Active filter synthesis based on nodal admittance matrix expansion

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1 Tan et al. EURASIP Journal on Wireless Communications and Networkin (1) 1:9 DOI 1.118/s RESEARCH Active filter synthesis based on nodal admittance matrix expansion Linlin Tan, Yunpen Wan and Guizhen Yu * Open Access Abstract Active network synthesis is important for circuit desiner to find new circuits with desired performance. In this paper, a method of Tow-Thomas (TT) Bi-quad band-pass filter circuit eneration methods is proposed usin nullor representation of the operational amplifier (OPA), and a method for synthesizin active band-stop filters is presented, both of which start from voltae 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 Åkerber-Mossber 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 desin realizable [1]. Admittance matrices of the active devices, such as the ideal operational amplifier (OPA), current mirror, voltae mirror [], do not exist. Throuh decades of painstakin research, D.G. Haih and A.M. Soliman enrich the theory of active network synthesis. D.G. Haih proposed the method of nodal admittance matrix (NAM) expansion and put it into practical applications, for example, obtainin the topoloy of a circuit from a iven transfer function based on the theory of nullors []. Alternative circuit topoloy structures can be obtained from the same transfer function, when takin account of the different performances [4]. On the basis of the nullors, two additional patholoical elements, current mirror (CM) and voltae mirror (VM) [], are proposed, which tremendously enrich the theory of active network synthesis and make the active circuit desin throuh theoretical method possible []. In comparison with the conventional eneration methods of circuits based on experience or buildin blocks, circuits desin via NAM expansion is a systematic methodoloy with no need to consider the * Correspondence: yuz@buaa.edu.cn School of Transportation Science and Enineerin, Beihan University, 1191 Beijin, China final topoloy of the obtained circuits durin the process of desin []. So, the method of NAM expansion provides a bride between the practical circuit desin and theory. And, it is already applied to filter [8] and oscillator [9] desin. Circuit desin usin NAM expansion is a novel subject in theory of active network synthesis. As OPA is prevalent devices in circuit, this paper firstly provides eneration method of the OPA-based Tow- Thomas (TT) Bi-quad band-pass filter circuit usin NAM expansion, in which the eneration process is based on a symbolic method of circuit desin, and the derived circuit oriinates 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 Åkerber-Mossber band-stop filter circuit are taken as examples to illustrate the synthesis method by nodal admittance matrix expansion on the same port admittance matrix. Theory of NAM expansion.1 Nullor and active device modelin The patholoical elements of nullor, that are the nullator and the norator, are specified accordin to the constraints they impose on their terminal voltae and current. For the nullator, shown in Fi. 1b, v = i =, The Author(s). 1 Open Access This article is distributed under the terms of the Creative Commons Attribution 4. International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you ive appropriate credit to the oriinal author(s) and the source, provide a link to the Creative Commons license, and indicate if chanes were made.

2 Tan et al. EURASIP Journal on Wireless Communications and Networkin (1) 1:9 Pae of 1 Fi. 1 a c Nullor equivalence (a) (b) (c) which enables it to be used for non-invertin voltae conveyin, while for the norator, shown in Fi. 1c, it imposes no constraints on its terminal voltae and the current flowin into is equal to the current flowin out, which enables it to be used for noninvertin current conveyin [1]. Nullors can be used to model various controlled sources as well as varieties of ideal active devices, such as OPA and the second eneration current conveyor (CCII-). Nullor equivalences of the OPA and the CCIIare shown in Fi... Introduction of nullors into NAM For a NAM with N nodes,thesamerowsandcolumns with zero terms can be added to the oriinal matrix. Considerin the row n is full of zero terms and node n is an internal node within the network, a norator can be connected between node n and an arbitrary node m (includin the port node and the reference node), for input current In =. While considerin the column k is full of zero terms and node k is an internal node within the network, a nullator can be connected between node k and an arbitrary node j (includin the port node and the reference node), which made the voltae of k equal to the voltae of node j. Process of introduction can be illustrated in Eq. (1). For the convenience of matrix processin, the variable- proposed in reference [11] has been introduced to represent the connected nullor. " y11 (1) y 1 y 1j y 1k y 1N y1 y y j y k y N y m1 y m y mj þ i y mk i y mn y n1 y n y nj i y nk þ i y nn y N1 y N y Nj y Nk y NN # ðþ In Eq. (), variable i indicates the ith nullor, i is a linked infinity parameter. Durin matrix manipulatin, terms in columns j and k connected between a nullator (a) Fi. a Nullor equivalence of OPA [1]. b Nullor Equivalence of CCII- [1] (b)

