Vc=+1v
M11
M31
M41
M13
M21
In order to build up the summing node of Fig. 1, a new
fully differential cell was used, minimizing the number
of components in this differential summing node.
M43
M33
M63
M61
Vp1p
M51
Vn1p
Vn1n
Ib_vc
10uA
Ibias_in
250uA
M32
Mb3
M62
M42
M34
M64
M14
Vn2n
Iop
Ion
+
Vss
-
Fig. 2 - FDMDA cell.
This cell is the recently proposed Fully Differential
Multiple Differences Amplifier (FDMDA), shown in
Fig. 2, with multiple differential voltage input ports and
a differential output[19].
This new amplifier increases the flexibility in the design
of complex analogue fully differential signal processing
structures, allowing the reduction of the number of
external components in the processing of multiple
differential signals.
Its functional structure (Fig. 3) is implemented using
linear V-I input transconductors with extended linear
input voltage range, like in the DDA and FDDA
amplifiers[1][8][17].
possibilities[9][10][14], because each of the input ports of
this amplifier are not virtually shorted when negative
feedback is applied, like in conventional differential
operational amplifiers.
The resulting summing current is converted to a
differential voltage using a regulated folded-cascode
differential load. The second stage is a class AB
amplifier for current buffering[17][19] (Fig. 5).
1st. Amplifier Gain Stage
Vpp1
Vnp1
Vpp2
Vnp2
Vpn1
Vnn1
Vpn2
Vnn2
VoVo+
CMFB1
Vnn1
Fig. 5 – Internal topology of the FDMDA cell.
Ip2-
+
+
I-V
A vd
Vod
-
In1+
Gmd_n1
In1In2+
Vpn2
Gmd_n2
Vnn2
In2-
Fig. 3 – Functional architecture of the FDMDA cell.
The open loop differential voltage transfer function is
given by
[
Vod = Avd (Vdp 1 + Vdp 2 ) − (Vdn 1 + Vdn 2 )
where Vd
p ,n k
CL
Linear Input Ports
Ip2+
+
Vpn1
RL
Vcm1
Ip1-
Gmd_p2
Vcmo
Cc
Cc
Vnp2
CMFB AB
Class AB Buffer Stage
Ip1+
Gmd_p1
Vss
Fig. 4 – FDMDA input structure based on four linear
transconductors.
Vod
Vpn1
Vnn1
Vpn2
Vnn2
Vpp2
M24
M54
Vp2n
M12
Vn2p
Vnp1
M44
M22
M52
Vp2p
Mb4
Vpp1
Vnp1
Vpp2
Vnp2
Vpp1
M23
M53
Vp1n
]
(1)
are the several differential input voltages
and Avd the open loop differential voltage gain. The
The main design parameters for this cell were[19]:
Supply voltages: Vdd=-Vss=+2.5V
Inputs differential voltage range: Vidk≤ ±0.5V
Avd (DC) ≥ 100 dB (Rload ≥ 1kΩ)
Gain-Bandwidth Product > 10 MHz (Cload ≤ 10pF);
Phase margin ≥ 70º (Cload ≤ 10pF)
CMR > 120 dB, PSR > 100 dB
The simulation results show an open loop differential
voltage gain of 120 dB, with a gain-bandwidth product
of 18 MHz and a phase margin of 85 degrees. These
results, shown in Fig. 6, were obtained with a load of
10kΩ in parallel with 5pF. In all Spice simulations the
BSIM 3V3 parameters, of a commercial available
of components besides the two integrator cells
structures, with the advantage of two high impedance
differential input ports, giving the possibility of filtering
the algebraic sum of both signals without extra
components.
The transfer function, with R1=R2=R and C1=C2=C is
CMOS 0.8µm N-well double metal technology, were
used.
Differential Gain and Phase ([dB] , [Degree])
200.0
Open loop differential Gain
Open loop differential Phase
100.0
H LP (s) =
[± V
Vod_LP (s)
id1_f
0.0
101
102
103
104
105
106
107
108
109
Fig. 6 – FDMDA cell open loop differential gain and phase
characteristics.
