INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS

Int. J. Circ. Theor. Appl. 2012; 40:405–438

Published online 5 November 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cta.733

Analytical synthesis and comparison of voltage-mode N th-order

OTA-C universal filter structures

Chun-Ming Chang1, ∗, † and M. N. S. Swamy2

1 Department of Electrical Engineering, Chung Yuan Christian University, 32023, Taiwan
2 Department of Electrical and Computer Engineering, Concordia University, Montreal, Que., Canada H3G 1M8

SUMMARY
Complementary single-ended-input operational transconductance amplifier (OTA)-based filter structures
are introduced in this paper. Through two analytical synthesis methods and two transformations, one
of which is to convert a differential-input OTA to two complementary single-ended-input OTAs, and
the other to convert a single-ended-input OTA and grounded capacitor-based one to a fully differential
OTA-based one, four distinct kinds of voltage-mode nth-order OTA-C universal filter structures are
proposed. TSMC H-Spice simulations with 0.35 m process validate that the new complementary single-
ended-input OTA-based one holds the superiority in output precision, dynamic and linear ranges than
other kinds of filter structures. Moreover, the new voltage-mode band-pass, band-reject and all-pass
(except the fully differential one) biquad structures, all enjoy very low sensitivities. Both direct sixth-order
universal filter structures and their equivalent three biquad stage ones are also simulated and validated
that the former is not absolutely larger in sensitivity than the latter. Finally, a very sharp increment of the
transconductance of an OTA is discovered as the operating frequency is very high and leads to a modified
frequency-dependent transconductance. Copyright 䉷 2010 John Wiley & Sons, Ltd.

Received 18 November 2008; Revised 9 July 2010; Accepted 14 August 2010

KEY WORDS: active filters; analog circuit design; continuous-time filters; analytical synthesis methods;
operational transconductance amplifiers

1. INTRODUCTION

Voltage- or current-mode nth-order operational transconductance amplifier (OTA)-C filter structures

have been investigated and developed for several years [1–19]. Recently, the analytical synthesis
methods (ASMs) [15–20] have been validated and demonstrated to be very effective for the design
of OTA-C filters [15–17, 19] and current conveyor-based filters [18]. A complicated nth-order
transfer function is manipulated and decomposed by a succession of innovative algebraic operations
until a set of simple and feasible equations are produced. The complete filter structure is constructed
by superposing the sub-circuitries realized from these simple equations. In fact, the new ASMs can
be used in the design of any kind of a linear system with a stable transfer function. In addition, all
the filter structures presented in [15–17, 19, 20] enjoy the following three important criteria [7]:
• filters use grounded capacitors, and thus can absorb equivalent shunt capacitive parasitics;
• filters employ only single-ended-input OTAs, thus overcoming the feed-through effects due
to finite input parasitic capacitances associated with differential-input OTAs;

∗ Correspondence to: Chun-Ming Chang, Department of Electrical Engineering, Chung Yuan Christian University,
32023, Taiwan.
† E-mail: chunming@dec.ee.cycu.edu.tw

Copyright 䉷 2010 John Wiley & Sons, Ltd.

406 C.-M. CHANG AND M. N. S. SWAMY

• filters have the least number of active and passive elements for a given order; thus, reducing
power consumption, chip areas, and noise.
In [16], it has been shown that the voltage-mode filter structure with arbitrary functions needs
2n +2, i.e. n more OTAs than the other voltage-mode filter structure with only low-pass (LP),
band-pass (BP) and high-pass (HP) functions. This led to the research work in [17]: a new ASM
for realizing the voltage-mode high-order OTA-C all-pass (AP) and band-reject (BR) filters using
only n +2 single-ended-input OTAs and n-grounded capacitors.
On the other hand, combining both the current-mode notch and inverting LP signals, a current-
mode HP signal can be obtained. Similarly, a current-mode AP signal can be obtained by connecting
current-mode notch and inverting BP signals. This well-known concept has been demonstrated in
the recently reported current-mode OTA-C universal filter structure [15]. However, the voltage-
mode circuit lacks this ability, unlike the current-mode circuit, of the arithmetic operations of direct
addition or subtraction of signals. Hence, although several voltage-mode OTA-C biquad filters
[21–26] have been presented recently, yet only two [25, 26] of them, using three [25] (resp., four
[26]) differential-input OTAs and two [25] (resp., three [26]) single-ended-input OTAs in addition
to two-grounded capacitors, can synthesize all the five different generic filtering signals, i.e. LP,
BP, HP, BR (or notch) and AP signals, simultaneously. Therefore, the problem as to how to bring
about the arithmetic superiority of the current-mode circuit to the voltage-mode counterpart and
still achieve the above three important criteria [7] for the design of OTA-C filters is an important
one. Such a problem has been solved for the biquad structure [20] with the additional valuable
advantage of ‘programmability’ using the recently reported ASM.
Although both the voltage-mode nth-order OTA-C LP, BP and HP filter structure of [16] and
the voltage-mode nth-order OTA-C AP and BR filter structure of [17] use the least number of
active and passive components, namely, n +2 single-ended-input OTAs and n-grounded capaci-
tors, yet none of the voltage-mode nth-order OTA-C universal filter structures employs such a
least number of active and passive components. Although the voltage-mode second-order OTA-C
universal filter structure of [20] is ‘programmable’ and uses 2+2(= 4) single-ended-input OTAs
and 2 grounded capacitors, none of the voltage-mode nth-order OTA-C universal filter structures
are ‘programmable’. Therefore, there does not exist any voltage-mode nth-order OTA-C universal
filter structure in the published literature that has both the least number of components and the
advantage of ‘programmability’. With these two properties in mind, a new voltage-mode nth-order
programmable, universal filter structure using n +2 single-ended-input OTAs and n-grounded
capacitors is developed. This is an extension of the recently reported voltage-mode second-order
OTA-C programmable, universal filter structure given in [20]. Its fully differential-input OTA-based
one can be easily obtained from the single-ended-input OTA and grounded capacitor structure
using the well-known transformation method.
A differential (or double) input OTA can be realized by two parallel single-ended-input
OTAs. It may be possible to synthesize an nth-order filter structure using n differential-input
OTAs instead of n +2 single-ended-input OTAs in addition to n capacitors. If it is possible
to do so, the following question is quite interesting: Which one is the better? Is the one with
n +2 single-ended-input OTAs or the one with n differential-input OTAs? The former uses
more OTAs with more non-ideal transconductance functions, but has lower parasitics for each
single-ended-input OTA, and the latter uses fewer OTAs with less non-ideal transconductance
functions, but has larger parasitics for each differential-input OTA. Therefore, it is really worth-
while to do such a comparison between the above-mentioned two cases. We then present the
second new ASM, different from the recently reported ones in [15–17, 19, 20], to realize a
voltage-mode nth order OTA-C universal filter structure using only n differential-input OTAs and
only n floating/grounded capacitors, the minimum number of active and passive components.
Moreover, since a differential-input OTA can be equivalent to two parallel and complementary
single-ended-input OTAs, the differential-input one can be transformed to a new complementary
single-ended-input OTA-based universal filter structure, which is validated to have the most precise
output signals among the four distinct kinds of synthesized universal filter structures: (i) single-
ended-input OTA-based one, (ii) fully differential OTA-based one, (iii) differential-input OTA-based

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
 ANALYTICAL SYNTHESIS AND COMPARISON 407

one and (iv) complementary single-ended-input OTA-based one, and two recently reported
biquad filters.
In addition to the output precision, the power consumption, noise, dynamic and linear ranges
of the proposed four new OTA-C filter structures and the recently reported two cases [25, 26] are
investigated using H-Spice simulations. The new complementary single-ended-input OTA-based
one is validated to enjoy the largest dynamic and linear ranges.
As to sensitivities, second-order and sixth-order filter structures are investigated using H-Spice
simulations. The realized band-pass, band-reject, and all-pass (except the fully differential one)
biquads enjoy very low sensitivities achieved by the well-known passive LC ladder network.
Both a direct sixth-order universal filter structure and its equivalent three-biquad-stage one are
also simulated. Although some three-biquad-stage filters have lower sensitivity than their direct
sixth-order one, yet some direct sixth-order filters have lower sensitivity than their equivalent
three-biquad-stage ones.
An output distortion with a sudden drop in the synthesized high-pass, band-reject and all-pass
amplitude–frequency responses is investigated. A very sharp increment of the transconductance of
an OTA is discovered using H-Spice simulation when the operating frequency is over a critical value.
The frequency-dependent transconductance function is then modified by adding an exponential-
like function.

