CHAPTER 6: INTRODUCTION TO PASSIVE FILTERS AHBMH Series & Parallel Resonance Passive Filter DEE2113 : Chap 6 - Introduction to Passive Filters 1 Resonance Resonance is a condition in an RLC circuit in which the capacitive and inductive reactances are equal in magnitude, thereby resulting in a purely resistive impedance.

The series resonant circuit AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 2 Series Resonance Input impedance: Vs 1 Z H() R jL I j C 1

Z R j L C Resonance occurs when imaginary part is 0 Resonant/center frequency: 1 0 rad / s LC AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters

3 Series Resonance At resonance: 1. The impedance is purely resistive, Z = R 2. The voltage and the current are in phase, pf=1 3. The magnitude of transfer function H(w) = Z(w) is minimum 4. The inductor voltage and capacitor voltage can be much more than the source voltage AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 4 Series Resonance

Average power dissipated by the RLC circuit: 1 2 P() I R 2 Where: I AHBMH Vm 2 R L 1 / C DEE2113 : Chap 6 - Introduction to Passive Filters 2

5 Series Resonance The current amplitude vs. frequency for the series resonant circuit Maximum power: 2 m 1V P(0 ) 2 R Power at certain frequency: 2 m V

P(1 ) P(2 ) 4R AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 6 Series Resonance Half power frequency: 2 R 1 R 1 2L

2L LC 2 R 1 R 2 2L 2L LC 0 12 AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 7 Series Resonance

The sharpness of the resonance in a resonant circuit is measured quantitatively by the quality factor Q 0 L 0 1 Q R 0 CR B The quality factor of a resonant circuits is the ratio of its resonant frequency to its bandwidth AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters

8 Series Resonance Relation between Q and bandwidth B: R 0 B 2 1 L Q The higher the circuit Q, the smaller the bandwidth AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 9 Series Resonance High Q circuit if,

Q 10 and half power frequency can be approximated as: B 1 0 2 B 2 0 2 AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 10 Example 1 R=2, L=1mH, C=0.4F. Determine :

a) The resonant frequency and the half-power frequency b) The quality factor and bandwidth c) The amplitude of the current at 0, 1 and 2 AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 11 Parallel Resonance The parallel-resonant circuit AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 12

Parallel Resonance Input admittance: I 1 1 Y H() jC V R jL 1 1 Y j C R L Resonant frequency: Resonance occurs when

imaginary part is 0 1 0 rad / s LC AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 13 Parallel Resonance Half power frequency: 2 1 1

1 1 2RC 2RC LC 2 1 1 1 2 2RC 2RC LC AHBMH

DEE2113 : Chap 6 - Introduction to Passive Filters 14 Parallel Resonance 1 B 2 1 RC 0 R Q 0 RC B 0 L AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 15

Parallel Resonance High Q circuit if, Q 10 and half power frequency can be approximated as: B 1 0 2 B 2 0 2 AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 16

Example 2 R=8 k, L=0.2 mH, C=8 F. Determine : a) The resonant frequency, quality factor and bandwidth b) The half-power frequencies c) The power dissipated at 0, 1 and 2 AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 17 AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 18

Filters A filter is a circuit that is designed to pass signals with desired frequencies and reject or attenuate others. 4 types of filters: 1. Lowpass filter: passes low frequencies and stops high frequencies 2. Highpass filter: passes high frequencies and rejects low frequencies 3. Bandpass filter: passes frequencies within a frequency band and blocks or attenuates frequencies outside the band 4. Bandstop filter: passes frequencies outside a frequency band and blocks or attenuates frequencies within the band AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 19

Filters Ideal frequency response of four types of filters: a) lowpass c) bandpass AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters b) highpass d) bandstop 20 Lowpass Filters A lowpass filter is designed to pass only frequencies from dc up to the cutoff frequency c

AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 21 Lowpass Filters Transfer function: V0 1 / j C 1 H() Vi R 1 / jC 1 jRC 1 1 H(C )

2 2 2 2 1 C R C Cutoff frequency: AHBMH 1 C RC DEE2113 : Chap 6 - Introduction to Passive Filters 22 Highpass Filter A highpass filter is designed to pass all frequencies above its cutoff frequency c

AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 23 Highpass Filters Transfer function: V0 R jRC 1 H ( ) Vi R 1 / jC 1 jRC 1 1 jRC

1 1 H (C ) 1 2 1 2 2 2 C R C Cutoff frequency: AHBMH 1 C RC DEE2113 : Chap 6 - Introduction to Passive Filters 24

Bandpass Filter A bandpass filter is designed to pass all frequencies within a band of frequencies, 1 < 0 < 2 AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 25 Bandpass Filters Transfer function: V0 R H() Vi R j L 1 / C Center frequency:

AHBMH 1 0 LC DEE2113 : Chap 6 - Introduction to Passive Filters 26 Bandstop Filter A bandstop filter is designed to stop or eliminate all frequencies within a band of frequencies, 1 < 0 < 2 AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters

27 Bandstop Filters Transfer function: V0 j L 1 / C H() Vi R j L 1 / C Center frequency: AHBMH 1 0 LC DEE2113 : Chap 6 - Introduction to Passive Filters

28 Example 3 Bandstop filter rejects 200 Hz while passing other frequencies. For R=150 and bandwidth 100 Hz, determine: a) L b) C AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 29 Exercise 1 For a series RLC bandstop filter, R=2 k, L=0.1 mH, C=40 pF. Determine : a) The center frequency

b) The bandwidth c) The half-power frequencies d) The quality factor AHBMH DEE2113 : Chap 6 - Introduction to Passive Filters 30