Lecture 21 Frequency Response: Nov. 21, 2011

Transcription

1 Lecture 21 Frequency Response: Resonance, 2 nd Order Filters and Active Filters Nov. 21, 2011 Material from Textbook by Alexander & Sadiku and Electrical Engineering: Principles & Applications, A. R. Hambley is used in lecture slides.

2 Frequency Response Chapter 14 in A & S Resonance Second Order Filters Active Filters 2

3 Series Resonance The relationship between the B, Q and ω o : The quality factor is the ratio of its resonant frequency to its bandwidth. If the bandwidth is narrow, the quality factor of the resonant circuit must be high. If the band of frequencies is wide, the quality factor must be low. 3

4 Series Resonance 4

5 Series Resonance 5

6 Series Resonance 6

7 Series Resonance 7

8 Parallel Resonance Resonance frequency: 8

9 Parallel Resonance The relationship between the B, Q and ω o : Q = Peak energy stored in the circuit Energy dissipated by the circuit in one period at resonance B = 1 RC = ω o Q = ω o RC = R ω o L The quality factor is the ratio of its resonant frequency to its bandwidth. If the bandwidth is narrow, the quality factor of the resonant circuit must be high. If the band of frequencies is wide, the quality factor must be low. 9

10 Series Resonance The relationship between the B, Q and ω o : The quality factor is the ratio of its resonant frequency to its bandwidth. If the bandwidth is narrow, the quality factor of the resonant circuit must be high. If the band of frequencies is wide, the quality factor must be low. 10

11 Resonance Summary characteristic Series circuit Parallel circuit ω o Q B ω 1, ω 2 Q 10, ω 1, ω 2 11

12 12 Ideal Filters

13 13 Noise Rejection

14 FDMA Wireless Systems

15 Relative Advantages of Passive and Active Filters Filter Type Advantages Limitations Passive Stable and Reliable Less vulnerable to environment Operates at high frequencies Requires inductors (size & weight) for pass and stop filters No gain Active Small and light and cheap, Easily implemented in ICs Can be designed in separate stages and cascade with voltage followers for isolation to avoid interaction between stages Can supply gain Doesn t require inductors Can t operate at high frequencies (> 100kHz) Less stable and reliable Less robust to environment

16 First order filters (R&C or R&L) Low pass Passive Filter Summary High pass Second order filters (all 3: R, L and C) Band pass Band stop (or reject) There are also 2 nd order high & low pass filters

17 Second Order Low-Pass Filter Just a voltage divider V out = Z C Z R + Z L + Z C V in = V out V in = H( f ) = 1+ j 2πf 0L R j 2πfC R + j2πfl j 2πfC j 2πfRC V in = 1+ j 2πf L 0 f 1 R f 0 2πff 0 LC j 2πfRC = jq S( f o / f ) f 1 f 1+ jq f 0 2πff 0 LC S f 0 f 0 f where Q s = V in and 2πf o = ω o 1 ω o CR = 1 2πf o CR 17

18 18 Second-Order Low-Pass Filter

19 2nd Order Low-Pass Filter 19

20 FDMA Wireless Systems

21 Second Order High-Pass Filter At low frequency the capacitor is an open circuit At high frequency the capacitor is a short and the inductor is open 21

22 Second Order Band-Pass Filter At low frequency the capacitor is an open circuit (j/ωc) At high frequency the inductor is an open circuit (jωl) 22

23 FDMA Wireless Systems

24 Second Order Bandstop Filter At low frequency the capacitor is an open circuit (j/ωc) At high frequency the inductor is an open circuit (jωl) 24

25 Example Design a filter that passes frequency components higher than 1 khz and rejects components lower than 1 khz. Chose L=50 mh f 0 =1 khz = 1 2π LC C = 1 (2π) 2 f 0 2 L = 1 (2π) 2 (1x10 3 ) 2 (50x10 3 ) = 0.507µF 25

26 High Pass Filter Example To avoid amplifying the signal at f 0 choose Q s =1 26

27 Relative Advantages of Passive and Active Filters Filter Type Advantages Limitations Passive Stable and Reliable Less vulnerable to environment Operates at high frequencies Requires inductors (size & weight) for pass and stop filters No gain Active Small and light and cheap, Easily implemented in ICs Can be designed in separate stages and cascade with voltage followers for isolation to avoid interaction between stages Can supply gain Doesn t require inductors Can t operate at high frequencies (> 100kHz) Less stable and reliable Less robust to environment

28 First-Order Low-Pass Filter 28 A low-pass filter with a dc gain of -R f /R i

29 First-Order High-Pass Filter 2πf B = ω c 29 A high-pass filter with a high frequency gain of -R f /R i

30 30 Higher Order Filters

31 Butterworth Transfer Function Butterworth filters are characterized by having a particularly flat pass-band. 31