Lecture 9
RC Filters
9-1
Outlines of Filter Design
input
Filter
output
Filtering:
Certain desirable features are retained
Other undesirable features are suppressed
9-2
Filters
Filters have the property of removing unwanted frequencies
from our signal.
Classes:
Passive (made of capacitors, resistors, inductors)
Active (involving an amplifier)
Types:
Low-Pass (remove high frequencies)
High-Pass (remove low frequencies or DC)
Band-Pass (remove a range of frequencies on
two sides)
Notch (removes frequencies in the middle)
9-3
Classification of Filters
Signal Filter
Analog Filter
Element Type
Active
Passive
Digital Filter
Frequency Band
Low-Pass
High-Pass
Band-Pass
All-Pass
Band-Reject
9-4
Filters – Type of filters
Passive
filters
9-5
http://www.ece.eps.hw.ac.uk/~pmr/teaching/ae/lectures/circuits1.htm
Terminology in Filter Design
• Signal-To-Noise Ratio (S/N)
S
⎛ WS
= 10 ⋅ log⎜⎜
N
⎝ WN
⎞
⎟⎟dB
⎠
• Bandwidth
¾the range of frequencies of |G(jw)|>0.707
• Cutoff Frequency
¾the end of pass-band frequency
• Break-point of a filter
¾the point with a gain of -3dB
9-6
RC Filters
In combination with a resistor, a capacitor’s variation in reactance with frequency can be
used to construct a simple low-pass or high-pass filter:
C
Vin
Vout
High-pass filter
Vin
R
Vout = Vin ⋅
C
Low-pass filter
R
Z
Vout = Vin ⋅
Z = R 2 + X C2
XC =
Vout
R
XC
Z
Z = R 2 + X C2
1
2πfC
XC =
1
2πfC
9-7
Passive Low-Pass Filter
H( jω)
Vout
Vin
ωp
ωs
C
Vout
ω
R
Vin
RL
• The pass-band is from
0 to some frequency
wp.
• Its stop-band extends
form some frequency
ws, to infinity.
• In practical circuit
design, engineers often
choose amplitude gain
of 0.95 for passive RC
filters:
9-8
Passive High-Pass Filter
H( jω)
• Its stop-band is form 0
to some frequency ws
• The pass-band is from
some frequency wp to
infinity.
Vout
Vin
ωs
ω
ωp
• In practical circuit
design,
engineers
choose amplitude gain
of 0.95 for passive CR
filters:
9-9
C
R
Vin
Vout
Design of Passive Filters
The amplitude response:
R
C
Vin
Vout
RL
V out
=
V in
1
1 + ( RC ω )
2
The amplitude gain:
Transfer Function
H ( jω ) =
1
jRCω + 1
1
H (s ) =
RCs + 1
G=
ZL
ZF + Z L
The 3dB break-point is at:
f3dB =
1
1
=
2πRC 2πτ
9-10
Guideline of Pass Filter Design
Select resistor based on amplitude gain:
R
C
Vin
Vout
RL
G=
ZL
= 0.95
ZF + ZL
ZF ≈ R =
Transfer Function
1
H (s ) =
τs + 1
0.05
Z L = 0.053⋅ RL
0.95
Select capacitor based on cut-off freq:
C=
Time Constant
τ
R
=
1
2πRf3dB
τ = RC
9-11
Higher Order Filters
R
Vin
R1
C
Vout
First Order RC Low Pass
Vin
C1
R2
C2
Vout
Second Order RC Low Pass
The higher the order of the filter,
the closer it approaches ideal characteristics.
9-12
Active Filters
• Active filters employ Op-Amps to attenuate
select frequencies and amplify signal during
filtering process.
• Q factor of a filter is defined as the ratio of
the center frequency fc to the bandwidth fH fL :
Q=
fC
( fH − fL )
9-13
Active filters- cascading low pass filters
First
order
Second order
3rd order
5th order
9-14
Op Amp for everyone, Ron Mancini, Ed, Texas instrument, 2001.
Low-Pass Active Filter
Passive filters take up lots of space in a circuit and
cause signal to be lost. Combining a passive RC
filter with an op amp for amplification creates
what is known as an active filter. By “active”
we mean that the filter requires power
R1
to operate.
C1
RF
-
Here is an example of an
R2
active low-pass filter. The signal
+
is provided to the noninverted
C2
input through an RC low-pass filter
made up of R2 and C2. Feedback to
limit gain comes through C1 and RF. The
parallel combination of C1 and RF presents
an impedance which decreases with increasing
frequency, meaning that more negative feedback is provided to the inverting input at
higher frequencies, reducing gain at those frequencies.
9-15
Design of Low Pass Active Filters
The -3 dB cut-off frequency:
C2
fH = 1
RF
R1
Vin
A
B
The DC gain:
-
K LP = −
Vout
+
Transfer Function:
T .F . = K LP
(2πRF C2 )
ω0
s + ω0
RF
R1
Example:
Design a low pass filter with
cut-off frequency of 5 kHz,
and DC gain of 10:
Two equations, three unknowns
9-16
High-Pass Active Filter
RF
R1
C1
C2
+
R2
Here is an example of an active high-pass
filter. C2 and R2 make up an RC high-pass
filter at the input of the op amp. R3 provides
a path for the input when the frequency is
R3
too low for C2 to freely conduct. When the
input signal passes through R3 instead of
into the amplifier, the output is tied directly to the input and the gain is reduced. So, this
amplifier has low gain at low frequencies and higher gain at high frequencies. C1 prevent
any DC at the input from being coupled to the output.
9-17
Design of High Pass Active Filters
The -3 dB cut-off frequency:
fH = 1
(2πR1C1 )
The DC gain:
R
K HP = F
RF
C1
Vin
R1
A
B
+
Transfer Function:
T .F . = K HP
s
s + ω0
Vout
R1
Two
equations,
unknowns
three
Select one component based
on other conditions, and
determine the values of the
other two components.
9-18
Filter Class
• A filter of a given order can be made to approximate to
ideal characteristics in a number of ways, depending on
the values of the filter components (or say: depending on
the filter class.
• Two useful classes are Butterworth (maximally flat) and
Chebyshev (equal-ripple) filters (n is the filter order)
Vout
1
Butterworth Filter
=
2n
Vin
f
⎛
⎞
1+ ⎜
⎟
⎝ fC ⎠
Vout
=
Vin
Chebyshev Filter
1
⎞⎟
1 + E 2Cn2 ⎛⎜ f
⎝ fC ⎠
9-19
Higher Order Active Filters
Filter Class
C1
Vin
+
R1
R2
-
R1
R2
C1
C2
K
Buterworth
3.01 dB at ωH
1.00 1.00 1.00 1.00 1.59
Chebyshev
1 dB ripple
1.00 1.00 0.94 0.97 2.00
Vout
C2
Rb
Ra
Gain=K
The above list gives the gain and
component valves for one of the
many choices for ωH=1. You may
find more combinations from
filter design handbook(s).
9-20
Active Filters– High Pass Filters
Low pass
High pass
9-21
Op Amp for everyone, Ron Mancini, Ed, Texas instrument, 2001.
Active Filters – Band Pass Filter
9-22
Op Amp for everyone, Ron Mancini, Ed, Texas instrument, 2001.
Active Filters – Band Reject Filter
Passive band reject filter
Active band
reject filter
9-23
Op Amp for everyone, Ron Mancini, Ed, Texas instrument, 2001.
References
• Op Amp for everyone, Ron Mancini, Ed,
Texas instrument, 2001.
9-24