Chemnitz University of Technology

Faculty of Electrical Engineering and Information

Technology

Professor Ship of Measurement and

Sensor technology

Practical

Smart Sensor System Active and passive filters

Objective of the experiment

The objective of this experiment is to get introduced to the working principles of different

filter topologies and their respective applications.

Dimensioning by hand or Program

Mechanical filter, digital, LC-Oscillator

It is assumed at the end of the experiment that one should be able to achieve sufficient

knowledge on the structure and operation of the filter. Independent study and learning about

the relevant instructions and additional literature are required to fulfill the objectives of the

experiment and successfully complete the Practical lab.

Literature

Williams, A.;Taylor, F.: Electronic Filter Design Handbook; 4. Edition; Verlag: McGraw-

Hill; 2006

Tietze, U.; Schenk, C.: Halbleiter-Schaltungstechnik; 10. Auage, Springer-Verlag Berlin

Heidelberg,1993

Seifart, M.: Analoge Schaltungen; 4. Auage; Verlag Technik GmbH, Berlin, 1994

Bernstein, H.: Analoge und digitale Filterschaltungen; VDE-Verlag, 1995

Gldner, K.: Systemanalytische Grundlagen der automatischen Steuerung; Gesellschaft

fr Wissens- und Technologietransfer der TU Dresden mbH, 1999

1. Background literature

1.1. Feedback

The principle of feedback: Under the conditions of linearity, with absence of feedback

signal flow applies in one direction:

1

a

e

X VG

X VK

(1)

Until | 1 + V K |> 1, there is negative feedback. If the amount is less than 1, it is positive

feedback.

In Equation (1), the reason behind using OPVs very often is: If | V | very large,

Figure 1: Principle of feedback

then | VK | >> 1, which is the case in a normal OPV, then equation (1) can be approximated

with,

1G

K (2)

This means that the behaviour of the overall circuit is solely due to the feedback branch.

The complex transfer function G is frequently illustrated using Bode Diagram, also known as

Bode plot. This diagram shows the amplitude response and the phase response over a

common frequency axis. The amplitude response is obtained from the magnitude of the

Transfer function:

2 2| ( ) | Im ReG s (3)

The phase response is calculated as follows:

Im( ) arctan( )

Res

(4)

1.2 Fundamentals of OPV

An OPV, an operational amplifier is a differential amplifier with specific properties. The most

important feature is the large open-loop gain, which is the differential gain without feedback.

For small frequencies, usually AD is assumed, but in reality it has a finite value.

. Ad P N A D D DD

UU U U U U A A

U

(5)

UP is referred to as a non-inverting input and UN as an inverting input. The Gain AD depends

on the frequency of each OPV. The frequency where the gain is dropped to 1, corresponds to

the frequency response of the OPVs, usually for a low-pass first order filter.

In addition to high differential gain, the gain-bandwidth product is, in practice, also important.

The Gain Bandwidth Product (GBWP) indicates at what frequency the open-loop gain of an

op-amp has dropped to AD = 1. An idealized Switcher CAD model has the following

frequency response: The basic gain is AD = 106 and the bandwidth at GBWP = 10 MHz.

Figure 2: voltages in OPV

Figure 3: Bode diagram of OPV model

To apply equation (2), the condition VK >> 1 must be fulfilled. Since the gain of the feedback

network K is less than 1, as a result, the condition V >> 1 must be met. One can realize V

with an OPV, so V = AD = f (). Equation (2) is therefore only valid for low frequencies. If

you move in the range of GBWP, the transfer function of the OPVs must be taken into

account! A rule of thumb states that the GBWP should be 100 times greater than the cut-off

frequency of the desired filter. For low quality filters, 10 times higher GBWP is sufficient.

OPVs have a very high input impedance. Due to this, it can be assumed for calculations that

no current in the input flows. The currents are compared with the signal currents of the circuit

and they are usually negligibly small.

