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Chapter 15

Analyze series and parallel ac circuits to find
◦
◦
◦
◦

Voltage
Current
Power
Total impedance, admittance
Apply known circuit theories
◦ Kirchhoff’s current, voltage laws
◦ Voltage or current divider rule

Find the frequency response of a series or parallel
combination of elements.
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15.1-15.4

How much an element impedes current flow
◦ Not a phasor itself, but represents the value
, ,
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= 
FIG. 15.2 Example
15.1.
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FIG. 15.3 Waveforms for
Example 15.1.
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FIG. 15.4 Example
15.2.
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FIG. 15.5 Waveforms for
Example 15.2.
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= 
= +
FIG. 15.7 Inductive ac
circuit.
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FIG. 15.8 Example
15.3.
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ELI
FIG. 15.9 Waveforms for
Example 15.3.
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FIG. 15.10 Example
15.4.
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ELI
FIG. 15.11 Waveforms for
Example 15.4.
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=
=
13
 
−
FIG. 15.13 Capacitive ac
circuit.
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FIG. 15.14 Example
15.5.
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ICE
FIG. 15.15 Waveforms for
Example 15.5.
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FIG. 15.16 Example
15.6.
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ICE
FIG. 15.17 Waveforms for
Example 15.6.
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FIG. 15.18 Phasor diagrams
for Examples 15.5 and 15.6.
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FIG. 15.19 Impedance diagram.
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=
FIG. 15.20 Series
impedances.
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FIG. 15.21 Example
15.7.
21
FIG. 15.22 Impedance
diagram for Example 15.7.
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FIG. 15.23 Example
15.8
FIG. 15.24 Impedance
diagram for Example 15.8.
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
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Given 15.26, find
◦E
◦ Zt
◦ Circuit current
◦ VR
◦ VL
◦ PT
◦ Fp
FIG. 15.26 Series R-L circuit.
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FIG. 15.28 Impedance
diagram for the series R-L
circuit in Fig. 15.26.
FIG. 15.29 Phasor diagram for
the series R-L circuit in Fig.
15.26.
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The basic format for the voltage
divider rule in ac circuits is exactly the
same as that for dc circuits:
Note the desired voltage drop and the
component used in the equation are
the same
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
Series
◦


=∑
◦
Parallel

◦
Series
= ∑
= ∑
Parallel
◦
=∑
◦ Opposite of resistors
◦ Same as resistors
Inductors
Capacitors
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Given Fig 15.41, find
◦ VR
◦ VL
◦ VC
◦ IT
◦ Fp
◦ Pav
FIG. 15.43 Example
15.11.
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15.5-15.6
FIG. 15.46 Reviewing the frequency response of the
basic elements.
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FIG. 14.21 Effect of low and high frequencies on the circuit
model of an inductor.
FIG. 14.23 Effect of low and high frequencies on the circuit model
of a capacitor.
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Behavior of circuit
changes with freq
◦ More resistive, reactive

F1 midpoint on graph
◦
=
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
Given Fig 15.47, find
◦
◦
◦
◦
ZC at f1
ZC at 100 hz
ZC at 1 kHz
ZC at 20 kHz
FIG. 15.47 Determining the frequency response of a series
R-C circuit.
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FIG. 15.51 The phasor diagram for the circuit in Fig. 15.47 for
f = 0 Hz and 1 kHz.
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FIG. 15.54 The magnitude of the voltage VC
versus frequency for the circuit in Fig. 15.47.
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FIG. 15.55 The phase angle between E and VC
versus frequency for the circuit in Fig. 15.47.
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
Given Fig 15.56,
find
◦ ZL at f1
◦ ZT at 100 hz
◦ VL at 10 kHz
FIG. 15.56 Circuit for
Example 15.12.
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For series ac circuits with reactive elements:
1.The total impedance will be frequency dependent.
2.The impedance of any one element can be greater than the
total impedance of the network.
3.The inductive and capacitive reactances are always in direct
opposition on an impedance diagram.
4.Depending on the frequency applied, the same circuit can
be either predominantly inductive or predominantly
capacitive.
5.At lower frequencies, the capacitive elements will usually
have the most impact on the total impedance, while at high
frequencies, the inductive elements will usually have the
most impact.
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6.
7.
8.
9.
10.
The magnitude of the voltage across any one element can
be greater than the applied voltage.
The magnitude of the voltage across an element compared
to the other elements of the circuit is directly related to the
magnitude of its impedance; that is, the larger the
impedance of an element, the larger is the magnitude of
the voltage across the element.
The voltages across a coil or capacitor are always in direct
opposition on a phasor diagram.
The current is always in phase with the voltage across the
resistive elements, lags the voltage across all the inductive
elements by 90°, and leads the voltage across all the
capacitive elements by 90°.
The larger the resistive element of a circuit compared to the
net reactive impedance, the closer is the power factor to
unity.
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15.8-15.11
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
Conductance
=∑

