An electric filter is usually a frequency-selective network that passes a specified band of frequencies and blocks or attenuates signals of frequencies outside this band.

Depending on the types of elements used in their construction, filters may be passive or active. A passive filter is built with passive components such as resistors, capacitors and inductors. Active filters, on the other hand, make use of transistors or op-amps (providing voltage amplification, and signal isolation or buffering) in addition to resistors and capacitors. The type of the elements used dictates the operating frequency range of the filter.

According to the operating frequency range, the filters may be classified as audio-frequency (AF) or radio-frequency (RF) filters.

Filters may also be classified as (i) low-pass, (ii) high-pass (iii) band-pass (iv) band-stop. The filter circuit may be so designed that some frequencies are passed from the input to the output of the filter with very little attenuation while others are greatly attenuated.

Figure 1.1 shown the frequency responses of the four types (mentioned above) of filters. These are ideal responses and cannot be achieved in actual practice.

A filters that provides a constant output from dc upto a cut-off frequency and then passes no signal above that frequency is called an ideal low-pass filter. The ideal response of a low-pass filter is illustrate in fig.1.1 (a) The voltage gain (the ration of output voltage and input voltage i.e. is constant over a frequency range from zero to cut off frequency, . The output of any signal having a frequency exceeding will be attenuated i.e there will be no output voltage for frequencies exceeding cut-off frequency . Hence output will be available faithfully from 0 to with constant gain, and is 0 from onward. The frequencies between 0 and are, therefore, called the passband frequencies, while the range of frequencies, those betyond , that are attenuated include the stopband frequencies. The bandwidth (BW) is, therefore .

Fig. (a) Low-pass Filter

Fig. (b) High-pass Filter

Fig. (c) Band-pass Filter

Fig. (d) Band-stop (or-Reject) Filter

Fig. 1.1 Ideal Response of Different Types of Filters

A filter that provides or passes signals above a cut-off frequency is a high-pass filter, as idealized in fig.1.1 (b) frequency , called the cut-off frequency, and above this frequency, the gain is constant, as illustrated in fig. 1.1(b). Thus signal of any frequency beyond is faithfully reproduced with a constant gain, and frequencies from 0 to will be attenuated.

When the filter circuit passes signals that are above one cut-off frequency and below a second cut-off frequency, it is called a band-pass filter, as idealized Fig.1.1 (c). Thus a band-pass filter has a passband between two cut-off frequencies and ,where and two stop-bands: and and are lower and higher cut-off frequencies respectively.

The band-stop or band-reject filter performs exactly opposite to the band-pass i.e. it has a bandstop between two cut-off frequencies and and two passbands : and . The ideal response of a band-stop filter is illustrates in Fig.1.1 (d). This is also called a band-elimation or notch filter.

The filter discussed above have ideal characteristics and a sharp cut-off but unfortunately, filter response is not practical because linear networks cannot produce the discontinuities. However, it is possible to obtain a practical response that appoximates the ideal response by using special design techniques, as well as precision component values and high-speed op-amps.

### Passive Filters

The simplest approach to build a filter is with passive components (resistors, capacitors, and inductors). In the R-F range it works quite well but with the lower frequencies, inductors create problems. AF inductors are physically larger and heavier, and therefore expensive. For lower frequencies the inductance is to be increased which needs more turns of wire. It adds to the series resistance which degrades the inductor’s performance.

Input and output impedances of passive filters are both a problem, specially below RF. The input impdance is low, that loades the source, and it varies with the frequency. The output impedance is usually relatively high, which restricts the load impedance that the passive filter can drive. There is no isolation between the load impedance and the passive filter. Thus the load will have to be considered as a component of the filter and will have to be taken into consideration while determining filters response or design. Any change in load impedance may significantly alter one or more of the filter response characteristics.

### Low-pass filters

Low-pass filters are of many types such as R-C, R-L inverted L-types, T-types and -type

#### 1. Low-pass R-C Filters Circuit

Low-pass R-C filter circuit is shown fig 1.2 . In this circuit output voltage is taken across the capacitor. Re-sistance offers fixed opposition. Since the reactance offered by the capacitors C falls with the increase in frequency, low frequency signal develops across the capacitor but signal of frequency above cut-off frequency develop negligible voltage across capacitor C, At zero frequency, the capacitor acts as an open-circuit and output in same as input. However, with the increases and so the out-put voltage. At infinity frequency, the capacitive reactance of the circuit will be zero, and therefore, output voltage will also be zero. Since it passes low-frequency signals and blocks the high-frequency signals, it is called the low-pass circuit.

