Active Filter Design by means of Switched Capacitors, Analog, and Digital Sachin Mehta Reno Nevada

1. Practical Active Filter Design by Means of Analog, Digital, and
Switched Capacitor Theory
Written By: Sachin Mehta
University of Nevada, Reno

2. 2
Abstract: Essentially all electronic signal-processing systems and networks incorporate filters of
one sort or another. Their applications in the world of electronics and most everything around
us in the world is an important area that needs to be studied and improved. Different types of
filters can give users specific responses and outputs that can be modified quite simply and with
ease given the transfer function of the network. The filters that will be studied and analyzed in
this paper can be grouped in the category of “active” filters. These types of circuits are
constructed with the use of operational amplifiers and can incorporate all kinds of resistors,
capacitors, or even none at all. In the following sections of this paper, we will discuss various
filter designs, their specifications, and of course the results & data of their outputs (cutoff
frequency, rise time, gain, etc.) A significant portion of this paper discusses the network in Fig.
1, which provided a total of three different filters when the output was simply moved to
different nodes of the circuit. Measuring the frequency response of these filters gave us
experimental data that we could then compare to desired outputs. Having these different plots
helped illustrate the discrepancy that arises when varying engineering techniques is
implemented. For example, Fig.5 shows the low-pass filter Bode Plot from the spectrum
analyzer compared to the ideal Bode Plot that was obtained with Matlab. This report discusses,
in detail similar, results comparing different methods of active filter design.
1) Transfer Function Determination
a) Low Pass
The transfer function
𝑉𝑜𝑢𝑡
𝑉𝑖𝑛
of Fig. 1 below is expressed in the s-domain as the following
The first part of any filter design is the derivation of the transfer function of the network in
question. For this report, three different transfer functions were determined by analyzing the
network depicted in Fig.1 below and by using the specifications shown in Table 1. The low-pass
transfer function was determined in the s-domain by means of mathematical and control
system theory. The circuit was represented by a block diagram (Fig. 2) and then the transfer
function was derived. We used the fact that in control theory blocks in series can be multiplied.
In addition, we used Eq. (1) which represents the closed loop feedback. If you examine the
block diagram, you can see the inner loop—which we simplified first.
𝐺𝑐𝑙 =
𝐺 𝑐𝑙
𝐺 𝑐𝑙+1
(1)

3. 3
Table 1: Butterworth Low Pass
Filter Specifications
Passband 0-2000 Hz
Minimum Power
Gain at fc
0.5 (-3 dB)
Start of Stop Band 3500 Hz
Maximum Power
Gain at fr
0.1 (-10 dB)
Figure 1: Practical Active Filter Circuit
Figure 2: Block Diagram of Main Network Studied in this Laboratory Experiment

4. 4
The inner loop (inside the dashed lines) was simplified using Eq. (1) and the multiplication rule,
mentioned previously. The result was as follows:
𝐺𝑐𝑙 =
𝐺𝑜𝑝𝑒𝑛
𝐺 𝑜𝑝𝑒𝑛 + 1
=
1
𝑅1 𝐶1 𝑠
1
𝑅1 𝐶1 𝑠
+ 1
=
1
𝑅1 𝐶1 𝑠 + 1
Next, we analyzed the remaining loop with the same rules and equations and obtained the
following expression:
𝐺𝑐𝑙 =
𝐺𝑐𝑙1 ×
1
𝑅2 𝐶2 𝑠
(𝐺𝑐𝑙1 ×
1
𝑅2 𝐶2 𝑠
) + 1
=
(
1
𝑅1 𝐶1 𝑠 + 1
)(
1
𝑅2 𝐶2 𝑠
)
[(
1
𝑅1 𝐶1 𝑠 + 1
) × (
1
𝑅2 𝐶2 𝑠
)] + 1
=
1
𝑅1 𝑅2 𝐶1 𝐶2 𝑠2
1
𝑅1 𝑅2 𝐶1 𝐶2 𝑠2 + 1
𝐻𝐿(𝑠) =
−1
𝑅1 𝑅2 𝐶1 𝐶2 𝑠2+𝑅2 𝐶2 𝑠+1
(2)
Where: 𝑇1 = 𝑅2 𝐶2 & 𝑇2 = 𝑅1 𝐶1
b) Band Pass
The transfer function
𝑉𝑜𝑢𝑡2
𝑉𝑖𝑛
was also determined by implementing block diagram rules, like
mentioned above regarding closed loop feedback systems. We were able to express the
following:

5. 5
𝐻 𝐵( 𝑠) =
𝐻 𝑜
1 + 𝐻 𝑜 𝐻𝑓
=
𝐻2
1 + 𝐻1 𝐻2
𝐻 𝐵( 𝑠) =
−𝑠𝑇
𝑠2 𝑇1 𝑇2 + 𝑠𝑇1 + 1
(3)
c) High Pass
The voltage transfer function HH(s) was determined, also by implementing controls systems
analysis and feedback control rules.

