⊲ 13: Filters
Filters
1st Order Low-Pass
Filter
Low-Pass with Gain
Floor
Opamp filter
Integrator
High Pass Filter
2nd order filter
Sallen-Key Filter
Twin-T Notch Filter
Conformal Filter
Transformations (A)
13: Filters
Conformal Filter
Transformations (B)
Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 1 / 13
Filters
13: Filters
Filters
1st Order Low-Pass
Filter
Low-Pass with Gain
Floor
Opamp filter
Integrator
High Pass Filter
2nd order filter
Sallen-Key Filter
Twin-T Notch Filter
Conformal Filter
Transformations (A)
⊲
A filter is a circuit whose gain varies with frequency. Often a filter aims to
allow some frequencies to pass while blocking others.
Conformal Filter
Transformations (B)
Summary
Radio/TV: a “tuning” filter blocks all frequencies
except the wanted channel
Loudspeaker: “crossover” filters send the right
frequencies to different drive units
Sampling: an “anti-aliasing filter” eliminates all
frequencies above half the sampling rate
–
Phones: Sample rate = 8 kHz : filter
eliminates frequencies above 3.4 kHz.
[Wikipedia]
Computer cables: filter eliminates interference
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 2 / 13
1st Order Low-Pass Filter
⊲
Conformal Filter
Transformations (B)
Summary
Y
X
=
1/jωC
R+1/jωC
=
1
jωRC+1
=
1
jω
p +1
b
Corner frequency: p = a =
Asymptotes: 1 and
1
RC
p
jω
Very low ω: Capacitor = open circuit
Very high ω: Capacitor short circuit
0
|Gain| (dB)
13: Filters
Filters
1st Order
Low-Pass Filter
Low-Pass with Gain
Floor
Opamp filter
Integrator
High Pass Filter
2nd order filter
Sallen-Key Filter
Twin-T Notch Filter
Conformal Filter
Transformations (A)
-10
-20
-30
0.1p
p
ω (rad/s)
10p
A low-pass filter because it allows low frequencies to pass but attenuates
(makes smaller) high frequencies.
The order of a filter: highest power of jω in the denominator.
Almost always equals the total number of L and/or C.
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 3 / 13
Low-Pass with Gain Floor
13: Filters
Filters
1st Order Low-Pass
Filter
Low-Pass with
Gain Floor
Opamp filter
Integrator
High Pass Filter
2nd order filter
Sallen-Key Filter
Twin-T Notch Filter
Conformal Filter
Transformations (A)
⊲
Conformal Filter
Transformations (B)
Summary
Y
X
=
R+1/jωC
4R+1/jωC
=
jωRC+1
jω4RC+1
Corner frequencies: p =
Asymptotes: 1 and
=
1
4RC ,
jω
q +1
jω
p +1
q=
1
RC
1
4
Very low ω:
Capacitor = open circuit
Resistor R unattached. Gain = 1
0
-5
-10
Very high ω:
Capacitor short circuit
Circuit is potential divider with gain 20 log10
E1.1 Analysis of Circuits (2015-7256)
0.1q
1
4
p
ω
q
10q
= −12 dB.
Filters: 13 – 4 / 13
Opamp filter
13: Filters
Filters
1st Order Low-Pass
Filter
Low-Pass with Gain
Floor
Opamp filter
Integrator
High Pass Filter
2nd order filter
Sallen-Key Filter
Twin-T Notch Filter
Conformal Filter
Transformations (A)
Inverting amplifier so
Conformal Filter
Transformations (B)
Summary
1. Can have gain > 1.
⊲
Y
X
/jωC )
3R(R+ /jωC )
= − 3R||(R+
=
−
R
R×(3R+R+1/jωC )
1
= −3 ×
R+1/jωC
4R+1/jωC
1
= −3 ×
jωRC+1
jω4RC+1
Same transfer function as before except × − 3 = +9.5 dB.
Advantages of op-amp crcuit:
2. Low output impedance - loading
does not affect filter
10
5
0
0.1q
p
ω
q
10q
3. Resistive input impedance - does
not vary with frequency
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 5 / 13
Integrator
13: Filters
Filters
1st Order Low-Pass
Filter
Low-Pass with Gain
Floor
Opamp filter
Integrator
High Pass Filter
2nd order filter
Sallen-Key Filter
Twin-T Notch Filter
Conformal Filter
Transformations (A)
⊲
Conformal Filter
Transformations (B)
Summary
Y
X
1/jωC
=−
1
= − jωRC
R
Capacitor: i = C dvdtC
x
= −C dy
i= R
dt
dy
dt
=
−1
RC x
Rt
dy
dt
0 dt
y(t) =
=
−1
RC
−1
RC
Rt
0
Rt
0
xdt
20
0
xdt + y(0)
-20
0.1
RNote: if x(t) =1cos ωt
cos(ωt)dt = ω sin(ωt) ⇒ gain ∝
1
ω RC
10
1
ω.