3 Tan et al. EURASIP Journal on Wireless Communications and Networkin (1) 1:9 Pae of 1 Table 1 Port admittance matrix descriptions of VCVS and CCCS [1] VCVS A = A v =N/D CCCS A = A i =N/D Type I Type II Type III Type IV N D D N N D Q 1 4 N 1 D D N D 4 N 1 Q N 1 D 1 p 1 N D P D 1 D N 1 N 4 1 P 1 1 P A N/D N/D -N/D N1P NP1 DP1 D1P can be moved between the two columns without affectin the equivalent circuit characteristics, while terms in rows m and n connected between a norator can be moved between the two rows. Term movin is called the element shift theorem. In particular, if there is i existin at the matrix, any terms can be added to the row and the column where i exists, while if there is ± i existin at the same row of the matrix, then any terms can be added to the row, and if there is ± i existin at the same columns, then any terms can be added to the column [1]. Term addin is called the arbitrary element theorem.. Theory of pivotal expansion and Gaussian elimination In an ideal transistor and OPA active circuit, passive linear elements can be described by a symmetric NAM, and the active elements can be described by nullors. If element t + rq/s is rearded as the pivotal element, then the matrix after pivotal expansion is equivalent to the oriinal 1-port matrix, in which the sins are selected that the number of minus sins is odd [1]. " h t þ rq i t þ rq # s t q ðþ s r s.4 Admittance matrix descriptions for circuits with prescribed voltae and current transfer functions Circuit analysis consists of solvin the transfer function of a circuit, such as the open-circuit voltae ain and the short-circuit current ain, throuh which the performance of the circuit can be analyzed. Circuit synthesis happens to be the inverse process of the circuit analysis. That is, iven the transfer function A v or A i,startinfromthenamof the voltae controlled voltae source (VCVS) or the current controlled current source (CCCS) shown in Table 1, the circuit satisfyin the requirements can be obtained. In Table 1, N and D respectively represent the numerator and denominator of the transfer function, Q represents an arbitrary admittance function parameter. The method of active network synthesis Just like passive filter synthesis, different orders of removin transmission zero result in different circuit topoloy. Different form of transfer function may lead to different structure of active filter. The process of passive-rc circuit synthesis consists of the followin six steps []: 1 Choose a suitable matrix from Table 1 accordin to the iven transfer function. Determine each element of the startin matrix referrin to the transfer function. Carry out expansion of N, D, P, orq terms until all terms become 1st-order admittance functions. 4 Introducin missin terms or shift oriinal matrix terms accordin to the arbitrary element theorem and element shift theorem proposed in Section.. If all matrix elements now correctly describe passive elements, the circuit is a sinle nullor circuit. Otherwise, add extra nullors to introduce missin matrix terms. 4 Active filter synthesis usin nodal admittance matrix expansion 4.1 Active band-pass filter synthesis Assumin the voltae transfer function is iven as b c d e f m Fi. Synthesized TT Bi-quad circuit a n

4 Tan et al. EURASIP Journal on Wireless Communications and Networkin (1) 1:9 Pae 4 of 1 Fi. 4 OPA realization of TT Bi-quad circuit ce bem þ bdn Av ¼ ace þ bdf ð4þ Accordin to the extended form of the voltae and current transfer functions provided in Table 1, the type III of the VCVS is selected. That is, N D Q 1 ace þ bdf Q Q 1 ce þ bem bdn Q 1 ðþ The first-order admittance function of Q 1 has been introduced in Eq. (). Appropriate selection of Q 1 makes the expansion process more efficient. Then, Q 1 is selected to equal to bd. 4 n þ em d ce bd f ace bd 1 ðþ Q After applyin the pivotal expansion to the elements of the n þ em d ce ace bd and f bd, Eq. () is obtained, where pivotal expansion is implemented twice. 1 n f Q e ðþ 4 m d c a b For any terms can be added to the second row and the third column, Eq. () can be transformed to Eq. (8), where there is an element of f in Q. Asafloatin element, f is connected between node and node. Viewin that the other elements can be described in thesamemannerasf, two columns of zero terms were Fi. TT Bi-quad circuit with forward feedback

