Int. J. Electron. Commun. (AEÜ) 64 (2010) 828–832

www.elsevier.de/aeue

New nullor–mirror equivalences

Hung-Yu Wanga,∗, Chih-Yi Liua, Sheng-Hsiung Changb

aDepartment of Electronic Engineering, National Kaohsiung University of Applied Sciences, 415 Chien Kung Road, Kaohsiung 807,Taiwan, ROCbDepartment of Optoelectronic Engineering, Far East University, Hsin-Shi, Taiwan, ROC

Received 27 December 2008; accepted 17 June 2009

Abstract

Several new nullor–mirror equivalences have been presented in this article. By virtue of the proposed approach, newtopologies possess the equivalent function as the initial circuit can be obtained. Moreover, the derived equivalent circuitsmight have simpler configurations than the original circuits. Some practical examples have been given to demonstrate thefeasibility of this study.� 2009 Elsevier GmbH. All rights reserved.

Keywords: Pathological element; Nullor; Nullor–mirror equivalence; Nullor equivalence; Mirror relocation

1. Introduction

The concept of nullors (nullator–norator pairs) has beenproven to be very valuable due to its capability of modelingactive elements independently of the particular implementa-tion. It provides a common framework for circuit analysis,network transformation and interrelating the realizations us-ing different active elements [1–4]. The use of nullor equiv-alences might simplify active networks and demonstrate theequivalences between different RC-active circuit configura-tions. By using the techniques of nullor equivalences andrelocation of nullators and norators, many new useful cir-cuits have been obtained [5–7]. Furthermore, the nullor isused for the realization of inverse transfer functions [8] andlow-sensitivity filters [9]. In the recent years, two new patho-logical elements, namely, the current mirror and the voltagemirror, are defined [10]. These two pathological elementsare basically used to represent active elements with current

∗ Corresponding author. Tel.: +886738145265640;fax: +88673811182.

E-mail address: hywang@cc.kuas.edu.tw (H.-Y. Wang).

1434-8411/$ - see front matter � 2009 Elsevier GmbH. All rights reserved.doi:10.1016/j.aeue.2009.06.007

or voltage reversing properties. They form a complete setanalogous to that formed by the nullator and the norator [11].The two mirrors elements are used extensively in generatingnew circuits by network transformation. To further derivenew useful circuits with the mirror elements, the continuedresearch investigates the characteristics of parallel and se-ries connections for the nullor and mirror elements. Theirusage is demonstrated by deriving simplified circuits withthe same function [12]. In this article, we present some newnullor–mirror equivalences. Their usages on some practicalexamples are given to demonstrate the presented theoreticalproperties.

2. New nullor–mirror equivalences

The symbols and definitions of nullor and mirror patho-logical elements are presented in Table 1 . Each of thevoltage mirror and current mirror symbols presented inTable 1 has a reference node which is set to ground. Themirror elements are bi-directional components and have the-oretical existences. Both the mirrors are two-port networkelements, but they can be used as two-terminal elements

H.-Y. Wang et al. / Int. J. Electron. Commun. (AEÜ) 64 (2010) 828–832 829

Table 1. Two terminal pathological elements.

Element Symbol Definition

Nullator V1 = V2, I1 = I2 = 0

Norator V1 and V2 are arbitrary, I1 = −I2 = arbitrary

Voltage mirror V1 = −V2, I1 = I2 = 0

Current mirror V1 and V2 are arbitrary, I1 = I2 = arbitrary

with the reference node unused. So the mirror elements canbe applied to the deriving of adjoint network easily [10].By nullor–mirror representation, some different currentconveyors (CCII−, CCII+, ICCII− and ICCII+) are suc-cessfully modeled, without the use of equal-value resistorsas their nullor representation [11]. From the element defini-tion in Table 1, the resemblance properties between nullorand mirror elements can be observed. Based on the ele-ment definitions and the nullor equivalences in [13], somenullor–mirror equivalences are revealed in the literature[11,12,14]. However, if we reconsider nullor equivalences in[13], we can derive some extra nullor–mirror equivalences,as presented in Table 2 . There are nine nullor–mirror equiv-alences in Table 2. Because the nullator and voltage mirrordraw no current, the derived equivalent active elements inTable 2 are only applicable in practical circuits if an exter-nal network provides for feedback from the outputs to theinputs [15].

