Harmonic Distortion Modeling of Fully-BalancedThird-order Butterworth Gm-C Filter by a Block Diagram Approach

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1 0 nternational Conference on Computer Science and nformation Technology (CCST 0) PCST vol. 5 (0) (0) ACST Press, Sgapore DO: 0.776/PCST.0.V5.40 Harmonic Distortion Modelg of Fully-BalancedThird-order Butterworth Gm-C Filter by a Block Diagram Approach Ygwu Miao and Yuxg Zhang School of Electronic Engeerg University of Electronic Science and Technology of Cha, Chengdu, Cha Abstract. Accurate distortion modelg of fully-balanced Gm-C filter is the ma subject of this paper. Based on the phasor method, a block diagram approach was developed this paper. To demonstrate the flexibility of this approach, a third-order Butterworth Gm-C filter was analyzed by usg the proposed approach. A good agreement is shon between theoretical analysis and transistor level simulation by Spectre. The approach is suitable to be used early steps of the design and it can give the designer a thorough understandg of the distortion of fully-balanced Gm-C filters. Keywords: harmonic distortion; nonlear distortion; Gm-C filter; block-diagram approach; phasor method.. ntroduction Contue-time filter is extensively used a large number of electronic systems such as audio, video and communication products. t is apparent to all that the contue time filters implemented with transconductance amplifiers and capacitors known as Gm-C are very popular for some reasons such as reducg power dissipation or obtag tunability. t is usually that the learity is sacrificed for satisfyg speed or power constrats. As a result, the distorted output of the filter contas nonnegligible high-order harmonic components. t is hard to control the learity without deep understandg of the distortion behaviors of the Gm-C filter. n recent years, several papers were engaged distortion or nonlear analysis of the Gm-C filters []-[5]. t worth notg that the method adopted paper [5] is development of the analysis used to deal with distortion modelg of other circuit such as operational amplifier or tegrators [6]-[8]. Followg the pioneerg works of [5]-[8] and explorg the phasor method, we obtaed a block diagram approach. The proposed approach regards the full-balanced filter as multi-stage and feedback structures. Therefore, it is easy to understand and use. n addition, the proposed approach is flexible to deal with other filter structures. n this paper, the distortion modelg of a fully-balanced third-order Butterworth filter as shown Fig. was given to demonstrate accuracy of the approach. The paper is organized as follows. n section, the third-order Butterworth Gm-C filter is divided to two stages and the nonlear coefficients, the third-order distortion factor are calculated different subsections. n section, the theoretical results are compared with the transistor level simulation by Spectre. Simple discussions are given this section. Section V concludes this paper.. Distortion Modelg of The Third-Order Butterworth Gm-C Filter The approach developed this paper is suitable for distortion modelg of commonly used Gm-C filters. To simplify the derivation process, we divided the filter to two nonlear stages. The first stage is a nonlear transconductance stage and it is modeled subsection A. The second stage is a nonlear resistance Correspondg author. address: ygwu.miao@yahoo.com. 87

2 stage and it is analyzed subsection B. n subsection C, the nonlear coefficients of the two stages are assembled to obta the nonlear coefficients of the Gm-C filter. nonlear stage nonlear stage V g g g g g g V out Fig.. Third-order Butterworth Gm-C filter v g v g v, e, e i m i i, m, m v m g v g v, m, m i o a i m a i Fig.. Block diagram of st nonlear stage m.. Coefficients of st Nonlear Stage The block diagram of the first nonlear stage is shown Fig.. With the negative feedback and parallel connected capacitor at output, the second transconductance amplifier of the first nonlear stage, i.e. g, can m j t serve as a nonlear resistance. Providg the put current of the nonlear resistance i x e and the m s j t output voltage v = x e j t x e, we can sert them to the KC equation m, s, s i g v jc v () m m m Omittg high order terms and equatg the terms with same exponential, we get the expressions of nonlear resistance coefficients as, ( j) jc g, g,,( j) ( j C g,)( jc g,) Then we can use the similar steps to assemble the equivalent second resistance and the third amplifier g. The current ga coefficients representg the two parts dashed box Fig. can be written as a ( j) g ( j) (4),, a ( j ) g ( j ) g [ ( j )],,,, (5) After obtag the current ga coefficients, we can assemble them with the transconductance coefficients of the first transconductance amplifier the first nonlear stage. epeatg the steps as above, immediately we get the coefficients of the first nonlear stage as h ( j) a ( j) g (6), ( ) ( ), ( )(,) (7) h j a j g a j g.. Coefficients of nd Nonlear Stage Observg the circuit topology of the third-order Gm-C Butterworth filter Fig., we can fd easily the second transconductance amplifier provides a feedback path from output to put of this nonlear stage which is marked with a dash dotted le. Therefore, the block diagram of the second nonlear stage is shown Fig.. () () 88

