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Resonance In
AC Circuits
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3.1 Introduction
MM
M
h
An example of resonance in the form ofmechanical : oscillation
Potential energy change to kinetic energy thankinetic energy will change back to potentialenergy.
If there is no lost of energy cause by frictionpotential energy is equal to kinetic energy.
mgh = mv It will oscillate for a long time.
Ep=mgh
Ek= mv v
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Resonance in electrical
circuit
C L
i
i
Ep= CV Em= LI
Potential energy stored in capacitor change tomagnetic energy that stored in inductor. Then
magnetic energy change back to potential energystored in capacitor.
If there is no lost of energy by resistor potentialenergy equal to magnetic energy
CV = LI It will oscillate for a long time.
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Characteristic of
resonancecircuit
The frequency response of acircuit is maximum
The voltage Vs andcurrent I are in phase
The impedances is purelyresistive.
Power factor equal to one
Circuit reactance equal zero because capacitive andinductive are equal in magnitude
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At frequency resonance,
(1)
V
I
jXL-jXC
CV = LI
We know
V = I * XL or V = I * XC = I * L = I * 1/C
= I * 2fL = I / 2fC
Substitute into 1
C (2fLI) = LI
f = L
C (2fL)
f = 1
LC2
CV = LI
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Ideal case ( noresistance)
Practical ( energy loss dueto resistance)
i
i
t
t
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Main objective we analysis resonance circuits to find five resonance parameters :
a) Resonance frequency, oAngular frequency when value of current or voltage is maximum
b) Half power frequency, 1 and 2Frequency where current (or voltage) equal Imax/2 (or Vmax/2 ).
c) Quality factor, QRatio of its resonant frequency to its bandwidth
d) Bandwidth, BWDifference between half power frequency
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3.3 Series Resonance
CircuitsR
VR VL
R j XL
- j XCV
By KVL : V = VR + VL + VC
= VR+ jVL jVC
At resonance XL = XC
Hence V = VR+ 0
= VR
= IR* R
Vc
Figure 1
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Series ResonanceCircuits
R
VR VL
j XL
- j XCV Vc
Figure 1
Z = R + j XL - jXC
= R + j (XL XC)
XL = 2fL
XC =1
2fC
where
XL
R
XCf0
f
|Z|
(XL-XC)
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f
f0
|I||Z|
|I| =|V|
|Z|
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Resonance parameter for
series circuita) Resonant frequency,oThe resonance condition is
oL = 1 / oC or o = 1 / LC rad/s
since o = 2fofo LC Hz2 /1 =
b) Half power frequencies
At certain frequencies = 1, 2, the half power frequencies are obtain by setting Z = 2R
R + (L 1/ C) = 2R
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Solving for, we obtain
1 = - R/2L + (R/2L) + 1/LC rad/s
2 = R/2L + (R/2L) + 1/LC rad/s
Or in term of resonant parameter,
1 = o [ - 1/ 2Q + (1/ 2Q) + 1 ] rad/s
2 = o [ 1/ 2Q + (1/ 2Q) + 1 ] rad/s
c) Quality factor, Q
Ratio of its resonant frequency to its bandwidth.
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Q = VLVS
= [ I ] x XL[ I ] x R
= L ; Q = XL
R R
frL 2 =R
Q = VC
V
= [ I ] x XC[ I ] x R
= 1 ; Q = XC
C RR
= 1 frCR2
or
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d) Bandwidth, BW
BW = 2 1
= o [ 1+ (1/ 2Q) + 1/ 2Q ] - o [ 1+ (1/ 2Q) - 1/ 2Q ]
= o [ 1/ 2Q + 1/2Q ]
= o [2/ 2Q]
= o / Q
Q = oL /R = 1/ oCR
thus,
BW = R / L = o / Q
or, BW = oCR
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3.4 Parallel Resonance
Circuits Resonance can be divided into 2:
a) Ideal parallel circuit
b) Practical parallel circuit
At least 3 important information that is needed to analyze to get the resonances
parameter: In resonance frequency, o the imaginary parts of admittance,Y must be equalto zero.
When in lower cut-off frequency, 1 and in higher cut-off frequency, 2 themagnitude of admittance,Y must be equal to 2/R.
i
+
v
-
RC L
Ideal Parallel RLC circuit
o
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Resonance parameter for
ideal RLC parallel circuitR-jXc jXL
Ideal Parallel RLC circuitYT
+
Lc XXj
R
111YT =
)()(11
jBGL
CjR
+=
+=
Whereas G() is the real part called the conductance
and B() is the imaginary parts called the susceptance.
