1

Chapter 14 Introduction to Frequency Selective Circuits

14.1 Some preliminaries14.2 Low-pass filters14.3 High-pass filters14.4 Bandpass filters14.5 Bandreject filters

2

We have seen that the response of a circuit depends on the types of elements, the way the elements are connected, and the impedance of the elements that varies with frequency.

In this chapter, we analyze the effect of varying source frequency on circuit voltages and currents. In particular, the circuits made of passive elements (R, L, C) that pass only a finite range of input frequencies.

Overview

3

Section 14.1 Some Preliminaries

1. Frequency response2. Four types of filters

4

Frequency response plot

The steady-state response due to a sinusoidal input Acost is determined by sampling the transfer function H(s) along the imaginary axis, i.e. H(j).

Since H(j)C, a frequency response plot consists of two parts: (1) magnitude plot |H(j)|, (2) phase angle plot (j).

5

Four types of ideal filters

LPFCutoff freq.

HPF

BPF BRF

6

Section 14.3 High-pass Filters (HPFs)

1. Frequency response of HPF2. Effects of load

7

Series RC circuit

Zc =1/( jC).

Zc as 0, vo 0 (no current flows through R).

Zc 0 as , vo vi

(short circuit).

The input voltage can pass through the circuit if the source frequency is high enough.

(0)

()

(0)

(vi )

8

Frequency response plot of a series RC circuit

By the equivalent circuit in the s-domain:

.1 where,)()(

)()( 1 RCss

sCRR

sVsVsH c

ci

o

.tan90)(

)(

,)(

1

22

c

c

c

j

jH

jjjH

71.02

1)( cjH

45)( cj

9

Example: Superposition of sinusoids

Let ).3sin()1.0cos()()()( 21 tttxtxtx cc

);723cos(95.0)(,1895.0901

);841.0cos(1.0)(,8410.001

1895.0)(

8410.0)(,

1)(

2

1

2

1

tty

tty

jH

jHj

jjH

c

c

c

c

Y

Y

2

1

(Tc = 2/c )

1 2

10

Example: Filtering in the spatial-frequency domain

An image is a function of (x,y), which can be decomposed into many “spatial-frequency” components ( fx , fy ). HPF enhances the contrast.

(wikipedia.org)

11

Example 14.4: RL circuit with load (1)

An RL circuit can be an HPF if vo

vL . (E.g. 14.3)

When a load resistor RL is in parallel with the output inductor L:

.1)(1

1, where,)(

L

cc RR

KLR

KsKssH

12

Example 14.4: RL circuit with load (2)

The effects of RL are: (1) reducing the passband magnitude by a factor of K, (2) lowering the cutoff frequency by the same factor of K.

Problem: the response varies with the load. Solution: use active filters (Ch15).

(RL =R)

13

Section 14.4 Bandpass Filters (BPFs)

1. Frequency response of BPF2. Relation with the poles, zeros

14

Series RLC circuit

Zc =1/( jC), ZL = jL.

Zc as 0, vo 0.

ZL as , vo 0.

At some frequency 0 (0,), Zc +ZL =0, vo =vi .

The input voltage can pass through the circuit if the source frequency is near 0 .

(0)

()

(0)

(0)

15

Frequency response plot of a series RLC circuit

By the equivalent circuit in the s-domain:

.1 , where,)(

)( 020

21 LCLR

sss

RsCsLRsH

.tan90)(

)()(

220

1

2220

2

j

jH

0)( 0 j

16

Calculations of parameters

Center (resonance) frequency is derived by:

.1 ,010 LCCj

Lj

Cutoff frequencies are derived by:

.22

,2

1)( 20

2

2,1

ccjH

Bandwidth .2

; 2121012

cccccc L

R

Quality factor (~ inversed relative bandwidth) is:

.0

RCL

Q

17

Example: White-noise rejection

If a signal of specific frequency f1 is buried in strong “white” noise, a BPF centered at f1 can be used to extract the signal.

(www.chem.vt.edu)

18

Frequency response visualized by poles, zeros

The poles and zeros of HPF and BPF are:

.0 , ,)(

zps

ssH cc

HPF

.0 ,2

4 ,)(

20

2

20

2

zpssssHBPF

Re

Im

-cRe

Im

-/2

19

Practical Perspective Pushbutton Telephone Circuits

20

Application of BPFs

Q: How to tell which button was pushed? How to tell the difference between the button tones and the normal vocal sounds (both are within 300- 3000 Hz) or ringing bell tones (20 Hz)?

(wikipedia.org)(jouielovesyou.wordpress.com)

21

Dual-tone-multiple-frequency (DTMF)

Each bottom is encoded with a unique pair of sinusoidal tones (fL , fH ), where the relative timing and amplitudes must be close enough.

BPFs: (1) identify whether both freq. groups are present, (2) select among possible tones.

(sohilp.blogspot.com)

22

BPF design for the low-frequency group

The transmission spectrum of a series RLC is:

,

)()(

2220

2

jH

.1 , where 0 LCLR

For the low-freq. group, c1 =2(697) =4379 rad/s, c2 =2(941) =5912 rad/s, c2 –c1 =1533 rad/s.

nF. 100)(1 ,1

. H39.01533) 600( ,

21210

cccc LCLC

RLLR

23

BPF performance (1)

The transmission of the 4 frequencies in the low- frequency group are imperfect (<1):

,707.02

1941Hz697Hz HH

.948.0

)( 2220

2852Hz770Hz

HH

24

BPF performance (2)

The transmission of the high-frequency group tones and the 20 Hz ringing tone are imperfect (>0):

25

BPF design for the high-frequency group

The transmission spectrum of a series RLC is:

,

)()(

2220

2

jH .1 , 0 LCLR

For the high-freq. group, c1 =2(1209) =7596 rad/s, c2 =2(1633) =10260 rad/s, c2 –c1 = 2664 rad/s.

Both L and C values are smaller:

nF. 57)(1 ,1

. H26.02664) 600( ,

21210

cccc LCLC

RLLR

26

BPF performance

The transmission of the low-frequency group tones and the 20 Hz ringing tone are imperfect (>0):