3 g v g v, o, o i f i i d d a i a i d i m j C v o i i, d, d v e g v g v, e, e Fig... Block diagram of nd nonlear stage The connected two parts dashed box this stage is the same as the counterpart first nonlear stage. Hence, we can easily obta the current ga coefficients by usg the results above subsection and gave the expressions as a ( j) g ( j) (8),, where ( ),,( ),[,( )] (9) a j g j g j, ( j) jc g, g,,( j) ( j C g, )( jc g, ) Because the loadg effect of capacitors at output of the filter, the feedforward path Fig. is represented by a nonlear resistance. The coefficients of this nonlear resistance can be obtaed by assemblg the two blocks at lower part Fig.. The equivalent nonlear resistance can be represented by coefficients a ( j) ( j) jc g, jc ( jc g ), (0) () () a ( j) ( j) jc g g g ( j C g ) j C ( j C g )( jc g ),,,,,, () After these preparations, we can derive the coefficients of the second nonlear stage by usg the phasor j 0t method aga. Providg that put i x e j0t j0t and output v = h x e h x e, sertg them to KC s o s s equation i d = i i f, we get i ( g h ) x e [ g h g ( h ) ] x e (4) j0t j0t d, s,, s nsertg the expressions of v and i o d to the equation v i i, we get an equation as o d d 89

4 h x e h x e j 0t j 0t s s ( j )( g h ) x e ( j )[ g h g ( h ) ] x e j 0t j 0t 0, s 0,, s ( j) ( g h ) x e [ g h g ( h ) ] x e j 0t j 0t, s,, s (5) Omittg high-order terms, we easily obta the first-order coefficient of this stage as h ( j) ( j) (6) ( j ) g, Equatg the terms with third-order exponential equation (5), we get the third-order coefficient h g ( j )[ ( j)] ( j), ( j) [ ( j ) g,][ ( j) g,].. rd Order Harmonic Distortion Factor of the Filter After obtag coefficients of the two nonlear stages above subsections, we can derive the nonlear coefficients of the filter by assemblg the coefficients of first and second stages. The task can be completed with the help of phasor method as above analyses and it is neglected this subsection. Then the coefficients of the filter can be written as h, ( j) h ( j ) h ( j) h ( j)[ h ( j)] h ( j) h ( j) h ( j), ( j) a ( j) g, ( j) g, ( j )[ a ( j ) g a ( j)( g ) ] [ a ( j) g ] { g ( j )[ ( j)] ( j)},,,, ( j ) g, [ ( j ) g,][ ( j) g,] (9) Fally, the third-order harmonic distortion factor is obtaed conveniently and it is written at the bottom of this page. n many cases, the same design is used for every transconductance amplifier. Therefore, we can use a transconductance g place of g and g (x=,,) above equations. n many practical cases, the thirdorder nonlear transconductance coefficient g x x is proportional to first-order one g. As a result, we use the product of ratio and lear transconductance g to present the third-order nonlear coefficient g. Hence, the equation (0) can be simplified as (7) (8) h ( j) HD v 4 h ( j),, v ( j )[ a ( j ) g, a ( j)( g, ) ][ ( j) g,] [ a ( j) g, ] { g, ( j )[ ( j)] ( j)} 4 ( j) a ( j) g,[ ( j ) g,] ( j)[ ( j ) g,][ ( j) g,] (0) HD ( j ) v jc( jc g) g ( jc g) j Cg g [ ( jc) g ] 4 ( jc g) [ jc ( jc g)g ] j C g [ jc( jc g) g ] () From above simplified expression, we fd that the third-order harmonic distortion is directly proportional to ratio of g to g (=,, x=,,). Although the third-order harmonic is maly contributed by thirdorder nonlearity, the frequency-dependent behavior is strongly affected by the lear transconductance of the x, x, amplifier and the load capacitance. 840

