ISTITUTO TECNICO INDUSTRIALE STATALE
GUGLIELMO MARCONI - Forlì
School Year 2009/2010
C.L.I.L. MODULE:
ANALYSIS AND DESIGN OF FIRST
ORDER RC FILTERS
Subjects: ELECTRONICS/ENGLISH
Teachers: Miria Bovino and Roberto Versari
Class: 4AEL
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Outline of the CLIL module




Lesson 1: RC low pass filter schematic and
transfer function (with C.A.D. example)
Lesson 2: frequency response graphical
representation (with excel example)
Lesson 3: Analysis of the CR high-pass
filter (with excel + C.A.D. example)
Lesson 4: An introduction to active filters
and circuit simulations (with C.A.D.
example)
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Lesson 1: RC low pass filter schematic
and transfer function (2h)
Lesson 1 Targets:





Learn how to draw the schematic of an RC low-pass
filter.
Learn how to calculate its transfer function G(jw)
Learn the filter constant time () definition
Learn how to calculate the RC low-pass filter output
for any sinusoidal input.
Learn how to realize an RC low pass filter with a
C.A.D. tool
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
RC low pass filter schematic
The first order passive RC low pass filter is
made of only two components:
R
VI(jw)
stimulus or
input sinusoidal
voltage
C
VO(jw)
output
sinusoidal
voltage
VI and VO can be represented by the corresponding
vector quantities: VI=VIej I, VO= VOej O
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Transfer function calculation (1/2)
The RC low pass filter can be easily seen as a simple
impedance divider with Z1=R and Z2=1/jwC
Z1=R
VI(jw)
Z2=1/jwC
VO(jw)
The impedance divider
formula provides:
VO/VI=Z2/(Z1+Z2).
VO/VI is the transfer function
G(jw) of the RC filter. Hence it
is: G(jw)=Z2/(Z1+Z2)
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Transfer function calculation (2/2)
Let’s substitute Z1=R and Z2=1/jwC in G(jw) expression
1
1
1
1
jwC
G
(
jw
)

 

1 jwC
1 jwRC

1
R

R

jwC
jwC
R.C= [s] is defined as the time constant of the
low-pass filter. Hence:
1
1
G
(
jw
)


1

jw
1

j

2

f



Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Output calculation (1/2)
G(jw) is a complex number with:

(jw
)
Amplitude: G

Phase:
1
2
1

(
w

)


artg
(
w

)
G
(jw
)
VO=G(jw). VI can be written using complex numbers properties:
V
1

G
(
jw
)

V


V
I
 VO Amplitude: V
O
I
I
2
2
1

(
w

)
1

(
w

)


VO Phase:







artg
(
w


)
O G
(
jw
)
I
I
G(jw) depends only on  and f (input signal frequency)

Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Output calculation (2/2)
RC low-pass filter
1)
2)
3)
4)
5)
VI, I, f
=R. C
sinusoidal input
G(jw)
G(jw)
VO, O, f
sinusoidal output
VI, I and f are given
the filter schematic is known   is calculated.
f and  are known  G(jw) and G(jw) are calculated
VO =|G(jw)| . VI is calculated
O = G(jw) + I is calculated
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Output calculation example
Exercise: calculate the output VO of
the circuit corresponding to the given
sinusoidal input VI
1) VI=1V, I=00, f=10KHz
remember: w2 f
2) =RC=320*50nF=16s
3) |G(10KHz)|=1/ 1+(2**10KHz*16s)2 =0,7
G(10KHz)=-artg(2**10KHz*16s)=-450
The output voltage is attenuated
4) VO=0,7*1V=0,7V
of a factor of 0,7 and delayed of
0
0
0
5) O=-45 +0 =-45
45 degrees when f=10KHz.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
LTspiceIV CAD tool
Download free from: http://www.linear.com/designtools/software


CAD tools: schematic implementation and real time simulation
needed to decrease electronic circuit time to market.
English language prerequisite for most CAD tools.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Lesson 2: RC low pass filter frequency
response graphical representation (2h)
Lesson 2 Targets:





Learn the graphical method to calculate any filter output.
Learn the low-pass filter cutoff frequency (fC) definition, its
bandwidth (Bw) and the relationship between fC and .
Learn how to read and draw first-order low pass filters
amplitude diagrams in logarithmic scale.
Learn how to design an RC low-pass filter for a given
bandwidth and/or fC.
Learn how to draw the amplitude and phase diagrams of
first-order RC low-pass filters with MS-excel.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
The graphical approach to the
calculation of filter output



Lesson 1 result: the filter output voltage can be derived by the
knowledge of the input voltage and the analitycal calculation of
the filter transfer function G(jw) – both amplitude and phase.
Disadvantages of the analitycal approach:
- two calculations (amplitude and phase) for each input
frequency.
- G(jw) analitycal expression depends on filter topology.
New graphical approach: two plots, namely |G(jw)| vs. f and
G(jw) vs. f are associated to each filter. The calculation of the
output voltage does not need G(jw) value calculation because
the values of amplitude and phase of G(jw) at the given
frequency can be simply read from the two given plots of the
filter.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Amplitude and Phase diagrams
G(jw) vs. f is the filter
phase diagram*
Amplitude diagram
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
Phase diagram
0
-15
-30
.
G(jw)
|G(jw)|
|G(jw)| vs. f is the filter
amplitude diagram*
-45
-60
-75
-90
1000000
100000
10000
frequency
1000
100
10
1
0,1
1000000
100000
10000
1000
100
10
1
0,1
frequency
Examples of amplitude and phase diagrams of an RC low
pass filter with R=1K and C=1F.
*=remember that: w2 f
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Output amplitude calculation with
graphical method (1/2)
VI, f
Amplitude diagram
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
VO, f
1000000
100000
10000
frequency
1000
100
10
1
0,1
sinusoidal input
|G(jw)|
RC low-pass filter amplitude diagram
sinusoidal output
1) VI, I and f are given
2) the filter amplitude diagram value at the frequency f
can be read from the plot.
3) VO =|G(jw)| . VI is easily calculated, since |G(jw)| is the
value read from the plot in step 2).
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Output amplitude calculation with
graphical method (2/2)


Case 1): VI=1,5V I=00 f=100Hz
From the filter amplitude diagram at f=100Hz we read
|G(jw)| =0,8.
Hence the output voltage amplitude at f=100Hz is
VO=0,8*1,5V=1,2V
Case 2): VI=1,5V I=00 f=1KHz
From the filter amplitude diagram at f=1KHz we read
|G(jw)| =0,18.
Hence the output voltage amplitude at f=1KHz is
VO=0,18*1,5V=0,27V
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Output phase calculation with graphical
method (1/2)
RC low-pass filter phase diagram
Phase diagram
0
-15
-30
O, f
G(jw)
I, f
-45
-60
-75
-90
1000000
100000
10000
frequency
1000
100
10
1
0,1
sinusoidal input
sinusoidal output
1) VI, I and f are given
2) the filter phase diagram value at the frequency f can
be read from the plot.
3) O = G(jw)+I is easily calculated, since G(jw) is the value
read from the plot in step 2).
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Output phase calculation with graphical
method (2/2)
Case 1): VI=1,5V I=00 f=100Hz
From the filter phase diagram at f=100Hz we read G(jw)=-350.
Hence the output voltage phase at f=100Hz is O=-350+0=350
 Case 2): VI=1,5V I=00 f=1KHz
From the filter phase diagram at f=1KHz we read G(jw)=-810.
Hence the output voltage phase at f=1KHz is O=-810+0=-810
Graphical method advantages:
- no need to calculate amplitude and phase: they are simply
read from the plots.
- it is a general approach valid for all kind of filters: it is only
needed the knowledge of the amplitude and phase plots.

Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
|G(jw)|
Amplitude diagram analysis
Amplitude diagram
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
fC=160Hz
stop band
pass band B
frequency scale is
logarithmic
1000000
100000
10000
1000
100
10
1
0,1
frequency
1) The cutoff frequency fC is defined as the frequency value at which |G(jw)|=0,7
In fact, all input signals with f< fC “pass”, i.e. we can find them at the filter output
without significant attenuation. On the contrary, all input signals with f> fC are “cut”,
i.e. the filter output has an amplitude significantly lower than the input.
2) The filter band B is defined as the range of input frequencies for which the filter
attenuation is  0,7. In practice B=[0, fC]. The bandwidth is the width of B=fC-0=fC
3) The relationship between fC and  is:
1
1
f


C
2


2

R
C
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Phase diagram analysis
fC=160Hz
Phase diagram
0
-15
-30

G(jw)
frequency scale is
logarithmic
-45
-60
-75
-90
1000000
100000
10000
1000
100
10
1
0,1
frequency
The cutoff frequency fC corresponds to the frequency value at which
G(jw)=-450. The filter introduces a maximum phase shift (a delay) of –900.
All input signals with frequencies within the filter pass band [0, fC] are delayed
less than 45/(360*f)=0,125/f. Input signals with frequencies outside the filter
pass band are delayed more than 0,125*f, with a maximum delay of 0,25/f
corresponding to the maximum phase shift of –900.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Amplitude diagram
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
frequency log. scale
|G(jw)|
|G(jw)|
Why a frequency logarithmic scale?
fC=160Hz
frequency linear scale
frequency
1) frequency logarithmic scale: the whole waveform is clearly
outlined and the amplitude values at different frequencies can be
read with reasonable accuracy
2) frequency linear scale: the amplitude values at different
frequencies cannot be read so the graph is useless.
The same considerations hold for the phase plot.
1000000
900000
800000
700000
600000
500000
400000
300000
200000
100000
0,1
1000000
100000
10000
1000
100
10
1
0,1
frequency
Amplitude diagram
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
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Whole logarithmic scale: the Bode plot
Analitycal formula for output calculation: VO = G(jw) . VI
Normalization (dimensionless): VO/1V= G(jw) . VI/1V
logarithmic: 20Log(VO/1V)=20Log G(jw) +20Log(VI/1V)
Output calculation in dB: VO[dBV]= G(jw) [dB]+VI[dBV]
Amplitude diagram in dB (Bode Plot)
|G(jw)| [dB]
0
-10
-20
-30
-40
-50
-60
fC=160Hz
20dB
1dec
-70
-80
1000000
100000
10000
1000
100
10
1
0,1
frequency
|G(jw)|[dB] vs. f is Bode plot
It is a full log. diagram.
f=fC  |G(jw)|=-3dB
(in fact, 20Log(0,7)=-3dB).
When f>fC Bode plot slope
is 20dB/dec.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Bode plot representation advantages


Very simple output calculation: VO=|G(jw)|+VI [dB] (only a
sum)
Very easy to draw!!! it can be approximated by:
|G(jw)|=0 for f<fC  horizontal line
|G(jw)|=-20*Log(f/fC)straight line with slope -20dB/dec
|G(jw)| [dB]
0dB
-20dB
-40dB
-60dB
fC
10fC 100fC 1000fC
f (log)
only approximation:
|G|=-3dB and not 0
when f=fC
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Design for a given fC or B
The design of a first order RC low pass filter for a given fC or B is
very simple:
 1  1
1) calculate the filter time constant:
2

fC 2

B
2) Choose a capacitor value among those available in the
laboratory: for instance, C=1F.
3) calculate the corresponding value of R:
R

C
4) if R is in the [100,100K] range  OK!!!, otherwise go back to
point 2) choosing another C value.
Example: design a low pass filter with fC=1KHz.
1

4
1) 


1
,
59

10
s2) C
2

f
C
 1F

4
1
,
59

10
s
3) R
 

159
4) OK!!!
6
10
F
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How to use MS-excel to draw
Amplitude and Phase diagrams
Electronic sheet tools are useful for automated calculations and diagram
representations  very useful for transfer function plot drawings.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Lesson 3: Analysis of the CR high-pass
filter (2h)
Lesson 3 Targets:






Learn the schematic of a high-pass CR filter and its
transfer function G(jw)
Learn how to draw the amplitude and phase plots of the
filter
Learn the relationship between , fC and B of the filter
Learn how to calculate the output of the filter
Learn how to design the filter
Learn how to simulate the filter with CAD tools in English
language.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
CR high pass filter schematic
First-order passive CR high-pass filter schematic:
C
O
1µ
VI(jw)
R
159
VI=VIeji
Same schematic as the
RC low pass filter but with
R and C interchanged.
VO(jw)
=RC=0,16ms
same formula as
VO=VOejo
the low pass filter
Using the same impedance divider model:
G(jw)=VO(jw)/VI(jw)=Z2/(Z1+ Z2), where Z1=1/jwC and Z2=R.
w

After a few mathematics we get:
G
(jw
)
jw

G
(jw
)
1
jw

2
1

(
w

)
0


90

artg
(
w

)
G
(jw
)
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Amplitude and Phase diagrams


Amplitude diagram in dB (Bode Plot)
0
-10
|G(fC)| = -3dB
-20
-30
20dB
-40
-50
-60
B=[fC,]
60
fC=1KHz
G(fc)=+450
45
30
fC=1KHz
15
0
1000000
100000
fC and B are defined in the same way as the RC low pass.
However, B=[fC,] because the filter attenuates only the
frequencies < fC, significantly.
10000
frequency
1000
100
10
1
0,1
1000000
100000
10000
1000
100
10
1
0,1
frequency
Phase diagram
75
1dec
-70
-80
0


90

artg
(
w

)
G
(jw
)
90
G(jw)
|G(jw)| [dB]
w

G
(
jw
)
[
dB
]

20

Log
(
)
2
1

(
w
)
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Diagrams characteristics

Amplitude:
 B=[fC,] because signals are not attenuated when f>fC.




Bw=.
|G(fC)|=-3dB (corresponding to a 0,7* attenuation)
when f<fC|G(f)| increases of 20dB for each decade of
increase of f (i.e. for a frequency increase of a factor of 10).
Phase:



f<<fCmaximum phase shift of +900, corresponding to an
output voltage anticipated of 0,25/f with respect to input.
f=fCG=+450; output voltage anticipated of 0,125/fC.
f>>fC G=00; output is in phase with input.
High-pass: input high frequencies are not attenuated nor shifted  “pass”. Input low
frequencies are attenuated and anticipated  “cut”. fC is the frequency cut threshold.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Filter design and Bode plot drawing

The filter design for a given fC can be done using the same
4 steps algorithm as the low pass filter:
1

2 fC
 1) calculate the filter time constant:



2) Choose a capacitor value among those available in the

laboratory: for instance, C=1F.
R
3) calculate the corresponding value of R:
C
4) if R  [100,100K]  OK!!!, otherwise go back to point 2)
choosing another C value available.
|G(jw)| [dB]
0,01fC 0,1fC fC
0dB
-20dB
-40dB
-60dB
0,001fC
f (log)
Bode plot drawing:
|G(jw)|=0 for f>fC  horizontal line
|G(jw)|=20*Log(f/fC)straight line with
slope +20dB/dec when f<fC
only approx. |G(fC)|=-3dB and not 0dB.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Output calculation examples
CR high pass filter with amplitude and phase diagrams of slide 27.
VO[dBV]=|G(f)|[dB]+VI[dBV]
O=G(f)+I



Case 1): VI=1V=0dBV I=00 f=100Hz
G(100Hz)=830. |G(100Hz)|=-20dB
 VO=-20dB+0dBV=-20dBV=0,1V O=830+00=830
Case 2): VI=1V=0dBV I=00 f=1000Hz=1KHz
G(1KHz)=450. |G(1KHz)|=-3dB
 VO=-3dB+0dBV=-3dBV=0,7V O=450+00=450
Case 3): VI=1V=0dBV I=00 f=10000Hz=10KHz
G(10KHz)=70. |G(10KHz)|=-0,09dB
 VO=-0,09dB+0dBV=-0,09dBV=0,99V O=70+00=70
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Circuit simulation: LTspiceIV
LTspiceIV download free from http://www.linear.com/designtools/software/
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Pros and cons of simulation


