A Contribution to the 2nd-order LP Filters DesignPravoslav MARTINEK1, Milan VALENTA1, Dasa TICHA2

1 Dept. of Circuit Theory, Czech Technical University, Technicka 2, 166 27 Praha, Czech Republic2 Dept. of Telecommunications and Multimedia, University of Zilina, Univerzitna 1, Zilina, Slovak Republic

martinek@fel.cvut.cz, valenta@fel.cvut.cz, ticha@fel.uniza.sk

Abstract. This paper discusses some aspects of design andrealization of the simple second-order low-pass buildingblocks (LP-SFB), suitable for less demanding anti-aliasingand reconstruction filters. The main attention is devotedto the ”single active device” solutions, based on CC-IItype current conveyor or transconductance amplifier (OTA).As the template, the Rauch’s multiple-feedback circuit wasused. Presented SFBs show very low ω0 and Q sensitivitiesand, in contrast to the known Sallen-Key circuits, save goodstop-band attenuation. Proposed design uses DE algorithmand allows almost independent tunning of ω0 and Q param-eters.

KeywordsActive filters, LP filters, Low-pass building blocks, sen-

sitivity, Evolutionary algorithms, optimization.

1. IntroductionAdvances in digital signal processing, CMOS technol-

ogy and their application in data transfer, mobile networks,

digital TV or digital video and other areas during last few

years gave rise to a higher attention to the integrable anti-

aliasing and reconstruction low-pass filters design. Such a

filter usually does not need to fulfill exacting requirements to

the pass-band and stop-band magnitude frequency response,

but, on the other hand, there are higher requirements to a

frequency range up to tens of MHz, low power consump-

tion, low supply voltage and compatibility to the collabora-

tive digital parts of the system.

Most of the attenuation requirements lead to the simple

polynomial approximations (some of them with additional

constraints to the group-delay response) with relatively low

Q−factors of complex poles of the resulting transfer func-

tion. In such case realizations using cascade connection of

the 2nd-order sections with minimum number of active el-

ements are usually preferred. As an example video-filters

produced by MAXIM [1], [2] and other producers of analog

integrated circuits (TI, AD, Fairchild,...) can be introduced.

The 2nd-order selective functional blocks (SFBs) are usually

realized as single Op-Amp structures, or using unity-gain

amplifiers. Here Sallen-Key and Rauch’s multiple feedback

structures dominate. Main problems of these SFBs usage

are relatively high Q− and ω0 sensitivities to the active el-

ement gain changes, request to very high GBW of the used

Op-Amps and problems with degradation of stop-band fre-

quency response caused by finite resistance of amplifier out-

put [3].

A solution of the problems presented was searched in

the use of current-mode design and usage of current convey-

ors, transconductance amplifiers and derived mixed-mode

building blocks, e.g. MIMO (Multiple Input Multiple Out-

put) current conveyors, CDBA or CDTA amplifiers [4], [5]

and others. Based on this, tens new biquads have been pub-

lished in journals and conference proceedings. Neverthe-

less, authors usually gave precedence to universality of the

developed SFBs, which lead to more complicated solutions.

The introduced circuits are mostly based on SFG- or state-

space principles, i.e. ”two integrators inside negative feed-

back loop”, which require more active devices and seem

to be unnecessarily complicated for the intended purpose.

Some examples are shown e.g. in Refs. [6], [7], [8]. Sim-

pler solutions using one active element were described in [9]

and [10], but with enhanced universality as well. These bi-

quads require output buffers for cascade realization and thus

they can hardly offer a better and more effective solutions.

With respect to the previous, our attention was turned

on the 2nd-order SFBs, derived from multiple-feedback cir-

cuits, where Op-Amps were replaced by CCII current con-

veyors or OTAs. In the following, some derived LP sections

are introduced and their properties and non-ideal behavior

discussed. Practical use in design of SFB for digital video

filter is demonstrated on the example. Design procedure is

based on DE algorithms, which allow suitable modeling of

ω0 and Q sensitivities to achieve partially independent tun-

ning of ω0 and Q.

2. Improved Low-pass SFBsAs mentioned, our aim was to derive CCII- and OTA-

based LP 2nd-order filters with simple circuit structure, min-

imum active devices and acceptable ω0 and Q sensitivi-

ties to the passive and active circuit elements together with

good high frequency behavior. As a prototype the multiple-

978-1-61284-324-7/11/$26.00 ©2011 IEEE

feedback structure has been chosen, with respect to the ap-

propriate stop-band properties (in contrast to Sallen-Key

structures - see e.g. [3], [11]). The original circuit schematic

is in Fig. 1. It shows good sensitivity properties, but the

”bottleneck” is hidden in the Op-Amp based integrator.

