Filters
Alan Murray
Agenda
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What is a filter?
Filter types (revision from Eng 1)
Capacitors and Frequency (revision)
Passive (RC) Filters
Active filter
Alan Murray – University of Edinburgh
Filters?? What are they
1kΩ
DC+AC
0.1μF
Vin
1kΩ
-
Vout
AC only
+
1Hz
AC only
THIS IS AN AMPLIFIER CIRCUIT,
WHOSE GAIN VARIES WITH
FREQUENCY
10Hz 100Hz1Hz
1kHz
-1
IT IS A FILTER!
GAIN
Alan Murray – University of Edinburgh
Frequency of
Vin
IDEAL
HIGH-PASS
FILTER
Frequency
IDEAL
BAND-PASS
FILTER
Frequency
BAND-STOP
("NOTCH") FILTER
IDEAL
Frequency
Clickers
Alan Murray – University of Edinburgh
Vout/Vin
Vout/Vin
Frequency
Vout/Vin
IDEAL
LOW-PASS
FILTER
Vout/Vin
Vout/Vin
Vout/Vin
Vout/Vin
Vout/Vin
Filter Types
Vin
FILTER
CIRCUIT
Vout
REAL
Frequency
REAL
Frequency
REAL
Frequency
REAL
Frequency
Voltage and Current (AC)
in a Capacitor
ω=2πf
V = V0 sin(ωt)
I
Vs
V
π/2
Animate this slide!
“simulation”
I = I0cos(wt) = I0sin(ωt+90°)
Alan Murray – University of Edinburgh
Potential Divider (again!)
Vin
R'
Vout
R
 Vin
R  R'
R
Now exchange R'
for a capacitor
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Vout
High-Pass RC Filter
Vin
C
Vout  Vin
R
1
R
j C
R
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Vout
High-Pass RC Filter
Vin
C
Vout
R
Capacitor blocks DC,
Impedes low frequencies
Discharged
to 0V
Vin
C
Vout
Capacitor passes AC,
Passes high frequencies
R
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High-Pass RC Filter
High Frequency, ω∞
Vin
C
Vout  Vin
R
1
R 0
j C
R
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Vout =Vin
High-Pass RC Filter
Low Frequency, ω0
Vin
C
Vout  Vin
R
1
R ∞
j C
R
Alan Murray – University of Edinburgh
Vout =0
High-Pass RC Filter
|Gain|
1
(1/ √2)
0
Phase
ωc = 1/(RC)
Frequency
ω>ωc
Vin
90°
C
Vout
|Gain| = 1,
Phase = 0°
R
45°
0°
ω<ωc
Vin
C
Vout
R
ωc is the “Critical
frequency”, 1/RC,
where Gain = 1/√2
and Phase difference
= 45° = π/4
ω=ωc
Vin
C
R
Alan Murray – University of Edinburgh
|Gain|≈ 0,
Phase = 90°
Vout
|Gain| = 1/√2,
Phase = 45°
Low-Pass RC Filter
Gain 
Vin
R
Vout
1
1  ωR C 
2
, Phase  tan 1 ω RC 
C
|Gain|
Clickers?
Frequency
ωc = 1/RC
Phase
0°
-45°
-90°
Alan Murray – University of Edinburgh
This is a "Passive Filter"
It has no power supplies
 No transistors, diodes etc.
 Gain is always ≤ 1
 Output load becomes part of the circuit ..

Vin
R
C
I
Vout
Rload
i.e. - this will not
work as
designed!
Alan Murray – University of Edinburgh
Inverting Circuit - Revision
Gain = - R
r
R
Vin
r
+
Vout
Add a capacitor ...
Alan Murray – University of Edinburgh
Inverting Circuit - Revision
X
C
Gain = - R
r
Gain = - R//C
r
R
Vin
r
+
Vout
What is R//C ...
R in parallel with C?
Alan Murray – University of Edinburgh
Handwaving analysis,
Active Filter ω → ∞
C
ZC = 1/(jωC), ω → ∞, ZC → 0
Vinv = Vnoninv = 0 = Vout
Blocks high-frequency,
therefore low-pass?
Vin
R
r
+
Alan Murray – University of Edinburgh
Vout
Handwaving analysis,
Active Filter ω → 0
X
C
ZC = 1/(jωC), ω → 0, ZC → ∞
Simple inverting
op-amp circuit.
Passes low
frequencies.
Definitely low-pass.
Vin
R
r
+
Alan Murray – University of Edinburgh
Vout
Active Filter Characteristics

Current into load comes from Op-Amp J
–NOT
from source
Filter is "buffered" from the load J
 Filter can have gain>1 J
 Needs power
L

Alan Murray – University of Edinburgh
Applications?

Passive

• Smoothing mains
power supply
• Loudspeaker
"crossover" circuit

Active
• Amplifier tone
controls


to woofer and
tweeter!
Alan Murray – University of Edinburgh
High-pass
Low-pass
= treble
= bass
Summary
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
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Why, what and how filters are
made.
Why Passive (RC) Filters work
Why Active (RC) Filters work and
are useful
Alan Murray – University of Edinburgh