Simulation advantages:
 Fast time to market: you do not need to wait physical
implementation of the circuit to know its behavior
 No hand calculations  no human error, no interconnection
errors
Simulation drawbacks:
 If components models are not accurate the circuit can
exhibit unexpected behavior
 Parasitic components are not included and are circuit layout
dependent  difficult to predict. That’s what differentiates
the good from the bad designer.
The Pros are exceedingly better than the Cons: that’s why
simulation is ALWAYS used to prototype new designs.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Lesson 4: An introduction to active
filters and simulation examples (2h)
Lesson 4 Targets:







Learn the main differences between RC passive and active
filters.
Learn the schematics of low-pass and high-pass RC active
filters.
Learn the amplitude and phase diagrams of active filters.
Learn the schematic and the frequency response diagrams of
the pass-band second order RC active filter.
Learn how to simulate active filters.
Learn some practical applications of filters.
Learn the difference between first-order and higher order filters.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Passive vs. active filters
Let’s consider an RC low pass filter connected to a 1K load RL:
R
159
O
C
VI(jw)
1µ
VI=VIeji
RL
1K
V
(
jw
) Z
O
G
(
jw
)

 2
V
jw
) Z

Z
I(
1
2
VO(jw)
VO=VOejo
The effect of RL is to change Z2: Z2=C//RL=RL/(1+jwCRL)
1
(jw
)
Consequently G(jw) changes too and becomes: G
R
1


jw

Consequences:
RL
1)
Maximum amplitude = 1+R/RL <1
2)
fC = (1+R/RL)/2
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Limits of passive filters
The main limits of passive filters are:
1)
the frequency response of the filter is dependent on the
RL load. In particular, the maximum amplitude of the
filter when connected to a load is <1 and load
dependent. fC value is load dependent too.
2)
the maximum amplitude of a passive filter is =1 (no
signal gain) even when not connected to any RL load.
Active filters are able to provide both a frequency response
independent of any RL load and a signal gain >1. They are
obtained adding an active component such as an OP.AMP.
(OPerational AMPlifier) to a passive filter.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
First order low-pass active filter
Inverting amplifier configuration 
R21.59K G(jw)=-Z2/Z1 where Z1=R1=159
and Z2=C//R2=R2/(1+jw2),
2=C*R2=0,159ms
VCC
Schematic:
C100n
R1
U1
VI
VO
159
-VSS
OP05
R
2 1
G
(
jw
)


R
1
1

jw


2
Advantages:

G(jw) independent of any RL (>100) connected to filter
output VO because of the low OPAMP output resistance
-VSS
VI

Programmable low frequency gain = R2/R1
V1
V3
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
low-pass active filter frequency
response



R
2
1
0
G
(
jw
)
[
dB
]

20

Log
()

20
Log
(
)

180

artg
(
w
)
2
G
(
jw
)
R
1
1

(
w

)
2
Amplitude diagram in dB (Bode Plot)
20
165
|G(fC)|=17dB
0
G(fC)=1350
150
G(jw)
|G(jw)| [dB]
10
Phase diagram
180
135
-20

-10
fC=1KHz
-30
120
105
90
-40
Amplitude: Same diagram as passive filter shifted up of 20*Log(R2/R1)
Phase: Same diagram as passive filter shifted up of 1800
fC is the frequency at which |G| is –3dB with respect to its low frequency
value (same definition used for amplifiers): |G(fC)|=|G(0)|-3dB.
When f= fC  G(fC) = G(0) – 450
1000000
100000
frequency
10000
1000
100
10
1
1000000
100000
10000
1000
100
10
1
frequency
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
First order high-pass active filter
Schematic:
R1
VI
C
U1
VO
OP05
-VSS
159
C 1µ
VCC
R21.59K
G(jw)=-Z2/Z1 where
Z2=R2=1,59K and
Z1=R1+1/jwC=(1+jw1)/jwC,
1=C*R1=0,159ms
2=C*R2=1,59ms
jw

