Download SIMPLE LOW PASS AND HIGH PASS FILTER

Summary of the lecture notes on simple frequency selective circuits
EE-201 (Hadi Saadat)
Low pass Filters are used to pass low-frequency sine waves and attenuate high
frequency sine waves. The cutoff frequency c is used to distinguish the passband (c)
from the stopband (c ). An elementary example of two passive lowpass filter is
given below.
j L
R
j
Vi
1
C
Vi
Vo
V0
1

Vi 1  jRC
R
V0

Vi
(1)
| Vo |
1

| Vi | 1  RC 2
Vo
1
1  j
L
R
| Vo |
1

1/ 2
2 1/ 2
| Vi | 
 L 
1     
  R  
L
q   tan 1 RC
(3)
q   tan 1 
R
At cutoff frequency Gain  1 / 2 . Substituti ng in (2) result in
1
R
c 
(4)
c 
RC
L
From (3), we see that the phase angle at cutoff frequency is  45 
V
The ratio o is shown by H ( j ) , and is called the frequency response transfer function.
Vi
The gain versus frequency, and the phase angle versus frequency known as the frequency
response is as shown.


(2)
1
0
0.8
0.707
-20
q°
|Vo/Vi|
0.6
-40
-45
0.4
-60
0.2
-80
0
0
c
2
4
6
, rad/s
8
0
1
c
2
4
6
, rad/s
8
High pass Filters are used to stop low-frequency sine waves and pass the high frequency sine waves. The
cutoff frequency c is used to distinguish the stopband (c) from the passband (c ). An elementary
example of two passive highpass filter is given below.
j
1
C
R
Vi
R
V0
RC

Vi RC  j1
Vo
j L Vo
Vi
V0

Vi
(5)
| Vo |
RC

| Vi | RC 2  1 1 / 2


L
R
L
 j1
R

L
R
| Vo |

1/ 2
2
| Vi | 

 L
    1

 R 
1
R
q  tan 1
(7)
q  tan 1
RC
L
At cut off frequency Gain  1 / 2 . Substituti ng in (6) result in
1
R
c 
c 
RC
L
From (7), we see that the phase angle at cutoff frequency is 45 
The gain versus frequency, and the phase angle versus frequency known as the frequency
response is as shown.


(6)
2
Bandpass Filters
(a) The Parallel RLC Resonance
IL
IC
j
Ii
1
C
j L
IR
R
Vo 
Vo
Ii
1
1
 j ( C 
)
R
L
A circuit is in resonance when the voltage and current at the input terminals are in phase.
1
1
) . At resonance Y is purely conductive and
The circuit admittance is Y   j ( C 
R
L
1
1
C
 0 , thus  o 
. The circuit admittance is minimum or the circuit
L
LC
impedance at resonance, given by Z ( o )  R , is maximum. Thus, the output voltage at
resonance is maximum and is given by Vo ( o )  RI i
100
|Vo| = Ii/|Y|, V
80
V0(0) = RIi
70.71
60
40
20
0
0
1
100 o
2 300
200
400
, R/s
500
The frequencies 1 and  2 at which the output power drops to one half of its values at
the resonant frequency are called the half-power frequencies. At these frequencies also
known as cutoff frequencies or corner frequencies, the output voltage is
| Vo ( c | 0.707 | Vo ( o ) | . This circuit which passes all the frequencies within a band of
frequencies ( 1     2 ) is called a bandpass filter. This range of frequency is known
as the circuit bandwidth.
   2  1
The half-power frequencies are obtained from
Ii
1
| Vo ( c ) |

RI i
1
2
2
2
 1  
1  2
     C 
 
 L  
 R  
Solving for  we obtain
2
2
1
1
1
1
 1 
 1 
, and  2 
1  
 
 
 
 
2 RC
LC
2 RC
LC
 2 RC 
 2 RC 
3
1
, or the circuit bandwidth is
RC
1

RC
The cutoff frequencies can be written in terms of  0 and  as follow:
From the above, we obtain,  2  1 



     o 2 , and  2       o 2
2
2
2
2
This shows that  o is the geometric mean of 1 and  2 , i.e.,
1  

2
2
 o  1 2
Notice that  is inversely proportional to R, i.e., smaller R results in a larger bandwidth.
1
The resonant frequency (  o 
) is a function of L and C. Therefore, by adjusting L
LC
and C a desired resonant frequency is obtained, whereas by adjusting R, the bandwidth
and the height of the response curve is adjusted. The sharpness of the resonance is
measured quantitatively by the quality factor Q. This is defined as the ratio of the
resonant frequency to the bandwidth.

