- Research
- Open Access
Active filter synthesis based on nodal admittance matrix expansion
- Lingling Tan^{1},
- Yunpeng Wang^{1} and
- Guizhen Yu^{1}Email author
https://doi.org/10.1186/s13638-017-0881-8
© The Author(s). 2017
- Received: 6 April 2017
- Accepted: 10 May 2017
- Published: 26 May 2017
Abstract
Active network synthesis is important for circuit designer to find new circuits with desired performance. In this paper, a method of Tow-Thomas (TT) Bi-quad band-pass filter circuit generation methods is proposed using nullor representation of the operational amplifier (OPA), and a method for synthesizing active band-stop filters is presented, both of which start from voltage transfer function and linked infinity variables to describe nullors in both nodal admittance matrix (NAM) and port admittance matrix of the circuit to be synthesized. The Tow-Thomas band-stop filter circuit and Åkerberg-Mossberg band-stop filter circuit are synthesized by nodal admittance matrix expansion on the same port admittance matrix.
Keywords
- Active filter synthesis
- Nullor
- Nodal admittance matrix (NAM) expansion
- Band-pass filter
- Band-stop filter
1 Introduction
Active network synthesis is the reverse process of the traditional active network analysis. Method of circuit synthesis makes circuit automatic design realizable [1]. Admittance matrices of the active devices, such as the ideal operational amplifier (OPA), current mirror, voltage mirror [2], do not exist. Through decades of painstaking research, D.G. Haigh and A.M. Soliman enrich the theory of active network synthesis. D.G. Haigh proposed the method of nodal admittance matrix (NAM) expansion and put it into practical applications, for example, obtaining the topology of a circuit from a given transfer function based on the theory of nullors [3]. Alternative circuit topology structures can be obtained from the same transfer function, when taking account of the different performances [4]. On the basis of the nullors, two additional pathological elements, current mirror (CM) and voltage mirror (VM) [5], are proposed, which tremendously enrich the theory of active network synthesis and make the active circuit design through theoretical method possible [6].
In comparison with the conventional generation methods of circuits based on experience or building blocks, circuits design via NAM expansion is a systematic methodology with no need to consider the final topology of the obtained circuits during the process of design [7]. So, the method of NAM expansion provides a bridge between the practical circuit design and theory. And, it is already applied to filter [8] and oscillator [9] design.
Circuit design using NAM expansion is a novel subject in theory of active network synthesis. As OPA is prevalent devices in circuit, this paper firstly provides generation method of the OPA-based Tow-Thomas (TT) Bi-quad band-pass filter circuit using NAM expansion, in which the generation process is based on a symbolic method of circuit design, and the derived circuit originates from a symbolic transfer function. As extra nullors are introduced, nullators and norators may be paired as operational amplifiers in alternative ways, then different active filter circuits can be derived from the same circuit network with nullors. Thus, the Tow-Thomas band-stop filter circuit and Åkerberg-Mossberg band-stop filter circuit are taken as examples to illustrate the synthesis method by nodal admittance matrix expansion on the same port admittance matrix.
2 Theory of NAM expansion
2.1 Nullor and active device modeling
2.2 Introduction of nullors into NAM
In Eq. (2), variable i indicates the ith nullor, ∞_{ i } is a linked infinity parameter. During matrix manipulating, terms in columns j and k connected between a nullator can be moved between the two columns without affecting the equivalent circuit characteristics, while terms in rows m and n connected between a norator can be moved between the two rows. Term moving is called the element shift theorem. In particular, if there is ∞_{ i } existing at the matrix, any terms can be added to the row and the column where ∞_{ i } exists, while if there is ±∞_{ i } existing at the same row of the matrix, then any terms can be added to the row, and if there is ±∞_{ i } existing at the same columns, then any terms can be added to the column [12]. Term adding is called the arbitrary element theorem.
