Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

Sensitivity and Selectivity 4-25 Introducing the normalized frequency V ¼ v v pv p v, one can find that Re S Ti(s) 1T i (s), ¼Q p Im S T 1i(s)T i (s), ¼Q p1Q pV 2 þ 1 Q p 2(4:131)VV 2 þ 1 Q p 2(4:132)pffiffiffiffiffiffiffiffiffiffiffiffiffiffiRe S T i(s)v p¼ V V 2 þ 4 V 2 1 2Q pV 2 þ 1 2(4:133)Q pIm S Ti(s)v p¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffi1Q pV 2 þ 4V 2 þ 1 2(4:134)Q pFigure 4.4 shows the graphs of these four functions. They allow the following conclusions [4]. Thefunctions reach high values in the vicinity of V ¼ 0, i.e., when v v p . This means that in the filter–Re S T i (s) [T i (s), 1/Qp](a)Re S T i (s)ωp(c)108642Q p = 1Q p = 2Q p = 100–2 –1.5 –1 –0.5 0 0.5 1 1.5 2Ω1197531–1Q p = 1Q p = 2Q p = 10–3–5–7–9–11–2 –1 0 1 2Ω54Q p = 1Q p = 23Q p = 10210–1–2–3–4–5–2 –1.5 –1 –0.5 0 0.5 1 1.5 2(b)ΩIm S T i (s) [T i (s), 1/Qp]Im S T iω (s)p(d)20161284Q p = 1Q p = 2Q p = 100–2 –1 0 1 2ΩFIGURE 4.4 Stage sensitivities: (a) real and (b) imaginary parts of the Q-factor sensitivity; (c) real and (d)imaginary parts of the pole frequency sensitivity.