EE 508 Lecture 6. Dead Networks Scaling, Normalization and Transformations
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1 EE 508 Lecture 6 Dead Network Scalig, Normalizatio ad Traformatio
2 Filter Cocept ad Termiology 2-d order polyomial characterizatio Biquadratic Factorizatio Op Amp Modelig Stability ad Itability Roll-off characteritic Ditortio Dead Network Root Characterizatio Scalig, ormalizatio, ad traformatio
3 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
4 Dead Network Example: i R R V OUT V IN C R C T = +RC Dead Network D = +RC
5 Dead Network i IN R V OUT C R C V i OUT IN R = T = +RC D = +RC Dead Network i IN R i OUT C R C i i OUT IN RC = T = +RC D = +RC Dead Network i IN R C V OUT R C V i OUT IN = T = C D = C Dead Network Note: Thi ha a differet dead etwork!
6 D() i the ame for ALL trafer fuctio of a give dead etwork D X OUT Thi i a importat obervatio. Why i it true? Plauibility argumet: V Coider a etwork with oly admittace elemet ad idepedet curret ource Y k V 2 At ode k, ca write the equatio i ik ki k i Y V -V I k V I k Yk V k Yk2 Yk3 V 3
7 D() i the ame for ALL trafer fuctio of a give dead etwork V D X OUT Y k V 2 I k Yk2 Plauibility argumet: V Yk V k Yk3 V 3 Doig thi at each ode reult i the et of equatio I matrix form Y Y... Y Y Y... Y.. Y Y... Y Y V = I The odal voltage are give by V V 2... V = I I 2... I - V = Y I
8 D() i the ame for ALL trafer fuctio of a give dead etwork V D X OUT Y k V 2 I k Yk2 Plauibility argumet: - V = Y I The odal voltage V k i thi olutio i give by the ratio of two determiate where the oe i the umerator i obtaied by replacig the kth colum with the excitatio vector ad the oe i the deomiator i the determiate of the idefiite admittace matrix Y Note the deomiator i the ame for all odal voltage ad i idepedet of the excitatio that i, it i depedet oly upo the dead etwork V k V Yk V k Yk3 Y Y.. I. Y 2 Y Y.. I. Y Y Y.. I. Y 2 Y Y... Y 2 Y Y... Y Y Y... Y 2 V 3
9 D() i the ame for ALL trafer fuctio of a give dead etwork V D X OUT Y k V 2 I k Yk2 Plauibility argumet: V Yk V k Yk3 V 3 Note the deomiator i the ame for all odal voltage ad i idepedet of the excitatio that i, it i depedet oly upo the dead etwork Thu all brach voltage ad all brach curret have the ame deomiator ad thi (after multiplyig through by the correct power of to make V k a ratioal fractio) i the characteritic polyomial D() Thi cocept ca be exteded to iclude idepedet voltage ource a well a depedet ource V k Y Y.. I. Y 2 Y Y.. I. Y Y Y.. I. Y 2 Y Y... Y 2 Y Y... Y Y Y... Y 2
10 Filter Cocept ad Termiology 2-d order polyomial characterizatio Biquadratic Factorizatio Op Amp Modelig Stability ad Itability Roll-off characteritic Ditortio Dead Network Root Characterizatio Scalig, ormalizatio, ad traformatio
11 Root characterizatio i -plae (for complex-cojugate root) Im -plae θ Q Re ω o - relatiohip betwee agle θ ad Q of root For low Q, θ i large For high Q, θ i mall
12 Root characterizatio i -plae (for complex-cojugate root) Im -plae θ Q ω o Re for θ=45 o, Q=/ 2 for θ=90 o, Q=/2 root located at Q 2 Q 2Q 2 Q ta 2 4Q
13 Filter Cocept ad Termiology 2-d order polyomial characterizatio Biquadratic Factorizatio Op Amp Modelig Stability ad Itability Roll-off characteritic Ditortio Dead Network Root Characterizatio Scalig, ormalizatio, ad traformatio
14 Scalig, Normalizatio ad Traformatio Frequecy calig Frequecy Normalizatio Impedace calig Traformatio LP to BP LP to HP LP to BR
15 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
16 Frequecy ormalizatio/calig example T Defie ω 0 = T jω 0 T 0 ω 0 ω Normalized trafer fuctio: T T jω ω
17 Frequecy ormalizatio/calig example T T jω Sythei of ormalized fuctio ω Vo V IN T
18 Frequecy ormalizatio/calig example T T jω Frequecy calig trafer fuctio by ω 0 = ω0 ω T ω0 Vo ω T 0 ω Frequecy calig circuit by ω 0 (actually magitude of ω 0 ) (cale all eergy torage elemet i circuit) C = C /ω 0 0 V IN /ω 0 ω T 0 ω 0 Frequecy caled trafer fuctio i that of the frequecy caled circuit!
19 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.
20
21 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
22 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
23 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
24 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
25 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!
26 Ed of Lecture 6
EE 508 Lecture 6. Scaling, Normalization and Transformation
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