Chapter 14
Introduction to Frequency
Selective Circuits
14.1
14.2
14.3
14.4
14.5
Some preliminaries
Low-pass filters
High-pass filters
Bandpass filters
Bandreject filters
1
Overview

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.
2
Section 14.1
Some Preliminaries
1.
2.
Frequency response
Four types of filters
3
Frequency response plot


The steady-state response due to a sinusoidal
input Acost 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).
4
Four types of ideal filters
HPF
LPF
Cutoff freq.
BPF
BRF
5
Section 14.3
High-pass Filters (HPFs)
1.
2.
Frequency response of HPF
Effects of load
6
Series RC circuit

Zc = 1/( jC).

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)
7
Frequency response plot of a series RC circuit

By the equivalent circuit in the s-domain:
1
Vo ( s )
R
s


, where c 
.
H ( s) 
1
Vi ( s ) R  ( sC )
s  c
RC
j
 H ( j ) 
,
j  c


 H ( j ) 
 2  c2



1
 c .



(
j

)
90
tan

H ( jc ) 
1
 0.71
2
 ( jc )  45
8
Example: Superposition of sinusoids

1
2
Let x (t )  x1 (t )  x2 (t )  cos(0.1c t )  sin(3ct ).
 H ( j1 )  0.1084
j  c 
H ( j ) 
, 
j  c   1
 H ( j2 )  0.9518
Y1  10  0.1084 ,  y1 (t )  0.1 cos(0.1c t  84 );
Y2  1  90  0.9518 ,  y2 (t )  0.95 cos(3c t  72 );
(Tc = 2/c)
9
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)
10
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
Ks
R
 H ( s) 
, where c  , K 
 1.
1  ( R RL )
s  Kc
L
11
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.
(RL = R)
Problem: the response varies with the load.
Solution: use active filters (Ch15).
12
Section 14.4
Bandpass Filters (BPFs)
1.
2.
Frequency response of BPF
Relation with the poles, zeros
13
Series RLC circuit

Zc = 1/( jC), ZL = j L.

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)
14
Frequency response plot of a series RLC circuit

By the equivalent circuit in the s-domain:
s
R
R
, where   , 0 
H ( s) 
 2
1
2
sL  ( sC )  R s  s  0
L
1
.
LC


 H ( j ) 
2
2 2
2







(
)

0

 


1 
 ( j )  90  tan   2   2 .
 0


 ( j 0 )  0 
15
Calculations of parameters

Center (resonance) frequency is derived by:
1
jL 
 0,  0 
jC

1
.
LC
Cutoff frequencies are derived by:

1
 
2
H ( j ) 
,  c1,c 2     0  .
2
2
2
c1  c 2
R






;





.
 Bandwidth c 2 c1
0
c1 c 2
2
L
2

Quality factor (~ inversed relative bandwidth) is:
LC
0
.
Q


R
16
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)
17
Frequency response visualized by poles, zeros

The poles and zeros of HPF and BPF are:
s
,  p  c , z  0.
H HPF ( s ) 
s  c
2
2






4
s
0
H BPF ( s )  2
,

p

, z  0.
2
s  s  0
2
Im
-c
Im
Re
-/2
Re
18
Practical Perspective
Pushbutton Telephone
Circuits
19
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 3003000 Hz) or ringing bell tones (20 Hz)?
(wikipedia.org)
(jouielovesyou.wordpress.com)
20
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.
(sohilp.blogspot.com)

BPFs: (1) identify whether both freq. groups are
present, (2) select among possible tones.
21
BPF design for the low-frequency group

The transmission spectrum of a series RLC is:
H ( j ) 


2


2 2
0
R
where   , 0 
L


 (  ) 2
,
1
.
LC
For the low-freq. group, c1 = 2(697) = 4379 rad/s,
c2 = 2(941) = 5912 rad/s,    c2–c1 = 1533 rad/s.
  R L ,  L  R   (600 ) 1533  0.39 H.
0  c1c 2  1
LC ,  C  1 ( Lc1c 2 )  100 nF.
22
BPF performance (1)

The transmission of the 4 frequencies in the lowfrequency group are imperfect (<1):
H 697Hz  H 941Hz
1

 0.707,
2
H 770Hz  H 852Hz 


2


2 2
0
 (  ) 2
 0.948.
23
BPF performance (2)

The transmission of the high-frequency group
tones and the 20 Hz ringing tone are imperfect
(>0):
24
BPF design for the high-frequency group

The transmission spectrum of a series RLC is:
H ( j ) 



R
,   , 0 
2
L
 2  02   (  )2
1
.
LC
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:
  R L ,  L  R   (600 ) 2664  0.26 H.
0  c1c 2  1
LC ,  C  1 ( Lc1c 2 )  57 nF.
25
BPF performance

The transmission of the low-frequency group
tones and the 20 Hz ringing tone are imperfect
(>0):
26