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Active simulation ofgrounded inductors withCCII+s and grounded passiveelementsMEHMET OGUZHAN CICEKOGLUPublished online: 09 Nov 2010.

To cite this article: MEHMET OGUZHAN CICEKOGLU (1998) Active simulation ofgrounded inductors with CCII+s and grounded passive elements, InternationalJournal of Electronics, 85:4, 455-462, DOI: 10.1080/002072198134003

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INT. J. ELECTRONICS, 1998, VOL. 85, NO. 4, 455± 462

Active simulation of grounded inductors with CCII+s and grounded

passive elements

MEHMET OGÏ UZHAN CË ÇICË EKOGÏ LU²

Grounded inductor simulation using op-amp or current conveyors is well known.Most of the presented topologies in the literature require certain element matchingconstraints, ¯ oating passive elements, or they employ active and passivecomponents more than necessary to implement a speci® c type of groundedinductor. In this study, four new topologies, all separately able to realize eightmain types of inductors employing only grounded passive components, arepresented. For the topologies, ideal and non-ideal impedance functions are givenand an application example is shown. Some experimental results are included toverify theory.

1. Introduction

Grounded inductor simulation using operational ampli® ers or current conveyorsas active elements is well known. The current conveyor as an active element o� ersseveral advantages, like greater linearity and wider bandwidth, over the voltagemode counterparts, op-amps. Actively simulated inductors ® nd application inareas like oscillator design, active ® lters and cancellation of parasitic inductances.Recent commercial availability of CCII+ type current conveyors has promoted agrowing interest in designs using them.

From the ¯ oating or grounded synthetic inductor topologies (Pal 1981a, 1981b,Singh 1981, Senani 1982, Higashimura and Fukui 1987, Paul and Patranabis 1981),some employ only grounded capacitors and resistors advantageous from the inte-grated circuit implementation point of view, but most su� er from an excessive num-ber of active elements or ¯ oating passive components. The circuits presented in Pal(1981a, 1981b), Singh (1981) and Senani (1982) employ four current conveyors andin Pal (1981a, 1981b) and Singh (1981) they are positive and negative types. Only thepositive-type current conveyor has become recently commercially available in ICform, the negative-type current conveyor may be implemented using two CCII+s.

A recent publication has presented a circuit suitable for inductor simulation thatemploys two third-generation current conveyor CCIIIs (Liu and Yang 1996). Thepresented grounded inductor simulating topology in Paul and Patranabis (1981)employs only a single current conveyor, six resistors and one capacitor to obtainsix types of inductors: that is, ideal L , L with a series positive resistance, L with aseries negative resistance, L with a parallel positive resistance, L with a parallelnegative resistance and the bilinear form. For the bilinear form, certain conditionsare to be satis® ed for the topology to be inductive. Furthermore, for each type ofinductor at least one condition and/or cancellation constraint is to be satis® ed torealize the inductor. Moreover, for each type of inductor all ® ve passive elements areused. The cancellation constraints require passive component matching, which is

0020± 7217/98 $12.00 Ñ 1998 Taylor & Francis Ltd.

Received 18 July 1997; accepted 5 December 1997.² BogÏ azicË i University, M.Y.O. Electronics Prog., 80815 Bebek-Istanbul, Turkey. e-mail:

cicekogl@boun.edu.tr.

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di� cult from the practical point of view. Similarly, the topologies presented inCË icË ekogÏ lu and Kuntman (1997) enjoy the advantage of a single current conveyoras an active element but su� er from a larger number and ¯ oating passive elementsand component matching constraint for pure L realization.

In this paper four di� erent inductor simulating topologies are presented whichare not derived from a bilinear form under certain matching conditions, on thecontrary they directly simulate the grounded inductors. The proposed topologiesrequire few passive components and no element matching conditions are necessary,including pure L realization. The forms are unconditionally inductive since they arenot derived from the bilinear form under certain constraints. For the topologies Land series/parallel conductance, G is independently adjustable, depending on thetype, to positive and negative values. The impedance functions for the non-idealcases are also given. All passive sensitivities are found to be no more than unity.All four circuits implement the same impedance function ideally but they representdi� erent physical circuits, therefore di� erent physical behaviour is expected forthem. These di� erences are observable when some non-idealities, like current orvoltage tracking errors, or the ® nite output resistance at the z output of the currentconveyor, are taken into account. Corresponding equations for these non-idealitiesare included. A suitable topology may be selected depending on application. Finally,an application example is included which gives a CCII+ based oscillator with allgrounded passive elements and quadrature property where CCII-based(Abuelma’atti et al. 1995) and CCI-based (Abuelma’atti and Al-Ghumaiz 1996)topologies are well known.

