Download Sinusoidal Steady

FREQUENCY SELECTIVE CIRCUITS
Osman Parlaktuna
Osmangazi University
Eskisehir, TURKEY
www.ogu.edu.tr/~oparlak
FILTERS



Varying the frequency of a sinusoidal source alters
the impedance of inductors and capacitors. The
careful choice of circuit elements and their
connections enables us to construct circuits that pass
a desired range of frequencies. Such circuits are
called frequency selective circuits or filters.
The frequency range that frequencies allowed to pass
from the input to the output of the circuit is called
the passband.
Frequencies not in a circuit’s passband are in its
stopband.
IDEAL FILTERS
|H(jw)|
|H(jw)|
1
1
Stopband
passband
Stopband
wc
w
Ideal low-pass filter
(LPF)
|H(jw)|
1
Stop
band
pass
band
wc1
wc
w
Ideal high-pass filter
(HPF)
|H(jw)|
1
Stopband
wc2
Ideal band-pass filter
(BPF)
passband
w
pass
band
Stop
band
wc1
passband
wc2
Ideal band-reject filter
(BRF)
w
LOW-PASS FILTERS
L
vi
vi
vi
+
+
+
+
R
vo
+
R
vo
+
R
vo
The input is a sinusoidal voltage source
with varying frequency. The impedance
of an inductor is jwL.
At low frequencies, the inductor’s impedance
is very small compared to that of resistor
impedance, and the inductor effectively
functions as a short circuit. vo=vi.
As the frequency increases, the impedance
of the inductor increases relative to that of
the resistor. The magnitude of the output
voltage decreases. As w ∞, inductor
approaches to open circuit. Output voltage
approaches zero.
QUANTITATIVE ANALYSIS
sL
vi(s) +
R
+
vo(s)
H ( s) 
H ( j ) 
R
L
s
R
L
R
L
 H ( j ) 
 2   RL 
2
,
R
L
j  RL
 ( )   tan 1 RL 
CUTOFF FREQUENCY
Cutoff frequency is the frequency at which the magnitude of
the transfer function is Hmax/√2.
For the series RL circuit, Hmax=1. Then,
R
1
R
L
H ( jc ) 

 c 
2
2
L
2
c   RL 
At the cutoff frequency θ(wc)=-450.
For the example R=1Ω, L=1H, wc=1 rad/s
EXAMPLE
Design a series RL low-pass filter to filter out any noise above 10 Hz.
R and L cannot be specified independently to generate a value for
wc. Therefore, let us choose L=100 mH. Then,
R  c L  (2 )(10)(100 10 3 )  6.28
Vo ( ) 
R
L
 2   RL 
2
Vi 
20
 2  400 2
Vi
F(Hz)
|V.|
|Vo|
1
1.0
0.995
10
1.0
0.707
60
1.0
0.164
A SERIES RC CIRCUIT
R
vi +
+
C
vo
1) Zero frequency w=0: The impedance of
the capacitor is infinite, and the capacitor
acts as an open circuit. vo=vi.
2) Increasing frequency decreases the impedance of the
capacitor relative to the impedance of the resistor, and the
source voltage divides between the resistor and capacitor. The
output voltage is thus smaller than the source voltage.
3) Infinite frequency w=∞: The impedance of the capacitor is
zero, and the capacitor acts as a short circuit. vo=0.
Based on the above analysis, the series RC circuit functions as a
low-pass filter.
QUANTITATIVE ANALYSIS
R
Vi(s) +
+
1/sC
Vo(s)
H ( s) 
1
RC
s
1
RC
 H ( j ) 
1
RC
 
2
H max  H ( j 0)  1
1
1
RC
H ( j c ) 
(1) 
1 2
2

c2   RC
1
c 
RC

1 2
RC
Relating the Frequency Domain
to the Time Domain


Remember the natural response of the firstorder RL and RC circuits. It is an exponential
L
with a time constant of   or   RC
R
Compare the time constants to the cutoff
frequencies for these circuits and notice that