The equivalence in Table 2(a) has been known before[16]. Other equivalences in Table 2, namely, Tables 2(b)–(i)can be derived in accordance with the terminal propertiesof pathological elements. To the authors’ knowledge, theseequivalences have not been clearly presented before.

The validity of each nullor–mirror equivalence in Table 2can be observed because of the identical electrical propertyfor the two connection circuits. In addition, each equivalentcircuit in Table 2 has the identical nodal equation represen-tation, so these equivalences exist clearly [5].

Therefore, in Table 2, the simpler configuration ofnullor–mirror representation might be helpful for the un-complicated circuit realization. Besides, due to the sameequivalence in Table 2, it provides the selectivity for the cir-cuit realization with different configurations. For instance,the equivalences in Tables 2(e) and (i) are equivalent to avoltage mirror and a norator, so either circuit realization isacceptable.

830 H.-Y. Wang et al. / Int. J. Electron. Commun. (AEÜ) 64 (2010) 828–832

Table 2. Nullor–mirror equivalences

3. Application examples

A CCII− can be represented using a nullator connected toa norator. The equivalence in Table 2(a) is well known that

the nullor can be implemented by using two CCII−s [13,16].A commercially available integrated circuit, the AD844from Analog Devices, is often used to implement a CCII+[17]. And the utilization of AD844 ICs on the realization

H.-Y. Wang et al. / Int. J. Electron. Commun. (AEÜ) 64 (2010) 828–832 831

Fig. 1. Realization of FTFN – with AD844 ICs.

Fig. 2. Simplification of CCII+s based impedance circuit us-ing nullor–mirror equivalence. (a) The original CCII+s basedimpedance circuit; (b) the obtained circuit by relocation of thenullators; (c) the impedance circuit with simpler configuration de-rived by applying the nullor–mirror equivalence in Table 2(c).

of a FTFN− (nullor) is familiar [15,18]. Its nullor–mirrorrepresentation is shown in Fig. 1. Apparently, it consistsof the equivalence in Table 2(c). Another example is aCCII+-based impedance circuit in Fig. 2(b) of [19], which is

redrawn in Fig. 2. The circuit in Fig. 2(b) can be obtainedby the relocation of the nullators [14] in Fig. 2(a). Afterapplying the nullor–mirror equivalence in Table 2(c), we canobtain the circuit in Fig. 2(c) which is composed of a CCII−and a CCII+. It is obvious that the derived circuit possessessimpler configuration than the original one. Routine analysisshows the equivalence for the circuits in Fig. 2(a) and (c)due to their identical input impedance Zi = y1/y3 y4.

4. Conclusion

In this article, several new nullor–mirror equivalenceshave been presented besides the previous reported ones[11,12,14]. Some practical applications have been given todemonstrate their usefulness. The proposed approach mighthelp to simplify the circuit topologies and interrelate therealizations using different active elements. Therefore, theversatility of pathological elements can be observed.

Acknowledgments

This work was supported by the National Science Coun-cil of the Republic of China (Grant no. NSC 97-2221-E-151-047). Technical support from the Chip ImplementationCenter is gratefully acknowledged.

References

[1] Chang SM, Wierzba GM. Circuit level decompositionof networks with nullors for symbolic analysis. IEEETransactions on Circuits and Systems I 1994;41:699–711.

[2] Celma S, Martinez PA, Sabadell J. A transformation methodfor equivalent infinite-gain op amp to unity-gain CCIInetworks. IEEE Transactions on Circuits and Systems I1996;43:61–3.

[3] Papazoglou CA, Karybakas CA. A transformation toobtain CCII-based adjoint of op.-amp.-based circuits. IEEETransactions on Circuits and Systems II 1998;45:894–8.

[4] Kumar P, Senani R. Bibliography on nullors and theirapplications in circuit analysis, synthesis and design. AnalogIntegrated Circuits and Signal Processing 2002;33:65–76.

[5] Svoboda JA. Current conveyors, operational amplifiers andnullors. Proceedings of the Institution of Electrical EngineersPart G 1989;136:317–22.

