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A network is in resonance (or resonant) when
the voltage and current at the network input
terminals are in phase.
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The resonant frequency is
I 1 
1 
Y    jC 

V R 
L 
1
o 
LC

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s  s / RC 1/ LC
Y (s)  C
s
(s    j d )(s    j d )
C
s
2

1

2RC
d    
2
o
1
o 
LC
2
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s /C
Z(s) 
(s    j d )(s    j d )
V ( j )  Z( j )I

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maximum energy stored
Q  quality factor  2
total energy lost per period
For the parallel RLC
circuit, the quality factor at
resonance is
Q0 = 2π f0RC = ω0RC
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The damping factor defined as ζ=1/2Qo
Now the quadratic factor can be written in several
equivalent ways:
2
2
s  2s  o
s2  2o s  o2


1
1
s 
s
RC
LC
2
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ω1: the lower half-power
frequency
ω2: the upper half-power
frequency.
The (half-power) bandwidth is defined as the difference of
these two half-power frequencies:
B ≡ ω2 − ω1
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Scaling allows us to design practical circuits at
realistic frequencies.
The following simple but unrealistic circuit will
serve to illustrate the method:
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Km=2000
R  KmR
L  KmL
C
C
Km
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Kf=5×10−6
RR
L
L
Kf
C
C
Kf
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


A Bode diagram or Bode plot is a useful tool
for visualizing transfer functions and
frequency responses.
A Bode plot shows either magnitude or phase
on a logarithmic scale for frequency ω.
Magnitude is shown on a decibel (dB) scale
defined as:
HdB = 20 log10 |H( jω)|
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s
H (s)  1
a
The two asymptotes intersect at ω = a, the
frequency of the zero.
This frequency is also described as the corner,
break, 3 dB, or half-power frequency.
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H(s)=20+0.2s=20(1+s/100)
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s
H (s)  1
a

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 s   s 
H (s)  1 2    
0  0 
2

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10s
H (s) 
(1 s)1 2(0.1)(s /100) (s /100)2 

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The transfer function is H(s)=1/(1+sRC) and the
corner frequency is ω=1/RC.
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The transfer function is H(s)=sRC/(1+sRC) and
the corner frequency is ω=1/RC.
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RCs
H (s) 
2
LCs  RCs 1

The bandwidth is R/L and the center
frequency is ω0 = 1/√LC
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The following circuit is an active lowpass filter
with a corner frequency of 1/R2C and a gain of
1+Rf / R1.
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
Filters can be designed to achieve various attenuation, stepresponse, and ripple goals
•
Butterworth and
Chebyshev are famous
classes of filters with
well-designed
characteristics, and can
be built using the
Sallen-Key amplifier
shown.
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