2. ANALYTICAL SYNTHESIS METHOD I

Although the ASM was applied to the design of recently reported filter structures [15–20], yet each
ASM is different from the others, and the realized filter structures are different from one another.
Therefore, the ASM is just one of the classifications of circuit design approaches. It is similar to the
signal flow diagram which is also one of the classifications of circuit design approaches. Different
signal flow diagram leads to distinct circuit structures. In this section, we would like to investigate
a new ASM using some differential-input OTAs and some floating capacitors, different from
the recently reported ones [15–20] with only single-ended-input OTAs and grounded capacitors.
Although this new kind of filter structure is without the aforementioned three important criteria
[7], a new complementary single-ended-input OTA-based filter structure produced from the new
differential-input OTA-based one will be validated to have much better output precision, even better
than the well-known single-ended-input OTA-based one. The voltage-mode nth-order universal
filter transfer function is given by:
n j
j =0 Vi( j ) (s a j )
Vout = n j
(1)
j =0 (s a j )

To realize the highest-order and lowest-order terms, s n an and a0 , we need a minimum numbers
of n capacitors and n transconductances (of OTAs), respectively; for consistence with the same
unit. Thus, the minimum number of passive and active components required for synthesizing
Equation (1) is n capacitors and n OTAs. In order to use the minimum number of active and passive
components, differential-input OTAs and floating capacitors are used in the following new ASM.
Cross-multiplying Equation (1), dividing by s n an and re-arranging the sequence of terms,
we obtain
     
ao a1 a2
Vout = n (Vi(o) − Vout )+ n−1 (Vi(1) − Vout )+ n−2 (Vi(2) − Vout )
s an s an s an
   
an−2 an−1
+ · · ·+ 2 (Vi(n−2) − Vout )+ (Vi(n−1) − Vout )+ Vi(n) (2)
s an san
Since
        
a0 an−1 an−2 a2 a1 a0
= ···
s n an san san−1 sa3 sa2 sa1

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
408 C.-M. CHANG AND M. N. S. SWAMY

and
       
aj an−1 an−2 a j +1 aj
= ··· for j = 0, 1, 2, 3 . . . , n −1 (3)
s n− j an san san−1 sa j +2 sa j +1
taking out the same common factor of the right side of (2),
⎧ ⎡ ⎡ ⎡ ⎡  ⎤ ⎤⎤⎤⎫
⎪
⎪   a0 ⎪
⎪
⎪
⎪ ⎢ ⎢  ⎢ a1 ⎢ sa (Vi(0) −Vout )⎥ ⎥⎥⎥⎪ ⎪
⎪
⎪ ⎢ ⎢ ⎢ ⎣ ⎦ ⎥⎥ ⎥⎪
⎪
⎪
⎪ ⎢ a ⎢ a2 ⎢ 1
⎥⎥⎥⎪⎪
⎪
⎪ ⎢ ⎢ . . . ⎢
sa 2 ⎥⎥ ⎥⎪
⎪
⎪
⎪ ⎢
n−3
⎢ sa ⎣ +Vi(1) −Vout ⎦⎥ ⎥⎪
⎪
 ⎪
⎨ n−2 ⎢ sa
a ⎢
3
⎥ ⎥⎪
⎬
an−1 ⎢ n−2
⎢ +V −V ⎥ ⎥
Vout =Vi(n) + san−1 ⎢ ⎣ i(2) out ⎦ ⎥ (4)
san ⎪⎪ ⎢ ⎥⎪
⎪
⎪ ⎢ +V −V ⎥⎪
⎪
⎪
⎪ ⎣ i(n−3) out ⎦⎪
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪ +V −V ⎪
⎪
⎪
⎪
i(n−2) out ⎪
⎪
⎪
⎩ ⎪
⎭
+Vi(n−1) −Vout

2.1. Part I: Equal capacitance approach

Observing Equation (4), we can let
 
(a0 /a1 )
V1 = (Vi(0) − Vout )+ Vi(1)
s
which is equivalent to
 
a0
(V1 − Vi(1) )s = (Vi(0) − Vout ) (5-1)
a1
and
 
(a j −1/a j )
Vj = (Vi( j −1) − Vout )+ Vi( j ) for j = 2, 3 . . . , n −1, n
s
which may be written as
 
a j −1
(V j − Vi( j ) )s = (Vi( j −1) − Vout ) for j = 2, 3 . . . , n −1, n (5-j)
aj
and Vout = Vn . Each of the above equations is simple and easy to realize using a differential-input
OTA, with a transconductance of a j −1/a j , and a floating capacitor with unit capacitance. The OTA-
C realizations of these simple first-order equations, (5-1), (5-2), (5-n-1), and (5-n), are presented
in the dashed line blocks from the left to the right, respectively, in Figure 1, which shows the
new voltage-mode nth-order OTA-C universal filter structure with the minimum number of active
and passive components, namely, n differential-input OTA and n floating capacitors. In particular,
all of the capacitances are of equal value. Equal-valued or simple-ratio-value capacitance design
overcomes the difficulty of precise variation of capacitances in IC fabrication. Note that when the
input impedance is not very large or the output impedance is not very small, as in the case of
the filter structure shown in Figure 1, a voltage buffer is needed at the input or the output stage,
respectively. Since the proposed voltage-mode nth-order OTA-C universal filter structure may need
a buffer to connect with the former or latter stage, the aforementioned minimum number of active
and passive components is without taking into consideration the requirement of buffers. If the
addition of the buffer is included in the filter core, we cannot positively say that the proposed filter
structure has the minimum number of active and passive components.
Note that since a differential-input OTA can be realized by two parallel and complementary
single-ended-input OTAs (please refer to Figure 2(a)), the differential-input OTA-based universal
filter structure shown in Figure 1 can be transferred to a complementary single-ended-input
OTA-based one shown in Figure 2(b).

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
 ANALYTICAL SYNTHESIS AND COMPARISON 409

(5-1) (5-2) (5-n-1) (5-n)

Vi(0)

+ a0 - + an −1-

a1 - an − 2 an
Vn-1
- a1 V2 Vn-2 a
V1 + n −1 Vout
+ a2
1 1
1 1
1
Vi(n-1) Vi(n)
Vi(2) Vi(n-2)
Vi(1)

Figure 1. Voltage-mode n-th order OTA-C universal filter structure with

differential-input OTAs and equal capacitance.

(a)

a0 a0 an −1 an −1
an −2
a1 a1 a1 an an
an−1
a2
an −2
a1 an−1
a2

(b)

Figure 2. (a) Transformation from a differential-input OTA to two parallel and

complementary single-ended-input OTAs and (b) complementary single-ended-input
OTA based universal filter structure derived from Figure 1.

