EECE 30 Signal & Syem Prof. ark Fowler oe Se #34 C-T Tranfer Funcion and Frequency Repone /4
Finding he Tranfer Funcion from Differenial Eq. Recall: we found a DT yem Tranfer Funcion Hz y aking he ZT of he Difference Eq. For a CT yem we do he ame kind if hing y aking he LT of he Differenial Eq: a y a y a y x... x x 0 0 ZT LT a y ay a0y... LT x x 0x Diff. Prop. & Algera Y a a a 0 X... 0 Algera 0 X a aa 0 Y... H So can ju wrie H y inpecion of D.E. coefficien! 2/4
Finding he Tranfer Funcion from he Circui Can find he freq. rep. of a circui uing impedance in erm of j. ow generalize: o find he TF he complex variale replace j: Ue -variale impedance: Z C = /C Z L = L Replace inpu ource ime funcion ymol x y LT ymol X Analyze circui o find Y hing ha muliplie X i T.F. H Example: x y X Y For hi circui he eaie approach i o ue he Volage Divider / C Y X R L / C A andard form: a raio of wo polynomial in w/ uniy coeff. on he highe power in he denom. / LC X 2 R / L / LC H 3/4
Pole and Zero of Tranfer Funcion H... 0 B aa A 0 Can alway normalize hi coeff. o Aume here are no common roo in he numeraor B and denominaor A. If no, aume hey ve een cancelled and redefine B and A accordingly Pole of H: The value on he complex -plane where H Zero of H: The value on he complex -plane where H = 0 The roo of he denominaor polynomial A deermine pole. The roo of he umeraor polynomial B deermine zero. H z z2... z p p... p 2 4/4
Example: Finding Pole and Zero y 6 y 0 y 8 y 2 x 2 x 20 x Conjugae Pair 2 2 2 20 H 3 2 6 0 8 2 3 j 3 j 4 j j zero: 3 j pole: 4, j Uing ATLAB: >> roo[2 2 20] >> roo[ 6 0 8] Pole-Zero Plo for hi H Conjugae Pair j - plane x denoe a pole o denoe a zero 5/4
Impule Repone of Syem Someime looking a how a yem repond o he impule funcion i.e., dela funcion can ell much aou a yem. Hiing a yem wih i lo like ringing a ell o hear how i ound oing ha he LT of = and uing he properie of he ranfer funcion and he LT ranform: L L h L H h H oe: If yem i caual, hen h = 0 for < 0 C-T LTI IC = 0 h The ymol h mean he impule repone. h F H From PFE and Pole/Zero we ee ha a TF like hi: will have an impule repone wih erm like hi: h ke u ke u k e u p p 2 p 2 H B A Some implifying aumpion made here! We almo alway wan hi o decay like a ell!: all pole Re{p i } < 0 6/4
Sailiy of Syem Definiion: A yem i aid o e ale if i oupu will never grow wihou ound when any ounded inpu ignal i applied and ha eem like a good hing!!! Wihou going ino all he deail a yem wih an impule repone ha decay fa enough i aid o e ale. From our exploraion of he effec of pole on he impule repone we ay ha: j - plane For a Sale Syem Pole mu e in Lef Half Plane Zero can e anywhere RLC circui are alway ale Bu Once you ar including linear amplifier wih gain > hi may no e rue epecially if here i feedack involved 7/4
Relaionhip: Tranfer Funcion and Freq. Rep. Recall: CTFT = LT evaluaed on j axi if j axi i inide ROC Fac: For caual yem j axi i inide ROC if all pole are in LHP H H j If all pole are in LHP Like freqz for DT frequency repone he ATLAB command freq can e ued o compue he Frequency Repone from he Tranfer Funcion coefficien: H... a a a 0 0 mu pu any zero i ino he vecor num... 0 mu pu any zero a den a... a a0 i ino he vecor Pick appropriae pacing omega -w_max:?:w_max H freqnum, denom, omega ploomega, ah ploomega, angleh 8/4
From he Pole-Zero Plo we can Viualize he TF funcion on he -plane: 9/4
From our Viualizaion of he TF funcion on he -plane we can ee he Freq. Rep.: 0/4
Can alo look a a pole-zero plo and ee he effec on Freq. Rep. A he pole move cloer o he j axi i ha a ronger effec on he frequency repone H. Pole cloe o he j axi will yield harper and aller ump in he frequency repone. By eing ale o viualize wha H will look like aed on where he pole and zero are, an engineer gain he ailiy o know wha kind of ranfer funcion i needed o achieve a deired frequency repone hen hrough accumulaed knowledge of elecronic circui require experience accumulaed AFTER graduaion he engineer can devie a circui ha will achieve he deired effec. /4
Cacade of Syem Suppoe you have a cacade of wo yem like hi: x H H y H 2 H 2 z Y H X Y H X Z H2 Y Z H Y 2 Z H H X 2 Z H H X 2 Thu, he overall frequency repone/ranfer funcion i he produc of hoe of each age: H H H oal 2 H H H oal 2 Oviouly, hi generalize o a cacade of yem: H H H H oal 2 H H H H oal 2 2/4
Differenial Equaion Tranfer Funcion Frequency Repone Impule Repone Fourier Tranform Inpecion in Pracice Time Domain Freq Domain Coninuou-Time Syem Relaionhip h 0 0 a a a H j H H = j 0 0 x d dx d x d d x d y a d dy a d y d a d y d LT in Theory Pole-Zero Diagram Thi Char provide a Roadmap o he CT Syem Relaionhip!!! Circui Diagram 3/4
In pracice you may need o ar your work in any po on hi diagram. From he differenial equaion you can ge: a. Tranfer funcion, hen he impule repone, he pole-zero plo, and if allowale you can ge he frequency repone 2. From he impule repone you can ge: a. Tranfer funcion, hen he Diff. Eq., he pole-zero plo, and if allowale you can ge he frequency repone 3. From he Tranfer Funcion you can ge: a. Diff. Eq., he impule repone, he pole-zero plo, and if allowale you can ge he frequency repone 4. From he Frequency Repone you can ge: a. Tranfer funcion, hen he Diff. Eq., he pole-zero plo, and he impule repone 5. From he Pole-Zero Plo you can ge: a. up o a caling facor Tranfer funcion, hen he Diff. Eq., he impule repone, and poile he Frequency Repone 6. From he Circui Diagram you can ge: a. Tranfer funcion, hen he Diff. Eq., he impule repone, and poily he Frequency Repone 4/4