A CMOS CONTINUOUS-TIME CURRENT MODE FILTER ...

A CMOS CONTINUOUS-TIME CURRENT MODE FILTER TECHNIQUE
FOR LOW SUPPLY VOLTAGE APPLICATIONS
Elena Doicaru*, Mircea Bodea**
*University of Craiova, Faculty of Automation, Computer Science and Electronics
Blvd. Decebal, Nr. 5, 1100 Craiova, ROMANIA, e-mail: dmilena@electronics.ucv.ro
** POLITEHNICA University of Bucharest, Faculty of Electronics, Telecommunication
and Information Technology
Blvd. Iuliu Maniu, Nr. 1-3, 061071 Bucharest, ROMANIA, e-mail: mircea.bodea@yahoo.com
Abstract: This paper presents a current mode (translinear) technique for low voltage,
continuous-time, analogue filter implementation. Firstly, the state-space description of
current-mode filter is presented. Then the current-mode integrator based on translinear
approach is presented and analyzed. Finally, as a test vehicle for the proposed technique,
a low voltage, third order, continuous-time filter, for low power applications structure
implementation is described.
Keywords: Analogue signal processing circuits, translinear circuits, CMOS analogue
filters, low power and low voltage circuits.
1. INTRODUCTION
The existing CMOS technologies provide ample
opportunity to integrate entire system on a single
chip. To date, the ability to integrate large digital
systems has far out weighted the ability to integrate
the analogue systems. The greatest impediment to
analogue CMOS VLSI design has been the difficulty
to get consistent circuit performance over the broad
range of requirements for signal gain, frequency
and/or phase response, delay, power consumption
and signal integrity. In the same time the analogue
design in mixed signal environments becomes more
difficult and challenging as the IC’s power supply
voltage scales down, to the values slightly higher
than the MOS threshold voltage.
The translinear circuits, due to their current domain
operation, are suitable for low supply voltage operation.
In translinear circuits the MOS transistors
usually operate in weak inversion (or sub-threshold)
region, (Enz et al., 1995), where the current-voltage
characteristic is exponential. The main problems
associated to the sub-threshold region operation are
the relatively low speed capability and inferior
matching. But these problems are relatively solved in
sub-micron technology.
This paper presents a synthesis technique for continuous
time current-mode filters operating at very
low supply voltage. The basic current-mode integrator
is analyzed and finally the structure of a third
order current-mode filter is presented. This technique
can be extended, (Doicaru, 2007), in order to optimize
performances like bandwidth, noise, errors due
to mismatching and dynamic range.
2. THE STATE-SPACE DESCRIPTION AND
SYNTHESIS OF CURRENT-MODE FILTER
This section presents the way the intermediate transfer
function synthesis method (Snelgrove et al.,
1986) can be used in current-mode domain filters.
The method was developed for AO-RC filter; basically
a given n-th order transfer function is realized
using n resistively interconnected integrators. The
design process comprises two steps: firstly a set of
intermediate transfer functions (IFs) is selected and
then the set is using in synthesis of circuit that realized
the given transfer function. The major advantage
of this method is that the filter’s performance evaluation
and optimization can be performed at the
abstract level of transfer function generation and not
at the circuit topology level.

The state-variable formulation of the AO-RC is
s ⋅ x(s)
= A ⋅ x(s)
+ b ⋅ u(s)
+ ε(s)
y(s)
= c
T
⋅ x(s)
+ d ⋅ u(s)
(1)
where the vector x(s) represents the circuit state
(integrators outputs), matrix A describes the interconnections
between the n integrators, vector b contains
the coefficients that multiply the input signal u(s) in
order to be applied to the integrators inputs, vector c
contains the coefficients required to form the output,
scalar d is the coefficient of the feedthrough component
from input to output, and ε(s) is the vector containing
the noise component at the integrator inputs.
The dual sets of IFs, {f i (s)} and {g i (s)} are given by:
∆ xi
( s)
−1
fi
(s) = ; f(
s) = (s ⋅ I − A ) ⋅ b
u( s)
∆ y( s) T T
−1
gi
(s) = ; g (s) = c (s ⋅ I − A )
εi
( s)
(2)
The set {f i (s)} contains the transfer function from the
filter input to the integrator outputs, and the set,
{g i (s)}, can be physically interpreted as the integrators
noise gains. Given a transfer function t(s), IF
synthesis is based on choosing a set of linearly
independent functions, {f i (s)}, having identical
denominator polynomials, e(s), and arbitrary
numerator polynomials of degree less than n. From
this set the {A, b, c, d} parameters can be obtained
using the following relation:
−1
T T −1
A = F⋅E⋅
F , b = F⋅1,
c = t ⋅F
,
T
d = tn+
1, t(s)
= t ⋅v(s)
+ tn+
1,
f (s) = F⋅
v(s)
, g(s)
= G ⋅v(s)
,
1
v
i
(s) = , i = 1,
n,
s − ei
T −1
G = H⋅F
(3)
where t is a vector containing the t(s) n residues at
the poles, t n+1 – the residue at s=∞, F – a matrix containing
the residues of the f functions evaluated at the
poles, G – a matrix of the residues of the g functions,
e i – the e(s) roots, E – the diagonal matrix having the
natural modes e i as its elements, H – a diagonal matrix
formed from the residues of t(s), and
1 = (1, 1, ...1) T .
