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Active and Passive Electronic Components
Volume 2011 (2011), Article ID 131546, 13 pages
http://dx.doi.org/10.1155/2011/131546
Research Article

Synthesis of Oscillators Using Limit Variables and NAM Expansion

Electronics and Communication Engineering Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt

Received 12 January 2011; Accepted 7 March 2011

Academic Editor: Jiun Wei Horng

Copyright © 2011 Ahmed M. Soliman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A systematic synthesis procedure for generating second-order grounded passive element canonic oscillators is given. The synthesis procedure is based on using nodal admittance matrix (NAM) expansion with the bracket method as well as using the infinity parameters. The resulting derived oscillators include circuits using various types of current conveyors. Two classes of oscillators are considered in this paper, and they have the advantages of having independent control on the condition of oscillation and on the frequency of oscillation by varying two different grounded resistors. The two classes of oscillators considered can be easily compensated for the parasitic element effects introduced by the current conveyors. This paper is considered to be continuation to the recently published paper on oscillators using NAM expansion D. G. Haigh et al. (2006). This is the first paper in the literature which uses limit-variables called infinity-variables D. G. Haigh et al. (2005) in the synthesis of oscillator circuits. Simulation results demonstrating the practicality of some of the generated circuits are included.

1. Introduction

The symbolic framework for systematic synthesis of linear active circuits based on NAM expansion was introduced and presented in [14]. The NAM expansion was limited to the use of nullators and norators as the two pathological elements [5].

For the nullator shown in Figure 1(a), 𝑉=𝐼=0. The norator shown in Figure 1(b) imposes no constraints on its voltage and current. Additional pathological elements called mirror elements were introduced in [68] to describe the voltage and current reversing actions.

fig1
Figure 1: The pathological elements: (a) nullator, (b) norator, (c) voltage mirror, (d) current mirror.

The voltage mirror (VM) shown in Figure 1(c) is a lossless two-port circuit element used to represent an ideal voltage reversing action, and it is described by𝑉1=𝑉2,𝐼(1a)1=𝐼2=0.(1b) The current mirror (CM) shown in Figure 1(d) is a two-port circuit element used to represent an ideal current reversing action, and it is described by:𝑉1and𝑉2𝐼arearbitrary,(1c)1=𝐼2,andtheyarealsoarbitrary.(1d)Recently, the systematic synthesis method based on NAM expansion using nullor elements has been extended to accommodate mirror elements. This results in a generalized framework encompassing all pathological elements for ideal description of active elements [911]. Accordingly, more alternative realizations are possible and a wide range of active devices can be used in the synthesis.

In this paper, the conventional systematic synthesis framework using NAM expansion is used to synthesize grounded passive element oscillator circuits. The active building blocks that are considered are the current conveyors (CCII) [12], inverting current conveyors (ICCII) [6], balanced output CCII (BOCCII), double output CCII (DOCCII), balanced output ICCII (BOICCII), double output ICCII (DOICCII), and the differential voltage current conveyor (DVCC) [13] also known as the differential difference current conveyor (DDCC) [14].

2. Formulation of the NAM Equation

The oscillators considered in this paper are grounded resistors and grounded capacitors second-order canonic (using two capacitors) oscillators having independent control on the condition of oscillation and on the frequency of oscillation by varying two different resistors.

The state equations are described by the following matrix equation:𝑑𝑣1𝑑𝑡𝑑𝑣2=𝑎𝑑𝑡11𝑎12𝑎21𝑎22𝑣1𝑣2.(2) The condition of oscillation and the radian frequency of oscillation are given by [15]𝑎11+𝑎22𝜔=0,o=𝑎11𝑎22𝑎12𝑎21.(3) If both 𝑎11 and 𝑎22 are zero, there will be no control on the condition of oscillation. For simplicity, it is assumed that either 𝑎22 (or 𝑎11) is zero so that the radian frequency of oscillation is controlled only by 𝑎12 and 𝑎21, and they must have opposite signs. In this case, the condition of oscillation is 𝑎11=0 (or 𝑎22=0).

