Chapter 11 Filters
Section 11.1
The Basic Principles of Filters
A filter allows only some signals to go through. A low pass filter only allows
signals with low frequencies to go through and a high pass filter only allows high
frequency signals to go through. A band pass filter is a circuit which allows signals
whose frequencies are within a certain range to go through.
Fig. 11.1-1 shows a low-pass filter.
R
C
vin
Fig. 11.1-1
vout
An RC low-pass filter
In this circuit, suppose that the frequency of the input signal is high.
will be almost short-circuited.
Thus vout  0 .
The capacitor
This means that no high frequency
signals can go through. On the other hand, if the frequency of the input is low, the
capacitor is almost open-circuited. Thus, vout  vin and the circuit is a low pass filter.
Fig. 11.1-2 shows a high pass filter.
vin
Fig. 11.1-2
C
R
vout
An RC high-pass filter
In this circuit, when the frequency of the input signal is low, the capacitor is almost
11-1
open-circuited and thus a very small current will flow. Thus, vout  iR  0 . On the
other hand, if the frequency is high, the capacitor will be almost short-circuited and
vout  vin . Thus, this is a high pass filter.
Fig. 11.1-3 shows a band pass filter.
It is a combination of a high pass filer and a
low pass filter. Since neither low nor high frequency signals can pass through, this is a
band pass filter.
R
vin
C
C
R
vout
Fig. 11.1-3 An RC band-pass filter
Filters can also be designed by using inductors, capacitors and resistors.
An
inductor is short-circuited when the frequency is low and open-circuited when the
frequency is high. The reader can thus easily see that the circuit in Fig. 11.1-4 is a low
pass filter, that in Fig. 11.1-4 is a high pass filter and that in Fig. 11.1-5 is a band pass
filter.
11-2
C
vin
vout
L
R
Fig. 11.1-4 An LCR low-pass filter
C
vin
L
vout
R
Fig. 11.1-5 An LCR high-pass filter
11-3
C
vin
L
R
Fig. 11.1-6
Section 11.2
vout
An LCR band-pass filter
The Transfer Functions of Some
Filters
In the above section, we briefly introduced the basic concept of filters.
We now discuss
the transfer functions of filters which describe the relationships between v out and vin in
terms of frequency.
Let us redraw the low pass filter in Fig. 11.1-1 again here as in Fig. 11.2-1.
R
C
vin
Fig. 11.2-1
vout
The redrawing of an RC low-pass filter
11-4
We shall denote j  2 f by S.
A( S ) 

vout
vin
The transfer function of the filer is as follows:
1
 SC
1
R
SC
(11.1-1)
1
1  SRC
A( j ) 

1
1  jRC
(11.2-2)
1
1   2 R 2C 2
From Equation (11.2-2), we can see why this is a low pass filter.
As
  , A( j)  0 , which means that high frequency signals cannot go through. It is
customary to denote 0 
1
1
and f 0 
. fc is called the critical frequency in
RC
2RC
this circuit. Note that
A max  A(  0)  1
and
A(  0 ) 
1
A
2 max
That is, when   0 , the gain of the filter is reduced to
1
of its maximum value.
2
We also say that it is reduced to its 3DB value.
Fig. 11.1-2 shows the transfer function of a low-pass filter.
11-5
| A( j ) |
1
1
2

0
Fig. 11.2-2
Let us redraw the high-pass filter in Fig. 11.1-2 as in Fig. 11.2-3.
vin
Fig. 11.2-3
C
vout
R
The redrawing of an RC high-pass filter
It is easy to derive the transfer function of this high pass filter to be as follows:
A( S ) 
and
From
SRC
1  SRC
A( j ) 
Equation
(11.2-3)
RC
1  R C
2
2
(11.2-4),
2

we
1
(11.2-4)
1
1 2 2 2
 RC
can
see
that
as
  0, A( j  0 ,
and
A(  )  1  A( j) max . Thus, this circuit is a high pass filter. We may again let
11-6
1
1
and f 0 
.
RC
2RC
0 
When   0 , the gain of the filter is reduced to
1
of
2
its maximum value.
The transfer function of this high-pass filter is now illustrated as in Fig. 11.2-4.
| A( j ) |
1
1
2
0
Fig. 11.2-4

