Finding the maximum and minimum magnitude responses (gains) of third-order filters without calculus

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1 Fining the maximum an minimum magnitue responses (gains) of thir-orer filters without calculus Kenneth V Cartwright 1, Patrick Russell 1 an Eit J Kaminsky 1 School of Mathematics, Physics, an Technology, College of The Bahamas, PO Box N491, Nassau, Bahamas Department of Electrical Engineering, EN 846 Lakefront Campus, University of New Orleans, New Orleans, LA 70148, USA kvcartwright@yahoocom (Receive 1 July 01, accepte December 01) Abstract The maximum an minimum gains (with respect to frequency) of thir-orer low-pass an high-pass filters are erive without using calculus Our metho uses the little known fact that extrema of cubic functions can easily be foun by purely algebraic means PSpice simulations are provie that verify the theoretical results Keywors: Filters, Maximum an minimum without calculus, PSpice simulation Resumen Derivamos las ganancias máxima y mínima (con respecto a la frecuencia) e filtros e tercer oren e paso bajo y e paso alto sin usar cálculo Nuestro métoo utiliza el hecho poco conocio que los extremos e funciones cúbicas pueen encontrarse fácilmente con métoos puramente algebraicos Verificamos los resultaos teoréticos con simulaciones en PSpice Palabras Clave: Filtros, Máximo y mínimo sin cálculo, simulaciones en PSpice PACS: 840Vn, 090+p, 0140Ha an 0140Fk ISSN I INTRODUCTION Thir-orer filters are common in electronics, an for a subgroup of thir-orer filters, two important features are the maximum an minimum of the magnitue responses (gains) attainable by the filters, whether they are low-pass or high-pass filters Conventionally, the mathematical expressions for these extrema values are foun with calculus, or inee simply state without erivation However, this puts the curious stuent who has not yet ha the chance to stuy calculus at a isavantage Presently, he or she has no choice but to accept the equations for the maximum an minimum gains without any unerstaning as to their origins Fortunately, as we show in this paper, it is straightforwar to erive these maximum an minimum gains without calculus To o this, we use the little known fact that extrema of cubic functions can be foun through algebraic means alone [1,,, 4] Furthermore, PSpice simulations will be use to verify the theoretical equations (PSpice is a popular electrical an electronic circuits simulation software package that is wiely use by electrical engineers an some physicists The latest emo version can be freely obtaine from [5]) II GAIN OF THIRD-ORDER FILTERS In this section, the gain of thir-orer filters is given A Low-Pass Filter The transfer function of a general thir-orer low-pass filter is given by Ts ( ), s as bs c where a, b, c, are constants an with an is the angular frequency of the applie sine-wave For example, the transfer function of a thir-orer 1-B Chebyshev low-pass filter with cut-off frequency of 1 ra/s is given by (pg1 of [6]) 0491 Ts ( ) s 0988s 184s 0491 (1) () Lat Am J Phys Euc Vol 7, No 4, Dec

2 Kenneth V Cartwright, Patrick Russell an Eit J Kaminsky The gain of the thir-orer low-pass filter is simply the magnitue of Eq (1), ie, T( ) a c b () 0491 Ts () s s s s s 506s 0117s 054 (5) A sketch of Eq () is shown in Fig 1, for the subgroup of thir-orer low-pass filters in which we are intereste, where the following are clearly ientifie: (i) maximum gain,, (ii) the frequency at which the maximum gain occurs,, (iii) minimum gain,, an (iv) the frequency at which the minimum gain occurs, The gain of the thir-orer high-pass filter is the magnitue of Eq (4), ie, T( ) a c b A sketch of Eq (6) is shown in Fig, for the subgroup of thir-orer high-pass filters in which we are intereste, where, again, (i) through (iv) are clearly ientifie: (6) FIGURE 1 Magnitue response (gain) low-pass filter with ripple in the pass-ban of a thir-orer FIGURE Magnitue response (gain) high-pass filter with ripple in the pass-ban of a thir-orer B High-Pass Filter The transfer function of a general thir-orer high-pass filter is given by s Ts ( ), s as bs c For example, the transfer function of a thir-orer 1-B Chebyshev high-pass filter with cut-off frequency of 1 ra/s can be erive from Eq () by using the low-pass to highpass transformation metho, ie, replacing s with 1/s (see Example 851 of [7]) to get (4) III DETERMINING THE MAXIMUM AND MINIMUM GAINS OF THIRD-ORDER FILTERS WITHOUT USING CALCULUS Now that the gains of thir-orer low-pass an high-pass filters have been state, we can show how the maximum an minimum gains can be etermine without using calculus We will use the following fact that is traitionally establishe with calculus, but which is easily verifie by purely algebraic means [1,,, 4], as shown in the Appenix Fact: Suppose f x Ax Bx Cx D, then the f x occurs when x is a minimum or maximum value of root of Ax Bx C, i e, Ax Bx C 0 Lat Am J Phys Euc Vol 7, No 4, Dec

