THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 10 | NUMBER: 2 | 2012 | JUNE
THIRD ORDER LOW-PASS FILTER USING SYNTHETIC
IMMITTANCE ELEMENTS WITH CURRENT CONVEYORS
Pa el BRANDSTETTER1, Lukas KLEIN1
1Depa men o Elec onics, Facul y o Elec ical Enginee ing and Compu e Science, VSB-Technical Uni e si y o
Os a a, 17. lis opadu 15, 708 33 Os a a, Czech Republic
pa el.b ands [email p o ec ed], lukas.klein@ sb.cz
Abs ac . The pape deals wi h a heo e ical p oposal o
he esul ing ci cui o he equency il e using syn he ic
immi ance elemen s o highe o de wi h cu en
con eyo s. The ex pays pa icula a en ion o design
p ocess o syn he ic immi ance elemen s, explains he
p inciple o inc easing o o de , which is hen e lec ed o
he equency il e o de . The ex hen deals less wi h he
heo y o cu en con eyo s, which has al eady been
discussed, in de ail, in p e ious pape s. Uni e sal cu en
con eyo (UCC) is discussed mo e. This ac i e elemen is
used o he heo e ical implemen a ion o he syn he ic
elemen solu ion used in he equency il e . The
heo e ical knowledge is hen demons a ed in he design
o 3 d o de low-pass equency il e . The inal
unc ionali y o he p oposed equency il e ci cui
solu ion is alida ed by PSpice simula ion.
Keywo ds
Cu en con eyo , equency il e , low-pass il e ,
syn he ic immi ance elemen .
1. In oduc ion
Me hod o he syn he ic elemen s is p ima ily based on
he needs o classical induc o s subs i u ion by syn he ic
equi alen s in elec onic ci cui s. I is no a new me hod,
bu i s use in combina ion wi h new mode n ac i e
componen s (cu en con eyo s), o he design and
ealiza ion o equency il e s, is a ela i ely new idea
[1], [2], [3], [4], [5], [6], [7], [8].
The basic ad an ages o using cu en con eyo s
[1] include a wide equency ange, ease o in eg abili y
o he esul ing ci cui solu ions, low supply ol age o
ac i e elemen s and he possibili y o use o ba e y
supply in esul ing ci cui , when i is used in he mobile
de ices.
Cu en con eyo s
[1] can be basically ega ded as
uni e sal elemen s, because ou basic unc ional block
s uc u es can be ealized by combina ions o hese ac i e
elemen s.
Howe e , he basic disad an age is low
comme cial a ailabili y o cu en con eyo s and
comme cial una ailabili y o uni e sal cu en con eyo
(UCC), which was selec ed o he ealiza ion o il e
p oposed in his pape . Selec ion o UCC has i s
impo an easons. This elemen seems o be e y
p omising, especially in he design o ci cui solu ions
wi h cu en con eyo s, because i could be used o he
implemen a ion o all cu en con eyo s a ia ions.
Uni e sal cu en con eyo cu en ly exis s only as a
labo a o y sample and he wide ex ension ye o come.
The pape ies o show i s impo ance and he
ad an ages.
2. Syn he ic Immi ance Elemen s o
Highe O de s
Syn he ic dipoles wi h immi ance o highe o de [2] a e
di ided in o ou g oups – DP, DS, EP, ES. They a e
consis ing o se ial o pa allel elemen a y dipoles
connec ions. Syn he ic elemen s DP, DS a e c ea ed by
elemen a y D ype dipoles o o de 0 o ND,max. Syn he ic
elemen s EP, ES a e c ea ed by elemen a y E ype dipoles
o o de 0 o NE,max.
The linea ci cui s heo y discusses ha s able
equency il e s can be ealized, i he elemen a y dipoles
o all o de s om he lowes (nd,min, ne,min) o highes
(nd,max, ne,max) will be connec ed in he syn he ic
elemen s. Mos o he alues o nd,min, ne,min equal o 0 o
1, alues o nd,max, ne,max se he o de o he syn he ic
elemen . In he mos cases, hese alues also se he o de
o equency il e ans e unc ion, espec i ely.
As s a ed abo e, he e a e ou connec ions o
syn he ic elemen s wi h immi ance o highe o de [2]:
© 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 89
THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 10 | NUMBER: 2 | 2012 | JUNE
 dipole DP ND,min ND,max consis s o a pa allel
connec ion o syn he ic elemen a y dipoles o ype
Dn o n = ND,min, ND,min + 1, ..., ND,max - 1, ND,max,
 dipole DS ND,min ND,max consis s o a se ial
connec ion o syn he ic elemen a y dipoles o ype
Dn o n = ND,min, ND,min + 1, ..., ND,max - 1, ND,max,
 dipole EP NE,min NE,max consis s o a pa allel
connec ion o syn he ic elemen a y dipoles o ype
En o n = NE,min, NE,min + 1, ..., NE,max - 1, NE,max,
 dipole ES NE,min NE,max consis s o a se ial
connec ion o syn he ic elemen a y dipoles o ype
En o n = NE,min, NE,min + 1, ..., NE,max - 1, NE,max.
2.1. The Condi ions o Syn he ic
Immi ance Elemen s Realiza ion
I sui able syn he ic immi ance elemen s o highe o de
should be designed, hen an app op ia e o m o inpu
impedance, espec i ely inpu admi ance is sea ched [3].
The equi ed inpu impedance espec i ely inpu
admi ance o m o syn he ic immi ance elemen s o DS
and ES ype should be:
VWV
U
IN YYY
Y
Z1
 ,
WU
WV
IN YY
YY
Y
. (1a, b)
Inc ease o syn he ic elemen o de is done by
epea edly eplace o admi ance YV wi h ci cui wi h
inpu admi ance (1b). Inc ease o o de o syn he ic
immi ance elemen can also be achie ed by eplacing he
admi ance YU wi h ci cui wi h he inpu admi ance (2b).
Fo syn he ic immi ance elemen s o DP and EP
ype, i is equi ed inpu impedance espec i ely
admi ance in o m:

WVU
W
IN YYY
Y
Z
, U
W
VU
IN Y
Y
YY
Y . (2a, b)
Admi ance YV is epea edly eplaced wi h ci cui
wi h he inpu admi ance (2b), i we wan o inc ease
o de o syn he ic immi ance elemen .
3. Uni e sal Cu en Con eyo
Uni e sal cu en con eyo (see Fig. 1) was c ea ed on
he base o he uni e sal elemen idea, which could easily
implemen e e y cu en con eyo gene a ion o a ia ion
in p ac ice. This elemen would allow a di e en
applica ion use o cu en con eyo s and hei wide
expansion. Uni e sal cu en con eyo has me hese
equi emen s [4].
I can ealize all he known ypes o cu en
con eyo s. I is he basic p inciple o i s e sa ili y.
Implemen a ion o he a ious ypes o cu en con eyo s
is p o ided by use o only ce ain inpu and ou pu
e minals o uni e sal cu en con eyo , he emaining
e minals a e p ope ly connec ed o g ounded.
Ano he ad an age o he elemen is ha i
con ains se e al inpu and ou pu e minals, which
p o ides an oppo uni y o sum signals a he inpu o
con eyo o o di ide he ou pu signals o mul iple
loca ions [4]. The ac i e elemen is no cu en ly mass
p oduced, and he e o e no widely a ailable.
Fig. 1: Schema ic symbol o uni e sal cu en con eyo (UCC).
Te minals Y1+, Y2-, Y3+ a e he ol age inpu s, X
is he cu en inpu (see Fig. 1). Te minals Z 1+ and Z2+
a e cu en ou pu s wi h a posi i e ans e o cu en , Z1-
and Z2- a e cu en ou pu s wi h a nega i e ans e o
cu en om e minal X.
The uni e sal cu en con eyo is gene ally
desc ibed by equa ions [4]:
321 YYYX uuuu