6. 6
(4)
2) Butterworth Low-Pass Filter Design
With the transfer function determined, we equated coefficients of the denominator to the
Butterworth second order filter prototype equation—Eq. (5).
𝐻( 𝑝) =
1
𝑝2+ √2𝑝+1
(5)
Although it looks as though we could equate coefficients of H(s) and H (p) at this point in time,
we did have to ensure that Eq. (3) was first standardized. This was required since the design
specifications called for a cutoff frequency of 2000 Hz. This Laplace operator (the variable ‘p’ in
Eq. 3) was put in terms of the s-domain as follows.
𝑝 =
𝑠
ѡ 𝑝
=
𝑠
2𝜋𝑓
=
𝑠
2𝜋(2000)
=
𝑠
12566.37
With a standardized form, H(p) now could be expressed in the s-domain and the new prototype
equation we used is shown next.
𝐻( 𝑝) = 𝐻(𝑠 𝑝𝑟𝑜𝑡𝑜𝑡𝑦𝑝𝑒 ) =
1
𝑠2
1.57 × 108 + √2(
𝑠
12566.37
)+ 1
Equating to the transfer function of the network (Eq. 2) and inputting C1 = C2 = 10 nF resulted in
two equations—with two unknown variables—R1 & R2.
1
1.57×108 = 𝑅1 𝑅2 𝐶1 𝐶2 = 6.33 x 10-9
𝑅1 𝑅2 = 6.33 × 107
𝑅1 =
6.33 × 107
𝑅2

7. 7
𝑅2 𝐶2 = 1.13 × 10−4
𝑅2 = 1.125 × 104
= 11.25 kΩ
Inputting the value of R2 into Eq. (4) we obtained R1 = 5.626 kΩ
Note, that precise components (𝑅1& 𝑅2) were not available for use so 𝑅1& 𝑅2
were adjusted to 5600 Ω and 11000 Ω respectively.
3) Butterworth Low-Pass Filter w/ Matlab
With values for each of the parameters of our system defined, we were able to simulate our
design with Matlab. The code below shows the transfer function being compiled in Matlab,
which enabled us to obtain a Bode Plot of the system.
The magnitude and phase plots are both depicted below in Fig. 3, which also displays some of
the plot characteristics—such as the cutoff frequency of the system; (1.26 x 104 rad/s).

8. 8
Figure 3: Matlab Bode Plot Output for 2nd Order Butterworth Low-Pass Filter
We analyzed the response above by comparing the cutoff frequency displayed in the plot to the
required design specifications of Table 1. An important representation that is used in many
scientific cases is known as percent discrepancy. This percentage describes the difference
between an experimental piece of data and its ‘ideal’ counterpart (Eq. 6).
(6)
For example, the percent discrepancy between the corner frequency which was required and
that which was obtained with Matlab is:
|2005.35 − 2000|
2000
× 100 = 0.268 %
The percentage determined, being below a mere 0.5 %, was readily acceptable. With such an
insignificant discrepancy, we were able to state that the design steps and transfer function we
determined earlier were both correct and extremely accurate. It is quite possible that the only
reason we had any discrepancy between our results was due to rounding errors in the following
scenarios:
 Rounding error when Laplace operator was configured to the s-domain.

