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8/17/2019 Comparison of Series and Parallel Resonance Circuits
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COMPARISON OF SERIES AND
PARALLEL RESONANCE
CIRCUITS
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ELECTRICAL RESONANCE
Electrical resonance occurs in an electric circuit at a particular resonance frequency where the imaginary parts of circuit element impedances or admittances cancel each
other. In some circuits this happens when the impedance between the input and output of the circuit is almost zeroand the transfer function is close to one.
Resonant circuits exhibit ringing and can generatehigher voltages and currents than are fed into them. They arewidely used in wireless (radio) transmission for bothtransmission and reception.
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RLC Circuit Introduction
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An RLC circuit (or LCR circuit) is an electrical circuitconsisting of a resistor, an inductor, and a capacitor,connected in series or in parallel. The RLC part of the name
is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively. Thecircuit forms a harmonic oscillator for current and willresonate in a similar way as an LC circuit will. The main
difference that the presence of the resistor makes is that anyoscillation induced in the circuit will die away over time if itis not kept going by a source.
This effect of the resistor is called damping. The
presence of the resistance also reduces the peak resonantfrequency somewhat. Some resistance is unavoidable in realcircuits, even if a resistor is not specifically included as acomponent. A pure LC circuit is an ideal which really onlyexists in theory.
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Resonance In Electric Circuits
Any passive electric circuit will resonate if it has an inductor
and capacitor.
Resonance is characterized by the input voltage and current
being in phase. The driving point impedance (or admittance)
is completely real when this condition exists.
In this presentation we will consider
(a) series resonance, and
(b) parallel resonance.
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SERIES RESONANCE
CONDITION
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Series Resonance
Consider the series RLC circuit shown below.
R L
C+
_ IV
V = VM 0
The input impedance is given by:
1
( ) Z R j wL
wC
The magnitude of the circuit current is;
2 2
| |1
( )
mV I I
R wL
wC
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Series Resonance
Resonance occurs when,1
wLwC
At resonance we designate w as wo and write;
This is an important equation to remember. It applies
to both series and parallel resonant circuits.
1 1,
2
o o f
LC LC
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Series Resonance
The magnitude of the current response for the series
resonance circuit is as shown below.
mV
R
2
mV
R
w
|I|
wow1 w2
Bandwidth:
BW = wBW = w2 – w1
Half power point
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Series Resonance
The peak power delivered to the circuit is;2
mV
P R
The so-called half-power is given when2
mV
I R
.
We find the frequencies, w1 and w2, at which this half-power
occurs by using;
2 212 ( ) R R wLwC
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Series Resonance
After some insightful algebra one will find two frequencies at which
the previous equation is satisfied, they are:
2
1
1
2 2
R Rw
L L LC
and 2
2
1
2 2
R Rw
L L LC
The two half-power frequencies are related to the resonant frequency by
1 2ow w w
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Series Resonance
The bandwidth of the series resonant circuit is given by;
2 1b
R BW w w w
L
We define the Q (quality factor) of the circuit as;
1 1o
o
w L LQ R w RC R C
Using Q, we can write the bandwidth as;
ow BW Q
These are all important relationships.
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jwL jwC
RV I
11
jwC jwL R I V
1
We notice the above equations are the same provided:
V I
R R
1
C L
If we make the inner-change,then one equation becomes
the same as the other.
For such case, we say the onecircuit is the dual of the other.
Series Resonance
Duality
If we make the inner-change,then one equation becomes
the same as the other.
For such case, we say the one
circuit is the dual of the other.
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The features of series resonance:
The impedance is purely resistive, Z = R;
• The supply voltage Vs and the current I are in
phase (cosq = 1)
• The magnitude of the transfer function H (ω) =
Z(ω) is minimum;
• The inductor voltage and capacitor voltage can
be much more than the source voltage.
QU L TY F CTOR OF ER E RE ON NCE
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QUALITY FACTOR OF SERIES RESONANCE
1 2
1
,
2 2
o
o
o
o o
LQ
R RC
R B
L Q
B B
oQ
B
Peak EnergyStored 2
Energy Dissipated in one Period at Resonance
1o
o
Q
LQ R RC
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PARALLEL RESONANCE
CONDITION
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The properties of the parallel RLC circuit can be obtained from theduality relationship of electrical circuits and considering that the
parallel RLC is the dual impedance of a series RLC. Considering this
it becomes clear that the differential equations describing this circuitare identical to the general form of those describing a series RLC.
For the parallel circuit, the attenuation α is given by
and the damping factor is consequently
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This is the inverse of the expression for ζ in the series circuit.Likewise, the other scaled parameters, fractional bandwidthand Q are also the inverse of each other. This means that a
wide band, low Q circuit in one topology will become anarrow band, high Q circuit in the other topology whenconstructed from components with identical values.
The Q and fractional bandwidth of the parallel circuitare given by
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The End