The sensitivities of filter directly depend of IFs set:
t(s)
Aij
S = gi
(s) ⋅ fi
(s) ⋅ ,
Aij
t(s)
t(s)
b t(s)
c
S = g (s) i , S f (s) i
i ⋅ = i ⋅
bi
t(s)
ci
t(s)
t(s)
d t(s)
s
Sd
= , S = f (s) g (s) ,
t(s)
γi
(s) i ⋅ i ⋅
t(s)
t(s)
t(s)
γi
(s)
S = S ⋅ S
µ i (s) γi
(s) µ i (s)
(4)
In the above equations γ i is the integrator gain and
µ i (s) is the operational amplifier gain.
Noise signals injected at integrator inputs can be
modelled by ε(s). Assuming white input noise, with
spectral density N
is given by
2
i
, the output noise power spectrum
2
2
n0 )
i
P
with a rms level of
where
( ω) = Ni
⋅ ∑ gi
( jω
(5)
2
i
P 0( ω) = N ∑ g ( jω)
(6)
n
g ( jω)
i
i
2
= ∫
∞ 2
i −∞
g ( jω) dω
2 i
(7)
This description can be adapted to the current-mode
filter. The current through the k-th capacitor of the
current-mode filter is
Ck
v&
Ck = iCk1
+ iC k 2 + ... + iC k n + iCbk
=
* * * * *
= a k 1i1
+ a k 2i2
+ ... + a kn v n + b k i in + ε k
2
2
(8)
where i Ckj , k, j=1÷n, is the k-th capacitor current component
dependent on output current i j , j=1÷n, i j is the
j-th current-mode integrator output current, i in is the
filter input current,
*
a kj , k, j=1÷n and
*
b k , k=1÷n,
coefficients describes the capacitor current components
dependence on the output currents of the
current-mode integrators, respectively, on the filter
* *
input current, and ε k = ∑ j ε kj is the total noise
current through capacitor.
Using convenient circuit technique to implement a
logarithmic dependence between capacitor voltage,
v Ck , and output current of current-mode integrator, i k ,
v = V ln ( i / I )
(9)
Ck
x
where V x and I y are scale factors, the capacitor voltage
derivative, v& Ck , becomes:
d
k
y
Vx
i
v& k
Ck = .
(10)
ik
dt
Using the translinear loops one get for each component
of the capacitor current
*
*
i Ckj ik
= akj
i j , iCbk
ik
= bk
iin
(11)
and equation (8) becomes
d
d
ik
t
where
akn
=
1 2 2
*
kn
ak1i
+ ak
i
+ ... +
*
b k
aknv
+
bk
n
i in
+ ε
k
(12)
= a /( C V ), b = /( C V ), ε = ε /( C V ) (13)
k
x
k
The circuit resulted by interconnection of n current
mode integrators with translinear loops can be
described by the same state-variable formulation (1):
k
x
k
*
k
s ⋅ x(s)
= A ⋅ x(s)
+ b ⋅iin(s)
+ ε(s)
T
iout
(s) = c ⋅ x(s)
+ d ⋅iin(s)
k
x
(14)
where the states x k are represented by the integrators
output current, matrix element A kj is implemented by
a translinear loop with input current x k and output the
component i Ckj of the current through k-th, the vector

element b k is implemented by translinear loop from
the input i in to state k, c k is multiplication coefficient
of state k required to form the output current of the
filter i out , d is multiplication coefficient of input current
i in and ε (s)
can model the current noise at the
input of the k-th current integrator.
1
I D /I F
5%
saturation
G
D
I R
B
I F
S
So the meaning of {f k (s)} IF's set is the same as in the
OA-RC filter synthesis (the state k and the input
signal ratio) and the physical meaning of {g k (s)} IF's
set is the noise gain to the input of integrator k at
output of filter. We can conclude that the all results
obtained in the OA-RC filter synthesis can be applied
to the current-mode filter synthesis.
3. MOS TRANSISTOR CURRENT-VOLTAGE
CHARACTERISTICS
This section presents a brief review of the MOS transistor
I-V characteristics, focused on weak inversion
operation.