The Tow-Thomas biquadratic section was developed
using two MOST-RC integrators[3][4][5][6][7], as shown in
Fig. 7. using a FDMDA cell to sum the input and
feedback signals without external components.
RQ
C2
W
WR
(5)
; β Q = µ n C ox Q )
LQ
LR
The use of these equivalent resistive structures allows
the continuous tuning of both cutoff frequency and
quality factor by adjusting the controlling voltage (V1RV2R). The final structure of this biquadratic cell is
depicted in Fig. 8.
Both integrators are controlled by the same differential
voltage (V1R -V2R) and the relationship between ohmic
values R’ and R’Q is then obtained by the relative
dimensions of both transistors structures.
(β R = µ n C ox
R2
R1
Vid2_f
Vod_f
R2
R1
(3)
quality factor Q.
Both differential resistors were implemented using
differential structures proposed by Banu-Tsividis[5] and
Czarnul[7] (Fig. 8). Each set of four transistors work in
the triode region and are considered matched. The
equivalent differential ohmic values R’ and R’Q depend
on the dimensions of the transistors and on the
difference of two controlling voltages (V1R-V2R) [5][7]:
1
1
(4)
R' =
R' Q =
β R (V1R − V2R )
β Q (V1R − V2R )
2.2 Filter implementation
Vid1_f
(2)
2
1
The resistor pairs R and RQ, keeping C constant, can
independently tune the cutoff frequency ω p and the
Frequency [Hz]
C1
]=
ωp
=
1
1
ω
s2 +
s + 2 2 s 2 + p s + ωp 2
R QC
R C
Q
RQ
with ω p = 1
;
Q=
RC
R
1
= 2 2
R C
-100.0
-200.0
100
(s) ± Vid2_f (s)
C1
C2
RQ
Fig. 7 – Filter structure using the FDMDA cell in the input summing
node.
This combination of a differential topology with the
inclusion of this new summing cell reduces the number
R’Q
M1
R’
M2
R’
C1
Vid1_f
M1
Vod_1
M2
Vid2_f
C2
M5
M6
Vod_LP
M4
M8
M3
M7
C2
C1
M4
Vod_BP
M3
V1R
V2R
Fig. 8 – Fully differential biquadratic filter using MOST-RC integrators and a summing FDMDA cell.
Considering that C1=C2=C then the second order
Butterworth low-pass transfer function is given by
=
β
2
R
Vod _ LP ( s )
(Vid1_f (s) + Vid2_f (s))
(V1R − V2 R )
C2
=
ω p2
s + 2ω p s + ω p
2
2
=
2
(6)
1
s2 +
β Q (V1R − V2 R )
C
s+
β 2 R (V1R − V2 R ) 2
C2
with
ωp =
Q=
β (V - V2R )
W (V - V2R )
1
= R 1R
⇔ ω p = µ p C ox R 1R
RC
C
LR
C
RQ
R
=
WQ
W
βR
1
=
⇔
= 2 R
LR
LQ
βQ
2
(7)
This topology has the flexibility of offering an extra
second order band-pass transfer function at the output
Vod_BP, after the loss integrator, as shown in Fig. 1. This
band-pass output is given by the expression
H BP ( s) =
=
Vod _ BP ( S )
(Vid1_f (S) + Vid2_f (S))
β R (V1R − V2 R )
C
=
ω ps
s 2 + 2ω p s + ω p
2
=
(8)
s
2
s +
β Q (V1R − V2 R )
C
s+
2.3 Simulation results
β 2 R (V1R − V2 R ) 2
C2
with a gain of 1/Q for the central frequency ω = ω p .
As can be seen from Fig. 8, the FDMDA performs a
differential voltage follower with
(9)
Vod_1 = (Vid_f + Vid2_f ) - Vod_LP
considering its high open loop gain Avd.
The two input signals, Vid1_f and Vid2_f, can be filtered
independently, or algebraically summed, by appropriate
multiplexing of both differential inputs, giving extra
flexibility in signal processing.
