Lect22EEE 2021 Passive Filters Dr. Holbert April 21, 2008

Lect22EEE 2022 Introduction We shall explore networks used to filter signals, for example, in audio applications –Today: passive filters: RLC components only, but gain < 1 –Next time: active filters: op-amps with RC elements, and gain > 1

Lect22EEE 2023 Filter Networks Filters pass, reject, and attenuate signals at various frequencies Common types of filters: –Low-pass: deliver low frequencies and eliminate high frequencies –High-pass: send on high frequencies and reject low frequencies –Band-pass: pass some particular range of frequencies, discard other frequencies outside that band –Band-rejection: stop a range of frequencies and pass all other frequencies (e.g., a special case is a notch filter)

Lect22EEE 2024 Bode Plots of Common Filters Frequency High Pass Frequency Low Pass Frequency Band Pass Frequency Band Reject Gain

Lect22EEE 2025 Passive Filters Passive filters use R, L, C elements to achieve the desired filter The half-power frequency is the same as the break frequency (or corner frequency) and is located at the frequency where the magnitude is 1/  2 of its maximum value The resonance frequency,  0, is also referred to as the center frequency We will need active filters to achieve a gain greater than unity

Lect22EEE 2026 First-Order Filter Circuits L +–+– VSVS C R Low Pass High Pass H R = R / (R + sL) H L = sL / (R + sL) +–+– VSVS R High Pass Low Pass G R = R / (R + 1/sC) G C = (1/sC) / (R + 1/sC)

Lect22EEE 2027 Second-Order Filter Circuits C +–+– VSVS R Band Pass Low Pass L High Pass Band Reject Z = R + 1/sC + sL H BP = R / Z H LP = (1/sC) / Z H HP = sL / Z H BR = H LP + H HP

Lect22EEE 2028 Higher Order Filters We can use our knowledge of circuits, transfer functions and Bode plots to determine how to create higher order filters For example, let’s outline the design of a third-order low-pass filter

Lect22EEE 2029 Frequency & Time Domain Connections First order circuit break frequency:  break = 1/  Second order circuit characteristic equation s  0 s +  0 2 [  = 1/(2Q) ] (j  )  (j  ) + 1[  = 1/  0 ] s 2 + BW s +  0 2 s 2 + R/L s + 1/(LC)[series RLC] Q value also determines damping and pole types Q 1) overdamped, real & unequal roots Q = ½ (  = 1) critically damped, real & equal roots Q > ½ (  < 1) underdamped, complex conjugate pair

Lect22EEE Time Domain Filter Response It is straightforward to note the frequency domain behavior of the filter networks, but what is the response of these circuits in the time domain? For example, how does a second-order band-pass filter respond to a step input?

Lect22EEE Types of Filters Butterworth – flat response in the passband and acceptable roll-off Chebyshev – steeper roll-off but exhibits passband ripple (making it unsuitable for audio systems) Bessel – yields a constant propagation delay Elliptical – much more complicated

Lect22EEE Class Examples Compare the frequency responses of fourth-order Butterworth and Chebyshev low-pass filters [use Excel to compute and produce Bode magnitude plots] –Butterworth: (s² s + 1)(s² s + 1) –Chebyshev: (2.488 s² s + 1)(1.08 s² s + 1) Drill Problem P10-1