A filter, be it an oil filter, a lint filter, a furnace filter, or an active filter, accepts a wide spectrum

of inputs, but only passes certain of these inputs through to the output. In some cases, it may pass

through the "good stuff" while it catches the "bad stuff." An oil filter in your car is one example.

Other applications require a filter to catch the "good stuff" and let the "bad stuff" pass through. A

gold prospector's sieve is an example of this type of filter action. In both of the preceding

examples, the filter discriminates between "good" and "bad" on the basis of physical size (i.e.,

size of the dirt particle). In the filters discussed in this review, the "good" and "bad" signals will

be classified on the basis of their frequency. The input will be a broad range of signal

frequencies. The filter will allow a certain range of them to pass and will reject others (Terrell,

1996).

The concept of filters has been an integral part of the evolution of electronics from the

beginning. Several technological achievements would not have been possible without electrical

filters. Because of this prominent role of filters, much effort has been expended on the theory,

1

CHAPTER IINTRODUCTION

TO FILTERS

1.1 INTRODUCTION

design, and construction of filters and many articles and books have been written on them. Our

discussions would be considered as introductory.

As a frequency-selective device, a filter can be used to limit the frequency spectrum of a signal

to some specified band of frequencies. Filters are circuits used in radio and TV receivers to allow

us to select on desired signal out a multitude of broadcast signals in the environment (Alexander

and Sadiku, 2001).

A filter is a passive filter if it consists of only passive elements R, L and C. it is said to be an

active filter if it consists of active elements (such as transistors and Op amps) in addition to

passive elements R, L, and C. we consider only the active filters in this section. Besides the two

listed filters, there are other kinds of filters such as Digital filters, electromechanical filters and

microwave filters which are beyond the scope of this text

There are three major limits to the passive filters, first, they cannot generate gain greater than 1;

passive elements cannot add energy to the network. Second, they may require bulky and

expensive inductors. Third, they perform poorly at frequencies below the audio frequency range

(300 Hz < f < 3000 Hz). Nevertheless, passive filters are useful at high frequencies (Alexander

and Sadiku, 2001).

Active filters consists of combinations of resistors, capacitors and op amps. They offer some

advantages over passive RLC filters. First, they are often smaller and less expensive, because

they do not require inductors. This makes feasible the integrated circuit realizations of filters.

Second, they can provide amplifier gain in addition to providing the same frequency response as

RLC filters. Third, active filters can be combined with buffer amplifiers (voltage followers) to

isolate each stage of the filter from source and load impedance effects. This isolation allows

2

designing the stages independently and then cascading them to realize the desired transfer

function (Bode plots, being logarithmic, may be added when transfer functions are cascaded.)

however active filters are less reliable and less stable. The practical limit of most active filters is

about 100KHz – most active filters operate well below that frequency (Alexander and Sadiku,

2001).

Electronic filters (whether active or passive) designed to discriminate as a function of frequency

can be broadly grouped into five classes:

Low pass: Allows frequencies below a specified frequency to pass through the

filter circuit.

High pass: Allows frequencies above a specified frequency to pass through the

filter circuit.

Bandpass: Allows a range or band of frequencies to pass through the filter circuit

while rejecting frequencies higher or lower than the desired band.

Band reject Rejects all frequencies within a certain band, but passes frequencies

higher or lower than the specified band. Also called a band-stop filter.

I. Notch essentially a band-stop filter with a very narrow range of

frequencies that are rejected.

Figure 1.1 shows the general frequency response curves for each of the basic filter types. The

exact nature of a given curve will vary with the type of circuit implementation. Most notably, the

slope of the curve between the "pass" and "reject" regions of the filter varies greatly with

different filter designs (Terrell, 1996).

3

There are seemingly endless ways to achieve the various filter functions listed. Each method of

implementation has its individual advantages and disadvantages for particular applications. In

this review, a representative filter design for each basic filter type would be selected. Band reject

and notch filters will be treated as one general class (Terrell, 1996).

An important consideration regarding active filters is how sharply the frequency response drops

off for frequencies outside of the passband of the filter. In general, the steeper the slope of the

4

Figure 1.1: Theoretical response curves for the five basic classes of filter circuits

curve, the more ideal the filter behavior. If the slope becomes too steep, however, the filter

becomes unstable and is prone to oscillations. It is common to express the steepness of the slope

in terms of dB per decade where a decade represents a factor of 10 increase or decrease in

frequency. For example, suppose a low-pass filter had a 20-dB-per-decade slope beyond the

cutoff frequency. This means that if the input frequency is increased by a factor of 10, then the

output will decrease by 20 dB. If the input frequency is again increased by a factor of 10, then

the output will decrease another 20 dB, or 40 dB from the first measurement (Terrell, 1996).

Typical filter circuits have slopes ranging from 6 to 60 dB per decade or more. In the case of

bandpass, bandstop, and notch filters, we often describe the steepness of the slopes in another

way. The ratio of the center frequency ( f c) to the bandwidth (bw) gives us an indication of the

sharpness of the cutoff region (Terrell, 1996).

The ratio f c /bw is called the Q of the circuit. The higher the Q, the sharper the cutoff slopes of

the filter.