2.2. Part II: Equal transconductance approach

Observing Equation (4), we can let
 
1
V1 = (Vi(0) − Vout )+ Vi(1)
s(a1 /a0 )

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
410 C.-M. CHANG AND M. N. S. SWAMY

which is equivalent to
 
a1
(V1 − Vi(1) ) s = (1)(Vi(0) − Vout )
a0
  (6-1)
1
Vj = (Vi( j −1) − Vout )+ Vi( j ) for j = 2, 3 . . . , n −1, n
s(a j /a j −1 )
which may be re-written as:
 
aj
(V j − Vi( j ) ) s = (1)(Vi( j −1) − Vout ) for j = 2, 3 . . . , n −1, n (6-j)
a j −1
Each of the above equations is simple and can easily be realized using a differential-input
OTA with unity transconductance and a floating capacitor of capacitance a j /a j −1. The OTA-C
realizations of these simple first-order equations, (6-1), (6-2), (6-n-1) and (6-n), are presented
in the dashed line blocks from the left to the right, respectively, in Figure 3, which shows the
new voltage-mode nth-order OTA-C universal filter structure with the minimum number of active
and passive components, namely, n differential-input OTA and n floating capacitors. In particular,
all the transconductances are of equal value. Equal transconductance design leads to a simpler
architecture in view of the fact that only a single biasing circuitry is needed for the entire filter
structure.
It is apparent that both Figures 1 (equal capacitance) and 3 (equal transconductance) have the
same circuit structure. Figures 1 and 3 show that the circuit structures with equal capacitance and
equal transconductance are realizable using analytical synthesis approaches, I and II, respectively.
As a matter of fact, the values of capacitances and tranconductances can be different. For example,
given a fourth-order universal biquad with the circuit structure same as Figures 1 and 3, and letting
the transconductances of the OTAs (resp., the capacitances of the capacitors) from left to right be
g1 , g2 , g3 , and g4 , (resp., C1 , C2 , C3 , and C4 ), circuit analysis yields
 4 
s C1 C2 C3 C4 Vi(4) +s 3 C1 C2 C3 G 4 Vi(3) +s 2 C1 C2 G 3 G 4 Vi(2)
+sC1 G 2 G 3 G 4 Vi(1) + G 1 G 2 G 3 G 4 Vi(0)
Vout = (7)
s 4 C1 C2 C3 C4 +s 3 C1 C2 C3 G 4 +s 2 C1 C2 G 3 G 4 +sC1 G 2 G 3 G 4 + G 1 G 2 G 3 G 4
Therefore, different values of capacitances and transconductances can be used in the circuit struc-
tures shown in Figures 1 and 3 for achieving the above transfer function.

Figure 3. Voltage-mode n-th order OTA-C universal filter structure with

differential-input OTAs and equal transconductance.

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
 ANALYTICAL SYNTHESIS AND COMPARISON 411

a2 an - 2 an - 1 an
a1 a1 an - 3 an - 2 an - 1
a0

Figure 4. Complementary single-ended-input OTA based universal filter structure derived from Figure 3.

A corresponding complementary single-ended-input OTA-based universal filter structure shown

in Figure 4 is obtained from the differential-input OTA-based one shown in Figure 3 using the
transformation shown in Figure 2(a). Note that a single-ended-input OTA is with much lower
parasitics than, and without feed-through effect unlike, a differential-input OTA. In view of this, a
very interesting question arises: ‘Which one has more accurate output signals? Is the differential-
input OTA-based one (Figures 1 and 3) with fewer OTAs or the complementary single-ended-
input OTA-based one (Figures 2(b) and 4) with more OTAs?’

2.3. Part III: Unequal capacitance and unequal transconductance approach

Observing Equation (4), we can let
 
a0
V1 = (Vi(0) − Vout )+ Vi(1)
sa1

which is equivalent to

(V1 − Vi(1) )(sa1 ) = (a0 )(Vi(0) − Vout ) (8-1)

and
 
a j −1
Vj = (Vi( j −1) − Vout )+ Vi( j ) for j = 2, 3 . . . , n −1, n
sa j

which may be written as

(V j − Vi( j ))(sa j ) = (a j −1)(Vi( j −1) − Vout ) for j = 2, 3 . . . , n −1, n (8-j)

and Vout = Vn . Each of the above equations is simple and easy to realize using a differential-input
OTA, with a transconductance of a j −1, and a floating capacitor with capacitance of a j . The OTA-C
realizations of these simple first-order equations, (8-1), (8-2), (8-n-1), and (8-n), are presented in
the dashed line blocks from the left to the right, respectively, in Figure 5. The unequal capacitance
and unequal transconductance approach is used to validate the feasibility of the filter when given
distinct values of capacitances and transconductances in the design.
In a similar way, the differential-input OTA-based universal filter structure shown in Figure 5
can be transferred to a complementary single-ended-input OTA-based one shown in Figure 6.

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
412 C.-M. CHANG AND M. N. S. SWAMY

Figure 5. Voltage-mode nth order OTA-C universal filter structure with differential-input OTAs and
unequal capacitances and transconductances.

Figure 6. Complementary single-ended-input OTA based universal filter structure derived from Figure 5.

3. ANALYTICAL SYNTHESIS METHOD II

This section presents an ASM for a multifunction filter, which can simultaneously realize nth-
order LP BP, HP, BR (or notch) and AP filtering signals programmably, using single-ended-input
OTAs and grounded capacitors. The method is based upon a succession of innovative algebraic
decomposition of an nth-order versatile filtering transfer function, which represents the LP BP, HP,
BR (or notch) and AP transfer functions, into n first-order simple and feasible equations and one
collateral constraint. The following shows the method which is an expansion of the second-order
one shown in [20].
The realized nth-order transfer function is
     
 a  V n (−1)i a s i 1 n (−1)i(aV )a s i 1 n [(−1)i a V ]a s i
in i in i (i) in i
Vout = ni=0
i
= i=0
n i
≡ i=0
n i
(9)
b i=0 ai s b i=0 ai s b i=0 ai s

The value of [(−1)i a(i) Vin ] may be zero or non-zero as controlled by a switch. Depending on
the type of filter to be realized, the position of the switch (whether it is to be open or closed) is
determined. The ASM for the realization of the filtering equation (9) with a gain (a/b) is given
below.

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
 ANALYTICAL SYNTHESIS AND COMPARISON 413

Cross multiplying Equation (9), dividing by an s n , and rearranging, we have

 

n−1 ai
bVout −(−1) a(n) Vin = [(−1) a(i) Vin −bVout ]
n i
(10)
i=0 an s n−i
We now provide an insight into the analytical decomposition of the first two and the last two terms
of the right side of Equation (10) as follows:
⎡ ⎤
  ((−1)n−1
a (n−1) Vin −bV out )+  

n−1 ai ⎢  ⎥ an−1
[(−1) a(i) Vin −bVout ]
i
=⎣ an−2 ⎦ a s (11)
i=n−2 an s n−i ((−1)n−2 a(n−2) Vin −bVout ) n
an−1 s
⎡ ⎤
  ((−1)a(1) Vin −bVout )+  
1 ai ⎢  ⎥ a1
[(−1) a(i) Vin −bVout ]
i
=⎣ a0 ⎦ (12)
i=0 an s n−i ((−1)0 a(0) Vin −bVout ) an s n−1
a1 s
Note
        
a0 a0 a1 an−3 an−2 an−1
= · · · (13)
an s n a1 s a2 s an−2 s an−1 s an s
Substituting (13) into (10) we obtain
bVout −(−1)n a(n) Vin
⎡ ⎤
(−1)n−1 a(n−1) Vin −bVout +
⎢ ⎡ ⎤⎥
⎢ (−1)n−2 a(n−2) Vin −bVout + ⎥
⎢ ⎥
 ⎢ ⎢ ⎡ ⎤ ⎥⎥
⎢
an−1 ⎢  ⎢ (−1) a(2) Vin −bVout +
2 ⎥⎥
= ⎢ ⎥⎥ (14)
an s ⎢
⎢
a n−2 ⎢ 
⎢
⎢
⎢
⎡ ⎤⎥⎥
⎥
⎥
⎥⎥
⎢ an−1 s ⎢ . . . a2 ⎢   (−1)a(1) Vin −bVout + ⎥ ⎥⎥
⎢ ⎢ a3 s ⎢ a1 ⎢   ⎥⎥⎥⎥
⎣ ⎣ ⎣ ⎣ a0 0 ⎦⎦⎦⎦
a2 s [(−1) a(0) Vin −bVout ]
a1 s