9. 9
 When equating coefficients of the prototype & closed loop equations, the results laid
out earlier in this paper were arrived at after rounding had been initiated.
 This then resulted in error in determining both of the resistances (R1 & R2).
4) Butterworth Low-Pass Filter w/ Breadboard
This section of the experiment required constructing the circuit from Fig. 1 on a breadboard
and analyzing an experimental output. Implementing this filter did take several components.
The components listed in Table 2 are actual parts that I could have used to build this low pass
filter. However, the parts used in lab were not the same as those listed, but close in tolerance
and other factors. Note that since we did not have an 11.25 kΩ resistor to use on our bread-
boarded design—we implemented an 11 kΩ resistor.
In order to determine the output of our low pass filter, we used both the spectrum analyzer
and the power supply provided in the laboratory room. Fig. 4 shows the output from the
display of the spectrum analyzer for the 2nd Order Butterworth filter.
Table 2: Part Supply List for Active Network shown in Fig. 1
Component
Type
Description
Mouser Part # Mfr. Part # Manufacturer Voltage
Rating or
Operating
Supply
Voltage(V)
Tolerance Power
Rating
Price
per
Unit($)
Units Total
Price ($)
Operational
Amplifier
595-LF347N LF347N Texas
Instruments
7 to 36 +/- 3.5 V to
+/- 18 V
N/A 0.75 1 0.75
Metal Foil
Resistor (10 kΩ)
71-
S102JT10K000TB
Y078510K0000T9L Vishay
Precision
Group
300 0.01% 600mW 15.25 6 91.50
Metal Foil
Resistor (5.6
kΩ)
71-
S102JT10K000TA
Y078510K00032T9L Vishay
Precision
Group
300 0.01% 600mW 15.25 1 15.25
Metal Foil
Resistor (11 kΩ)
71-
S102JT10K000TAR
Y078510K83000T9L Vishay
Precision
Group
300 0.01% 600mW 15.25 1 15.25
Capacitor
(0.01µF)
598-
715P10356KD3
715P10356KD3 Cornell
Dubilier
600 5% N/A 1.26 2 2.52
140.52

10. 10
Figure 4: Spectrum Analyzer Output of Bode Plot for 2nd Order Butterworth Low Pass Filter
Unfortunately, the output above fails to detail the -3dB point of the system. However, by using
the AASCI file from the spectrum analyzer, we were able to plot the same data shown above in
a different manner and determine the cutoff frequency that way. With a few lines of code
(shown below) we plotted the data from the AASCI file, and then superimposed that plot onto
the magnitude plot that was obtained previously using the Matlab software. The result was the
plot shown in Fig. 5, where it is evident that there is a slight difference between the two
methods of analysis—experimentally and with use of Matlab.

11. 11
Figure 5: Bode Plots of Matlab Output vs. Spectrum Analyzer Data
The fact that our Matlab design used the exact resistor (R2 = 11.25 kΩ) value that was
calculated in the design process makes the Bode Plot in Fig. 3 (and therefore the “Matlab Bode
Plot” in the above output) the more accurate plot.
We were able to roughly estimate the percent error between our bread-boarded design, and
our Matlab design by implementing Eq. (6) as follows.
|1.262 − 1.26|
1.26
× 100 = 0.159 %
was indeed the more precise second order Butterworth low pass filter design.
As discussed previously, the circuit constructed used an 11 kΩ resistor rather than the more
precise 11.25 kΩ resistor because of lack of available hardware. This could have been among
the various factors of error that occurred.

12. 12
5) Butterworth Low-Pass Filter w/ MultiSim
Implementing the circuit of Fig. 1 in MultiSim, allowed us to compare various design methods
and their respective accuracies. The circuit that was constructed with proper resistors and
capacitors is shown next, in Fig. 6.
Figure 6: Circuit Schematic of 2nd order Butterworth Low Pass Filter
With the circuit above, we were able to run an AC analysis of the network—which gave the
plots depicted in Fig. 7.

13. 13
Figure 7: Multisim Bode Plot of Low Pass Filter
The plot shows the trace and cursor measuring the -3dB point and says that at this occurred at
2.00 kHz—exactly what our design called for. This filter design behaved just as expected
because at a gain of -3dB—the frequency is the same as that of the corner frequency we
needed.
6) Butterworth High-Pass Filter w/ Matlab
Using Matlab, we took the time to analyze the high pass filter. The code shown below is what
was used in order to simulate our design.

