EE 508 Lecture 6. Scaling, Normalization and Transformation

Transcription

1 EE 508 Lecture 6 Scalig, Normalizatio ad Traformatio

2 Review from Lat Time Dead Network X IN T X OUT T X OUT N T = D D The dead etwork of ay liear circuit i obtaied by ettig ALL idepedet ource to zero. Replace idepedet curret ource with ope Replace idepedet voltage ource with hort Depedet ource remai itact D() i characteritic of the dead etwork ad i idepedet of where the excitatio i applied or where the repoe i meaured D() i the ame for ALL trafer fuctio of a give dead etwork

3 Review from Lat Time Root characterizatio i -plae (for complex-cojugate root) Im -plae θ Q ω o Re for θ=90 o, Q=/ 2 root located at Q 2 Q 2Q 2 Q 2 ta 4Q

4 Scalig, Normalizatio ad Traformatio Frequecy calig Frequecy Normalizatio Impedace calig Traformatio LP to BP LP to HP LP to BR

5 Scalig, Normalizatio ad Frequecy ormalizatio: Traformatio 0 Frequecy calig: 0 Purpoe: ω 0 idepedet approximatio ω 0 idepedet ythei Simplifie aalytical expreio for T() Simplifie compoet value i ythei Ue igle table of ormalized filter circuit for give ormalized approximatig fuctio Note: The ormalizatio ubcript i ofte dropped

6 Frequecy ormalizatio/calig example T Defie ω 0 = T jω 0 T 0 ω 0 ω Normalized trafer fuctio: T T jω ω

7 Frequecy ormalizatio/calig example T T jω Sythei of ormalized fuctio ω Vo V IN T

8 Frequecy ormalizatio/calig example T T jω Frequecy calig by ω 0 (of trafer fuctio) = ω 0 T ω0 ω T 0 ω ω 0 Frequecy calig by ω 0 (actually magitude of ω 0 ) (cale all eergy torage elemet i circuit) V IN C = C /ω 0 /ω 0 Vo ω T 0 ω 0

9 Frequecy ormalizatio/calig example T T jω 0 T T jω 0 ω ω 0 ω Frequecy calig / ormalizatio doe ot chage the hape of the trafer fuctio, it oly cale the frequecy axi liearly The frequecy caled circuit ca be obtaied from the ormalized circuit imply by calig the frequecy depedet impedace (up or dow) by the calig factor Thi make the ue of filter deig table for the deig of lowpa filter practical whereby the circuit i the table all have a ormalized bad edge of rad/ec.

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11 Frequecy ormalizatio/calig Example: Table for paive LC ladder Butterworth filter with 3dB bad edge of rad/ec ad reitive ource/load termiatio T T jω ω

12 Frequecy ormalizatio/calig The frequecy caled circuit ca be obtaied from the ormalized circuit imply by calig the frequecy depedet impedace (up or dow) by the calig factor Compoet deormalizatio by factor of ω 0 Normalized Compoet Deormalized Compoet R R C C/ω o L L/ω o Other Compoet Uchaged Compoet value of eergy torage elemet are caled dow by a factor of ω 0

13 Degi Strategy Theorem: A circuit with trafer fuctio T() ca be obtaied from a circuit with ormalized trafer fuctio T ( ) by deormalizig all frequecy depedet compoet. C L C/ω o L/ω o

14 Example: Deig a V-V paive 3 rd -order Lowpa Butterworth filter with a 3-db bad-edge of K rad/ec ad equal ource ad load termiatio. 3 2 (from the BW approximatio which will be dicued later:) T = R S L L 3 V OUT V IN C 2 R L Filter architecture L =H L 3 =H V OUT V IN C 2 =2F Normalized filter C L C/θ L/θ T =K L =mh L 3 =mh V OUT V IN C 2 =2mF Deormalized filter 9 0 T =K

15 Example: Deig a V-V paive 3 rd -order Lowpa Butterworth filter with a bad-edge of K Rad/Sec ad equal ource ad load termiatio. L =mh L 3 =mh V OUT V IN C 2 =2mF 9 0 T =K I thi olutio practical? Some compoet value are too big ad ome are too mall!

16 Filter Cocept ad Termiology Frequecy calig Frequecy Normalizatio Impedace calig Traformatio LP to BP LP to HP LP to BR

17 Impedace Scalig Impedace calig of a circuit i achieved by multiplyig ALL impedace i the circuit by a cotat R C L A θr C/θ Lθ θa for trareitace gai A for dimeiole gai A/θ for tracoductace gai

18 Impedace Scalig Theorem: If all impedace i a circuit are caled by a cotat θ, the a) All dimeiole trafer fuctio are uchaged b) All trareitace trafer fuctio are caled by θ c) All tracoductace trafer fuctio are caled by θ -

19 Impedace Scalig Example: V IN V OUT T = + T() i dimeiole Impedace caled by θ=0 5 00K V IN 0uF V OUT T = + Note ecod circuit much more practical tha the firt

20 Example: Deig a V-V paive 3 rd -order Lowpa Butterworth filter with a bad-edge of K Rad/Sec ad equal ource ad load termiatio. L =mh L 3 =mh V OUT V IN C 2 =2mF 9 0 T =K I thi olutio practical? Some compoet value are too big ad ome are too mall! Impedace cale by θ=000 R C L θr C/θ θl K L =H L 3 =H V OUT V IN C 2 =2uF K 9 0 T =K Compoet value more practical

21 Typical approach to lowpa filter deig. Obtai ormalized approximatig fuctio 2. Sytheize circuit to realize ormalized approximatig fuctio 3. Deormalize circuit obtaied i tep 2 4. Impedace cale to obtai acceptable compoet value

22 Ed of Lecture 6