transducers and sensors - philadelphia university

5/9/2019

1

Advanced Measurement Systems and Sensors

Dr. Ibrahim Al-Naimi

Chapter one

Signal conditioning and Processing

5/9/2019

2

Signal Conditioning and Processing

• Signal conditioning and processing refers to operations performed on signals to convert them to a form suitable for interfacing with other elements in control system.

• It is possible to categorize signal conditioning and processing into several general types: - Signal level and bias changes

- Convertors

- Comparators

- Filtering

- Linearization

Operational Amplifiers

• Many special circuits and general purpose amplifiers are now contained integrated circuit packages producing a quick solution to signal conditioning problems together with small size, low power consumption, and low cost.

5/9/2019

3


• Op amp typically requires connection to bipolar power supplies, i.e. Both +Vs and –Vs with respect to ground.

• When considered as a functional element of some larger circuit, however, all we are concerned with are its input and output signals.

• For that reason, the power supply connections are not shown in it its own symbol, only two input terminals and an output terminal are shown.


Where: (+) : Noninverting terminal (-) : Inverting terminal

5/9/2019

4


• Transfer function.

• This devise is always used with feedback of output to input. Such feedback permits implementation of many special relationships between input and output voltage.

Characteristics for Ideal Op Amp

• Infinite open loop gain Ad .

• Infinite input impedance (No current can flow through the op amp input terminals).

• Zero output impedance.

• The frequency bandwidth extends from zero to infinity.

• The output voltage is zero when the input voltage is zero.

5/9/2019

5

Characteristics for practical Op Amp

• Finite open loop gain varies from 104 to 106.

• Input impedance is very high, typically exceeding 1MΩ.

• Output impedance is very low, typically less than 100 Ω.

Rules to Analyze Op Amp operations

• There are two rules that can be applied to analyze the ideal operation of any op amp circuit. In most cases, such an analysis will provide the circuit transfer function with little error. The design rules are:

Rule 1: Assume that no current flows through the op amp input terminals, i.e. i+ = i- = 0

Rule 2: Assume that there is no voltage difference between the op amp input terminals, i.e. V+ = V-

5/9/2019

6

Ideal Inverting Amplifier

Op Amp Specifications

• Input offset voltage

In most cases, the op amp output voltage may not be zero when the voltage across the input is zero, i.e. When the input terminals are kept floating without being connected or being shorted and grounded. The dc voltage that must be applied across the input terminals to drive the output to zero is the offset input voltage.

5/9/2019

7

Offset Voltage


• Input bias and offset currents

each op amp has a small dc bias current flowing out of both the input terminals to the ground. The offset current is the difference in theses bias currents. Bias current produces an effect similar to that of offset voltage and it will not be zero even if the offset voltage is zero.

5/9/2019

8


• Temperature drift

A drift in the output voltage per unity change in the temperature (e.g. μV/ oC)

• Slew rate

If the voltage is suddenly applied to the input of op amp, the output will saturate to the maximum. The slew rate is the maximum output voltage change per unit time expressed in V/μs.


• Unity gain frequency bandwidth

Although the gain of the op amp is assumed to be infinity, it is normally between 104 and 108 and it is a function of frequency. The bandwidth extends from dc to about 5 or 10 Hz and then falls off at a uniform rate of 20 dB/decade. The frequency at unity gain is about 1 MHz and the amplifier is said to possess a 1 MHz unity gain frequency bandwidth product.

5/9/2019

9

Op Amp Practical Issues • Op amp generally requires connection to bipolar

power supplies, i.e. Both +Vs and –Vs with equal magnitude. Typically, the value of these supply voltages is in the range of 9 to 15 volts.

• Approximate input offset current compensation can be provided by making the resistance feeding both input terminals approximately the same. As shown in the following figure for the inverting amplifier, this has been provided by a resistor on the noninverting terminal whose value is the same as R1 and R2 in parallel, since that is the effective resistance seen by the inverting terminal. Bias current is smaller in FET op amp and is normally ignored.

Op Amp Practical Issues

5/9/2019

10


• The effect of offset voltage should be eliminated. Why? Compensation of input offset voltage can be provided in one of two ways. 1: Many modern IC op amps provide terminals to allow input offset voltage compensation. This is shown in the previous figure as a variable resistor connected to two terminals of the op amp. The wiper of the variable resistor is connected to the supply voltage, either +Vs or –Vs according to the specifications of the op amp. This resistor need to be adjusted only one time, unless the particular op amp is changed or when the temperature changes.


