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Unformatted text preview: lters Filters Fred Terr 1 © Fred Terry Fall, 2008 ommon Types of Filters Common Types of Filters • Low ‐ Pass • igh ‐ ss High Pass • Band ‐ Pass • Notch or Band ‐ reject or Band ‐ Stop 2 Filters and Time elay ( ) ( ) ( ) cos o g V t H V t ω ω φ = + Delay ( ) ( ) ( ) cos g H V t H V t φ ω φ ω ω = ∠ ⎡ ⎤ ⎛ ⎞ = + ⎟⎥ • For Most Filters, We Want Signals in the ( ) ( ) c o s o g g p H V t t ω ω ω ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ ⎡ ⎤ = + Δ ⎣ ⎦ Pass ‐ Band to Propagate Through with the Same me Delay NOT Phase Delay p p t t φ ω φ = Δ Δ = Time Delay NOT Constant Phase Delay is Means Linear Group Delay p g t ω φ ω ∂ Δ = ∂ • This Means Linear Phase in the Pass ‐ Band for a single frequency sinusoid p g t t Δ = Δ 3 Single Pole LPF ω c =2 π (50KHz) 4 LPF ω c =2 π (50KHz) 5 etter Filters? Better Filters? • Flatter Response in Pass ‐ band? • Constant Group Delay in Pass ‐ Band? • Shaper Cut ‐ off? • Answer is more poles in the right places – Poles closer together in magnitude – Complex poles, resonance effects • Discussion can lead to lots of interesting ath stems Theory (EECS 216 +) math T Systems Theory (EECS 216 +) • We will concentrate on circuits implementations 6 esonance Resonance • Any Sharp Feature in a Response Function Is Often Called Resonance • In a series or parallel RLC Circuit, Resonance Occurs when the Capacitive and Inductive Impedances Are Equal in Magnitude Resulting a Purely Resistive Impedance (phase = 0) – In general RLC networks, the phase = 0 definition may or may not be useful t Resonance the Capacitor and Inductor Exchange Energy • At Resonance, the Capacitor and Inductor Exchange Energy with Each Other with relatively little power dissipation in the rest of the circuit ey to sharp filters rong resonance w energy loss per • Key to sharp filters T strong resonance T low energy loss per cycle T high Q 7 Series Resonance L i(t) + V C + V L What is Z(s), H(s)=V /V g What Kind of Filter is this? C R + V g (t) Z V + 8 Series Resonance L i(t) + V C + V L At Resonance ( ω = ω n ), the Series Condition Yields V C (t)+V L (t)=0 C R + V g (t) Z V + 9 Power and Bandwidth C ⎡ ⎤ Since we have a resistor (V&I in phase), we can find the ½ power points by find ω ’s for which: ( ) 2 2 1 ( ) 1 1 1 1 1 ) 1 2 jRC H j LC jRC L R C j H L ω ω ω ω ω ω ω ⎡ ⎤ ⎢...
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This note was uploaded on 01/31/2011 for the course EECS 215 taught by Professor Phillips during the Spring '08 term at University of Michigan.
 Spring '08
 Phillips

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