3.1. Part I: Equal capacitance approach

Observing Equation (14), we let
     
a0 a0 /a1 a0
[(−1)0 a(0) Vin −bVout ] = [(−1)0 a(0) Vin −bVout ] ≡ Vout(1)
a1 s s a1
       (15-0)
ai−1 ai /ai+1 ai
(−1) a(i) Vin −bVout +
i
Vout(i) ≡ Vout(i+1)
ai s ai+1
for i = 1, 2 . . . , n −2, n −1 (15-i)
 
an−1
bVout −(−1) a(n) Vin =
n
Vout(n) (15-n)
an
Equation (15-0) can be realized using the OTA-C structure with the switch S(0) closed, as
shown in Figure 1 of [20], in which we use three single-ended-input OTAs with the three different
transconductances, a(= a(0)), b and a0 /a1 , and one grounded capacitor with unity capacitance.
Since a(0) Vin is the output current of the OTA with the transconductance a, a(0) Vin = 0 or a(0) Vin = 0
can be controlled by the switch S(0) .
The OTA-C implementation of both Equations (15-0) and (15-1), when both two switches S(0)
and S(1) are closed and both a(0) and a(1) are equal to a, is shown in Figure 2 of [20]. We note
that −a(1) Vin = 0 or −a(1) Vin = 0 can be controlled by the second switch S(1).

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
414 C.-M. CHANG AND M. N. S. SWAMY

Figure 7. Single-ended-input OTA-grounded C universal filter structure with equal capacitance.

Implementing Equations (13-0) to (13-n) using circuits similar to that of Figures 1 and 2 in
[20] yields the nth order OTA-C universal filter structure and is shown in Figure 7. We note that
a(i) Vin = 0 or a(i) Vin = 0 can be controlled by the ith switch S(i) , as shown in Figure 7. All of
the n +2 active elements are single-ended-input OTAs and all of the n capacitors are grounded,
and are of equal value. Equal capacitance design overcomes the difficulty of accurately tuning the
capacitances in IC fabrication.
Then, the five different generic filtering functions can be obtained from the universal filter
structure shown in Figure 7 by the following specifications.
(i) Low-pass: Close S(0) , but open all other S(i) .
(ii) Band-pass: Close S(n/2) when n is even or close S(n±1)/2 when n is odd, but open all other
S(i).
(iii) High-pass: Close S(n) but open all other S(i).
(iv) Notch: Close both S(0) and S(n) but open all other S(i).
(v) All-pass: Close all of the switches.
The OTA-C universal filter structure shown in Figure 7 is then digitally programmable. No compo-
nent matching is used in the analytical synthesis.
The filter structure shown in Figure 7 with single-ended-input OTAs and grounded capacitors
can be easily transformed (referring to [27, p. 618]) to a fully differential structure shown in
Figure 8.

3.2. Part II: Equal transconductance approach

Observing Equation (12), we let
   
a0 1
[(−1) a(0) Vin −bVout ]
0
= [(−1) a(0) Vin −bVout ]
0
≡ (1)Vout(1)
a1 s (a1 /a0 )s
  (16-0)
1
[(−1) a(i) Vin −bVout +(1)Vout(i) ]
i
≡ (1)Vout(i+1)
(ai+1 /ai )s

for i = 1, 2 . . . , n −2, n −1 (16-i)

bVout −(−1)n a(n) Vin = (1)Vout(n) (16-n)

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 ANALYTICAL SYNTHESIS AND COMPARISON 415

a0 a1 an - 2 an -1
a1 a2 an - 1 an

Figure 8. Fully differential OTA-C universal filter structure with equal capacitance.

Figure 9. Single-ended-input OTA-grounded C universal filter structure with equal transconductance.

Equation (16-0) can be realized using the OTA-C structure with the switch S(0) closed as shown
in Figure 1 of [20], but in which we use three single-ended-input OTAs with the three different
transconductances, a(= a(0) ), b, and unity, and one grounded capacitor with the capacitance a1 /a0 .
Since a(0) Vin is the output current of the OTA with a transconductance a, a(0) Vin = 0 or a(0) Vin = 0
can be controlled by the switch S(0) .
The OTA-C implementation of both Equations (16-0) and (16-1), when both of the switches
S(0) and S(1) are closed and both a(0) and a(1) are equal to a, is similar to Figure 2 of [20]. We note
that −a(1) Vin = 0 or −a(1) Vin = 0 can be controlled by the second switch S(1).
Implementing Equations (16-0) to (16-n) using circuits similar to those of Figures 1 and 2
of [20] yields a different nth-order OTA-C universal filter structure, as shown in Figure 9 with
a = b = 1. We note that all of the n +2 active elements are single-ended-input OTAs, all of the
n capacitors are grounded, and all the transconductance are of equal value. Equal transconductance
design leads to a simpler architecture, since only a single sub-circuitry is used in the entire filter
structure for varying the bias current.

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416 C.-M. CHANG AND M. N. S. SWAMY

Figure 10. Fully differential OTA-C universal filter structure with equal transconductance.

The filter structure shown in Figure 9 with single-ended-input OTAs and grounded capacitors
can be easily transformed (referring to page 618 of [27]) to a fully differential structure shown in
Figure 10.

3.3. Part III: Unequal capacitance and unequal transconductance approach

Observing Equation (14), we let
 
a0
[(−1) a(0) Vin −bVout ]
0
≡ (a0 )Vout(1)
a1 s
  (17-0)
ai
[(−1)i a(i) Vin −bVout +(ai−1 )Vout(i) ] ≡ (ai )Vout(i+1)
ai+1 s
for i = 1, 2 . . . , n −2, n −1 (17-i)
bVout −(−1)n a(n) Vin = (an−1 )Vout(n) (17-n)
Equation (17-0) can be realized using the OTA-C structure with the switch S(0) closed, as
shown in Figure 1 of [20], in which we use three single-ended-input OTAs with the three different
transconductances, a(= a(0) ), b, and a0 , and one grounded capacitor with capacitance of a1 . Since
a(0) Vin is the output current of the OTA with a transconductance a, a(0) Vin = 0 or a(0) Vin = 0 can
be controlled by the switch S(0).
The OTA-C implementation of both Equations (17-0) and (17-1), when both of the switches
S(0) and S(1) are closed and both a(0) and a(1) are equal to a, is similar to Figure 2 of [20]. We
note that −a(1) Vin = 0 or −a(1) Vin = 0 can be controlled by the second switch S(1) .
Implementing Equations (17-0)–(17-n) using circuits similar to those of Figures 1 and 2 of [20]
yields a different nth-order OTA-C universal filter structure, as shown in Figure 11 with a = b = 1.
We note that all of the n +2 active elements are single-ended-input OTAs, all of the n capacitors
are grounded.
The filter structure shown in Figure 11 with single-ended-input OTAs and grounded capacitors
can be easily transformed (referring to page 618 of [27]) to a fully differential structure shown in
Figure 12.
The unequal capacitance and unequal transconductance approach (Part III) is used to validate
the feasibility of the filter when given distinct values of capacitances and transconductances in the
design.

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 ANALYTICAL SYNTHESIS AND COMPARISON 417

Figure 11. Single-ended-input OTA-grounded C universal filter structure with

unequal capacitances and transconductances.

S(n)
S(1) S(n-1)
S(0)
+Vin
+ -
+/- +
-/+ +
+ + + + + +
+- +
+ +
an-2 an-1
+ +Vout
a - b -- a1 a0 a2 a1 an
+ - - - - - - - -Vout
-
+/- --
-/+
- +
-Vin S(0)
S(1) S(n-1)
S(n)

Figure 12. Fully differential OTA-C universal filter structure with unequal
capacitances and transconductances.