3 A Low-Pass Filter From Eq (), the square of the gain of the thir-orer lowpass filter is T ( ) a c b 6 4, x Bx Cx D a b b ac c where x, B a b, C b ac, an D c Clearly, the minimum/maximum value of Eq () or Eq (7) occurs when the enominator of Eq (7) is a maximum/minimum value, ie, at the solutions of B C x x 0 Hence, min or max Fining the maximum an minimum magnitue responses (gains) of thir-orer filters without calculus (7) B B C (8) Clearly, the minimum/maximum value of Eq (6) or Eq (9) occurs when the enominator of Eq (10) is a maximum/minimum value, ie, at the solutions of B C x x 0 Hence, minor max 1 B B C (11) Note again that we must have B C in orer for there to be a maximum an minimum gain Substituting Eq (11) into Eq (9) gives the square of the maximum or minimum gain of the thir orer high-pass filter IV PSPICE SIMULATIONS In this section, we will use PSpice simulations to verify the equations for the extrema of the gains an the frequencies at which they occur for the thir-orer low-pass an high-pass filters Note that we must have B C in orer for there to be a maximum an minimum gain Substituting Eq (8) into Eq (7) gives the square of the maximum or minimum gain of the thir orer low-pass filter B High-Pass Filter From Eq (6), the square of the gain of the thir-orer highpass filter is T ( ) Letting 6 a c b 6 a b b ac c a b b ac c x D 1/ c, we get 4 6, B b ac / c, c T( x) x Bx Cx D (9) C a b / c, an (10) A Low-Pass Filter For the low-pass filter example use in Eq (), a 0988, b 184, c 0491, an c Hence, B 1500 an C 0565 So, from Eq (8), min or max ra/s (1784 mhz) or ra/s (79577 mhz) Furthermore, substituting these values into Eq (7) an taking the square root gives T ( ) (0 B) or ( B) max or min To check these theoretical values, a PSpice simulation of the thir-orer Chebyshev 10 B low-pass filter was one using the iagram shown in Fig 1Vac V1 NUM=0491 IN FIGURE PSpice implementation of a thir-orer Chebyshev 10 B low-pass filter with cut-off frequency of 1 ra/s OUT DENOM=s*s*s+988*s*s+184*s Vo R1 1k Lat Am J Phys Euc Vol 7, No 4, Dec

4 Kenneth V Cartwright, Patrick Russell an Eit J Kaminsky Upon completion of the simulation, the PSpice magnitue response was plotte in Fig 4, using 1000 points per ecae From this figure, it is clear that (i) Eq () is in fact the transfer function of a thir-orer Chebyshev 10 B lowpass filter; otherwise, the magnitue response woul be ifferent from that observe, an (ii) the theoretical values are very close to the simulate values given in the graph, as summarize in Table I TABLE I Comparison of theoretical an PSpice simulate values Parameter Theoretical Value Simulate Value f cutoff mhz mhz f min mhz mhz f max 1784 mhz 175 mhz T( ) min B B T( ) 0 B 450x10-4 B max 4 (17544m,44999u) 0-4 (796868m, ) ( m, ) mHz 0mHz 100mHz 00mHz 10Hz DB(V(Vo)) Frequency FIGURE 4 PSpice magnitue response (gain in B) of a thir-orer Chebyshev 10 B low-pass filter, as simulate with Fig B High-Pass Filter For the high-pass filter example use in Eq (5), a 506, b 0117, c 054 an 1 Hence, B 1499 an C 0564 So, from Eq (11), min or max ra/s (1878 mhz) or 0001 ra/s (18 mhz) Furthermore, substituting these values into Eq (9) an taking the square root gives T ( ) (0 max or min B) or ( B) To check these theoretical values, a PSpice simulation of the thir-orer Chebyshev 10 B high-pass filter was one using the iagram shown in Fig 5 Upon completion of the simulation, the PSpice magnitue response was plotte in Fig 6 From this figure, it is clear that (i) Eq (5) is in fact the transfer function of a thir-orer Chebyshev 10 B low-pass filter; otherwise, the magnitue response woul be ifferent from that observe, an (ii) the theoretical values are very close to the simulate values given in the graph, as summarize in Table II 1Vac V1 NUM=s*s*s IN FIGURE 5 PSpice implementation of a thir-orer Chebyshev 10 B high-pass filter with cut-off frequency of 1 ra/s OUT DENOM=s*s*s+506*s*s+0117*s Vo R1 1k Lat Am J Phys Euc Vol 7, No 4, Dec