Two classes of oscillator circuits are considered in this paper. The class I oscillator is a five-node oscillator using four resistors one of them shares a node with one of the capacitors as shown in Figure 2(a). The class II oscillator is a four-node oscillator using three grounded resistors, one of them shares a node with one of the capacitors as shown in Figure 2(b).

fig2
Figure 2: (a) Class-I four grounded resistor five node oscillator. (b) Class-II three grounded resistor four node oscillator.

3. Class I Oscillators

The generalized class I oscillator configuration can be described by the following state equation:𝐶1𝑑𝑣1𝐶𝑑𝑡2𝑑𝑣2=𝐺𝑑𝑡4𝐺1𝐺3±𝐺20𝑣1𝑣2.(4) The admittance matrix 𝑌 of the two-port oscillator circuit taking the capacitors 𝐶1 and 𝐶2 as external elements at ports 1 and 2 is formulated from the above equation by interchanging the signs of the admittance parameters. There are two types that belong to class I oscillators.

The type-A for which the NAM is given by 𝐺𝑌=1𝐺4𝐺3𝐺20,(5a) and the NAM for type-B is given by 𝐺𝑌=1𝐺4𝐺3𝐺20.(5b)Table 1 includes the admittance matrix 𝑌 of the two types of class I. To limit the paper length only type A is considered in this paper. The above NAM equations can be expanded in several alternative ways resulting in different oscillator circuits as described next.

tab1
Table 1: A summary of the parameters of the two classes of oscillators.

It should be noted that after the synthesis procedure is completed the pathological elements are paired to realize the proper CCII- [12] or ICCII- [6] based oscillator circuit as follows:

The nullator, and norator with a common terminal realizes a CCII−, the nullator and CM with a common terminal realizes a CCII+, the VM, and norator with a common terminal realizes an ICCII−, the VM, and CM with a common terminal realizes an ICCII+.

3.1. Realization I

The expansion of the matrix 𝑌 is demonstrated in several steps as follows. Starting from (5a), adding two blank rows and columns, and connecting a nullator between node 2 and node 4 to move 𝐺3 to position 1, 4 as follows: 6(6) Next a norator from node 1 to node 4 is introduced to move 𝐺3 to the diagonal position 4, 4 as follows:7(7) A second nullator from node 1 to node 3 is introduced to move 𝐺2 to positions 2, 3 as follows:8(8) Next, a CM is connected between node 2 and node 3 to move 𝐺2 to become 𝐺2 at the diagonal positions 3, 3 as follows: 9(9) Adding an infinity term to the position 1, 1 and subtracting an equivalent term as demonstrated in [1], it follows that10(10) Adding a fifth blank row and column and applying pivotal expansion [2] to the fourth term in the 1, 1 position it follows that 11(11) The infinity parameters added will move 𝐺4 to become 𝐺4 at the diagonal positions 5, 5 as follows:12(12) Figure 3(a) represents the pathological element realization of the above equation after connecting the two capacitors at nodes 1 and 2.

fig3
Figure 3: (a) Realization I of Class-I oscillator. (b) Realization II of Class-I oscillator, (c) Realization III of Class-I oscillator.
3.2. Realization II

A new circuit that can be generated from (5a) by alternative matrix expansion is given next.

Starting from the 𝑌 matrix given by (7), A VM is connected between nodes 1 and 3 to move 𝐺2 to become 𝐺2 at the position 2, 3 as follows:13(13) A norator is added next between nodes 2 and 3 to move 𝐺2 to the diagonal position 3, 3 as follows: 14(14) Adding a fifth blank row and column, adding infinity parameter element to the 1, 1 position, subtracting an equivalent term, and apply pivotal expansion in alternative way from (11), it follows that15(15) The infinity parameters added will move 𝐺4 to become 𝐺4 at the diagonal positions 5, 5 as follows:16(16) Figure 3(b) represents the realization of (16) after connecting the two capacitors at nodes 1 and 2, which is realizable using two ICCII− and one CCII−.

