Keywords

The concept of pathological nullor and mirror elements has proven to be a very valuable tool for circuit analysis, synthesis, circuit transformation and interrelation in the realizations using different active elements. The pathological elements provide a common framework for linear and nonlinear circuits. In this paper, the nullor and nullor-mirror equivalents are reviewed, and several new nullor-mirror equivalents are proposed. The nullor-mirror equivalents are conductive to reducing circuit complexity and synthesizing new functional circuits. Combining the nullor-mirror equivalents with circuit transformation techniques, more useful functional circuits with simpler circuit structures can be generated. The circuit synthesis using the proposed nullor-mirror equivalents is illustrated to demonstrate the usefulness and feasibility.

1 Introduction

The concept of nullors, i.e. combinations of nullators and norators, was first implicitly introduced by Tellegen [1] under the name of ideal elements and given its present name by Carlin [2]. The terminal behaviors of the nullors cannot be described by any of the existing network descriptions such as Z, Y and H matrices. Such networks usually called pathological networks, are degenerate forms and are needed to complete the domain of network elements. The pathological nullor elements have been proven to be very valuable due to their capability of modeling active devices independently of the particular implementation of the active devices and the simplicity with which nodal or loop analysis of these circuits can be carried out [3, 4]. They provide a common framework for circuit analysis, synthesis, circuit transformation and interrelating the realizations using different active devices [5–9]. Despite the ability of nullors to represent all active elements, without the use of resistors, they fail to represent some important analog elements. Therefore, two new pathological elements, i.e. the current mirror and voltage mirror have been defined in 1999 [10]. The new defined pathological mirror elements are basically used to represent active devices with current or voltage reversing properties. Their usefulness to circuit synthesis has been demonstrated in the literature [11–14]. In this article, the properties of pathological elements are reviewed and their usage of modeling active devices is illustrated. All the available nullor equivalents are presented and many new nullor and nullor-mirror equivalents are proposed. The nullor-mirror equivalences are conductive to reducing circuit complexity, realizing circuit using convenient structure and synthesizing new functional circuits. Combining the nullor-mirror equivalents with circuit transformation techniques, more useful functional circuits with simpler circuit structures can be generated. The circuit synthesis using the new proposed nullor-mirror equivalents is illustrated to demonstrate their usefulness and feasibility.

2 Pathological Elements and Equivalents

The symbols and definitions of the nullor and mirror elements are shown in Table 26.1. The pathological elements in Table 26.1a–d are bi-directional pathological elements that possess ideal characteristics and are specified on the basis of the constraints they impose on their terminal voltages and currents. The pathological nullor elements comprise the nullator and norator, as shown in Table 26.1a and b. The pathological voltage mirror (VM) and current mirror (CM), as shown in Table 26.1c and d, are lossless two-port network elements used to represent an ideal voltage reversing property and an ideal current reversing property, respectively. Each of the symbols of the pathological mirror elements has a grounded reference node [11]. Although these two mirror elements are two-port network elements, they can be used as two-terminal elements with the reference node unused [10]. The symbols and definitions of the floating mirror elements are shown in Table 26.1e–f [15]. It is clear that when the reference node of each mirror element in Table 26.1e and f is connected to ground, they can be regarded as the mirror elements in Table 26.1c and d, respectively. In [16], two floating voltage mirrors with a common reference node are used to represent the pathological differential voltage cell and differential voltage conveying cell, as shown in Table 26.2a and b. Also, two floating current mirrors with a common reference node are used to express the pathological current replication cells in Tables 26.2c and d. These two pathological cells can be used to model some popular ideal active devices with differential or multiple single-ended features in concise forms [16]. The terminal properties for the cells consist of two floating mirror elements with a common reference node are shown in Table 26.2. According to the definitions of pathological elements in Table 26.1 and considering the labeled terminals only, the cells in Tables 26.2e,f,g and h can be respectively regard as a nullator, a pathological VM, a norator and a pathological CM.

Table 26.1 Symbols and definitions of nullor and mirror elements
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Table 26.2 Some pathological sections and their terminal properties
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The nullor elements have been used to model many ideal active devices before. With the new defined mirror elements, the pathological representations of some active devices possess the simpler circuit structures (i.e. circuits with less node number) compared to their nullor representations. The simpler circuit structure is conductive to achieving high-performance symbolic nodal analysis since the less nodal equations are needed to build [17, 18]. Table 26.3 shows the pathological representations of some ideal active devices, including the MOS transistor, the operational amplifier (OP AMP), the balanced-output second generation current conveyor (BOCCII), the balanced-output second generation inverting current conveyor (BOICCII), the fully differential second generation current conveyor (FDCCII) and differential difference current conveyor (DDCC). All the pathological elements in Table 26.1 have been used.