Note that two multiple +/− output OTAs are used in the filter structures shown in Figures 7,
9 and 11. The current mirror and the inverting-type current mirror are applied to produce an
in-phase and an out-of-phase output current, respectively, from a multiple output OTA. Since
current mirrors constructed by transistors are not ideal, it is quite interesting for us to know as to
how much is the difference in magnitude between two ++, two −−, or two +− output signals
is. A five output, having three +’s and two −−‘s, OTA with the CMOS implementation [27] is
used to do this investigation. H-Spice simulations exhibit that the five +++−− output currents
have the magnitudes: 100.0005628 A, 100.0005628 A, 100.0005628 A, 100.0005458 A and
100.0005458 A, respectively, when we bias the OTA with a bias current 9.742 A at 1 MHz.
It can be seen that the error of using a current mirror is almost zero, and the error of applying an
inverting-type current mirror is also very small just about 17 %. In summary, such an error due
to duplicating an output current is negligible.
We note that the proposed filter structures (Figures 7–12) have a high input impedance (which is
cascadable with the former voltage-mode stage), although a voltage buffer is needed at the output
terminal due to high output impedance.
The analog switch shown in Figures 7–12 can be realized using the transmission gate [20].
In a conducting transmission gate, since the source-drain voltage swing of the MOS transistors is

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418 C.-M. CHANG AND M. N. S. SWAMY

small, they work in the linear region and can be modeled as an effective resistance Reff , which
is a function of the input transition time and load capacitance, bounded by two parallel grounded
capacitances C T , which include interconnect parasitic capacitances, capacitances from source and
drain-to-substrate, and half the channel capacitance of the transistors. The circuit analysis for
the non-ideal second-order filter structure derived from Figure 7 with the replacement of the
transmission gate by its equivalent model produces a complicated sixth-order denominator D(s)
and fifth-order numerator N (s) [20]. We notice that the addition of a non-ideal transmission gate
increases the parasitic effect be. If the precision of the output signals is critical, the analog switch
is not necessary. It is also acceptable to replace an open switch with an open circuit and a closed
switch with a short circuit for realizing a filter with a single function. However, if the switches
were not used in the design, the filter structure could not be called a digitally programmable one.
Now, we synthesize four different kinds of OTA-C universal filter structures which are (i) three
double-input OTA and floating/grounded capacitor based ones (Figures 1, 3, and 5), (ii) three
complementary single-ended-input OTA and floating/grounded capacitor-based ones (Figures 2, 4,
and 6), (iii) three single-ended-input OTA and grounded capacitor-based ones (Figures 7, 9 and 11),
and (iv) three fully differential OTA and floating capacitor-based ones (Figures 8, 10 and 12).
A comparison among the above four distinct universal filter structures and two recently reported
OTA-C universal biquad filters [25, 26] is made in the next section.

4. PERFORMANCE COMPARISONS

The second-order OTA-C universal filters derived from Figures 1, 2, 7 and 8 use two differential-
input OTAs and two floating/grounded capacitors, four complementary single-ended-input OTAs
and two floating/grounded capacitors, four single-ended- input OTAs and two grounded capacitors,
and four fully differential OTAs and two floating capacitors, respectively. The two recently reported
universal biquads [25, 26] employ three [25] (resp., four [26]) differential-input OTAs and two
[25] (resp., three [26]) single-ended-input OTAs in addition to two grounded capacitors. Thus, we
would like to know which one in the above six biquads is the most precise one.
The TSMC035 level-49 H-Spice simulation is used for finding this solution using the CMOS
implementation of the OTA [27] with supply voltages VDD = 1.65 V, VSS = −1.65 V and W/L =
5/0.35  and 10 /0.35  for NMOS and PMOS transistors, respectively. Element values are given
by ga = gb = 100 S, g1 = 222.144 S, g2 = 444.288 S, and C1 = 50 pF, C2 = 50 pF for the biquads
derived from Figures 1, 2, 7 and 8, with the resonant frequency, 1 MHz. All the transconductances
are given by 314.159 S, and C1 = 50 pF, C2 = 50 pF for the two recently reported universal biquads
[25, 26]. The amplitude or phase-frequency responses of the above six universal biquads are shown
in Figures 13–19. The simulated 3 dB or center frequencies and percentage errors are shown in
Table I. As can be seen, except the universal biquad derived from Figure 2, the one derived from
Figure 7 with all single-ended-input OTAs and all grounded capacitors is much more precise than
the other four ones, all of which are with some differential-input OTAs and some single-ended-
input OTAs. The reason is that filters that employ only single-ended-input OTAs overcome the
feed-through effects due to finite input parasitic capacitances associated with differential-input
OTAs. However, we need to notice that although the differential-input OTA-based universal biquad
derived from Figure 1 has a large amount of errors, when the differential-input OTA is replaced
by two parallel and complementary single-ended-input OTAs, one of which has for its input signal
at the + terminal and the other one of which has its input signal entering from the – terminal,
leading to a perfect cancellation of their corruptions to output signals. Therefore, as can be seen,
the output precision of the universal biquad derived from Figure 2 is much better than that of
the ones with some differential-input OTAs and even a little bit better than the one derived from
Figure 7 with all single-ended-input OTAs and all grounded capacitors.
Another question that may be asked is: ‘If the differential-input OTA shown in [25, 26] is
replaced by a couple of complementary single-ended-input OTAs, is the output precision improved
further as in the case of the one shown in Figure 2(b) transformed from Figure 2(a)? The answer

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 ANALYTICAL SYNTHESIS AND COMPARISON 419

Figure 13. LP, BP, HP, and NH amplitude-frequency responses of the

universal biquad derived from Figure 1.

Figure 14. LP, BP, HP, and NH amplitude-frequency responses of the

universal biquad derived from Figure 2.

is in the positive. The output responses of the filters obtained by replacing the differential OTAs
by single-ended-input OTAs in [25] and [26] are shown in Figure 20 and Figure 21, respectively,
these have the output errors improved from 37.33 to 32.64% for LP, from 42.46 to 0.69% for
BP, from 45.83 to 25.05% for HP, from 42.46 to 0% for BR, and from 41.78% to 0.346% for
AP, and improved from 33.63 to 26.25% for LP, from 25.87 to 0.46% for BP, from 46.67 to
20.78% for HP, from 0.23 to 0.23% for BR, and from 59.74 to 1.17% for AP, respectively. These
results indicate that: ‘the replacement of a differential-input OTA with a couple of complementary
single-ended-input OTAs is very effective for the output precision of an OTA-C filter structure.’
For further reconfirmation, four sixth-order universal filter structures derived from Figures 1, 2, 7
and 8 are used to compare their performances. Component values are given by g1 = 16.262 S, g2 =
32.524 S, g3 = 51.302 S, g4 = 76.953 S, g5 = 121.382 S, g6 = 242.764 S and C1 = C2 = 10 pF
for the ones derived from Figures 1 and 2 and g1 = g2 = 100 S, g3 = 8.418 S, g4 = 121.382 S,
g5 = 26.556 S, g6 = 76.953 S, g7 = 121.382 S, g8 = 242.764 S and C1 = C2 = 10 pF for the ones
derived from Figures 7 and 8. In addition to these, a sixth-order filter structure can also be realized
by using three second-order biquad filter in cascade. The above four sixth-order filter structures

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420 C.-M. CHANG AND M. N. S. SWAMY

Figure 15. LP, BP, HP, and NH amplitude-frequency responses of the universal biquad derived from Figure 7.

Figure 16. LP, BP, HP, and NH amplitude-frequency responses of the universal biquad derived from Figure 8.

Figure 17. LP, BP, HP, and NH amplitude-frequency responses of the universal biquad shown in [25].

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DOI: 10.1002/cta
 ANALYTICAL SYNTHESIS AND COMPARISON 421

Figure 18. LP, BP, HP, and NH amplitude-frequency responses of the universal biquad shown in [26].

Figure 19. AP phase-frequency responses of the universal biquads shown in Figures 2 (case 1), 1 (case 2),
7 (case 3), and 8 (case 4), and in [25] (case 5) and [26] (case 6), respectively.

