Series & Parallel Resonance Passive Filter

The series resonant circuit 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

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

Series Resonance At resonance: The impedance is purely resistive, Z = R The voltage and the current are in phase, pf=1 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

Series Resonance Average power dissipated by the RLC circuit: Where:

The current amplitude vs. frequency for the series resonant circuit Series Resonance The current amplitude vs. frequency for the series resonant circuit Maximum power: Power at certain frequency:

Series Resonance Half power frequency:

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

The higher the circuit Q, the smaller the bandwidth Series Resonance Relation between Q and bandwidth B: The higher the circuit Q, the smaller the bandwidth

Series Resonance High Q circuit if, and half power frequency can be approximated as:

Example 1 R=2Ω, L=1mH, C=0.4μF. Determine : The resonant frequency and the half-power frequency The quality factor and bandwidth The amplitude of the current at ω0, ω1 and ω2

The parallel-resonant circuit Parallel Resonance The parallel-resonant circuit

Parallel Resonance Input admittance: Resonance occurs when imaginary part is 0 Resonant frequency:

Parallel Resonance Half power frequency:

Parallel Resonance

Parallel Resonance High Q circuit if, and half power frequency can be approximated as:

Example 2 R=8 kΩ, L=0.2 mH, C=8 μF. Determine : The resonant frequency, quality factor and bandwidth The half-power frequencies The power dissipated at ω0, ω1 and ω2

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

Ideal frequency response of four types of filters: a) lowpass b) highpass d) bandstop c) bandpass

Lowpass Filters A lowpass filter is designed to pass only frequencies from dc up to the cutoff frequency ωc

Lowpass Filters Transfer function: Cutoff frequency:

Highpass Filter A highpass filter is designed to pass all frequencies above its cutoff frequency ωc

Highpass Filters Transfer function: Cutoff frequency:

Bandpass Filter A bandpass filter is designed to pass all frequencies within a band of frequencies, ω1 < ω0 < ω2

Bandpass Filters Transfer function: Center frequency:

Bandstop Filter A bandstop filter is designed to stop or eliminate all frequencies within a band of frequencies, ω1 < ω0 < ω2

Bandstop Filters Transfer function: Center frequency:

Example 3 Bandstop filter rejects 200 Hz while passing other frequencies. For R=150 Ω and bandwidth 100 Hz, determine: L C

Exercise 1 For a series RLC bandstop filter, R=2 kΩ, L=0.1 mH, C=40 pF. Determine : The center frequency The bandwidth The half-power frequencies The quality factor