[6] Wang GH, Fukui Y, Kubota K, Watanabe K. Voltage-mode tocurrent-mode conversion by an extended dual transformation.In: Proceedings of the IEEE international symposium oncircuits and systems, 1991. p. 1833–6.

[7] Palomera-Garcia R. Generation of equivalent circuits byFTFN relocation. In: Proceedings of the internationalsymposium on circuits and systems, vol. 1, 2005. p. 252–5.

[8] Leuciuc A. Using nullors for realisation of inverse transferfunctions and characteristics. Electronic Letters 1997;33:949–51.

832 H.-Y. Wang et al. / Int. J. Electron. Commun. (AEÜ) 64 (2010) 828–832

[9] Ozoguz S, Acar C, Toker A, Gunes EO. Derivation of low-sensitivity current-mode CCII-based filters. IEE Proceedingsof Circuits, Devices and Systems 2001;148:115–20.

[10] Awad IA, Soliman AM. Inverting second generation currentconveyors: the missing building blocks, CMOS realizationsand applications. International Journal of Electronics1999;86:413–32.

[11] Awad IA, Soliman AM. On the voltage mirrors and the currentmirrors. Analog Integrated Circuits and Signal Processing2002;32:79–81.

[12] Wang HY, Lee CT, Huang CY. Characteristic investigationof new pathological elements. Analog Integrated Circuits andSignal Processing 2005;44:95–102.

[13] Bruton LT. RC active circuits: theory and design, vol. 1.Englewood Cliffs, NJ: Prentice-Hall; 1980.

[14] Wang HY, Chang SH, Jeang YL, Huang CY. Rearrangementof mirror elements. Analog Integrated Circuits and SignalProcessing 2006;49:87–90.

[15] Cam U, Toker A, Cicekoglu O, Kuntman H. Current-modehigh output impedance sinusoidal oscillator configurationemploying single FTFN. Analog Integrated Circuits andSignal Processing 2000;24:231–8.

[16] Cabeza R, Carlosena A. Analog universal active device:theory, design and applications. Analog Integrated Circuitsand Signal Processing 1997;12:153–68.

[17] Svoboda JA. Applications of a commercially available currentconveyor. International Journal of Electronics 1991;70:159–64.

[18] Shan NA, Malik MA. Voltage/current-mode universal filterusing FTFN and CFA. Analog Integrated Circuits and SignalProcessing 2005;45:197–203.

[19] Abuelma MT. Comment on active simulation of groundedinductors with CCII+ s and grounded passive elements.International Journal of Electronics 2000;87:177–81.

Hung-Yu Wang was born in Kaoh-siung, Taiwan, Republic of China, in1969. He received his Ph.D. degreefrom the Institute of Optical Sciencesof National Central University, Taiwan,in January 2002. In 1993, he workedon promoting the prototyping IC im-plementation of academic researchesand propelling the collaboration of theacademia and industries in National

Chip Implementation Center (CIC), National Applied ResearchLaboratories. Since 2006 he has joined Kaohsiung University ofApplied Sciences, as an associate professor in Electronic Engi-neering Department. His main research interests are in the area ofanalog circuits design and signal processing.

Chih-Yi Liu was born in Tainan, Tai-wan, in 1976. He received the B.S. de-gree in Physics from National TaiwanNormal University, Taipei, in 1998,and the M.S. and Ph.D. degrees inElectronic Engineering from NationalChaio-Tung University, Hsinchu, in2002 and 2005, respectively. In 2005,he joined the Center R&D of UnitedMicroelectronics Corp. (UMC), Tainan,

working in the area of 65 nm device engineering. Since 2006, hehas been an associate professor in the Department of ElectronicEngineering, National Kaohsiung University of Applied Sciences.His research interests include nonvolatile memory, semiconductorphysics, passive microwave device, and applied electronics.

Sheng-Hsiung Chang was born inKaohsiung, Taiwan, Republic of China,in 1967. He received his Ph.D. degreefrom the Institute of Optical Sciencesof National Central University, Taiwan,in August 2000. Since 2000, he hasbeen working as an associate professorin the Far East University. His mainresearch interests are in the area of theapplications of optoelectronic engineer-ing and diffractive optical element.