1. Chapter 9
Sinusoids and Phasors
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2. Sinusoids
A sinusoid is a signal that has the form of the sine or cosine function.
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3. t 
radians/second
(rad/s)
f is in hertz(Hz)
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4. Phase difference:
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5. Complex Number 
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6. Phasor
a phasor is a complex number representing the amplitude and phase angle of a sinusoidal voltage or current.
Eq.(8-1)
Eq. (8-2)
and
Eq.(8-3)
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7. When Eq.(8-2) is applied to the general sinusoid we obtain
The phasor V is written as
Eq.(8-5)
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8. Two features of the phasor concept need emphasis:
Fig. 8-1 shows a graphical representation commonly called a phasor diagram.
Two features of the phasor concept need emphasis:
Phasors are written in boldface type like V or I1 to distinguish them from signal waveforms such as v(t) and i1(t).
A phasor is determined by amplitude and phase angle and does not contain any information about the frequency of the sinusoid.
Fig. 8-1: Phasor diagram
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9. Phase-domain representation Time domain representation
In summary, given a sinusoidal signal                      , the corresponding phasor representation is            . Conversely, given the phasor            , the corresponding sinusoid is found by multiplying the phasor by        and reversing the steps in Eq.(8-4) as follows:
Eq.(8-6)
Phase-domain representation
Time domain representation
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10. Properties of Phasors additive property Eq.(8-7) Eq.(8-8) Eq.(8-9)
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11. Time domain representation Phase-domain representation
derivative property
Eq.(8-10)
Time domain representation
Phase-domain representation
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12. Time domain representation Phase-domain representation
Integral property
Time domain representation
Phase-domain representation
The differences between v(t) and V:
V(t) is the instantaneous or time-domain representation, while V is the frequency or phasor-domain representation.
V(t) is a real signal which is time dependent, while V is just a supposed value to simplify the analysis
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13. Fig. 8-2: Complex exponential
The complex exponential is sometimes called a rotating phasor, and the phasor V is viewed as a snapshot of the situation at t=0.
Fig. 8-2: Complex exponential
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15. Construct the phasors for the following signals:
EXAMPLE 8-1
Construct the phasors for the following signals:
(b) Use the additive property of phasors and the phasors found in (a) to find v(t)=v1(t)+v2(t).
SOLUTION
(a) The phasor representations of v(t)=v1(t)+ v2(t) are
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16. The waveform corresponding to this phasor sum is
(b) The two sinusoids have the same frequent so the additive property of phasors can be used to obtain their sum:
The waveform corresponding to this phasor sum is j V2 1 V V1
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17. Construct the phasors representing the following signals:
EXAMPLE 8-2
Construct the phasors representing the following signals:
(b) Use the additive property of phasors and the phasors found in (a) to find the sum of these waveforms.
SOLUTION:
(a) The phasor representation of the three sinusoidal currents are
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18. (b) The currents have the same frequency, so the additive property of phasors applies. The phasor representing the sum of these current is
Fig. 8-4
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19. The sinusoid corresponding to the phasor jV is
EXAMPLE 8-3
Use the derivative property of phasors to find the time derivative of v(t)=15 cos(200t-30° ).
SOLUTION:
The phasor for the sinusoid is V=15∠-30 °  . According to the derivative property, the phasor representing the dv/dt is found by multiplying V by j .
The sinusoid corresponding to the phasor jV is
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20. Device Constraints in Phasor Form
Resistor: Re
jIm I V
Voltage-current relations for a resistor in the: (a) time domain, (b) frequency domain.
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21. Device Constraints in Phasor Form
Inductor: 
Voltage-current relations for an inductor in the: (a) time domain, (b) frequency domain.
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22. Device Constraints in Phasor Form
Capacitor: 
Voltage-current relations for a capacitor in the: (a) time domain, (b) frequency domain.
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23. Connection Constraints in Phasor Form
KVL in time domain
Kirchhoff's laws in phasor form (in frequency domain)
KVL: The algebraic sum of phasor voltages around a loop is zero.
KCL: The algebraic sum of phasor currents at a node is zero.
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24. The Impedance Concept V=ZI or Z= V/I Eq.(8-16)
The IV constraints are all of the form
V=ZI or Z= V/I Eq.(8-16)
where Z is called the impedance of the element
The impedance Z of a circuit is the ratio of the phasor voltage V to the phasor current I, measured in ohms()
The impedance is inductive when X is positive
is capacitive when X is negative
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25. The Impedance Concept
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26. EXAMPLE 8-5
Fig. 8-5
The circuit in Fig. 8-5 is operating in the sinusoidal steady state with                           and                     . Find the impedance of the elements in the rectangular box.
SOLUTION:
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28. The Admittance Concept
The admittance Y is the reciprocal of impedance, measured in siemens (S)
Y=G+jB
Where G=Re Y is called conductance and B=Im Y is called the susceptance
How get Y=G+jB from Z=R+jX ?
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30. Basic Circuit Analysis with Phasors
Step 1: The circuit is transformed into the phasor domain by representing the input and response sinusoids as phasor and the passive circuit elements by their impedances.
Step 2: Standard algebraic circuit techniques are applied to solve the phasor domain circuit for the desired unknown phasor responses.
Step 3: The phasor responses are inverse transformed back into time-domain sinusoids to obtain the response waveforms.
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31. Series Equivalence And Voltage Division
where R is the real part and X is the imaginary part
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32. (a) Transform the circuit into the phasor domain.
EXAMPLE 8-6
Fig. 8-8
The circuit in Fig is operating in the sinusoidal steady state with
(a) Transform the circuit into the phasor domain.
(b) Solve for the phasor current I.
(c) Solve for the phasor voltage across each element.
(d) Construct the waveforms corresponding to the phasors found in (b) and (c)
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33. SOLUTION:
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34. PARALLEL EQUIVALENCE AND CURRENT DIVISION
Rest of the circuit Y1 Y2 YN I V I1 I2 I3
phasor version of the current division principle
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35. (a) Transform the circuit into the phasor domain.
EXAMPLE 8-9
Fig. 8-13
The circuit in Fig is operating in the sinusoidal steady state with iS(t)=50cos2000t mA.
(a) Transform the circuit into the phasor domain.
(b) Solve for the phasor voltage V.
(c) Solve for the phasor current through each element.
(d) Construct the waveforms corresponding to the phasors found in (b) and (c).
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36. SOLUTION:
(a) The phasor representing the input source current is Is=0.05∠0° A. The impedances of the three passive elements are
Fig. 8-14
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37. And the voltage across the parallel circuit is
The current through each parallel branch is
The sinusoidal steady-state waveforms corresponding to the phasors in (b) and (c) are
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38. EXAMPLE 8-10
Fig. 8-15
Find the steady-state currents i(t), and iC(t) in the circuit of Fig (for Vs=100cos2000t V, L=250mH, C=0.5  F, and R=3k ).
SOLUTION:
Vs=100∠0°
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41. Y←→△ TRANSFORMATIONS The equations for the △ to Y transformation are
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42. The equations for a Y-to- △ transformation are
when Z1=Z2=Z3=ZY or ZA=ZB=ZC=ZN.
ZY=ZN /3 and ZN =3ZY
balanced conditions
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43. EXAMPLE 8-12
Use a △ to Y transformation to solve for the phasor current IX in Fig
SOLUTION:
ABC △ to Y
Fig. 8-18
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