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a) Resonant frequency,o
Angular resonance frequency is when B()=0.
b) Lower cut-off angular frequency, 1
Produced when the imaginary parts = (-1/R)
sradLC
LC
/1
;01
210
==
=
+
+=
=
LCRCRC
RLC
1
2
1
2
1
11
2
1
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c) Higher cut-off angular frequency, 2.
Produced when the imaginary parts = (1/R)
d) Quality Factor, Q
e) Bandwidth, BW
sradLCRCRC
RLC
/1
2
1
2
1
11
2
1
+
+=
+=
RCQ
L
CR
L
RQ
0
0
=
==
QRCBW 012
1 ===
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Duality Concept
R
VR VL
j XL
- j XCV Vc
Figure 1
Series circuit Parallel circuit
i
+
v
-
RC L
Ideal Parallel RLC circuit
Z = Z1 + Z2 + Z3
Z = R + j XL - jXC
Y = Y1 + Y2 + Y3
Y =
+
Lc XXj
R
111
+=L
Cj
R
11
YZ = R + j (L )C
1
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Duality Concept
R
VR VL
j XL
- j XCV Vc
Figure 1
Series circuit Parallel circuit
i
+
v
-
RC L
Ideal Parallel RLC circuit
+=
LCj
R
11YZ = R + j (L )
C
1
R R
1
L C
C L
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Duality Concept
R
VR VL
j XL
- j XCV Vc
Figure 1
Series circuit Parallel circuit
i
+
v
-
RC L
Ideal Parallel RLC circuit
1 = - R/2L + ((R/2L) + 1/LC) rad/s2 = R/2L + ((R/2L) + 1/LC) rad/s
1 = - 1/2RC + ((1/2RC) + 1/LC) rad/s2 = 1/2RC + ((1/2RC) + 1/LC) rad/s
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Resonance parameter for
practical RLC parallel circuit
Practical Parallel RLC circuit
i
+
V
-
R1
C
L
I1 IC
I1
IC
Z1 = R1 + jXL = |Z1|/
|I1|cos
|I1
|sin
I
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Resonance parameter for
practical RLC parallel circuit
Practical Parallel RLC circuit
i
+
V
-
R1
C
L
I1 IC
I1
IC
Resonance occur when |I1|sin = IC
|I1|cos
|I1
|sin
I
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Resonance occur when |I1|sin = IC
|I1|sin = IC
|V||Z1|
x XL
|Z1||V|=XC
XL
|Z1|2 = XC
1
2frL
R2
+ (2frL)2
= 2frC
R2 + (2frL)2L
=
C
(2frL)2 =LC
- R2
=
21
2
1
L
R
LCr
f
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Q factor
XL
R
2frL
=
=
Q = current magnification IC
=
|I1|sin
I = |I1|cos
tan
=
R
I1
IC
|I1|cos
|I1
|sin
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Resonance parameter for
practical RLC parallel circuit
Ideal Parallel RLC circuit
i
+
V
-
R1
C
L
Second approach to analyze this circuit is by changing the seriesRL to parallel RL circuit.
The purpose of this transformation is to make it much more easierto get the resonance parameter.
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RL in series
Rl
L
Rl jXp
RL in parallel
pp
T
l
ll
l
llll
T
llT
jXRY
X
XRj
R
jXRjXRY
jXRZ
11
1112222
+=
++
+=
+=
+=
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By matching equation ZT and YT above, we can
get:
Or
By defining the quality factor,
l
lp
l
llp
R
LRR
RXRR
22
22
)(+=
+= and
and
l
lp
l
llp
X
LRX
XXRX
22
22
)(+=
+
=
ll
l
R
L
R
XQ
=
Rp
and Xp
can be write as:
2
2
2
2
2
1
)1(
l
lp
ll
l
lllp
Q
QLL
QRR
XRRR
+=
+=+=
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Resonance parameter
a) Angular resonance frequency, o
b) Lower cut-off angular frequency, 1
Produced when the imaginary parts = (1/R)
L
CR
LCX
R
LC
l
l
l
2
02
2
111
==
sradCLCRCR
RLC
ppp
/1
2
1
2
1
11
2
1
+
+=
=
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c) Higher cut-off angular frequency, 2.
d) Quality Factor, Q
e) Bandwidth, BW
Produced when the imaginary parts = (1/R)
sradCLCRCR
RLC
ppp
/1
2
1
2
1
11
2
2
+
+=
+=
RCQ
L
CR
L
RQ
p
p
p
=
==
QCRBW
p
012
1 ===