- 211 -

Chapter 13

Filters

13.0 Introduction Filter is the circuit that capable of passing signal from input to output that has frequency within a specified band and attenuating all others outside the band. This is the property of selectivity. They are four basic types of filters. They are low-pass, high-pass, band-pass, and band-stop.

The basic filter is achieved with various combinations of resistors, capacitors, and sometimes inductors. These are named as passive filters. Active filters besides using passive element, it also uses active element such as transistors or operational amplifiers to provide desired voltage gains or impedance characteristics.

Each type of filter response can be tailored by circuit component values that

have Butterworth, Chebyshev, or Bessel characteristics. Each of these characteristics is identified by the shape of its response curve and each has an advantage in certain application.

In this chapter, we shall explore the basic concepts of designing passive and

active filters namely the low-pass, high-pass, band-pass, and band-stop filters. The characteristics of each filter type in terms of it bandwidth, critical

frequency, and gain would be discussed in details. We shall not be discussing the methods to design filter that has Butterworth, Chebyshev, or Bessel characteristics. This part of the lecture shall be left to be discussed in the analogue circuit design subject.

13.1 Ideal Filter Ideally, filter should have the characteristics as shown in Fig. 13.1. In practice, such characteristics are not possible to be achieved. In practice, the attenuation of the signal after the critical frequency is either exponentially increased or decreased. It is not abruptly decreased or increased as shown in Fig. 13.1.

13 Filter

- 212 -

Figure 13.1: Ideal filter characteristic – low-pass, high-pass, band-pass, and band-stop

13.2 Passive Filters Four basic passive filters namely the low-pass, high-pass, band-pass, and band-stop are be discussed in this section. 13.2.1 Low-Pass Filter A basic passive low-pass circuit is shown in Fig. 13.2.

Figure 13.2: Passive low pass filter

The transfer function H(s) of the filter is

13 Filter

- 213 -

H(s) = C/1R

C/1

V

V

i

0

ω+ω=

j

j (13.1)

The magnitude of the function shall be 2)RC(1

1

ω+. The critical frequency fC of

the filter shall be

fC = RC2

1

π (13.2)

It can determined either by setting function by setting ωRC = 1 and the – 3dB

frequency is determined by setting 2)RC(1

1

ω+=

2

1 .

The phase of the filter shall be φ = - tan-1(ωRC).

Based on equation (13.1), as frequency increases, the function H(s) is

approaching zero. As frequency decreases, the function H(s) is approaching one. The characteristic of the filter is shown in Fig. 13.3.

Figure 13.3: The characteristics of a passive low pass filter

13.2.2 High-Pass Filter A basic passive high-pass filter circuit is shown in Fig. 13.4.

13 Filter

- 214 -

Figure 13.4: Passive high-pass filter

The transfer function of the filter H(s) is

H(s) = C/1R

R

V

V

i

0

ω+=

j (13.3)

The magnitude of the function shall be 2)RC(1

RC

ω+

ω . The critical frequency fC of

the filter shall be

fC = RC2

1

π (13.4)

It can determined either by setting function ωRC = 1 and the -3dB frequency can

be determined by setting 2)RC(1

RC

ω+

ω = 2

1 .

The phase of the filter shall be φ = 900 - tan-1(ωRC).

Based on equation (13.3), as frequency increases, the function H(s) is

approaching one. As frequency decreases, the function H(s) approaches zero. The characteristic of the filter is shown in Fig. 13.5.

13 Filter

- 215 -

Figure 13.5: The characteristics of a passive high pass filter

13.2.3 Passive Band-Pass Filter A basic passive band-pass filter circuit is shown in Fig. 13.6.

Figure 13.6: Passive band-pass filter

The transfer function of the filter H(s) is

H(s) = )C/1L(R

R

V

V

i

0

ω−ω+=

j (13.5)

The magnitude of the function shall be22 )RC/1L(R

R

ω−ω+. The resonant

frequency fo of the filter shall be

fo = LC2

1

π (13.6)

13 Filter

- 216 -

It can be determined by setting the imaginary part of the impedance to zero. The –

3dB frequencies are determined by solving expression 22 )RC/1L(R

R

ω−ω+=

2

1 .