The term Q is also used with reference to low-pass and high-pass filters, but it must be

interpreted differently. The output of some filters peaks just before the edge of the passband. The

Q of the filter indicates the degree of peaking. A Q of 1 has only a slight peaking effect. A Q of

less than 1 reduces this peaking, while a Q greater than 1 causes a more pronounced peaking.

There is usually a trade-off between peaking (generally undesired) and steepness (generally

desired) of the slope (Terrell, 1996).

5

The following terms described below are terms commonly used with filters

Active filter A frequency-selective circuit consisting of active devices such as

transistors or op-amps coupled with reactive components (Floyd, 2005).

The passband/Stopband: These are the ranges of frequency that are nominally

passed through and blocked respectively by the filter (inst.eecs.berkeley.edu).

Slew Rate: The maximum rate of change of the output voltage in response to a step

input voltage is the slew rate of an op-amp (Floyd, 2005).

The cutoff frequency: the critical frequency is determined by the values of resistors

and capacitors of a selective RC circuit, for a single-pole (first order) filter (Floyd,

2005), the critical frequency is given by

f c=1

2 πRC

Stopband attenuation is the minimum attenuation of all stopband frequencies.

Ripple: This is how much the magnitude response ripples in passband or stopband.

Pole: This is a term used with filters to describe the number of RC circuits contained

in the filter (Floyd, 2005).

Order of a Filter: number of poles

Roll-off rate: This is the rate of decrease in gain, below or above the critical

frequencies of a filter (Floyd, 2005).

Damping factor: This is a filter characteristic that determines the type of response.

6

1.2 DEFINITION OF FILTER TERMINOLOGIES

(1.1)

Roll-off

Passband

Stopband

Stopband attenuation

Cutoff

Ripple

Group Delay: This is simply the derivative of the phase response (both are plotted

vs. frequency), it is a measure of the time it takes for a signal to pass through a filter

(inst.eecs.berkeley.edu).

A filter is a circuit that passes certain frequencies and attenuates or rejects all other frequencies.

The passband of a filter is the range of frequencies that are allowed to pass through the filter with

minimum attenuation (usually defined as less than -3 dB of attenuation). The critical frequency,

f c (also called the cutoff frequency) defines the end of the passband and is normally specified at

the point where the response drops - 3 dB (70.7%) from the passband response. Following the

passband is a region called the transition region that leads into a region called the stopband.

There is no precise point between the transition region and the stopband (Floyd, 2005).

7

Figure 1.2: Example filter magnitude response and general terminology

1.3 BASIC FILTER RESPONSES

1000 fc

-3 dB

0.001fc

0 dB

-60 dB --

-60 dB --

-60 dB --

Transition region

Gain (normalized to 1)

- 20 dB/decade

0.1fc fc 10 fc 100fcf

Stopband region

Passband

BW

1000fcfc0.001fc

- 20 dB/decade

- 40 dB/decade

- 60 dB/decade

-60 dB --

-60 dB --

-60 dB --

0 dB

Gain (normalized to 1)

0.1fc 10 fcf

100fc

8

1.3.1 Low-Pass Filter Response

Figure 1.3 (a) Comparison of an ideal Low-pass filter response (shaded area) with actual response

Figure 1.3 (b) Idealized Low-pass filter responses

A low-pass filter is one that passes frequencies from dc to f cand significantly attenuates all other

frequencies. The passband of the ideal low-pass filter is shown in the shaded area of Figure

1.3(a); the response drops to zero at frequencies beyond the passband. This ideal response is

sometimes referred to as a "brick-wall" because nothing gets through beyond the wall. The

bandwidth of an ideal low-pass filter is equal to f c (Floyd, 2005).

f c=BW

The ideal response shown in Figure 1.3(a) is not attainable by any practical filter. Actual filter

responses depend on the number of poles, a term used with filters to describe the number of RC

circuits contained in the filter. The most basic low-pass filter is a simple RC circuit consisting of

just one resistor and one capacitor; the output is taken across the capacitor. The basic RC filter

has a single pole, and it rolls off at -20 dB/decade beyond the critical frequency. The actual

response is indicated in Figure 1.3(a). The response is plotted on a standard log plot that is used

for filters to show details of the curve as the gain drops. Notice that the gain drops off slowly

until the frequency is at the critical frequency; after this, the gain drops rapidly.

The -20 dB/decade roll-off rate for the gain of a basic RC filter means that at a frequency of 10 f c

, the output will be - 20 dB (10%) of the input. This roll-off rate is not a particularly good filter

characteristic because too much of the unwanted frequencies (beyond the passband) are allowed

through the filter.

The critical frequency of a low-pass RC filter occurs when X c=R , where f c=1

2πRC. From basic

dc/ac studies that the output at the critical frequency is 70.7% of the input. This response is

equivalent to an attenuation of -3 dB.

9

(1.2)

Figure 1.3(c) illustrates three idealized low-pass response curves including the basic one pole

response (-20 dB/decade). The approximations show a flat response to the cutoff frequency and a

roll-off at a constant rate after the cutoff frequency. Actual filters do not have a perfectly flat

response to the cutoff frequency but have dropped to -3 dB at this point as described previously.