The transfer of the RC low-pass filter (Equation 11.2-4)
Let us now consider the band pass filter in Fig. 11.1-3. We redraw Fig. 11.1-3 as in
Fig. 11.2-5.
C
{
vin
Z2
C
11-7
{
R
R
Z1
vout
Fig. 11.2-5
Z 1 and Z 2 in the RC band-pass filter
vout
Z1

vin
Z1  Z 2
1
R
SC 
Z1 
1
1  SRC
R
SC
1
1  SRC
Z2  R 

SC
SC
R
Thus,
R
1  SRC
A( S ) 
1  SRC
R

SC
1  SRC
SRC
 2 2 2
S R C  3SRC  1
S
RC

3S
1
2
S 
 2 2
RC R C
Let 0 
1
.
RC
Then
A( S ) 
0 S
2
S  30 S  0
and
(11.2-5)
(11.2-6)
2
A( j ) 
0
(0   2 )2  9 20
2
2
(11.2-7)
It can be easily shown that in this case, the gain is maximized when   0 , as shown in
Fig. 11.2-6. Perhaps it should be noted that the term 0 has different meanings for
different filters.
For both low pass and high pass filters, when   0 , the gain is
11-8
reduced to
1
of its maximum value while in the band pass case, when   0 , the
2
gain is maximized.
A
0
Fig. 11.2-6
The transfer of the RC band-pass filter

(SHOULD BE A(JW))
In summary, we have the following three transfer functions:
1. Low Pass:
2. High Pass:
3. Band Pass:
By letting 0 
1
A( S )  RC
1
S
RC
A( S ) 
(11.2-8)
S
(11.2-9)
1
S
RC
S
RC
A( S ) 
3S
1
S2 
 2 2
RC R C
1
, we have:
RC
11-9
(11.2-10)
1. Low Pass:
A( S ) 
0
S  0
(11.2-11)
2. High Pass:
A( S ) 
S
S  0
(11.2-12)
3. Band Pass:
A( S ) 
0 S
2
S  30 S  0
(11.2-13)
2
Finally, we have:
1. Low Pass:
A( j ) 
2. High Pass:
A( j ) 
3. Band Pass:
A( j ) 
Section 11.3
1
(11.2-14)
2
1 2
0
1
(11.2-15)
2
1  02

0
(0   2 )2  9 20
2
2
(11.2-16)
An RLC Filter
In the above section, we introduced low pass, high and band pass filters.
In the
following, we shall introduce a circuit containing resistor, capacitor and inductance, as
shown in Fig. 11.3-1.
11-10
C
vin
L
R
Fig. 11.3-1
An RLC circuit
By choosing the output terminals in different ways, this circuit can be used as low
pass, high pass and band pass filter as shown in Fig. 11.3-2.
C
vin
vout
C
vin
L
L
R
C
vout
vin
L
R
(a)
(b)
Fig. 11.3-2
R
(c)
Three filters out of the RLC circuit
11-11
vout
Let us see why the circuit will behave differently when different output terminals are
chosen.
Case 1:
The capacitor is chosen as the output.
Then the output voltage will be almost
0 when the frequency is large and will be equal to the input voltage as the frequency is
low. Thus this is a low pass filter.
Case 2:
The inductor is chosen as the output.
Since the behavior of an inductor is just
opposite to a capacitor, the circuit becomes a high pass filter.
Case 3:
The resistor is chosen as the output. We can imagine that neither low frequency
signals, nor high frequency signals, can pass the filter. Thus this is a band pass filter.
In the following, we shall derive transfer functions for these filters.
Case 1. The Low Pass Filter.
C
vin
vout
L
R
11-12
vout

vin
1
1
1
SC
LC
 2

1
R
1
S LC  1  SRC
SL 
R
S2  S 
SC
L LC
Case 2: The High Pass Filter
C
vin
vout
L
R
vout

vin
Case 3: The Band Pass Filter
C
vin
L
R
SL
S 2 LC
S2
 2

1
R
1
S LC  1  SRC
SL 
R
S2  S 
SC
L LC
vout
11-13
vout
vin
R
R
SRC
L

 2

1
R
1
S LC  1  SRC
SL 
R
S2  S 
SC
L LC
S
In Summary, we have the following three transfer functions:
1. Low Pass:
2. High Pass:
3. Band Pass:
Let 0 
1
vout
LC

R
1
vin
S2  S 
L LC
vout
S2

R
1
vin
S2  S 
L LC
R
S
vout
L

R
1
vin
S2  S 
L LC
(11.3-1)
(11.3-2)
(11.3-3)
R 0
1
1 L

and Q 
. Then
. Thus, we have the following three
L Q
R C
LC
transfer functions:
v
0
A( S )  out 