2
G
(jw
)

1

jw

1
High frequency gain =2/1=R2/R1
-VSS
V1
1
VI
fC 
V3
21
.tran 4m
Independent of load
RL (if RL>100)
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
high-pass active filter frequency
response
 

0

270

artg
(
w


)
G
(
jw
)
1
Amplitude diagram in dB (Bode Plot)
20
10
0
-10
-20
-30
-40
-50
-60
Phase diagram
270
255
240
G(jw)
|G(fC)|=17dB

|G(jw)| [dB]
w

R
2
1
G
(
jw
)
[
dB
]

20

Log
()

20
Log
(
)
2
R
1
1

(
w

)
1
fC=1KHz
G(fC)=2250
225
210
195
180
Amplitude: Same diagram of passive filter shifted up of 20*Log(R2/R1)
Phase: Same diagram of passive filter shifted up of 1800
fC is the frequency at which |G| is –3dB with respect to its high frequency
value: |G(fC)|=|G(f=)|-3dB.
When f= fC  G(fC) = G(0) – 450
1000000
100000
10000
frequency
1000
100
10
1
0,1
1000000
100000
10000
1000
100
10
1
0,1
frequency
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
What’s a pass-band active filter?
Low-pass filter: only low frequency inputs are not attenuated.
High-pass filter: only high frequency inputs are not attenuated.
Pass-bandonly frequency inputs [f1,f2] are not attenuated.
B=[f1,f2] is the pass-band of the filter.
Simpler pass-band filter: cascade of low-pass + high-pass
filters:
fC1
fC2
OUTPUT
INPUT
low-pass
high-pass
SIGNAL
SIGNAL
with fC=fC1
with fC=fC2
frequencies above
fC1 are attenuated
frequencies above
fC1 and below fC2
are attenuated
Works only if fC2>fC1; It doesn’t matter if low-pass is first or
second in the cascade if they are active filters.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Second order pass-band active filter
C130n
VI
C
VCC
Schematic:
R1
G(jw)=-Z2/Z1 where
Z2=R2//C2=R2/(1+ jw2)
R21.59K Z1=R1+1/jwC1=(1+jw1)/jwC1,
1=C1*R1=30s
2=C2*R2=9.54s
12=C1*R2=47,7s
U1
C2 6n
VO
1K
-VSS



j

w
12
G
(
jw
)


(
1

j

w
2
)

(
1

j

w
1
)
OP05
Filter pass-band B=[fC1,fC2], with fC1<fC2:
1
1
f

16
,
7
KHz
f


5
,
31
KHz
C
2
C
1
2



2



1
2
-VSS
1 VI
V1
V3
f

max
2


9
,
4
K

1 2
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Pass-band active filter frequency
response






w

0
R
2
1

270

artg
(
w
1
)

art
(
w
2
)
G
(
jw
)
[
dB
]

20

Log
(
)