• 2- Some op amps do not provide terminals for input offset compensation. In this case, a small bias voltage must be placed on the input to provide the required compensation. The following figure shows one way to do this in the case of inverting op amp.

5/9/2019

11



• General purpose IC op amps can source or sink no more than about 20 mA, which includes the current in the feedback circuit. This leads to a general design criterion to be applied to design with op amps. Think of mA and KΩ when designing circuits that use op amps.

5/9/2019

12

Voltage Follower

• Op amp circuit with unity gain and very high input impedance.

• The input impedance is the is essentially the input impedance of the op amp itself, which can be greater than 10 MΩ.

• The output impedance is less than 100 Ω.

• The voltage output tracks the input over a range defined by the plus and minus saturation voltage outputs.

• Current output is limited to the short circuit current of the op amp.

Voltage Follower

5/9/2019

13

Voltage Follower

• In many cases, the manufacturer will market the an op amp voltage follower whose feedback is provided internally.

• The unity gain voltage follower is essentially an impedance transformer (isolating buffer) in the sense of converting a voltage at high impedance to the same voltage at low impedance.

Inverting Amplifier

• This circuit inverts the input signal and may have either attenuation or gain, depending on the ratio of input and feedback resistance.

• The input impedance of this circuit is essentially equal to the input resistance (R1). In general, this resistance is not large, and hence the input impedance is not large.

• The output impedance is low.

5/9/2019

14

Inverting Amplifier

Inverting Summing Amplifier (Span and Zero)

• A common modification of the inverting amplifier is an amplifier that adds two or more applied voltages as shown in the following figure.

• The transfer function of this amplifier is given by:

• The sum can be scaled by proper selection of resistors. For example, if we make R1 = R2 = R3, then the output is simply the inverted sum of V1 and V2.

][ 2

3

21

1

2 VR

RV

R

RVout

5/9/2019

15



Example:

When the temperature in a process is at its minimum, the transducer outputs 2.48 V. At maximum temperature, it outputs 3.9 V. The A/D converter used to input these data into computer has the range 0 to 5 V. To provide maximum resolution, design a zero and span circuit that can modify the transducer signal so that it fills the entire range of the converter.

5/9/2019

16

Noninverting Amplifier

• A noninverting amplifier can be constructed from one op amp as shown in the following figure.

• The noninverting amplifier has a gain that depends on the ratio of the feedback resistor and the ground resistor, but this gain can never be used for voltage attenuation because the ratio is added to 1.

• Because the input is taken directly into the noninverting input of the op amp, the input impedance is very high, since it is effectively equal to the op amp input impedance.

• The output impedance is very low.


5/9/2019

17


Differential Amplifier • There are many instances in measurement and

control systems in which the difference between two voltages needs to be conditioned (amplified).

• A good example is the Wheatstone bridge, where the offset voltage ΔV = Va – Vb is the quantity of interest.

• Another example is conditioning (amplifying) the difference between biomedical signals.

• An ideal differential amplifier provides an output voltage with respect to ground that is some gain times the difference between two input voltages:

Vout = A(Va – Vb)

5/9/2019

18

Differential Amplifier

Differential Amplifier By setting R1 = R2, and R3 = R4, gives:

5/9/2019

19


• However, this amplifier suffers from two main disadvantages:

1. It has relatively low input impedance. This could load the sensor that is feeding this amplifier and consequent distortion of the signal.

2. In order to achieve a high value of CMRR (common mode rejection ratio), the values of the resistors have to be exactly matched, and this is very difficult to achieve with discrete components

• Common Mode Input Voltage (Vcm): Is the average of voltage applied to the two input

terminals Vcm = (Va + Vb)/2

• Common Mode Gain(Acm): Is the amplification factor of the common mode input

voltage • Common Mode Rejection Ratio(CMRR): Is the ratio of the difference mode gain to the common

mode gain CMRR = A/Acm

• Common Mode Rejection (CMR) Is the CMRR expressed in dB

CMR = 20log10(CMRR)

5/9/2019

20


• Ideally, the amplifier used to amplify the weak signal coming out of the sensor needs to have the following characteristics:

1- Have high input impedance so that it does not load the sensor (and to provide the conditions for maximum voltage transfer).