To achieve a voltage tuning range of one full decade in
frequency, V1R is made constant and equal to
V1R=+2.2V and the frequency adjustment process uses
V2R, ranging in the interval [+1.5V, +2.15V].
The global specified range for the cutoff frequency is
10Hz to 10kHz, covering 3 frequency decades.
The capacitor values C were then dimensioned
accordingly to the following table:
The Fig. 9 shows the obtained frequency characteristics
of this low-pass biquadratic section, using
Vid1_f=+100mV and Vid2_f=0 as differential input signals.
The results shown are for V2R=+1.5V and V2R= +2.13V,
with C=10nF, 1nF, 100pF and 10pF. It can be seen that
it is possible to cover 4 frequency decades in the cutoff
frequency, between 10Hz and 100kHz, choosing the
appropriate value C of the equal capacitors and varying
the tuning voltage V2R.
Within each decade for a fixed value of the capacitors,
the cutoff frequency can be continuously adjusted by
20.0
V2R=+2.13V, C=10nF
V2R=+1.5V, C=1nF
V2R=+1.5V, C=100pF
V2R=+1.5V, C=10pF
0.0
-40.0
-60.0
-80.0
-100.0
-120.0
-140.0
-160.0
10Hz ≤ fp ≤100Hz
→
C= 10nF
100Hz ≤ fp ≤1KHz
→
C=1nF
-200.0
1KHz ≤ fp ≤10KHz
→
C=0.1nF
-220.0 0
10
Both equivalent resistive structures R’ and R’Q have all
the transistors with same length LQ=LR=L=40µm, in
order to minimize channel length modulation effects in
the triode region. The width dimensions were
WR=4.8µm and WQ=6.8µm.
V2R=+1.5V, C=10nF
-20.0
Filter Gain [dB]
H LP ( s) =
The two fully differential operational amplifiers were
designed using a two-stage topology, with gain
bandwidth product of 15MHz and open-loop DC gain of
90dB.
The differential input range of both inputs is limited to
±200mV in order to minimize distortion in the input
transconductors of the FDMDA cell and in the
differential MOST-R structures.
One differential input can be used to cancel the filter
output offset, caused by internal component
mismatches, summing a DC differential voltage at the
input to null that offset voltage.
One important issue in CMOS analogue time
continuous filters, in particular considering the
application of this biquadratic section in a
programmable sensor interface, is the ‘in-circuit’ tuning
capability to adjust the frequency characteristics of the
filter. This feature can be achieved controlling the
voltage V2R with an extra tuning circuit, using negative
feedback topologies.
-180.0
10 1
10 2
10 3
10 4
10 5
10 6
10 7
10 8
10 9
Frequency [Hz]
Fig. 9 – Filter low-pass cutoff frequency variation range for
several values of capacitance C.
V2R, in the range [+1.5V, +2.15V], as can be seen in
Fig. 10.
20.0
V2R=+1.75V
V2R=+2.0V
V2R=+2.1V
V2R=+2.13V
V2R=+1.5V
C1=C2=C=100pF
10.0
0.0
C11=105pF; C12=95pF; C21=90pF; C22=110pF
C11=90pF; C12=95pF; C21=109pF; C22=94pF
C11=95pF; C12=98pF; C21=108pF; C22=99pF
C11=94pF; C12=102pF; C21=97pF; C22=105pF
-20.0
0.0
-60.0
Filter Gain [dB]
Filter Gain [dB]
-40.0
-80.0
-100.0
-120.0
-140.0
-160.0
-10.0
-20.0
-30.0
-40.0
-180.0
-200.0
10 0
10 1
10 2
10 3
10 4
10 5
10 6
-50.0
10 0
10 7
Frequency [Hz]
Here a value of C=100pF has been used in the
simulations, allowing an adjustment of the cutoff
frequency between 1kHz and 10kHz.
