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Chapter 20



Resonant and cut-off frequencies
Tuned network quality, bandwidth, and
power levels
Quality factor
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20.1-20.7
“A condition established by the
application of a particular frequency to
a series or parallel RLC network. The
transfer of power is at maximum, and
power drops off for frequencies above
and below this frequency.”
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Must have resistive
elements
◦ Internal resistances (reality)
◦ Helps shape curve

Must have reactive
elements
◦ Both capacitive and inductive
required of equal impedance
◦ Energy level absorbed by one
is released by another
FIG. 20.1 Resonance
curve.
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

Must have capacitive and inductive element
Resistive elements
◦
◦
◦
- source internal resistance
- inductor internal resistance
- added resistance to shape the response curve
FIG. 20.2 Series resonant
circuit.





=
+
+
= + ( −
=
=
=
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)
FIG. 20.2 Series resonant
circuit.
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AT RESONANCE

E and I in phase

Power triangle
P= Real power
Q= Reactive power
S= Apparent power
FIG. 20.3 Phasor diagram
for the series resonant
circuit at resonance.
FIG. 20.4 Power triangle
for the series resonant
circuit at resonance.
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P (Real Power, Watts)
◦ Power delivered to resistive elements in circuit
◦ =
◦ Resistances only

Q (Reactive Power, Volt-amps Reactive VAR)
◦
sin
=
◦ Reactances only

S (Apparent Power)
◦ Accounts for phase angle
◦
S
Q
±
P
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
counters
FIG. 20.5 Power curves at resonance for the series
resonant circuit.
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Level of power stored (L or C)
compared to the level of power
dissipated (R)
=
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
High Q systems (communication)
◦
or
may be higher than source
◦ Requires special insulation for “Q rise”
FIG. 20.7 High-Q series resonant
circuit.
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Total Z – function of frequency
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FIG. 20.8 Resistance versus
frequency.
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FIG. 20.9 Inductive reactance
versus frequency.
15
FIG. 20.10 Capacitive reactance
versus frequency.
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FIG. 20.11 Frequency response
of the inductive and capacitive
reactance of a series R-L-C
circuit on the same set of axes.
FIG. 20.12 ZT versus frequency
for the series resonant circuit.
At
→
=
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FIG. 20.13 Phase plot for the series
resonant circuit.
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
Find
◦
◦
◦
◦
◦
XL
IT
V R, V L , V C
Q
L and C at 5 kHz
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Bandwidth (-3 dB)
◦ Freq. where
◦ −
◦ At is near max
 0.707
 0.707

FIG. 20.14 I versus frequency for
the series resonant circuit.
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
Freq. must be “selected”
to fall within BW
◦ Large BW, small selectivity

Component values
shape curve
◦ Higher R, L, C, tighter the
curve
FIG. 20.15 Effect of R, L, and
C on the selectivity curve for
the series resonant circuit.
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Quality factor ( )
proportional to BW
◦ Small BW, high

≥ 10
◦ Resonance Freq bisects BW
◦ Curve is symmetrical on RF

Ideal situation
FIG. 20.16 Approximate series resonance
curve for Qs ≥ 10.
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
=

=
+
+
+
=


−
−
+
=
=
Fractional Bandwidth
◦
=
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For Ex. 20.1, Find
◦ I
◦
at resonance
◦ at resonance
◦
at resonance
◦
◦ BW when = 5 kHz
◦
+
(
) at
FIG. 20.19 Example 20.1.
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20.8-20.12

For Ex. 20.4, find
◦
◦ L and R when C=100nF
 100 × 10
◦ E
FIG. 20.20 Example 20.4
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FIG. 20.21 Ideal parallel resonant
network.
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Practical
◦ More realistic
◦ Rl -inductor internal resistance
◦
=
(no phase angles)
◦
=
(no phase angles)
FIG. 20.23 Equivalent parallel network for a
series R-L combination.
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FIG. 20.24 Substituting the equivalent parallel network for the series R-L
combination in Fig. 20.22.
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Final result
◦ Same look as ideal circuit
◦ Correct, realistic values
◦ R = Rs ║ Rp
FIG. 20.25 Substituting R = Rs ║ Rp for the network
in Fig. 20.24.
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

Unity Power Factor, fp
Maximum Impedance, fm
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Unity power factor ( )
◦ Total reactive element = 0
◦
=
◦
=
◦
>
or
=
1−
FIG. 20.26 ZT versus frequency for the
parallel resonant circuit.
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
Maximum Impedance (
)
◦ Actual freq. input Z is the
highest
◦ Highest power output
◦ Based on
◦
=
◦
=
1−
=

FIG. 20.26 ZT versus frequency for the
parallel resonant circuit.

=
◦ When
=

‖
=
=
≫
−
→
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‖
=
=
=

=
− +

=
+
+
+
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FIG. 20.28 Effect of Rl, L, and C on the parallel
resonance curve.
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FIG. 20.29 Phase plot for the parallel
resonant circuit.
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
Inductive Reactance, XLP
≅
◦

Resonant Frequency, fp (Unity Power Factor)
◦

≅
≅
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≅
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≅
ZTp
◦ ◦ If

≅
Rp
◦

=
Resonant Frequency, fm (Maximum VC)
◦

≅
=
= ∞Ω or
≫
→
= ∞Ω or
≫
→
=
Qp
◦ If
≅
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FIG. 20.30 Approximate equivalent circuit for Ql ≥ 10 .
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BW
=
◦


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−
≅
=
=
FIG. 20.31 Establishing the relationship
between IC and IL and the current IT.
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ECET 207 AC Circuit Analysis, PNC
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Use the summery tables
◦ Multiple versions of equations listed
◦ If no value given, it is skipped
 Ex- No Rs given, exclude it from equation

Given Prob. 15 (Pg. 909), find
◦
◦
◦
◦
Resonance frequency
Vtank
Power delivered by source at resonance
Power loss in the tank coil
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