We can limit the LF gain to 20 dB:
Y
X
10R× /jωC
= − 10R||R/jωC = − R(10R+
1/jωC )
10
0.1
= − jω10RC+1
ωc = RC
E1.1 Analysis of Circuits (2015-7256)
1
1
Filters: 13 – 6 / 13
High Pass Filter
13: Filters
Filters
1st Order Low-Pass
Filter
Low-Pass with Gain
Floor
Opamp filter
Integrator
High Pass Filter
2nd order filter
Sallen-Key Filter
Twin-T Notch Filter
Conformal Filter
Transformations (A)
⊲
Conformal Filter
Transformations (B)
Summary
Y
X
=
R
R+1/jωC
=
jωRC
jωRC+1
Corner Freq: p =
1
RC
Asymptotes: jωRC and 1
Very low ω: C open circuit: gain = 0
Very high ω: C short circuit: gain = 1
We can add an op-amp to give a
low-impedance output. Or add gain:
jωRC
RB
Z
X = 1 + RA × jωRC+1
0
-10
-20
-30
0.1p
E1.1 Analysis of Circuits (2015-7256)
ω
p
10p
Filters: 13 – 7 / 13
2nd order filter
13: Filters
Filters
1st Order Low-Pass
Filter
Low-Pass with Gain
Floor
Opamp filter
Integrator
High Pass Filter
2nd order filter
Sallen-Key Filter
Twin-T Notch Filter
Conformal Filter
Transformations (A)
⊲
Conformal Filter
Transformations (B)
Summary
Y
X
=
R2 +jωL
1/jωC +R1 +R2 +jωL
=
LC(jω)2 +R2 Cjω
LC(jω)2 +(R1 +R2 )Cjω+1
=
jωC(jωL+R2 )
LC(jω)2 +(R1 +R2 )Cjω+1
Asymptotes: jωR2 C and 1
Corner frequencies:
+20 dB/dec at p =
−40 dB/dec at q =
0
-20
R2
L = 100 rad/s
√1
= 1000 rad/s
LC
q
√b
=
2 (R1 + R2 ) C
4ac
-40
p
100
q
ω
1k
10k
Damping factor: ζ =
= 0.6.
1
= Q = 0.83 = −1.6 dB (+0.04 dB due to p)
Gain error at q is 2ζ
Compare with 1st order:
2nd order filter attenuates more rapidly than a 1st order filter.
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 8 / 13
Sallen-Key Filter
13: Filters
Filters
1st Order Low-Pass
Filter
Low-Pass with Gain
Floor
Opamp filter
Integrator
High Pass Filter
2nd order filter
Sallen-Key Filter
Twin-T Notch Filter
Conformal Filter
Transformations (A)
⊲
Conformal Filter
Transformations (B)
Summary
0
-20
-40
100
ω
Asymptotes:
1k
10k
jω
p
2
and 1
−X
−Z
−Z
KCL @ Y : Y1/jωC
[assume V+ = V− = Z]
+ 1Y/jωC
+ YR
=0
⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC
KCL @ V+ :
Z
mR
+
Z−Y
1/jωC
= 0 ⇒ Z(1 + jωmRC) = Y jωmRC
Sub Y : Z (1+jωmRC)
(1 + 2jωRC) − Z (1 + jωRC) = XjωRC
jωmRC
⇒
Z
X
=
m(jωRC)2
m(jωRC)2 +2jωRC+1
Corner freq: p =
√ 1
mRC
=
(jω/p)2
(jω/p)2 +2ζ(jω/p)+1
= 996 rad/s, ζ =
1
2Q
= pRC =
√1
m
= 0.6
Sallen-Key: 2nd order filter without inductors. Can easily have gain >1.
Designing: Choose m = ζ −2 ; C any convenient value; R =
E1.1 Analysis of Circuits (2015-7256)
ζ
pC .