In order to produce a filter that has a steeper transition region (and hence form a more effective

filter), it is necessary to add additional circuitry to the basic filter. Responses that are steeper than

-20 dB/decade in the transition region cannot be obtained by simply cascading identical RC

stages (due to loading effects). However, by combining an op-amp with frequency-selective

feedback circuits, filters can be designed with roll-off rates of -40, -60, or more dB/decade.

Filters that include one or more op-amps in the design are called active filters. These filters can

optimize the roll-off rate or other attribute (such as phase response) with a particular filter

design. In general the more poles the filter uses, the steeper its transition region will be. The

exact response depends on the type of filter and the number of poles.

A high-pass filter is one that significantly attenuates or rejects all frequencies below f c and

passes all frequencies abovef c. The critical frequency is, again, the frequency at which the output

is 70.7% of the input (or - 3 dB) as shown in Figure 1.4(a). The ideal response, indicated by the

shaded area, has an instantaneous drop at f c, which, of course, is not achievable. Ideally, the

passband of a high-pass filter is all frequencies above the critical frequency. The high frequency

response of practical circuits is limited by the op-amp or other components that make up the

filter (Floyd, 2005).

10

1.3.2 High-Pass Filter Response

0 dB

0.001fc-60 dB --

-60 dB --

-60 dB --

Passband

-3 dB

Gain (normalized to 1)

- 20 dB/decade

0.01fc 0.1fc fc 10 fc 100fcf

0.001fc-60 dB --

-60 dB --

-60 dB --

0 dB

Gain (normalized to 1)

- 20 dB/decade

0.01fc 0.1fc fc 10 fc 100fcf

- 60 dB/decade

- 40 dB/decade

11

Figure 1.4 (a) Comparison of an ideal High-pass filter response (shaded area) with actual response

Figure 1.4 (b) Idealized High-pass filter responses

As in the case of the low-pass filter, the basic RC circuit has a roll-off rate of - 20 dB/decade, as

indicated by the in Figure 1.4(a). Also, the critical frequency for the basic high-pass filter occurs

when X c=R, where the critical frequency is same as that of the low-pass filter.

Figure 1.4(b) illustrates three idealized high-pass response curves including the basic one-pole

response (-20 dB/decade) for a high-pass RC circuit. As in the case of the low- pass filter, the

approximations show a flat response to the cutoff frequency and a roll-off at a constant rate after

the cutoff frequency. Actual high-pass filters do not have the perfectly flat response indicated or

the precise roll-off rate shown. Responses that are steeper than -20 dB/decade in the transition

region are also possible with active high-pass filters; the particular response depends on the type

of filter and the number of poles (Floyd, 2005).

A band-pass filter passes all signals lying within a band between a lower frequency limit and an

upper-frequency limit and essentially rejects all other frequencies that are outside this specified

band. A generalized band-pass response curve is shown in Figure 1.5. The bandwidth (BW ) is

defined as the difference between the upper critical frequency f c2 and the lower critical

frequency f c1 (Floyd, 2005).

BW=f c 2−f c1

The critical frequencies are, of course the points at which the response curve is 70.7% of its

maximum. The frequency about which the passband is centered is called the center frequency, f 0,

defined as the geometric mean of the critical frequencies (Floyd, 2005).

f 0=√ f c1 f c2

12

1.3.3 Band-Pass Filter Response

(1.3)

(1.4)

Another category of active filter is the band-stop filter, also known as notch, band-reject, or

band-elimination filter. You can think of the operation as opposite to that of the band-pass filter

because frequencies within a certain bandwidth are rejected, and frequencies outside the

13

Figure 1.5: General Band-pass response curve

Figure 1.6: General Band-stop filter response

1.3.4 Band-Stop Filter Response

bandwidth are passed. A general response curve for a band-stop filter is shown in Figure 1.6.

Notice that the bandwidth is the band of frequencies between the 3 dB points, just as in the case

of the band-pass filter response (Floyd, 2005).

Active filters can realize the same filter characteristics as their passive counterpart; however,

they have several advantages over their passive counterparts:

Active filters can provide gain, and are frequently used to simultaneously match filtering

(frequency-determining) and gain specifications.

They are readily implemented in integrated-circuit technology, whereas the inductor

element of passive filters is not readily realized. As a result, the active filter is

inexpensive, and is attractive for its small size and weight. In addition, it is readily

included with other signal-processing functions on a single integrated circuit.

The design of active filters is considerably simpler than that of passive ones. In addition,

it is easy to provide for variability, which can be used to change filter characteristics by

electrical input signals. (www.answers.com)

Since the active filter contains electronic components, it requires a power supply, which

adds to the complexity of the realization. The electronic components also place

restrictions on the level of the signals that can be applied to the filter and on the noise

component that the filter may add to the filtered signal.

14

1.4 ADVANTAGES OF ACTIVE FILTERS

1.5 DIS-ADVANTAGES OF ACTIVE FILTERS

The mathematical process by which the active filter produces filtering characteristics in

general requires the use of internal feedback. When this feedback is positive, the resulting

filter may be very sensitive to lack of precision in component values, and the effects of

aging and environmental conditions.