3.3. Realization III

Another new circuit that can be generated from (5a) by alternative matrix expansion is given next.

Starting from the 𝑌 matrix given by (7) and by successive NAM expansion steps to move 𝐺2 and 𝐺3 to the diagonal positions 3, 3 and 4, 4, respectively, then using infinity parameters to move 𝐺4 to the diagonal position 5, 5, the following NAM is obtained:17(17) Figure 3(c) represents the realization of the above equation after connecting the two capacitors at nodes 1 and 2. The circuit is realizable using one ICCII−, one CCII+ and one CCII−.

Figure 4(a) represents a two CCII+ one CCII− circuit realizing Figure 3(a) [16]. Figure 4(b) represents a two ICCII− one CCII− circuit realizing Figure 3(b). This circuit represents a new oscillator with flotation property, that is 𝐼𝐺=0.

fig4
Figure 4: (a) Realization of Figure 3(a) using two CCII+ and one CCII− [16]. (b) Realization of Figure 3(b) using two ICCII− and one CCII−.

There are a total of sixteen circuits having the same circuit topology as that of Figure 4(a). Eight circuits belong to type-A, and are generated from (5a) and the other eight circuits belong to type-B and are generated from (5b).

The topology of the class I oscillators generated in this paper has the advantage that the parasitic resistance 𝑅𝑋1 can be easily compensated by subtracting its value from the design value of 𝑅2, the parasitic resistance 𝑅𝑋2 can be easily compensated by subtracting its value from the design value of 𝑅3, and the parasitic resistance 𝑅𝑋3 can be easily compensated by subtracting its value from the design value of 𝑅4. Similarly, the parasitic capacitance 𝐶𝑍1 can be easily compensated by subtracting its value from the design value of 𝐶2. Similarly, the parasitic capacitances (𝐶𝑍2+𝐶𝑍3) can be easily compensated by subtracting their sum value from the design value of 𝐶1.

4. Adjoint of Class I Type-A Oscillators

The two classes of oscillators considered in this paper can lead to additional oscillator families based on the adjoint transformation [18, 19].

The NAM equations of the two adjoint classes are given in Table 1, and they are the transposition of the original NAM equations.

To limit paper length only realization I of class I type-A defined as class I-Ad where Ad stands for adjoint will be considered.

The NAM equation for the class I-Ad oscillator is given by𝐺𝑌=1𝐺4𝐺2𝐺30.(18) First two blank rows and columns are added, and a nullator is connected between nodes 1 and 4 in order to move 𝐺3 to position 2, 4 as follows: 19(19) Next, a norator is connected between nodes 2 and 4 to move 𝐺3 to the diagonal position 4, 4 as follows:20(20) Next, a VM is connected between nodes 2 and 3 to move 𝐺2 to become 𝐺2 at the position 1, 3 as follows:21(21) Next, a norator is connected between node 1 and node 3 to move 𝐺2 to the diagonal position 3, 3 as follows: 22(22) Adding a fifth blank row and column and using infinity parameters to move 𝐺4 to the diagonal position 5, 5 the following NAM is obtained:23(23) Figure 5(a) represents the pathological realization of the above equation after connecting the capacitors 𝐶1 and 𝐶2 at nodes 1 and 2, respectively.

fig5
Figure 5: (a) Realization I-of Class I-A oscillator. (b)Realization of Figure 5(a) using two ICCII− and one CCII−.

