Dr. Kasra Eteadi February 7, 7
Chapter 4:Transients Chapter 5: Sinusidal Surces Chapter 6: nnsinusidal surces Furier Trasr Transer Functin Filters Lwpass Filters Highpass Filters andpass Filters
Surce esistrs, Capacitrs, and nductrs ne capacitr r ne inductr d CrL C CrL( dt ( t t s
C i t c (t? dc ( t C C ( t dt ( t C Ke C ( i Ke K st d ( t dt c (t t s C C C ( C ( t K K t e C ( s Ke s K st s sin(t (t 5k C F c (t d( t C ( t 4C cs(t dt Particular ( t Acs(t sin(t d C Particular Cpleentary dt ( t Cpleentary cs(t 45 ( t ( t Cpleentary ( t Ke t / C Ke t /τ C ( t e i t / C C ( t s s e t /τ ( t ( t ( t Particular Cpleentary ( t cs(t sin(t 4e t / C µ A C (t s i (7% initial value.63 s 4 (36% initial value.368 i 4 6 8 Tie cnstant C t Tie cnstant C t 4
Circuits with tw energystrage eleents Fr Exaple ne capacitr and ne inductr Series d ( t d( t ds ( t ( t dt L dt CL L dt Parallel d ( t d ( t d n ( t C ( t dt dt L dt
General slutin the SecndOrder Equatin CL in series d ( t d( t ds ( t ( t dt L dt CL L dt CL in parallel d ( t d ( t d n ( t C ( t dt dt L dt d X dt ( t dx ( t α ω X dt ( t ( t General slutinparticular slutin (x P Cpleentary slutin (x C (e.g. DC Surce ( (t ζ ζ > ζ < α ω ζ Daping ati Overdaped Critically daped Underdaped s X ( t K e C s X ( t K e C X ( t K e C t t αt K e K e cs( ω t αt K te st st n sin( ω t n Assue a slutin: Characteristic Equatin: X ( t s C Ke st αs ω Daping Ceicient Undaped esnant Frequency
Z L jlj 9 L.H C S (tsin(t S (tcs(t9 S C ZC 9 Z C (t Z C C F jω jωc Z Z Z LC L C Z S LC
Current lags ltage θ L L θ θ 9 e θ Current in phase with ltage θ θ e θ 9 C Current leads ltage θ C θ e 9 9 9 L cs( θ 9 L sin( θ d L L Lω dt t cs( θ θ 9 cs( θ θ t cs( ωt θ cs( ωt θ C cs( θ θ dc C C Cω sin( θ dt t cs( θ 9 Pure nductance Pure esistance Pure Capacitance θ 9
Pure esistance Pure nductance Pure Capacitance (t (t (t (t t (t t (t t P(t P(t P(t P avg t t t ( t ( t cs( ωt cs( ωt P( t ( t ( t P( t cs Average pwer is absrbed by the resistr cs ( ωt ( ωt ( t ( t cs( ωt cs( ωt 9 P( t ( t ( t sin( ωt cs( ωtsin( ωt P( t sin(ωt Average pwer is zer ( t ( t cs( ωt cs( ωt 9 P( t ( t ( t P( t sin(ωt Average pwer is zer sin( ωt cs( ωtsin( ωt
Pwer Calculatin r a General Lad t cs( ωt θ ( ( t cs( ωt θ LC Lad Z P( t ( t ( t cs( ωtcs( ωt θ P avg T T p( t dt P avg cs( θ Pwer angle rs θ θ θ Apparent Pwer rs cs( θ Pwer actr, (PF Phase the vltage Phase the current
WyeCnnected Surce DeltaCnnected Surces an cn bn a b c n ca bc ab a b c an bn cn ( t ( t ( t Y Y Y cs( ωt ab bc ca cs( ωt cs( ωt
WyeWye Cnnectin Phase A the surce Line current Phase A the lad a aa A an cn bn b c b cc C Z θ Z θ Z θ Lad pedances are equal Line n nn N Neutral Nn ( 3 Y p t L cs( θ P p( t 3 cs( θ avg Yrs Lrs Q 3 Yrs Lrs sin( θ
DeltaDelta Cnnectin aa a 3 3 aa A CA c A b A C A C CA A A 3 Z θ ( θ L 3
Mst signals are nnsinusidal Nn sin usidalsignal a an cs( nωt bn sin( nωt n n All realwrld signals are cpsed sinusidal cpnents Furier Analysis Hepage
in (t Twprt netwrk ut (t nput Prt Output Prt H Transer Functin: ( ut H ( H ( in
Given: Exaple Transer Functin a ilter ut H ( H ( H ( in H ( H ( 4 6 3 3 3 (Hz 3 (Hz ( t cs(πt in 4 Find: ut ( t?