In general, active filters are designed with the assumption that the active elements are

ideal (not in practical). For example, the operational amplifier is assumed to have infinite

gain, infinite input impedance, zero output impedance, and an infinite frequency range

(gain-bandwidth). All practical filter realizations must be evaluated for the effect that the

non-ideal practices have on actual filter performance (www.answers.com).

15

Figure 2.1: magnitude response of an ideal filter

Each type of filter response (low-pas, high-pass, band-pass. or band-stop) can be tailored by

circuit component values to have a Butterworth, Chebyshev/Inverse Chebyshev, Elliptic, or

Bessel characteristic. Each of these characteristics is identified by the shape of the response

curve, and each has an advantage in certain applications (Floyd, 2005).

16

CHAPTER IIFILTER RESPONSE

TOPOLOGIES

2.1 INTRODUCTION

ButterworthBessel

Chebyshev and Inverse ChebyshevElliptic

The Butterworth filter is the easiest of the 5 topologies to implement in analog circuits It was

first described by the British engineer Stephen Butterworth in his paper entitled "On the Theory

17

2.2.1 Butterworth

2.2 ACTIVE FILTER TOPOLOGIES

Figure 2.2: Basic Active Filter Topologies

of Filter Amplifiers". Its magnitude response is characterized by a maximally flat passband, -3dB

attenuation at the cutoff frequency, and a roll-off of –20N dB/decade beyond cutoff, where N is

the order. This roll-off is the slowest, with the exception of the Bessel filter. Stopband

attenuation and ripple have no effect on the Butterworth (inst.eecs.berkeley.edu).

The Butterworth characteristic provides a very flat amplitude response in the passband and a roll-

off rate of -20 dB/decade/pole. The phase response is not linear. However, and the phase shift

(thus, time delay) of signals passing through the filter varies nonlinearly with frequency.

Therefore. a pulse applied to a filter with a Butterworth response will cause overshoot on the

output because each frequency component of the pulse's rising and falling edges experiences a

different time delay (Floyd, 2005).

Filters with the Butterworth response are normally used when all frequencies in the passband

must have the same gain. The Butterworth response is often referred to as a maximally flat

response.

Filters with these response characteristics roll off more steeply than the Butterworth, but exhibit

ripple in passband (Chebyshev) or stopband (Inverse). The value of stopband attenuation has no

effect on the Chebyshev filter; ripple has no effect on the Inverse Chebyshev (Alexander and

Sadiku, 2001).

Filters with the Chebyshev response characteristic are useful when a rapid roll-off is required

because it provides a roll-off rate greater than - 20 dB/decade/pole. This is a greater rate than that

of the Butterworth, so filters can be implemented with the Chebyshev response with fewer poles

and less complex circuitry for a given roll-off rate. This type of filter response is characterized

18

2.2.2 Chebyshev and Inverse Chebyshev

by overshoot or ripples in the passband (depending on the number of poles) and an even less

linear phase response than the Butterworth (Floyd, 2005).

The elliptic filter rolls off more steeply than all other filters, but exhibits ripple in both passband

and stopband (inst.eecs.berkeley.edu).

The Bessel filter has the slowest roll-off of all 5 filters. Its chief advantage is its maximally flat

group delay in the passband, The Bessel’s flat group delay means all frequencies in the passband

are delayed the same amount, which means there is no signal distortion at the output due to

different frequencies being delayed different amounts. The stopband attenuation and ripple have

no effect on this filter (inst.eecs.berkeley.edu).

19

2.2.3 Elliptic

2.2.4 Bessel

Figure 3.1(a) shows one of the most common implementations of the low-pass filter circuit. This

particular configuration is called a Butterworth filter and is characterized by a very flat response

in the passband portion of its response curve (Terrell, 1996).

Ideally, a low-pass filter will pass frequencies from DC up through a specified frequency, called

the cutoff frequency, with no attenuation or loss. Beyond the cutoff frequency, the filter ideally

offers infinite attenuation to the signal. In practice, however, the transition from passband to

stopband is a gradual one. The cutoff frequency is defined as the frequency that passes with a

70.7-percent response. This, of course, is the familiar half-power point referenced in basic

electronics theory (Terrell, 1996).

A stiff physical barrier tends to reflect higher sound frequencies, and so acts as a low-pass filter

for transmitting sound. When music is playing in another room, the low notes are easily heard,

while the high notes are attenuated. Low-pass filters are used to drive subwoofers and other types

of loudspeakers, to block high pitches that they can't efficiently broadcast. Radio transmitters use

20

CHAPTER IIIACTIVE FILTERS AND

THEIR OPERATION

3.1 ACTIVE LOW-PASS FILTER

R1 R2

R3

C1

C2

V1

Vo

Figure 3.1(a): A low-pass Butterworth filter circuit.

low-pass filters to block harmonic emissions which might cause interference with other

communications. The tone knob found on many electric guitars is a low-pass filter used to reduce

the amount of treble in the sound (en.wikipedia.org/wiki/Low-pass_filter).