,
=
=
Direct Current


Admittance
=∑
,
=
=

siemens (S)

Susceptance =
Alternating Current
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
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Given Ex. 15.14, find
◦
◦
◦
◦
◦
ZT
YR
YL
YC
YT
FIG. 15.63 Example 15.14.
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FIG. 15.67 Parallel ac
network.
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For Fig 15.68, find
◦
◦
◦
◦
◦
IT
IR
IL
PT
FP ,lead or lag
FIG. 15.68 Parallel R-L
network.
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FIG. 15.70 Admittance diagram for
the parallel R-L network in Fig.
15.68.
FIG. 15.71 Phasor
diagram for the parallel RL network in Fig. 15.68.
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FIG. 15.78 Applying phasor notation to the network
in Fig. 15.77.
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FIG. 15.79 Admittance diagram
for the parallel R-L-C network in
Fig. 15.77.
FIG. 15.80 Phasor diagram for
the parallel R-L-C network in
Fig. 15.77.
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FIG. 15.81 Waveforms for the parallel R-L-C
network in Fig. 15.77.
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=
FIG. 15.83 Example
15.16.
+
NOT Zt like
VDR
FIG. 15.84 Example
15.17.
Note the desired voltage drop and the component
used in the equation are NOT the same. Use the
other component in the pair
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Smallest resistance or impedance has
largest influence
FIG. 15.85 Frequency response for parallel R-L-C
elements.
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=
2
FIG. 15.87 The frequency response of the individual
elements of a parallel R-L network.
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FIG. 15.86 Determining the frequency response of a
parallel R-L network.
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FIG. 15.88 The magnitude of the input impedance
versus frequency for the network in Fig. 15.86.
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FIG. 15.89 The phase angle of the input
impedance versus frequency for the network in
Fig. 15.86.
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FIG. 15.93 The phasor diagram for
the parallel R-L network in Fig.
15.86 at f = 40 kHz.
FIG. 15.92 The phasor diagram
for the parallel R-L network in
Fig. 15.86 at f = 1 kHz.
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For parallel ac networks with reactive elements:
1. The total admittance or impedance will be frequency
dependent.
2. Depending on the frequency applied, the same network
can be either predominantly inductive or predominantly
capacitive.
3. The magnitude of the current through any one branch can
be greater than the source current.
4. The inductive and capacitive susceptances are in direct
opposition on an admittance diagram.
5. At lower frequencies, the inductive elements will usually
have the most impact on the total impedance, while at
high frequencies, the capacitive elements will usually have
the most impact.
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The impedance of any one element can be less than the total
impedance (recall that for dc circuits, the total resistance must
always be less than the smallest parallel resistor).
7. The magnitude of the current through an element, compared to the
other elements of the network, is directly related to the magnitude
of its impedance; that is, the smaller the impedance of an element,
the larger is the magnitude of the current through the element.
8. The current through a coil is always in direct opposition with the
current through a capacitor on a phasor diagram.
9. The applied voltage is always in phase with the current through the
resistive elements, leads the voltage across all the inductive
elements by 90°, and lags the current through all capacitive
elements by 90°.
10. The smaller the resistive element of a network compared to the net
reactive susceptance, the closer is the power factor to unity.
6.
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15.12
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
Simpler may be better
◦ Same circumstances
FIG. 15.94 Defining the equivalence between two networks at a
specific frequency.
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FIG. 15.95 Finding the series
equivalent circuit for a parallel RL network.
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FIG. 15.96 Defining the
parameters of
equivalent series and
parallel networks.
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
=

=

=

=
Parallel to Series
61
Series to Parallel
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FIG. 15.97 Example 15.18.
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FIG. 15.99 Example
15.19.
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

Do not draw the waveform sketches when
asked
Q2
◦ Answers should be in phasor form

Q6
◦ Find the individual component values (R+jX)

Q10
◦ Power DELIVERED – Uses only resistors
◦ (i) Find I, Vr, Vl, and Vc

Q22 C and E
◦ Find the TOTAL impedance (Zt)
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