Practical Active Filter Design by Means of Analog, Digital, and Switched Capacitor Theory

Written By: Sachin Mehta

University of Nevada, Reno

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 V out

V ¿ 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=Gcl

G cl+1 (1)

2

Table 1: Butterworth Low Pass Filter Specifications

Passband 0-2000 HzMinimum Power

Gain at fc 0.5 (-3 dB)

Start of Stop Band 3500 HzMaximum Power

Gain at fr0.1 (-10 dB)

Figure 1: Practical Active Filter Circuit

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Figure 2: Block Diagram of Main Network Studied in this Laboratory Experiment

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=

1R1C1 s

1R1C1 s

+1= 1R1C1s+1

Next, we analyzed the remaining loop with the same rules and equations and obtained the following expression:

Gcl=G cl1×

1R2C2 s

(Gcl1×1

R2C2 s )+1=¿

( 1R1C1 s+1 )( 1

R2C2 s )[( 1R1C1 s+1 )×( 1

R2C2 s )]+1

¿

1

R1R2C1C2 s2

1R1 R2C1C2 s

2+1

H L(s)=−1

R1R2C1C2 s2+R2C2s+1

(2)

Where: T 1=R2C2∧T 2=R1C1

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b) Band Pass

The transfer function V out 2

V ¿ was also determined by implementing block diagram rules, like

mentioned above regarding closed loop feedback systems. We were able to express the following:

HB ( s )=H o

1+H oH f

=H 2

1+H 1H 2

HB ( s )= −sTs2T 1T 2+sT 1+1

(3)

c) High Pass

The voltage transfer function HH(s) was determined, also by implementing controls systems analysis and feedback control rules.

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(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+√2 p+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= sѡ p

= s2πf

= s2π (2000)

= s12566.37

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 ( sprototype )= 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.

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1

1.57×108 = R1R2C1C2 = 6.33 x 10-9

R1R2=6.33×107

R1=6.33×107

R2

R2C2=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.

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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).

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:

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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 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-S102JT10K000TA

R

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

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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.

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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.

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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.

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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.

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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%

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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.

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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.

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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.

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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.

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Figure 12 : Matlab Simulation of Band Pass Filter

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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%

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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.

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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)= −1τ1 τ2 s

2+τ1 s+1

which reads τ to be a time constant of the system.

We can let the quantity τ1 τ2 = 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 =( 2T )( z−1z+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:

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H (Z)= −1

B( 2T )( z−1z+1 )2

+A ( 2T )( z−1z+1 )+1Setting this equal to “output/input” and cross-multiplying gives:

[B(( 2T )( z−1z+1 ))2

+A ( 2T )( z−1z+1 )+1] = −X

Simplifying gives:

[(( 4 BT 2 )( z−1z+1 ))2

+( 2 AT )( z−1z+1 )+1] = −X

Y (4 B ( z−1 )2

T 2 ( z+1 )2+2 AT ( z−1 ) ( z+1 )

T 2 ( z+1 )2+T2 ( z+1 )2

T2 ( z+1 )2)=−X

Y (4B ( z−1 )2+2 AT ( z−1 ) ( z+1 )+T 2 ( z+1 )2 )=−X (T2(Z+1)2)

If each polynomial is expanded, we obtain an equation like:

−X T 2(z2+2 z+1)=Y (4 B ( z2−2 z+1 )+2 AT ( z2−1 )+T 2 ( z2+2 z+1 ))

Rewriting transfer function in inverse powers of z:

−X T 2(1+2 z−1+z−2)=Y (4B (1−2 z−1+z−2 )+2 AT (1−z−2 )+T2 (1+2 z−1+z−2) )

4 BY−8BYz−1+2 ATY +4 BYz−2−2 ATYZ−2+T2Y +2T2Yz−1+Yz−2T 2=−XT 2−XT 2 z−2−2 X T 2 z−1

If we transform each X term, Z term, and Y term:

4 BY n−8BY n−1+4 BY n−2+2 AT Y n−2 ATY n−2+T2Y n+2T

2Y n−1+T2Y n−2=−T 2X n−2T

2 Xn−1−T2 Xn−2

One more manipulation collecting like terms we obtain the following:

Y n (4 B+2 AT+T2 )+Y n−1 (−8 B+2T 2 )+Y n−2 (4 B−2 AT +T 2)=−T 2X n−2T2 Xn−1−T

2 Xn−2

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Y n=Y n−1 (8B−2T2 )+Y n−2 (−4B+2 AT−T 2 )−T 2 Xn−2T2 Xn−1−T 2 Xn−2

(4 B+2 AT+T 2)

Finally, substituting back in the quantity τ1 τ2 = B and the quantity τ1 = A:

Y n=Y n−1 (8 τ1 τ2−2T 2 )+Y n−2 (−4 τ1 τ2+2 τ1T−T 2 )−T2 Xn−2T 2 Xn−1−T 2Xn−2

(4 τ1 τ2+2 τ1T +T 2)

Inputting values:

Y n−1 (8×0.113ms×56.27 µs−202 )+Y n−2 (−4×0.113ms×56.27µs+2×0.113ms×10µs−10 µs2)−10µs2 Xn−20 µs2X n−1−10µs2 Xn−2

(4×0.113ms×56.27 µs+2×0.113ms×10 µs+10 µs2)

Y n=1.822Y n−1−0.837Y n−2−0.0036 X n−0.0072 Xn−1−0.0036 Xn−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

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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.

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:

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f o=√(1− 1

2Q2 )+√(1−1

2Q2)2

+1

Solving for the center frequency gave Q = 0.707

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 clkf o

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.

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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.

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

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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 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

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switch of a ‘mode’. There is only one mode with an analog—and that is the resistor or capacitor you are using.

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=wo

B=RL

Another equation was for Q as follows:

Q= 1R √ LC

Q was calculated to be:

Q=10.7MHz600Khz

= 17.83

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L= 100600∗2π

= 26.5 µH

Rearranging the equation above, we get the capacitance:

C= L

QR2= 26.5µH

17.83∗1002=8.33µF

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

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14) Dismissed

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

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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.

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:

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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. 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 seem to 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.

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