To use the OPV in analog filters, it must always have a negative feedback path due to its

open-loop gain. Thus, there will be a direct current path from the output to the inverting input

of the op amps. Due to this negative feedback and the high gain of the op amps, the

differential voltage at the input UD 0V is maintained, because the voltage at the output is

phase shifted by 180 degrees to UN. The output voltage thus counteracts its own original

cause. Depending on which input the input signal is applied, one can distinguish between

inverting and non-inverting mode operation.

1

2

( ) A

E

U ZA j

U Z (6)

Figure 4: inverting amplifier

1

2

( ) 1A

E

U ZA j

U Z (7)

In a non-inverting amplifier, the gain is always greater than 1. Other important characteristics

of OPVs in practice are: noise performance, offset, CMRR, slew rate and common mode

rejection. For consideration of these properties with respect to the realization of an analog

filter, reference is made at this point to [2].

1.3 Fundamentals of filters

A filter is primarily a frequency selective, that is, the attenuation or the gain change with the

frequency. Such circuits can be realized in various forms, for example, passive filters, active

filters, digital filters, switched-capacitor filter, mechanical filter, CTD-filter, etc. In spite of

the different forms of implementation, all filters can be classified according to type,

characteristics and border frequency.

Filter types are lowpass, highpass, bandpass, bandstop and allpass. The known filter

characteristics are Bessel, Butterworth, Cauer and Tschebyscheff. The mathematical

description of a filter is carried out by the transfer function. This generally has the form

2

0 1 2

2

0 1 2

...( )( )

( ) ...

n

n

n

n

P P PZ PA P

N P P P P

(8)

Or as a special case the polynomial filter with the form:

02

0 1 2

( )... nn

A PP P P

(9)

The polynomial N (P) can be written as a product of functions of 2nd order:

2 2

1 1 2 2

( )(1 )(1 )...

A Pa P b P a P b P

(10)

Here the order of the denominator polynomial N(P) indicates the order of the filter and the

coefficients a and b determine the filter characteristics. 0 is a factor that indicates the basic

gain of the filter. Also the gain in a particular frequency range, in which the filter does not

have an influence on the signal. For all further considerations that factor should always be 1.

Figure 4: Non-inverting amplifier

The amplitude response is obtained from the magnitude of the transfer function:

2 2| ( ) | Re[ ( )] Im[ ( )]A P A P A P (11)

The phase transition is given by:

Im[ ( )]( )

Re[ ( )]

A PP

A P

(12)

And from the phase transition the total response time can be calculated as:

( )

Gr

d jT

d

(13)

The diagram below gives an introduction to the mathematical modeling of filters, based on the

low pass filter. From the voltage divider rule, the transfer function is calculated as:

Figure 5: RC low pass filter

1

1( )

1 1

A

E

CU j

A jU j RC

R Cj

(14)

By defining P = j = j(w/wg) mit wg = 1/RC = 2fg; fg : 3dB break frequency, results the

term:

1

( )1

A PP

(15)

This is the transfer function of a 1st order (-40dB / dec) low-pass filter (TP). For comparison,

the overall transfer function of such a filter:

0

1

( )1

AA P

P

(16)

Where A0 is the gain at frequency f = 0 Hz.

A comparison of coefficients gives A0 = a1 = 1, i.e. the filter is completely determined by P,

so eventually by wg. For a first-order filter, also no characteristics can be selected. It is always

a filter with critical damping.

RLC-TP: The voltage divider rule gives:

Figure 6: RLC low-pass

2

2 2

1

1( )

1 1 ( )

1( )

1

A

E

g g

U j CA j

U j RC j LCR j L Cj

A PRCP LCP

(17)

It is a TP 2nd order (-40dB / dec). The general transfer function is:

02

1 1

( )1

AA P

P b P

(18)

In order to be able to determine the component values we perform a simple comparison of

coefficients by fixing any value of R, C or L and set according to the sizes of this order.

Obviously, a further degree of freedom in the dimensioning of filter has been added in filters

of 2nd order. It now has the filter characteristic, which is determined by the coefficients a1

and b1. In the Appendix, such tables from which these coefficients can be taken for

dimensioning ends, can be found.