20
Log
( 1
)

20
Log
(
)
G
(
jw
)
2
2
R
1 1

(
w

)
1

(
w

)
1
2
Amplitude diagram in dB (Bode Plot)
10
|G(jw)| [dB]
0
G(jw)
-10

-20
-30
fC1=5,3K
-40
fC2=16,7K
-50
G(fC1)=2050
G(fC2)=1550
The Bode plot of the pass-band active filter is exactly the sum of the
Bode plots of the low-pass and high-pass active filters with the same 2
and 1, respectively.
The phase diagram of the pass-band active filter is the sum decreased of
1800 of the phase diagrams of the low-pass and high-pass active filters
with the same 2 and 1, respectively.
1000000
100000
frequency
10000
1000
100
10
1
1E+07
1000000
100000
10000
1000
100
10
frequency
Phase diagram
270
255
240
225
210
195
180
165
150
135
120
105
90
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Filter simulation: passive vs. active
C1
C2
The RL load does not
affect active filter
response while it affects
both amplitude and
phase response of the
passive filter. Without
RL the two responses are the same.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Filter simulation: pass-band
C1
C2
Average value of input
square waveform “cut”.
High frequency
components “cut” too.
Output is almost
sinusoidal (depends on
filter selectivity).
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Filter applications: ADSL modem
6th order low-pass
You can phone while browsing the
Internet!!! The phone signal doesn’t
disturb the ADSL signal because of the
high-pass filter embedded in the ADSL
module. The ADSL signal doesn’t disturb
the phone signal because of the low-pass
filter inserted just before the phone
receiver.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Filter applications: audio systems
audio signal
[0,20KHz]
cross-over
low-pass
[0,3KHz]
woofer (bass)
cross-over
high-pass
[3K,20K]
tweeter (high)
A Hi-Fi (High Fidelity) audio system is composed of at least
two different speakers: a woofer to reproduce bass sounds and
a tweeter for the high sounds. A cross-over filter is placed
before each speaker so that the speaker can be reached only
by the frequency components of the input audio signal it is
able to reproduce. Cross-over filters are typically 2nd or 3rd
order low-pass or high-pass filters.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Higher order filters.
In practical applications 2nd or higher order filters are typically used
because of their higher selectivity:
|G(jw)| [dB]
0dB
Note: the order of a
filter is equal to the
number of reactive -20dB
components in its
schematic (i.e. the
-40dB
number of poles of its
-60dB
transfer function).
fC
10fC 100fC 1000fC
1st
f (log)
order: -20dB/dec
2nd order: -40dB/dec
3rd order: -60dB/dec
Low-pass filter attenuation at f=10*fC is:
1st order  0.1
2nd order  0.01
3rd order  0.001
The higher the order of the filter the better its ability to select only
the frequencies within its pass band B.
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Bibliography (1/3)
The following list of references can be used for an in-depth
investigation of the subject of the presentation:
1. The Analysis and Design of Linear Circuits, 6th Edition, R. E.
Thomas, A. J. Rosa, G. J. Toussaint, ISBN: 978-0-470-38330-8,
2009. (basic textbook on linear circuits analysis with a complete
chapter on AC analysis and Bode plots of analog filters).
2. http://www.animations.physics.unsw.edu.au/jw/RCfilters.html
(basic analysis of RC filters with animations).
3. http://www.play-hookey.com/ac_theory/filter_basics.html (basic
theory of active and passive filters).
4. http://www.educypedia.be/electronics/analogfil.htm (link to
several resources on analog filters, both passive and active).
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Bibliography (2/3)
5. http://www.usna.edu/Users/ee/kintzley/ee303sp09/lectures/EE3
03Sp09_L05_Filters.pdf (lecture on basic filter topologies and
solutions).
6. http://dev.emcelettronica.com/performing-ac-analysis-ltspice
(method to perform an AC analysis of an RC filter with Ltspice)
7. http://www.swarthmore.edu/NatSci/echeeve1/Ref/DataSheet/Int
roToFilters.pdf (basic introduction to analog filters from
National Semiconductor Corp., with examples of real analog
filters IC's and criteria to select among filter topologies).
8. Active Filter Cookbook, D. Lancaster, 2nd ed. Thatcher,
Synergetics Press, 1995 (a comprehensive textbook
on active filter design and analysis)
Istituto Tecnico Industriale Statale “G. Marconi” - Forlì
Bibliography (3/3)
9. http://www.wisc-online.com/objects/index_tj.asp?objID=ACE2803
(analysis of the bode plots of an RC low pass filter).
10. http://www.che.ttu.edu/pcoc/software/ppt/Chap09.ppt (chapter on
Bode plot analysis of linear systems with stability criteria).
11. http://eprints.iisc.ernet.in/13500/1/lec_5_web.pdf (lecture on Bode
plots and second order linear systems)