ECE 3 Mh le Sprig, 997. Fucio d Operor, (. ic( i( π (. ( β,, π (.3 Im, Re (.4 δ(, ; δ( d, < (.5 u( 5., (.6 rec u( + 5. u( 5., > rc( β /, π + rc( β /, <. rigoomeric Ideiie: j (. co( co( (. i( i( (.3 e co( + ji( (.4 co ± i( (.5 i co( (.6 co( π ± π j j ± e + e j j (.7 co( ± π co( (.8 i( ± π i( (.9 i( e e j (. co ( + i ( (. co ( i ( co( (. co ( co( (.3 i ( co( + (.4 co( ± β co( co( β i( i( β (.5 i( ± β i( co( β ± co( i( β (.6 co( co( β co( β + co( + β (.7 i( i( β co( β co( + β (.8 i( co( β i( β + i( + β (.9 co( + i( + co (, 3. Derivive β d co β dβ d i β dβ de (3. i β (3. co β (3.3 e d d d d d d d dx (3.4 ( x( β dβ x( (3.5 xu ( ( u ( x( δ( d c d d + 4. i-derivive (4. co( d i( (4. i( d co( (4.3 co( d co( + i( (4.4 i( d i( co( β d β d Revied //98 Pge
ECE 3 Mh le Sprig, 997 (4.5 e d (4.6 e e d e co( co( + i( + e (4.7 e i( d i( co( + 5. Iegrl (5. ic( (5. ic ± jπ d ( d (5.3 e d ic( (5.4 + + x ( δ ( cd xc ( u ( c u ( c xuu ( ( ( d uu ( ( x ( d (5.5 xu ( ( d x ( d (5.6 xud ( ( u ( xd ( (5.7 x ( u ( u ( d u ( x ( d (5.8 x ( δ ( d x ( (5.9 6. Summio Formul N i( λn / λ N (6. < (6. λ i( λ / co ( (6.3 xum [ ] [ ] x [ ] (6.4 xu [ ] [ ] um [ ] x [ ] m N N i( λn / λ N (6.5 i( λ (6.6 i( λ / i ( m co( m N, N, (6.7 xu [ ] [ m] x [ ] (6.8 xuuk [ ] [ ] [ ] uk [ ] x [ ] m 7. Micelleou Relio k k k (7. x( d x( d (7. x[ k] x[ k] k k k k (7.3 x ( h ( d x ( h ( d (7.4 k xkh [ ] [ k] x [ khk ] [ ] k Revied //98 Pge
ECE 3 Mh le Sprig, 997 8. Serie Expio (8. e 3 (8. 3 4 + + + + l( + + +! 3! 3 4 4 6 (8.3 + + (8.4 3 + 5 7 co( i( +! 4! 6! 3! 5! 7! (8.5 + 3 + 5 7 + 7 + 3 + 5 ( (8.6 ( 3 5 35 3 5 9. Fourier Serie (9. Recgulr Pule ri x( τ ic τ - τ/ τ/ (9. rigulr Pule ri τ τ ic x( - τ/ τ/ (9.3 Swooh Pule ri τ jπτ j + e 4 πτ/ π τ x( - τ Revied //98 Pge 3
ECE 3 Mh le Sprig, 997 (9.4 Hl-Wve Reciied Siuoid x( + ( + ( ic ic 4 - -/ / (9.5 Full-Wve Reciied Siuoid + + ( ( + ( ic ic 4 - (9.6 Impule ri regh (re o ech impule x( - - 3. Operiol Properie o Fourier Coeicie (. x ( [ k] e i k jπk/ i + jπk/ [ k] xe ( (. x( + x( + [ k] + [ k] + (. x ( e j π k / [ k] jπ jπk/ (.3 e [ k] e ke [ ] j k π / (.4 co( π ke [ ] j k k k Periodic oly i i ieger (.5 x ke j π ( [ ] k / k k k π + d k [ ] e e + x ke j π k / ( [ ] k (.6 x( [ k] *[ k] k j j k π + π Periodic oly i i ieger >, / Revied //98 Pge 4
ECE 3 Mh le Sprig, 997 dx( jπk (.7 k [ ] d (.8 x ( d jπk k [ ] ee oe elow he iegrl exi oly i x( h o dc compoe; i.e., oly i []. (.9 + x ( d k [ ] k Prevl Ideiy. Fourier rorm Fucio Fourier rorm jπ (. ( e d xe ( j π d (. δ ( (. δ ( (.3 u ( / τ (.4 e u( ( τ > (.5 e j π (.6 co( π (.7 i( π (.8 r τ δ ( + τ + jπτ δ ( jπ δ( + δ( + δ( δ( + j j τ icτ See (. d (.6 Fic( F r F See (. d (.6 k δ k jπk k kδ k (.9 (. δ ( (. e k k. Operiol Properie o he Fourier rorm (. x( d ( d Ryleigh Ideiy (. x( + x( + ( + ( + (.3 x( * x( ( ( (.4 x( x( ( ( * Revied //98 Pge 5
