ECE30710
Active Filter Circuits
Z. Aliyazicioglu Electrical and Computer Engineering Department Cal Poly Pomona
Active Filter Circuits Introduction Filter circuits with RLC are passive filter circuit Use op amp to have active filter circuit Active filter can produce bandpass and bandreject filter without using inductor. Passive filter incapable of amplification. Max gain is 1 Active filter capable of amplification The cutoff frequency and bandpass magnitude of passive filter can change with additional load resistance This is not a case for active filters We look at few active filter with op amps. We look at that basic op amp filter circuits can be combined to active specific frequency response and to attain close to ideal filter response ECE 30710 2
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Active Filter = Circuits FirstOrder Lowpass Filters C
Zf
R2
Zi
R1 Vi
OUT
+ Vo
OUT +
+
Transfer function of the circuit
H (s ) =
− Zf Zi
ωc H ( s ) = −K (s + ω c )
−R2 H (s ) = R1(sR2C + 1)
The Gain K=

Vi

Cutoff frequency
R2 R1
ωc =
H (s ) =
R2 1 − sR SC = 2C + 1 R1 R1
−R2 
Transfer function in jω H ( j ω ) = −K
1 R2C
+ Vo
1 (1 + j
ω ) ωc
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Active Filter Circuits Example
• Find R2 and C values in the following active Lowpass filter for gain of 1 and cutoff frequency of 1 rad/s.
C 1F R1 2 R1 Vi
1
From the gain
1 OUT +
K=
+ Vo
R2 =1 R1
R2 = R1 = 1Ω
From the cutoff frequency ωc =
H ( jω ) =
1 (1 + j
ω 1
1 =1 R2C
C=
1 = 1F R2
)
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Active Filter Circuits Example >> w=0.1:.1:10; >> h=20*log10(abs(1./(1+j*w))) ; >> semilogx(w,h) >> grid on >> xlabel('\omega(rad/s)') >> ylabel('H(j\omega) dB') >>
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Active Filter Circuits A first order highpass filter R2 R1
C

Vi
OUT +
+ Vo
Transfer function of the circuit H (s ) =
H (s ) =
−R2s s 1 H ( s ) = −K R1(s + ) (s + ω c ) R1C
The Gain K=
R2 R1
− Zf Zi
H (s ) =
−R2 −R2sC = 1 R 1sC + 1 R1 + sC
Transfer function in jω
Cutoff frequency
ωc =
1 R1C
H ( j ω ) = −K
jω ωc (1 + j
ω ) ωc
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Active Filter Circuits Example R1 Vi
R2 C
20 K 0.1 uF
200K OUT +
+ Vo
• Find R2 and R1 values in the above active Highpass filter for gain of 10 and cutoff frequency of 500 rad/s. From the cutoff frequency From the gain
K=
ωc =
1 = 500 R12C
R1 =
1 = 20 K Ω 500C
R2 = 10 R2 = R110 = 200 K Ω R1
Transfer function in jω
H ( jω ) = −10
jω 500
(1 + j
ω
500
) ECE 30710 7
Active Filter Circuits Example >> w=1:10000; >> h=20*log10(10*(abs((j*w/500 )./(1+j*w/500)))); >> semilogx(w,h) >> grid on >> xlabel('\omega(rad/s)') >> ylabel('H(j\omega) dB') >>
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Active Filter Circuits Scaling
• In filter design, we can transform RLC values in to realistic values, this process is called scaling • Two types of scaling, magnitude and frequency scaling • In magnitude scaling, we multiply all L and R by scaling factor km, multiplying all C by 1/km R ' = kmR
L ' = km L
C' =
C km
• km,is positive real number
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Active Filter Circuits Scaling
• frequency scaling, we multiply all L, C by 1/kf where kf is scaling factor. R' = R
L' =
L kf
C' =
C kf
• A circuit can be scaled in both magnitude and frequency in simultanously R ' = kmR
k L' = m L kf
C' =
C kmkf
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Active Filter Circuits Example
• Example 1 , Find R2 and R1 values in the active Lowpass filter for gain of 5 and cutoff frequency of 1Khz and c=0.01 µF
C 1F R1 2 R1 Vi
1
1 OUT +
km =
+ Vo
kf =
ωc ' 2π 1000 = = 6283.185 ωc 1
1 C 1 = = 15915.5 kf C ' 6283.185(10−8 )
R2 ' = kmR2 = 15915.5(1) = 15.9 K Ω
• For gain specification, we need to change R1 R1 =
R2 15.9K = = 3.18 K Ω K 5
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Active Filter Circuits Example
>> f=1:10000; >> w=2*pi*f; >> h=20*log10(5*abs(1./(1 +j*w/(2*pi*1000)))); >> semilogx(f,h) >> grid on >> xlabel(‘f(Hz)') >> ylabel('H(jf) dB')
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Active Filter Circuits Op Amp BandPass Filters
• • • •
Three components A unity gain lowpass filter, cutoff frequency is ωc2 A unity gain highpass filter , cutoff frequency ωc1 A gain component to provide the desired level
ωc 2 ≥2 ω c1 Vi
Lowpass filter
Highpass filter
Inverting amp.
Vo
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Active Filter Circuits Op Amp BandPass Filters CL RL RH RL
Rf