2- Have a high common mode rejection ratio (CMRR).

Instrumentation Amplifier • Two voltage followers are often used on the

input of the differential amplifier to provide high input impedance. The result is called instrumentation amplifier.

5/9/2019

21

Instrumentation Amplifier

• Differential amplifiers with high input impedance and low output impedance are given the special name of instrumentation amplifier.

• Two voltage followers are simply placed on the inputs of the differential amplifier constructing one type of instrumentation amplifier commonly used.

• The transfer function is still the same as normal differential amplifier.

• One disadvantage of this circuit is that changing gain requires changing two resistors and having them carefully matched in value.


5/9/2019

22



5/9/2019

23


• A more common configuration of instrumentation amplifier is the circuit shown in the following figure.

• This circuit Allows for selection of gain by adjustment of a single resistor, RG.

• It can be shown that the CMR of this circuit, although still dependent in careful matching of the differential amplifier resistors, does not depend on matching of the two R1 resistors


5/9/2019

24


• The only way to ensure that the resistors are matched is to implement this amplifier as an integrated circuit.

• Examples of commercially available instrumentation amplifiers are AD623, and INA114.

• RG can be externally connected to the IC to set the required gain.

Voltage to Current Converter

• Because signal in process control are most often transmitted as a current, specifically 4 to 20 mA, it is often necessary to employ a linear voltage to current converter.

• Such a circuit must be capable of sinking a current into a number of different loads without changing the voltage to current transfer characteristics.

• An op amp circuit that provides this function is shown in the following figure.

5/9/2019

25



• An analysis of this circuit shows that the relationship between current and voltage is given by:

Provided that the resistances are selected so that:

5/9/2019

26


• The transfer function in the previous equation is independent of load resistance as long as op amp specifications are not exceeded.

• The maximum load resistance and maximum current are related and determined by the condition that the amplifier output saturates in voltage.

• Analysis of the circuit shows that when the op amp output voltage saturates, the maximum load resistance and maximum current are related by:


• A study of the previous equation shows that the maximum load resistance is always less than Vsat/Im. The minimum load resistance is zero

5/9/2019

27


Current to Voltage Converter

• At the receiving end of the process control signal transmission system, we often need to convert the current back into a voltage.

• This can be done most easily with circuit shown in the following figure.

• This circuit provides an output voltage given by:

Vout = -IR

provided the op amp saturation voltage has not been reached.

5/9/2019

28

Current to Voltage Converter

Linearization (Logarithmic Amplifier)

• The op amp can implement linearization. Generally this is achieved by placing nonlinear element in the feedback loop of the op amp, as shown in the figure.

5/9/2019

29


• Such a component is the semiconductor diode, which can be used as a feedback element.

• The semiconductor diode has the property that the current through it increases exponentially as the applied voltage increases linearly. This means that the voltage will increase logarithmically if we control the applied current rather than the voltage. This is the normal behavior of the operational amplifier, so this is a highly practical method of generating a logarithm.


5/9/2019

30

Charge Amplifier

• The basic theory behind piezoelectricity is based on the electrical dipole. At the molecular level, the structure of a piezoelectric material is typically an ionic bonded crystal.

• At rest, the dipoles formed by the positive and negative ions cancel each other due to the symmetry of the crystal structure, and an electric field is not observed.

• When stressed, the crystal deforms, symmetry is lost, and a net dipole moment is created. This dipole moment forms an electric field across the crystal.

Charge Amplifier

• In this manner, the materials generate an electrical charge that is proportional to the pressure applied.

• If a reciprocating force is applied, an ac voltage is seen across the terminals of the device.

• Piezoelectric sensors are not suited for static or dc applications because the electrical charge produced decays with time due to the internal impedance of the sensor and the input impedance of the signal conditioning circuits. However, they are well suited for dynamic or ac applications.

5/9/2019

31

Charge Amplifier

• As a conclusion, Piezoelectric transducers are composed of high impedance material that generates electric charge in response to varying load.

• A piezoelectric sensor is modelled as a charge source with a shunt capacitor and resistor, or as a voltage source with a series capacitor and shunt resistor. These models are shown in the following Figure along with a typical schematic symbol.

• The charge produced depends on the piezoelectric constant of the device. The capacitance is determined by the area, the width, and the dielectric constant of the material. As previously mentioned, the resistance accounts for the dissipation of static charge.