Fig.1.2 Low pass R-C

The current through the circuit is given as

Output voltage,

So gain

From the above equation it is obvious that for a circuit, the gain A depends upon the frequency of the input signal.

Fig. 1.3 Frequency Response Curve For Low-Pass R-C Circuit

The frequency response curve for the circuit is given inFig.1.3 From frequency response curve it is obvious that all frequencies below cut-off frequency are passed while those above are attenuated, At cut-off frequency , the phase angle is and output power is half of the input power, The output voltage is 0.707 times the maximum value of voltage . The cut-off frequency is given as

Substituting

Gain

Now magnitude of A and phase angle are given as

and phase angle

As and

When

is also known as high 3 dB frequency

also implies that

or

So is the frequency at which capacitive reactance Xc is equal to the resistance of the circuit,

#### 2. Low-Pass R-L Filter Circuit

Such as filter circuit is illustrated in Fig.2.4 in which the output voltage is taken across R. Since the reactance offered by the inductor L increases with the increase in frequency so it allows the frequency so it allows the frequencies up to cut-off frequency to pass through the coil without much opposition but offers high reactance to frequencies above cut-off frequency. The output voltage developed across resistor R is given by the equation

Fig. 1.5

and cut-off frequency,

Cut-off frequency occurs when (i) output voltage times input voltage (ii) (iii) and (iv) impedance phase angle is (n0t- as in low: pass R-C filter).

#### 3. Inverted L-Type Filter Circuit

Such as circuit employs a choke and a capacitor and is illustrated in Fig.2.5 Choke L blocks higher frequencies as it offers high reactance to high frequencies and capacitor C short them to ground because capacitor offers negligible reactance to the high frequencies. Thus only low frequencies below cut-off frequencies are allowed to pass through without significant attenuation. The output voltage is taken across the capacitor

Fig.1.5

Fig. 1.6

#### T-Type Filter Circuit

Such a filter circuit consists of a second choke connected on the output side in order to improve the filtering action. Such a filter circuit is shown in Fig.1.6.

#### Type Filter Circuit

Such a filter circuit is shown in Fig.2.7. The capacitors is added in the circuit to improve the filtering action by grounding higher frequencies

Fig.1.7

Note: Choke is always connected in series between the input and the output and the capacitor/capacitors is/are grounded in parallel. The output voltage is taken across the capacitor .

**Example Problem**:

A simple low-pass R-L filter having a cut-off frequency of 2.0 kHz ia connected to a source of supply of constant voltage of 7.5 V but variable frequency. Determine (a) the value aof L if R=5,000 (b) output voltage and its decible level when the frequency is (i) (ii)

Solution: Cut-off frequency, =2.0 kHz=2.000 Hz

(a) From L=

= 0.398 mH Ans.

(b) (i) When f=

Output voltage,

V Ans.

Decibel decrease=

=3dB Ans.

(ii) When

Output voltage,

=1.47 Ans

Decibel decrease

=-14.15 dB Ans.

#### HIgh-Pass Filters

High pass filter are aiso of R-C,R-L, inverted L,T-and types as illustrated in Figs. 1.8,1.10,1.11.1.12 and 1.13 respectively.

#### 1. High-Pass R-C Filter Circuit

High-pass R-C filter circuit is shown in Fig.1.8. In this, circuit output voltage is taken across the resistor. This reactance of the capacitor C is given as

or

i.e Reactance of the circuit capacitor decreases with the increase in frequency, Thus at low frequencies the capacitor C offers considerable reactance and so blocks them, but at higher frequencies it offers little reactance and allows them to pass through it. The output voltage is developed across resistor R. Since the circuit blocks and attenuates low frequencies but allows high frequency signals to pass through it, the circuit is called high-pass R-C circuit.

Fig. 1.8 High-Pass R-C Circuit

With the increase in frequency, the reactance of the capacitors decreases and,therefore. the output and gain increase. At very high frequencies, the capacitive reactance becomes very small, so the output becomes almost equal to input and gain becomes equal to unity. Since this circuit attenuates the low-frequency signals and allows transmission of high-frequency signals with little or no attenuation, it is called a high-pass circuit.