14. 14
The code shown above allowed us to simulate the high pass filter and obtain its corresponding
Bode Plot, which was then studied to make sure design specifications were met and correct
design was met. The frequency response is actually shown below in Fig. 8, which shows the
trace at -3.03 dB at a frequency of 1.26 × 104
rad/s. At first glance, it is easy to fret and state
that this design is wrong, since the cut off frequency we desired was 2000 Hz. However, closer
examination reveals that if units are taken into account—we can see that 1.26 × 104
rad/s is
actually equivalent to 2005.35 Hz. The percent discrepancy that we see, using Eq. (6) comes to:
|2005.35 − 2000|
2000
× 100 = 0.268 %

15. 15
This error could have been a result of using 11.25 kHz in the simulation instead of the 11 kHz.
However this error, or lack thereof, just shows how truly correct our high pass filter Matlab
simulation was.
Figure 8: Frequency Response of High Pass Filter
6)-----#4 High Pass Filter w/ Breadboard
Implementing this filter on the breadboard in lab allowed us to gain experience with the
necessary material and tools. Using the spectrum analyzer, we were able to obtain the
experimental frequency response of our circuit, which can be seen below in Fig. 9.

16. 16
Fig. 9: Spectrum Analyzer Output of High Pass Filter
The output above displays that the -3dB point occurs at a frequency of 2.112 kHz—a frequency
close to the design specifications. In fact, this experimental cutoff frequency and the one
desired had a percent error between them of:
|2.112 − 2|
2
× 100 = 5.6 %
This percent discrepancy is somewhat minimal and therefore insignificant in our results.
Possibilities for its occurrence come down to the fact that actual resistors and capacitors were
used—which increase the chance of error because they are not 100% precise values. In
addition, the voltage that was set through the power supply probably was not an exact 5 V due
to lack of calibration and percent tolerance.
6)-----#5 High-Pass Filter w/ MultiSim
Using MultiSim, we were able to design and analyze the high pass filter which we had built on
the breadboard. In doing this, we could see firsthand what was occurring—and what should
occur. The circuit schematic of the high pass filter was really exactly the same as that of the
Butterworth low pass—except for the placement of the probe (for frequency response
analysis). This schematic can be viewed in Fig. 10.

17. 17
Figure 10: Schematic for High Pass Filter w/ Probe
The circuit above gave an output just of what a high pass filter—with the given design
specifications—should look like. This Bode Plot (Fig. 11) shows where the cutoff frequency
occurs and also what the phase looks like.

18. 18
Figure 11: MulitSim Frequency Response of High Pass Filter
The plot above shows that at the -3dB point, the frequency is 2 kHz—which is what the design
specifications required.
There shows no percent discrepancy between this frequency and design—meaning that the
simulation truly shows what the high pass filter with the given specs should look like.
6) Band Pass Filter w/ Matlab
Using the code below we were able to simulate the band pass filter. The plot in Fig. 12 depicts
the Matlab simulation of the Band Pass filter. It shows that at nearly 0 dB, the frequency is
around 2005 Hz—which the specifications call for.

19. 19
Figure 12 : Matlab Simulation of Band Pass Filter

20. 20
6)-----#4 Band Pass Filter w/ Breadboard
Using the breadboard, wire kit, resistors, capacitors, and an operational amplifier—we were
able to build a band pass filter to meet the specifications of Table 1. Then, with the
implementation of the spectrum analyzer, the output shown in Fig. 13 is the frequency
response of the circuit.
Figure 13: Spectrum Analyzer Frequency Response of BandPass Filter
The above output shows that the -3dB point occurs at 2.112 kHz—very close to the 2kHz
desired in the specifications. The percent error between is
|2.112 − 2|
2
× 100 = 5.6 %

21. 21
This could have resulted from:
 Rounding error when Laplace operator was configured to the s-domain.
 When equating coefficients of the prototype & closed loop equations, the results laid
out earlier in this paper (such as Eq. 4) were arrived at after rounding had been initiated.
 This then resulted in error in determining both of the resistances (R1 & R2).
6)-----#5 Band Pass Filter w/ MultiSim
Using MultiSim, we designed the bandpass filter shown in Fig. 14.
Figure 14: Schematic of BandPass Filter in MultiSim
It is interesting to note that this bandpass filter is almost exactly the same circuit that was used
for both the low pass and high pass filters. What is different? The placement of the output of
the circuit.
Measuring from this output, Fig. 15 shows the following.