Figure 5(b) represents the circuit realization using two ICCII− and one CCII−. This is the adjoint circuit to that of Figure 4(a), noting that the adjoint of CCII+ number 1 in Figure 4(a) is the ICCII− number 2 in Figure 5(b), the adjoint of CCII− number 2 in Figure 4(a) is the CCII− number 1 in Figure 5(b), and the adjoint of CCII+ number 3 in Figure 4(a) is the ICCII− number 3 in Figure 5(b). It should also be noted that the circuit of Figure 5(b) has a similar topology to the circuit of Figure 4(b) except for the interchange of the two branches of 𝐺2 and 𝐺3. It is seen that the topology of the class I oscillators generated in this paper is self-adjoint, that is, eight of the generated circuits are the adjoints of the other eight circuits.

5. Class II Oscillators

The two types of class II oscillators are considered next. Generation method of this class of oscillators was given recently in [20] based on using three single output CCII and ICCII, two grounded capacitors, and four grounded resistors two of them are equal. Then, the two CCII having equal resistors connected to their port 𝑋 are combined to realize a BOCCII or a DOCCII depending on the 𝑍 port polarity of the two CCII. An alternative and much simpler approach in the generation of this class of oscillators without expanding the NAM equation to a 5×5 matrix equation (five-node circuit) is given next.

It should be noted that in the bracket method, the brackets representing a nullator, norator, and a CM with a common node realize a BOCCII. The brackets representing a nullator, two CM with a common node realize a DOCCII++. The brackets representing a nullator, two norator with a common node realize a DOCCII−−.

The brackets representing a VM, norator, and a CM with a common node realize a BOICCII. The brackets representing a VM, two CM with a common node realize a DOICCII++. The brackets representing a VM, two norator with a common node realize a DOICCII−−.

5.1. Class II Type-A

The nodal admittance matrix for the four node class II type-A oscillator is given by𝐺𝑌=1𝐺2𝐺3𝐺20.(24)

5.1.1. Realization I

The NAM expansion starts by adding two blank rows and columns and using a nullator to link nodes 2 and 4 and a CM to link nodes 1 and 4 in order to move 𝐺3 to become 𝐺3 at the diagonal position 4, 4 as follows:25(25) A nullator is added between nodes 1 and 3 to move 𝐺2 and 𝐺2 to positions 1, 3 and 2, 3, respectively as follows:26(26) A CM is added between nodes 1 and 3 to move 𝐺2 to the diagonal position 3, 3 and a norator is added between nodes 2 and 3 to move 𝐺2 to the diagonal position 3, 3 as follows:27(27) The above equation is obtained after applying the scaling of row rules demonstrated in [2] to row three and with a scaling factor of one half.

It should be noted the brackets representing the nullator between nodes 1, 3, the norator between nodes 3, 2 and the CM between nodes 3, 1 realize a BOCCII.

The nullator between nodes 2, 4 and the CM between nodes 4, 1 realize CCII+. Figure 6 realizes the above equation using a BOCCII and a CCII+ [17].

131546.fig.006
Figure 6: Realization I of class II type-A oscillator [17].

Three more circuits that belong to class II type-A can be generated in a similar way and are not included to limit paper length.

5.2. Class II-Type B

The NAM equation for the class II type-B oscillator is given by𝐺𝑌=1𝐺2𝐺3𝐺20.(28)

5.2.1. Realization I

Starting from (28) and adding two blank rows and columns then following successive NAM expansion steps the following NAM is obtained:29(29) It should be noted that the brackets representing the nullator joining nodes 1, 3, the CM joining nodes 3, 2 and the CM joining nodes 3, 1 realize a DOCCII++. The nullator between nodes 2, 4 and the norator between nodes 4, 1 realize CCII− as shown in Figure 7(a).

fig7
Figure 7: (a) Realization I of class II type B oscillator [17]. (b) Realization II of class II type-B oscillator. (c) Realization III of class II type-B oscillator. (d) Realization IV of class II type-B oscillator.
5.2.2. Realization II

Following successive NAM expansion steps starting from (28) after adding two blank rows and columns, the following NAM equation is obtained:30(30) Figure 7(b) realizes the above equation using a DOCCII++ and an ICCII+.