2
vin
S 2  S 0  0
Q
2
1. Low Pass:
11-14
(11.3-4)
A( S ) 
2. High Pass:
vout
S2


2
vin
S 2  S 0  0
Q
S
A( S ) 
3. Band Pass:
(11.3-5)
0
vout
Q


2
vin
S 2  S 0  0
Q
(11.3-6)
Finally, we have the following:
0 2
A( j 
1. Low Pass:
(0   2 ) 2 
2
(0   2 ) 2 
2
(11.3-7)
Q2
2
A( j 
2. High Pass:
 20 2
 20 2
(11.3-8)
Q2
 0
Q
A( j 
3. Band Pass:
( 0   ) 
2
2 2
The physical meaning of Q is now given:
Q  2
 2 0 2
(11.3-9)
Q2
Q is defined as follows:
maximum energy stored in an LCR circuit
A

energy lost per cycle
B
(11.3-10)
A
1
2
LI m
2
(11.3-11)
B
1
1
2
2 2
RI m T  RI m
2
2

(11.3-12)
Thus, we have
1
2
LI m
L
Q  2 2

1
2 2
R
RI m
2

(11.3-13)
11-15
Section 11.4
The Significance of the Second
Order Transfer Functions:
In Section 11.2, we have three transfer functions, expressed in Equations (11.2-11),
(11.2-12) and (11.2-13).
Both (11.2-11) and (11.2-12) are first order transfer functions
while (11.2-13) is a second order transfer functions. The transfer functions we obtained
in Section 11.3 are all second order transfer functions, as shown in Equations (11.3-4) to
(11.4-6).
In this section, we shall explain why we need second order transfer functions.
Let us consider the low pass filter as an example. Equation (11.2-11) is a first
order transfer function for low pass filters.
We rewrite its magnitude expressed in
Equation (11.2-14) as follows:
A( j ) 
1
(11.4-1)
2
1 2
0
Fig. 11.4-1 is a plot of the above function.
11-16
|A(j? )|
1
0.707
0

Fig. 11.4-1 The transfer function of an RC low-pass filter (Equation 11.4-1)
Equation (11.3-4) expresses a second order transfer function of a low pass filter.
Its magnitude function, expressed in Equation (11.3-7), is now rewritten as follows:
0 2
A j  
( 0   2 ) 2 
2
 0
2
2
(11.4-2)
Q2
From (11.4-2), we can prove the following:
Case 1:
0Q
1
. There is only one maximum:
2
max  0
(11.4-3)
Amax  1
(11.4-4)
11-17
1
.
2
Case 2:
Q
(1)
max  0
In this case, there are two cases:
Amax  1
(2)
and
max  0 1 
Amax 
1
2Q 2
(11.4-5)
Q
(11.4-6)
1
1
4Q 2
We further have:
A 0  Q
(11.4-7)
The proof of the above equations can be easily obtained and will not be given in this
book. In Fig. 11.4-2, we plot the second order transfer of the low pass filter expressed in
Equation (11.4-2) for different Q’s as follows:
11-18
Fig. 11.4-2 Equation 11.4-2 for different Q’s
If we compare the second order transfer function for low pass filter, as expressed in
Equation (11.4-2) with the corresponding first order transfer function, we can easily see
the difference between these two transfers. The second order transfer function provides
an additional parameter to control the maximum magnitude and the sharpness of the
transfer function of the low pass filter.
be used to control the bandwidth.
For the first order low pass filter, only 0 can
For the second order transfer function, Q plays a
critical role. As discussed above, a very small Q gives a rather narrow bandwidth and
we usually assume that Q 
1
.
2
We have seen the significance of Q.
meaning of Q.
It is now appropriate to explain the physical
For this RLC circuit,
11-19
Q
1 L
R C
(11.4-8)
Note that the circuit is a series connection of resistor, inductance and capacitor.
If the
circuit only contains inductance and capacitor, it will cause resonance. The existence of
the resistor will dampen the oscillation caused by the resonance.
resistance is, the more the circuit will tend to oscillate.
the circuit has a small tendency to oscillate.
The smaller the
If the resistance is very large,
But, as seen in Equation (11.4-6), a small
resistance corresponds to a large Q. Thus a high Q means that 0 is closer to max .
This will be made clearer when we examine the band pass filter.
As for the high pass filter, the situation is the same as that of the low pass filter.
We shall now discuss the band pass filter.
Let us rewrite the transfer function of the
RLC band pass filter, expressed in Equation (11.3-6), as below:
S
A( S ) 
0
vout
Q


2
vin
S 2  S 0  0
Q
(11.4-9)
Its magnitude function, as expressed in Equation (11.3-9), is as follows:
 0
Q
A( j ) 
( 0   2 ) 2 
2
 2 0 2
(11.4-10)
Q2
In this case, we can easily prove that
max  0
(11.4-11)
11-20
and
Amax  1
(11.4-12)
We would like to know the frequencies where A drops to 1
2
of its maximum
value. To do this, we have to solve the following equation:

0
2

2
 2 
 20 2
2
Q2
 20 2
Q2
Thus, we have to solve two equations:
2 
0
2 
0
 0  0
(11.4-13)
 0  0
(11.4-14)
2
Q
2
Q
There are four solutions for the above two equations. The following two satisfy the
condition that they have to be positive:
2 
0
2Q
1  
 0 1 
0
2Q
2  1 
1
4Q 2
 0 1 
(11.4-15)
1
4Q 2
(11.4-16)
0
(11.4-17)
Q
The above discussion is illustrated in Fig. 11.4-3.
11-21
Fig. 11.4-3
Equation (11.4-10) for different Q’s
From the above discussion, we can see that the larger Q, the smaller the bandwidth.
We may conclude that the second order transfer function gives us more flexibility to
design a filter than the first order transfer function.
Section 11.5
Experiments with the LCR Filter
ALL FIGURES NEED TO BE LABELED
Experiment 11.5-1 The Low Pass Filter
The filter circuit is as shown in Fig. 11.5-1.
11-22
1
L=40mH
2
R=1k
vin
3
C=25nF
vout
Fig. 11.5-1 The LCR low-pass filter circuit for Experiment 11.5-1
The critical frequency
f0 
0
1
1


 5 KHz.
2 2 LC 2 40  10 3  25  10 9
The program is shown in Table 11.5-1 and the gain vs frequency curve is shown in Fig.
11.5-2.
Table 11.5-1 Program for Experiment 11.5-1
LCR
11-23
.PROTECT
.OPTION POST
.LIB 'c:\mm0355v.l' TT
.UNPROTECT
.op
L
1
2
40mH
C
3
0
25nF
R
2
3
1k
Vin
1
0
AC
.AC DEC
100 1
1
10000k
.PLOT AC VDB(3)
.END
In the above, there is an instruction as follows:
AC DEC
100 1
10000k
The meaning of the above instruction as follows:
DEC means in decimal, 100 means sampling for every 100Hz and 1
10000k means that we sample from frequencies 1 to 10000k.
11-24
Fig. 11.5-2
The LCR low-pass filter transfer function in log scale
From the above figure, we can see that f 0  10 3.87  5KHz which is correct.
that in this experiment, Q 
1
103
Note
40  103
1
 1. 3 
 0.707 and there is a maximum
9
25  10
2
point close to the critical frequency.
Besides, it should be noted that both gain and
frequency are in log-scale. This is so for all experiments presented in this section.
Experiment 11.5-2 The Increasing of the Q-Value of the Low Pass Filter
In this experiment, we reduced the value of R from 1K ohms to 100 ohms.
Q was
thus increased and the critical frequency remains the same, but the gain vs frequency is
changed.
Table 11.5-2 displays the program while Fig. 11.5-3 shows the gain vs
frequency curve.
11-25
Table 11.5-2 Program for Experiment 11.5-2
LCR
.PROTECT
.OPTION POST
.LIB 'c:\mm0355v.l' TT
.UNPROTECT
.op
L
1
2
40mH
C
3
0
25nF
R
2
3
100
Vin
1
0
AC
.AC DEC
100 1
1
10000k
.PLOT AC VDB(3)
.END
11-26
Fig. 11.5-3
The LCR low-pass filter transfer function with Q enlarged
As explained in the above section, the increasing of the Q-value will make the curve have
a sharp maximum.
Experiment 11.5-3 The Decreasing of the Q-Value for the Low Pass Filter
In this experiment, we decrease the value of Q-value by increasing the value of
resistor from 100 ohms to 3K ohms. In this case, it can be shown that Q is around 0.43
which is smaller than
1
. The program is in Table 11.5-3 and the gain vs frequency
2
curve is in Fig. 11.5-4.
As can be seen, this Q-value creates a flat curve.
usually called a maximally flat curve.
Table 11.5-3 Program for Experiment 11.5-3
LCR
11-27
In fact, this is
.PROTECT
.OPTION POST
.LIB 'c:\mm0355v.l' TT
.UNPROTECT
.op
L
1
2
40mH
C
3
0
25nF
R
2
3
3k
Vin
1
0
AC
.AC DEC
100 1
1
10000k
.PLOT AC VDB(3)
.END
Fig. 11.5-4
The LCR low-pass filter transfer function with a small Q
11-28
Experiment 11.5-4 The LCR Band Pass Filter
In this experiment, we tested the performance of the LCR band pass filter. The
circuit is shown in Fig. 11.5-5. The program is displayed in Table 11.5-4 and the gain vs
frequency curve is shown in Fig. 11.5-6.
1
L=40mH
2
C=25nF
vin
3
R=3k
Fig. 11.5-5 The LCR band-pass filter for Experiment 11.5-4
Table 11.5-4 Program for Experiment 11.5-4
LCR
11-29
vout
.PROTECT
.OPTION POST
.LIB 'c:\mm0355v.l' TT
.UNPROTECT
.op
L
1
2
40mH
C
2
3
25nF
R
3
0
3k
Vin
1
0
AC
.AC DEC
100 1
1
10000k
.PLOT AC VDB(3)
.END
Fig. 11.5-6
The LCR band-pass filter transfer function
