Finding the Minimum Input Impedance of a Second-Order Unity-Gain Sallen-Key Low-Pass Filter without Calculus

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1 University f New Orleans SchlarWrks@UNO Electrical Engineering Faculty Publicatins Department f Electrical Engineering -3 Finding the Minimum Input Impedance f a Secnd-Order Unity-Gain Sallen-Key Lw-Pass Filter withut Calculus Kenneth V Cartwright The Cllege f the Bahamas Edit J Kaminsky University f New Orleans, ejburge@unedu Fllw this and additinal wrks at: Part f the Electrical and Electrnics Cmmns Recmmended Citatin Cartwright, KV, and E J Kaminsky, Finding the minimum input impedance f a secnd-rder unity-gain Sallen-Key lw-pass filter withut calculus, Lat Am J Phys Educ, vl 7, n, Dec 3, pp This Article is brught t yu fr free and pen access by the Department f Electrical Engineering at SchlarWrks@UNO It has been accepted fr inclusin in Electrical Engineering Faculty Publicatins by an authrized administratr f SchlarWrks@UNO Fr mre infrmatin, please cntact schlarwrks@unedu

2 Finding the minimum input impedance f a secndrder unity-gain Sallen-Key lw-pass filter withut calculus Kenneth V Cartwright and Edit J Kaminsky Schl f Mathematics, Physics and Technlgy, Cllege f The Bahamas, PO Bx N9, Nassau, Bahamas Department f Electrical Engineering, EN 8 Lakefrnt Campus, University f New Orleans, New Orleans, LA 78, USA kvcartwright@yahcm (Received 3 July 3, accepted 8 December 3) Abstract We derive an expressin fr the input cmplex impedance f a Sallen-Key secnd-rder lw-pass filter f unity gain as a functin f the natural frequency, quality factr and the rati f the resistrs f the filter Frm this expressin, it is shwn that the filter behaves like a single capacitr fr lw frequencies and as a single resistr at high frequencies Furthermre, the minimum input impedance magnitude is fund withut using calculus We discvered that the minimum input impedance magnitude is inversely prprtinal t and can be substantially less than its highfrequency value Apprximatins t the minimum input impedance and the frequency at which it ccurs are als prvided Additinally, PSpice simulatins are presented which verify the theretical derivatins Keywrds: Sallen-Key lw-pass filter, Minimum withut calculus, Input impedance Resumen Derivams una expresión para la impedancia de entrada cmpleja de un filtr Sallen-Key de pas baj de segund rden en función de la frecuencia natural, el factr de calidad, y el cciente de ls resistres del filtr Cmenzand cn esta expresión, mstrams que el filtr se cmprta cm un sól capacitr para frecuencias bajas y cm un sól resistr para altas frecuencias Es más, encntrams la magnitud de la impedancia mínima sin usar cálcul Descubrims que la magnitud de la impedancia mínima es inversamente prprcinal a y puede ser significativamente menr que a frecuencias altas Prveems aprximacines para la impedancia de entrada mínima y para la frecuencia a la que curre Presentams también simulacines en PSpice que verifican las derivacines teréticas Palabras Clave: Filtr Sallen-Key de pas baj, Mínim sin cálcul, Impedancia de entrada PACS: 83Vn, 9+p, Ha and Fk ISSN I INTRODUCTION Fig shws the circuit diagram f an active secnd-rder unity-gain Sallen-Key lw-pass filter, which is widely used in electrnics One imprtant parameter f such a filter is its transfer functin, which has been widely studied and which relates its utput vltage t its input vltage Anther imprtant parameter is its input impedance Unfrtunately, very little has been written abut this input impedance, even thugh designers need t knw its minimum value t ensure that the filter des nt lad dwn the surce r a previus stage Inspectin f Fig wuld suggest t the naive designer that the minimum input impedance is R as it is in series with the rest f the circuit Unfrtunately, this is nt the case: the input impedance can be very much lwer than R as we shw belw Indeed, the authrs recent design f a secnd-rder unity-gain Sallen-Key filter failed t achieve its design criteria because the filter did nt interface well with its surce, as we initially used the naive assumptin given abve This experience prmpted us t fully investigate the input impedance and its minimum value Furthermre, we find the minimum value f the input impedance withut using calculus, which shuld be f benefit t the student wh has nt yet had the pprtunity t study math at this level In this paper, we reprt ur theretical findings, which are als verified by PSpice simulatins (PSpice is a ppular electrical and electrnic circuits simulatin sftware package that is widely used by electrical engineers and sme physicists The latest dem versin can be freely btained frm []) Lat Am J Phys Educ Vl 7, N, Dec