2. The proposed circuit topolog ies

An ideal second generation current conveyor of positive type CCII+ shown in® gure 1 is characterized by the following equations

iy (t) = 0, vx (t) = vy (t), iz (t) = ix (t) (1)

and if CC non-idealities are taken into account

iy (t) = 0

vx (t) = (1 + e v)vy (t)

iz (t) = (1 + e i) ix (t)

(2)

where |e i| ! 1 and |e v| ! 1 represent the current and voltage tracking errors of thecurrent conveyor.

The proposed inductance simulators are shown in ® gure 1. The input impedanceof the circuits is given as

Zi =vi

ii=

y1 - y2

y3y4(3)

If we choose y1 = sC1, y2 = 0, y3 = G3 and y4 = G4 then the input impedancebecomes

Zi =sC1

G3G4= sL eq (4)

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Active simulation of grounded inductors with CCII+s 457

Figure 1. Inductance simulating topologies (a) circuit 1; (b) circuit 2; (c) circuit 3; (d) circuit 4.

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where L eq º C1 /(G3G4) . On the other hand, an inductor with a positive or negativeseries resistance can be implemented by choosing y1 = sC1 + G1, y2 = G2, y3 = G3

and y4 = G4. In this case L eq = C1 /(G3G4) and Req = (G1 - G2) /(G3G4) . Similarly,an inductance with a parallel resistance can be obtained by choosing y1 = sC1,y2 = 0, y3 = sC3 + G3 and y4 = G4. For this case L eq = C1 /(G3G4) andReq = C1 /(C3G4) . The realizable forms are given in table 1.

The sensitivities of L eq and Req to passive components are given by SL eq

C1,C2=

SReq

G1,G2,C1,C2= 1 and S

L eq

G3,G4= S

Req

G3,G4,C3= - 1 thus all passive sensitivities are no

more than unity.For some circuits, easy derivation leading to (3) is possible simply by visual

inspection of the ® nal topology and combining well-known circuits. Consider thecircuit in ® gure 1(c) for example: starting from the Sedra± Smith gyrator (Sedra andSmith 1970) which employs two CCIIs of opposite polarity, and using threegrounded admittances y3, y4 and y0 gives Zi = y0 /(y3y4) . If one uses both CCII+showever one will obtain Zi = - y0 /(y3y4) . Now, if y0 is realized as a sum of twoadmittances y2 and - y1 and for - y1 using an NIC realizable with a single CCII+without any passive components, we obtain Zi = (y1 - y2) /(y3y4) .

The features of the presented circuits can be summarized as follows:

(a) All passive elements are grounded, which permits integrability and easy digitalcontrollability of the realized inductance and resistance values by weightedelements.

458 M. O. CË icË ekogÏ lu

Type y1 y2 y3 y4 Zi L eq Req

1 pure +L sC1 0 G3 G4sC1

G3G4

C1

G3G40

2 +L with series +R sC1 + G1 0 G3 G4sC1 + G1

G3G4

C1

G3G4

G1

G3G4

3 +L with series - R sC1 G2 G3 G4sC1 - G2

G3G4

C1

G3G4

- G2

G3G4

4 +L with parallel + R sC1 0 sC3 + G3 G4sC1

sC3G4 + G3G4

C1

G3G4

C1

C3G4

5 +L with parallel - R ± ± ± ± ± ± ±

6 pure - L 0 sC2 G3 G4- sC2

G3G4

- C2

G3G40

7 - L with series +R 0 sC2 + G2 G3 G4- sC2 - G2

G3G4

- C2

G3G4

- G2

G3G4

8 - L with series - R G1 sC2 G3 G4G1 - sC2

G3G4

- C2

G3G4

- G1

G3G4

9 - L with parallel + R ± ± ± ± ± ± ±

10 - L with parallel - R 0 sC2 sC3 + G3 G4- sC2

sC3G4 + G3G4

C2

G3G4

C2

C3G4

Table 1. The actively realizable inductance forms.

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(b) Independently adjustable L and G for all forms (both for negative and posi-tive values).