V2R=+2.1V
V2R=+2.13V
V2R=+2.0V
V2R=+1.75V
V2R=+1.5V
0.0
Filter Gain [dB]
-20.0
-40.0
-60.0
-80.0
-100.0
10 0
10 1
10 2
10 3
10 4
10 2
10 3
10 4
10 5
Frequency [Hz]
Fig. 10 – Tuning of the cutoff frequency between 1kHz and 10kHz
using several values of the controlling voltage V2R with C=100pF.
20.0
10 1
10 5
10 6
10 7
Frequency [Hz]
Fig. 11 – Frequency characteristics of biquadratic bandpass
output for several values of V2R with C=100pF.
The filter module as also a second order band-pass
output that may be used in low-frequency sensor data
acquisition circuits. The results of this extra second
order filtering function are in Fig. 11, for C=100pF and
sweeping V2R between [+1.5V, +2.13V].
As the result of the Butterworth type of implementation
in the low-pass section of the filter, with Q= 1 , the
2
gain for ω =ωp is –3dB, as can be seen in Fig. 11.
For this low frequency application the values of the
capacitors C are not suitable for monolithic integration,
and they have to be external to the filter module.
The influence of the tolerance of the capacitor values in
the transfer function is not relevant in the filter
characteristics, and the deviations can be tuned by
automatic tuning methods. Anyway, the developed
module is mainly intended for anti-aliasing filtering,
without high precision frequency requirements. In Fig.
12 several simulation runs with different values for all
capacitors of the structure are shown, keeping the
Fig. 12 –Frequency transfer characteristics using different values for
all the capacitors in the structure, considering deviations within ±10%
from the nominal value C=100pF (V2R=+2.13V).
capacitance values within ±10% deviation from the
nominal C=100pF value. It can be seen that the
deviations from the nominal cutoff frequency are not
significative for this kind of application.
The main specifications for this fully differential
tunable biquadratic Butterworth section are as follows:
Supply voltages: Vdd=-Vss=+2.5V
Inputs differential voltage range:
Vidk≤ ±0.2V (max. allowed: Vidk ≤±0.5V)
Inputs common mode voltage range:
-1V ≤ Vick ≤ +1V
Filtering function:
2nd order Butterworth low-pass and band-pass
frequency characteristics;
Range of low-pass cutoff frequencies:
10Hz ≤ fp ≤ 100kHz
Frequency range setting capacitances:
10pF ≤ C ≤ 100nF
Range of frequency tuning voltage V2R:
+1.5V ≤ V2R ≤ +2.15V
CMR > 120 dB, PSR > 100 dB
3 Conclusions
A new approach to build up a fully differential
frequency tunable biquadratic Tow-Thomas resonant
filter was proposed, using a new analogue fully
differential cell, the Fully Differential Multiple
Differences Amplifier. Using this multiple differential
input cell in the summing node of this topology the
following benefits are achieved:
.
.
two differential high impedance inputs. Two voltage
signals can be filtered independently or algebraically
summed using flexible signal multiplexing;
minimum number of extra components to implement
the filter functional architecture, based on two
MOST-RC integrators;
The fully differential topology enhances the linearity
and harmonic distortion characteristics of the developed
Butterworth low-pass filter, due to its symmetry.
The use of voltage controlled MOST-R differential
structures, allows the continuous voltage tuning of both
low-pass and band-pass frequencies of the section
transfer functions, within a full decade frequency
variation range. This key functionality makes it possible
to implement automatic ‘in-chip’ tuning circuits.
The simulation results referred previously show the
feasibility and flexibility of this approach, making this
filter module suitable to be included in smart
microsystems with programmable analogue processing
interfaces,
where
flexibility
and
functional
programmability are important issues.
Further work is being carried out to optimize the
characteristics of the fully differential cells in order to
design a test prototype.