1

2

1

g

g

a RC

b LC

(19)

In order to realize a higher-order filter, you can cascade several 1st and 2nd order filters

together. However, it should be noted that the cutoff frequency fges of the cascaded filter is

not equal to the cut-off frequency of the individual filters.

ges xf f (20)

The corresponding coefficients can also be found in the tables.

Each filter characteristic is achieved with a particular set of coefficients. These coefficients

can be determined by so-called approximations. The aim of these approximations is to

replicate the transmission behavior of an ideal filter with different properties with respect to

the transition slope, ripples in the pass / stop band and the phase response. An ideal filter does

not influence the signal up to the cut-off frequency and after the cutoff frequency, the signal

has infinitely strong attenuation. This property cannot be implemented in practice, however,

in most cases, it is not necessary.

The following summary provides an overview of the different filter characteristics and their

properties:

1. Polynomial filter according to equation (7)

Bessel filter

- No ripple in the pass and stop bands

- Lower slope (high-order filter needed)

- Constant group delay

- Butterworth filter

Butterworth filter

- No ripple in the pass and stop bands

- Good edge steepness

- No Constant group delay

Tschebysche I - Ripple in the pass band

- High slope

- No Constant group delay

2. Filter according to equation (6)

Tschebyscheff II

- Ripple in the stopband

- High slope

- No constant group delay (slightly better than Tschebyscheff I)

Cauer filter

- Ripple in the pass and stop bands (separately adjustable)

- Very high slew rate

- No Constant group delay

The high pass filter is considered below. This is easy to explain and it is done by replacing P

with 1 / P. This mathematical method is called Lowpass-Highpass-Transformation. The

general form of the polynomial filter is therefore:

0

1 1 2 22 2

( )1 1 1 1

(1 )(1 )...

A P

P P P P

(21)

1

( )1 1

1

RA j

Rj C j RC

(22)

1

( )1

1

A P

P

(23)

Figure 7: RC high-pass filter

All other considerations follow analogous to LP and should not be listed here.

A similar approach is also applied for the band pass. Here P is replaced by 1/. (P+1/P) with

= w/wg. Thus, For a 1st order LP, the following expression is created:

2

1( )

1 1 11 ( )

PA P

P PP

P

(24)

Although you will find a 2nd-order filter, the band pass has still only an attenuation of 20 dB /

dec.

Figure 8: RLC band pass

This transformation can, however, apply only for narrowband band pass filters.

The transfer function of a band pass filter of 4th order is in the form:

2

1

22 3 41 1

1 1 1

( )

( )( )

1 [2 ]

P

bA P

a aP P P P

b b b

(25)

When dimensioning, make sure that the coefficient is derived from LP of 2nd order, although

this is a band-pass of 4th order.

There are two ways to realize a band pass. The first way is to cascade a low pass and a high

pass. The cut-off frequencies of the partial filter must not be selected accordingly and

individually dimensioned. However, this means a double circuit complexity for the filter and

it is particularly suitable for broadband filters. The second way is using resonant circuits and

oscillators. The particular resonance frequency will be selective damped or amplified

(bandpass) thereby.

Wobei die jeweilige Resonanzfrequenz wahlweise verstrkt (Bandpass) ober gedmpft

(Bandsperre) wird. Mit dieser Variante knnen gute schmalbandige Filter aufgebaut werden,

also Filter hoher Gte.

The quality Q is defined as:

,max ,min

1r r

g g

fQ

B

(26)

B is the 3dB bandwidth, and wr is the resonance frequency of the filter or the center

frequency.