, ,
0
321

YYY
iii
XZZ iii


 21 , . (3a, b, c, d)
XZZ iii   21
4. Realiza ion o Syn he ic
Immi ance Elemen s wi h
Cu en Con eyo s
I he app op ia e s uc u e o syn he ic immi ance
dipoles o ype DP, DS, EP and ES ype is sea ched, hen
he pa icula o m o inpu admi ance o impedance is
p ima ily equi ed. I is e y di icul o ind he exac
exp essions ha equal o he o mulas (1a, b) espec i ely
(2a, b). We can mo e o en ind he exp essions
app oaching hese o ms han exac o ms. They can be
also used.
Howe e , i is necessa y o heo e ically e i y
hei sui abili y o use in equency il e s. The main
a ge is o ge o ms o equa ions ha will be mos
simila o he equa ions (1a, b) espec i ely (2a, b).
The gene al ci cui ne wo k o sea ching o
sui able ci cui s uc u es was used o implemen
syn he ic immi ance elemen . The gene al ci cui
ne wo k consis s o nine passi e elemen s (admi ances)
and one gene al ou -po cu en con eyo [2]. Final
© 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 90
THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 10 | NUMBER: 2 | 2012 | JUNE
s uc u e is shown in Fig. 2.
Fig. 2: De aul admi ance ne wo k o sea ch he syn he ic elemen
solu ion.
The syn he ic elemen sea ching p ocedu e is qui e
simple. The selec ed admi ances a e emo ed om he
gene al ci cui ne wo k. The esul ing ci cui s uc u e
should consis o h ee admi ances and gene al ou -po
cu en con eyo . Th ee admi ances a e enough o ealize
he syn he ic immi ance elemen . Any o he gene al
cu en con eyo e minals Y, X, Z1 o Z2 can be used as
he inpu o he esul ing ci cui s uc u e [5]. The inpu
admi ance o m is hen edi ed again by he sui able
choice o he gene al cu en con eyo coe icien s a, b,
c11, c22. I is inal change o he inpu admi ance. The
inal inpu admi ance o he esul ing ci cui is hen
deduc ed, i i equals (1a, b), (2a, b) o o ms close o
hese equa ions. Se e al dozens o combina ions we e
examined his way.
Figu e 3 shows one o he ound s uc u es ha
can be used o he ealiza ion o syn he ic elemen . The
ci cui is sui able o implemen ing il e s o highe o de .
Fig. 3: Ci cui s uc u e sui able o ealiza ion o syn he ic elemen .
The inpu admi ance gene al o m o he ci cui
shown in Fig. 3 has he o m:
22711771122
9722929222722272
1172117272727272
IN cYcYYcYY
abYYacYYabYYacYYcYY
acYYcYYabYYbYYaYYYY
Y


