CHAPTER 6: INTRODUCTION TO PASSIVE FILTERS

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CHAPTER 6 INTRODUCTION TO PASSIVE FILTERS

• Series Parallel Resonance
• Passive Filter

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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
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Series Resonance
Input impedance
Resonance occurs when imaginary part is 0
Resonant/center frequency
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Series Resonance

• At resonance
• The impedance is purely resistive, Z R
• The voltage and the current are in phase, pf1
• The magnitude of transfer function H(w) Z(w) is
  minimum
• The inductor voltage and capacitor voltage can be
  much more than the source voltage

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Series Resonance
Average power dissipated by the RLC circuit
Where
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Series Resonance
The current amplitude vs. frequency for the
series resonant circuit
Maximum power
Power at certain frequency
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Series Resonance
Half power frequency
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Series Resonance
The sharpness of the resonance in a resonant
circuit is measured quantitatively by the quality
factor Q
The quality factor of a resonant circuits is the
ratio of its resonant frequency to its bandwidth
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Series Resonance
Relation between Q and bandwidth B
The higher the circuit Q, the smaller the
bandwidth
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Series Resonance
High Q circuit if,
and half power frequency can be approximated as
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Example 1

• R2O, L1mH, C0.4µF. Determine
• The resonant frequency and the half-power
  frequency
• The quality factor and bandwidth
• The amplitude of the current at ?0, ?1 and ?2

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Parallel Resonance
The parallel-resonant circuit
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Parallel Resonance
Input admittance
Resonance occurs when imaginary part is 0
Resonant frequency
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Parallel Resonance
Half power frequency
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Parallel Resonance
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Parallel Resonance
High Q circuit if,
and half power frequency can be approximated as
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Example 2

• R8 kO, L0.2 mH, C8 µF. Determine
• The resonant frequency, quality factor and
  bandwidth
• The half-power frequencies
• The power dissipated at ?0, ?1 and ?2

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Filters
A filter is a circuit that is designed to pass
signals with desired frequencies and reject or
attenuate others.

• 4 types of filters
• Lowpass filter passes low frequencies and stops
  high frequencies
• Highpass filter passes high frequencies and
  rejects low frequencies
• Bandpass filter passes frequencies within a
  frequency band and blocks or attenuates
  frequencies outside the band
• Bandstop filter passes frequencies outside a
  frequency band and blocks or attenuates
  frequencies within the band

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Filters
Ideal frequency response of four types of filters
a) lowpass
b) highpass
d) bandstop
c) bandpass
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Lowpass Filters
A lowpass filter is designed to pass only
frequencies from dc up to the cutoff frequency ?c
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Lowpass Filters
Transfer function
Cutoff frequency
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Highpass Filter
A highpass filter is designed to pass all
frequencies above its cutoff frequency ?c
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Highpass Filters
Transfer function
Cutoff frequency
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Bandpass Filter
A bandpass filter is designed to pass all
frequencies within a band of frequencies, ?1 lt ?0
lt ?2
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Bandpass Filters
Transfer function
Center frequency
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Bandstop Filter
A bandstop filter is designed to stop or
eliminate all frequencies within a band of
frequencies, ?1 lt ?0 lt ?2
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Bandstop Filters
Transfer function
Center frequency
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Example 3

• Bandstop filter rejects 200 Hz while passing
  other
• frequencies. For R150 O and bandwidth 100 Hz,
• determine
• L
• C

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Exercise 1

• For a series RLC bandstop filter, R2 kO, L0.1
  mH,
• C40 pF. Determine
• The center frequency
• The bandwidth
• The half-power frequencies
• The quality factor