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GIC Based Third-order Active Low-pass Filters

Nino Stojković, Dražen Jurišić and Neven Mijat

University of Zagreb, Faculty of Electrical Engineering and Computing

Unska 3, 10000 Zagreb, Croatia

Tel. +385 1 6129 911, Fax. +385 1 6129 652, e-mail: nino.stojkovic@fer.hr

Abstract

Third order low-pass filters are analyzed. Two different

configuration of General Impedance Converter (GIC)

based filters are compared to two Sallen and Key (SAK)

based structures. For GIC based third-order filters

realization procedures are given. The transfer functions

and the component values for the third-order filters with

Butterworth response are presented. The sensitivity

analysis is done, and the results of Monte Carlo runs, as

well as Schoeffler sensitivities are shown. The best results

are obtained with the GIC based filter, which use four

capacitances for the realization.

1. Introduction

Many practical filtering problems in electronic design

can be solved by low cost, low order, easy to realize active

filters. Second- and/or third-order active filter sections are

usually suitable enough for such applications. Various

second-order single or multiple amplifier filter sections are

widely used as building blocks of high-order filters as

well, and significant efforts in the past have been paid to

their analysis and design [1]. On the other hand third-order

filter sections attracted considerably less attention of filter

designers, mostly thanking to the fact that their application

in high-order filters is rather unusual. It is well known that

high-order LP or HP filters are usually realized as a

cascade of second-order blocks, if the order n of a filter is

even, and one first-order section for odd filter transfer

function order. The properties of an odd-order filter can be

improved if one second-order section and a first-order

section are replaced by some less sensitive third-order

section. Some authors have even proposed the

approximation technique suitable for realization of high-

order filters by cascading third-order filter blocks [2].

Third-order filter structures commonly used in filter

realizations, are considered

configurations, which are suitable from many points such

as the power consumption, ease of realization procedure,

low cost etc. In this paper we introduce a less sensitive

two-amplifier third-order active filter sections, which can

as single-amplifier

be applied for low-order filtering applications, as well as

for above mentioned replacement in high-order filters. The

filter sections are based on a general impedance converter

(GIC) circuit used for realizations of second-order filter

blocks [3].

2. Third-order low-pass filters

GIC based second-order active filter sections are well

known low sensitive filter structures, suitable for

realization of high-order filters. The third-order low-pass

(LP) filters can be easily realized by cascading such

sections with the first-order LP filter realized by a simple

RC circuit and a voltage follower. By connecting the RC

circuit without the voltage follower, to the input of a

second-order GIC based filter section, a third-order

section can be realized using one of the filter structures

shown in Fig. 1.a) and 1.b) denoted as GIC1 and GIC2

respectively. For such circuit new values of filter

components must be calculated. The corresponding

realization procedures are presented in the sequel, and

compared to the Sallen and Key based third-order filters

shown in Fig 1.c) and 1.d) denoted by SAK1 and SAK2,

[1], [4].

2.1 Third-order GIC based filter with 3

capacitances (GIC1)

The first GIC based third-order filter which is analyzed

is a filter structure with three capacitances (GIC1), shown

in Fig. 1.a). The transfer function is shown in Table 1.

Calculation of the filter elements can be performed by

comparison of the transfer function of GIC1 to the general

LP third-order transfer function

( )

T s

s a s a s a

++

21

giving the following identities

GG

C

0

b

2

=

+

0

3

0

,(1)

a

06

2

+

=

(2)

Page 2

G0

C0

G4

G3

G2

G1

C2

C1

Vo

Vi

G0

C0

G2

G6

C3

Vo

Vi

C5

G4

G1

G0

C0

G5

G3

C2

Vo

Vi

C5

G4

G1

C6

a)

c)

b)

d)

G0

C0

G2

G1

C2

C1

Vo

Vi

Figure 1. Third-order active filters: a) GIC1, b) GIC2,

c) SAK1, d) SAK2

G G G

G C C

1 3 5

G G G G

G C C C

1 0 3 5

G

Since there are 4 equations with 8 unknowns, the

additional conditions are introduced

CCCGG

354

===

,

and the components can be expressed as functions of

capacitance C.

GC a

=

1

ba

a ab

2 10

−

a

246

1

=

(3)

a

0 246

0

=

(4)

1

2

1

0 4 6

G G G

C C C

0 3 5

0

+

=

G

b.(5)

G

6

=

, (6)

(7)

GC

0

01

=

(8)

CC

a

a

b

a

0

1

2

0

1

=

−

(9)

GG

12

=

(10)

Choosing initial value for the capacitance C all elements

are defined except the resistors G1 and G2, which can be

chosen freely. The calculated element values for the GIC1

third-order Butterworth low-pass filter are presented in

Table 2.