( t cs(πt in 4 πt H ( H ( 4 6 3 3 3 (Hz 3 (Hz H ( 3 ( 3 3 ut H in H ( 3
( 3 3 ut H in ( t cs(πt in 4 ut 3 3 in 4 in ut 3 3 4 6 7 ( t 6cs(πt ut 7
nput signal Using the transer Functin with Several nput Cpnents. The input signal is separated int cpnents. The aplitude and phase each cpnent are altered by the transer unctin 3. Cnvert the phasrs back int tiedependent signals * 4. The altered cpnents are added Furier Transr ai bi cs( ω t ci sin(ωt a a i b i cs( ω t c i sin( ωt a3i b3i cs( 3ω t c3i sin(3ωt ni b ni i i H (... cs( nω t c ni sin( nωt i i H ( 3i 3 i H ( 3 ni H ( ni n a b cs( ω t c sin(ωt a a b cs( ω t c sin( ωt a3 b3 cs( 3ω t c3 sin(3ωt n b n... cs( nω t c n sin( nωt additin Output signal
Experiental deterinatin the transer unctin.measure in and ut (aplitude and phases. Divide the utput phasr by the input phasr ( ut / in in (t H( ut (t 3. This is repeated r each requency interest
Lwpass Filters Pass lwrequency cpnents and reject highrequency cpnents Highpass Filters Pass highrequency cpnents and reject lwrequency cpnents andpass Filters Pass requency cpnents within a requency band and reject cpnents utside the band
Firstrder lwpass ilter t c (t dc ( t C dt C ( t s Firstrder dierential equatin highpass Lwpass Lwpass
Lwpass C j Z in LC in π / C Z C ut π C j C in ut π π / C j H in ut π ( ( in ut j H / ( πc halpwer requency in C
H ( ut in j ( / Lwpass H ( ( / H ( arctan..77.5 45 3 9 3
Firstrder Highpass ilter highpass Highpass Lwpass
in / jπc ut Highpass ut H ( H ( ut in in ut in / jπc jπc jπc j ( / j( / πc halpwer requency
( / / ( H H arctan 9 ( 45 9 3 ( ( in ut j j H / / ( Highpass 3.5.77.
Decibels H ( lg H ( d H( 6 / 3 / 3 /.. H ( 4 6 4 d H ( H (..8.6.4.. d 4 6 8.5 (Hz 4 5 6 8 (Hz 4 5 6 8
Cascade TwPrt Netwrks in in H ( H ( ut ut in nput Prt Output Prt ut ut ut ut ut H ( H ( H( in ut in in in H ( H( H ( H ( ( H ( H d d d
de Plt (Netwrk unctin in decibels versus requency in lgarithic scale H ( db H (..
de Plt r FirstOrder Lwpass Filter H ( ( / H ( arctan H ( lg H ( d H ( d lg >> H ( d lg
H ( d. Lwrequency asyptte Actual respnse curve Lwpass Filter 4 Highrequency Asyptte ( d/decade slpe H ( 45 arctan. Actual phase curve Apprxiatin 9
de Plt r FirstOrder Highpass Filter ( lg ( H H d lg lg ( d H d H lg ( << ( / / ( H H arctan 9 (
4 H ( d. Lwrequency asyptte Actual respnse curve Slpe d/decade 3 d Highrequency Asyptte Highpass Filter H ( 9 9 arctan. Actual phase curve 45 Apprxiatin
Series esnance Fr the basis r ilters jωl S Z S Z S j ω C ( jπl j πc Z S ( j πl Pure resistance (reactance is zer j πc π LC esnant requency
Z S ( jπl esnant requency Quality actr π L j πc π LC π C Q S jωl S Z S j ω C The quality actr is the rati reactance the inductance (r reactance capacitance at the resnance requency t the resistance. Z S ( jq s
jω L S Z S j ω C Nralized agnitude and Phase r the ipedance the series resnant circuit versus requency Z S ( jq s Z S 5 Q5 Q Q Z S 9 45 Q Q Q5 Q 5 Q.8.9... 45 9.8.9... pedance agnitude is iniu at the resnance requency. As the quality actr bece larger, the iniu bece sharper.
Q S Q S 5 Q S Q S.8.9.....4.6.8. S g Series esnant Circuit as a andpass Filter S Z S jωl C j ω jq Z s S ( Z S S ( ( jq S S / /
S andpass ilter..77 Q S bandwidth L H Hal pwer requency Q S >> H L
Parallel esnance Z S j ω C C L jωl ut Z P ( / jπc j(/ πl Z S ( j πc Pure resistance (reactance is zer j πl π LC esnant requency
Z S j ω C C L jωl ut Quality actr Q π C P π L Z P ( / jπc j(/ πl Z P ( jq P ( / /
Z S j ω C C L jωl ut Z P ( jq P ( / / ut Z P jq P ( / /
Series esnant Circuit as a andpass Filter..8.6 Q S.4. Q S Q S 5 Q S.8.9... H L Q P
SecndOrder Lwpass Filters Firstrder Lwpass Filter Lwpass Secndrder Lwpass Filter Lwpass
H ( d Q.5 Q5 Q Q Firstrder ilter Secndrder ilter 4 6 deal ilter (Hz / esnant requency π LC Quality actr π L π C Q S