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CHAPTER 4
RESONANCE CIRCUITS
Tunku Muhammad Nizar Bin Tunku MansurPegawai Latihan Vokasional
Pusat Pengajian Kejuruteraan Sistem Elektrik
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Content Series Resonance Parallel Resonance Important Parameters
Resonance Frequency, o
Half-power frequencies, 1 and 2
Bandwidth, Quality Factor, Q
Application
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Introduction
Resonance is a condition in an RLC circuit in which the capacitive and reactive reactance are equal in magnitude, thereby resulting in a purely resistive impedance.
Resonance circuits are useful for constructing filters and used in many application.
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Series Resonance Circuit
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At Resonance
At resonance, the impedance consists only resistive component R.
The value of current will be maximum since the total impedance is minimum.
The voltage and current are in phase. Maximum power occurs at resonance
since the power factor is unity.
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Series Resonance
CLTotal jX-jXRZ
R
V
Z
VI m
Total
sm
Total impedance of series RLC Circuit is
At resonance
The impedance now reduce to
CL XX
R ZTotal
)X-j(XRZ CLTotal
The current at resonance
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Resonance Frequency
Resonance frequency is the frequency where the condition of resonance occur.
Also known as center frequency.
Resonance frequency
rad/sLC
1ωo
HzLC2
1
of
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Half-power Frequency
rad/sLC
1
2L
R
2L
Rω
2
2
Half-power frequencies is the frequency when the magnitude of the output voltage or current is decrease by the factor of 1 / 2 from its maximum value.
Also known as cutoff frequencies.
rad/sLC
1
2L
R
2L
Rω
2
1
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Bandwidth,
rad/s)( 12 cc ωωβ
Bandwidth, is define as the difference between the two half power frequencies.The width of the response curve is determine by the bandwidth.
rad/sL
Rβ
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Current Response Curve
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Voltage Response Curve
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Quality Factor (Q-Factor)
The ratio of resonance frequency to the bandwidth
The “sharpness” of response curve could be measured by the quality factor, Q.
R
LQ oo
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High-Q
It is to be a high-Q circuit when its quality factor is equal or greater than 10.
For a high-Q circuit (Q 10), the half-power frequencies are, for all practical purposes, symmetrical around the resonant frequency and can be approximated as
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o 22
o
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Q-Factor Vs Bandwidth
Higher value of Q, smaller the bandwidth. (Higher the selectivity)
Lower value of Q larger the bandwidth. (Lower the selectivity)
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Maximum Power Dissipated
The average power dissipated by the RLC circuit is
The maximum power dissipated at resonance where
R
V
2
1)P(ω
m2
o
R
VI m
RI2
1)P(ω 2
o
Thus maximum power dissipated is
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Power Dissipated at 1 and 2
At certain frequencies, where ω = ω1 and ω2, the dissipated power is half of maximum power
Hence, ω1 and ω2 are called half-power frequencies.
4R
V
R
)2/(V
2
1)P(ω)P(ω
m22
m21
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Example 14.7If R=2Ω, L=1mH and C=0.4 F, calculate
Resonant frequency, ωo
Half power frequencies, ω1 and ω2
Bandwidth, Amplitude of current at ωo, ω1 and ω2.
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Practice Problem 14.7
A series connected circuit has R=4Ω and L=25mH. Calculate Value of C that will produce a quality
factor of 50. Find 1 , 2 and . Determine average power dissipated
at = o , 1 and 2. Take Vm = 100V
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Parallel Resonance
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Parallel Resonance
The total admittance
ωLωC
1
Resonance occur when
)ωω L1/Cj(R
1YTotal
321Total YYYY
C)(-j/
1
L)(j
1
R
1YTotal
CjωL
j-
R
1YTotal ω
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At Resonance
At resonance, the impedance consists only conductance G.
The value of current will be minimum since the total admittance is minimum.
The voltage and current are in phase.
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Parameters in Parallel Circuit
rad/sLC
1
2RC
1
2RC
1ω
2
1
Parallel resonant circuit has same parameters as the series resonant circuit.
rad/sLC
1ωo
rad/sLC
1
2RC
1
2RC
1ω
2
2
Resonance frequency
Half-power frequencies
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Parameters in Parallel Circuit
RCβ
ωQ o
o
RC
112 ωωβ
Bandwidth
Quality Factor
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Example 14.8If R=8kΩ, L=0.2mH and C=8F, calculate
ωo
Q and ω1 and ω2
Power dissipated at ωo, ω1 and ω2.
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Practice Problem 14.8
A parallel resonant circuit has R=100kΩ, L=25mH and C=5nF. Calculate o
1 and 2 Q
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APPLICATION
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PASSIVE FILTERS A filter is a circuit that is designed to
pass signals with desired frequencies and reject or attenuates others
A filter is a Passive Filters if it consists only passive elements which is R, L and C.
Filters that used resonant circuit Bandpass Filter Bandstop Filter
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BANDPASS FILTER
A bandpass filter is designed to pass all frequencies within
ω1 ωo ω2
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BANDPASS FILTER
LC
1
2L
R
2L
Rω
2
1
SERIES RLC CIRCUIT LC
1ωo
2o
CR
L
β
ωQ
L
Rωωβ 12
LC
1
2L
R
2L
Rω
2
2
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BANDPASS FILTER
LC
1
2RC
1
2RC
1ω
2
1
PARALLEL RLC CIRCUIT LC
1ωo
L
CR
β
ωQ
2o
RC
1ωωβ 12
LC
1
2RC
1
2RC
1ω
2
2
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BANDSTOP FILTER
A bandstop or bandreject filter is designed to stop or reject all frequencies within
ω1 ωo ω2
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BANDSTOP FILTER
LC
1
2L
R
2L
Rω
2
1
SERIES RLC CIRCUITLC
1ωo
2o
CR
L
β
ωQ
L
Rωωβ 12
LC
1
2L
R
2L
Rω
2
2
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BANDSTOP FILTER
LC
1
2RC
1
2RC
1ω
2
1
PARALLEL RLC CIRCUIT LC
1ωo
L
CR
β
ωQ
2o
RC
1ωωβ 12
LC
1
2RC
1
2RC
1ω
2
2
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EXERCISE