Table I. The resonant frequencies and errors of five generic filtering signals for six different biquads.
Filter
f 3dB (LP) f 3dB (HP) f c (BP) fc (BR) f c (AP)
Biquad error (%) error (%) error (%) error (%) error (%)
Figure 1 575.3 k 344.6 k 714.5 k 474.2 k 151.5 k
42.47 65.54 28.55 52.58 84.85
Figure 2* 1003.0 k 998.4 k 1002.3 k 1000.0 k 996.6 k
0.30 0.16 0.23 0.00 0.34
Figure 7 1005.0 k 997.2 k 929.0 k 1000.0 k 970.5 k
0.50 0.28 7.10 0.00 2.95
Figure 8 821.6 k 825.3 k 891.3 k 1000.0 k 988.5 k
22.53 17.48 10.87 0.005 1.15
Horng [25] 626.7 k 541.6 k 575.4 k 575.4 k 582.2 k
37.33 45.84 42.46 42.46 41.78
Abuelma’atti and Bentrcia [26] 663.7 k 306.3 k 741.3 k 1002.3 k 136.2 k
33.63 69.37 25.87 0.23 86.38
The ∗ means with the best.

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422 C.-M. CHANG AND M. N. S. SWAMY

Figure 20. LP, BP, HP, and NH amplitude-frequency responses of the universal biquad with all
single- ended-input OTAs transformed from [25].

Figure 21. LP, BP, HP, and NH amplitude-frequency responses of the universal biquad with all
single-ended-input OTAs transformed from [26].

have their own corresponding three-biquad-stage structures. Then, we would like to know which
one in the above total eight different kinds of sixth-order filter structures has the best output
precision? Their amplitude/phase responses are shown in Figures 22–37. The errors are shown in
Table II. From these figures and Table II, we may make the following observations. (i) The output
precision of the one derived from Figure 2 with complementary single-ended-input OTAs is still
in the range of an 1.5% error, and is far more accurate than any of the other types, including
even the single-ended-input OTA-based ones shown in Figures 26–27, and 34, 35. (See Table II.)
Therefore, we may make important conclusion that if we replace a differential-input OTA by two
parallel and complementary single-ended-input OTAs from a structure which is composed of only
differential-input OTAs and capacitors, the transformed circuit structure enjoys the most precise
output signals, even better than the ones with all single-ended-input OTAs and all grounded
capacitors. (ii) We cannot conclude that the direct sixth-order universal filter structure is better in
output precision than its counterpart of the three-biquad-stage sixth-order universal filter structure
or vice versa. (iii) Only the three-biquad-stage universal filter structure derived from Figure 7 with
single-ended-input OTAs has its output precision better than its counterpart of the direct sixth-order

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 ANALYTICAL SYNTHESIS AND COMPARISON 423

Figure 22. LP, HP, BP, and NH amplitude responses of the sixth-order universal filter derived from Figure 1.

Figure 23. AP phase-frequency responses of the sixth-order universal filter derived from Figure 1.

Figure 24. LP, HP, and NH amplitude-frequency responses of the sixth-order

universal filter derived from Figure 2.

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424 C.-M. CHANG AND M. N. S. SWAMY

Figure 25. AP phase-frequency responses of the sixth-order universal filter derived from Figure 2.

Figure 26. LP, HP, BP, and NH amplitude responses of the sixth-order universal filter derived from Figure 7.

one. (iv) Although the output precision of the direct band-pass and band-reject sixth-order universal
filters derived from Figure 2 is lower than 1.15% error, yet the lowest error is obtained from
the ones with three biquad stages derived from Figures 2 and 7. (v) The non-ideal effect, i.e.
Io /(V+ − V− ) = G(s) = G o /(1+s/o ) for an OTA, is apparent for all the four three-biquad-stage
all-pass phase-frequency responses shown in Figures 31, 33, 35 and 37. For example, the numerator
of the transfer function of the two-biquad-stage all-pass filter structure derived from Figure 7 can
be analyzed and obtained as

N (s) = [(b2 b2 )s 4 −(2b1 b2 )s 3 +(2b2 b0 +b1 b1 )s 2 −(2b1 b0 )s +b0 b0 ]

+[(2a3 b2 −2a4 b1 )s 5 +2(a4 b0 −a3 b1 −a2 b2 )s 4 +2(a3 b0 +a2 b1 )s 3 −(2a2 b0 )s 2 ]
+[(a4 a4 )s 8 +(2a4 a3 )s 7 +(2a4 b2 +a3 a3 −2a4 a2 )s 6 +(−2a3 a2 )s 5 +(a2 a2 )s 4 ]
= A(s)+ B(s)+C(s)

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 ANALYTICAL SYNTHESIS AND COMPARISON 425

Figure 27. AP phase-frequency responses of the sixth-order universal filter derived from Figure 7.

Figure 28. LP, HP, BP, and NH amplitude responses of the sixth-order universal filter derived from Figure 8.

Figure 29. AP phase-frequency responses of the sixth-order universal filter derived from Figure 8.

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426 C.-M. CHANG AND M. N. S. SWAMY

Figure 30. LP, HP, BP, and NH amplitude responses of the 3-biquad-stage
universal filter derived from Figure 1.

Figure 31. AP phase-frequency responses of the 3-biquad-stage universal filter derived from Figure 1.

where

b2 = C1 C2 G a0
b1 = C2 G 10 G a0
b0 = G 10 G 20 G a0
a4 = (C1 C2 G a0 /10 20 )
a3 = [C1 C2 G a0 (10 +20 )/(10 20 )]
and
a2 = (C2 G 10 G a0 /20 ) (18)

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 ANALYTICAL SYNTHESIS AND COMPARISON 427

Figure 32. LP, HP, BP, and NH amplitude responses of the 3-biquad-stage
universal filter derived from Figure 2.

Figure 33. AP phase-frequency responses of the 3-biquad- stage universal filter derived from Figure 2.

As we operate the filter at a frequency lower than the 3 dB frequency of the transconductance
function G(s) of an OTA, i.e. 10 and 20 are very large, the term A(s) of N (s) is the most dominant
one, B(s) is the main non-ideal part, and the term C(s) is minor and negligible. Consider the
various terms in B(s): (i) (2a3 b2 −2a4 b1 ) is positive since a4  a3 ,a2 and (ii) (a4 b0 −a3 b1 −a2 b2 )
is negative for the same reason. It is evident that the sign sequence of the coefficients of the
polynomial B(s) are +, −, + and − confirms that the all-pass phase-frequency response due to
the other terms in addition to the main phase response resulting from dominant term A(s) is very
minor. Thus, the additional phase-frequency responses shown in Figures 31, 33, 35 and 37 agree
with the theory.
From (18), we note that the ASMs, shown in [15–20] and in this paper, do not consider the
effect of parasitic elements and mismatches in the design. It would be interesting and useful to

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428 C.-M. CHANG AND M. N. S. SWAMY

Figure 34. LP, HP, and NH amplitude-frequency responses of the

3-biquad-stage universal filter derived from Figure 7.

Figure 35. AP phase-frequency responses of the 3-biquad-stage universal filter derived from Figure 7.

investigate the effect of the parasitic elements and mismatches in the ASM through some examples.
This would be undertaken in a future work.
In addition to the output precision, nonlinearity, noise and dynamic range of an OTA-C filter are
of concern in the design [28–31]. H-Spice simulation results for the six biquads under consideration
are shown in Tables III–VI. It is seen from these tables that (i) the biquad derived from Figure 1
has the lowest power consumption and the second lowest noise level due to the use of the minimum
number of components, (ii) except for the one derived from Figure 1, the single-ended-input OTA-
based one derived from Figure 7 has the lowest power consumption in comparison with the ones
with differential-input OTAs [25, 26] and the one with complementary single-ended-input OTAs (it
means that a single-ended-input OTA has lower power consumption than a complementary single-
ended-input OTA), (iii) a fully differential OTA-based biquad has the lowest noise, is much lower
than the other kinds of biquad structures, (iv) the complementary single-ended-input OTA-based
one derived from Figure 2 enjoys the largest dynamic range and the largest average linear range
(with output signals having the THD lower than 1%), (v) both the single-ended-input OTA-based
one (Figure 7) and the fully differential OTA-based one (Figure 8) have very small linear range and

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 ANALYTICAL SYNTHESIS AND COMPARISON 429

Figure 36. LP, HP, BP, and NH amplitude responses of the 3-biquad-stage
universal filter derived from Figure 8.