To go from the transfer function of the LP to a band top we replace P by /(P+1/P). A 2nd

order band-stop filter therefore has the form:

21 1( )

11

1

PA P

P P

PP

(27)

Figure 9: RLC band stop

1.4 Active filters

Until now all circuits were considered purely as passive RLC - networks. They have the

disadvantage that they are reactively coupled and the signal only dampens, but cannot

increase. In addition, the coils, in the low frequency range, are very large and susceptible to

electromagnetic interference. These disadvantages can be overcome by the use of active

filters. The easiest way to build such a filter is shown here: The RC filter is followed by a

non-inverting amplifier. One might as well combine a RLC filter with an amplifier circuit. It

still has the absence of feedback and gives a freely selectable gain. But the aim should be to

replace the inductors through other circuit elements. For this purpose one uses the possibility

of feedback of the amplifier.

Figure 10: active 1st order low pass filter

For polynomial filter, this results in two variants: positive and negative feedback.

The positive feedback variant is also referred to as Sallen-Key filter, the negative feedback as

MFB-variant filter. All filter characteristics can be realized with two variants, but there are

some properties which somewhat differ from each other.

For Sallen-Key topology, the following features should be considered in the filter selection:

There is no amplification less than 1 realisable

The real transfer function has a term in the open-loop gain of the op amps in the denominator. Since this gain decreases with increasing frequency, the gain

of the circuit increases. This effect is observed in the stop band. This can be

observed with an increase of the amplitude response.

With suitable dimensioning, the cut off frequency and the filter characteristics can be adjusted separately. The characteristics alone can be set via a voltage

divider, which causes no change in the cut-off frequency.

The output must not be loaded capacitively or inductively, as this would affect the filter behaviour.

For an MFB filter the following applies:

Basic gain, cut-off frequency and filter characteristics cannot be adjusted separately.

The phase of the signal is shifted by the inverting operation by 180 degrees.

a) Sallen-Key

This is the positive feedback variant. Such 2nd order LP looks like Figure 12 as shown.

R2 and C1 form a 1st order LP, also the gain of the system using C2 will be reduced with

increasing frequency. R3 and R4 must be set properly for the gain, because a positive

feedback has always a risk of instability in the circuit.

3 4( 1)R k R

(28)

Figure 11: Sallen-Key low pass filter

2 2

1 1 2 1 1 2 1 2 1 2

( )1 [ (1 ) ]g g

kA P

P RC R C k RC P R R CC

(29)

For the circuit, there are two special cases:

Case 1: R1 = R2 = R, and k = 1

2 2 2

1 1 2

( )1 2 g g

kA P

P RC P R CC

(30)

Matching the coefficients from the same equation, we get,

11

4 g

aC

f R

(31)

12

14 g

bC

f Ra

(32)

Case 2: R1 = R2 = R and C1 = C2 = C,

2 2 2 2

( )1 (3 )g g

kA P

P RC k P R C

(33)

1

2 g

bRC

f

(34)

10

1

3a

k Vb

(35)

b) MFB (Multiple Feedback)

This is the negative feedback variant and illustrated in Figure. 13. R1, R3 and C2 form again a

1st order low pass filter and C1 reduces the gain with increasing frequency. Without R2, the

circuit would act as integrator and R3 provides a potential difference between the first node of

C2 and ground.

2

1

2 22 31 2 3 2 3 1 2

1

( )

1 ( )g g

R

RA P

R RC R R P R R CC P

R

(36)

2 2

1 2 1 2 1 2 1 0

2

1 2

4 (1 )

4 g

a C a C CC b AR

f C C

(37)

Figure 12: MFB low pass filter

21

0

RR

A

(38)

13 2 21 2 24 g

bR

f C C R (39)

1 02

2

1 1

4 (1 )b AC

C a

(40)

The values of the capacitors C1 and C2 are to be set such that the ratio is not much higher

than the above criterion.

c) Sallen-Key high-pass filter

Figure 13: Sallen-Key high-pass filter

3 4( 1)R k R

(41)

2 1 2 1 2

2 2

1 2 1 2 1 2 1 2

( )( ) (1 ) 1 1 1

1g g

kA P

R C C RC k

R R CC P R R CC P

(42)

With k = 1 and C1 = C2 = C:

1

1

1

g

Rf Ca

(43)

12

14 g

aR

f Cb

(44)

d) MFB high pass filter

Figure 14: MFB high pass filter

For this filter we have C1 = C3 = C:

2

2

2 2

1 2 1 2 2

( )2 1 1 1

1g g

C

CA P

C C

RCC s R R CC s

(45)

0

2

CA

C

(46)

21

1 2

2

g

C Ca

RCC

(47)

21

1 2

2

g

C Cb

RCC

(48)

01

1

1 2

2 g

AR

f Ca

(49)

12

2 1 02 (1 2 )g

aR

f C b A

(50)

e) Active band pass filter

As already mentioned, such a filter can be realized by the combination of high- and low-pass

filters as well as by oscillatory circuits. For both versions, there are active solutions. However,

this is not to be considered in much detail here, because it can be easily deduced from the

preceding circuits.

All these filters have one thing in common that the gain at center frequency Ar is proportional

to Q. So it is necessary to do a level adjustment. This can be achieved by a downstream gain

or attenuator. In addition, the tendency to oscillation increases with increasing quality. This

characteristic should be considered, especially with circuits using positive feedback.

Figure 15: Sallen-Key band pass filter

f) Sallen-Key band pass filter

1 2( 1)R k R

(51)

2 2 2 2

( )1 (3 )

r

r r

kRC PA P

RC k P R C P

(52)

1

2rf

RC

(53)

3r

kA

k

(54)

1

3Q

k

(55)

There is obviously a special case, in which all the frequency determining elements, that is, the

resistors and capacitors on the non-inverting input, are dimensioned in each case with the

same values. The low-pass filter characteristic is now solely dependent on the adjusted

internal gain, i.e. the ratio of R1 to R2. With such narrow-band filters, a change of this ratio

alone results in a change in the quality and the gain at the center frequency. This should be

taken into account in the dimensioning. Because if the gain and this signal amplification is the

greater than the quality, then this increase must be either corrected or taken into account in the

analysis of the signal. In addition, it must be ensured that the OPV used, is not brought to its

operating limits, otherwise there will be signal distortions.

g) MFB Band pass filters

A band pass in the MFB variant is shown in Fig. 17. It is simplified by putting C1 = C2 = C:

2 3

1 3

2 2 21 3 1 2 3

1 3 1 3

( )2

1

r

r r

R RC P

R RA P

R R R R RC P C P

R R R R

(56)

1 3

1 2 3

1

2r

R Rf

C R R R

(57)

Figure 16: MFB band pass filter

2

12r

RA

R

(58)

2 1 32

1 3

( )1

2r

R R RQ f R C

R R

(59)

2

1rfBQ R C

(60)

It can be seen that no parameter can be chosen independently, without having an effect on

other parameters.

h) Band pass filter with Wien-Robinson bridge

Narrow-band filter can also be constructed with the aid of resonant circuits. In this example, a

Wien-Robinson bridge is used.

Figure 17: Wien-Robinson band pass filter

1

2rf

RC

(61)

1 2

1 22r

R RA

R R

(62)

1

1 22

RQ

R R

(63)

This circuit is also known as a Wien bridge oscillator in a similar manner. However, as long

as the oscillation condition K * V = 1 is not satisfied, the circuit operates as a band pass.

Theoretically, an infinite quality is reached, however, the circuit then oscillates and is thus no

longer be used as a filter.

i) I-band stop filter

A simple variant, to build a band-stop filter, is the so-called active double-T filter. Generally

it can be developed from each band pass and band stop. Here, the band-pass is included in the

negative feedback of an amplifier. The signal is then attenuated maximum at maximum gain

of the band pass filter.

Figure 18: Double T band-stop

1 2( 1)R k R

(64)

2

2

(1 )( )

1 2(2 )

k PA P

k P P

(65)

1

2rf

RC

(66)

0A k

(67)

1

2(2 )Q

k

(68)

j) MFB Universal circuit

Table 1

Filter Characteristics Z1 Z2 Z3 Z4 Z5

Low pass R C R R C

High pass C R C C R

Band pass R R C C R

Figure 19: MFB Universal circuit

1.5 Filter Application

A filter is used primarily to remove unwanted signal components from the measured signal.