. (4)
The e he alues o he gene al cu en con eyo
coe icien s om he possible combina ions we e chosen.
Bes sui ed combina ion is a=-1, b=0, c11 =-1, c22=1. This
combina ion o alues simpli ies he inpu admi ance
gene al o m and b ings i o i s ideal o m. Gene al
ci cui s uc u e is sui able o implemen he syn he ic
immi ance elemen s DP o EP (1a, b). Selec ed
coe icien s co espond o he in e ing cu en con eyo
o second gene a ion ICCII+/-. A e his adjus men , he
inpu admi ance o he ci cui is ans o med in o he
o m:
2
7
92
IN 2Y
Y
YY
Y ,
)2( 792
7
IN YYY
Y
Z
. (5a, b)
Ta ge ed selec ion o passi e elemen s and hei
subs i u ion o gene al admi ance in he ci cui s uc u e
c ea e he syn he ic immi ance elemen DP o second
o de . The ci cui s uc u e is shown in Fig. 4.
Fig. 4: Ci cui s uc u e o second o de syn he ic elemen DP.
The esul ing inpu admi ance o syn he ic
immi ance elemen shown in Fig. 4 has he o m:
. (6)
2927
2
IN 2CpCCRpY 
4.1. Syn he ic Immi ance Elemen O de
Inc easing
Inc easing o he syn he ic elemen o de will be
demons a ed on he syn he ic elemen shown in Fig. 4.
Inc easing o he syn he ic elemen o de is cascade, i.e.
he e is a epea ed eplacemen o a passi e elemen o a
whole ci cui s uc u e o iden ical syn he ic elemen in
he s uc u e o syn he ic elemen . Inc easing o o de o
syn he ic elemen in Fig. 4 o hi d o de shows Fig. 5.
As can be seen om Fig. 5, he capaci o ma ked
in Fig. 4 as C9 was eplaced by ci cui s uc u e o
syn he ic elemen DP shown in Fig. 4.
Fig. 5: Thi d o de syn he ic immi ance elemen DP.
© 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 91
THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 10 | NUMBER: 2 | 2012 | JUNE
The numbe o capaci o s in he s uc u e o a
syn he ic elemen was inc eased. The esul is also
inc easing o he syn he ic elemen o de . The inpu
admi ance o he newly c ea ed syn he ic elemen o
hi d o de has o m as ollows:
. (7)
1211
2
32121
32CpCCRpCCCRRpYIN 
The new inc easing o o de o syn he ic elemen
would be made by eplacing he capaci o C3 o he
whole syn he ic elemen s uc u e shown in Fig. 4. Fo ms
o o he inpu admi ance caused by inc easing o he
syn he ic immi ance elemen o de a e shown in Tab. 1.
Tab.1: Inpu admi ance o ms.
O de o
Syn he ic
Elemen Inpu Admi ance Fo m
4. YIN = p4R1R2R3C1C2C3C4 + p32R1R2C1C2C3 +
p2R1C1C2+ p2C1
5.
YIN = p5R1R2R3R4C1C2C3C4C5 +
p42R1R2R3C1C2C3C4 +
p32R1R2C1C2C3 + p2R1C1C2 + p2C
1
6.
YIN = p6R1R2R3R4 R5C1C2C3C4C5C6 +
p52R1R2R3R4C1C2C3C4C5 +
p42R1R2R3C1C2C3C4 +
p32R1R2C1C2C3 + p2R1C1C2 + p2C1
7.
p52R1R2R3R4C1C2C3C4C5 +
p42R1R2R3C1C2C3C4 +
p32R1R2C1C2C3 + p2R1C1C2 + p2C1
YIN = p7R1R2R3R4 R5R6C1C2C3C4C5C6C7+
p62R1R2R3R4 R5C1C2C3C4C5C6 +
5. F equency Fil e s Using Syn he ic
Immi ance Elemen s
The e was used syn he ic immi ance elemen DP shown
in Fig. 5 o he ealiza ion o low-pass il e o hi d
o de . Ci cui solu ion o hi d o de low-pass is shown in
Fig. 6.
FThi d o de low-pass using syn he ic elemen wi h cu en
con eyo s.
Two uni e sal cu en con eyo s we e used in he
inal ci cui solu ion o ac i e equency il e . Wi h i s
help, ICCII+/- was implemen ed. The e was also used
passi e low-pass a he inpu o ac i e il e o smoo h he
inal equency esponses a highe equencies. Resis o
R
ig. 6:
he wi h capaci o C1. T ans e unc ion o
ac i e hi d o de low-pass shown in Fig. 6 has o m as
ollows:
1 was di ided in o wo pa s, and i c ea es passi e low-
pass oge
1442 213221
2
432321
3
CRpCCRRpCCCRRRp
KU
Speci ic alues o passi e componen s we e
calcula ed wi h gen
1
)( p. (8)
e al design ela ions o ac i e hi d
de low-pass [6]:
o
432321
3
0
33 2
cCCCRRR

, (9)
3221
2
0
32 4
cCCRR

, (10)
21
0

Bu e wo h app oxima ion was chosen. The
coe icien s o his app oxima ion o il e s o hi d o de
ha e alues c
31 4
cCR. (11)
esis o s (12), (13), (14) a e
exp essed om equa ions (9), (10), (11):
31=c32=2, c33=1 [6]. As al eady men ioned
in he p e ious ex , he alues o capaci o s a e chosen
and alues o esis o s a e calcula ed. The alues o
passi e elemen s o equency il e a e also based on he
chosen alue o cu -o equency 1 MHz and he
coe icien s o chosen app oxima ion. The o ms o
calcula ion o alues o
20
31
14
c
C
R