Let us try to understand the operation of the low-pass filter circuit shown in Figure 3.1(a) from

an intuitive or logical standpoint. First, mentally open-circuit the capacitors. This modified

circuit is shown in Figure 3.0(b), which is essentially how the circuit will look at low frequencies

when the capacitive reactance of the capacitors is high. We can see that this amplifier is

connected as a simple voltage follower circuit. Resistor R3 is included in the feedback loop to

compensate for the effects of bias currents flowing through R1 and R2. For low frequencies, then,

we expect to have a voltage gain of about unity (Terrell, 1996).

21

3.1.1 Operation

VoR1 R2

R3

V1

Figure 3.1(c): A high-frequency equivalent circuit for the low-pass filter shown in Figure 3.1(a)

VoR1 R2

R3

V1

Figure 3.1(b): A low-frequency equivalent circuit for the low-pass filter shown in Figure 3.1(a)

Now let us mentally short-circuit the capacitors in Figure 3.1(b) to get an idea of how the circuit

looks to high frequencies where the capacitive reactance is quite low (Terrell, 1996). This

equivalent circuit is shown in Figure 3.1(c).

First, notice that the (+) input of the amplifier is essentially grounded. This should eliminate any

chance of signals passing beyond this point. The junction of R1 and R2 is effectively connected

to the output of the Op Amp. This, you will recall, is a very low impedance point, so for high

frequencies, the junction of R1 and R2 also has a low impedance to ground (Terrell, 1996).

As our preliminary analysis indicates, the low frequencies should receive a voltage gain of about

1, and the high frequencies should be severely attenuated

22

Figure 3.2(a) (inst.eecs.berkeley.edu) shows the schematic diagram of a high-pass filter circuit

that provides a theoretical roll-off slope of 40 dB per decade. The circuit configuration is

obtained by changing positions with all of the resistors and capacitors (except R3) in the low-

pass equivalent (figure 3.2(b)). As a high-pass filter, we will expect it to severely attenuate

signals below a certain frequency and pass the higher frequencies with minimal attenuation

(Terrell, 1996).

High-pass filters have many applications. They are used as part of an audio crossover to direct

high frequencies to a tweeter while attenuating bass signals which could interfere with, or

damage, the speaker. When such a filter is built into a loudspeaker cabinet it is normally a

passive filter that also includes a low-pass filter for the woofer and so often employs both a

capacitor and inductor (although very simple high-pass filters for tweeters can consist of a series

capacitor and nothing else). An alternative, which provides good quality sound without inductors

(which are prone to parasitic coupling, are expensive, and may have significant internal

resistance) is to employ bi-amplification with active RC filters or active digital filters with

separate power amplifiers for each loudspeaker. Such low-current and low-voltage line level

crossovers are called active crossovers (en.wikipedia.org/wiki/High-pass_filter).

An intuitive feel for the operation of the circuit in Figure 3.2(a) can be gained by picturing the

equivalent circuit at very low and very high frequencies. At very low frequencies, the capacitors

will have a high reactance and will begin to appear as open circuits. Figure 3.2(b) shows the

23

3.2.1 Operation

3.2 ACTIVE HIGH-PASS FILTER

R3

V1

Vo

R2

R1

C2C1

Figure 3.2(a): A 40-dB-per-decade high-pass filter circuit.

R3

V1

Vo

R1

Open Circuit

Figure 3.2(b): A low-frequency equivalent circuit for the high-pass filter shown in Figure 3.2

R3

V1

Vo

R1

Figure 3.2(c): A High-frequency equivalent circuit for the high-pass filter shown in Figure 3.2(a)

low-frequency equivalent circuit for the high-pass filter shown in Figure 3.2(a). As you can

readily see, the amplifier acts as a unity gain circuit, but it has no input signal (Terrell, 1996).

24

At low frequencies, we expect little or no output signal and at high frequencies, the capacitors

will have a low reactance and will begin to appear as short circuits. The high-frequency

equivalent circuit is shown in Figure 3.2(c). Here we see that the capacitors have been replaced

with direct connections. Also, resistor R2 has been removed, because it is connected between

two points that have the same signal amplitude and phase (i.e., input and output of a voltage

follower). Because it has the same potential on both ends, it will have no current flow and is

essentially open. The resulting equivalent circuit indicates that for high frequencies, our high-

pass filter will act as a simple voltage follower (Terrell, 1996).

Figure 3.3(a) (Terrell, 1996) shows the schematic diagram of a bandpass filter. This circuit

provides maximum gain (or minimum loss) to a specific frequency called the resonant, or center,

frequency (even though it may not actually be in the center). Additionally, it allows a range of

frequencies on either side of the resonant frequency to pass with little or no attenuation, but

severely reduces frequencies outside of this band (Terrell, 1996).

The edges of the passband are identified by the frequencies where the response is 70.7 percent of

the response for the resonant frequency (Terrell, 1996).

Outside of electronics and signal processing, one example of the use of band-pass filters is in the

atmospheric sciences. It is common to band-pass filter recent meteorological data with a period

range of, for example, 3 to 10 days, so that only cyclones remain as fluctuations in the data fields

(en.wikipedia.org/wiki/Bandpass_filter).