(c) The impedance functions are not derived form the bilinear form under certainmatching conditions therefore all circuits are unconditionally inductive forall positive values of passive components that is,

" G ³ 0 ^ " C ³ 0 Þ Im[Zi]= x L

where L is a real number.(d) No element matching conditions are imposed.(e) The topologies, except type 4 and 10, are canonic and they employ a single

capacitance.( f ) Large inductors can easily be simulated without the need for large capacitors.(g) By the appropriate choice of y1 . . . y4 the circuits can be used to simulate large

values of positive and negative capacitances using a small capacitance, wherethe enlargement factor depends on a resistance ratio.

(h) All passive sensitivities are no more than unity.(i) Only plus-type current conveyors are used which are commercially available.

More complicated forms are also possible. For example, choosing y1 = sC1 + G1,y2 = sC2 + G2, y3 = sC3 + G3 and y4 = G4, equation (3) becomes

Zi =vi

ii=

s(C1 - C2) + (G1 - G2)sC3G4 + G3G4

(5)

Equation (5) is in a bilinear form as

Zi =vi

ii=

as + bcs + d

(6)

and the coe� cients a . . . d can appropriately be de® ned comparing (5) and (6). Using(5), the types 2± 3 or 8± 9 can be realized together. Special care has to be takenhowever since, in this case, the passive sensitivity factor deteriorates. For the bilinearform given with (6) all coe� cients are orthogonally adjustable.

2.1. Non-ideal e� ectsTaking non-idealities as given in (2), equation (3), representing the ideal input

impedance of the circuits, becomes:

for circuit 1

Zi =y1 - y2 (1 + e i2 + e v2 + e i2 e v2 + e v3 + e i2 e v3 + e v2 e v3 + e i2 e v2 e v3)

(1 + e i1) (1 + e i2) (1 + e i3) (1 + e v1) (1 + e v3)y3y4(7)

for circuit 2

Zi =y1 - y2 (1 + e i2 + e v2 + e i2 e v2)

(1 + e i1) (1 + e i2) (1 + e i3) (1 + e v1) (1 + e v3)y3y4(8)

for circuit 3

Zi = (1 + e i2)y1 - y2 (1 + e i3 + e v3 + e i3 e v3)

(1 + e i1)2 (1 + e i3) (1 + e v1) (1 + e v2) (1 + e v3)y3y4

(9)

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for circuit 4

Zi =y1 (1 + e i3 + e v2 + e i3 e v2 + e v3 + e i3 e v3 + e v2 e v3 + e i3 e v2 e v3) - y2

(1 + e i1) (1 + e i2) (1 + e v1) (1 + e v2)y3y4(10)

thus, the non-ideal impedance functions of the circuits are di� erent. Equations (7) ±(10) can be used with table 1 to determine the realized equivalent inductance andresistance values for the non-ideal case.

3. Experimental veri® cation, discussion and an application example

The circuit in ® gure 1(a) (circuit 1) is experimentally tested to verify the theory.The test circuit was constructed with AD844/AD. The circuit was supplied withsymmetrical voltages of 6 12 V. For circuit 1, pure L is realized with C1 = 100 nF,R2 = ¥ , R3 = 1 kV , R4 = 1 kV thus L eq = 100mH. DC measurements show anReq = 3.9 V instead of a 0 V ideal resistance. It is possible to shift this value tozero with ® nite R2. To test the performance of the circuit 1, a 10Vp-p triangularvoltage waveform is applied through a 10 kV resistor (to represent a triangularwaveform current source of 1 mAp-p with 10 kV inner resistance) to the input ofcircuit 1. Next R2 = 10kV is added to the circuit and ® nally R1 = 10 kV is added,removing R2. These three cases represent pure L , L with a series - R, L with a series+R. The voltage waveforms observed on the oscilloscope are shown in ® gures 2(a± c)respectively. The measured values show that the circuit performs the inductancesimulation well. To illustrate an application of circuit 1 an oscillator is formedconnecting a 100 nF capacitor between the input of the circuit and ground, to imple-ment an LC circuit. The resistor R2 generates a negative damping coe� cient and theamplitude of the growing oscillations are limited by the nonlinearity of the activeelement. The element values are as follows: C1 = 1 m F, R2 = 5kV , R3 = 1kV ,R4 = 1 kV , thus L eq º 1 H and f = 503 Hz. The oscillator exhibits quadrature prop-erty and the resulting waveforms on C1 (larger amplitude signal) and R4 are shownin ® gure 3. The experimental results conform the theory well.