5 Fining the maximum an minimum magnitue responses (gains) of thir-orer filters without calculus TABLE II Comparison of theoretical an PSpice simulate values Parameter Theoretical Value Simulate Value f cutoff mhz mhz f min 18 mhz 1858 mhz f max 1878 mhz 1851 mhz T( ) min B B T( ) max 0 B x10-6 B V CONCLUSIONS We have shown that the minimum an maximum gains of thir-orer low-pass an high pass filters can be foun without using calculus The metho uses the little known mathematical fact that the extrema of cubic functions can be foun by purely algebraic methos Furthermore, PSpice simulations were shown to verify the theoretical calculations 0 (185148m,808017u) ( m, ) -5 ( m, ) mHz 00mHz 10Hz 0Hz 10Hz DB(V(Vo)) Frequency FIGURE 6 PSpice magnitue response (gain in B) of a thir-orer Chebyshev 10 B high-pass filter, as simulate with Fig 5 REFERENCES [1] Lennes, N J, Note on maxima an minima by algebraic methos, The American Mathematical Monthly 17, Issue 1, 9-10 (1910) [] Peterson, Max an min of functions without erivative, The Math Forum, April n, 00 accesse on 10 th May, 01 [] Cherkas, B, Fining polynomial an rational function turning points in precalculus, International Journal of Computers 8, 15-4 (00) [4] Taylor Jr, R D an Hansen, R, Optimization of cubic polynomial functions without calculus, The Mathematics Teacher 101, (008) [5] Orca PSpice emo ownloa: spx#emo [6] Stensby, J, Chebyshev filters, : accesse on 10 th May, 01 [7] Practical filters, wwwmrcuiahoeu/~ennis/ece450- F11/85%0Practicaloc: accesse 10th May, 01 APPENDIX Let r1, b be the coorinate of one of the extrema of the cubic f(x) as shown in Fig A1 Then, to fin an extremum of the cubic f(x), we seek the intersection of f(x) with the line g(x) = b, with g(x) tangent to f(x) at r 1, b If the line is to be tangent to the curve at r 1, b, two of the three roots r1, r1 an r of f(x)-b must be coincient [1], as shown in Fig A Hence, FIGURE A1 Plot of the function f () x Ax Bx Cx D intersecting with the line g( x) b Note that the line is tangent to the cubic function at the extremum point Lat Am J Phys Euc Vol 7, No 4, Dec

6 Kenneth V Cartwright, Patrick Russell an Eit J Kaminsky (Note that the necessity of Ax Bx Cx D b A x r x r 1 was also proven by purely algebraic means in Eq (1) of []) Equating coefficients of Eq (A1) gives: 1 A r r B, (A) 1 1 Ar r r C, (A) FIGURE A Vertical translation of the cubic function of Fig A1 Note that there is now a ouble root of the translate cubic f ( x) b at the extremum point From Eq (A), Ar r D b (A4) Ax Bx Cx D b A x r x r A x xr r x r Ax x r r xr r r r r (A1) B r r 1 A (A5) Using Eq (A5) in Eq (A) an simplifying yiels the esire equation: Ar r B C 0 (A6) 1 1 Lat Am J Phys Euc Vol 7, No 4, Dec