Q o

1
1
Substituting for  
and  o 
the quality factor can be expressed as
RC
LC
Q   o RC 
At resonance IL and IC are given by
Vo
V
IL 
  j o   jQI i and
j o L
R/Q
R
C
R
o L
L
I C  j o CVo  j
Q
Vo  jQI i
R
I C  jQI i
Vo
Ii
I L   jQI i
As it can be seen at resonance depending on the Q factor, IL and IC can be many times the
supply current (current amplification).
4
(b) The Series RLC Resonance
1
j
C I
j L
Vi
 VC 
V 
L
R

Vo
Vi
I
R  j ( L 

1
)
C
A circuit is in resonance when the voltage and current at the input terminals are in phase.
1
) . At resonance Z is purely resistive and
The circuit impedance is Z  R  j ( L 
C
1
1
 L
 0 , thus  o 
. The circuit impedance at resonance, given
C
LC
by Z ( o )  R is minimum, and the current is maximum. Thus, the output voltage at
resonance is maximum and is given by Vo ( o )  RI ( o )
10
|Vo| = R|I( )|, V
8
V0(0) = RI(0)
7.071
6
4
2
0
0
1
100 o
2
200
300
400
, R/s
500
The frequencies 1 and  2 at which the output power drops to one half of its values at
the resonant frequency are called the half-power frequencies. At these frequencies also
known as cutoff frequencies or corner frequencies, the output voltage is
| Vo ( c ) | 0.707 | Vo ( o ) | . This circuit which passes all the frequencies within a band of
frequencies ( 1     2 ) is called a bandpass filter. This range of frequency is known
as the circuit bandwidth.
   2  1
The half-power frequencies are obtained from
Vi
1
| Vo ( c ) | R

Vi
1
2
2
 2 
1  2
 R     L 
 
 C  


5
Solving for  we obtain
2
2
R
1
R
1
 R
 R
, and  2 
1      
   
2L
LC
2L
LC
 2L 
 2L 
R
From the above, we obtain,  2  1 
, or the circuit bandwidth is
L
R

L
The cutoff frequencies can be written in terms of  0 and  as follow:



1        o 2 , and  2       o 2
2
2
2
2
This shows that  o is the geometric mean of 1 and  2 , i.e.,

2
2
 o  1 2
Notice that  is proportional to R, i.e., larger R results in a larger bandwidth. The
1
resonant frequency (  o 
) is a function of L and C. Therefore, by adjusting L and
LC
C a desired resonant frequency is obtained, whereas by adjusting R, the bandwidth and
the height of the response curve is adjusted. The sharpness of the resonance is measured
quantitatively by the quality factor Q. This is defined as the ratio of the resonant
frequency to the bandwidth.

Q o

R
1
Substituting for   and  o 
the quality factor can be expressed as
L
LC
o L
1
1 L

R
 o RC R C
At resonance VL and VC are given by
VL  j o LI ( 0 )  jQVi
1
VC   j
I ( 0 )   jQVi
 oC
Q

VL  jQVi
I
Vi
VC   jQVi
As it can be seen at resonance depending on the Q factor, VL and VC can be many times
the supply voltage (voltage amplification).
For a circuit with a very high quality factor Q, the corner frequencies may be


approximated to 1   o  , and 1   o 
2
2
6
Bandreject Filter
A bandreject filter is designed to stop all frequencies within a band of frequencies
( 1     2 ). In the series RLC circuit consider the output across the series
combination L and C.
R
I

 VR 
j L
1
j ( L 
)
Vo

C
V

o
Vi
1
1
Vi
j
R  j ( L 
)
C
C

The voltage gain magnitude is
1
 L
| Vo |
C

2 1/ 2
| Vi | 
1  

2
 R   L 
 
 C  


At   0 , the inductor behaves like a short circuit, and the capacitor behaves like an
open circuit, I  0 and Vo  Vi , and the voltage gain is unity. At    , the inductor
behaves like an open circuit and the capacitor behaves like a short circuit, I  0 and again
Vo  Vi , and the voltage gain is unity. At resonance Z is purely resistive and
1
1
 L
 0 , thus  o 
. Since the numerator of the voltage gain is zero, the
C
LC
gain drops to zero at  o .
1
|Vo| = ( L-1/( C)|I|, V
0.8
.7071
0.6
0.4
0.2
0
0
1
100 o
2
200
300
400
, R/s
500
The cutoff frequencies, the bandwidth, and the quality factor are the same as the series
R




, 1        o 2 , and  2       o 2
L
2
2
2
2
2
RLC bandpass filter,  
7
2
8