22. 22
Figure 15: MultiSim Output of Bandpass Filter
The output shows that at nearly 2000 Hz frequency, the magnitude is essentially at 0 dB. Since
MultiSim is not a completely accurate and precise software—this was most likely the reason for
the variance.
7) Digital Filter Design—Realizing 2nd Order Low Pass Butterworth Filter from (2)—With
Recursive Equation
In order to complete the recursive equation, we rewrite Eq. (2) with the parameters shown
𝐻(𝑠) =
−1
𝜏1 𝜏2 𝑠2+𝜏1 𝑠+1
which reads τ to be a time constant of the system.
We can let the quantity τ1τ2 = B and the quantity 𝜏1 = A. This simplifies calculations and lets
the bilinear transform be implemented in order to transform the analog filter into our digital
design. It is important to note that for the bilinear transform—we let s =(
2
𝑇
)(
𝑧−1
𝑧+1
).
Substituting this into the equation above, as well as B & A, lets us write the low pass filter in
terms of the Z-domain as follows:

23. 23
𝐻(𝑍) =
−1
𝐵 (
2
𝑇
)(
𝑧 − 1
𝑧 + 1
)
2
+ 𝐴 (
2
𝑇
)(
𝑧 − 1
𝑧 + 1
) + 1
Setting this equal to “output/input” and cross-multiplying gives:
[𝐵 ((
2
𝑇
)(
𝑧−1
𝑧+1
))
2
+ 𝐴 (
2
𝑇
)(
𝑧−1
𝑧+1
) + 1] = − 𝑋
Simplifying gives:
[((
4𝐵
𝑇2
)(
𝑧−1
𝑧+1
))
2
+ (
2𝐴
𝑇
)(
𝑧−1
𝑧+1
) + 1] = − 𝑋
𝑌(
4𝐵( 𝑧−1)2
𝑇2( 𝑧+1)2
+
2𝐴𝑇( 𝑧−1)( 𝑧+1)
𝑇2( 𝑧+1)2
+
𝑇2( 𝑧+1)2
𝑇2( 𝑧+1)2
) = −𝑋
𝑌(4𝐵( 𝑧 − 1)2
+ 2𝐴𝑇( 𝑧 − 1)( 𝑧 + 1) + 𝑇2( 𝑧 + 1)2) = −𝑋(𝑇2
(𝑍 + 1)2
)
If each polynomial is expanded, we obtain an equation like:
−𝑋𝑇2
(𝑧2
+ 2𝑧 + 1) = 𝑌(4𝐵( 𝑧2
− 2𝑧 + 1) + 2𝐴𝑇( 𝑧2
− 1) + 𝑇2( 𝑧2
+ 2𝑧 + 1))
Rewriting transfer function in inverse powers of z:
−𝑋𝑇2
(1 + 2𝑧−1
+ 𝑧−2
) = 𝑌(4𝐵(1 − 2𝑧−1
+ 𝑧−2) + 2𝐴𝑇(1 − 𝑧−2) + 𝑇2(1 + 2𝑧−1
+ 𝑧−2))
4𝐵𝑌 − 8𝐵𝑌𝑧−1
+ 2𝐴𝑇𝑌 + 4𝐵𝑌𝑧−2
− 2𝐴𝑇𝑌𝑍−2
+ 𝑇2
𝑌 + 2𝑇2
𝑌𝑧−1
+ 𝑌𝑧−2
𝑇2
= −𝑋𝑇2
− 𝑋𝑇2
𝑧−2
− 2𝑋 𝑇2
𝑧−1
If we transform each X term, Z term, and Y term:
4𝐵𝑌 𝑛 − 8𝐵𝑌 𝑛−1 + 4𝐵𝑌 𝑛−2 + 2𝐴𝑇𝑌 𝑛 − 2𝐴𝑇𝑌 𝑛−2 + 𝑇2
𝑌 𝑛 + 2𝑇2
𝑌𝑛−1 + 𝑇2
𝑌𝑛−2 = −𝑇2
𝑋𝑛 − 2𝑇2
𝑋𝑛−1 − 𝑇2
𝑋𝑛−2
One more manipulation collecting like terms we obtain the following:
𝑌 𝑛
(4𝐵 + 2𝐴𝑇 + 𝑇2) + 𝑌 𝑛−1
(−8𝐵 + 2𝑇2) + 𝑌 𝑛−2
(4𝐵 − 2𝐴𝑇 + 𝑇2) = −𝑇2
𝑋𝑛 − 2𝑇2
𝑋𝑛−1 − 𝑇2
𝑋𝑛−2
𝑌𝑛 =
𝑌𝑛−1(8𝐵 − 2𝑇2)+ 𝑌𝑛−2(−4𝐵 + 2𝐴𝑇 − 𝑇2) − 𝑇2
𝑋 𝑛 − 2𝑇2
𝑋 𝑛−1 − 𝑇2
𝑋 𝑛−2
(4𝐵 + 2𝐴𝑇 + 𝑇2)
Finally, substituting back in the quantity τ1τ2 = B and the quantity 𝜏1 = A:

24. 24
𝑌𝑛 =
𝑌𝑛−1(8τ1τ2 − 2𝑇2)+ 𝑌𝑛−2(−4τ1τ2 + 2𝜏1 𝑇 − 𝑇2) − 𝑇2
𝑋 𝑛 − 2𝑇2
𝑋 𝑛−1 − 𝑇2
𝑋 𝑛−2
(4τ1τ2 + 2𝜏1 𝑇 + 𝑇2)
Inputting values:
𝑌𝑛−1(8× 0.113ms × 56.27 µs − 202)+ 𝑌𝑛−2(−4 × 0.113ms × 56.27µs + 2 × 0.113 𝑚𝑠 × 10µs− 10µs2) − 10µs2
𝑋 𝑛 − 20µs2
𝑋 𝑛−1 − 10µs2
𝑋 𝑛−2
(4 × 0.113ms × 56.27 µs + 2 × 0.113𝑚𝑠 × 10µs+ 10µs2)
𝑌𝑛 = 1.822𝑌𝑛−1 − 0.837𝑌𝑛−2 − 0.0036𝑋 𝑛 − 0.0072𝑋 𝑛−1 − 0.0036𝑋 𝑛−2 (7)
8) Use Matlab for Digital Filter Design
The code we used for Matlab for the digital filter was the following and gave the output shown
in Fig. 16.
DLP = c2d (HLPF, 0.00000001,‘tustin’)
bode(DLP)
Figure 16: Digital Filter Frequency Response
The Bode Plot above is very comparable to that of Fig. 3. The cutoff frequency shows to be 1.97
kHz, and 3470 Hz at -10 dB. The specifications in Table 1 request that the power gain at the
stop band of 3500 Hz be -10 dB. These results depict that this digital filter that was designed
was in fact the same low pass filter—constructed in a different setting and by a different
mechanism.

25. 25
9) Switched Capacitor Filter Design
In order to construct the switched capacitor filter with the digital chip, no resistors or capacitor
components had to be used. The pin out of the chip used can be seen below in Fig. 17.
Figure 17: Maxim 263 Digital Chip
This chip was designed for absolute precision filtering without the use of external components,
like capacitors and resistors. The programmable parameters of the chip and its various modes
were set by means of digital logic. In order to meet the specifications from Table 1, a
Butterworth low pass filter, the design steps that needed to be followed started from the basis
of the chip’s ‘Mode 1’. The center frequency, Q, was calculated quite effortlessly by
implementing the following equation:
𝑓𝑜 = √(1 −
1
2𝑄2
) + √(1 −
1
2𝑄2
)2 + 1
Solving for the center frequency gave Q = 0.707

26. 26
In order to continue with the design, we had to determine which frequency we wanted to use
from Table 3 since 0.707 was not an entry. Since 0.703 was close to 0.707, we noted down the
correct Q0 – Q6 for the filter design.
Table 3: Q Program Selection Table
The ratio
𝑓 𝑐𝑙𝑘
𝑓𝑜
was needed in order to determine the program code F0=F4. Since the ratio
previously stated was 100.53, the code used was 0 0 0 0 0.
Two last pin configurations which needed to be determined were thee M0 and M1 pins—which
were both set to ‘0’ for our design, since Mode 1 was being used.
The end configurations for the design can be seen below in Fig. 18.

27. 27
Figure 18: Resulting Pin-Out of Switched Capacitor Filter
It is important to remember that in designing and implementing this circuit—that the setting
low (or 0) for the chip meant that we applied a -5 V. Vice versa, the setting 1 (or high) resulted
in us putting a +5 V to the necessary pin.
10) Using the Spectrum Analyzer—obtain a Bode Plot for the Filter
The switched capacitor filter chip was simply another way to implement the Butterworth low
pass filter. The output of the chip’s filter (or its frequency response) should, in theory, have a
corner frequency of 2000 Hz. If you look closely at the output below (Fig. 19), you can see that
at a -2.968 dB gain—the frequency that corresponds is 2.144 kHz. Although not precisely at 2