Thus a filter always influence the signal. This stands, however, in contrast to the claim of

measurement. This should be done without retroactive effect and without distortion of the

relevant measurement signal. For this reason, care must be taken when designing a filter and it

should be noted that at the cutoff frequency, the signal is already attenuated by 3dB. It is thus

decreased to about 70% of the original amplitude. In addition, filters can also influence the

signal away from the cut-off frequency in the pass band. An example of this is the

Tschebyscheff characteristic in which a particular ripple in the pass band exists.

It is important to ensure while using active filters in metrology that the influence of these

additional noise source does not exceed by a maximum degree. This may require a noise

analysis of the filter. This can be done by measuring, by calculation, or by an estimate. In case

of excessive noise entry into the signal path, the resistance values can be reduced, made the

bandwidth smaller or poorer noise OPV can be used.

In any case, the application of an active filter is a compromise between the improvement of

the signal quality and the influence of the signal. This assessment must be carried out every

time for each application.

A filter is quite often used in metrology to prevent aliasing. In principle it should be

maintained that no frequencies reach higher than half the sampling frequency of the ADC.

However in reality, this is not feasible, because the noise voltages alone bring higher

frequencies with it. Therefore, the maximum amplitude for each application must be

determined and it should not be exceeded. Accordingly, then, a filter must be selected and

dimensioned. As a guideline for this dimensioning, the noise can be used. If the signal is

attenuated so much at half the sampling frequency that it is in the range of noise amplitude, so

there is no benefit with respect to the anti-aliasing feature with further attenuation.

1.6 Noise

Noise is a problem in many measurement applications. When superimposed on the measuring

signal, it leads to the lowering of the detection limit because the measuring signal can only be

distinguished from the noise signal when the amplitudes of the two signals differ sufficiently.

Which amplitude ratio is sufficient for detecting, cannot be defined universally and must be

decided case by case. In order to make this decision, the noise has to be assessed

quantitatively.

Rms noise voltage at thermally rushing resistors:

4rtU kTBR (69)

with k = 1, 38 10-23

Ws/K

In order to analyze a circuit with respect to the noise, but a noise equivalent circuit diagram is

introduced.

Figure 20: Noise equivalent circuit resistance

This method is also used to simulate the noise performance of OPV (Figure 22). In most

cases, the noise current can be neglected because it is relatively small.

In the datasheet of OPV, the RMS voltage is rarely found, but the so-called Equivalent Input

Noise Voltage in nV per Hz is there. This is the noise voltage density or spectral noise

voltage Ur. It represents the rms noise voltage in a frequency band of width B = 1Hz.

Figure 21: Noise equivalent circuit diagram OPV

4r rV

u kTRuB

(70)

The exact definition is:

22 rr

dUu

df

(71)

As already mentioned, the noise voltage has little explanatory power alone. The decisive

factor is the ratio of useful signal to noise signal, the signal-to-noise ratio SNR:

2

2

s s

r r

P US

N P U

(72)

It should be noted that the noise voltages caused by resistors and OPV, only one part of the

total interference in the signal path. So EMI and mismatches have a much greater influence on

the signal quality than the noise. Reducing these parasitic effects can improve the

measurement signal may more clearly, as lower-noise op amp or smaller resistors.

If there are several sources of noise in a signal path exists, then the noise voltages

geometrically added:

2 2 2 21 2 3 ...N N N N NU U U U U n (73)

1.7 Filter Designing softwares

In practice, the dimensioning of filters is the most important objective of specific programs

and they calculate the component values of a filter circuit. Even if the problem should be

solved by hand mathematically, it outweighs the benefits of computer-aided dimensioning.