5.2.3. Realization III

Starting from (28) and adding two blank rows and columns then following successive NAM expansion steps, the following 𝑌 matrix is obtained: 31(31) It should be noted that the brackets representing the VM joining nodes 1, 3, the norator joining nodes 3, 2, and the norator joining nodes 3, 1, realize a DOICCII−−. The nullator between nodes 2, 4 and the norator between nodes 4, 1 realizes CCII− as shown in Figure 7(c).

5.2.4. Realization IV

Starting from (28) and adding two blank rows and columns then following successive NAM expansion steps, the following 𝑌 matrix is obtained:32(32) It should be noted that the brackets representing the VM joining nodes 1, 3, the norator joining nodes 3, 2, and the norator joining nodes 3, 1 realize a DOICCII−−. The VM between nodes 2, 4 and the CM between nodes 4, 1 realize an ICCII+ as shown in Figure 7(d).

6. Realization of Class II Oscillators Using Infinity Parameters

The class II circuits generated in the previous section can also be obtained using combination of the bracket method and infinity parameters method as demonstrated in this section.

6.1. Class II-Type A

The circuit shown in Figure 6 can also be obtained using the infinity parameters describing the NAM of the BOCCII as explained next.

The NAM representation of the BOCCII is given by [11, 21] 𝑋𝑍𝑍+𝑋𝑌𝑖𝑖𝑖𝑖𝑖𝑖.(33) From (25), adding and subtracting infinity parameter terms to the positions 1, 1 and 2, 1 as follows: 34(34) Apply pivotal expansion to the third term in the 1, 1 position to move 𝐺2 from 1, 1 position to the diagonal position 3, 3 as follows: 35(35) Next, apply also pivotal expansion to the third term in the 2, 1 position to move 𝐺2 from 2, 1 position to the diagonal position 3, 3 as follows: 36(36) Applying the scaling of row rules demonstrated in [2] to row three and with a scaling factor of one half it follows that: 37(37) The brackets realize a CCII+; comparing with (33) it is seen that the infinity parameters realize a BOCCII as shown in Figure 6.

The additional three equivalent class II-type-A oscillator circuits can be generated in a similar way using infinity parameters to realize the first building block and are not included here to limit paper length.

6.2. Class II Type-B
6.2.1. Realization I

The circuit shown in Figure 7(a) can also be obtained using the infinity parameters describing the NAM of the DOCCII as explained next.

The NAM representation of the DOCCII++ is given by𝑋𝑍+𝑍+𝑋𝑌𝑖𝑖𝑖𝑖𝑖𝑖.(38) After adding two blank rows and columns to (28), brackets are used to represent the nullator between nodes 2 and 4 and the norator between nodes 1 and 4 that are combined to move 𝐺3 from 1, 2 position to the diagonal position 4, 4. The infinity parameters are used next to move both of 𝐺2 from 1, 1 position and 𝐺2 from 2, 1 position to the diagonal position 3, 3 as follows: 39(39) The brackets are realizing the CCII− and the infinity parameters are realizing the DOCCII++. The above equation is realized as shown in Figure 7(a).

6.2.2. Realization II

The circuit shown in Figure 7(b) can also be obtained using the infinity parameters describing the NAM of the DOCCII as explained next.

After adding two blank rows and columns to (28), brackets are used to represent the VM between nodes 2 and 4 and the CM between nodes 1 and 4 that are combined to move 𝐺3 from 1, 2 position to the diagonal position 4, 4. The infinity parameters are used next to move both of 𝐺2 from 1, 1 position and 𝐺2 from 2, 1 position to the diagonal position 3, 3 as follows:40(40) The brackets are realizing the ICCII+ and the infinity parameters are realizing the DOCCII++. The above equation is realized as shown in Figure 7(b).

