Download The frequency – domain version of a Norton equivalent circuit

JAMES W. NILSSON
&
SUSAN A. RIEDEL
ELECTRIC
CIRCUITS
EIGHTH EDITION
CHAPTER 9
SINUSOIDAL
STEADY –
STATE
ANALYSIS
© 2008 Pearson Education
CONTENTS
9.1 The Sinusoidal Source
9.2 The Sinusoidal Response
9.3 The Phasor
9.4 The Passive Circuit Elements in the
Frequency Domain
© 2008 Pearson Education
CONTENTS
9.5 Kirchhoff’s Laws in the Frequency
Domain
9.6 Series, Parallel, and Delta-to-Wye
Simplifications
9.7 Source Transformations and ThéveinNorton Equivalent Circuits
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CONTENTS
9.8 The Node-Voltage Method
9.9 The Mesh-Current Method
9.10 The Transformer
9.11 The Ideal Transformer
9.12 Phasor Diagrams
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9.1 The Sinusoidal Source
A sinusoidal voltage
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9.1 The Sinusoidal Source
A sinusoidal voltage source (independent or
dependent) produces a voltage that varies
sinusoidally with time.
A sinusoidal current source (independent or
dependent) produces a current that varies
sinusoidally with time.
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9.1 The Sinusoidal Source

The general equation for a sinusoidal source is
V  Vm cos(t   )
(voltage source)
or
I  I m cos(t   )
(current source)
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9.1 The Sinusoidal Source
Vrms
Vm

2
rms value of a sinusoidal voltage source
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9.1 The Sinusoidal Source
Example: Calculate the rms value of the
periodic triangular current shown below.
Express your answer in terms of the peak
current Ip.
Periodic triangular current
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9.2 The Sinusoidal Response
The frequency, , of a sinusoidal
response is the same as the frequency of
the sinusoidal source driving the circuit.
The amplitude and phase angle of the
response are usually different from those
of the source.
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9.3 The Phasor
The phasor is a complex number that
carries the amplitude and phase angle
information of a sinusoidal function.
V  Vm e
j
P {Vm cos(t   )
Phasor transform (from the time domain to
the frequency domain)
P
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9.3 The Phasor
P
1
j
j
jt
{Vm e }  R {Vm e e }
The inverse phasor transform (from the
frequency domain to the time domain)
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9.4 The Passive Circuit Elements
in the Frequency Domain
The V-I Relationship for a Resistor
V  RI
Relationship between phasor voltage and phasor
current for a resistor
The frequency-domain equivalent circuit of a resistor
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9.4 The Passive Circuit Elements
in the Frequency Domain
A plot showing that the voltage and current at the
terminals of a resistor are in phase
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9.4 The Passive Circuit Elements
in the Frequency Domain
The V-I Relationship for an Inductor
Relationship between phasor voltage and phasor
current for an inductor
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9.4 The Passive Circuit Elements
in the Frequency Domain
A plot showing the phase relationship between
the current and voltage at the terminals of an
inductor (θi = 60°)
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9.4 The Passive Circuit Elements
in the Frequency Domain
The V-I Relationship for a Capacitor
1
V {
}.I
jC
Relationship between phasor voltage and phasor
current for a capacitor
The frequency domain equivalent circuit of a capacitor
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9.4 The Passive Circuit Elements
in the Frequency Domain
A plot showing the phase relationship between
the current and voltage at the terminals of a
capacitor (θi = 60o)
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9.4 The Passive Circuit Elements
in the Frequency Domain
Impedance and Reactance
Definition of impedance
V  ZI
Z = the impedance of the circuit element
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9.4 The Passive Circuit Elements
in the Frequency Domain
Impedance and reactance values
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9.5 Kirchhoff’s Laws in the
Frequency Domain
 Kirchhoff’s Voltage Law in the Frequency
Domain
V1  V2  ...Vn  0
 Kirchhoff’s Current Law in the Frequency
Domain
I1  I 2  ...I n  0
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9.6 Series, Parallel, and
Delta-to-Wye Simplifications
Impedances in series
Vab
Z ab 
 Z1  Z 2  ...  Z n
I
The equivalent impedance between terminals a and b
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9.6 Series, Parallel, and
Delta-to-Wye Simplifications
Admittance and susceptance values
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9.6 Series, Parallel, and
Delta-to-Wye Simplifications
Delta-to-Wye transformations
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9.7 Source Transformations and
Thévenin-Norton Equivalent Circuits
A source transformation
in the frequency domain
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9.7 Source Transformations and
Thévenin-Norton Equivalent Circuits
The frequency – domain version
of a Thévenin equivalent circuit
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9.7 Source Transformations and
Thévenin-Norton Equivalent Circuits
The frequency – domain version
of a Norton equivalent circuit
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9.8 The Node – Voltage Method
Example: Use the node-voltage method to find
the branch currents Ia, Ib, and Ic in the circuit
shown below.
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9.9 The Mesh-Current Method
Example: Use the mesh-current method to
find the voltages V1, V2, and V3 in the circuit
shown below.
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9.10 The Transformer
 The
two-winding linear transformer is a
coupling device made up of two coils
wound on the same nonmagnetic core.
 Reflected
impedance is the impedance of
the secondary circuit as seen from the
terminals of the primary circuit or vice
versa.
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9.10 The Transformer
 The
reflected impedance of a linear
transformer seen from the primary side
is the conjugate of the self-impedance
of the secondary circuit scaled by the
factor (ωM / |Z22|)2.
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9.11 The Ideal Transformer
 An
ideal transformer consists of two
magnetically coupled coils having N1 and N2
turns, respectively, and exhibiting these 3
properties:
1. The coefficient of coupling is unity (k=1).
2. The self-inductance of each coil is infinite
(L1 = L2 = ∞).
3. The coil losses, due to parasitic resistance, are
negligible.
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9.11 The Ideal Transformer
Determining the Voltage and Current Ratios
V1 V2

N1 N 2
Voltage relationship for
an ideal transformer
I1 N1  I 2 N 2
Current relationship for
an ideal transformer
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9.11 The Ideal Transformer
Determining the Polarity of the Voltage and
Current Ratios
Circuits that show the proper algebraic signs for
relating the terminal voltages and currents of an
ideal transformer
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9.11 The Ideal Transformer
Three ways to show that the turns
ratio of an ideal transformer is 5
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9.11 The Ideal Transformer
The Use of an Ideal Transformer for
Impedance Matching
Using an ideal transformer to couple
a load to a source
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9.12 Phasor Diagrams
A graphic representation of phasors
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9.12 Phasor Diagrams
Example: Using Phasor Diagrams to Analysis a Circuit.
For the circuit shown at below, use a phasor diagram
to find the value of R that will cause the current
through that resistor, iR, to lag the source current, is,
by 45° when  = 5 krad/s.
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THE END
© 2008 Pearson Education