References
[1] E. Säckckinger, W. Guggenbühl,“A Versatile Building
Block: The CMOS Differential Difference Amplifier”,
IEEE J. of Solid-State Circuits, vol. SC-22. No. 2, April
1987, pages 287-294
[2] Phillip P. Allen, Douglas R. Holberg, CMOS Analog
Circuit Design, Oxford Univ. Press, 1987
[3] K. Laker, W. Sansen, Design of Analog Integrated
Circuits and Systems, McGraw-Hill, 1994
[4] D. A. Johns, K. Martin, Analog Integrated Circuit
Design, Wiley, 1997
[5] Y. Tsividis, M. Banu, J. Khoury, Continuous-Time
MOSFET-C Filters in VLSI, IEEE Transactions on
Circuits and Systems, Vol. CAS-33, No. 2, February
1986, pp. 125-139
[6] T. Georgantas, Y. Papananos, Y. Tsividis, A
Comparative Study of Five Integrator Structures for
Monolithic Continuous-Time Filters –A Tutorial, Proc.
1993 IEEE Int. Symp. On Circuits and Systems, 1993, pp.
1259-1262
[7] Z. Czarnul, Modification of Banu-Tsividis ContinuousTime Integrator Structure, IEEE Transactions on Circuits
and Systems, Vol. CAS-33, No. 7, July 1986, pp. 714716
[8] S.R.Zarabadi, F. Larsen, M. Ismail,“A Reconfigurable
Op-Amp/DDA CMOS Amplifier Architecture”, IEEE
Trans. on Circuits and Systems-I: Fund. Theory and
Applications, vol. 39, No. 6, June 1992, pages 484-487
[9] S. Szczepansky, R. Schaumann, P. Wu, “Linear
Transconductor Based On Cross coupled CMOS pairs”,
Electron. Lett., vol. 27, April 1991, pages 783-785
[10] S. Szczepanski, A. Wyszynski, R. Schaumann,“Highly
Linear Voltage-Controlled CMOS Transconductors”,
IEEE Trans. on Circuits and Systems-I: Fund. Theory
and Applications, vol. 40, No. 4, 1993, pages 258-262
[11] Z. Wang, W. Guggenbühl, “A Voltage-Controllable
Linear MOS Transconductor Using Bias Offset
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
Technique”, IEEE J. of Solid-State Circuits, Vol. 25,
Feb. 1990, pages 315-317
Eduard Säckinger, Walter Guggenbühl, “A High-Swing,
High-Impedance MOS Cascode Circuit”, IEEE J. of
Solid-State Circuits, vol. 25, No.1, Feb. 1990, pages 289297
Zhong Y. Chang, W. M. C. Sansen, Low-Noise Wide
Band Amplifiers in Bipolar and CMOS Technology,
Kluwer Academic Publishers, 1991
C.-C. Hung, C. Hwang, M. Ismail,, K Halonen, V. Porra,
“Low-Voltage CMOS Rail-to-Rail V-I Converters”,
Anal. Int. Circ. and Signal Processing, vol. 13, Kluwer
Academic Publishers, 1997, pages 261-274
C.-H. Lin, T. Pimenta, M. Ismail, “A Low-Voltage LowPower CMOS V-I Converter with Rail-to-Rail
Differential Input for Filtering Applications”, 5th. IEEE
Int. Conference on Circuits and Systems, Lisbon, 1998,
pages 165-168
A.Assi, M. Swan, “High Performance CMOS
Transconductor for Mixed Signal Analog-Digital
Applications”, Anal. Int. Circ. and Signal Processing,
vol. 19, Kluwer Academic Publishers, 1999, pages 303317
C. Shi, Y. Wu, H. Elwan, M. Ismail, ‘A Low-Power
High Linearity CMOS Baseband Filter for Wideband
CDMA Applications’, ISCAS’2000, IEEE Int.
Symposium on Circuits and Systems, May 28-31, 2000,
Genéve, Switzerland, pp. II152-IV155
António J. Gano, José E. Franca, “Fully Differential
Variable Gain Instrumentation Amplifier Based on a
Fully Differential DDA Topology”, 6th. IEEE Int.
Conference on Circuits and Systems, Paphos, 1999,
pages 785-788
António J. Gano, José E. Franca, ‘New Multiple Input
Fully Differential Variable Gain Instrumention
Amplifier’, ISCAS’2000, IEEE Int. Symposium on
Circuits and Systems, May 28-31, 2000, Genéve,
Switzerland, pp. IV449-IV452