11-30
Experiment 11.5-5 The Increasing of the Q-Value for the Band Pass Filter
In this experiment, we increased the Q-value of the circuit by reducing the value of
R from 3K ohms to 100 ohms.
The program is in Table 11.5-5 and the gain vs
frequency curve is shown in Fig. 11.5-7. As can be seen, the bandwidth is decreased.
Table 11.5-5 Program for Experiment 11.5-5
LCR
.PROTECT
.OPTION POST
.LIB 'c:\mm0355v.l' TT
.UNPROTECT
.op
L
1
2
40mH
C
2
3
25nF
R
3
0
100
Vin
1
0
AC
.AC DEC
100 1
1
10000k
.PLOT AC VDB(3)
.END
11-31
Fig. 11.5-7
The LCR band-pass filter transfer function with a larger Q
NEEDS TO BE LABELED
Experiment 11.5-6 The Decreasing of the Q-value for the Band Pass Filter
In this experiment, we decreased the Q-value by increasing the value of R to 10K
ohms. The program is in Table 11.5-6 and the gain frequency curve is shown in Fig.
11.5-8. As shown, the bandwidth is larger now.
Table 11.5-6 Program for Experiment 11.5-6
LCR
.PROTECT
.OPTION POST
.LIB 'c:\mm0355v.l' TT
.UNPROTECT
.op
11-32
L
1
2
40mH
C
2
3
25nF
R
3
0
10k
Vin
1
0
AC
.AC DEC
100 1
1
10000k
.PLOT AC VDB(3)
.END
Fig. 11.5-8
The LCR band-pass filter transfer function with a smaller Q
Experiment 11.5-7 The Decreasing of the Critical Frequency for the Band Pass
Filter
In this experiment, we decreased the critical frequency by increasing the value of C
11-33
from 25nF to 300nF.
The program is shown in Table 11.5-7 and the gain frequency
curve is shown in Fig. 11.5-9. As shown, the critical frequency is decreased.
Table 11.5-7 Program for Experiment 11.5-7
LCR
.PROTECT
.OPTION POST
.LIB 'c:\mm0355v.l' TT
.UNPROTECT
.op
L
1
2
40mH
C
2
3
300nF
R
3
0
100
Vin
1
0
AC
.AC DEC
100 1
1
10000k
.PLOT AC VDB(3)
.END
11-34
Fig. 11.5-9
The decreasing of the critical frequency of the LCR band-pass filter
Section 11.6
Some Active Filters
In the above sections, we only used passive components to design filters. But filters
with reactive components only will have attenuation. To avoid attenuation, we will
employ active filters.
Fig. 11.6-1 shows a typical low pass filter which employs a
non-inverting operational amplifier.
It is easy to see why this is a low pass filter
because the RC circuit itself is a low pass filter.
11-35
R
+
vin
vout
A
C
Ra
Rb
Fig. 11.6-1 A low-pass filter built upon an operational amplifier
Fig. 11.6-2 shows another low pass filter with negative feedback.
11-36
R2
C
R1
-
vin
+
A
vout
Fig. 11.6-2 A low-pass filter with a capacitor connected the –terminal and the output
terminal
Why is this circuit a low pass filter?
Note that the capacitor is open-circuited when
the frequency is low and the circuit becomes that shown in Fig. 11.6-3.
frequency signals may get through.
11-37
Thus the low
R2
C
R1
-
vin
+
Fig. 11.6-3
A
vout
The filter in Fig. 11.6-2 in low frequency
But the capacitor becomes nearly short-circuited when the frequency is high as
shown in Fig. 11.6-4. The operational amplifier becomes a voltage follower and thus
there is no gain.
That is, the high frequency signals cannot go through and the circuit is
a low pass filter.
11-38
C
R1
vin
+
A
Fig. 11.6-4 The filter in Fig. 11.6-2 in high frequency
Two high pass filters are shown in Fig. 11.6-5 and Fig. 11.6-6.
11-39
vout
C
+
-
vout
A
vin
Ra
R
Rb
Fig. 11.6-5 A high-pass filter based upon an operational amplifier
R2
R1
C
-
vin
+
Fig. 11.6-6
A
Another high-pass filter based upon an operational amplifier
11-40
vout
Section 11.7
A General Case for Second-Order
Active Filters
Fig. 11.7-1 shows a general case for second order active filters. We may obtain low
pass, high pass and band pass filters by giving different components to Zi ' s.
A
Z4
Z1
Vin
Z5
Z3
V
-
Z2
+
Vout
.
Fig. 11.7-1 A general case of filters based upon an operational amplifier
VIN and VOUT not correct
Let us now find the transfer function for this general case circuit. Note that the
voltage at the inverting terminal is almost 0 for small signals.
Thus, for Node A, we
have:
(v  vin ) (v  vout ) v
v



