Circuitos Eletricos 2 - 5a Lista de Exercícios - UFPA

UNIVERSIDADE FEDERAL DO PARÁ INSTITUTO DE TECNOLOGIA FACULDADE DE ENGENHARIA DA COMPUTAÇÃO E DE TELECOMUNICAÇÕES CIRCUITOS ELÈTRICOS 2 5ª LISTA PROFESSOR JOÃO WEYL Entrega xxx Exercícios As questões abaixo são copiadas do livro texto Introduction to Electric Circuits James A Svoboda and Richard C Dorf I Table 1632 Denominators of Butterworth LowPass Filters with a Cutoff Frequency ωc 1 rads ORDER DENOMINATOR Ds 1 s 1 2 s² 1414s 1 3 s 1s² s 1 4 s² 0765s 1s² 1848s 1 5 s 1s² 0618s 1s² 1618s 1 6 s² 0518s 1s² 1414s 1s² 1932s 1 7 s 1s² 0445s 1s² 1247s 1s² 1802s 1 8 s² 0390s 1s² 1111s 1s² 1663s 1s² 1962s 1 9 s 1s² 0347s 1s² s 1s² 1532s 1s² 1879s 1 10 s² 0313s 1s² 0908s 1s² 1414s 1s² 1782s 1s² 1975s 1 Table 1641 SecondOrder RLC Filters FILTER TYPE CIRCUIT TRANSFER FUNCTION DESIGN EQUATIONS Lowpass Highpass Bandpass Bandstop notch Hs 1LC s² RL s 1LC Hs s² s² RL s 1LC Hs RL s s² RL s 1LC Hs s² 1LC s² RL s 1LC ω₀ 1LC Q 1R LC k 1 ω₀ 1LC Q 1R LC k 1 ω₀ 1LC Q 1R LC k 1 ω₀ 1LC Q 1R LC k 1 Table 1642 Continued FILTER TYPE CIRCUIT DESIGN EQUATIONS Highpass Bandpass Bandstop notch ω₀ 1RC Q 13 A k A ω₀ 1RC Q 13 A k AQ ω₀ 1RC Q 14 2A k A Table 1651 Measuring the Parameters of a Filter Stage PARAMETER DEFINITION MEASUREMENTS Input impedance Zis Vis ITs Output impedance Zos VTs ITs Transfer function Hs Vos Vis Table 1652 FirstOrder Filter Stages FILTER TYPE FIRSTORDER CIRCUIT DESIGN EQUATION Lowpass where p 1 R2 C and k 1 R1 C Highpass where p 1 R1 C and k R2 R1 P 1631 Obtain the transfer function of a thirdorder Butterworth lowpass filter having a cutoff frequency equal to 100 hertz Answer HLs 6283 s 628s2 628s 6282 P 1633 Highpass Butterworth filters have transfer functions of the form HHs k sn Dns where n is the order of the filter Dns denotes the nth order polynomial in Table 1632 and k is the passband gain Obtain the transfer function of a thirdorder Butterworth highpass filter having a cutoff frequency equal to 100 rads and a passband gain equal to 5 Answer HHs 5 s3 s 100s2 100s 10000 P 1641 The circuit shown in Figure P 1641 is a secondorder bandpass filter Design this filter to have k 1 ω0 1000 rads and Q 1 P 1642 The circuit shown in Figure P 1642 is a secondorder lowpass filter Design this filter to have k 1 ω0 200 rads and Q 0707 P 1643 The circuit shown in Figure P 1643 is a secondorder lowpass filter This filter circuit is called a multipleloop feedback filter MFF The output impedance of this filter is zero so the MFF lowpass filter is suitable for use as a filter stage in a cascade filter The transfer function of the lowpass MFF filter is HLs 1 R1 R3 C1 C2 s2 1 R1 C1 1 R2 C1 1 R3 C1 s 1 R2 R3 C1 C2 Design this filter to have ω0 2000 rads and Q 8 What is the value of the dc gain Hint Let R2 R3 R and C1 C2 C Pick a convenient value of C and calculate R to obtain ω0 2000 rads Calculate R1 to obtain Q 8 P 1645 The circuit shown in Figure P 1645 is a lowpass filter The transfer function of this filter is HLs 1 R1 R2 C1 C2 s2 1 R1 C1 s 1 R1 R2 C1 C2 Design this filter to have k 1 ω0 1000 rads and Q 1 Figure P 1641 R C L circuit with vst input and vot output Figure P 1642 R C L circuit with ist input and iot output Figure P 1643 R2 R3 R1 C1 and C2 with vit input and vot output operational amplifier circuit Figure P 1645 R1 R2 C1 C2 with vit input and vot output operational amplifier circuit Atividade P1631 A equação do filtro passabaixa Butterworth de 3ª ordem é Hms 1 s1s²s1 Ajustando a frequência de corte Wc 2π f 2π 1000 628 rads Logo Hms 1 s628 1 s628² s628 1 Hms 628² 628² 1 s628 1 s628² s628 1 Hms 628³ s628s²628s628² P1633 Filtro passaalta Butterworth H Ksm Dms m ordem do filtro Dm polinômio K ganho Considerando Wc 1000 rads e ganho de 5 a ordem 3 Hs 5 s³ s1s²s1 polinômio tabela 1632 Ou Hs 5 s100³ s100 1 s100² s100 1 5 s³ s100s²100s100² P1641 A função de transferência deste circuito é Hs Vos Vss sL 1Cs sL 1Cs sL 1 Cs sL 1 Cs R Hs sL s²LC 1 sL s²LC 1 R Hs sL s²LC sLR Hs sRC s² sRC 1LC K1 Wo1000 rads Q1 Considerando C 1µF L 1 C Wo² 1H R Q LC 1 kΩ O filtro consta de R1kΩ L1H C 1µF P1642 Função de transf Ts Ios Iss 1LC s² sRC 1LC τs LC s2 sRC 1LC k1 W0 200 rads Q0707 Definindo C14μF L 1C W02 25H R Q LC 3535 Ω O filtro conta de R3535KΩ L25H C14μF P 16 43 Considerando W02000 rads Q8 R2R3R1 C1C2C A função de transferência levando em conta essas considerações fica τs 1R1 R2 C2 s2 1RC 2RR1 s 1R2 C2 04 Supondo C1pf R 1W0 C 12000C 500 KΩ Sabemos que 1RC 2RR1 W0 Q 2 RR1 RCW0 Q RR1 RCW0 Q 2 R1 R RCW0Q 2 250 KΩ P 1645 Digamos que R1R2C14 μF Sabse que W0 106 R1 R2 05 1R1C W0 Q Q R1R2 R2 R1 Q2 Neste caso considerando Q1 R2R1 R1 106 1000 1 KΩ P 1649 A eq geral que descreve a resposta em freq para o filtro τ Hs K W0Q s s2 W0Q s W02 Pelos dados fornecidos W0 2π10106 628 x 106 rads K 10 dB 316 06 Porém sabese que BW wo Q 27π 02x106 126x106 rads Substina equação Hs K wo Q s s2 wo s Q wo2 316 126x106 s s2 126x106 s 02x1062 Hs 398 x 106 s s2 126 x 106 s 3944 x 105 P 1671 Para checar as especificações identificamos wo 10000 100 rads Hs 75 s s2 55 s 10000 wo Q Com isso Q wo 25 100 25 4 Q precisa ser 5 Q 5 4 oi Não atende as especificações 07