Charge Amplifier

5/9/2019

32

Charge Amplifier

• Piezoelectric signals cannot be read using low-impedance devices. The two primary reasons for this are:

–High output impedance in the sensor results in small output signal levels and large loading errors.

– The charge can quickly leak out through the load and connecting leads.

Charge Amplifier

5/9/2019

33

Charge Amplifier

5/9/2019

34

Charge Amplifier

Charge Amplifier

5/9/2019

35

Charge Amplifier

• The charge amplifier just transfers the input charge to another reference capacitor and produces an output voltage equal to the voltage across the reference capacitor. Thus the output voltage is proportional to the charge of the reference capacitor and, respectively, to the input charge; hence the circuit acts as a charge-to-voltage converter

Charge Amplifier

5/9/2019

36

Comparator

• The Op-amp comparator compares one analogue voltage level with another analogue voltage level, or some preset reference voltage, VREF and produces an output signal based on this voltage comparison.

• In other words, the op-amp voltage comparator compares the magnitudes of two voltage inputs and determines which is the largest of the two.

Comparator

• Voltage comparators either use positive feedback or no feedback at all (open-loop mode) to switch its output between two saturated states.

• Then due to the high open loop gain, the output from the comparator swings either fully to its positive supply (+Vcc) or fully to its negative supply (-Vcc) on the application of varying input signal which passes some preset threshold value.

5/9/2019

37

Comparator

• The open-loop op-amp comparator is an analogue circuit that operates in its non-linear region as changes in the two analogue inputs, V+ and V- causes it to behave like a digital bistable device as triggering causes it to have two possible output states, +Vcc or -Vcc.

• The voltage comparator is essentially a 1-bit analogue to digital converter, as the input signal is analogue but the output behaves digitally.

Comparator

5/9/2019

38

Comparator

Comparator

5/9/2019

39

Comparator

Comparator with Hysteresis (Schmitt Trigger)

• If the input signal, VIN is slow to change or electrical noise is present, then the op-amp comparator may oscillate switching its output back and forth between the two saturation states, +Vcc and -Vcc as the input signal hovers around the reference voltage, VREF level.

• One way to overcome this problem and to avoid the op-amp from oscillating is to provide positive feedback around the comparator.

5/9/2019

40


• positive feedback is a technique for feeding back a part or fraction of the output signal that is in phase to the non-inverting input of the op-amp via a potential divider set up by two resistors with the amount of feedback being proportional to their ratio.


• The use of positive feedback around an op-amp comparator means that once the output is triggered into saturation at either level, there must be a significant change to the input signal VIN before the output switches back to the original saturation point.

• This difference between the two switching points is called hysteresis producing what is commonly called a Schmitt trigger circuit.

5/9/2019

41



5/9/2019

42


Applications:

• Analog to digital conversion

• Level detection

• Changing a sine wave to square wave

Filters

• A filter is a device that passes electric signals at certain frequencies or frequency ranges while preventing the passage of others.

• Filter Classifications – Passive (RC, RLC)

– Active (RC with op-amps)

• Filter Types: – Low Pass RC Filters

– High Pass RC Filters

– Band Pass RC Filters

– Band Reject RC Filters

– All Pass Filters

5/9/2019

43

Low Pass RC Filters

• Low pass filters block the high frequencies and pass low frequencies.

• It would be most desirable if a low pass filter had a characteristic such that all signals with frequency above some critical value are simply rejected.

• Practical filter circuits approach that ideal with varying degrees of success.

Low Pass RC Filters

5/9/2019

44

Low Pass RC Filters

• Critical frequency is that frequency for which the ratio of the output to the input voltage is approximately 0.707

• The output to input voltage ratio for any signal frequency can be determined graphically from the figure or computed by:

Low Pass RC Filters

5/9/2019

45

2. Calculate the required resistance value. If it is below 1 KΩ or above 1 MΩ, try a

Low Pass Filter

5/9/2019

46

High Pass Filter

• High pass filter passes the high frequencies (no rejections) and blocks (rejects) low frequencies.

• Similar to low pass filter, the rejection is not sharp in frequency but distributed over a range around a critical frequency.

5/9/2019

47

High Pass Filter

High Pass Filter

5/9/2019

48

High Pass Filter

5/9/2019

49

High Pass Filter

5/9/2019

50

High Pass Filter

High Pass Filter

5/9/2019

51

5/9/2019

52

Band Pass Filter • Band pass filter blocks frequencies below a low limit

and above a high limit while passing frequencies between the limits.