0
Z1
Z4
Z 2 Z3
(11.7-1)
At the inverting terminal,
v vout

0
Z3 Z5
(11.7-2)
11-41
Based upon Equations (11.7-1) and (11.7-2), we have:
vout

vin
1
Z5
1
Z1 Z 3
 1
1
1
1 
1
 
 


 Z1 Z 2 Z 3 Z 4  Z 3 Z 4
(11.7-3)
.
A Low Pass Filter Derived from the General Case Filter
Fig. 11.7-2 shows a low pass filter derived from the general case filter. If the
frequency is high, C5 becomes short-circuited.
voltage follower.
The operational amplifier becomes a
If the frequency is low, signals can go through. Thus the circuit is a
low pass filter.
C5
R4
Vin
R1
R3
+
C2
Fig. 11.7-2
A low-pass filter derived from the general case
NEEDS MODIFIED
From Equation (11.7-3), we have:
11-42
Vout
vout

vin
1
R1 R3C 2 C5
1
S S
C2
2
 1
1
1 
1
 
 

 R1 R3 R4  R3 R4 C 2 C5
(11.7-4)
By letting
0 
Q
and
A0 
1
R3 R4C2C5
(11.7-5)
C2
C5 R3 R4
(11.7-6)
1
1
1 
 
 
 R1 R3 R4 
Vout
Vin

f 0
R4
,
R1
A00
we will have A( S ) 
S 
2
0
Q
(11.7-7)
2
S  0
.
(11.7-8)
2
Note that Equation (11.7-8) is almost exactly the same as Equation (11.3-4) which is
the transfer function of a low pass filter..
Experiment 11.7-1 The Low Pass Filter Derived from the General Case Filter
The operating amplifier circuit used throughout the experiments discussed in this
section is shown in Fig. 11.7-3.
11-43
VDD!
VBIAS3=0.75
VDD!
M3
V+
M1
M2
M10
M11
M8
M9
V-
vout
VBIAS67=0V
VBIAS45=-0.737V
M6
M7
M4
M5
VSS!
Fig. 11.7-3
The low-pass filter with an operational amplifier for Experiment 11.7-1
In this experiment,
C2  0.0048F , C5  0.0048F , R3  R4  3.3K .
program is in Table 11.7-1 and the gain vs frequency curve is shown in Fig. 11.7-4.
Table 11.7-1 Program for Experiment 11.7-1
Experiment 11.7-1
.PROTECT
.OPTION POST
.LIB 'c:\flexlm\model\tsmc\MIXED035\mm0355v.l' TT
.UNPROTECT
.op
11-44
The
VDD
VDD!
0
1.5V
VSS
VSS!
0
-1.5V
.GLOBAL
VDD!
VSS!
M1 5
Vi-
6
6
PCH
W=10U L=2U m=3
M2 4
Vi+
6
6
PCH
W=10U L=2U m=3
M3 6
VB3
VDD!
VDD!
PCH
W=100U L=2U m=7
M4 5
VB45
VSS!
VSS!
NCH
W=10U L=2U
M5 4
VB45
VSS!
VSS!
NCH
W=10U L=2U
M6 3
VB67
5
VSS!
NCH
W=10U L=2U
M7 VO
VB67
4
VSS!
NCH
W=10U L=2U
M8 3
3
1
1
PCH
W=10U L=2U m=3
M9 VO
3
2
2
PCH
W=10U L=2U m=3
M10 1
1
VDD!
VDD!
PCH
W=10U L=2U m=3
M11 2
1
VDD!
VDD!
PCH
W=10U L=2U m=3
Vi+
Vi+
0
0v
VBIAS3
VB3
0
0.75v
VBIAS45
VB45
0
-0.737v
VBIAS67
VB67
0
0v
Vin1
11
0
.AC DEC 100
AC 1
1
5000k
R1
11
10
C2
10
0
R3
Vi-
10
3.3k
R4
10
Vo
3.3k
C5
Vi-
Vo
0.0048u
.PLOT
3.3k
0.0048u
AC VDB(Vo)
11-45
.END
Fig. 11.7-4 The transfer function of the low-pass filter in Experiment 11.7-1
Experiment 11.7-2 The Decreasing of 0