7. Adjoint of Class II-Oscillators

7.1. Adjoint of Class II Type-A

Consider the NAM equation of the adjoint of the class II-type-A defined as class II-Ad-type-A oscillators given by𝐺𝑌=1𝐺2𝐺2𝐺30.(41) The NAM expansion is carried out using both the brackets method and the infinity parameters. The active building block to be used here is the DVCC [13, 14]. The NAM representation of the DVCC− is given by [11, 21]𝑋𝑍𝑋𝑌1𝑌2𝑖𝑖𝑖𝑖𝑖𝑖.(42) Add two blank rows and columns to (41) a VM between nodes 1 and 4 and a norator between nodes 2 and 4 are combined together to move 𝐺3 from the position 2, 1 to become 𝐺3 in the diagonal position 4, 4. 43(43) The infinity parameters are used next to modify the above NAM equation by moving both of 𝐺2 from 1, 1 position and 𝐺2 from 1, 2 to the diagonal position 3, 3, thus;44(44) Figure 8(a) realizes the above equation using ICCII− and DVCC−. This is the adjoint circuit to that shown in Figure 6.

fig8
Figure 8: (a) Class II-Ad oscillator circuit realizing (44). (b) Class II-Ad oscillator circuit realizing (48).
7.2. Adjoint of Class II Type-B

Consider the NAM equation of the adjoint of the class II type-B defined as class II-Ad-type-B oscillators given by𝐺𝑌=1𝐺2𝐺2𝐺30.(45) The active building block to be used here is the DVCC+ [13, 14].

The NAM representation of the DVCC+ with two noninverting 𝑌 inputs (𝑌1 and 𝑌3) is given by: 𝑋𝑍+𝑋𝑌1𝑌3𝑖𝑖𝑖𝑖𝑖𝑖.(46)Adding two blank rows and columns to (45); connecting a VM between nodes 1 and 4 and a CM between nodes 2 and 4 to move G3 to the diagonal position 3, 3 as follows:47(47) The infinity parameters modify the above NAM to be:48(48) The above equation is realized by an ICCII+ and a DVCC+ as shown in Figure 8(b).

8. Simulation Results

The active building block used in all simulations included in this paper is the DVCC [13]. The DVCC is defined as a five-port building block with a describing matrix of the form𝑉𝑋𝐼𝑌1𝐼𝑌2𝐼𝑍+𝐼𝑍=𝐼0110000000000001000010000𝑋𝑉𝑌1𝑉𝑌2𝑉𝑍+𝑉𝑍.(49) The DVCC is a very powerful building block as it realizes each of CCII+, CCII−, ICCII+ and ICCII− as special cases.

Figure 9 represents the CMOS DVCC circuit [13], and the transistor aspect ratios are given in Table 2 based on the 0.5 μm CMOS model from MOSIS. The supply voltages used are ±1.5 V and 𝑉B1=0.52V and 𝑉B2=0.33V.

tab2
Table 2: Dimensions of the MOS transistors of the circuit of Figure 9.
131546.fig.009
Figure 9: CMOS circuit of the DVCC [13].

Figure 10(a) represents the output voltage waveform of the oscillator of Figure 4(b) designed for oscillation frequency equal to 1 MHz by taking 𝐶1=𝐶2=40pF, 𝑅1=𝑅2=𝑅3=𝑅4=4kΩ. To start oscillations 𝑅1 is increased to 4.2 kΩ. It should be noted that the simulations given are based on the above values of circuit components with no compensation. The simulation results indicate an oscillation frequency slightly lower than 1 MHz due to the addition of the parasitic elements 𝑅𝑋1 added to 𝑅2, 𝑅𝑋2 is added to 𝑅3, and 𝑅𝑋3 added to 𝑅4. The parasitic capacitances are also affecting the oscillation frequency since 𝐶𝑍1 is added to 𝐶1 and 𝐶𝑍2 is added to 𝐶2. The total power dissipation is given by 2.8731 mW.

fig10
Figure 10: (a) Simulated output waveform of the circuit of Figure 4(b). (b) Simulated output waveform of the circuit of Figure 5(b). (c) Simulated output waveform of the circuit of Figure 8(a).