Table 26.3 Pathological representations of some ideal active devices
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Nullor equivalents have received much attention as they can be used to obtain new alternative circuits or reduce the circuit complexity with the identical function [19–22]. Table 26.4 shows seven common nullor equivalents in [19]. The nullor equivalents in Tables 26.4e and f are very useful and they have been applied to many circuit transformations techniques, such as the inverse transformation in [23, 24]. Besides, the nullor equivalents of nullor relocations shown in Table 26.5 are also the popular techniques to obtain different circuit structures with identical transfer function and perform circuit transformations [20–22]. Due to the appearance of new pathological mirror elements, nullor-mirror equivalents were further investigated recently [25–27]. For the completeness of pathological equivalents, we try to present the available nullor-mirror equivalents and proposed some new equivalents.

Table 26.4 Common nullor equivalents
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Table 26.5 Nullor relocation
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Table 26.6 shows some nullor-mirror equivalents. The equivalents in Table 26.6a,b,e and f were proposed in [28] and the equivalents in Table 26.6c and d are the new proposed ones. To investigate the properties of the series and parallel connections of nullor-mirror elements, their equivalences or terminal characteristics according to the element definitions in Table 26.1 are given in Table 26.7 [25]. All the series connections of pathological elements in Table 26.7a–d are equivalent to an open circuit. For the structures in Table 26.7g–j, the V1 and I1 beside node 1 are used to denote respectively the voltage from node 1 to ground and the current flowing out of node 1. The relative voltage and current for node 2 are shown beside node 2, according to the element properties. If node 2 in Tables 26.7g–j is adopted as grounded node, all the nodes 1 of parallel conditions can be seen to be a short circuit to ground.

Table 26.6 Nullor-mirror equivalents
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Table 26.7 Series and parallel connections of nullor-mirror elements
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Other new proposed nullor-mirror equivalents of nullor-mirror relocations are presented in Table 26.8. The validity of voltage mirror relocation in Table 26.8a can be observed since the identical electrical property for the three connection circuits according to the element definitions. In addition, from the super-node concept in circuit analysis, the three schemes in Table 26.8b are equivalent as they can be represented by the same nodal equations. Furthermore, the nullor equivalent in Table 26.4g was extended to obtain other new nullor-mirror equivalents, as redrawn in Table 26.9 [27]. Their usefulness was also illustrated in [27].

Table 26.8 Proposed nullor-mirror relocations
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Table 26.9 Nullor-mirror equivalents
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3 The Application Example of the New Proposed Nullor-Mirror Equivalent

To demonstrate the application of new proposed nullor-mirror relocation in Table 26.8, one circuit example is presented below. We consider the current-mode filter circuit in Fig. 1c of [26], which is redrawn in Fig. 26.1a. According to the nullor relocation in Table 26.5a, one can obtain the circuit given in Fig. 26.1b. Applying the equivalence in Table 26.8b, the equivalent circuit in Fig. 26.1c is derived. It can be found that we can realize this circuit with three identical CCII+s. The simplified circuit in Fig. 26.1d can be obtained after applying the equivalent in Table 26.7i. Namely, only two CCII+s are needed to implement the circuit in Fig. 26.1(d). Therefore, the usefulness of the new proposed nullor-mirror equivalent is demonstrated.

Fig. 26.1
figure 1

The simplified procedure of a filter circuit applying nullor-mirror equivalent: a the original circuit, b the derived circuit applying nullor relocation, c the derived circuit applying nullor-mirror relocation, d the obtained circuit with simpler circuit structure

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4 Conclusion

In this paper, the properties of available pathological elements and cells are reviewed. The ideal behaviors of some active devices are modeled with different pathological elements. The available nullor equivalents in the literature are presented and some new pathological equivalents are proposed. The nullor-mirror equivalences are conductive to reducing circuit complexity and facilitating circuit realization. Combining the nullor-mirror equivalences with circuit transformation techniques, more useful functional circuits with simpler circuit structures can be generated. One circuit example using the presented nullor-mirror equivalence is illustrated to demonstrate the usefulness and feasibility.