Overview of LectureIdeal Lowpass FilterIdeal Bandpass FilterIdeal Highpass FilterIdeal Notch FilterIdeal Delay FilterFilter ApplicationsI/O Impedance RestrictionsPositive Real Function TestStrictly Hurwitz TestReal Part TestExample Of Positive Real TestUltimate Positive Real TestParameter NormalizationNormalization . . . Cont’dNormalization ExampleShort Circuit Y-ParametersAlternative Y-Parameter ModelingBilateral Pi—Type ModelPlausible Design StrategySecond Order Lowpass FilterLowpass Filter Design . . . Cont’dLowpass Filter Design . . . Cont’dLowpass Filter Design . . . Cont’dHSPICE Verification Of DesignHSPICE Simulation ResultsOpen Circuit z-ParametersAlternative z-Parameter ModelingBilateral Tee-Type ModelFilter Design StrategyTransmission Parametersc-Parameter Network PropertiesComments On Transfer PropertiesCascade Network InterconnectionSeries And Shunt BranchesEE 541Class LectureWeeks 1 & 2Prof. John Choma, ProfessorDepartment of Electrical Engineering-ElectrophysicsUniversity of Southern CaliforniaUniversity Park; MC: 0271; PHE #604Los Angeles, California 90089-0271213-740-4692 [USC Office]213-740-7581 [USC Fax][email protected] Port Filter Network Models And AnalysisFall 2006 SemesterUSC Viterbi School of EngineeringEE 536 Fall 2005 Lecture #02/Choma2Overview of LectureOverview of Lecturez Fundamental Types Of Filters Frequency ResponseLowpassBandpassHighpassNotch Constant Delayz Common Filter Characteristics Maximum Voltage Or Current Transfer Maximum Power TransferLossless Branch ElementsInput And Output Port Impedance Matchingz Two Port Network Models y-Parameters z-Parameters c-ParametersUSC Viterbi School of EngineeringEE 536 Fall 2005 Lecture #02/Choma3z System Level Diagram And Responsez Basic Filter Properties Constant “Gain” Within Filter Passband: Zero “Gain” In Stopband: Designable Passband Gain Or Attenuation, |H(0)| Designable Bandwidth, Bz Positive Real Impedance Requirement (All Filters)+−ZsZlVsLowpassFilterZinZoutVo|H(j )|ω|H(0)|ωB0osV(jω)H(jω)V(jω)H(jω)H(j0),0ω B=≤<H(jω)0,ω B=>inRe Z (jω)0,0ω≥≤<∞outRe Z (jω)0,0ω≥≤<∞Ideal Lowpass FilterIdeal Lowpass FilterUSC Viterbi School of EngineeringEE 536 Fall 2005 Lecture #02/Choma4z System Level Diagram And Responsez Basic Filter Properties Constant “Gain” Within Filter Passband: Zero “Gain” In Low And High Frequency Stopbands Designable Passband Gain Or Attenuation, |H(jωo)| Designable Bandwidth, B, And Center Or Tuned Frequency, ωo Resultantly Designable Quality Factor:+−ZsZlVsBandpassFilterZinZoutVo|H(j )|ω|H(j )|ωoωωo+B/20ωo−B/2BωoosV(jω)H(jω)V(jω)oo oBBH(jω)H(jω ), ωωω22=−<<+oωQBIdeal Bandpass FilterIdeal Bandpass FilterUSC Viterbi School of EngineeringEE 536 Fall 2005 Lecture #02/Choma5z System Level Diagram And Responsez Basic Filter Properties Constant “Gain” Within Filter Passband: Zero “Gain” In Stopband: Designable Passband Gain Or Attenuation, |H(j∞)| Designable Cutoff Frequency, ωcz Positive Real