The MOS transistor drain current equation valid in
all operating regions is (Enz et al., 1995):
with
I
D
VD
∫
= β ( −Q
/ C ) dV
(15)
VS
i
ox
β = µC ox ( W / L)
(16)
where W and L are the channel width and length,
C ox – gate capacitance per unit area, µ – charge
carrier mobility, Q i – induced mobile charge in
channel, V D , V S – drain, source voltages referred to
the local substrate, V – channel potential. Equation
(15) may be decomposed, (Vittoz, 1994), into:
I
D
∞
= β ∫
( −Qi
/ Cox
)dV
− β∫
∞
VS V D
= IF − I R
( −Q
/ C
i
ox
) dV
(17)
where I F is called forward current (controlled by
source voltage V S ) and I R is called reverse current
(controlled by drain voltage V D ). Taking into account
weak inversion channel charge dependency on
channel potential
I F and I R are given by
( VP − V ) / V
Q
T
i / Cox
~ e
(18)
I F ~ β e
( R)
( V −V
P
S(
D)
) / V
And the drain current equation results:
V
ID
= IS
e
P
/ V
T
⎡ −V
⎢e
⎣
S
/ V
T
T
−V
− e
D
/ V
In terms of V GS and V GD equation (20) becomes
( V
ID
= IS
e
V ) / V
P − G
T
⎡ −V
⎢e
⎣
GS
/ V
T
−V
− e
T
GD
⎤
⎥
⎦
/ V
T
⎤
⎥
⎦
(19)
(20)
(21)
where I S is a specific current (limit of weak
inversion), proportional to W/L
0
0 1 2 3 4 5 6 V G
DS /V T
I R
V S =const.
S
(a)
(b)
Fig. 1. (a) The operation regions in weak inversion of
the MOS transistor; (b) Non-saturated MOS transistor
is equivalent to two saturated transistors
anti-parallel connected.
I
S
e
( VP
−VG
) / VT
=
W
L
I
◊
( V )
G
(22)
I ◊ ( V G ) is the square transistor zero-bias ( V GS = 0 )
current.
Using (22) I F and I R equations become:
I
I
non-saturation
F
R
W
= ⋅ I
L
W
= ⋅ I
L
◊
◊
( V )
G
( V )
G
⋅ e
⋅ e
V
V
GS
GD
/ V
T
/ V
T
(23)
If I R

(a)
(b)
Fig. 2. The current mode integrator with translinear loop for capacitor current component I Cij .
(a) Circuit schematic; (b) Circuit symbolic representation.
Therefore, using the decomposition technique
described above (see Fig. 2.b.), the fictitious
transistors M 2* , M 4* , M 6* and M 13* (dashed line in
Fig. 2.b) were added in order to account the nonsaturated
operation of these transistors. This way all
transistors can now be regarded as saturated, I R

But
and
I
I
I
I
D4*
D6*
D2*
D13*
I0
+ I
I
0
( iin
+ I IN ) I0
=
I
0
+ I
( iout
+ IOUT
) I
=
I0
+ I
( I0
+ I ) I0
=
iout
+ IOUT
I I IN
=
i + I
I
out
D10
= I
OUT
D9
0
0
i out + I
I
0
OUT
I D10
= I D4
− I D4*
− I D5
+ I D6
− I
( iin
+ I IN ) I0
= iin
+ I IN −
− I0
+
I0
+ I
( iout
+ IOUT
) I0
i out + IOUT
−
=
I0
+ I
( iin
+ I IN ) I ( iout
+ IOUT
) I
=
+
− I
I + I I + I
=
i
out
0
= ( I − I
IN
I I
+ I
IN
) − ( I −
i
OUT
out
I I
+ I
IN
OUT
− I
D6*
I D9
= iC
+ I x − I0
+ I D2
− I D2*
=
( I0
+ I ) I0
= iC
+ I x − I0
+ I + I0
−
iout
+ IOUT
( I0
+ I ) I0
= iC
+ I
x
+ I −
iout
+ IOUT
I
x
= ( I − I IN ) − ( I D13
− I D13*
− I
IN
0
D16
=
) =
=
) =
(33)
(34)
(35)
Substituting equations (33) and (34) into general
translinear loop equations (32) yields:
( i in + I IN ) I = ( iC
+ I x ) ( iout
+ IOUT
)
( i + I ) I = i ( i + I ) + I I
in
IN
in
C
out
i I = i +
OUT
C ( iout
IOUT
)
IN
(36)
Noting that the capacitor voltage, v C , is the difference
between the gate-source voltages of M 7 and M 8
v
C
i i I
v
out + OUT
= (37)
I
D7
GS7 − vGS8
= VT
ln = VT
ln
I0
and replacing (37) into the capacitor current equation
dvC
iC
= C
(38)
dt
and taking into account the lower equation (36)
results the capacitor current
i
C
C VT
diout
= (39)
iout
+ IOUT
dt
Finally we get a linear integrator function:
0
Fig. 3. The third order filter structure.