2.3 Theory of pivotal expansion and Gaussian elimination
2.4 Admittance matrix descriptions for circuits with prescribed voltage and current transfer functions
Port admittance matrix descriptions of VCVS and CCCS [12]
Type I | Type II | Type III | Type IV | |
---|---|---|---|---|
VCVS A = A _{ v } =N/D | \( \left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\infty}_1\hfill & \hfill -{\infty}_1\hfill \\ {}\hfill - N\hfill & \hfill 0\hfill & \hfill D\hfill \end{array}\right] \) | \( \left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {\infty}_1\hfill & \hfill 0\hfill & \hfill -{\infty}_1\hfill \\ {}\hfill 0\hfill & \hfill - D\hfill & \hfill N\hfill \end{array}\right] \) | \( \left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {\infty}_1\hfill \\ {}\hfill - N\hfill & \hfill - D\hfill & \hfill Q\hfill \end{array}\right] \) | \( \left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {\infty}_1\hfill & \hfill -{\infty}_1\hfill \\ {}\hfill -{N}_1\hfill & \hfill -{D}_1\hfill & \hfill {p}_1\hfill & \hfill 0\hfill \\ {}\hfill -{N}_2\hfill & \hfill -{D}_2\hfill & \hfill 0\hfill & \hfill {P}_2\hfill \end{array}\right] \) |
CCCS A = A _{ i } =N/D | \( \left[\begin{array}{ccc}\hfill {\infty}_1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill - N\hfill \\ {}\hfill -{\infty}_1\hfill & \hfill 0\hfill & \hfill D\hfill \end{array}\right] \) | \( \left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill - D\hfill \\ {}\hfill {\infty}_1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill -{\infty}_1\hfill & \hfill 0\hfill & \hfill N\hfill \end{array}\right] \) | \( \left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill - D\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill - N\hfill \\ {}\hfill {\infty}_1\hfill & \hfill 0\hfill & \hfill Q\hfill \end{array}\right] \) | \( \left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill -{D}_1\hfill & \hfill -{D}_2\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill -{N}_1\hfill & \hfill -{N}_2\hfill \\ {}\hfill {\infty}_1\hfill & \hfill 0\hfill & \hfill {P}_1\hfill & \hfill 0\hfill \\ {}\hfill -{\infty}_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {P}_2\hfill \end{array}\right] \) |
A | N/D | N/D | -N/D | \( \frac{N_1{P}_2-{N}_2{P}_1}{D_2{P}_1-{D}_1{P}_2} \) |
3 The method of active network synthesis
- 1
Choose a suitable matrix from Table 1 according to the given transfer function.
- 2
Determine each element of the starting matrix referring to the transfer function.
- 3
Carry out expansion of N, D, P, or Q terms until all terms become 1st-order admittance functions.
- 4
Introducing missing terms or shift original matrix terms according to the arbitrary element theorem and element shift theorem proposed in Section 2.2.
- 5
If all matrix elements now correctly describe passive elements, the circuit is a single nullor circuit.
- 6
Otherwise, add extra nullors to introduce missing matrix terms.
4 Active filter synthesis using nodal admittance matrix expansion
4.1 Active band-pass filter synthesis
The position of the introduced nullors indicates that the norator and nullator of the second nullor are connected respectively between node 6 and the reference node, node 4 and the reference node, while the norator and nullator of the third nullor are connected respectively between node 7 and the reference node, node 5 and the reference node. The admittance function Q contains a term of e.
Component values of the designed band-pass filter
C _{1} = C _{2} = 1 nF | R _{1} = R _{2} = R _{3} = R _{4} = R _{5} = R _{7} = R _{8} = 15.91 5kΩ | R _{6} = 7.957 KΩ |
4.2 Active band-stop filter synthesis
Component values of the designed band-stop filter
C _{1} = 3.17 nF | R _{1} = R _{2} = R _{3} = R _{4} = 1 kΩ | R _{5} = 159.15 Ω |
---|---|---|
C _{2} = 1 uF | C _{3} = 2 uF | R _{6} = 2 KΩ |
5 Conclusions
The generation method of the TT Bi-quad band-pass circuit, the TT band-stop filter circuit, and Åkerberg-Mossberg band-stop filter circuit is presented using the theory of NAM expansion, which is a new design method for circuit design. The active circuit topologies of the TT band-stop filter and the Åkerberg-Mossberg band-stop filter are synthesized from the same transfer function. The analysis in the paper verifies the effectiveness of the theory of NAM expansion for circuit design in theory and practice. At the same time, further research needs to be conducted to enrich the method of NAM expansion.
Declarations
Acknowledgements
There is no other one to acknowledge in this section.
Funding
This research was funded partially by the National Science Foundation of China under grant no. 61371076.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
LT as the first author wrote the manuscript and carried out all the simulation of examples in this manuscript. YW as the supervisor of Lingling Tan provided some guidance on the method of circuit design base on nodal admittance matrix expansion. GY as the corresponding author provided some suggestions on the English writing. All authors read and approved the final manuscript.
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