CHAPTER 8
AC Sinusoidal Steady-State Signals,
Phasors, and Impedance
CHAPTER OUTLINE
8.1
The Need for Complex Numbers
(review Ch. 4)
8.2
Capacitive Reactance and Impedance
(review Ch. 5)
8.3
Inductive Reactance and Impedance
(review Ch. 6)
8.4
Response of RL and RC Series Circuits to AC
8.1 THE NEED FOR COMPLEX NUMBERS
What is Eq. (8.1)?_________________________________________________________________________
The sinusoidal voltage below is express in what domain ?__________________________________________
v(t) = VP sin(t +)
(8.1)
Identify each symbol:
v(t)=
VP =
=
=
Determine and write the expression for the sine wave shown in Figure 8.1 below:
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Class Notes Ch. 8
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Check your answer: v(t) = 155 sin(377t + 35°)
Determine vT (t) = v1(t) + v2(t) in Figure 8.2 for the sine wave shown in Figure 8.3 (v1(t) and v2(t) are given).
Figure 8.2
Figure 8.3
v1(t) = 180 sin(62,832t + 2.356)
v2(t) = 180 sin(62,832t + 1.571)
Hint: you may need the following trigonometric identity:
1

1

sin( A)  sin( B)  2 sin  ( A  B)  cos  ( A  B) 
2

2

Check: vT (t) = 333 sin(62,832t + 112.5°)
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Class Notes Ch. 8
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What does Figure 8.4 below illustrate about the sum of the two sine waves from Figure 8.3?
Was this a significant amount of work?______Is it a valid approach?______Is it an efficient approach?______
Rhetorical question: Is there a more efficient means to perform the mathematics of AC circuit analysis? Yes!
Identify Eq. (8.2):___________________________________________________________________________
c = a + bi
(8.2)
What is a? _______________________________________________________________________________
What is b?________________________________________________________________________________
What is the i?_____________________________________________________________________________
In electricity and electronics, i is the symbol for current. A different symbol is used for the imaginary part: j.
c = a + jb
(8.3)
8.1.1 Complex Numbers
A complex number is NOT a vector!
A complex number has two pieces of information: a real part and an imaginary part.
complex number = real part + j(imaginary part)  a + jb  R + jX
Why are two pieces of information needed in AC circuits?
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Class Notes Ch. 8
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Identify the real and imaginary parts on the plot on the complex number plane in Figure 8.5 below:
What angle is the imaginary part plotted with respect to the real part?_________________________________
What is j in the context of angles? ____________________________________________________________
Notation: the tilde over the symbol indicates that the number is complex. Z in rectangular form is
Z  R  jX
Identify:
(8.4)
Z =
R=
X=
j=
Compare the rectangular form to the polar form of a complex number in Figure 8.6:
Z  Z 
Identify:
(8.5)
Z =
Z=
=
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Class Notes Ch. 8
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Why are two dimensions needed in AC circuits?________________________________________________
Can an AC signal can be represented by one complex number? ______________Why?
How can one convert between the rectangular and polar forms of complex numbers? Explain each equation:
Z  R  jX  Z 
(8.6)
Z  R2  X 2
(8.7)
X
X
  tan 1    arctan  
R
 
R
(8.8)
R = Z cos 
(8.9)
X = Z sin 
(8.10)
Modern calculators perform the calculations in Equations (8.7), (8.8), (8.9), and (8.10) internally.
If complex expressions with variables are being used (Ch. 15 and 16)  manual complex number operations.
Example 8.1.1 Explain each step:
Convert 9 – j6 into polar form manually. Check the result with your scientific calculator.
_____________________:
Z  R  jX  9  j 6
_____________________:
Z  Z 
_____________________:
X
Z  R 2  X 2 ,   tan 1  
R
Solution:
Z  R2  X 2  92  62  10.817 _________________________________________________
X
 6 
  tan 1    tan 1    33.69 ________________________________________________
R
 9 
Were the signs of the numerator and the denominator important? (Hint: quadrant ) Explain.
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Z  Z     ________________________________________________________
Check this conversion on your scientific calculator.
Example 8.1.2
Convert 10.817 –33.690° into rectangular form manually. Check the result with your scientific calculator.
_____________________:
Z  Z  = 10.817 33.690
_____________________:
Z  R  jX
_____________________:
R = Z cos , X = Z sin 
Solution:
R = Z cos  = 10.817 cos(–33.690°) = +9.0003___________________________________________
X = Z sin  = 10.817 sin(–33.690°) = –6.0002____________________________________________
Z  R  jX = +9.0003  j 6.0002  9.00  j 6.00 __________________________________________
Check this conversion on your scientific calculator.
8.1.2 Complex-Number Operations (addition, subtraction, multiplication, and division)
 Operation: addition and subtraction
 Process: Use rectangular form; add or subtract the corresponding real and imaginary parts.
Z1  Z2  ( R1  jX1 )  ( R2  jX 2 )  ( R1  R2 )  j ( X1  X 2 )
Example 8.1.3
(8.11)
If Z1  5  j 7 and Z2  2  j9, find Z1  Z 2 .
Solution:
Answer: Z1  Z 2  3  j 2
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Class Notes Ch. 8
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 Operation: multiplication
 Process: Use polar form; multiply the magnitudes and algebraically add the angles.
Z1Z2  (Z1 )(Z2  )  Z1Z 2 (   )
Example 8.1.4
(8.12)
If Z1  5  j 7 and Z2  2  j9, find Z1Z 2 .
Solution:
Answer: Z1Z 2  79.3131.9
 Operation: division
 Process: Use polar form; divide the magnitude of the numerator by the magnitude of the denominator;
algebraically subtract the angle of the denominator from the angle of the numerator.
Z1 Z11 Z1


