ESE- First order low pass filter First order high pass filter Leture 4 ontinued sawyes@rpi.edu www.rpi.edu/~sawyes A v in v A v in v As (D): Z ; v = vin As : Z ; v = Let's look at this in the s-domain V (s) Find H(s) = V (s) in No Initial Stored Energy
FIST ODE FILTES H(s) V (s) τ V(s) s s τ s = = = = θ j H(j ) = = H(j ) / H(j ) = = Gain θ = tan = Phase Shift st ODE LOW PASS FILTE H(j ) = = Gain At =, Gain ; log (Gain) db At =, Gain, log (Gain) 4 db At =, Gain, log (Gain) 6 db Gain dereases at a Slope = db deade st ODE LOW PASS FILTE log H Gain in db 3 4 Asymptotes H(j ) = = Gain db/deade st ODE LOW PASS FILTE log H Gain in db 4 Asymptotes Passband H(j ) = = Gain db/deade Stopband.. Bode Plot of Gain log sale.. Bode Plot of Gain log sale
st ODE LOW PASS FILTE θ = tan = Phase Shift For "low frequenies" ( << ): θ For "high frequenies" ( >> ): θ 9 For = : θ = 45 Most of the hange in θ ours for. orner Frequenies at., = θ 45 9 st ODE LOW PASS FILTE. Asymptotes θ tan = =. Bode Plot of θ Phase Shift log sale st ODE LOW PASS FILTE BETTE st ODE LOW PASS H(s) v in V (s) V (s) s s out = = = in v out => Low Pass Filter Low Frequenies "Pass"; High Frequenies "Stopped" v in V v out V F F H(s) = ( ) s = F ( ) s 3
st ODE HIGH PASS FILTE H(s) V (s) s s V (s) s s out = = = in v in v out => High Pass Filter High Frequenies "Pass"; Low Frequenies "Stopped" st ODE HIGH PASS FILTE H(j ) = = Gain As, Gain, log (Gain). At =., Gain =. (.) log (Gain) = 4 db st ODE HIGH PASS FILTE H(j ) = = Gain At =., Gain.; log (Gain) 4 db At =., Gain., log (Gain) db Gain inreases at a Slope = db deade At =, Gain =, log (Gain) = 3 db st ODE HIGH PASS FILTE log H Gain in db 3 4. db/deade H(j ) = = Gain. Bode Plot of Gain Asymptotes log sale 4
st ODE HIGH PASS FILTE log H Gain in db 4. db/deade Stopband H(j ) = = Gain Passband. Bode Plot of Gain Asymptotes log sale st ODE HIGH PASS FILTE θ = = 9 tan Phase Shift For "low frequenies" ( << ): θ 9 For "high frequenies" ( >> ): θ For = : θ = 45 Most of the hange in θ ours for. orner Frequenies at., = θ 9 45 st ODE HIGH PASS FILTE. Asymptotes θ = = 9 tan Phase Shift. Bode Plot of Phase Shift log sale BETTE st ODE HIGH PASS v in F V v out V F H(s) = ( ) s s F s = ( ) s 5
. Find poles, zeros. In eah region between sequential poles/zeros, an use H(j) or H(s) a) Draw line in first region based on three possibilities based on frequeny range b) epeat for every region 3. Add orretions at poles/zeros a) n*pole n(-3db) orretion at that pole b) n*zero n(3db) orretion at that zero We have three possibilities Note: an ALSO analyze using H(s) instead of H( j) n db n de slope H( j) onstant, K onstant log K H( j) db n n de slope Hj ( ) sawyes@rpi.edu www.rpi.edu/~sawyes sawyes@rpi.edu www.rpi.edu/~sawyes. rossing a n*pole is a hange of phase of n*9 deg (absolute hange) a) hanging over approximately two deades b) Speifially,. and times. rossing a n*zero is a hange of phase n*9 deg (absolute hange) a) hanging over approximately two deades b) Speifially,. and times. rossing a n*pole is a hange of phase of n*9 deg (absolute hange) a) hanging over approximately two deades b) Speifially,. and times. rossing a n*zero is a hange of phase n*9 deg (absolute hange) a) hanging over approximately two deades b) Speifially,. and times SLA have slopes of /- n*45 deg per deade slope ( is a zero, - is a pole) sawyes@rpi.edu www.rpi.edu/~sawyes 3 sawyes@rpi.edu www.rpi.edu/~sawyes 4 6
ESE- Leture 9. asaded filters/parallel filters First order Bandpass filter First order Bandstop (Noth filter) sawyes@rpi.edu www.rpi.edu/~sawyes 6 BANDPASS FILTE s H(s) K s s L = H L H(j ) = K L H L High Pass Low Pass Let's Design Suh that L > > H H L L H Gain in db K db KdB db/deade Passband db/deade Stopband Stopband. H H L L Bandwidth = log sale L L > H H 7
BANDGAP O NOTH FILTE s = = A L H(s) H L(s) H H (s) B s L s H Gain in db H > L For Low Frequenies For High Frequenies Let's Design Suh that st Looks Like a Order Low Pass st Looks Like a Order High Pass > L L H H H > L K db db de Passband L Stopband LH Bandwidth = H db de Passband H L log sale a) H ( s) = ( s ) a) H ( s) = ( s ) sawyes@rpi.edu www.rpi.edu/~sawyes 3 sawyes@rpi.edu www.rpi.edu/~sawyes 3 8
>> sys=tf([],[ ]); >> h=bodeplot(sys);grid >> Bode Diagram b) H ( s) = ( s ) Magnitude (db) - - -3-4 Phase (deg) -45-9 3 4 5 Frequeny (rad/s) sawyes@rpi.edu www.rpi.edu/~sawyes 33 sawyes@rpi.edu www.rpi.edu/~sawyes 34 b) H ( s) = ( s ) >> sys=tf([],[ ]); >> h=bodeplot(sys);grid Bode Diagram Magnitude (db) - -4-6 Phase (deg) -8-45 -9-35 -8 3 4 5 Frequeny (rad/s) sawyes@rpi.edu www.rpi.edu/~sawyes 35 sawyes@rpi.edu www.rpi.edu/~sawyes 36 9
) s H ( s) = ( s )( s ) ) s H ( s) = ( s )( s ) sawyes@rpi.edu www.rpi.edu/~sawyes 37 sawyes@rpi.edu www.rpi.edu/~sawyes 38 ) s H ( s) = ( s )( s ) ) s H ( s) = ( s )( s ) sawyes@rpi.edu www.rpi.edu/~sawyes 39 sawyes@rpi.edu www.rpi.edu/~sawyes 4
>> sys=tf([ ],[ ]); >> h=bodeplot(sys);grid >> setoptions(h,'maglowerlimmode','manual','maglowerlim',-6) ) Bode plot-multiple stages Phase (deg) Magnitude (db) 4 - -4-6 8 35 9 45 Bode Diagram Vin L. U - 3 k OPAMP 4 9k L. H(s) H(s) H3(s) OUT k Vout 3 4 5 6 Frequeny (rad/s) sawyes@rpi.edu www.rpi.edu/~sawyes 4 sawyes@rpi.edu www.rpi.edu/~sawyes 4 sawyes@rpi.edu www.rpi.edu/~sawyes 43 sawyes@rpi.edu www.rpi.edu/~sawyes 44
>> sys=tf([ ],[ ]); >> h=bodeplot(sys);grid Bode Diagram M a g n i t u d e ( d B ) - - -3 9 45 P h a s e ( d e g ) -45-9 3 4 5 6 7 8 Frequeny (rad/s) sawyes@rpi.edu www.rpi.edu/~sawyes 45