5 Tan et al. EURASIP Journal on Wireless Communications and Networkin (1) 1:9 Pae of 1 Table Component values of the desined band-pass filter C 1 = C =1nF R 1 = R = R = R 4 = R = R = R 8 = 1.91 kω R =.9 KΩ added next to the third column, two rows of zero terms were added under the fifth row. Then Eq. (9) equivalent to Eq. (8) is obtained. f f þ 1 n f Q e ð8þ 4 m d c a b f f þ 1 n f Q e m d c ð9þ a b 4 Infinity variables and are introduced to the newly added rows and columns, then correspondin terms are added to make elements appear in the form of floatin or roundin. a þ f f þ 1 a n f Q e m c þ d d c a a þ b b 4 e d þ d þ e c b þ b þ c ð1þ The position of the introduced nullors indicates that the norator and nullator of the second nullor are connected respectively between node and the reference node,node4andthereferencenode,whilethenorator and nullator of the third nullor are connected respectively between node and the reference node, node and the reference node. The admittance function Q contains a term of e. The terms m, m and n, n are moved to the first row from node. The admittance function Q equals to e + f + n. þ m þ n n m a þ f f þ 1 a n f eþ f þ n e m m þ c þ d d c a þ a þ b b 4 e d þ d þ e c b þ b þ c ð11þ Then, Eq. (11) obtained described the TT Bi-quad circuit in Fi.. Obviously, Fi. can be realized by OPA combined with passive elements. The nullors in Fi. can be respectively replaced by nullor equivalence of the OPA, as shown in Fi. 4. For realization, a is selected as a resistor, b is selected as a resistor combined with a capacitor, that is C 1 s + 1, c is selected as a resistor, d is selected as a resistor 8, e is selected as a resistor, f is selected as a capacitor, that is C s, is selected as a resistor 4, m is selected as a resistor, n is selected as a resistor, then Eq. (4) yields A v ¼ 8ð C 1s þ 1 Þþ 4 ðc 1 s þ 1 Þ 8 C sc ð 1 s þ 1 Þþ ð1þ R R1 C C1 R8 R4 X1 R X R X vdb1 v V1 R R Fi. The simulation of TT Bi-quad band-pass filter

6 Tan et al. EURASIP Journal on Wireless Communications and Networkin (1) 1:9 Pae of 1 vdb1 Results of TT Bi-quad band-pass filter circuit k 1.k 1.k freq Fi. The simulation results of TT Bi-quad band-pass filter 1.M A v ¼ 4 þ þ sc C 1 C s þ 1 8 C s þ 4 þ A v ¼ þ sc C 1 C s þ 1 C s þ 8 ð1þ ð14þ Then, the circuit of Fi. is obtained as the TT Biquad circuit with forward feedback. A desin example is provided to illustrate the desin procedure. Desin a band-pass filter accordin to the parameters below: fp ¼ 1 khz; Qp ¼1; K ¼ 1 where fp represents the pass-band central frequency, Qp represents the quality factor, K represents the stop-band voltae manification. A v ðþ¼ s K ω p Q p s s þ ω p Q p s þ ω p ð1þ In this desin example, for simplicity, capacitor is chosen to 1 nf. Component values which are calculated accordin to Eq. (1) are listed in Table. 4. Active band-stop filter synthesis The transfer function of a band-stop filter may be in the form the TT Bi-quad circuit is simulated accordin to the component values presented above respectively, which are shown in Fis. and A v ¼ k s þ ω z s þ ω Q s þ ω ð1þ Assumin the elements C 1, C, C, 1,,, 4,, and are used to build the band-stop filter (Fi. ), where 1 is used for tunin of ω z,,,and 4 may be used to form the ain of the amplifier, is used for tunin of ω, is used for tunin of Q. The transfer function is then in the form Fi. 8 The circuit topoloy synthesized from Eq. (18)