ECE 3 Mh le Sprig, 997 dx( (.5 jπ ( d (.6 x ( d ( + ( δ ( jπ jπ (.7 e x( ( jπ (.8 x ( e ( (.9 x( 3. Bilerl Lplce rorm Fucio c+ j (3. x ( ( e d jπ c j (3. δ ( (3. u ( / τ (3.3 e u( ( τ > (3.4 co( π u ( (3.5 i( π u ( rorm ( x( e d τ + τ + ( π π + ( π 4. Operiol properie o he Lplce rormio Fucio rorm (4. x i i (4. x( + x( + ( + ( + (4.3 x( * x( ( ( dx( (4.4 ( d (4.5 x ( d ( (4.6 e x( ( + (4.7 x ( e ( (4.8 x( Revied //98 Pge 6
ECE 3 Mh le Sprig, 997 5. Bilerl rorm Fucio (5. x [ ] ( jπ C d (5. δ[ ] (5. u [ ] (5.3 u [ ] (5.4 u[ ] (5.5 co( λ u [ ] (5.6 i( λ u [ ] (5.7 co( λu [ ] (5.8 i( λu [ ] rorm ( x[] ( co( λ co( λ + i( λ co( λ + co( λ co( λ + i( λ co( λ + 6. Operiol Properie o he rorm Fucio rorm x i i (6. x[ ] + x[ ] + ( + ( + (6. x[ ] * x[ ] ( ( (6.3 xm [ ] ( m j (6.4 e λ x[ ] e j λ (6.5 x [ ] ( Revied //98 Pge 7
ECE 3 Mh le Sprig, 997 7. Ordiry Lier Co-Coeicie Diereil Equio Soluio re give or ipu x ( co( ω u (. o oi oluio or x( u(, e ω. 7. Fir Order: dy τ + y x, x( co( ω u( d he iiil vlue o he oluio i y( +. Clcule 7. Secod Order: H( jω, B H( jω, θ H( jω + jωτ + y ( Bco( ω + θ, y ( Bco( θ, C y( y ( / y( Ce τ + y ( u( d y + dy + cy x; x( co( ω u ( ( d d + dy + he iiil codiio re y(, y(, d + he complee repoe i y ( y( + y ( u ( u where y ( i he orced repoe d y (i he uorced repoe, deermied elow. ( Forced Repoe: Clcule u d H( jω, B H( jω, θ H( jω ( jω + jω + c y ( Bcoω + θ Uorced Repoe: Clcule Revied //98 Pge 8
ECE 3 Mh le Sprig, 997 D c, y ( Bco( θ, y ( ω Bi( θ, 4 K y( y (, K y( y ( 7.. D (criiclly-dmped ce: Clcule, C K, C K C he complee repoe i y ( ( C + Ce + y ( u ( 7.. D > (over-dmped ce: Clcule + D, K K K C, C he complee repoe i D C y ( Ce + Ce + y ( u ( 7..3 D < (uder-dmped ce: Clcule σ, ω D, K σ K + jω K E, C E, φ E jω he complee repoe i y ( Ce σ co ω+ φ + y ( u ( 8. Ordiry Lier Co-Coeicie Dierece Equio Soluio re give or ipu x [ ] co[ λ ] u [ ], wih >. o oi oluio or x[ ] u[ ], e λ. 8. Fir Order: he iiil codiio i y[ ] Clcule y [ ] + y [ ] x [ ], x [ ] co[ λ] u [ ] Revied //98 Pge 9
ECE 3 Mh le Sprig, 997 He (, B H( e, θ H( e + e he orced repoe i y [ ] B co[ λ + θ] Clcule y [ ] Bco[ θ λ], C y[ ] y [ ] he complee repoe i y [ ] C( + + y [ ] u [ ] 8. Secod Order: Iiil codiio re y[ ], y[ ]. y [ ] + y [ ] + cy [ ] x [ ], x [ ] co( λ u [ ] Clcule He (, B H( e, θ H( e + e + ce he complee repoe i he um o he orced d uorced repoe: he orced Repoe i Clcule y [ ] y [ ] + y[ ] u [ ] y [ ] B co[ λ + θ] D c, y[ ] Bco( θ λ, y[ ] Bco( θ λ 4 K y[ ] y [ ], K y[ ] y [ ] 8.. D (criiclly-dmped ce: Clcule + D, C K K, C C K u he complee repoe i y [ ] ( C + C + y [ ] u [ ] 8.. D > (over-dmped ce: Clcule Revied //98 Pge
ECE 3 Mh le Sprig, 997 + D, β D ( βk K β ( K C C, C β he complee repoe i y [ ] C + Cβ + y [ ] u [ ] 8..3 D < (uder-dmped ce: Clcule + j D, σ, γ ( K * K E, * C E, φ E, he complee repoe i y [ ] Cσ co( γ + φ + y [ ] u [ ] Revied //98 Pge