1
c
HIGH-PASS FILTERS
C
vi
vi
vi
+
+
+
+
R
vo
vo
As the frequency increases, the impedance
of the capacitor decreases relative to the
impedance of the resistor and the voltage
across the resistor increases.
+
When the frequency of the source is infinite,
the capacitor behaves as a short circuit and
vo=vi.
+
R
R
At w=0, the capacitor behaves like an open
circuit. No current flows through resistor.
Therefore vo=0.
vo
QUANTITATIVE ANALYSIS
1/sC
Vi(s)
+
+
R
Vo(s)
s
j
H ( s) 
 H ( j ) 
1
1
s  RC
j  RC
H ( j ) 

 
2

1 2
RC
 ( j )  900  tan 1 RC
SERIES RL CIRCUIT
R
Vi(s) +
sL
+
Vo(s)
s
j
H ( s) 
 H ( j ) 
R
s L
j  RL
H ( j ) 

 2   RL 
2
H max  H ( j)  1
c
1
 H ( j c ) 
2
2
c2   RL 
R
c 
L
LOADED SERIES RL CIRCUIT
R
Vi(s) +
sL
+
Vo(s)
H (s) 
RL
RL sL
RL  sL
RL sL
RL  sL
R
Ks
H (s) 
,
s  c


s

 
s
,   KR / L
K 
RL
R  RL
RL
R  RL
RL
R  RL
R
L
c
The effect of the load
resistor is to reduce the
passband magnitude by the
factor K and to lower the
cutoff frequency by the
same factor.
BANDPASS FILTERS
L
vi
+
+
R
L
vi
+
vo
C
+
+
R
L
vi
C
vo
C
+
R
vo
At w=0, the capacitor behaves like an
open circuit, and the inductor behaves
like a short circuit. The current through
the circuit is zero, therefore vo=0.
At w=∞, the capacitor behaves like a
short circuit, and the inductor behaves
like an open circuit. The current through
the circuit is zero, therefore vo=0.
Between these two frequencies, both the
capacitor and the inductor have finite
impedances. Current will flow through the
circuit and some voltage reaches to the
load.
CENTER FREQUENCY
At some frequency, the impedance of the capacitor and
the impedance of the inductor have equal magnitudes and
opposite signs; the two impedances cancel out, causing
the output voltage to equal the source voltage. This
special frequency is called the center frequency wo. On
either side of wo, the output voltage is less than the
source voltage. Note that at the center frequency, the
series combination of the inductor and capacitor appears
as a short circuit and circuit behaves as a purely resistive
one.
QUANTITATIVE ANALYSIS
sL
Vi(s)
+
1/sC
R
+
Vo(s)
H ( s) 
s2  
H ( j ) 

R
L
R
L
1
LC
s
s  LC1
 RL

  2   RL 
2
2
  RL 

 ( j )  90  tan  1
2 
 LC   
0
1
CENTER FREQUENCY AND
CUTOFF FREQUENCIES
At the center frequency, the transfer function will be real,
that means:
1
1
j o L 
j o C
 0  o 
LC
At the cutoff frequencies, the magnitude of H(jw) is Hmax/√2.
H max  H ( jo ) 


o
1
LC

  
2 2
o
1 R
LC L
 LC1  LC1 2  
R
L
1 R
LC L

2
o R 2
L
1
1

2
 1  c

c
1
LC

R
L
   
2 2
c
R 2
c L


1
R
c L

 c1RC  1
2
L
1

 c2 L  c R  1 / C  0
R c RC
c1   2RL 
c 2  2RL 
 2RL 2   LC1 
 2RL 2   LC1 
It is easy to show that the center
frequency is the geometric mean of the
two cutoff frequencies
o  c1  c 2
BANDWIDTH AND THE QUALITY FACTOR
The bandwidth of a bandpass filter is defined as the
difference between the two cutoff frequencies.
  c 2  c1
  2RL 

R

L
  
R 2
2L
1
LC
The quality factor is defined as
the ratio of the center frequency
to bandwidth
   2RL 
 