Vi
OUT
RH
CH

+
Rf
OUT
OUT
+ +
−ωc 2 −s Rf H (s ) = − s + ωc 2 s + ωc1 Ri H (s ) =
2
−K ω c 2 s
s + (ωc1 + ωc 2 )s + ωc1ωc 2
ωc 2 ωc 1
ωc 2 =
1 RLCL
H (s ) =
H (s ) =
ωc1 =
+ Vo
−K ω c 2 s (s + ωc 2 )(s + ωc1)
βs s + β s + ω02 2
R 1 H ( j ω0 ) = −K = − f Ri max RHCH ECE 30710 14
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Active Filter Circuits Example:
• Design a bandpass filter for a graphical equalizer that has gain 2 within the frequency between 100 and 10,000 Hz. Use 0.1 µF capacitors • For upper cutoff frequency from LP filter
ωc 2 =
1 RLCL
RL =
1
ωc 2CL
=
1 2π 10000(0.1)10−6
= 80 Ω
• For Lower cutoff frequency from HP filter
ωc1 =
1 RHCH
RH =
1 1 = = 7958 Ω ωc1CL 2π 100(0.1)10−6
• For gain, choose Ri=1KΩ K=
Rf Ri
Rf = Ri K = 1000(2) = 2 K Ω ECE 30710 15
Active Filter Circuits From transfer function −2π 1000 2000 − jω H ( jω ) = − A = 20log10  H ( jω )  ω π ω π + + j 2 1000 j 2 100 1000 dB
>> f=10:80000; >> w=2*pi*f; >> H=((2*pi*10000)./(j*w+2*pi* 10000)).*((j*w)./(j*w+2*pi*100))*( 2); >> A=20*log10(abs(H)); >> semilogx(f,A) >> grid on; >> ylabel ('A_{dB}') >> xlabel ('F (Hz)')
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Active Filter Circuits Op Amp BandReject Filters
• • • •
Three components A unity gain lowpass filter, cutoff frequency is ωc1 A unity gain highpass filter , cutoff frequency ωc2 A gain component to provide the desired level
Lowpass filter
Vi
Vo
Inverting amp. Highpass filter
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Active Filter Circuits Op Amp BandReject Filters CL
−ωc1 −s Rf + H (s ) = − + + s s ω ωc 2 Ri c1
RL RL
OUT
Rf
+
Rf
Vi 
RH RH
CH
OUT +
OUT
For ωc 2 >> ωc1
Rf
+
H (s ) =
Rf Ri
s 2 + 2ω s + ω ω c1 c1 c 2 (s + ωc1)(s + ωc 2 )
+ Vo
ωc1 =
1 RLCL
H ( jω )
ωc 2 =
max
1 RHCH
=K =
Rf Ri
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Active Filter Circuits Example:
• Design an active bandreject filter that has gain 5 and the stop frequency between 100 and 2000 Hz. Use 0.5 µF capacitors Fc1 = 100Hz and Fc 2 = 2000Hz
ωc1 =
1 RLCL
ωc 2 =
1 RHCH
RL =
For ωc 2 >> ωc1
1 1 = = 3.18 K Ω ωc1CL 2π 100(0.5)10−6
RH =
1 1 = = 159 Ω ωc 2CH 2π 2000(0.5)10−6
• For gain, choose Ri=1KΩ K=
Rf Ri
Rf = Ri K = 1000(5) = 5 K Ω ECE 30710 19
Active Filter Circuits −ωc1 − jω Rf H ( jω ) = + − jω + ωc1 jω + ωc 2 Ri
AdB = 20log10  H ( jω ) 
>> f=10:80000; >> w=2*pi*f; >> H=(((2*pi*100)./(j*w+2*pi*100)) +((j*w)./(j*w+2*pi*2000)))*(5); >> A=20*log10(abs(H)); >> semilogx(f,A) >> grid on; >> xlabel ('F (Hz)') >> ylabel ('A_{dB}')
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