Here especially the speed and flexibility in adapting the proposed filter to the desired

properties is crucial. Last but not the least, the risk of incorrect sizing reduces calculation

errors. There are various ligands programs for both analog and digital filters in the market.

Some vendors of commercial software have free versions but they are limited in their capacity

Real resistor

(noise)

Ideal resistor

Noise source

to available programs. Under certain circumstances this program are sufficient for simple

requirements for order and cut-off frequency. An example of such a program is Filter Free,

the free version to filter Solution of Nuhertz. This program will also be used for the

experiment, since it provides all the necessary functionalities for the experiment already in the

free version. Other notable programs are FilterCad, FiltersCad, FilterPro and FilterLab. To

Operate Filter Free the following notes are helpful. By default, the frequency axis is set to ,

not to f. In addition, the start and end values of the frequency axis must always be entered

manually. You do not change automatically. The key for the test topologies are not referred to

as MFB and Sallen-Key, but as a negative and positive SAB. This term refers to the type of

feedback and the circuit variant because SAB is the abbreviation for Single amplification he

biquad.

The biggest drawback of the program are the fixed resistance values. In practice, it is

customary to determine the capacitance values and adjust the resistance values. These are

usually required for a finer gradation.

This is also generally be observed when using dimensioning of filter programs. The calculated

component values are frequently not available, so there are slight variations of calculated

filter behavior in the physical deployment.

2. Experimental setup

To be able to practically implement theory, the experimental set-up is very important. It offers

the possibility of the tasks to solve by applying the acquired knowledge. Also room for

creativity in solving the tasks set is given. It is desirable to make mistakes, to identify and

resolve.

2.1 Test circuit

Figure 22: Block Diagram of the experimental circuit

To be able to select the different impedances, it is composed of a Z diagram of the elements

shown in Figure 24.

With this arrangement, filter with MFB and Sallen-Key structure and an active Wien-

Robinson Band filter can be built.

Noise voltage source

Noise current source

Figure 23: Connection of the impedances

According to the desired circuit, the individual components can be connected together. Since

the signal comes from a digital source, so time and value is obtained discretely, a 1st order

filter is placed at the entrance of the test circuit. This is primarily as a reconstructive filter. On

antialiasing filter, at the inputs the acquisition card has been omitted because the circuit does

not produce additional harmonics. Only interference and the produced switching noise could

cause aliasing. This would cause harmonics but with respect to the source, it has a negligible

amplitude.

Figure 24: Wiring of the impedances

AO0 is the input signal. AI0 is the output signal for the so-called Stimulus - signal, ie the

signal that serves as a reference. AI1 is the output signal for the so-called response - signal,

also the signal which has influence on the filter.

If one uses AO0 as a stimulus and can keep AI0 open, the realization of a 3rd order filter

would be possible. Inverting and non-inverting operation can be realized via the plug-in

connections. Inverting e.g. for MFB filter, for non-inverting Sallen-Key filter. The analysis of

the circuit can be done using a PC. The corresponding program was written in LabVIEW. For

the AD and DA conversion, the data acquisition card PCI-6221 comes with the BNC-2110

terminal block of National Instruments to use. The inputs of the DAQ board can convert 16-

bit with 250kS / s.

2.2 Test Program

The PC is used in the experiment as a signal generator and meter. The measurement includes

the mapping of the time signals and the display of a Bode diagram showing the transfer

characteristics of the filter map. To calculate the Bode diagram two measurement signals are

necessary. The input signal of the filter and the output signal. The input signal of the filter is

identical with the output signal of the signal generator. Under ideal conditions, this signal

could be connected to the internal program evaluation. However, since the DA and AD

conversion and the non-ideal properties of the transmission path influence the signal, the input

signal of the filter is measured. This ensures that the difference between the two signals are

caused by the filter.

The test program itself is divided into two levels. A measuring level and an evaluation level.

Only one is visible at one time. The levels are selectable on two riders with appropriate

labeling. This structure was chosen because it simplifies program enhancements. So

additional tabs can be used to insert additional program options or to improve the clarity of

each level.