3 Kenneth V Cartwright and Edit J Kaminsky III INPUT IMPEDANCE FOR THE SALLEN- KEY LPF OF FIGURE As we shw in the Appendix, the nrmalized cmplex input impedance Z( s) / R fr the circuit f Fig is given by FIGURE Circuit diagram f secnd-rder unity-gain Sallen- Key lw-pass filter II TRANSFER FUNCTION FOR THE SALLEN- KEY LPF OF FIGURE In this sectin, the transfer functin fr the circuit f Fig will be given, s that the key parameters such as natural frequency and quality factr can be defined Indeed, it is straightfrward t shw, as demnstrated in the Appendix, that the transfer functin is given by Eq (9) f [] (remembering that the gain f the p-amp is unity as it is in a vltage fllwer cnfiguratin), ie, V, ut Vin s R RC C R R Cs where s i, with i and (rad/sec) is the angular frequency f the applied sine-wave (The reader shuld be aware that Fig 5 f [] differs frm ur Fig : C and C are interchanged This means that Eq (9) f [] will als require that C and C be interchanged in Eq () Many authrs, eg, [3], als use ur Fig, rather than Fig 5 f []) The denminatr f Eq () can be written as s R RCC R RCC s R R C C s s, R R C where the natural frequency is / R RC C and, R RC C the quality factr f the filter, is R R C () () Zs () s R R C C R R C s R s R R C C R C s = s R R R C R R RC RC s s s s s =, s k s R where k R R R R Nte that the input impedance depends upn the rati f the resistrs, r R/ R: in rder t have lw sensitivity filters, this rati shuld be unity (ie, (3) k / ) as r pinted ut in [] Nevertheless, fr varius reasns, designers might nt heed t this chice f k :hence, we will give results fr general k values Interestingly, Eq (3) becmes unity fr large frequencies, ie, the input impedance lks simply as R, and the phase is On the ther hand, fr lw frequencies (as appraches zer), Eq (3) becmes, apprximately, k s, ie, the input impedance lks like a capacitr k whse value is given by Ce R Using the expressins fr, and k just given, it is clear that Ce C Hence, fr lw frequencies, the magnitude f the input impedance is just the reactance f this capacitr, ie, R, and the phase appraches 9 C C k e A Magnitude f the Input Impedance Frm Eq (3), the magnitude f the nrmalized impedance becmes Zs () R p ip /, k p ip where p / is the nrmalized frequency () Lat Am J Phys Educ Vl 7, N, Dec 3 5