The range of frequencies that make up the passband is called the bandwidth of the filter, this can

be stated as bw=f 1−f 2.

25

3.3 ACTIVE BAND-PASS FILTER

R1

R3

C2

C1

V1R2 R4

Vo

Figure 3.3(a): A band-pass fitter

Where f 2 and f 1 are the frequencies that mark the edges of the passband. The Q of the circuit is a

way to describe the ratio of the resonant (centre) frequency (f r) to the bandwidth (bw). That is,

Q=f r

bw

If the Q of the circuit is 10 or less, we call the filter a wide-band filter. Narrow-band filters have

values of Q over 10. In general, higher Q’s produce sharper, more well defined responses

(Terrell, 1996). If the application requires a Q of 20 or less, then a single Op Amp. Filter circuit

can be used. For higher Qs, a cascaded filter should be used to avoid potential oscillation

problems.

To help us gain an intuitive understanding of the circuit's operation, let us draw equivalent

circuits for the filter shown in Figure 3.3(a) (inst.eecs.berkeley.edu) at very high and very low

frequencies. To obtain the low-frequency equivalent where the capacitors have a high reactance,

we simply open-circuit the capacitors, Figure 3.3(b) shows the low frequency equivalent; it is

26

3.3.1 Operation

(3.1)

R1

R3

V1R2 R4

Vo

Open Circuit

Figure 3.3(b): A low-frequency equivalent circuit for the bandpass filter shown in Figure 3.3(a).

obvious from it that the low frequencies will never reach the amplifier's input and therefore

cannot pass through the filter (Terrell, 1996).

At high frequencies, the reactance of the capacitors will be low and the capacitors will begin to

act like short circuits. Figure 3.3(b) shows the high frequency equivalent circuit, which was

obtained by short-circuiting all of the capacitors. From this equivalent circuit, we can see that the

high frequencies will be attenuated by the voltage divider action of R1 and R2. Additionally, the

amplifier has 0 resistance in the feedback loop, which causes our voltage gain to be 0 for the op

amp (i.e., no output) (Terrell, 1996).

At some intermediate frequency (determined by the component values), the gain of the amplifier

will offset the loss in the voltage divider (R1 and R2) and the signals will be allowed to pass

through. The circuit is frequently designed to have unity gain at the resonant frequency, but may

be set up to provide some amplification (Terrell, 1996).

27

R1

V1R2 R4

Vo

Figure 3.3(c): A high-frequency equivalent circuit for the bandpass fitter shown in Figure 3.3(a).

A band reject filter is a circuit that allows frequencies to pass that are either lower than the lower

cutoff frequency or higher than the upper cutoff frequency. That is, only those frequencies that

fall between the two cutoff frequencies are rejected or at least severely attenuated (Terrell, 1996).

For instance, when measuring non-linearities of power amplifiers a very narrow notch filter

could be very useful to avoid the carrier so maximum input power of e.g. a spectrum analyser

used to detect spurious content will not be exceeded (en.wikipedia.org/wiki/Bandstop_filter).

Figure 3.4 shows an active filter that is based on the common twin "T" configuration. The twin T

gets its name from the two RC T networks on the input. For purposes of analysis, let us consider

the lower ends of R3 and C3 to be grounded. This is a reasonable approximation, since the output

impedance of an op amp is generally quite low. The T circuit consisting of Q, C2, and R3 is, by

itself, a high pass filter (Terrell, 1996).

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3.4.1 Operation

3.4 ACTIVE BAND-REJECT FILTER

Vo

R4

C2C1

R3

R1 R2

C3

V1

Figure 3.4: A band-reject filter circuit

That is, the low frequencies are prevented from reaching the input of the op amp because of the

high reactance of Q and C2. The high frequencies, on the other hand, find an easy path to the op

amp because the reactance of Q and C2 is low at higher frequencies (Terrell, 1996).

The second T network is made up of R1, R2 and C3 and forms a low-pass filter. Here the low

frequencies find C3's high reactance to be essentially open, so they pass on to the op amp input.

High frequencies, on the other hand, are essentially shorted to ground by the low reactance of C3.

It would seem that both low and high frequencies have a way to get to the (+) input of the op

amp and therefore to be passed through to the output. If, however, the cutoff frequencies of the

two T networks do not overlap, there is a frequency ( f r) that results in a net voltage of 0 at the

(+) terminal of the op amp (Terrell, 1996).

To understand this effect, we must also consider the phase shifts given to a signal as it passes

through the two networks. At the center, or resonant, frequency ( f r), the signal is shifted in the

negative direction while passing through one T network. It receives the same amount of positive

phase shift while passing through the other T network. These two shifted signals pass through

29

equal impedances (R2∧XC 2) to the (+) input. Thus, at any instant in time (at the center

frequency), the effective voltage on the (+) input is 0. The more the input frequency deviates

from the center frequency, the less the cancellation effect. Thus, as we initially expected, this

circuit rejects a band of frequencies and passes those frequencies mat are higher or lower than

the cutoff frequencies of the filter (Terrell, 1996).