Based on equation (13.5), the characteristic of the filter is shown in Fig. 13.7.

Figure 13.7: The characteristics of a passive band pass filter

13.2.4 Passive Band-Stop Filter A basic passive band-stop filter circuit is shown in Fig. 13.8.

Figure 13.8: Passive band pass filter The transfer function of the filter H(s) is

H(s) = )C/1L(R

)C/1L(

V

V

i

0

ω−ω+ω−ω=

j

j (13.7)

13 Filter

- 217 -

The magnitude of the function shall be22 )RC/1L(R

)RC/1L(

ω−ω+

ω−ω . The resonant

frequency fo of the filter shall be

fo = LC2

1

π (13.8)

It can be determined by setting the imaginary part of the impedance to zero. The –

3dB frequencies are determined by solving 22 )RC/1L(R

)RC/1L(

ω−ω+ω−ω =

2

1 .

Based on equation (13.7), the characteristic of the filter is shown in Fig. 13.9.

Figure 13.9: The characteristics of a passive band-pass filter

13.3 Active Filter Four basic active filters employing operational amplifier namely the low pass, high pass, band pass, and band reject are be discussed in this section. 13.3.1 Active First-Order Low-Pass Filter A basic first-order low-pass filter is shown in Fig. 13.10.

13 Filter

- 218 -

Figure 13.10: An active first-order low pass filter

The transfer function, which the voltage gain function H(s) is

H(s) = ff

f

RCj1

1

R

R

i ω+⋅− (13.9)

The magnitude of the gain function is 2

i )RC(1

1

R

R

ff

f

ω+⋅ . The critical frequency is

ff CR

1C =ω , while the phase is – tan-1(ωCfRf).

13.3.2 Active First-Order High Pass Filter A basic first order low pass filter is shown in Fig. 13.11. The transfer function, which the voltage gain function H(s) is

H(s) = ff

fff

j

j

RC1

RC

R

R

i ω+ω

⋅− (13.10)

The magnitude of the gain function is 2

i )RC(1

RC

R

R

ff

fff

ω+

ω⋅ . The critical frequency is

ff CR

1C =ω , while the phase φ is φ = 900 – tan-1(ωCfRf).

13 Filter

- 219 -

Figure 13.11: An active first-order high pass filter

13.3.3 Active Band-Pass Filter The basic circuit of an active band-pass filter is shown in Fig. 13.12. The circuit comprises of a low-pass filter stage, a high-pass filter stage, and an inverting stage that provides voltage gain.

Figure 13.12: An active band-pass filter

13 Filter

- 220 -

13.3.4 Active Band-Stop Filter The basic circuit of an active band-pass filter is shown in Fig. 13.13. The circuit comprises of a low-pass filter stage, a high-pass filter stage, and summing amplifier stage that provides voltage gain and band-reject.

Figure 13.13: An active band-stop filter

13 Filter

- 221 -

Exercises 13.1. Show that circuit below is a low-pass filter. Calculate the corner frequency

fc if L = 2mH, C= 10 µF, and R = 10 kΩ.

13.2. Find the transfer function Vo/Vs of the circuit in the figure. Show that the

circuit is a low-pass filter.

13.3. Determine the type of filter for the circuit shown in the filter and find its

corner frequency.

13.4. Design an RL low-pass filter that uses a 50 mH coil and has a cutoff

frequency of 5.0 kHz. 13.5. Design a series RLC type band-pass filter with cutoff frequencies of 10 kHz

and 11 kHz. Assuming that C = 80.0 pF, find R, L, and Q. 13.6. Find the transfer function for each of the active filters.

13 Filter

- 222 -

13.7. Obtain the transfer function of the active filter in the figure and determine

the type.

13.8. A high-pass filter is shown in the figure, show that the transfer function is

H(ω) = RC1

RC

R

R1

i ω+ω

+

j

jf .

Bibliography 1. Charles K. Alexander and Matthew N.O Sadiku, “Electric Circuits”, 2nd

edition, McGraw-Hill 2004.