4 Nrmalized Input Impedance (db) Phase f Input Impedance (deg) Finding the minimum input impedance f a secnd-rder unity-gain Sallen-Key lw-pass filter withut calculus Actually, it will be mre cnvenient t wrk with the magnitude squared f the nrmalized impedance Hence, p p tan 9 tan Eq () becmes (7) p k Zs () p p / Fr p, the phase becmes tan and fr R k k p p p, Eq (7) can be written as (5) p / p p p 9 tan tan (8) k p k p p Nte frm Eq (7) that fr and frm Eq (8) that as, These facts are als cnfirmed Furthermre, Eq (5) can be rewritten as by a plt f Eq (7) and Eq (8) shwn in Fig 3 where Z( s) x Ax, R x Bx x p, A / and k B Taking the square rt f Eq () allws us t make a plt f the nrmalized impedance in db (ie, lg Z(s) / R as a functin f the nrmalized frequency, as shwn in Fig fr k 5 and varius values Als shwn are straight-line plts f the reactance f the equivalent capacitr Ce C, cnfirming ur earlier statement that the magnitude f the input impedance fr lw frequencies is simply the reactance f C Clearly, Fig als verifies the high-frequency value f the input impedance nted earlier 5 3 () ( ) =5 = = = Nrmalized Frequency =5 = = = FIGURE 3 Phase f input impedance (deg) as a functin f the nrmalized frequency fr k = 5 Clearly, Fig 3 verifies the lw-frequency and highfrequency values f the phase f the cmplex input impedance nted earlier Nrmalized Frequency FIGURE Magnitude f the nrmalized input impedance (in db) as a functin f the nrmalized frequency fr k 5 The straight-lines are plts f the reactance (in db) f Ce C B Phase f the Input Impedance The phase f the input impedance is easily fund frm Eq (3) t be IV FINDING THE MINIMUM INPUT IMPEDANCE MAGNITUDE WITHOUT CALCULUS Nw that the nrmalized input impedance has been fund, it can be shwn hw its minimum value can be fund withut calculus Using lng divisin, Eq () can be written as a prper ratinal functin, ie, Zs () R x Bx A B x Zm Clearly, fr Eq (9) t be a minimum, must be a maximum Zm (9) A B x x Bx Lat Am J Phys Educ Vl 7, N, Dec

5 Nrmalized Minimum Frquency Kenneth V Cartwright and Edit J Kaminsky Let y A B x Hence, y A B Zm y B A B y B A B () min k k k k Frtunately, it is straightfrward t find the maximum value f Eq () withut using calculus Indeed, the methd we used in a previus paper [5] will be utilized here as well T begin, first divide the numeratr and denminatr f Eq () by y t get A B Zm y B A B y B A B A B B A B y B A B B A B y () Clearly, Eq () is maximized when y B A B s that y B A B y is minimized, ie, made t be zer, y Recall that x and x p / Hence, A B the frequency at which Zm is a maximum and Eq (9) is a minimum is given by min B A B A B Ntice that when y B A B f Eq () becmes Zm, the maximum value A B max () (3) B A B B A B Hence, frm Eq (9), the minimum value f the input impedance becmes A B Z( s) R min () B A B B A B k Recall that A / and B Hence, the nrmalized frequency at which the minimum input impedance ccurs is, frm Eq (), k k k k k k, k (5) and, frm Eq (), the nrmalized minimum input impedance magnitude is k Z( s) min R k k k k k k k R k k k k () T illustrate hw the nrmalized minimum frequency depends upn the quality factr f the filter, Eq (5) is pltted in Fig fr three values f k uality Factr FIGURE Nrmalized minimum frequency as a functin f the quality factr f the filter In rder fr Eq (5) t be valid, k=5 k=5 k=75 k Hence, k in rder fr there t be a minimum in the Lat Am J Phys Educ Vl 7, N, Dec

6 Perentage Errr Perentage Errr Nrmalized Impedance k input impedance magnitude Fr, the input impedance decreases mntnically frm infinity t R as the frequency increases frm zer t infinity This is illustrated in Fig 5 fr k 5 Ntice frm Fig 5 that if there is a minimum, then there is als a frequency at which the nrmalized impedance is unity Frm Eq (), this frequency is determined t be 5 3 unity k Finding the minimum input impedance f a secnd-rder unity-gain Sallen-Key lw-pass filter withut calculus (7) = = Als, fr large (5) becmes, k k Hence, Eq min k By the way, cmparing Eq (7) with Eq () reveals that () min unity () A plt f the percentage errr, (-Aprximate Value/True Value), f Eq (8) is shwn in Fig, where it is clearly seen that the percentage errr depends upn the value f k, fr small values f ; hwever, as increases this dependence becmes smaller and smaller 8 k=5 k=5 k= Nrmalized Frequency FIGURE 5 Nrmalized impedance as a functin f the nrmalized frequency, fr k 5, fr values n either side f min k / 3 / 8 In rder fr a minimum impedance t exist, min Nte als, frm Eqs (7) and (5), that lwer bund n min unity A Apprximatins t the Minimum Frequency behaves as a Using the first line f Eq (5) and the fact that r rr z rz z fr small z [], it is straightfrward t shw that fr large, min, (8) r mre accurately, as btained with Matlab s Taylr functin perating n Eq (5): min 8k 3 (9) uality Factr FIGURE Percentage errr f Eq (8) apprximatin t the minimum frequency fr varius values f k k=5 k=5 k= uality Factr FIGURE 7 Percentage errr f Eq (9) apprximatin t the minimum frequency fr varius values f k Lat Am J Phys Educ Vl 7, N, Dec