I
iout = ∫iin
d t
(40)
CV
T
The integrator operates correctly as long as the
quiescent values of the currents I, I 0 , I IN , I OUT is
chosen so that the integrator’s transistors drain
currents to be strictly positive for the input voltage
range, (Seevinck, 1988).
5. THE THIRD ORDER
CURRENT MODE FILTER
As an example of the application of the synthesis
technique presented in Section 2 a third order current
mode filter have been synthesised. The transfer
function of the filter is:
1
t (s) =
3 2
(41)
s + 2s + 2s + 1
The system parameters {A,b,c,d} resulted using the
SSAF program (Doicaru et. al., 2007) for orthonormal
IFs are:
⎡ 0
⎢
A =
⎢
0
⎢
⎣−
0,707
⎡ 0 ⎤
⎢ ⎥
b =
⎢
0
⎥
⎢
⎣0,798
⎥
⎦
0,707 0 ⎤
⎥
−1,225
− 2
⎥
0 1,225⎥
⎦
⎡ 0 ⎤
⎢ ⎥
c =
⎢
0
⎥
⎢
⎣1,447⎥
⎦
d = 0
(42)
We choose this type of IFs to be generated by SSAF
because the structures to be synthesised using orthonormal
intermediate transfer functions generally have
good dynamic range, good signal swing and low sensitivity.
The structure of the current mode filter characterised
by parameters (42) is presented in Fig. 3 and the
system equation are:
i
i
i
i
C1
C2
C3
out
= iC12
= iC
22 + i
= iC
31 + i
= c i
3 out3
C23
C33
+ i
Cb3
(43)

Using the relations developed in Section 3 the system
(43) becomes:
out3
C1V
T diout1
iout2
I
= 12
iout1
+ IOUT1
dt
iout1
+ IOUT1
C3VT
iout3
+ I
iout1I
−
i + I
C2VT
diout2
=
iout2
+ IOUT
2 d t
iout2
I22
iout3
I
= −
− 23
iout2
+ IOUT
2 iout2
+ IOUT
2
OUT 3
31
OUT 3
di
dt
out3
+
i
=
iout3
I
+ I
out3
i = c i
out
33
OUT 3
3 out3
+
i
out3
i I
+ I
in b3
OUT 3
and the state-space description of this filter is:
di
d t
di
dt
out3
out2
= −
It is obvious that:
diout1
iout
I
= 2 12
d t C1VT
== −
iout1I
C V
3
iout2
I
C V
31
T
2
22
T
iout3
I
+
C V
iout3
I
−
C V
3
i out = c3iout3
33
T
2
23
T
i I
+
C V
in b3
I12
a12
= ;
C1VT
I 22
I23
a22
= , a23
= ;
C2VT
C2VT
I31
I33
a31
= , a33
= ;
C3VT
C3VT
Ib3
b3
= ;
C3VT
( W / L)
outMOS
c3
=
.
( W / L)
TLb3
3
T
(44)
(45)
(46)
In the future work the current-mode filters synthesis
method presented in this paper will be refined being
focused on optimization of specific performances
like bandwidth, noise, errors du mismatching and
dynamic range.
6. CONCLUSIONS
Future analogue circuits will have to operate successfully
at supply voltages slightly higher than the MOS
transistor threshold voltage. So, the suitable topologies
for signal processing at such low values of
supply voltages are the translinear circuits because
are operating in current domain and in this way the
very small voltage swings are avoided.
This paper presented a technique for very low supply
voltage, continuous time, current-mode filters synthesis
based on the intermediate transfer functions
method. This method has the distinct advantage of
filter performance optimization at the abstract level
of IFs, not at the topological level. In paper is also
presented the basic cell of the filter – the current
mode integrator, suitable for static and dynamic
analogue signal processing and very low supply
voltage operation. The minimum value of supply
voltage required for this circuit is given by the sum
of the MOS transistor threshold voltage and the
drain-source saturation voltage.
All transistors of these networks operate in weak
inversion due to the requirements of translinear
principle to have an exponential I-V characteristic
and the very low power supply voltage. Therefore,
bandwidth will be limited and the circuits will be
sensitive to the threshold voltage matching. Finally,
as a test vehicle for the proposed synthesis method,
the structure of a third order continuous-time filter
for low power applications was is presented and
analyzed.
ACKNOWLEDGMENT
This work was supported by the CNCSIS Research
Project no. 29C/08.05.2007, theme no. 5, CNCSIS
code no.602.
REFERENCES
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A Powerful Tool for High-Order Analogue
Continuous Time Filter Synthesis, WSEAS
European Computing Conference, Athens,
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transfer function algorithm, unpublished
works.
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