(1  2 )
Z 2 Z 2 2 Z 2
Example 8.1.5
(8.13)
Z
If Z1  5  j 7 and Z 2  2  j9, find 1 .
Z 2
Solution:
Answer:
Z1
 0.933055  23.009  0.933  23.0
Z 2
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Class Notes Ch. 8
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8.1.3 The Meaning of j
What from is the complex number 0 + j1? _______________________________________________________
What from is the complex number: 1  90 ?_____________________________________________________
Justify j  1  90
(8.14)
Explain Z  R0  X 90
(8.15)
Example 8.1.6
Evaluate
1
.
j
Evaluate
1
.
j10
Solution:
Answer:
1
j
j
Example 8.1.7
Solution:
Answer:
1
  j 0.1
j10
Now we are ready to answer “Why does j  1 ?” Explain the following equations:
j 2  (1  90)(1  90)
(8.16)
j 2  1  180
j 2  1
j 2  j  1
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(8.17)
Class Notes Ch. 8
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8.1.4 Phasors
So far: (1) Working with AC voltages and currents in the time domain is cumbersome.
(2) Complex numbers contain two pieces of information.
(3) Complex number math is algebra-like with specific rules.
How are complex numbers used in AC circuit analysis? Explain each step:
v(t) = VP sin(t + V)
(8.18)
i(t) = IP sin(t + I)
(8.19)
What is V ?______________________________________________________________________________
What is I ? ______________________________________________________________________________
Freeze time at t = 0. How are v(t) and i(t) represented in the complex number plane (Figure 8.7 below)?
A phasor is defined as _____________________________________________________________________
Is the magnitude of the complex number peak or RMS (effective)? [“trick” question] Explain.
The angle of the complex number is ___________________________________________________________
Notation:
V  VP V
I  I P I
or
V  VRMS V
I  I RMS I
(8.20)
Does the variable of time does occur in a phasor? Why are why not?
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When using phasors in a given problem, can signals be at different frequencies? Why or why not?
A phasor is a complex number representation of an AC sinusoidal steadystate signal with the sinusoidal time dependence removed but assumed.
Notice how the phasor diagram shows the relative phases of the signals.
Example 8.1.8 (fill in the steps or explanations not provided)
a.
Determine and plot the voltage and current phasors in the complex number plane for the signals shown in
Figure 8.8. Note that the x-axis is labeled in degrees, not time.
b.
Determine which signal is leading and by how many degrees.
c.
Determine the mathematical expressions for v(t) and i(t).
Given:
Figure 8.8 below
Desired:
a. V  VP V , I  I P I , plot in complex number plane
b. which signal is leading, and the phase difference  between v and i
c. v(t), i(t)
Strategy:
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Class Notes Ch. 8
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Solution:
Answers:
a.
V  110  12 V
I = 3.2+95° A
b.
 = –107°, i(t) leads v(t).
c.
Not enough information to determine
frequency  leave  as a variable
v(t) = VP sin(t + V) = 110 sin(t – 12°)
i(t) = IP sin(t + I) = 3.2 sin(t + 95°)
Example 8.1.9 Find the expression for the total voltage in the circuit shown in Figure 8.2 (copied below).
Given: v1(t) = 180 sin(62,832t + 2.356), v2(t) = 180 sin(62,832t + 1.571)
Desired:
vT (t)
Strategy:
Solution:
Explain each step:
v1(t) = 180 sin(62,832t + 2.356), V1  180  135
v2(t) = 180 sin(62,832t + 1.571), V2  180+90
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Class Notes Ch. 8
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VT  V1  V2  180135 + 18090
VT = (–127.3 + j127.3) + (0 + j180) = –127.3 + j307.3
VT  332.6 + 112.5
vT (t) = 333 sin(62,832t + 112.5°)
Is the third-to-last step necessary with modern calculators? (Try it!)__________________________________
Thus, phasors are used to add and subtract AC voltages or currents.
Name two important calculations for AC circuits where one adds and subtracts voltages and currents:
General principle for AC circuit analysis
AC circuit analysis is performed as it was in DC except one must use phasors for the voltages and the
currents and complex numbers for the V/I ratio (impedance) for R, L, and C.
8.2 CAPACITIVE REACTANCE AND IMPEDANCE
Rhetorical question: Why does phase shift occur in a circuit? Consider Figures 8.10 and 8.11 below.
What is the phase relationship for the voltage across and the current through the resistance? Explain.
How can the voltage across the capacitor change? (hint: charge)
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Current is the ______________of charge flow, not just the number of charges.
Explain the current  charge  voltage relationship for a capacitor:
Thus, does current lead or lag the voltage for a capacitor?______________________________________
Define lead: __________________________________________________________________________
If the capacitor AC voltage has leveled out at a maximum point (in either polarity), what is the current? Why?
If the capacitor AC current is maximum, how is the voltage changing? Explain.
How far (in terms of angle) should the AC current lead the voltage for a capacitor? _________________
Note: the reason for this value comes from power and energy for the capacitor (Chapter 10)
A capacitor always opposes an instantaneous change of _______________________________________
Explain why the following i–v relation is not valid for capacitors:
v(t)  C i(t)
(8.21)
What are the two aspects of how a capacitor responds to an AC signal? ________________________________
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Explain the following magnitude relationship between the voltage across and the current through a capacitor:
XC 
VP VRMS