7 Perentage Errr Perentage Errr Perentage Errr Kenneth V Cartwright and Edit J Kaminsky On the ther hand, the percentage errr f Eq (9), shwn in Fig 7, depends upn k nly in a minr way Surprisingly, Eq (8) is mre accurate than Eq (9) fr sme and k values, eg, 7 and k 75 Nnetheless, Eq (9) is clearly mre accurate than Eq (8) fr greater than 5 apprximately, fr all k values studied here Furthermre, recall that Eq () was derived assuming large ; hence, it is smewhat surprising that the percentage errr is quite gd fr all values, as shwn in Fig k=5 k=5 k= k=5 k=5 k= uality Factr FIGURE 9 Percentage errr f Eq () apprximatin t the minimum magnitude f the impedance fr varius values f k Pltting the percentage errr f Eq (3) in Fig shws that it prvides a mre accurate estimatin fr the minimum input impedance than Eq () des uality Factr FIGURE 8 Percentage errr f Eq () apprximatin t the minimum frequency fr varius values f k B Apprximatins t the Minimum Input Impedance k=5 k=5 k=75 Wrking with the secnd line f Eq (), it is straightfrward t btain the fllwing apprximatin t the nrmalized minimum input impedance magnitude fr large : R Zs ( ), () min uality Factr FIGURE Percentage errr f Eq (3) apprximatin t the minimum magnitude f the impedance fr varius values f k r mre accurately (as fund with Matlab sftware): R k Zs ( ) min 8 (3) Frm Fig, it is apparent that the minimum input impedance ccurs apprximately when p, fr large values Hence, substituting p int Eq (5) gives A plt f the percentage errr f Eq () is shwn in Fig 9, where it is clearly seen that the percentage errr depends upn the value f k Hwever, as increases the percentage errr decreases fr all values f k Z( s) R min k () The percentage errr fr Eq (), shwn in Fig, is clearly much better than that f Eq () and is mre easily derived than Eq (3) Lat Am J Phys Educ Vl 7, N, Dec

8 Perentage Errr Perentage Errr FIGURE Percentage errr f Eq () apprximatin t the minimum magnitude f the impedance fr varius values f k Finding the minimum input impedance f a secnd-rder unity-gain Sallen-Key lw-pass filter withut calculus k=5 k=5 k= uality Factr V PSPICE SIMULATIONS In rder t verify the theretical derivatins, we perfrmed PSpice simulatins f the filter in Fig Fr all the simulatins, we set rad/s r f 595 Hz,, k 5 and R A Design f the Filter T simulate the filter, we must select values fr the capacitrs R rr k R / k and that Recall that / R R C C ; hence, R, and R k C C k () Surprisingly, by trial and errr, we fund that Eq () can be imprved significantly by simply adding a cnstant in its denminatr t get Z( s) R min k 3 (5) As shwn in Fig, the percentage f errr fr the apprximatin f Eq (5) is less than % fr all values The cnstant was chsen s that the psitive and negative peak percentage errrs are apprximately the same magnitude fr 5 k 75 If k values utside this range are used, the cnstant will have t be mdified fr best results k=5 k=5 k= uality Factr FIGURE Percentage errr f Eq (5) apprximatin t the minimum magnitude f the impedance fr varius values f k Als, recall that R R C C R R C ; therefre, k R CC k C k k k C R R C k Slving Eq () and Eq (7) simultaneusly gives C F and C 5 F (7) k Furthermre, recall that Ce R Using Eq () and Eq (7), it is straightfrward t shw that Ce C, as we claimed earlier B Verificatin f the Design f the Filter The first thing we want t d with ur simulatins is t verify that ur design has met ur specificatins fr and T d this, we find the maximum gain f the filter, T max, by pltting the magnitude respnse f the simulated filter, as shwn in Fig 3 Frm this graph, it is clear that the maximum gain is 9 db r 57 Hwever, frm Eq () f [5], T / / ( ) 57; hence, the max simulated Als, frm Fig 3, the frequency at which the maximum gain ccurs is fund t be max rad/s Hence, the simulated natural frequency is Lat Am J Phys Educ Vl 7, N, Dec