An interesting feature of the presented circuits is that they all realize the sameimpedance function although they are topologically di� erent. The passive sensitiv-ities are equal since they are derived from the same impedance function.

Considering table 1, the condition for some conductances to be equal to zero is avery strict condition, which is impossible to realize in practice. For example, for alltopologies to the node were y2 is connected, z terminals of one or more currentconveyors are also connected. Thus, if G2 = 0 then the current conveyor outputconductances have to be taken into account to replace y2. Therefore, the suggestionof the possibility of pure L realization (type 1) without an element matching con-dition needs further discussion. To obtain sC1 /(G3G4) from sC1 /(G3G4)+(G1 - G2) /(G3G4) derived from (3), the condition is G1 = G2 or G1 = G2 = 0; thatis, the absence of a conductance. The topologies, ® gure 1(a± d), therefore do notshow equivalent characteristics to implement a pure inductor. If more accuratevalues are desired, input impedance functions have to be derived for circuits 1± 4directly from ® gures 1(a± d) considering the output conductances of the active ele-ments. Similarly, the response of the circuits to current or voltage tracking errors isdi� erent, as given in (7) ± (10). These however represent minor e� ects for most appli-cations.

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Active simulation of grounded inductors with CCII+s 461

(a)

(b)

(c)Figure 2. The applied triangular voltage waveforms and the voltage signal on the input of

the inductance simulator. (1 cm º 0.5 ms; 0.2 V) (a) pure inductance; (b) inductancewith a series positive resistance; (c) inductance with a series negative resistance.

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4. Conclusions

In this paper four di� erent inductor active simulating topologies are proposed.The presented topologies employ all grounded passive components attractive fromthe integrated circuit implementation point of view. They employ the same type ofcurrent conveyor, CCII+, which has recently become commercially available. Theydo not require passive component matching to obtain the desired type of inductance.Four di� erent topologies are presented and all can realize eight di� erent types ofinductors. The circuits allow orthogonal control of the simulated inductance andresistance values. An application example is given to illustrate the practical use of thetopologies. Experimental results are included to verify theory.

References

Abuelma ’atti, M. T., and Al-Ghumaiz , A. A., 1996, Novel CCI-based single element con-trolled oscillators employing grounded resistors and capacitors. IEEE Transactions onCircuits and Systems I. Fundamental Theory and Applications, 43, 153± 156.

Abuelma ’atti, M. T., Al-Ghumaiz , A. A., and Khan, M. H., 1995, Novel CCII-basedsingle-element controlled oscillators employing grounded resistors and capacitors.International Journal of Electronics, 78, 1107± 1112.

CË ÇICË EKOGÏ LU, O., and Kuntman, H., 1997, Single CCII+ based simulation of grounded induc-tors. European Conference on Circuit Theory and Design, ECCTD 97, pp. 105± 109, 30August± 2 September, Budapest, Hungary.

Higashimura, M., and Fukui, Y., 1987, Novel method for realizing lossless ¯ oating immit-tance using current conveyors. Electronics Letters, 23, 498± 499.

Liu, S. I., and Yang, Y. Y., 1996, Higher-order immittance function synthesis using CCIIIs.Electronics Letters, 32, 2295± 2296.

Pal, K., 1981a, Novel ¯ oating inductance using current conveyors. Electronics Letters, 17,638; 1981b, New inductance and capacitor ¯ oating schemes using current conveyors.Electronics Letters, 17, 807± 808.

Paul, A. N., and Patranabis, D., 1981, Active simulation of grounded inductors using asingle current conveyor. IEEE Transactions on Circuits and Systems, 28, 164± 165.

Sedra, A., and Smith, K. C., 1970, A second-generation current conveyor and its applica-tions. IEEE Transactions on Circuit Theory, 17, 132± 134.

Senani, R., 1982, Novel lossless synthetic ¯ oating inductor employing a grounded capacitor.Electronics Letters, 18, 413± 414.

Singh,V., 1981, Active RC single-resistance-controlled lossless ¯ oating inductance simulationusing single grounded capacitor. Electronics Letters, 17, 920± 921.

Figure 3. Quadrature oscillator voltage waveforms on C1 (larger amplitude signal) and onR4 (1 cm º 1 ms; 2 V).

462 Active simulation of grounded inductors with CCII+sD

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