5 V d d Cadence, V =0.V MATAB, V =0.V Cadence, V =0.V -50 MATAB, V =0.V Cadence, V =0.V o V V o HD (db) MATAB, V =0.V V s s Fig.4. Fully-differential OTA with herent CMF nput Frequency (Hz) Fig. 5. Third-order distortion factor curve. Simulation The accuracy of the proposed approach is verified this section by comparg results between theoretical analysis and transistor level simulation. The transconductance amplifier used this paper is a fullydifferential OTA with herent common mode feedback [9] and it is illustrated Fig.4. The third-order Gm-C Butterworth filter shown Fig. used a 0.5μm CMOS process and was simulated Spectre with susoid put signals with different amplitude. The tolerances of the simulator were set to relative small value to avoid excess numerical error. The output data of the transient simulation was post processed by Fourier analysis to get the harmonic components and the simulated third-order distortion factors. The curves of the simulated distortion factor are drawn Fig.5. The nonlear coefficients of the transconductance amplifier were extracted by usg Fourier analysis when a susoid excitation signal is used as put of the amplifier. With these steps, we obta the nonlear coefficients g = 8μA/V and g x, x, = 5μA/ V (=,, x=,,). The ratio of third-order nonlear coefficient to lear transconductance is calculated by the expression = g g. All the load capacitors were sized equal to 0 pf, which yieldg a cut-off frequency of. MHz. The equations derived previous section were plotted MATAB and drawn Fig.5. Excellent agreement has been observed between results of theoretical analysis and circuit simulation with EDA tool the frequency range of terest. 4. Conclusion n this paper, a block diagram approach based on phasor method was developed to model the distortion of a fully-balanced third-order Gm-C Butterworth filter. The theoretical results are accordance with the circuit simulation at transistor level. n addition, the approach used the paper shows flexibility for that it can be used to evaluate distortion of tegrator, biquad and other high-order Gm-C filter. More important, the proposed approach gives more sights and understandg of the distortion behaviors of fully-balanced filters. t can serve as a helpful tool to predict the distortion early steps of the contue time filters design and avoid time-consumg transient simulation. 5. eferences [] Zhaonian Zhang, Celik, A., Sotiriadis, P.P., A fast state-space algorithem to estimate harmonic distortion fullydifferential weakly nonlear Gm-C filters, Proc. SCAS 006, pp [] Zhaonian Zhang, Celik, A., Sotiriadis, P.P., State-space harmonic distortion modelg weakly nonlear, fully balanced Gm-C filters-a modular approach resultg closed-form solutions, EEE, Trans. Circuits Syst., Fundam. Theory Appl., vol.5, no., pp.48-59, Jan 006. [] Celik, A., Zhaonian Zhang, Sotiriadis, P.P., A State-Space Approach to ntermodulation Distortion Estimation Fully Balanced Bandpass Gm C Filters With Weak Nonlearities, EEE, Trans. Circuits Syst., Fundam. Theory Appl., vol.54, no.4, pp , April 007. [4] Sotiriadis, P.P., Celik, A., oizos, D., Zhaonian Zhang, Fast State-Space Harmonic-Distortion Estimation Weakly Nonlear Gm C Filters, EEE, Trans. Circuits Syst., Fundam. Theory Appl., vol.54, no., pp.8-8, Jan

6 [5] Palumbo, G., Pennisi, M., Pennisi, S., Distortion analysis the frequency doma of a Gm-C biquad, Proc. 8th European Conference on Circuit Theory and Design (ECCTD 007), Sevilla, Spa, pp.-5, Aug 007. [6] G.Palumbo, S.Pennisi, High-frequency harmonic distortion feedback amplifiers: analysis and application, EEE, Trans. Circuits Syst., Fundam. Theory Appl., vol.50, no., pp.8-40, Mar. 00. [7] S.O.Cannizzaro, G.Palumbo, S.Pennisi, Accurate estimation of high-frequency harmonic distortion two-stage Miller OTAs, EE Proc. Circuits, Devices Syst., vol.5, no.5, pp.47-44, Oct [8] S. O. Cannizzaro, G.Palumbo, S.Pennisi, An approach to model high-frequency distortion negative-feedback amplifiers, nternational Journal of Circuit Theory and Applications, vol.6, no., pp.-8, Jan/Feb 008. [9] Sanchez-Sencio, E., Silva-Martez, J.: CMOS transconductance amplifiers, architectures and active filters: a tutorial, EE Proc. Circuits, Devices Syst., vol.47, no., pp.-, Feb