Filters: 13 – 9 / 13
Twin-T Notch Filter
13: Filters
Filters
1st Order Low-Pass
Filter
Low-Pass with Gain
Floor
Opamp filter
Integrator
High Pass Filter
2nd order filter
Sallen-Key Filter
Twin-T Notch
Filter
Conformal Filter
Transformations (A)
⊲
Conformal Filter
Transformations (B)
Summary
After much algebra:
p=
(1+m)((2jωRC)2 +1)
(2jωRC)2 +4(1−m)jωRC+1
Z
X
=
=
(1+m)((jω/p)2 +1)
(jω/p)2 +2ζ(jω/p)+1
1
2RC
= 314, ζ = 1 − m = 0.1
Very low ω: C open circuit
Z
Non-inverting amp, X
=1+m
m+1 = 5.6dB
2ζ p
0
-20
Very high ω: C short circuit
Z
-40
Non-inverting amp, X
=1+m
200
300
500
ω (rad/s)
2
= −1: numerator = zero resulting in infinite attenuation.
At ω = p, jω
p
The 3 dB notch width is approximately 2ζp = 2(1 − m)p.
Used to remove one specific frequency (e.g. mains hum @ 50 Hz)
Do not try to memorize this circuit
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 10 / 13
Conformal Filter Transformations (A)
13: Filters
Filters
1st Order Low-Pass
Filter
Low-Pass with Gain
Floor
Opamp filter
Integrator
High Pass Filter
2nd order filter
Sallen-Key Filter
Twin-T Notch Filter
Conformal Filter
Transformations
(A)
⊲
Conformal Filter
Transformations (B)
Summary
Y
, can always be written using dimensionless
A dimensionless gain, VVX
jωL ZL
ZL
ZR
2
=
jωRC,
=
,
=
−ω
LC.
impedance ratio terms: Z
Z
R
Z
C
R
C
Impedance scaling:
0
♥
-10
Scale all impedances by k:
R′ = kR, C ′ = k−1 C, L′ = kL
Impedance ratios are unchanged
so graph stays the same.
(k is arbitrary)
Frequency Shift:
♠
-20
-30
10
100
1k
ω rad/s
10k
k = 20
Scale reactive components by k:
R′ = R, C ′ = kC, L′ = kL
⇒ Z ′ (k−1 ω) ≡ Z(ω)
Graph shifts left by a factor of k.
k=5
Must scale all reactive components in the circuit by the same factor.
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 11 / 13
Conformal Filter Transformations (B)
13: Filters
Filters
1st Order Low-Pass
Filter
Low-Pass with Gain
Floor
Opamp filter
Integrator
High Pass Filter
2nd order filter
Sallen-Key Filter
Twin-T Notch Filter
Conformal Filter
Transformations (A)
Conformal Filter
Transformations
(B)
Summary
0
Change LR circuit to RC:
♣
-10
1
Change R′ = kL, C ′ = kR
ZR′
jωL
′ ′
=
jωR
C
=
⇒Z
R =
′
C
♦
-20
ZL
ZR
Impedance ratios are unchanged
at all ω so graph stays the same.
(k is arbitrary)
⊲
Reflect frequency axis around ωm :
-30
1k
10k
ω rad/s
100k
1M
k = 106
Change R′ = ωmk C , C ′ = ωm1kR
2 ∗
ωm
ZR′
C
⇒Z ′ ω = Z
ZR (ω)
C
k = 0.1, ωm = 20 k
(a) Magnitude graph flips
(b) Phase graph flips and negates since ∠z ∗ = −∠z.
(k is arbitrary)
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 12 / 13
Summary
13: Filters
Filters
1st Order Low-Pass
Filter
Low-Pass with Gain
Floor
Opamp filter
Integrator
High Pass Filter
2nd order filter
Sallen-Key Filter
Twin-T Notch Filter
Conformal Filter
Transformations (A)
Conformal Filter
Transformations (B)
Summary
⊲
• The order of a filter is the highest power of jω in the transfer function
denominator.
• Active filters use op-amps and usually avoid the need for inductors.
◦ Sallen-Key design for high-pass and low-pass.
◦ Twin-T design for notch filter: gain = 0 at notch.
• For filters using R and C only:
◦ Scale R and C: Substituting R′ = kR and C ′ = pC scales
−1
frequency by (pk) .
◦ Interchange R and C: Substituting R′ = ω0kC and C ′ = kω10 R flips
the frequency response around ω0 (∀k).
Changes a low-pass filter to high pass and vice-versa.
For further details see Hayt et al. Chapter 16.
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 13 / 13