I/O Impedance Requirement Remains For This And All Other Passive Filters+−ZsZlVsHighpassFilterZinZoutVo|H(j )|ω|H(j )|∞ω0ωcosV(jω)H(jω)V(jω)cH(jω)H(j), ω > ω=∞cH(jω)0,ω < ω=Ideal Highpass FilterIdeal Highpass FilterUSC Viterbi School of EngineeringEE 536 Fall 2005 Lecture #02/Choma6z System Level Diagram And Responsez Basic Filter Properties Constant Attenuation Within Filter Stopband: Constant “Gain” In Low And High Frequency Passbands Designable Stopband Attenuation Factor, |H(jωo)| / |H(0)| Designable Stopband Bandwidth, B, And Notch Frequency, ωo Resultantly Designable Quality Factor:+−ZsZlVsNotchFilterZinZoutVo|H(j )|ω|H(j )|ωoωωo+B/20ωo−B/2Bωo|H(0)|osV(jω)H(jω)V(jω)oo oBBH(jω)H(jω ), ω < ω < ω22=−+oωQBIdeal Notch FilterIdeal Notch FilterUSC Viterbi School of EngineeringEE 536 Fall 2005 Lecture #02/Choma7z System Level Diagram And Responsez Filter Delivers Linear I/O Phase Response Significance Is Constant Time Delay Of Steady State Sinusoids Cursory Analysis: Note Time DelayIn Steady StateIs FrequencyIndependent()()[]()ssmossmosmosmdV(jω)V ωtV(jω)H(jω)V (jω)H(jω)V ωtV(jω)H(jω)V ωt φ(ω)V(jω)H(jω)V t Tcoscoscoscos ω===+=−+−ZsZlVsDelayFilterZinZoutVoφω()ω0φω−ω() = Tdjφ(ω)osV(jω)H(jω)H(jω)V(jω)e=Ideal Delay FilterIdeal Delay FilterUSC Viterbi School of EngineeringEE 536 Fall 2005 Lecture #02/Choma8Filter ApplicationsFilter Applicationsz Lowpass Filter Mitigate Interference From High Frequency Signals Integrate Applied Input Signal Serve As Prototype For Other Filter Typesz Bandpass Filter Selective Signal Processing Over Stipulated Passband Tuning In Communication Receiversz Highpass Filter Mitigate Interference From Low Frequency Signals Differentiate Applied Input Signalz Notch Filter  Mitigate In Band Signal Interferencez Delay Filter  Incur Nominally Constant Time Delay In Sinusoidal Steady StateUSC Viterbi School of EngineeringEE 536 Fall 2005 Lecture #02/Choma9+−ZsZlVsLinearPassive FilterZinZoutVoz Positive Real Impedances Required For Stability Required For PhysicallyPossible Realization Mathematical Constraintsz Signal Processing Voltage Transfer  |Zin(jω)| >> |Zs(jω)| ; |Zl(jω)| >> |Zout(jω)|  Current Transfer  |Zin(jω)| << |Zs(jω)| ; |Zl(jω)| << |Zout(jω)| Maximum Power Transfer  Zin(jω) = Zs(–jω) ; Zl(jω) = Zout(–jω)Complex Conjugate Impedance Matching At I/O Network PortsMaximum Power Transfer  Critical In Radio Frequency (RF) Applications Were Signal Power Levels Are AnemicLossless Network Desirable To Mitigate Any Signal Power Loss In FilterinoutZ(jω) 0, for all ω 0Z(jω) 0, for all ω 0ReRe≥≥≥≥I/O Impedance RestrictionsI/O Impedance RestrictionsUSC Viterbi School of EngineeringEE 536 Fall 2005 Lecture #02/Choma10eoeoN (s) + N (s)N(s)Z(s) K KD(s) D (s) + D (s)==z Generalized Impedance Function (Constant K > 0)z Definitions Ne(s) / No(s)  Even/Odd Polynomial Components Of N(s) De(s) / Do(s)  Even/Odd Polynomial Components Of D(s)z Positive Real (PR) Tests (3-Step