The op amp offers a high impedance to the T networks, thus reducing the loading effects and

therefore increasing the Q of the circuit. Additionally, by connecting the "ground" point of C3

and R3 to the output of the op amp, we have another increase in Q as a result of the feedback

signal. At or very near the center frequency, very little signal makes it to the (+) input of the op

amp. Therefore, very little signal appears at the output of the op amp. Under these conditions the

output of the op amp merely provides a ground (i.e., low impedance return to ground) for the T

networks. For the other frequencies, though, the feedback essentially raises the impedance

offered by C3 and R3 at a particular frequency. Therefore, they don't attenuate the off-resonance

signals as much, which has the effect of narrowing the bandwidth or, we could say, increasing

the Q (Terrell, 1996).

Resistor R4 is to compensate for the voltage drops caused by the op amp bias current flowing

through R1 and R2. It is generally equal in value to the sum of R1 and R2 (Terrell, 1996).

30

It is sometimes desirable to have circuits capable of selectively filtering one frequency or range

of frequencies out of a mix of different frequencies in a circuit. A circuit designed to perform this

frequency selection is called a filter circuit, or simply a filter. A common need for filter circuits

is in high-performance stereo systems, where certain ranges of audio frequencies need to be

amplified or suppressed for best sound quality and power efficiency. You may be familiar with

equalizers, which allow the amplitudes of several frequency ranges to be adjusted to suit the

listener's taste and acoustic properties of the listening area. It is also used in crossover networks,

which block certain ranges of frequencies from reaching speakers. A tweeter (high-frequency

speaker) is inefficient at reproducing low-frequency signals such as drum beats, so a crossover

circuit is connected between the tweeter and the stereo's output terminals to block low-frequency

signals, only passing high-frequency signals to the speaker's connection terminals. This gives

better audio system efficiency and thus better performance. Both equalizers and crossover

networks are examples of filters, designed to accomplish filtering of certain frequencies

(Kuphaldt et al, 1996).

31

CHAPTER IVAPPLICATIONS OF ACTIVE

FILTERS

4.1 INTRODUCTION

Another practical application of filter circuits is in the "conditioning" of non-sinusoidal voltage

waveforms in power circuits. Some electronic devices are sensitive to the presence of harmonics

in the power supply voltage, and so require power conditioning for proper operation. If a

distorted sine-wave voltage behaves like a series of harmonic waveforms added to the

fundamental frequency, then it should be possible to construct a filter circuit that only allows the

fundamental waveform frequency to pass through, blocking all (higher-frequency) harmonics

(Kuphaldt et al, 1996).

Active filters have been designed and developed for compensation of current

and voltage harmonics; yet, now they have many functions and many

applications. Active filters, based on topology and control scheme, can

compensate current harmonics, voltage harmonics, reactive power, load

imbalance, neutral current, voltage imbalance, voltage regulation, voltage

flicker, and voltage sag and swell, as mention above. In this section, some

special applications of active filters are explained (Bekiarov et al, 2005).

A cycloconverter or a cycloinverter converts an AC waveform, such as the

mains supply, to another AC waveform of a lower frequency, synthesizing the

output waveform from segments of the AC supply without an intermediate

direct-current link (en.wikipedia.com)

32

4.2 APPLICATIONS

4.2.1 Cycloconverter

In variable-speed cycloconverter drive systems, active filters are used in the

AC side to reduce the variable frequency current harmonics, which are

produced by both the converter and inverter as shown in Figure 4.1(Bekiarov

et al, 2005). Passive filters cannot work properly because the frequency is

variable (Bekiarov et al, 2005).

In induction motor drives, there is interference between the harmonic and

signaling system, leading to improper working of the system. Simple low-

pass passive filters can be used to suppress these harmonics; however, they

will be large and heavy (Bekiarov et al, 2005) For example, a low-pass LC passive filter

with a corner frequency of 5 kHz for an induction motor of 1.5 MW has a total weight of 5 tons.

Another application of active filters is the cancellation of bulky and unreliable capacitors with a

current-fed small size, reliable, costly, and resonance free shunt active filter, as shown in Figure

4.2. This active filter cancels voltage harmonics in the DC-link of the DC/AC converter and

33

Figure4.1: Application of an active filter with a cycloconverter

4.2.2 Adjustable Speed Drives

4.2.3 DC Capacitor Cancellation

makes the voltage of this DC-link smooth; therefore, the quality of the output AC will be better

(Bekiarov et al, 2005).

Active filters are used in both DC and AC sides of the HVDC systems to

cancel the voltage harmonics in the DC side and current harmonics in the AC

side. Orders of voltage harmonics in the DC side are of 12n (n = 1, 2, 3,…)

and orders of current harmonics in the AC side are 12 n ±1 (n = 1, 2, 3,…).

(Bekiarov et al, 2005).

In high-power locomotives, current harmonics produced by variable-

frequency DC/AC inverters cause excessive losses in AC/DC converters

34

Fgure4.2Cancellation of a capacitor in DC/AC converter

4.2.4 HVDC Systems

4.2.5 High-Power Locomotives

(Figure 4.3) and transformers. Active filters offer the best choice to mitigate

these harmonics (Bekiarov et al, 2005).