• The band pass filter shown in the figure is simply a high pass filter followed by a low pass filter.

• The lower critical frequency is for the high pass filter, whereas the high critical frequency for the low pass filter.

• Care must be taken that the second filter does not load the first filter.

• If the low and high critical frequencies are too close together, the pass band region never reaches unity (i.e. The output is attenuated for all frequencies)

Band Pass Filter

5/9/2019

53

Band Pass Filter

• To provide a good pass band, it is essential that the critical frequencies be as far as apart as possible and that the resistor ratio be kept below 0.01.

Band Pass Filter

Quality Factor Q

• For band-pass filters, Q is defined as the ratio of the mid frequency, fm, to the bandwidth at the two –3 dB points:

5/9/2019

54

Band Pass Filter

5/9/2019

55

Band Reject Filter • Filter that block a specific range of frequencies.

• Such filter is used to reject a particular frequency or a small range of frequencies that are interfering with a data signal.

• The definitions are much the same as the band pass filter in that fL is a critical frequency above which the signal is attenuated by 0.707, whereas fH is a critical frequency below which signals are attenuated by 0.707.

• It is difficult to realize such filters with passive RC combination. It is possible to construct band reject frequencies using inductors and capacitors, but the most success is obtained using active circuits.

Band Reject Filter

5/9/2019

56

Notch Filter

• One very special band reject filter, which can be realized with RC combinations, is called a notch filter because it blocks a very narrow range of frequencies.

• Bridge - T Filter

• Twin -T Filter

• Much more improved band reject and notch filters can be realized using active circuits.

Twin - T Filter

5/9/2019

57

Twin - T Filter • The characteristics of this filter are determined

strongly by the value of the grounding resistor and capacitor labeled R1 and C1.

• For particular combination of R1 = πR/10 and C1 = 10C/ π, the filter response versus frequency is shown in the following figure

• The critical “notch” frequency occurs at a frequency given by:

fn = 0.785fc where fc = 1/(2 πRC)

• The frequencies for which the output is down 3 dB from the pass band are given:

fL = 0.187fc and fH = 4.57fc

Twin - T Filter

5/9/2019

58

Bridge –T Filter

5/9/2019

59

Active Filters

• n passive filter stages can be connected in series as shown in Figure to have steeper response.

• To avoid loading effects, op amps, operating as impedance converters, separate the individual filter stages

Fourth-Order Passive RC Low-Pass with Decoupling Amplifiers

filter stages.

5/9/2019

60

Active Filters

• In comparison to the ideal low-pass, the RC low-pass lacks in the following characteristics:

1- The passband gain varies long before the corner frequency, fc, thus amplifying the upper passband frequencies less than the lower passband.

2- The transition from the passband into the stopband is not sharp, but happens gradually.

3- The phase response is not linear, thus increasing the amount of signal distortion significantly.

5/9/2019

61

Active Filters

• The gain and phase response of a low-pass filter can be optimized to satisfy one of the following three criteria:

1) A maximum passband flatness,

2) An immediate passband-to-stopband transition,

3) A linear phase response.

Active Filters

• For that purpose, the transfer function must allow for complex poles and needs to be of the following type:

where A0 is the passband gain at dc, ai and bi are the filter coefficients.

5/9/2019

62

Active Filters

• The transfer function represents a series of cascaded second-order low-pass stages, with ai and bi being positive real coefficients.

• These coefficients define the complex pole locations for each second-order filter stage, thus determining the behaviour of its transfer function.

Active Filters

• The following three types of predetermined filter coefficients are available listed in table format:

1- The Butterworth coefficients, optimizing the passband for maximum flatness

2- The Tschebyscheff coefficients, sharpening the transition from passband into the stopband

3- The Bessel coefficients, linearizing the phase response up to fc

5/9/2019

63

5/9/2019

64

Active Filters

• The transfer function of a passive RC filter does not allow further optimization, due to the lack of complex poles.

• The only possibility to produce conjugate complex poles using passive components is the application of LRC filters. However, these filters are mainly used at high frequencies. In the lower frequency range (< 10 MHz) the inductor values become very large and the filter becomes uneconomical to manufacture.

• In these cases active filters are used. Active filters are RC networks that include an active device, such as an operational amplifier (op amp).

