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Abstract This paper presents a unified symbolic analysis

of immittance inverters and revisits their realizations with

lumped and/or distributed elements. Various equivalent

realizations are symbolically analyzed with Mathematica to

generate the exact representation by means of the ABCD

parameters. Matrix representation of an immittance inverter

is used to get better insight into its operation when used as a

building block in bandpass or bandstop filter design.

Keywords immittance inverters, symbolic analysis, filter

design

I. INTRODUCTION

MMITTANCE inverters are reciprocal four-terminal two-

port linear time-invariant electrical networks that are

typically used to convert a ladder network, which uses

both series and shunt immittances, to equivalent forms that

use only series or only parallel immittances [1-6].

An idealized impedance inverter operates like an ideal

quarter-wave lossless transmission line of characteristic

impedance KZ c at all frequencies. If it is terminated in

an impedance Z on one end, the impedance seen looking

in at the other end is ZKZ 2inv .

An idealized admittance inverter operates like an ideal

quarter-wave lossless transmission line of characteristic

admittance JY c at all frequencies. If an admittance Y

is attached at one end, the admittance seen looking in at

the other end is YJY 2inv .

The general inverter parameters, also referred to as the

cascade or ABCD parameters, Fig. 1, can be derived from

the equations

)(

)(

221

221

IDVCI

IBVAV

assuming standard (associated) voltage-current reference

directions. The immitance inversion property implies

0,0 DA

The immittance inverter is a reciprocal network, so the

following holds:

11 CBCBDA

In this paper we present symbolic analysis of inverter

realizations with ideal lumped elements, or transmission

line sections, or both. In addition, we illustrate the analysis

with portions of the corresponding Mathematica code [7].

D. V. Tosic and M. Potrebic are with the School of Electrical

Engineering, University of Belgrade, Bulevar kralja Aleksandra 73,

11120 Belgrade, Serbia; (e-mail: tosic@etf.bg.ac.yu).

2

2

1

1

I

V

DC

BA

I

V

1V 2V

+ +K

(J)

1I 2I

Fig. 1. General inverter parameters also referred to as the

cascade or ABCD parameters.

II.LUMPED ELEMENT REALIZATION OF IMMITTANCE

INVERTERS

Immittance inverters can be realized with ideal lumped

elements if negative element values are allowed. Inductive

inverter realizations are shown in Fig.2 and the capacitive

realizations are presented in Fig. 3. The parameters L and

C are assumed to be positive. The realizations are

approximately frequency independent in a narrow

frequency range Z'rZ .

-L -L

L -L -L

L

Z

Z

0j

j0

L

L

Z

Z

0j

j0

L

L

Fig. 2. Inductive realizations of immittance inverters;matrices above the schematics are the ABCD matrices.

-C-C

C -C-C

C

ZZ0j

j0

CC

ZZ

0j

j0

CC

Fig. 3. Capacitive realizations of immittance inverters;

matrices above the schematics are the ABCD matrices.

Symbolic analysis of immittance inverters

Dejan V. Tosic and Milka Potrebic

I

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III. DISTRIBUTED REALIZATION OF INVERTERS

Ideal quarter-wave lossless transmission line, Fig. 4, is a

distributed element realization of the immittance inverter.

It approximates the desired frequency behavior within a

narrow range of frequencies around the frequency at

which the electrical length of the line is 2S .

JY cKZ c

2S4 2S4

01

j

j0

c

c

Z

Z

0j

1j0

c

c

Y

Y

Fig. 4. Distributed realizations of immittance inverters;

matrices above the schematics are the ABCD matrices.

IV. LUMPED-DISTRIBUTED REALIZATION OF IMMITTANCE

INVERTERS

Networks with ideal lumped elements and ideal lossless

transmission line sections of negative electrical length can

be used to realize immittance inverters, Figs. 5 and 6. The

networks approximate the desired frequency behavior

within a narrow range of frequencies around the frequency

at which the electrical length of the line is 4 .

LZc Zc

4 4

Z

4

2

)2tan(cZL )tan(c 4ZK

4

4

0)tan(

j

)tan(j0

c

c

Z

Z

Fig. 5. Lumped-distributed realization of the impedance

inverter; matrix above the schematic is the ABCD matrix.

V.SYMBOLIC VERIFICATION OF INVERTER REALIZATIONS

Symbolic algebra system (CAS), such as Mathematica,

can be efficiently used to verify different realizations of

the immittance inverters. Here is a set of functions that

compute the ABCD matrix of shunt admittance, series

impedance, and transmission line section.

ParallelAdmittanceABCD[Y_] := {{1, 0}, {Y, 1}};

SeriesImpedanceABCD[Z_] := {{1, Z}, {0, 1}};

LineABCD[Z_, _] := {{Cos[], I Z Sin[]},{I Sin[]/Z, Cos[]}}

C

Yc Yc

4 4

Z

4

2

)2tan(cYC )tan(c 4YJ

44

0)tan(j

)tan(

j0

c

cY

Y

Fig. 6. Lumped-distributed realization of the admittance

inverter; matrix above the schematic is the ABCD matrix.

First, let us verify the capacitive 3-network from Fig. 3.We can represent the network as a cascade of three

simpler networks, shunt capacitor, series capacitor, shunt

capacitor, and multiply the corresponding ABCD matricesto compute the overall matrix.