28. 28
kHz, this output clearly shows that the filter that we designed was in fact a correct mechanism
to meet the desired result.
Figure 19: Switched Capacitor Bode Plot
Determining a percent discrepancy between the cut-off frequency above, and the desired 2 kHz
would not be that helpful in comparison because the cursor is not even exactly at -3 dB. If,
however, the trace resulted at a frequency at exactly -3 dB, then implementing Eq. (5) would
have been useful. Regardless, the information shown in the output above shows the Pin-Out of
Fig. 18 that we determined was the correct programmed design with this switched capacitor
filter chip.
11) Comparisons
The difference between the analog and digital filters we designed was really not different in
their end result. The process of design was completely 180 degrees out of phase, but that is
typical and is expected. Since the end outputs of the frequency response and Bode Plots were
so similar, then one must choose which route they want to take in the design and construction
phase. Whether active or passive, an analog filter will be easier to implement than the digital
filter. However, the analog filter cannot be “reprogrammed” like the digital filter can by the
switch of a ‘mode’. There is only one mode with an analog—and that is the resistor or
capacitor you are using.

29. 29
The process or determining the recursive algorithm of the digital filter earlier in this paper
shows how time consuming it is to implement this kind of design. Digitizing an analog design
with bilinear transformation can weigh heavily and is not ideal. When end results are in fact
the same for practical applications, it can be advantageous to use analog design instead of
digital. In fact, practical recursive filters will usually be IIR types (or infinite-impulse response
filters) are based upon their analog equivalents.
Since filters are used and found everywhere in the world around us, especially in audio and
music devices, deciding which type of filter to use in any given application is an important
decision. Recently in the past 20 years, switched capacitor filters have been designed into
“telecom” circuits which have provided much lower distortion. The advantages of this
capacitor filter are that it lies on one chip and is therefore not sensitive to component
tolerances. It is common sense that when fewer parts are used—less room for error arises. A
disadvantage of the switched capacitor filter vs. their RC competitors has to do with the noise
that can occur in some circuit configurations—due to clock feedthru. Digital filters, on the other
hand, can have problems with latency and the difference between input and output—but all
real physical filters can have this problem; however the extent and severity to which this occurs
is what varies.
12) LC Bandpass Filter Design
The design of this LC filter had to meet certain criteria, such as the center frequency being 10.7
Mhz, the passband being 600 kHz, and the Source and Load resistance being 50 Ω.
In order to calculate correct components and design we had to use the following equation:
𝑄 =
𝑤 𝑜
𝐵
=
𝑅
𝐿
Another equation was for Q as follows:
𝑄 =
1
𝑅
√
𝐿
𝐶
Q was calculated to be:
𝑄 =
10.7 𝑀𝐻𝑧
600 𝐾ℎ𝑧
= 17.83
𝐿 =
100
600∗2 𝜋
= 26.5 µH

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Rearranging the equation above, we get the capacitance:
𝐶 =
𝐿
𝑄𝑅2
=
26.5µ 𝐻
17.83 ∗ 1002
= 8.33 µ𝐹
The circuit was designed using MultiSim and is shown in Fig. 20.
Figure 20: LC Bandpass Filter Schematic
13) PSPICE Frequency Response of LC Filter
Fig. 21 shows the frequency response of the LC filter and shows the cursor at the center
frequency of 10 MHz—which is what the requirement was.
Figure 21: Frequency Response of LC Bandpass Filter
14) Dismissed

31. 31
15) Dismissed
16) Dismissed
17) Dismissed
18) LC Filter Applications
The resonance effect of LC filters has important applications in signal processing and
communications systems. The most common application is seen every day when driving in the
car—when the radio is on. Tuning the channel takes heavily into account the resonance
frequency of an LC filter. These filters are needed when frequency mixer hardware is put
together. In addition, oscillators have the need for such criteria. These filters are also quite
important in the application of harmonics because attenuation has to be accomplished quickly.
19) Questions
a) Analog filers have fixed-values for their component values. This results in the fact that if
output changes are needed—then components need to be varied as well which can be a costly
thing. On the other hand, digital filters need to be programmed for certain outputs and can be
reprogrammed conveniently without component changes. These sample signals at intervals—in
a discrete time manner. The dependency on component tolerances is virtually eliminated when
the switched capacitor is implemented because the switched capacitor filter’s integrator
depends on capacitor ratios and not on absolute values (like RC filters). This ratio mechanism
provides very good accuracy regarding center frequencies and Q values. Since existing
electronics and integrated circuit technology can implement capacitor ratios much more
accurately than resistor ratios—the switched capacitor filter can provide filter capability with
detailed accuracy.
b) Analog design is limited by the fixed values needed for each different filter. This results in a
time consuming and money consuming fate when changes need to be implemented. The
power supplies also affect the analog filters in performing to design requirements when high
power is needed. The analog filter designed and analyzed in this paper—the Butterworth
filter—provides very little control over the resulting design since it is essentially a maximally-flat
design. These analog filters are limited in their dynamic range—both the amplitude and
frequency range. This is why analog filters are almost always plotted on a logarithmic scale.