7 Tan et al. EURASIP Journal on Wireless Communications and Networkin (1) 1:9 Pae of 1 Fi. 9 Tow-Thomas band-stop filter circuit C 1 C s þ 1 4 A v ¼ C 1 C s þ C 1 s þ 4 ¼ C 1 C s þ 1 4 C 1 C s þ C 1 s þ 4 ¼ C 1 C s þ 1 4 C 1 sc s þ þ 4 ð1þ For simplicity, let a stand for 1, b stand for, c stand for, d stand for 4, e stand for C s +, stand for, h stand for C 1 s, i stand for C s, yields ihc þ abd A v ¼ ð18þ ehc þ bd From the iven transfer function, type III in Table 1 is selected accordin to the content description in Section.4, that is, N ¼ ihc þ abd D ¼ ehc þ bd 4 1 Q ¼ ðd þ e þ iþhc N D Q 1 4 ihc þ abd ehc þ bd ðd þ e þ iþhc Q Q Q ðþ An arbitrary 1st-order function Q is introduced to Eq. () in order to carry out pivotal expansion, where Q = hc. After pivotal expansion, another function Q = b is introduced to the riht side matrix of Eq. (1). (1) A v ¼ N D ð19þ Allocate every terms of the matrix after choosin transfer function Fi. 1 The reconstructed circuit topoloy

8 Tan et al. EURASIP Journal on Wireless Communications and Networkin (1) 1:9 Pae 8 of 1 Table Component values of the desined band-stop filter C 1 =.1 nf R 1 = R = R = R 4 =1kΩ R = 19.1 Ω C =1uF C =uf R =KΩ row. Then, the equivalent matrix of Eq. () is obtained as Eq. (). () Fi. 11 Åkerber-Mossber band-stop filter circuit Carry pivotal expansion on the terms in the fourth row of the riht side matrix of Eq. (1), we et Eq. () (Fi. ). () Accordin to the arbitrary element theorem, ±e elements are introduced to the row where 1 exists. The ±i elements are moved to the first row from the zeroth Extra nullors need to be introduced since the elements -a, b, c, d, h, and cannot be represented by floatin terms. The matrix of Eq. (4) can be obtained by introducin all zeroes in the 4th and th rows and all zeroes in the th and th columns; addin a nullator between node and the rounded node, and another one between node and the rounded node; addin a norator between node 4 and the rounded node and another one between node and the rounded node. i i e 1 e i e d þ e þ i d a h 4 c b ð4þ Accordin to the arbitrary element theorem, Eq. () is obtained. R C C1 R R R1 X1 R X R4 X vdb1 v V1 C Fi. 1 The simulation of TT band-stop filter

9 Tan et al. EURASIP Journal on Wireless Communications and Networkin (1) 1:9 Pae 9 of 1 R R C1 X1_ R R R1 X1_ R4 C X1_4 vdb1_1 v C V1_ Fi. 1 The simulation of Åkerber-Mossber band-stop filter a þ i i a e þ 1 e i e d þ e þ i d d cþ d c a a þ þ h h 4 h bþ h b c b bþ c ðþ It can be seen that the circuit includes three pairs of nullors from Eq. (). For one pair, the nullator is connected between node and the rounded node, and the norator between node and the rounded node. For another pair, the nullator is connected between node and the rounded node, and the norator between node and the rounded node. And for the third one, the nullator is connected between node and the rounded node, and the norator between node 4 and the rounded node. Therefore, the new active circuit topoloy is obtained as shown in Fi. 8. As assumed before, the elements a, b, c, d, and are selected as the resistor; h and i as the capacitor; e as the combination of a resistor and a capacitor; and nullors are in pair in succession from the input node to the output node. Then, the Tow-Thomas band-stop filter circuit shown in Fi. 9 is obtained. Alternatively, if we reorder the nullors in Fi. 8 as the followin sequence shown in Fi. 1, then the reconstructed circuit topoloy with nullors replaced by OPA is synthesized to the Åkerber-Mossber band-stop filter, which is shown in Fi. 11. Band-stop filter can be obtained by Eq. (), and a desin example is provided to illustrate the desin procedure. Desin a band-stop filter accordin to the parameters below: fp¼ khz; Qp ¼ ; K ¼ Where fp represents the stop-band central frequency, Qp represents the quality factor, K represents the passband voltae manification. vdb_tow -Thomas band-stop filter circuit vdb_akerber-mossber band-stop filter circuit Results of Tow-Thomas band-stop filter circuit and Akerber-Mossber band-stop filter circuit k 1.k freq Fi. 14 The simulation results of TT band-stop filter circuit and Åkerber-Mossber band-stop filter circuit