2.2 Third-order GIC based filter with 4

capacitances (GIC2)

Another configuration of the GIC based third-order

low-pass filter (GIC2), needs four capacitances for the

realization as shown in Fig. 1.b). The transfer function of

this filter is shown in Table 1. It can be obtained only if

the condition

C GC G

5 3 6 4

=

,

is satisfied. After comparison of the GIC2 transfer

function to the general low-pass third-order transfer

function (1) the following equations are obtained:

GG

CC

02

G G

C CC C

0 22 5

G G G

C C C

0 2 5

G G G

C C C

0 2 5

In order to simplify the calculation procedure we introduce

a new variable τ0:

0

0

G

It can be shown that τ0 must be in range

1

0

a

σ

where σ represents negative real pole of the third-order

transfer function (1) and a1 and a2 are the denominator

coefficients of the same function. Choosing capacitances

C0 and C5 the rest of filter elements can be calculated as

follows:

=C

G

(

0

10201

1

τ−τ

aa

(11)

G

a

011

2

+

+=

(12)

G G

1 5

a

0 1

1

+=

(13)

a

0 1 5

0

=

(14)

b

0 1 5

0

=

.(15)

0

C

=τ

.(16)

0

1

a

<τ<−

(17)

0

0

0

τ

(18)

) 1

3

0

0

2

−τ+τ−τ=

aaaGG

(19)

2

0

001

12

=

GC (20)

Page 3

001

0

a

55

τ−

=

a

a

CG. (21)

Setting

43

GG =

(22)

from (11) follows

C =

Calculated element values for the GIC2 third-order

Butterworth low-pass filter are presented in Table 2.

56

C.(23)

2.3 Third-order SAK based filters (SAK1 and

SAK2)

The GIC based filters are compared to SAK based

filters. For that purpose, the third-order low-pass SAK

filter with one operational amplifier, shown in Fig. 1.c)

and with two operational amplifiers shown in Fig. 1.d) are

analyzed. Transfer functions of these filters are shown in

Table 1. and calculated elements for the filter with

Butterworth response are given in Table 2.

3. Sensitivity analysis

In order to compare the influence of filter parameters

variations to the filter amplitude response for various

structures, the sensitivities of the filter amplitude response

T(jω) to the variation of elements xi are analyzed. The

sensitivity function S is defined with

( )

T j

i

(

)

()

S

d T j

d x

x

T j

x

i

i

ω

ω

ω

=⋅

. (24)

If the gain of the amplitude response α(ω) is expressed

by

( )

α ωω= 20logT j

dB

the gain sensitivity function S can be defined as

( )

( )

[

Sx

i

i

d x

We can define the gain variation as

( )

[ ]

x

i

i

where xi represents all passive filter elements. As a

sensitivity measure the

sensitivity function Is

( )

Is

x

i

is used [5]. It can be shown that if all network component

variations are uncorrelated random variables with zero

expectations and the same standard deviation σ, than the

standard deviation of gain variation is approximately

proportional to Is(ω). Therefore from

(

x

i

it follows

( )()( )

ωδ≅ωα∆σ

s I

() []

, (25)

]

i

x

d

dB

α ω

α ω

=

. (26)

( )

xi

dB

x

i

∆

≅ωα∆

∑

ωα

S

, (27)

Schoeffler multiparameter

2(ω)

( )

i

2

2

ω

α ω

=

∑ S

(28)

)

σδ∆x

i =

(29)

.(30)

Table 1. Transfer functions of presented third-order filters

T s ( )

V

V s

i

s

o

( )

( )