9 Kenneth V Cartwright and Edit J Kaminsky max / rad/s (See Eq () f [5]) Clearly, the parameters f the simulatin match the theretical design quite well C Verificatin f the Magnitude f the Input Impedance PSpice measures the magnitude f the input impedance by dividing Vin by IR ( ), the current thrugh R Furthermre, the PSpice cmmand db( Vin / I( R )) measures lg Z, max ie, the magnitude f the input impedance in db, a plt f which is shwn in Fig Als shwn in this figure is the straight-line plt f the reactance f the equivalent capacitr fr lw-frequencies, which is lg lg C f Clearly, the simulated plt cincides with the reactance straight-line at lw frequencies, as expected Furthermre, the simulated plt shws that the magnitude f the input impedance appraches db r, as expected fr high-frequencies Additinally, frm Eq (7), the theretical value f rad/s r 35 Hz Frm Fig unity, PSpice simulates this as Hz r 8935 rad/s (95,9) Hz 3Hz Hz 3Hz KHz DB(V(Vut)) Frequency FIGURE 3 Simulated magnitude respnse (db) f the filter (7,85) (8733,589879) 5 Hz 3Hz Hz 3Hz KHz DB(V(Vin)/I(R)) DB(e/pi/ Frequency) Frequency FIGURE Simulated magnitude (db) f the input impedance f the filter The straight-line is the reactance f the equivalent capacitr at lwfrequencies, ie, the reactance f C Lat Am J Phys Educ Vl 7, N, Dec

10 Finding the minimum input impedance f a secnd-rder unity-gain Sallen-Key lw-pass filter withut calculus Als, frm Fig, fmin 873 Hz r min 35 rad/s The theretical value fr this is given by Eq (5), ie, / 3 rad/s r 5 given apprximately by Eq (), ie, 8/5 9 rad/s Furthermre, frm Fig, the minimum magnitude f the input impedance is db r 878 On the ther hand, Eq () gives the theretical value fr this as Alternatively, Eq (5) gives the apprximate value f d -d -d -d (598,-3859) (593989,-38358) -8d -d -d Hz 3Hz Hz 3Hz Hz 3Hz KHz 3KHz KHz P(V(Vin)/I(R)) Frequency FIGURE 5 Simulated phase respnse (deg) f the input cmplex impedance f the filter D Verificatin f the Phase f the Input Impedance The PSpice cmmand P( Vin / I( R )) measures the phase f the input impedance in degrees, a plt f which is shwn in Fig 5 As can be seen, the phase becmes 9 at lwfrequencies and at high frequencies, as expected Furthermre, recall (frm Eq (7) with p =) that the theretical phase at the natural frequency Hz) is given by tan / k (/( ) 5955 tan () 339 On the ther hand, the PSpice simulatin gives a phase shift f -38 at a frequency f 59 Hz and a phase shift f -383 at a frequency f 5939 Hz Therefre, interplating between these tw pints gives a PSpice simulated value f 399 at the natural frequency E Summary f Theretical-PSpice Cmparisn Fr cnvenience, the theretical and PSpice results given abve are summarized in Table I As can be seen, there is excellent agreement between the tw TABLE I Summary Theretical-PSpice cmparisn Item Theretical Value PSpice Value rad/s rad/s unity 897 rad/s rad/s min 3 rad/s 3 rad/s Phase at Min Input Impedance Lat Am J Phys Educ Vl 7, N, Dec