Figure 37. AP phase-frequency responses of the 3-biquad- stage universal filter derived from Figure 8.

dynamic range. In summary, the new complementary single-ended-input OTA-based one derived
from Figure 2 (resp., the single-ended-input OTA-based one derived from Figure 7) has very good
(resp. bad) dynamic and linear ranges, acceptable noise level, and power consumption. Of course,
the fully differential OTA-based one derived from Figure 8 enjoys the lowest noise but suffers from
very bad dynamic and linear ranges. Another important circuit parameter is relevant to component
sensitivities, and the largest sensitivities are shown in Table VII. In the four new second-order
universal biquads derived from Figures 1, 2, 7 and 8, band-pass, band-reject, and all-pass (except
the one derived from Figure 8) sensitivities of their center frequencies to each capacitance and
conductance have been simulated and illustrated to have the absolute value not larger than 0.620
(the all-pass sensitivity of f o to g3 or g4 for the one derived from Figure 7), which is smaller
than 0.630, the largest one of the sensitivities of the second-order passive all-pass filter with a
lattice topology [18]. Thus, we may say that the three new BP, BR and AP biquads derived from
Figures 1, 2, 7 and 8 (except for the all-pass signal) enjoy very low sensitivities, which have been
achieved by passive LC ladder circuits. The BP, BR and AP sensitivity-frequency diagrams for the
corresponding biquad derived from Figure 2 are shown in Figures 38–40, respectively.

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430 C.-M. CHANG AND M. N. S. SWAMY

Table II. The resonant frequencies and errors of five generic filtering signals
for eight distinct sixth-order filters.
Type
f 3dB (LP) f 3dB (HP) fc (BP) f c (BR) f c (AP)
Filter error (%) error (%) error (%) error (%) error (%)
Figure 1 441.8 k 484.8 k 490.9 k 481.9 k 330.0 k
55.82 51.52 50.91 51.81 67.00
Figure 1 748.4 k 380.6 k 714.5 k 474.2 k 108.8 k
3-biquad 25.16 61.94 28.55 52.58 89.12
Figure 2* 1013.5 k 985.3 k 988.6 k 997.7 k 996.5 k
1.35 1.47 1.14 0.23 0.35
Figure 2 3-biquad 717.0 k 1784 k 1000 k 1000 k 1567 k
28.30 78.40 0.00 0.00 56.7
Figure 7 150.8 k 1477.8 k 1180.3 k 1096.5 k 22 647 k
84.92 47.78 18.03 9.65 2164
Figure 7 3-biquad 719.8 k 1389 k 1004.6 k 10 000 k 4603 k
28.02 38.92 0.46 0.00 360
Figure 8 153.9 k 1635 k 905.7 k 889.2 k 2599.8 k
84.61 63.56 9.43 11.08 156.0
Figure 8 3-biquad 524.6 k 649.1 k 621.2 k 312.3 k 359.0 k
47.54 35.09 37.88 68.77 63.68
The ∗ means with the best.

Table III. Power consumptions of the six universal biquads (the bold data mean with the smallest quantity).
Filter
LP HP BP BR AP
Biquad (mW) (mW) (mW) (mW) (mW)
Figure 1* 0.279 0.255 0.377 0.271 0.294
Figure 2 1.780 1.593 1.593 1.812 1.980
Figure 7 1.192 1.150 1.160 1.129 1.122
Figure 8 1.637 1.584 1.632 1.604 3.076
Horng [25] 1.344 1.344 1.344 1.344 1.344
Abuelma’atti and Bentrcia [26] 2.157 8.251 2.181 2.149 2.496
The ∗ means with the best.

Table IV. Noise of the six universal biquads (the bold data mean with the smallest quantity).
Filter
LP HP BP BR AP
Biquad (V ) (V ) (V ) (V ) (V )
Figure 1 80.0  74.2  74.2  83.3  90.0 
Figure 2 148.4  74.2  74.2  132.4  148.0 
Figure 7 147.5  82.4  82.4  164.4  164.3 
Figure 8* 2.4 n 8.44 n 0.97 n 2.15 n 18 n
Horng [25] 532.0  75.9  71.1  450.2  1430 
Abuelma’atti and Bentrcia [26] 224.8  523.7  217.1  217.3  233.3 
The ∗ means with the best.

It is of interest to know whether the sensitivity of a high-order filter structure is the same as that
of a second-order filter structure. In addition to this, since a sixth-order filter structure can also be
realized by using a three-stage biquad structures in cascade, it is also of interest to compare the
sensitivities of a direct sixth-order one with that of the corresponding three-biquad-stage one.
The component sensitivities of the four different sixth-order filter structures derived from
Figures 1, 2, 7 and 8 are shown in Table VIII in which the largest sensitivity is presented.

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
 ANALYTICAL SYNTHESIS AND COMPARISON 431

Table V. Linear ranges (THD < 1.0 %) of the six universal biquads (the bold data mean with the largest range).
Filter
LP HP BP BR AP
Biquad V V V V V
Figure 1 74.2 m 24.4 m 94.8 m 117.8 m 2.0 m
Figure 2* 108.1 m 171.5 m 86.5 m 198.9 m 16.3 m
Figure 7 24.5 m 1.2 m 1.7 m (1.05 m) (1.7 m)
Figure 8 (22.8 m) (0.65 m) (1.42 m) 1.61 m 13.8 m
Horng [25] 113.8 m 34 m 60 m 71.1 m 28 m
Abuelma’atti and Bentrcia [26] 96.0 m 21 m 55 m 97.5 m 9.9 m
The ∗ means with the best.

Table VI. Dynamic ranges of the six universal biquads (the bold data mean with the largest range).
Filter
LP HP BP BR AP
Biquad V V V V V
Figure 1 Unlimited 10 0.5 20 2
Figure 2* Unlimited 46 1 70 6
Figure 7 34 m (1.41 m) (1.85 m) (3 m) (1.8 m)
Figure 8 (30 m) 1.5 m 60 m 30 m 42 m
Horng [25] Unlimited 70 m 80 m 800 m 60 m
Abuelma’atti and Bentrcia [26] Unlimited 300 m 150 m 250 m 50 m
The ∗ means with the best.

Table VII. Largest component sensitivities of the six universal biquads for the five generic filtering signals
(the * means with very low sensitivity).
Filter
Biquad LP HP BP BR AP
Figure 1 0.500∗ 0.850 0.542∗ 0.386∗ 0.298∗
Figure 2 0.999 1.031 0.542∗ 0.533∗ 0.542∗
Figure 7 0.999 1.130 0.568∗ 0.464∗ 0.620∗
Figure 8 1.017 0.828 0.000∗ 0.000∗ 4.814
Horng [25] 0.895 0.777 0.777 0.457∗ 0.762
Abuelma’atti and Bentrcia [26] 1.000 0.920 0.452∗ 0.480∗ 0.490∗
The ∗ means with the best.

0.6

0.4
c1+3%
0.2
C2+3%
0 g1-1+3%
-0.2 g2-1+3%
g2-2+3%
-0.4

-0.6
100 1k 10k 50k 100k 500k 1M

Figure 38. Component sensitivities of the band-pass biquad derived from Figure 2.

Based upon Table VIII, we can conclude that (i) the component sensitivity of the three-biquad-
stage sixth-order using the new complementary single-ended-input OTA-based one is equal or
lower than that of the direct sixth-order one, (ii) the other three types do not have this property
as (i), i.e. some direct sixth-order ones enjoy component sensitivity lower than its corresponding

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
432 C.-M. CHANG AND M. N. S. SWAMY

0.6
0.4 c1+3%
0.2 C2+3%
g1-1+3%
0
g1-2+3%
-0.2
g2-1+3%
-0.4 g2-2+3%
-0.6

0k

0k

1M
0

k
1k
10

10

50

10

50
Figure 39. Component sensitivities of the band-reject biquad derived from Figure 2.

0.6

0.4 c1+3%
0.2 C2+3%
g1-1+3%
0
g1-2+3%
-0.2 g2-1+3%
-0.4 g2-2+3%

-0.6
100 1k 10k 50k 100k 500k 1M

Figure 40. Component sensitivities of the all-pass biquad derived from Figure 2.