In the measurement plane both reference signal and the filtered signal are shown in separate

graphs. In addition, amplitude and phase response are also available. At this level, all the

necessary adjustments should be made. The sequence control of the measurement can be

found here. By pressing the "Evaluate" button in the current Bodeplot, if desired, is written in

two files, and then transferred to the evaluation plane. The information on Cursor key data for

the Bode plots can then be determined relatively accurately. In addition, there the amplitude

spectra of the two signals, in addition to this plot nor shown, as well as the SNR of these

signals. The generated files are saved in .txt format, the last two entries of this text file, the

start frequency and the end frequency include resolution to allow an evaluation with an Office

program.

3. Preparation Tasks

1. A sensor signal needs to be recorded. However, this is very weak and it has to overcome a great distance from the sensor to the transmitter. List all known ways to

keep the noise as low as possible or to suppress in order to increase the SNR and thus

the detection accuracy.

2. Give typical filter characteristic and briefly describe essential properties. 3. Which filter characteristic does a 1st order filter have? 4. In a situation, a SNR of 25 dB is required. The useful signal has a frequency of 10 Hz

1 kHz. The transmitter generates a spectral noise voltage of 5 nV /Hz. The sensor can be approximated for the noise analysis by a thermally active resistor 1k. How big

the useful signal must be to meet the given conditions at 20 C and 100 C? The noise

coupling can be neglected.

5. In order to suppress higher-frequency disturbances, a 2nd order low pass filter can be used. Which filter characteristics (Bessel, Butterworth or Tschebyshev) would you

choose to influence as little as possible, the useful signal and to suppress interference?

Justify this decision! Calculate the cut-off frequency needed to obtain at 1 kHz at least

95% of the output amplitude, if the gain in the Pass band is A = 1 ! (H (j) | H (j) | g)

6. Dimension the filter of 3.5. for MFB topology. C1 = 10 nF; C2 = 100 nF. Note sign! 7. Why no dedicated inductors are used in active filters? 8. How can you tell that a circuit and in particular a structured filter oscillates? 9. A weak, analog sensor signal to be digitized, processed and passed on as a control

signal to an actuator. The bandwidth of the signal is known. Which filters are

johnnySticky NoteMaking repeated measurements of one item,Increasing sample size,Randomizing samples,Randomizing experiments,Repeating experiments, andIncluding covariates.

johnnySticky Notea filter is an electrical network that altersthe amplitude and/or phase characteristics of a signal withrespect to frequency. Ideally

johnnySticky Noteit has only one reactive component, the capacitor, in the circuit.

necessary and / or useful? Justify, at which point the signal path you would use (digital

/ analog) for any purpose that filter!

10. A low-pass first order is frequently used as an antialiasing filter for measurement cards. Explain this observation. Dimension this filter for a sampling frequency of

125 kHz so that after sampling certainly no aliasing occurs! In order to preserve the

value of the required attenuation, the following data are needed:

Resolution: 16 bits

C=270pF Is such a filter practicable? Explain!

4. Practical tasks

1. Familiarize yourself with the program and take a "no-load characteristic", the two OPVs can be operated as a voltage follower! Use the white noise as a reference signal!

2. Determine a resistance value of the input filter, where the filter has the best influence on the measurement! Explain!

3. Generate a sinusoidal signal with the measurement program! Check the signal with and without input filter! (R = 100k, C = 10nF, C = 100 nF)

4. Inspect the influence of the circuit components. Vary the frequency between 100Hz and 5 kHz with R = 100k and C = 10nF. Determine the cut-off frequency.

5. Build the filter from the Preparation Exercise 3.6. And assume its characteristic. Draw the Bode plot without phase.

6. Make yourself familiar with the operation of the "Filter Free" program!

Let a Butterworth high pass 2nd order filter with a cutoff frequency fg = 2 kHz get charged.

Build on the filter and compare the measurement result with the filter program!

The supervisor gives you the detailed task at the workplace!