Figure 10(b) represents the output voltage waveform of the oscillator of Figure 5(b) designed for oscillation frequency equal to 1 MHz by taking the same design values as above. The circuit operates well with the design value of 𝑅1 without increasing its value as in the circuit of Figure 4(b). The total power dissipation is given by 2.92668 mW.

Figure 10(c) represents the output voltage waveform of the oscillator of Figure 8(a) designed for oscillation frequency equal to 1 MHz by taking 𝐶1=𝐶2=40pF, 𝑅1=𝑅2=𝑅3=4kΩ. To start oscillations 𝑅1 is increased to 4.4 kΩ. The total power dissipation is given by 1.90799 mW.

9. Conclusions

A new approach in the systematic synthesis of grounded passive elements canonic oscillators is given. The synthesis procedure is based on the generalized systematic synthesis framework using NAM expansion and infinity parameters. The suggested use of the NAM expansion method in the synthesis of oscillator circuits provided many new oscillator circuits to complete the families of known oscillator circuits. The active building blocks that have been considered are the CCII, ICCII, BOCCII, DOCCII, BOICCII and DOICCII. This is the first paper in the literature which uses infinity parameters in the synthesis of oscillator circuits. The oscillators generated in this paper enjoy the advantages of having independent control on the condition of oscillation by varying 𝑅1 and on the frequency of oscillation by varying 𝑅3. The oscillator circuits that belong to the topology of the class I generated in this paper and to class II have the advantage of being easily compensated for the parasitic effects of the current conveyors. The adjoint to class I oscillators is also considered, and it is found that the class I oscillator is self-adjoint. The adjoint of the class II oscillators using the DVCC as the basic building block is also given. Simulation results demonstrating the practicality of the circuits are included. The advantage of using the nullators, norators, VM and CM in symbolic circuit analysis has been most recently demonstrated in the literature [22]. It is worth noting that the generic algorithms given in [23] can also be extended to different types of pathological elements.

Acknowledgment

The author thanks the reviewers for their useful comments.