Table VIII. Largest component sensitivities of the eight sixth-order universal filter structures for the five
generic filtering signals (the * means with three biquad stages in cascade).
Type
LP HP BP BR AP
Filter sixth-order sixth-order sixth-order sixth-order sixth-order
Figure 1 0.78 18.87 0.45 35.52 16.95
Figure 1* 1.08 0.48 0.22 0.00 0.91
Figure 2 2.16 2.24 2.33 0.00 2.51
Figure 2* 0.62 1.07 0.00 0.00 2.28
Figure 7 1.12 0.79 1.36 0.00 1.15
Figure 7* 0.51 0.83 0.48 0.49 0.19
Figure 8 1.16 1.07 0.69 12.72 2.35
Figure 8* 1.12 1.36 2.33 0.00 2.40
The ∗ means with the best.

three-biquad-stage one, (iii) the case, Figure 7*, shown in Table VIII, is with the lowest average
component sensitivity (in which low-pass, band-pass, band-reject, and all-pass types even enjoy
very low sensitivities achieved by passive LC ladder circuits), and (iv) although the direct sixth-
order differential-input OTA-based one (Figure 1) has very large sensitivities, such as 18.87 and
16.95, for high-pass and all-pass cases, respectively, yet its three biquad stages in cascade have low
sensitivities, 0.48 and 0.91, leading to the importance of the usage (direct sixth-order or three biquad
stages in cascade) of the proper filter structures. Figures 41and 42 show the component sensitivities
of the direct sixth-order low-pass and high-pass filters derived from Figures 1 and 7, respectively.
In order to validate the feasibility of the synthesized filter structures, the second-order low-pass
biquads derived from Figures 1 and 3 are used to do the validation. Component values are given
by (i) C1 = C2 = 330 pF (equal capacitance), g1 = 160 S, and g2 = 320 S and (ii) g1 = g2 = 320 S
(equal transconductance), C1 = 660 pF, and C2 = 330 pF for realizing a second-order Butterworth
low-pass filter with f 3 dB at 109.2 kHz. We use the LM13700 as the practical OTA. The experimental
results are shown in Figures 43–44 with the input-and-output signals at 120 kHz (the former
having 57.983 mV input and 41.945 mV output signals and the latter having 58.456 mV input

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
 ANALYTICAL SYNTHESIS AND COMPARISON 433

Figure 41. Component sensitivities of the straight sixth-order low-pass filter derived from Figure 1.

Figure 42. Component sensitivities of the straight sixth-order high-pass filter derived from Figure 7.

Figure 43. Input and output signals of the low-pass biquad derived from Figure 1.

Figure 44. Input and output signals of the low-pass biquad derived from Figure 3.

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
434 C.-M. CHANG AND M. N. S. SWAMY

Measured Frequency Responses

1.2
Equal g
1 Equal C

Voltage Gain (lin)

0.8
0.6
0.4
0.2
0
10 100 1000
Log Frequency (kHz)

Figure 45. Measured amplitude-frequency responses of equal C or equal g low-pass

biquads derived from Figures 1 and 3.

and 41.665 mV output signals); Figure 45 shows the measured amplitude–frequency responses
and the f 3 dB is the frequency range of 120–125 kHz. It is seen that the output signals from the
equal capacitance biquad and the equal transconductance biquad are nearly the same. Thus, the
synthesized filter structures shown in Figures 1 and 3 have been validated to be feasible using this
experiment.

5. MODIFIED FREQUENCY DEPENDENT TRANSCONDUCTANCE

Figures 15–18, 20–21, 28, 30, 32, 34 and 36 show that a major distortion (sudden magnitude fall)
occurs in the frequency range from 200 to 1 GHz. This apparent distortion is investigated below.
The four transconductances, G 1 , G 2 , G a and G b , of the second-order universal biquad derived
from Figure 7 are functions of frequency and are shown in Figures 46–49, respectively. We note that
(i) the initial part of the curves shown in Figures 46–49 is of a low-pass nature and is in
agreement with the traditional representation, G(s) = [g/(1+s/o )] [7], but the latter part
shows an exponential-like increase and this has not been mentioned earlier in the literature.
The frequency-dependent transconductance G(s) of an OTA needs to be modified as follows:
⎧ g
⎪
⎨
⎪
s
 , f < f lowest
G(s) = 1+ o (19)
⎪
⎪
⎩ b
a10 2 s , f  f lowest
where f lowest is the frequency with the lowest value of transconductance in the amplitude–
frequency response of G(s). For example, f lowest ≈ 660, 72 MHz, 10 GHz and 3.5 GHz for
Figures 46–49, respectively. The exponential-like function is with a = 26.46 and b = 0.07
for G 2 (s) (Figure 50).
(ii) Figure 47 shows a sharp increase in the transconductance G 2 (s) when the operational
frequency is over 100 MHz. The value of G 2 is 459 S at 100 MHz and abruptly jumps up
to 304 mS at 1000 MHz. This situation may be called ‘the collapse of the transconductance
of the OTA’. On the other hand, we notice that the f lowest is 660 MHz, 72 MHz, 10 GHz
and 3.5 GHz for Figures 46–49, respectively. Note that the f lowest of G 2 is much lower than
that of other three ones. It leads to the value of G 2 to be much larger than G 1 , G a , and
G b , at 200 MHz and then dominant for the output responses. As a consequence, distortions
have occurred, as shown in Figure 15.
(iii) Both the differential-input OTA-based universal biquad derived from Figure 1 and the
new complementary single-ended-input OTA-based one derived from Figure 2 have no
collapses of transconductances. However, the universal biquads derived from Figures 7
and 8 with all single-ended-input or fully differential OTAs and shown in [25, 26] with
some single-ended-input OTAs and some differential-input OTAs have distortion due to the
collapse of transconductances.

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
 ANALYTICAL SYNTHESIS AND COMPARISON 435

Figure 46. Amplitude-frequency response of G1 .

Figure 47. Amplitude-frequency response I of G2 .

Figure 48. Amplitude-frequency response of Ga .

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
436 C.-M. CHANG AND M. N. S. SWAMY

Figure 49. Amplitude-frequency response of Gb .

Figure 50. Several functions of the exponential-like curve G2 (s) when the
operational frequency is over 72 MHz.

6. CONCLUSIONS

A new type of operational transconductance amplifier (OTA)-based circuit structure, called a

complementary single-ended-input OTA-based one, is introduced. Four distinct kinds of voltage-
mode nth-order structures, namely, (i) differential-input, (ii) single-ended-input, (iii) complementary
single-ended-input and (iv) fully differential, OTA-based universal filter structures are synthesized
using two different analytical synthesis methods (ASMs) and two transformations. A compar-
ison study with two recently reported works is made using 0.35 m TSMC H-Spice simulations.
Simulation results show that the complementary single-ended-input OTA-based one is the best
one in terms of output precision, and dynamic and linear ranges. A modified frequency-dependent
transconductance G(s) for an OTA is given for presenting the situation of collapse of the transcon-
ductance of an OTA. The band-pass, band-reject and all-pass (except the fully differential one)
biquad structures enjoy very low sensitivities achieved by passive LC ladder networks. In addition,
depending on the kind of structure and the type of filter, the sensitivity of a direct sixth-order filter
may be greater than or less than that of its equivalent sixth-order one with three biquad stages in
cascade.

Copyright 䉷 2010 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2012; 40:405–438
DOI: 10.1002/cta
 ANALYTICAL SYNTHESIS AND COMPARISON 437

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DOI: 10.1002/cta