32. 32
The maximally flat digital filter applications can be expensive in their implementation. They are
also limited in the power supply they can use. In addition, it can be quite easy to design an op-
amp circuit to simultaneously handle frequencies between 1 kHz and 100 kHz, but a digital
system would be overloaded with data if this was tried. This is why digital filters need to be
plotted on a linear scale to show their ideal performance.
Switched capacitor filters, on the other hand, can be very easy to use—because of their
programmable nature—but they will sometimes have more noise than RC configurations. In
addition, environmental factors can be a great limitation on these filters.
c) Digital filters are more applicable in digital signal processing which is a huge field of study and
in industry. They are used in music industry and all personal electronics as well. The switched
capacitors are used when resistors and their ratios are not an appropriate design. A switched
capacitor filter would best be used in an environment when the design specifications are
changing—because the cutoff frequency can be changed with a simple change in the clock
frequency. Capacitor ratios, on the other hand, are in integrated circuits that are ratio based—
not like active or passive filters. Analog filters are inexpensive and mass producible in certain
systems. When the design or use of a product does not need to change, then these analog
filters are the ideal choice.
d) The stability of a filter depends on the feedback path to be closed continuously. An analog
RC filter, designed properly, will have this continuously closed feedback path—discussed earlier
in this paper. When switched capacitor filters are discussed—the circuit is always going to be
stable if the poles are restricted to the real axis between 0 and 1. Digital filters, since they are
transformed into the z-domain have to be considered in a different setting: all the poles of the
transfer function in question have to be located within unit circle in the z-plane. Stability of
each of these designs can be accomplished through control system analysis with Nyquist
criterion and Root locus techniques. This process gives the certain stability and can be
accomplished in Matlab easily—which provides good analysis. By plotting root locus, zeroes
and poles can be recognized and if in the left half of the s-plane—then stability is confirmed.
Adjusting steady state response and transients can help in changing systems if stability of a
system is not good.
Summary:
This laboratory experiment discussed the relationship between discrete time (FIR) and digital
(IIR) filters. We saw that with end results essentially being equivalent, implementation of
certain designs use different methods. Although digital filters are efficient in their design
requirements—and can usually be accomplished with a lower order—calculations can be dense.

33. 33
It was quite clear that comparisons among different methods (Matlab, MultiSim, and
experimental) the same results were obtained. For example the Butterworth Low Pass filter
Matlab simulation shown in Fig. 3 depicts the cutoff frequency to be at nearly the same gain as
that of the MultiSim Output (Fig. 7). The percent error between the experimental design and
the Matlab simulation amounted to a mere 0.159%--which was insignificant. This negligible
amount of error meant that the experimental data collected from the breadboard and
spectrum analyzer was quite precise. The discrepancy could be accounted for by the fact that
resistors and capacitors used are never 100% accurate component values. Also, equipment
used such as the power supply was most likely not calibrated completely so some error could
have arisen from this factor. In addition, Fig. 5, the superimposed Frequency Responses of both
the experimental data and the Matlab design seemto be almost 100% overlapping—meaning
that the applications and the cutoff frequencies obtained were very accurate. Both the high
pass and band pass filters were also designed to specifications as the percent discrepancies
display. Design of these filters gave great insight into the process or active filter design and
what it takes for a system to be stable. In the latter parts of this lab, the z-transform (a newly
learned method) was applied and a recursive equation was determined. This equation
represented the same Butterworth Low Pass Filter from earlier in the report by a different
mechanism. On the other hand, the switched capacitor filter rely on the fact that these filters’
capabilities lie on one chip and is therefore not sensitive to component tolerances. This makes
the switched capacitor an ideal choice for electronics and telecommunications and an industry
standard.