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Freq Resp v4

# Freq Resp v4 - Frequency Response DePiero EE 201...

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Unformatted text preview: Frequency Response DePiero EE 201 Learning Objec;ves •  Find the steady ­state frequency response via phasor techniques •  Analyze 1st and 2nd order ﬁlters and determine ﬁltering proper;es •  ( No Fourier No Bode ) •  Note: Filters are common & useful –  Eliminate unwanted frequencies from a signal –  Crossover network for speakers, equalizer Filter Characteris;cs Described by Frequency Response •  Deﬁni;on: H ( w) = Vout (w ) /Vin (w ) •  H(w) describes ra;o of output vs input •  |H(w)| & <H(w) predict steady state response, any w € –  |Vout| = |H(w)| |Vin| –  <Vout = <H(w) + <Vin Vin ( w ) H ( w) Pass band € H (w ) € Stop € band w wc € ‘Low Pass’ Filter wc Cutoff frequency € € Vout ( w ) Ideal Filters Look Like Brick Walls H (w ) Perfectly flat < H ( w) = 0 1.0 No phase shift at any frequency 0.0 € € € w Perfectly vertical € € •  Ideal ﬁlters cannot be implemented •  Prac;cal, simple, ﬁlters have gradual transi;ons and non zero phase shiY Prac;cal Filters Have a Gradual Transi;on H (w ) 1.0 1 € € 2 0.0 € w wc € •  Cutoﬀ frequency, wc, deﬁnes boundary between € ‘pass’ and ‘stop’ bands. € •  At cutoﬀ the output is at ½ of its maximum power 2 Vout R H ( w) 2 = 2 Vin R € H (wc )2 = 1/ 2 € Many Types of Filters – Many Uses H (w ) Low Pass High Pass Band Pass (Two cutoffs) Band Reject ‘Notch’, Narrow version •  Audio Applica;ons: –  Low, Band, High Pass connect speaker elements to receiver. –  Notch (narrow ﬁlter) to remove 60 Hz from audio –  Equalizer has many adjustable ﬁlters •  AM Radio: –  Band pass + ‘Absolute value’ + Low pass + Gain What is the Cutoﬀ for a Simple Low Pass Filter? H (w ) + Vin(t) - R C + Vout(t) - 1.0 1 € € € 2 0.0 wc € ZC 1 jwC 1 Vout = Vin = Vin = Vin Z R + ZC R + 1 jwC 1 + jwRC € € V 1 1 H ( w ) = out = H (w c ) = ⇔ w c RC = 1 Vin 1 + ( wRC ) 2 2 •  Cutoﬀ frequency –  wc = 1/RC €ads/sec r –  fc = 1/(2pi RC) Hz w Resonance Enables 2nd Order Band Pass Filter + L Vin(t) - C R + Vout(t) - •  Natural (resonant) frequency w n = 1 / LC Vout ZR R€ jwRC = Vin = Vin = Vin Z R + ZC + Z L R + 1 jwC + jwL 1 + jwRC + ( jw ) 2 LC Vout jwA = Vin (1 + jw / w1 )(1 + jw / w 2 ) Vout jwB = Vin 1 + (1 / Qw n ) jw + ( jw / w n ) 2 V Aw H ( w ) = out = Vin 1 + ( w / w1 ) 2 1 + ( w / w 2 ) 2 This form reveals cutoffs, w1 and w2 This form reveals Q, ‘quality’ High Q <-> Narrow band pass Bandwidth = w2 – w1 ≈ wn / Q Convenient approximation for Q € Resonance Enables 2nd Order Band Pass Filter Is + L Vin(t) - + Vout(t) - C R VR= Is R VL= Is (jwL) VL Im KVL : −Vin + VL + VC + VR = 0 VL + VC + VR = Vin VR Vin € VC= Is (-j 1/wC) Vc Re Also Is direction H ( w ) = Vout /Vin = VR /Vin •  Input frequency determines VL and VC thus aﬀec;ng H(w) € •  Resonance occurs with series L,C or parallel L,C ...
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