Active Filters

There are many advantages of active filters, compared with traditional passive filters: 1- Isolation (high input impedance) 2- Cascadeability 3- Gain 4- Small size and weight

5/9/2019

65

Butterworth Low-Pass Filters

• The Butterworth low-pass filter provides maximum passband flatness. Therefore, a Butterworth low-pass is often used as anti-aliasing filter in data converter applications where precise signal levels are required across the entire passband.

• As shown in the following figure, the higher the filter order, the longer the passband flatness.

Butterworth Low-Pass Filters

5/9/2019

66

Tschebyscheff Low-Pass Filters • The Tschebyscheff low-pass filters provide an even

higher gain roll-off above fC. However, as the following figure shows, the passband gain is not monotone, but contains ripples of constant magnitude instead.

• For a given filter order, the higher the passband ripples, the higher the filter’s roll-off.

• Each ripple accounts for one second-order filter stage. Filters with even order numbers generate ripples above the 0-dB line, while filters with odd order numbers create ripples below 0 dB.

• Tschebyscheff filters are often used in filter banks, where the frequency content of a signal is of more importance than a constant amplification.

Tschebyscheff Low-Pass Filters

5/9/2019

67

Bessel Low-Pass Filters

• The Bessel low-pass filters have a linear phase response over a wide frequency range, which results in a constant group delay in that frequency range.

• The passband gain of a Bessel low-pass filter is not as flat as that of the Butterworth low-pass, and the transition from passband to stopband is by far not as sharp as that of a Tschebyscheff low-pass filter.

Quality Factor Q

• For band-pass filters, Q is defined as the ratio of the mid frequency, fm, to the bandwidth at the two –3 dB points:

• For low-pass and high-pass filters, Q represents the pole quality and is defined as:

5/9/2019

68

Quality Factor Q

• High Qs can be graphically presented as the distance between the 0-dB line and the peak point of the filter’s gain response.

• An example is given in Figure, which shows a tenth-order Tschebyscheff low-pass filter and its five partial filters with their individual Qs.

• The higher the Q value, the more a filter inclines to instability.

Quality Factor Q

5/9/2019

69

First Order Low Pass Filter

First Order High Pass Filter

5/9/2019

70

Band Pass/Reject

Second Order Low Pass Filter

There are two topologies for a second-order low-pass filter:

• Sallen-Key topology.

• Multiple Feedback (MFB) topology.

5/9/2019

71


Sallen-Key topology

• The general Sallen-Key topology in the following Figure allows for separate gain setting via A0 = 1+R4/R3.

• However, the unity-gain topology in the next Figure is usually applied in filter designs with high gain accuracy, unity gain, and low Qs (Q < 3).

• Note: the transfer function is normalized according to the corner frequency.


General Sallen-Key Filter

5/9/2019

72


Unity Gain Sallen-Key Filter



• The coefficient comparison between this transfer function and the general equation yields:

• Given C1 and C2, the resistor values for R1 and R2 are calculated through:

5/9/2019

73



• In order to obtain real values under the square root, C2 must satisfy the following condition:



Example: The task is to design a second-order unity gain Tschebyscheff low-pass filter with a corner frequency of fC = 3 kHz and a 3-dB passband ripple.

Solution:

• From the table of Tschebyscheff coefficients for 3-dB ripple, obtain the coefficients a1 and b1 for a second-order filter with a1 = 1.0650 and b1 = 1.9305.

• Specifying C1 as 22 nF yields in a C2 of:

5/9/2019

74


• Inserting a1 and b1 into the resistor equation for R1,2 results in:


Multiple Feedback Topology

• The MFB topology is commonly used in filters that have high Qs and require a high gain.

5/9/2019

75


Multiple Feedback Topology

• Through coefficient comparison with the general equation obtains the relation:

• Given C1 and C2, and solving for the resistors R1–R3:

Second Order Low Pass Filter Multiple Feedback Topology

• In order to obtain real values for R2, C2 must satisfy the following condition:

5/9/2019

76

High Order Active Filter

Bessel Coefficients Table

5/9/2019

77

Butterworth Coefficients Table

Tschebyscheff Coefficients (0.5 dB) Table

5/9/2019

78

Design Guidelines

5/9/2019

79

5/9/2019

80

5/9/2019

81

5/9/2019

82

5/9/2019

83

transducers and sensors - philadelphia university

Documents