Remote generation circuits have a higher efficiency when working with variable frequency.

Current harmonics caused by the nonlinear loads in these systems cause higher losses in the

generator and distortion in voltage, which cannot be mitigated by passive filters (Terrell, 1996).

In a typical wind power generation circuit, when it is connected to a power network, active filters

should be used to prevent harmonics from flowing into the power system (Bekiarov et al, 2005).

Commercial buildings, wherein many nonlinear loads such as computers, laser printers,

fluorescent lights, and other electronic equipment are used, can benefit from active filters to

compensate current harmonics, reactive current, neutral current, and load imbalance. Active

filters can also prevent the effect of the voltage harmonic and voltage sag and swell from the

network on loads (Bekiarov et al, 2005).

35

Figure4.3Variable-speed high-power induction motor for a locomotive

4.2.6 Remote Generations

4.2.7 Commercial Loads

4.2.8 Other Applications

Filters find application in audio systems and televisions etc. bandpass filters are used to select

frequency ranges corresponding to desired radio or television station channels. Similarly,

bandstop filters are used to reject undesirable signals that may contaminate the desirable signal.

For example, low-pass filters are used to eliminate undesirable hum in D.C power supplies

(Theraja, 1999).

Suppression of voltage flicker in arc furnace systems can be done by using high-frequency

power active filters. Another application is current harmonic compensation in magneto-

hydrodynamic power generation systems. A high precision DC magnet, which is used for

particle acceleration, needs a well regulated DC current input. This current can be achieved by

using a shunt power active filter (Bekiarov et al, 2005).

The following are other Apps of electric filters

To eliminate signal contamination such as noise in communication systems

To separate relevant from irrelevant frequency components

To detect signals in radios and TV’s

To demodulate signals

To bandlimit signals before sampling

To convert sampled signals into continuous-time signals

To improve the quality of audio equipment, e.g., loudspeakers

In time-division to frequency-division multiplex systems

In speech synthesis

In the equalization of transmission lines and cables

In the design of artificial cochlea’s (Chen, 2009)

36

The first chapter introduces readers to Active filters and their various kinds and the definition of

various terminologies involved with filters and the responses of each of the groups (Low-pass,

High-pass, Band-pass, Band-Reject) of filters.

The second chapter now dwells into the details of the classification (Butterworth, Chebyshev and

Inverse Chebyshev, Elliptic, or Bessel characteristics) of the response characteristics

The third chapter now looks back into the various groups of filters (discussed in chapter one)

with the circuits and details of their operation to see how filtering is really achieved.

37

CHAPTER VSUMMARY AND

CONCLUSION

5.1 SUMMARY

The last chapter no looks at various areas of electronics where the application (e.g. HVDC

systems, High power locomotives, remote generation and other applications like suppression of

noise in communication systems, detection of signals in radios and TV’s, separation of

frequencies and lots more) of active filters becomes necessary

As lengthy as this review has been up to this point, it only begins to scratch the surface of filter

technology. A quick perusal of any advanced filter textbook is sufficient to prove my point. The

mathematics involved with component selection and frequency response prediction is daunting

to say the least -- well beyond the scope of this review. It has been my intent here to present the

basic principles of active filters with as little mathematics as possible.

A program called “ SPICE circuit analysis program “ can be used to explore filter performance

and understand their various response , the benefit of such computer simulation software cannot

be understated, for the beginners (student) or for the working class (engineer).

38

5.2 CONCLUSION

1. Alexander C.K., Sadiku M.N.O, 2001, Fundamentals of Electric Circuits, McGraw-Hill,

pages 608-626

2. Bekiarov, S.B, Nasiri A. and Emadi, A., 2005, Uninterruptable power supplies, and

Active filters, CRC Press, New York, pages 73 – 79.

3. Chen W.K., Passive, Active and Digital Filters, 2009, CRC Press (Taylor & Francis

Group), Chicago (U.S.A) pages 11.1 – 11.32.

4. Floyd T.L., 2005, Electronic Devices- Conventional current version, 7th edition, Pearson

Education International, New Jersey. Pages 736-734

5. Kuphaldt T.R, Jason S., Dennis C., Bill S., 1996, Fundamental of Electrical Engineering

and Electronics, unpublished e-book.

6. Theraja B.L., Theraja A.K, 1999, A Textbook of Electrical Technology S. Chand and

Company Limited.

39

REFERENCES

7. Terrell D.L., 1996, Op Amps- Design, Application, and Troubleshooting, 2nd Edition,

Elsevier Science, Burlington, pages 212-238.

8. http://en.wikipedia.org/wiki/bandpass_filter

9. http://en.wikipedia.org/wiki/low-pass_filter

10. http://en.wikipedia.org/wiki/high-pass_filter

11. http://en.wikipedia.org/wiki/bandstop_filter

12. http://inst.eecs.berkeley.edu/~ee40/su10/labs/Active%20filter%20lab/

FilterLabPrototype.pdfhttp://www.answers.com/topic/electric-filter

13. http://en.wikipedia.org/wiki/cycloconverter

40