10 Tan et al. EURASIP Journal on Wireless Communications and Networkin (1) 1:9 Pae 1 of 1 A V ðsþ ¼ K S þ ω p S þ ω p Q p S þ ω p ðþ In this desin example, for simplicity, capacitor C is chosen to 1uF, and R 1 = R = R = R 4 =1 kω. Component values which are calculated accordin to Eq. () are listed in Table. The TT band-stop filter and the Åkerber-Mossber band-stop filter are simulated accordin to the component values presented above respectively, which are shown in Fis. 1 and 1. The simulation results are shown in Fi. 14, which explains that the results of the two band-stop filter are consistent. Conclusions The eneration method of the TT Bi-quad band-pass circuit, the TT band-stop filter circuit, and Åkerber- Mossber band-stop filter circuit is presented usin the theory of NAM expansion, which is a new desin method for circuit desin. The active circuit topoloies of the TT band-stop filter and the Åkerber-Mossber 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 desin in theory and practice. At the same time, further research needs to be conducted to enrich the method of NAM expansion.. DG Haih, A method of transformation from symbolic transfer function to active-rc circuit by admittance matrix expansion. IEEE Trans. Circuits Syst. I, Re. Papers (1), 1 8 () 4. RA Saad, AM Soliman, Generation, modelin, and analysis of CCII-based yrators usin the eneralized symbolic framework for linear active circuits. International Journal of Circuit Theory and Applications (), 89 9 (8). IA Awad, AM Soliman, On the voltae mirrors and the current mirrors. Analo Interated Circuits and Sinal Processin (1), 9 81 (). RA Saad, AM Soliman, Use of mirror elements in the active device synthesis by admittance matrix expansion. IEEE Trans. Circuits Syst. I, (8). Linlin Tan, Yu Bai, Jianfu Ten, Kaihua Liu, Wenqin Men, Transimpedance filter synthesis based on nodal admittance matrix expansion, circuits, Systems & Sinal Processin. (1). dio:1.1/s y 8. AM Soliman, Two interator loop filters: eneration usin NAM expansion and review. J. Electr. Comput. En 188(1), 1 (1). doi:1.11/ 1/ AM Soliman, Generation of current conveyor based oscillators usin nodal admittance matrix expansion. Analo Interated Circuits and Sinal Processin 1, 4 9 (1) 1. AM Soliman et al., Applications of voltae and current unity ain cells in nodal admittance matrix expansion. Circuit and System Maazine 9(4), 9 (9) 11. DG Haih, PM Radmore, Admittance matrix models for the nullor usin limit variables and their application to circuit desin. IEEE Trans. Circuits Syst. I, Re. Papers (1), 14 (1) 1. DG Haih, FQ Tan, C Papavassiliou, Systematic synthesis of active-rc circuit buildin-blocks. Anal. Inter. Circuits Sinal Process, Netherlands 4(), 9 1 () 1. RA Saad, AM Soliman, A new approach for usin the patholoical mirror elements in the ideal representation of active devices. International Journal of Circuit Theory and Applications 8(), (1) Acknowledements There is no other one to acknowlede in this section. Fundin This research was funded partially by the National Science Foundation of China under rant no. 11. Competin interests The authors declare that they have no competin 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 Linlin Tan provided some uidance on the method of circuit desin base on nodal admittance matrix expansion. GY as the correspondin author provided some suestions on the Enlish writin. All authors read and approved the final manuscript. Publisher s Note Spriner Nature remains neutral with reard to jurisdictional claims in published maps and institutional affiliations. Received: April 1 Accepted: 1 May 1 References 1. T Yanaisawa, N Kambayashi, Realization of arbitrary conductance matrix usin operational amplifiers. IEICE Trans. Part A J9-A(), (19). DG Haih, P Radmore et al., Systematic synthesis method for analoue circuits.part I. Notation and synthesis toolbox. ISCAS I, 1 4 (4)

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