We increased the values of capacitors and thus decreased 0 according to Equation
(11.7-5).
C2  C5  1F .
The program is shown in Table 11.7-2 and the gain vs
frequency curve is shown in Fig. 11.7-5.
As can be seen, 0 was significantly
decreased.
Table 11.7-2 Program for Experiment 11.7-2
11-46
Experiment 11.7-2
.PROTECT
.OPTION POST
.LIB 'c:\flexlm\model\tsmc\MIXED035\mm0355v.l' TT
.UNPROTECT
.op
VDD
VDD!
0
1.5V
VSS
VSS!
0
-1.5V
.GLOBAL
VDD!
VSS!
M1 5
Vi-
6
6
PCH
W=10U L=2U m=3
M2 4
Vi+
6
6
PCH
W=10U L=2U m=3
M3 6
VB3
VDD!
VDD!
PCH
W=100U L=2U m=7
M4 5
VB45
VSS!
VSS!
NCH
W=10U L=2U
M5 4
VB45
VSS!
VSS!
NCH
W=10U L=2U
M6 3
VB67
5
VSS!
NCH
W=10U L=2U
M7 VO
VB67
4
VSS!
NCH
W=10U L=2U
M8 3
3
1
1
PCH
W=10U L=2U m=3
M9 VO
3
2
2
PCH
W=10U L=2U m=3
M10 1
1
VDD!
VDD!
PCH
W=10U L=2U m=3
M11 2
1
VDD!
VDD!
PCH
W=10U L=2U m=3
Vi+
Vi+
0
0v
VBIAS3
VB3
0
0.75v
VBIAS45
VB45
0
-0.737v
VBIAS67
VB67
0
0v
Vin1
.AC DEC
11
0
100 1
AC 1
5000k
11-47
R1
11
10
3.3k
C2
10
0
1u
R3
Vi-
10
3.3k
R4
10
Vo
3.3k
C5
Vi-
Vo
1u
.PLOT
AC VDB(Vo)
.END
Fig. 11.7-5 The transfer function of the low-pass filter in Fig. 11.7-3 with values of
capacitors increased
A High Pass Filter Derived from the General Case Filter
11-48
Fig. 11.7-6 shows a high pass filter derived from the general case filter.
R5
C4
vin
C1
C3
R2
vout
+
Fig. 11.7-6 A high-pass filter derived from the general case
A needs to be added.
The transfer function of the high pass filter is as follows:
C1
C4
A( S ) 
1  C
1
1 
1
S 2  S  1 
  
R5  C3C4 C4 C3  R2 R5C3C4
S2
By letting
0 
Q
1
R2 R5C3C4
(11.7-9)
(11.7-10)
C3C4 
R5 

R2  C1  C3  C4 
(11.7-11)
11-49
A0 
and
vout
vin

 
C1
,
C4
(11.7-12)
we have:
A0 S 2
A( S ) 
S 
2
0
Q
S  0
(11.7-13)
2
Equation (11.7-13) is almost exactly the same as Equation (11.3-5), which is the transfer
function of another high pass filter.
A Band Pass Filter Derived from the General Case Filter
Fig. 11.7-14 shows a band pass filter derived from the general case filter.
C4
vin
R5
R1
C3
+
vout
R2
Fig. 11.7-14 A band-pass filter derived from the general case
It can be easily seen that this circuit is a band pass filter. Note that high frequency
11-50
signals cannot go through because of C3 and low frequency signals cannot go through
because of C4. The transfer function of this band pass filter is:
1
R1C4
A( S ) 
1  1
1 
1 1
1 
  
S 2  S    
R5  C3 C4  R5C3C4  R1 R2 
S
(11.7-14)
By letting
0 
1 1
1 
  
R5C3C4  R1 R2 
(11.7-15)
1
1   C3C4 
Q  R5    
 R1 R2   C3  C4 
A0 
and
vout
vin
(11.7-16)
R5 C3
R1 C3  C4