11 Kenneth V Cartwright and Edit J Kaminsky VI CONCLUSIONS We have derived an expressin fr the input cmplex impedance fr the secnd-rder unity-gain Sallen-Key lwpass filter, which is given in Eq (3) Frm this expressin, we have shwn that the input cmplex impedance is R fr high-frequencies, whereas fr lw-frequencies it is simply i C Furthermre, we have fund the minimum f the magnitude f the input impedance withut calculus, as given in Eq () and its apprximatins in Eq ()-Eq (5) We have als discvered an expressin fr min, as given in Eq (5) and its apprximatins in Eq (8)-Eq () Finally, we prvided PSpice simulatins which verified the theretical results In future wrk, we intend t study the input impedance fr arbitrary gain Sallen-Key lw-pass and high-pass filters REFERENCES [] Orcad PSpice dem dwnlad: px#dem [] Karki, J, Analysis f the Sallen-Key architecture, Texas Instruments Applicatin Reprt SLOAB, () Available frm: fthesallen-keyarchitecturepdf Accessed: th April, 3 [3] Zumbahlen, H, Sallen-Key filters, Analg Devices Mini Tutrial MT-, () Available frm: pdf Accessed: th April, 3 [] Minimizing cmpnent-variatin sensitivity in single p amp filters, Maxim Integrated Tutrial 738, () Accessed: th April, 3 [5] Cartwright, K V, Russell, P, and Kaminsky, E J, Finding the maximum magnitude respnse (gain) f secndrder filters withut calculus, Lat Am J Phys Educ, () [] Mungan, C E, Three imprtant Taylr series fr intrductry physics, Lat Am J Phys Educ 3, (9) and V (s) sc V (s) (A) R+ sc I (s)= =, scr + V ( s) Vut ( s) I( s) V ( s) Vut ( s) sc (A) sc Nticing that the vltage acrss C is Vut () s because the input t a vltage fllwer is equal t its utput vltage, and using vltage divisin, we btain: r sc V s V s R sc ut ( ) ( ), V (s)= sc R + V (s) ut Using (A) in (A) and (A) gives, respectively: and Using KCL, I () s (A3) (A) I ( s) sc V ( s), (A5) ut scr Vut ( s) Vut ( s) sc s C C R V ut in ( s) I (s) = I (s)+ I (s) = sc sc R + V ut (A) (A7) Writing the lp equatin fr the leftmst lp using KVL, we btain: APPENDIX V in(s)= I in(s)r + I (s) R + sc (A8) In this appendix, we derive Eqs () and (3) Let Iin () s be the current thrugh the surce V I () s be the current thrugh C, frm tp t bttm, and I () s be the current thrugh R frm left t right Lking at the nde marked with vltage V (s) we btain: Finally, using (A7) and (A5) in (A8), we get the expressin relating the utput vltage t the input vltage: Vin ( s) sc sc R R Vut ( s) sc R Vut ( s) sc Vut ( s) sc R sc R scr (A9) Lat Am J Phys Educ Vl 7, N, Dec

12 Finding the minimum input impedance f a secnd-rder unity-gain Sallen-Key lw-pass filter withut calculus Eq (A9) can clearly be written as the transfer functin shwn in Eq () In rder t find the input cmplex impedance, we simply divide the input vltage by the input current, using (A9) and (A7), t get: Vin () s Zs () I () s in V s s R R C C sc R R sc sc R V ( s) ut ( ) s R RCC sc R R s CC R sc ut (A) Finally, dividing by R we btain the nrmalized cmplex impedance f Eq (3) Lat Am J Phys Educ Vl 7, N, Dec