=

GIC1

1

2

1

sG G G

G C C

0 4 6

G G G

C C C

0 3 5

32

06

0

2 4 6

1 3 5

0 2 4 6

G G G G

G C C C

1 0 3 5

+

+

+

+

+

G

G

ss

GG

C

GIC2

G G G

C C C

sG G

+

ss

GG

C

G

CC C

0 2

G G

C C

2 5

G G G

C C C

0 2 5

0 1 5

0 2 5

32

01

0

1

2

0 11 50 1 5

+

+

+

+

+

, C GC G

6 45 3

=

SAK1

1

4

3

G G

C C

0 1 2

0 1 2

+

G G

32

01

0

12

1

2 4

2 3

C G

0 10 2

0 1

1 21 2

1 2

C C

0 2 4

G G G

1 2 4

G G G

0 2 3

C C G

0 1 2

0 1 2

C C C

+

+

+

+

+

C

−

+

+

+−

+

+

G

G

G G G

C C C

ss

GG

C

GGG G

sG G

G GG G G

SAK2

G G G

C C C

G G

0 1

ss

G

C

GG

C

s

G G

0 2

C C

0 1

G G

C C

1 2

G G G

C C C

0 1 2

0 1 2

0 1 2

320

0

12

1

1 20 1 2

++

+

+

+

+

+

Page 4

Table 2. Filter elements of analyzed 3rd-order filters

GIC1GIC2

0.638897

0.680551

SAK1

0.638897

0.680552

2.299900

1

1

SAK2

1

1

1

G0 0.471405

G1 1.414214

G2 1.414214

G3

G4 1.414214

G5

G6 1.414214

C0

0.942809

C1

C2

C3

C5

C6

1

1

2.299899

11

1

1

1

1

11

1

11

1

The standard deviation of amplitude response according to

(30) is calculated assuming 1% standard deviations of

elements. The results are obtained using programming tool

Mathematica and shown in Fig. 2. Schoeffler sensitivity

function gives results shown in Fig. 3.

The sensitivities can be compared numerically, as well,

if multiparameter measure M defined by

(

MS

x

i

001.

is used [4]. It represents the area below the correspondent

sensitivity function in Fig. 3. The results are given in the

Fig. 3. for all filters which have been considered. The

result shows significant improvement in sensitivities for

GIC based filters. Significant part of the result is placed

left from the cut-off frequency, in the pass-band. In the

stop-band, the sensitivities are less important because of

the filter attenuation. The significant influence on signals

can be done in the pass-band where amplitude response

variation should be the smallest. It can be seen that the

number M gives picture of merit, which is not completely

reliable. Although the smallest M is obtained for GIC1

filter, in the pass-band the lowest sensitivities has GIC2

filter. The worst sensitivities

Configuration GIC1 shows good results in the

neighborhood of the cut-off frequency but in pass-band its

sensitivities are close to the sensitivities of SAK1 and

SAK2 filters.

The same relations between the sensitivities of the

considered third-order filters are obtained using Monte

Carlo analysis, Fig. 4. This analysis completes filter

sensitivity picture. Assuming that the component values

have some finite tolerances, the amplitude response is

calculated. In the presented analysis the components are

assumed to have 1% tolerance. The obtained results

confirm previously described relations between presented

filters.

)

d

T j

i

=

∑

∫

ω

ω

2

10

(31)

has SAK1 filter.

0.6σ(α(ω)) [dB]

0.2

00.0016

0.010.11.6

f [Hz]fg

GIC2

GIC1

SAK2

SAK1

1

0.4

0.5

0.3

0.1

Figure 2. Standard deviation of the gain σ σ σ σ(α α α α(ω ω ω ω)) for

the filters presented in Fig. 1.

40Schoeffler Sensitivity

SAK1: M=14.169

SAK2: M=10.537

GIC1: M=10.13

GIC2: M=11.321

30

20

10

00.0016

0.010.1 1.6

f [Hz]

fg

GIC2

GIC1

SAK2

SAK1

1

Figure 3. Schoeffler sensitivity for the filters presented

in Fig. 1.

3. Conclusions

In this paper, third-order LP filters, which can be used

for low-order, low-cost applications, as well as building

blocks for high-order applications, are presented and

analyzed. It is shown that low sensitivities of the filter

amplitude response can be obtained by using GIC based

third-order structures instead of single-amplifier SAK

based configuration.

Disadvantage of GIC based filters can be the use of

two operational amplifiers resulting with increased power

consumption in comparison to single amplifier structures.

Therefore these filter configurations should be used in the

applications where power consumption is not the primary

interest. The design procedure is straight-forward, and

relatively simple. The sensitivity analysis is performed

using Schoeffler sensitivity measure and confirmed by

Monte-Carlo runs.

Page 5

Figure 4. Monte Carlo response plots of the filters presented in Fig. 1.

4. References

[1] G.S. Moschytz, Linear Integrated Networks: Design, New

York, Van Nostrand-Reinhold, 1975.

[2] M. Biey, A. Premoli, Design of Low-Pass Maximally Flat

RC active filters with Multiple Real Pole:The MURROMAF

Polynomials, IEEE Transactions on Circuits and Systems,

vol. CAS-25, no. 4, April 1978.

[3] N. Fliege, A New Class of Second-order RC-Active Filters

with Two Operational Amplifiers, Nachrichtentech Zeitung,

Vol. 26, pp. 279-282, June 1973.

[4] R. Schauman, M. S. Ghausi, K. R. Laker, Design of Analog

Filters - Passive, Active RC and Switched Capacitor,

Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1990.

[5] J.D. Schoeffler, The Synthesis of Minimum Sensitivity

Networks, IEEE Transactions on Circuit Theory, pp. 271-

276, June 1964.