Fig. 1. The original multiple-feedback LP.

A significant improvement, especially in wider fre-

quency band, can be achieved using modified ”active inte-

grator”, as shown in Fig. 2. Here the CCII- current conveyor

is used to realize integrator without limitations caused by

real Op-Amp behavior. Transfer function of the idealized

circuit is expressed by Eq. (1), and parameters ω0 and Q by

Eq. (2). Here A1 means gain of the output voltage follower,

and β the CCII current transfer X-to-Z.

Fig. 2. The CCII- based LP circuit.

H = −R3

R1

β A1

s2 C1 C2 R2 R3 + sC2 R2R1+R3

R1+ β A1

(1)

ω0 =

√1

C1 C2 R2 R3; Q =

R1

R1 +R3

√C1 R3

C2 R2. (2)

Gain constant is given by a ratio h = R3/R1. Cir-

cuit presented in Fig. 2 has its counterpart in the schematic

in Fig. 3, where current conveyor is replaced by transcon-

ductance amplifier (OTA) under Fig. 4. The relationship

gm = 1/R2 is evident.

Transfer function of the idealized circuit is expressed

by formula (3) and the corresponding parameters ω0 and Qby (4). Gain constant is the same as in the previous circuit.

HOTA = −R3

R1

gmA1

s2 C1 C2 R3 + sC2R1+R3

R1+ gmA1

(3)

ω0 =

√gmA1

C1 C2 R3; Q =

R1

R1 +R3

√gmA1 R1 C1

C2.

(4)

The similarity to the first version in Fig. 2 is evident.

Both the introduced circuits show excellent sensitivity

properties – absolute values of ω0 sensitivities to the passive

and active elements are 1/2, Q-factor sensitivities to the ac-

tive devices are 1/2 as well, to the passive circuit elements

±1/2 or less.

2.1 Analysis of Non-ideal CircuitA real behavior of the mentioned circuits is mainly in-

fluenced by imperfections of active devices. In the case of

current conveyor, the main problems cause finite resistance

Rxi of terminal X, output Z parasitic conductance goz and

X-to-Z current transfer β less then 1. The simplified equiv-

alent circuit for CCII is shown in Fig. 5. The parasitic ca-

pacitance CT will be neglected (it can be added to C2),

goz = 1/RT .

Transconductance amplifier main imperfection is in fi-

nite output conductance go. Assuming C-MOS implemen-

tation, the input impedances can be neglected.

For the output buffer with voltage gain A1 = 1 in the

ideal case, A1 ≤ 1 and non-zero output resistance Ro have

to be supposed.

To simplify results, frequency-dependent gain and in-

put/output immittances of active devices are not assumed

in the following. Simultaneously, with respect to the large

symbolic transfer function, the parameters ω0s, Qs and gain

constant h will be introduced only.

Considering CCII-based version, the main parameters

are as follows:

ω0 =

√(R2 +Rxi)(R1 +R3 +Ro)goz +A1βR1

C1C2R1(R2 +Rxi)(R3 +Ro), (5)

Q =

√C1C2R1(R3+Ro)((R2+Rxi)(R1+R3+Ro)goz+R1βA1

R2+Rxi

C1R1(R3 +Ro)goz + C2(R1 +R3 +Ro),

(6)

h =R3 A1 β − goz Ro(R2 +Rxi)

R1 A1 β + goz(R2 +Rxi)(R1 +R3 +Ro). (7)

Comparing these results to the ω0, Q and h parameters

for idealized circuit defined by Eqs. (1) and (2), it is possible

to make following conclusions:

Fig. 3. The OTA based LP circuit.

Fig. 4. Equivalent circuit to the resistively loaded CCII.

Fig. 5. The CCII simplified equivalent circuit.

• Internal resistance Rxi of the CCII port X only in-

creases value of R2 without any other changes;

• Output resistance Ro of the output buffer increases

value of R3 in comparison to the ideal circuit param-

eters. But, in addition, it causes parasitic forward

transfer creating unadvisable transfer zero, given by

expression (8)

sz = − βA1 R3

C2 Ro(R2 +Rxi)+

goz

C2. (8)

• CCII Z-output parasitic conductance goz = 1/Roz

added additional terms to the ω0 and Q expressions,

and, in fact, decreases both these parameters.

• Sensitivities of ω0 and Q do not exceed absolute val-

ues computed for idealized circuit. Some are equal,

particularly Sω0

C1= Sω0

C2= −0.5, Sω0

β = Sω0

A1= -Sω0

R1

< 0.5, Sω0

R1, Sω0

R3< 0.5. Similarly SQ

C1= −SQ

C2,

SQβ = SQ

A1= SQ

R2. Some of Q-sensitivities can be

minimized or set to required values – especially SQC1,2

,

SQR1,R3,Roz

.