    LC1 
R 2
2L

Q o 

1
LC
R
L


L
CR 2
CUTOFF FREQUENCIES IN TERMS OF CENTER
FREQUENCY, BANDWIDTH AND QUALITY FACTOR

 
c1       o2
2
2

2
 
c 2      o2
2
2
2
2

 1  
1


c1  o 
 1  
 2Q
2Q  



2

 1  
1


c1  o
 1  
 2Q
2Q  



EXAMPLE
Design a series RLC bandpass filter with cutoff frequencies 1 and
10 kHz.
Cutoff frequencies
f o  f c1 f c 2  (1000)(10000)  3162.28Hz
give us two equations
1
1
but we have 3
L 2 
 2.533 mH
2
6
parameters to
o C 2 (3162.28) (10 )
choose. Thus, we
fo
3162.28
need to select a value Q 

 0.3514
f c 2  f c1 10000  1000
for either R, L, and C
and use the
L
2.533(10 3 )
R

 143.24
equations to find
2
6
2
CQ
(10 )(0.3514)
other values. Here,
we choose C=1μF.
EXAMPLE
R
+
Vi(s)
+
1/sC
Vo(s)
L
sL ( sC1 )
C
Z eq ( s ) 

sL  sC1 sL  sC1
sL
H ( s) 
s2 
s
RC
s
RC
 LC1
This transfer function and the transfer function for the series RLC
bandpass filter are equal when R/L=1/RC. Therefore, all
parameters of this circuit can be obtained by replacing R/L by
1/RC. Thus,
1
o 
LC
1

RC
Q
R 2C
L
1
c1   2 RC

c 2 
1
2 RC

1 2
 2 RC
   LC1 
1 2
 2 RC
   LC1 
BANDPASS FILTER WITH
PRACTICAL VOLTAGE SOURCE
Ri
Vi(s)
sL
1/sC
+
R
H ( j ) 
o 


R
L
1
LC
+
Vo(s)

 
  2   R LRi
2
H ( s) 

2
1
R
 H max  H ( jo ) 
R  Ri
LC
R  Ri
L
L
Q
C
R  Ri
R
L
R  Ri
L
s
s2 
 s 
1
LC
R  Ri
1
 R  Ri 
c1  
 


2L
 2 L  LC
2
R  Ri
1
 R  Ri 
 
 
2L
 2 L  LC
2
c 2 
The addition of a
nonzero source
resistance to a series
RLC bandpass filter
leaves the center
frequency unchanged
but widens the
passband and reduces
the passband
magnitude.
Relating the Frequency
Domain to the Time Domain
The natural response of a series RLC circuit is related to
the neper frequency   2RL and the resonant frequency o 
1
LC
o is also used in frequency domain as the center frequency.
The bandwidth and the neper frequency are related by   2
The natural response of a series RLC circuit may be underdamped, overdamped, or critically damped. The transition from
overdamped to criticallydamped occurs when o2   .2 The
transition from an overdamped to an underdamped response
occurs when Q=1/2. A circuit whose frequency response contains
a sharp peak at the center frequency indicates a high Q and a
narrow bandwidth, will have an underdamped natural response.
R=200Ω,
L=1H, C=1μF
R=200Ω,
L=2H, C=0.5μF
R=200Ω,
L=0.2H, C=5μF
BANDREJECT FILTERS
R
vi +
R
+
L
C
vo
vi
+
R
+
L
C
vo
vi
+
+
L
vo
C
At w=0, the capacitor behaves as an open circuit and inductor
behaves as a short circuit. At w=∞, these roles switch. The output
is equal to the input. This RLC circuit then has two passbands, one
below a lower cutoff frequency, and the other is above an upper
cutoff frequency.
Between these two passbands, both the inductor and the capacitor
have finite impedances of opposite signs. Current flows through the
circuit. Some voltage drops across the resistor. Thus, the output
voltage is smaller than the source voltage.
QUANTITATIVE ANALYSIS
R
Vi(s) +
sL  sC1
s 2  LC1
H ( s) 
 2 R
1
R  sL  sC s  L s  LC1
+
sL
Vo(s)
1/sC
H ( j ) 
1
LC

1
LC
2

  2  LR 

 ( j )   tan 

2
Q
1
LC
L
R 2C

R
L


2 
 
R
1
o 
2
L
1
LC
c1   2RL 
c 2  2RL 
 2RL 2   LC1 
 2RL 2   LC1 