The current through the circuit is given as

and output voltage, =Voltage drop across resistor R

So again

From above equation it is obvious that for a given circuit, the gain A depends upon the frequency of the input signal.

Fig. 1.9 Frequency Response Curve For High-pass R-L Circuit

The frequency response curve for the circuit is given in Fig.1.9. From frequency response curve it is obvious that all frequency above cut-off frequency , are passed while those below are attenuated. At cut-off frequency , the phase angleis and output power is half of the input power. The output voltage is 0.707 times the maximum value of voltage . The cut-off frequency is given as

Substituting

Gain

|A|=

and phase angle

As ;

When ;

is also known as low 3 dB frequency.

also implies that

or

So is the frequency at which capacitive reactance is equal to the resistance of the circuit.

The maximum possible value of gain (unity)is approached asymptotically at high frequencies’

Fig. 1.10

Fig. 1.11

Fig. 1.12

Fig. 1.13

#### 2. High-Pass R-L Filter Circuit

In such a filter resistance R offers fixed opposition. Since the reactance offered by the inductor L increases with the increase in frequency, so high frequencies signal develops across L but signal frequency below cut=off frequency develops negligible voltage across inductor L. High frequency output voltage developed across inductor is given by the equation

and

The frequency response curve for high pass R-L filters are the same as for high pass R-C filters (Fig, 1.9).

#### Inverted L-Type Filter Circuit

At low frequencies, reactance offered by capacitors is large but reactance offered by inductor L is small so output voltage developed across inductor L is small. But at higher frequencies becomes much large and is reduced so considerable output voltage is developed across L. Thus lower frequencies are attenuated and higher frequencies are passed.

#### T-Type Filter Circuit

It employs an additional capacitor, as illustrated in Fig. 1.12 to improve the filtering action.

#### -Type Filter Circuit

Such a filter circuit employs one capacitor and two inductors as illustrated in Fig. 1.13. The inductors shunt out the lower frequencies.

Note: In high-pass filters the capacitors are connected in series between the input and output and the inductor/inductors is/are grounded in parallel. The output voltage is developed across the inductor.

**Example Problem**:

A simple high-pass R-C filter having a cut-off frequency of 5 kHz is connected to a source of supply of constant voltage of 10 v and variable frequency. Determine (a) the capacitance C if R=2,000 (b) output voltage and its decibel level when the supply frequency is (i) cut-off frequency (ii) zero (iii) 50 kHz.

Solution: Cut-off frequency, =5kHz = 5,000 Hz

(a) From Eq. (2.13)

Ans.

(b) (i) When supply frequency

Output voltage,

=7.07

Decibel decrease

= -3 dB Ans.

(ii) When supply frequency f=0

The capacitor C acts as an open -circuit so no voltage is developed across the resistor R. Hence output voltage is zero.

(iii) When supply frequency f=50 kHz

Output voltage,

Ans.

Decibel decrease

= -0.043 dB Ans.

#### R-C Band-Pass Filter

Sometime it is desirable to pass a certain band of frequencies and to attenuate other frequencies on both sides of this passband. The pass-band is called the bandwidth of the filter. It can be achived by cascading low-pass filter capable of transmitting all frequencies upto to a high pass filter capable of transmitting all frequencies higher than , with . The system devloped will be capable of transmitting frequencies between and and attenuate all other frequencies below and above . The passband of this filters is given by the band of frequencies lying between and . The values are given by the equation.

and

Fig. 1.14 (a)

The ratio of output and input voltages are given by the equation

…from

…from

The R-C band-pass filter and its response curve are shown in Fig. 1.14 (a) and (b) respectively. This arrangement would give rise to the desired characteristic but is not very economical.

Fig. 1.14 (b)

#### R-C Band-Stop (or Band-Elimination) Filter

R-C band-stop or band-elimination filter is similar to a band-pass filter in which the shunt arm is replaced by the shunt type. As stated earlier, such a filter attenuates the electric signal over a specified frequency range (say from to ) and below above has all pass-band region. The This figs. 1.15 (a) and (b) respectively. The stopband is represented by the group of frequencies that lie between and where response is below -60 dB. The output voltage and cut-off frequencies are given by the expressions

from to

from to

(a)

(b)

Fig.1.15

Note: In practice several low-[ass R-C filter circuits are cascaded to several high-pass R-C filter circuits as R-C filters can be produced in the from of LSIC (large scale integrated circuits) and provide almost vertical roll off* and rises. R-L circuits are hardly employed for this purpose.