As a summary, it can be stated:

1. Influence of Rxi and Ro to the main parameters can

be suppressed by a correction of R2, respective R3.

2. Influence of the parasitic transfer zero does not

dramatically degrade stop-band filter frequency re-

sponse, only changes slope from −40 dB/dec to

−20 dB/dec. Parasitic zero position can be kept at

least two decades from the pass-band corner, with re-

spect to the ratio R3/Ro.

3. Influence of goz (Roz) cannot be excluded, but it can

be exploited for improved design without deteriora-

tion in quality of the resulting circuit.

4. Some sensitivities of Q-factor significantly differ

from the idealized case and can be set to required val-

ues in interval (0....0.5). This means important change

against the ideal circuit and a benefit to design opti-

mization of ”real” SFBs.

The presented conclusions are valid for both the versions in

Fig. 2 and Fig. 3.

2.2 Design ProcedureIn contrast to the ”standard” approach, our idea is to

design directly ”real” filter, respecting circuit devices im-

perfections and leading to the optimum properties achieve-

ment. This is hardly realizable using standard exact proce-

dures, thus, our solution leads via Differential Evolutionary

algorithms which were found very friendly in our previous

filter designs – see e.g. [12], [13].

Design requests:

1. A correct setting of the main parameters ω0, Q, h.

2. ω0- and Q-sensitivities minimization.

3. Creating conditions for partially independent tunning

of ω0 and Q. This was carried out by selection of two

pairs of sensitivities under the rule

Sω0x1→ 0, and SQ

x1→ ±0.5 ;

Sω0x2→ ±0.5, and SQ

x2→ 0 .

These conditions are fulfilled by combinations SQgoz and

SQC1,2

as will be evident from an example in Section 3.

The algorithm used is the same as in the mentioned

papers [12] and [13]. Objective function fit is defined as

follows

fit = δ2om+δ2Q+δ2h+δ2SQg+(SQC1

)2+(Sω0goz)

2+Par2 , (9)

where δom, δQ, δh denote relative errors of ω0, Q and h of

the tested individual, δSQg and Par are defined by expres-

sions (10)

δSQg = 0.5− |SQgoz| ; Par =

1

sz. (10)

The used trial vector �v is in the form

�v = �X3 +K ( �Xbest − �X3) + F ( �X1 − �X2) .

The algorithm was implemented in MAPLE mathematical

program.

2.3 A Comparison to the Similar LP-SFBsSimilar circuit, insensitive to the stop-band fre-

quency response degradation, was presented by Michal and

Fig. 6. The Michal-Sedlacek LP.

Sedlacek in [11], but this SFB suffers from non-ideal dy-

namic properties, with respect to the higher voltage at CCII

terminal Z, exceeding output voltage – see Fig. 6.

3. An ExampleAs an example the LP-section design for cascade re-

alization of video-filter under Ref. [14] is presented. Given

frequency scaled parameters:

ω0 = 1.043329713, Q = 3.189672628, h = 1.0.

The chosen parameters were:

β = 1.0, A1 = 1.0, Ro = 0.242, C1 = 1.0, C2 = 0.22. Pa-

rameter computation using aforementioned algorithm gave

the following results:

R1 = 12.111928, R2 = 0.338086, R3 = 12.419981,

goz = 0.4411494.

Evaluated ”critical” sensitivities:

Sω0

C1,2= −0.500000, SQ

C1,2= 0.006144,

Sω0goz = 0.012284, SQ

goz = −0.493860.

The resulting filter parameters after simulation:

ω0 = 1.043329877, Q = 3.189673272, h = 1.000481,

parasitic transfer zero: sz = −109.684Note that the obtained results, especially ω0- and Q-

sensitivities, are not influenced by the C1/C2 ratio. This

only changes the relation R2/R1.

Real values of filter elements after recalculation for

fN = 10MHz: C1 = 31 pF , C2 = 6.8 pF , R1 =6.236 kΩ, R2 = 174Ω, R3 = 6.395 kΩ, Roz = 14.137 kΩ,

Ro = 125Ω.

The corresponding frequency response is shown in

Fig. 7

4. ConclusionsIn this paper we have tried to show non-standard ap-

proach to the simple single-purpose 2nd-order SFBs multi-

criteria design. As shown, Differential Evolutionary al-

gorithms are very suitable for this task, and usage of

”weighted” sensitivities can contribute to more effective

practical realization and tunning of the designed filters.

Fig. 7. Simulated frequency response

AcknowledgementResearch described in the paper was financially sup-

ported by the research program ”Research in the Area of the

Prospective Information and Navigation Technologies” No.

MSM6840770014.

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