PiC = Fold[Dot, IdentityMatrix[2],{ ParallelAdmittanceABCD[I (-C)],

SeriesImpedanceABCD[1/(I C)],ParallelAdmittanceABCD[I (-C)] }

];

TraditionalForm[PiC]

L

N

MMM0

CZ

CZ 0

\

^

]]]

Similarly, the realization of Fig. 6 can be derived as

follows:

ThetaC = Fold[Dot, IdentityMatrix[2],{ LineABCD[1/Yc, -],

SeriesImpedanceABCD[1/(I C)],LineABCD[1/Yc, -] }

] // Simplify

Cos#2 4' YcCos#4' Sin#4'

C Z,

Cos#4' +YcCos#4' 2 C Z Sin#4'/

C Yc Z,

YcSin#4' +2 C Z Cos#4' YcSin#4'/C Z

,

Cos#2 4' YcCos#4' Sin#4'

C Z

Diagonal elements of the ABCD matrix should be zero,

so we solve the equation

CRule = Solve[ThetaC[[1, 1]] == 0, C]// First // Simplify

C Yc Tan#2 4'2 Z

The overall ABCD matrix of the realization is

ThetaCJ = ThetaC /. CRule // Simplify;TraditionalForm[ThetaCJ]

L

N

MMMM

0 cot+4/

Yc Yc tan+4/ 0

\

^

]]]]

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VI. IMMITTANCE INVERSION APPLICATION

Impedance inverters, for a real positive inversion factor

K, are described by a general ABCD matrix of the form

r

r

0j

j0

K

K

KABCD

Similarly, for real positive J, admittance inverters arecharacterized by

r

r

0j

j0

JJJABCD

Consider a series impedance and a shunt (parallel)

admittance as shown in Fig. 7.

Y Z

1

01

YYABCD

10

1 ZZABCD

Fig. 7. Series impedance and shunt admittance networks.

Immittance inverters can be used to transform a series

impedance into a shunt admittance, Fig. 8, and vice versa,

Fig. 9.

ZJY 2

ZJ J

1:1

Fig. 8. Transformation of series impedance.

YKZ 2

Y

K K

1:1

Fig. 9. Transformation of shunt admittance.

The ideal resistive transformer with unit turn ratio andphase inversion, in Figs. 8 and 9, is required, which can be

verified by symbolic analysis. For example, for the

transformation of Fig. 8, the overall ABCD matrix of the

cascade inverter-impedance-inverter is as follows

JZJ = Fold[Dot, IdentityMatrix[2],{ {{0, I/J}, {I*J, 0}},

SeriesImpedanceABCD[Z],{{0, I/J}, {I*J, 0}} }

];

TraditionalForm[JZJ]

L

NMM1 0

J2Z 1

\

^]]

which is of the form

YJZJ ABCDABCD

10

01

where ZJY 2 .

The ABCD matrix of the ideal resistive (frequency

independent) transformer with a turn ratio ofn and phase

inversion is

n

n

n 10

0ABCD

so we obtain

1

nYnJZJABCDABCDABCD

The ideal resistive transformer is a network, but it can

be approximately realized with an ideal lossless

transmission line section of the electrical length S4 ,

Fig 10.

4,cZ

44

44

4 )cos()sin(j

)sin(j)cos(

c

c

Y

Z

ABCD

Fig. 10. Ideal lossless transmission line section.

S4

10

01ABCD ,

cc

1

ZY

The transmission line section approximates the desired

behavior within a narrow range of frequencies around the

frequency at which the electrical length of the line is S.At microwave frequencies, the ideal resistive

transformer cannot be implemented, so the transmission

line section approximation is a practical issue. Similar

reasoning applies for implementation of practical

inverters.

Many excellent books [1-4] that develop theory and

applications of immittance inverters do not clarify the

inversion mechanism detailed in this section. Even good

textbooks [5,6] drop out the ideal resistive transformer

when explaining the operation of the inverters.

Typical application of immittance inverters is to

transform a ladder LC bandpass filter realization of Fig.

10 to realizations with only series resonators, Fig. 12, or

only shunt resonators, Fig. 13.

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Vg

Rg

RloadL1

L2 C2

C1L3 C3

L4 C4

. . .

Fig. 11. Ladder LC realization of a bandpass filter.

Vg

R0g

R0,load

L01 C01 K1L02 C02

. . .

K2Kn+1

Fig. 12. Impedance inverter realization of the bandpass filter from Fig. 10.

Vg

R0g

R0,loadL01 C01

J1

L02 C02. . .

J2Jn+1

Fig. 13. Admittance inverter realization of the bandpass filter from Fig. 10.

VII. CONCLUSION

We revisited definition, operation, and realization of

immittance inverters. By using Mathematica as a

computer algebra system we symbolically analyzed the

inversion mechanism and presented exact equivalentnetworks for transformation of series impedance to shunt

admittance and vice versa. The equivalent networks are

important for understanding practical implementations of

bandpass and bandstop filters, particularly their phase

response, when starting from the realizations that use

immittance inverters.

ACKNOWLEDGMENT

We thank Ministry of Science and Environmental

Protection of the Republic of Serbia for partial support of

our research on this topic (Project TR6154).

REFERENCES

[1] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters,

Impedance-Matching Networks, and Coupling Structures, McGraw-

Hill, New York, 1964. Reprinted by Artech House, Norwood, MA,

1980.

[2] J. A. G. Malherbe, Microwave Transmission Line Filters, ArtechHouse, Dedham, Massachusetts, 1979.

[3] R. W. Rhea, HF Filter Design and Computer Simulation, Noble

Publishing, Norcross, GA, USA, 1994.

[4] Jia-Sheng Hong, M. J. Lancaster, Microstrip Filters for

RF/Microwave Applications, John Wiley, New York, 2001.

[5] R. E. Collin, Foundations For Microwave Engineering, 2nd Ed.,

McGraw-Hill, New York, 1992.

[6] D. M. Pozar, Microwave Engineering, 3rd Ed., John Wiley, New

York, 2005.

[7] S. Wolfram, The Mathematica Book, Cambridge University Press,

Cambridge, 2003.

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