, (12)
3310
32
2
c2
Cc
R

, (13)
4320
3Cc

Values o esis o s a e calcula ed a e subs i u ing
o speci ic alues. Op imiza ion in PSpice was also used
o de e mine he ideal alues o passi e elemen s. The
esul ing alues o he passi e elemen s we e de e mined
R
33
c2
R. (14)
1=2,7 kΩ, R2=5,6 kΩ, R3=1,2 kΩ, C1=C2=C3=22 pF,
C4=100 pF. The esul ing ampli ude equency esponse
and phase esponse o a eal model o hi d o de low-
pass is shown in Fig. 7. The e a e wo aces he e. Fi s
ace is o ci cui solu ion o ac i e low-pass and second
ace desc ibes he beha io o ci cui solu ion o ac i e
low-pass using also passi e low-pass. As can be seen, use
o passi e low-pass in he ci cui s uc u e o hi d o de
ac i e low-pass changes he posi ion o cu -o equency.
Howe e , his solu ion also gi es good esul s a highe
equencies, whe e he gain inc ease is educed. Cu -o
equency o he il e is de ined as a dec ease o
magni ude abou 3 dB [6]. Cu -o equency in his case
was chosen 1 MHz. The simula ed cha ac e is ic shown
in Fig. 7 does no exac ly ma ch hose equi emen s. The
selec ed cu -o equency o 1 MHz is de ined as a
dec ease o ampli ude equency esponse abou 4,43 dB.
Such a la ge de ia ion can be ole a ed. Some o
© 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 92
THEORETICAL AND APPLIED ELECTRICAL ENGINEERING NUMBER: 2 | 2012 | JUNE VOLUME: 10 |
© 2012 IN ELECTRICAL AND ELECTRONIC ENGINEERING 93
ideal
ope
hange o cu -o equency o ac i e low-pass depends
cula aces depending on he a iable ole ance o
C2.
he ac o s ha could cause he de ia ion a e he
p ies o he used uni e sal cu en con eyo model
and he passi e low-pass a he inpu o ac i e il e .
Also, sensi i i y analysis was ca ying ou . The
mos ly on capaci o C2. The ampli ude equency
esponse and phase esponse shown in Fig. 8 desc ibe
pa i
c
Fig. 7: Ampli ude equency esponse and phase esponse o hi d o de low-pass using syn he ic elemen wi h cu en con eyo s.
Fig. 8: Mon e Ca lo analysis o ampli ude equency esponse and phase esponse o hi d o de low-pass wi h passi e il e .
he inal ci cui solu ion. This
y is a disad an age and
ADVANCES
6. Conclusion
The pape ies b ie ly desc ibe he ealiza ion o
equency il e s wi h syn he ic immi ance elemen s o
highe o de wi h cu en con eyo s. Applica ion use
o syn he ic immi ance elemen s b ings a lo o
ad an ages. One o he mos impo an ad an ages is
he easy in eg abili y o
is a e y impo an ac o especially in elec o-
indus ial p oduc ion.
I is ad an ageous o use uni e sal cu en
con eyo in design. UCC is able o subs i u e all
p e ious gene a ions o cu en con eyo s bu i s
comme cial una ailabili

THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 10 | NUMBER: 2 | 2012 | JUNE
un sal cu en con eyo can be used only as a
heo e ical elemen now.
The desc ibed heo y and he inal ci cui
solu ion o hi d o de low-pass indica e ha he
desc ibed me hod p oduces ela i ely good esul s. The
a icle desc ibes only one o o al h ee ound syn he ic
elemen s. Despi e u he a emp s no o he elemen s
we e ound. I can he e o e be assumed o ind ano he
solu ion o syn he ic immi ance elemen s wi h cu en
con eyo s is e y di icul . Howe e , o he solu ions o
syn he ic immi ance elem
i e
en s can be ound wi h
combina ions o di e en ypes o cu en con eyo s o
wi h o he ac i e elemen s.
SP2012/85, which was suppo ed by S uden G an
Compe i ion o VSB-Technical Uni e si y o Os a a.
u en
c ions using
Uni e sal Cu en Con eyo . In: Con e ence
3868-6.
ISSN 0925-1030. DOI: 10.1007/s10470-
onics. 2009, ol. 96, iss. 8,
iwice: IEEE,
ional
ampli ie s. P aha: BEN, 2002. ISBN 80-7300-047-4.
n s,
Acknowledgemen s
In he pape , he e a e he esul s o he p ojec
Re e ences
[1] SMITH, K. C. and A. S. SEDRA. The cu en con eyo : a
new ci cui building block. IEEE P oceedings o he CAS.
1968, ol. 56, iss.1, ISSN 0018-9219.
[2] SPONAR, R. and K. VRBA. Syn he ic dipole elemen s wi h
highe -o de immi ances in equency il e s wi h c
con eyo s. Elek o e ue [online]. 2004, ol 2004, no. 13.
ISSN 1213-1539. A ailable a :
h p://www.elek o e ue.cz/clanky/04013/index.h ml.
[3] HORNG, J.-W., Ch.-L. HOU, Ch.-M. CHANG, H. YANG
and W.-T SHYU. Highe -o de immi ance un
cu en con eyo s. Jou nal Analog In eg a ed Ci cui s and
Signal P ocessing. 2009, ol. 61, no. 2, pp. 205-209. ISSN
0925-1030. DOI: 10.1007/s10470-009-9298-6.
[4] ZHANG, Q., Ch. WANG and Y. LENG. A No el Building
Block:
P oceedings o IEEE 8 h In e na ional Con e ence on ASIC.
Changsha: IEEE, 2009, pp 1109-1112. ISBN 978-1-4244-
[5] PSYCHALINOS, C. and G. SOULIOTIS. Low-Vol age
Cu en Con olled Cu en Con eyo . Jou nal Analog
In eg a ed Ci cui s and Signal P ocessing. 2010, ol. 63, no.
1, pp. 129-135.
009-9416-5.
[6] KOTON, J., K. VRBA and HERENCSAR. Tuneable Fil e
Using Vol age Con eyo s and Cu en Ac i e Elemen s.
In e na ional Jou nal o Elec
pp. 787-794. ISSN 0020-7217.
[7] BRANDSTETTER, P. and L. KLEIN. Applica ions o Non-
In e ing Posi i e Second Gene a ion Cu en Con eyo as a
Comme cially A ailable Ve sa ile Ac i e Elemen . In:
Con e ence P oceedings o In e na ional Con e ence on
Signals and Elec onic Sys ems - ICSES´10. Gl
2010, pp. 157-160. ISBN 978-1-4244-5307-8.
[8] PUNCOCHAR, J. His o y and p esen o ope a
Abou Au ho s
Pa el BRANDSTETTER was bo n in Os a a, Czech
Republic, 1955. He ecei ed he M.Sc. and Ph.D.
deg ees in Elec ical Enginee ing om B no Uni e si y
o Technology, Czech Republic, in 1979 and 1987,
espec i ely. He is cu en ly ull p o esso in Elec ical
Machines, Appa a us and D i es and ice dean o
Facul y o Elec ical Enginee ing and Compu e
Science a VSB-Technical Uni e si y o Os a a. His
cu en esea ch in e es s a e applied elec onics,
mic ocompu e con ol sys ems and mode n con ol
me hods o elec ical d i es.
Lukas KLEIN was bo n in Uhe ske H adis e in 1984.
He ob ained Bachelo ’s deg ee a B no Uni e si y o
Technology in ield o Telein o ma ics in 2007 and
hen he ob ained his Mas e ’s deg ee in ield o Mobile
echnologies in 2009 a VSB-Technical Uni e si y o
Os a a. He is cu en ly pu suing Ph.D. s udy a
Depa men o Elec onics on Facul y o Elec ical
Enginee ing and Compu e Science. His esea ch a ea
includes applica ion o mode n ac i e eleme
especially cu en con eyo s, in elec onic ci cui s.
© 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 94