#### Band-Pass and Band-Stop Resonant Filter Circuits

Frequency resonant circuits (both series and parallel resonant circuits) are employed in electronic systems for developing band-pass and band-stop filters because of their voltage of current magnification characteristics at resonant frequency. At resonant frequency (i) the impedance offered by the series R-L-C circuit, being equal to R, is minimum and (ii) the current drawn, being equal to R, is minimum and (ii) the current drawn, being equal to V/R, is maximum.

L-C parallel circuit being equal to , is maximum and (ii) the current drawn, being equal to , is minimum.

#### Series Resonant Band-pass Filter Circuit

Series resonant band-pass filter circuit as illustrated in Fig. 1.16 (a), essentially consists of a series resonant R-L-C circuits shunted by an output resistance . Output voltage, obviously, will be maximum at resonant frequency since series resonant impedance is equal to R, which is negligible in comparison to output risistance . The output voltage will reduce to 0.707 times of maximum voltage at cut-off frequencies and as illustrated in the response curve shown in Fig. 1.16(b). The phase angle is positive for frequencies exceeding and negative for frequencies below . The equations for output voltage and Q-factor at resonance, and bandwidth are give

(a)

Fig. 1.16

(b)

Output voltage at resonance,

Q-factor at resonance.

Bandwidth,

#### Parallel Resonant Bandpass Filter Circuit

Parallel resonant bandpass filter, as illustrated in Fig. 1.17 , consist of a series resistance shunted by a parallel resonant circuit. The output voltage is taken across parallel resonant circuit and obviously, will be maximum at resonant frequency since at resonance the parallel circuit offers maximum impedance equal to L/CR. The amplitude-response curve for this filter is similar to that for a series-resonant bandpass filter discussed above. The expressions for output voltage and bandwidth are given below.

Fig.1.17

Output voltage at resonance,

Bandwidth

#### Series Resonant Band-stop Filter

Series resonant band-stop filter, as illustrated in Fig.1.18 (a), essentially consists of a series resistance shunted by series resonant R-L-C circuit. The output voltage is taken across the series resonant circuit and, obviously minimum at resonate

(a)

(b)

Fig. 1.18

frequency since at resonance equal to R. the equations for output voltage at resonance, Q-factor at resonance, bandwidth and output at any other frequency are given below:

Output voltage at resonance,

Q-factor at resonance,

Bandwidth,

Output voltage at any other frequency f,

Such filters are commonly employed for rejecting a particular frequency such as 50 Hz hum produced by inductor or transformer in recording equipment.

#### Parallel Resonant Band-stop Filter

Parallel resonant band-stop filter, as shown in Fig.1.19, essentially consist of a parallel resonant circuit shunted by an output resistance . At resonance or at nearby frequencies the parallel circuit offers extremely high impedance in comparision to output resistance and, therefore output voltage developed across is negligibly small in comparision to that developed across resonant circuit. This various relations are given below:

Fig. 1,19

At resonant frequency,

Output voltage, where . R

Bandwidth

Output voltage at any frequency f,

where

**Example Problem**:

A series resonant bandpass filter consider consists of a series resonant circuit containing a coil of resistance 5 [/latex size="3"]\Omega[/latex] an inductance 3.52 H and a capacitor of 0.0018 shunted by a resistor of 2.5k . If the applied signal voltage is 7.5 v. determine (a) resonant frequency (b) Q-factor at resonance (c) resonance (c) edge frequencies and (d) passband width and (e) output voltage at resonant and cut-off frequencies.

Solution: Output resistance

L=3.52 H and

(a) Resonant frequency,

= 2,ooo Hz or 2 kHz. Ans.

(b) Q-factor at resonance.

=17.66 Ans.

(c)Passband width .

=113 Hz or 0.113 kHz. Ans.

(d)Side frequencies

=1,943.5 Hz or 1.9435 kHz. Ans.

=2,056.5 Hz or 2,0565 kHz. Ans.

(e) Output voltage at resonant frequency.

=7.485 V Ans.

Output voltages at cut-off frequencies.

=0.707 7.485=5.29 V. Ans.

Output voltage at cut-off frequency can also be determined as

Ans. same as before

Output voltage at cut-off frequency ,

Ans.

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