References

  1. D. G. Haigh, T. J. W. Clarke, and P. M. Radmore, “Symbolic framework for linear active circuits based on port equivalence using limit variables,” IEEE Transactions on Circuits and Systems I, vol. 53, no. 9, pp. 2011–2024, 2006. View at Publisher · View at Google Scholar · View at Scopus
  2. D. G. Haigh, F. Q. Tan, and C. Papavassiliou, “Systematic synthesis of active-RC circuit building-blocks,” Analog Integrated Circuits and Signal Processing, vol. 43, no. 3, pp. 297–315, 2005. View at Publisher · View at Google Scholar · View at Scopus
  3. D. G. Haigh and P. M. Radmore, “Admittance matrix models for the nullor using limit variables and their application to circuit design,” IEEE Transactions on Circuits and Systems I, vol. 53, no. 10, pp. 2214–2223, 2006. View at Publisher · View at Google Scholar · View at Scopus
  4. D. G. Haigh, “A method of transformation from symbolic transfer function to active-RC circuit by admittance matrix expansion,” IEEE Transactions on Circuits and Systems I, vol. 53, no. 12, pp. 2715–2728, 2006. View at Publisher · View at Google Scholar
  5. H. J. Carlin, “Singular network elements,” IEEE Transactions on Circuit Theory, vol. 11, pp. 67–72, 1964.
  6. I. A. Awad and A. M. Soliman, “Inverting second generation current conveyors: the missing building blocks, CMOS realizations and applications,” International Journal of Electronics, vol. 86, no. 4, pp. 413–432, 1999. View at Scopus
  7. I. A. Awad and A. M. Soliman, “On the voltage mirrors and the current mirrors,” Analog Integrated Circuits and Signal Processing, vol. 32, no. 1, pp. 79–81, 2002. View at Publisher · View at Google Scholar · View at Scopus
  8. I. A. Awad and A. M. Soliman, “A new approach to obtain alternative active building blocks realizations based on their ideal representations,” Frequenz, vol. 54, no. 11-12, pp. 290–299, 2000. View at Scopus
  9. R. A. Saad and A. M. Soliman, “Generation, modeling, and analysis of CCII-based gyrators using the generalized symbolic framework for linear active circuits,” International Journal of Circuit Theory and Applications, vol. 36, no. 3, pp. 289–309, 2008. View at Publisher · View at Google Scholar · View at Scopus
  10. R. A. Saad and A. M. Soliman, “Use of mirror elements in the active device synthesis by admittance matrix expansion,” IEEE Transactions on Circuits and Systems I, vol. 55, no. 9, pp. 2726–2735, 2008. View at Publisher · View at Google Scholar · View at Scopus
  11. R. A. Saad and A. M. Soliman, “On the systematic synthesis of CCII-based floating simulators,” International Journal of Circuit Theory and Applications, vol. 38, no. 9, pp. 935–967, 2010. View at Publisher · View at Google Scholar
  12. A. S. Sedra and K. C. Smith, “A second generation current conveyor and its applications,” IEEE Transactions on Circuit Theory, vol. 132, pp. 132–134, 1970.
  13. H. O. Elwan and A. M. Soliman, “Novel CMOS differential voltage current conveyor and its applications,” IEE Proceedings, vol. 144, no. 3, pp. 195–200, 1997.
  14. W. Chiu, S. I. Liu, H. W. Tsao, and J. J. Chen, “CMOS differential difference current conveyors and their applications,” IEE Proceedings, vol. 143, no. 2, pp. 91–96, 1996.
  15. A. M. Soliman, “Simple sinusoidal active RC oscillators,” International Journal of Electronics, vol. 39, no. 4, pp. 455–458, 1975. View at Scopus
  16. A. M. Soliman, “Synthesis of grounded capacitor and grounded resistor oscillators,” Journal of the Franklin Institute, vol. 336, no. 4, pp. 735–746, 1999. View at Publisher · View at Google Scholar · View at Scopus
  17. A. M. Soliman, “Current mode CCII oscillators using grounded capacitors and resistors,” International Journal of Circuit Theory and Applications, vol. 26, no. 5, pp. 431–438, 1998. View at Publisher · View at Google Scholar · View at Scopus
  18. B. B. Bhattacharyya and M. N. S. Swamy, “Network transposition and its application in synthesis,” IEEE Transactions on Circuit Theory, vol. 18, pp. 394–397, 1971.
  19. A. M. Soliman, “Adjoint network theorem and floating elements in the NAM,” Journal of Circuits, Systems and Computers, vol. 18, no. 3, pp. 597–616, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. A. M. Soliman, “Generation of current conveyor based oscillators using nodal admittance matrix expansion,” Analog Integrated Circuits and Signal Processing, vol. 65, no. 1, pp. 43–59, 2010. View at Publisher · View at Google Scholar
  21. A. M. Soliman, “On the DVCC and the BOICCII as adjoint elements,” Journal of Circuits, Systems and Computers, vol. 18, no. 6, pp. 1017–1032, 2009. View at Publisher · View at Google Scholar · View at Scopus
  22. E. Tlelo-Cuautle, M. A. Duarte-Villaseñor, and I. Guerra-Gómez, “Automatic synthesis of VFs and VMs by applying genetic algorithms,” Circuits, Systems, and Signal Processing, vol. 27, no. 3, pp. 391–403, 2008. View at Publisher · View at Google Scholar · View at Scopus
  23. C. Sánchez-López, F. V. Fernández, E. Tlelo-Cuautle, and S. X. D. Tan, “Pathological element-based active device models and their application to symbolic analysis,” IEEE Transactions on Circuits and Systems I, vol. 58, no. 6, pp. 1382–1395, 2011. View at Publisher · View at Google Scholar