  0
(11.7-17)
we have:
A0
A( S ) 
S 
2
0
Q
0
Q
S
S  0
(11.7-18)
2
Experiment 11.7-3 The Band Pass Filter Derived from the General Case Filter
In this experiment, we set C3  C4  50 pF , R1  R2  R5  13K . The program is
in Table 11.7-3 and the gain vs frequency curve is in Fig. 11.7-15.
Table 11.7-3 Program for Experiment 11.7-3
11-51
Experiment 11.7-3
.PROTECT
.OPTION POST
.LIB 'c:\flexlm\model\tsmc\MIXED035\mm0355v.l' TT
.UNPROTECT
.op
VDD
VDD!
0
1.5V
VSS
VSS!
0
-1.5V
.GLOBAL
VDD!
VSS!
M1 5
Vi-
6
6
PCH
W=10U L=2U m=3
M2 4
Vi+
6
6
PCH
W=10U L=2U m=3
M3 6
VB3
VDD!
VDD!
PCH
W=100U L=2U m=7
M4 5
VB45
VSS!
VSS!
NCH
W=10U L=2U
M5 4
VB45
VSS!
VSS!
NCH
W=10U L=2U
M6 3
VB67
5
VSS!
NCH
W=10U L=2U
M7 VO
VB67
4
VSS!
NCH
W=10U L=2U
M8 3
3
1
1
PCH
W=10U L=2U m=3
M9 VO
3
2
2
PCH
W=10U L=2U m=3
M10 1
1
VDD!
VDD!
PCH
W=10U L=2U m=3
M11 2
1
VDD!
VDD!
PCH
W=10U L=2U m=3
Vi+
Vi+
0
0v
VBIAS3
VB3
0
0.75v
VBIAS45
VB45
0
-0.737v
VBIAS67
VB67
0
0v
Vin1
.AC DEC
11
0
AC 0.00001
100 1
1000000000k
11-52
R1
11
10
13k
R2
10
0
13k
C3
Vi-
10
50p
C4
10
Vo
50p
R5
Vi-
Vo
13k
.PLOT
AC VDB(Vo)
.END
Fig. 11.7-15
The transfer function of the band-pass filter in Fig. 11.7-3
Experiment 11.7-4 The Decreasing of 0 and Enlarging of the Bandwidth
11-53
We decreased 0 by setting C3  C4  5 pF .
Q-value and an enlargement of the bandwidth.
This caused a decreasing of the
The program is in Table 11.7-4 and the
gain vs frequency is in Fig. 11.7-16. As can be seen, 0 is made smaller and the
bandwidth is now larger.
Table 11.7-4 Program for Experiment 11.7-4
Experiment 11.7-4
.PROTECT
.OPTION POST
.LIB 'c:\flexlm\model\tsmc\MIXED035\mm0355v.l' TT
.UNPROTECT
.op
VDD
VDD!
0
1.5V
VSS
VSS!
0
-1.5V
.GLOBAL
VDD!
VSS!
M1 5
Vi-
6
6
PCH
W=10U L=2U m=3
M2 4
Vi+
6
6
PCH
W=10U L=2U m=3
M3 6
VB3
VDD!
VDD!
PCH
W=100U L=2U m=7
M4 5
VB45
VSS!
VSS!
NCH
W=10U L=2U
M5 4
VB45
VSS!
VSS!
NCH
W=10U L=2U
M6 3
VB67
5
VSS!
NCH
W=10U L=2U
M7 VO
VB67
4
VSS!
NCH
W=10U L=2U
M8 3
3
1
1
PCH
W=10U L=2U m=3
M9 VO
3
2
2
PCH
W=10U L=2U m=3
M10 1
1
VDD!
VDD!
PCH
W=10U L=2U m=3
M11 2
1
VDD!
VDD!
PCH
W=10U L=2U m=3
Vi+
Vi+
0
0v
11-54
VBIAS3
VB3
0
0.75v
VBIAS45
VB45
0
-0.737v
VBIAS67
VB67
0
0v
Vin1
11
0
AC
.AC DEC
100 1
1000000000k
R1
11
10
13k
R2
10
0
13k
C3
Vi-
10
0.005u
C4
10
Vo
0.005u
R5
Vi-
Vo
13k
.PLOT
AC VDB(Vo)
.END
11-55
0.00001
Fig. 11.7-16
Section 11.8
The band-pass filter with  0 decreased
The Sallen and Key Filters
There are other active filters.
In this section, we shall introduce the Sallen and Key
filters. The Sallen and Key filters employ positive feedback. Fig. 11.8-1 shows a
Sallen and Key low pass filter.
11-56
C1
R1
R2
+
vin
vout
-
C2
Ra
Rb
Fig. 11.8-1 A Sallen and Key low-pass filter
It is obvious that this is a low pass filter because the high frequency will short-circuit the
capacitor C2.
Fig. 11.8-2 shows a Sallen and Key high pass filter.
It is obvious that the low
frequency signals cannot go through as they will be blocked by the capacitors.
11-57
R1
+
vin
C1
C2
vout
-
R2
Ra
Rb
Fig. 11.8-2
A Sallen and Key high-pass filter
11-58