I P I RMS
(8.22)
XC = capacitive reactance (): the magnitudes of the voltage and current form an Ohm’s law–like relationship
Does this ratio represent resistance (note the units of ohms)? Why or why not?
Then explain why the term reactance is used to describe the voltage-to-current ratio in a component that stores
energy, such as a capacitor (and an inductor, too, as covered in the next section):
If the frequency of the AC signal changes in a circuit with a capacitor, what happens to the voltage-to-current
ratio? Why?
Explain the following equations:
XC 
XC 
1
f
(8.23)
1
1

2fC C
(8.24)
How can both the magnitude and phase relationships for capacitance be incorporated into one equation?
Impedance is defined as the complex number ratio of the phasor V to the phasor I in AC circuits:
V
Z 
I
(8.25)
where Z is the impedance in ohms. What does the tilde indicate?___________________________________
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Class Notes Ch. 8
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Explain the following equations:
V V V V
Z  
 (V  I )  Z 
I
I I I
(8.26)
V
V
,  
 ( V  I )
I
I
(8.27)
Z
where:
Z =
Z=
=
Is impedance equal to resistance in general? ________ Why or why not?
Note: Equation (8.25) is still often referred to as Ohm’s law (AC form of Ohm’s law).
Notation:  will normally be reserved for the angle (phase shift) between voltage and current   = V – I
Explain the following equations:
ZC 
V V
V
1
1
  90  X C  90  0  jX C 
j
I (V  90) I
j 2fC
2fC
(8.28)
V
1
1
1
ZC  C  X C  90   jX C 
j
j
I
j 2fC
2fC
C
C
(8.29)
What is the name of ZC ?___________________________________________________________________
What is the magnitude of the impedance of a capacitor?___________________________________________
What is the real part of the impedance of a capacitance? __________________________________________
What is the imaginary part of the impedance of a capacitor? _______________________________________
What does an imaginary impedance represent in terms of energy or power?____________________________
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Class Notes Ch. 8
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For comparison, consider the i–v relation for resistance. Explain each equation:
R
VRMS VP

I RMS I P
(8.30)
V V V
V
Z R  R 
 (V  I )  Z 
I I
I
IR
(8.31)
V
V
Z R  R  0  R  R  j 0  R0
IR I
(8.32)
Is the impedance of a resistance purely real or imaginary?__________________________________________
What does it represent in terms of electrical energy?_______________________________________________
What does the imaginary part of the impedance of a resistance equal? ________________________________
What does the real part of the impedance of a resistance equal? _____________________________________
Why is the angle zero?___________________________________________________________
Now consider the i–v relation for capacitance:
V
1
1
1
ZC  C  X C  90   jX C 
j
j
I
j 2fC
2fC
C
C
Why is the negative sign required for the angle? _________________________________________________
What does the difference in angle suggest as a fundamental difference between a resistance and a reactance?
Example 8.2.1
Determine the (a) reactance and (b) impedance of a capacitor if the effective current
is 25 mA when the peak-to-peak voltage is 18 V.
Given:
Desired:
Strategy:
Solution:
Answer: ZC  255 90    j 255 
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Class Notes Ch. 8
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8.3 INDUCTIVE REACTANCE AND IMPEDANCE
Rhetorical question: Why does phase shift occur in a circuit with an inductor? Consider Figures 8.12 and 8.13
below.
How can the current through the inductor change? (hint: magnetic field)
What is the voltage induced across an inductor proportional to? What law states this?
Explain the current  magnetic field  voltage relationship for an inductor with an AC signal:
Thus, does current lead or lag the voltage for an inductor?______________________________________
If the inductor AC current has leveled out at a maximum point (in either polarity), what is the voltage? Why?
If the inductor AC voltage is maximum, how is the current changing? Explain.
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How far (in terms of angle) should the AC current lead the voltage for an inductor? _________________
Note: the reason for this value comes from power and energy for the capacitor (Chapter 10)
An inductor always opposes an instantaneous change of _______________________________________
Explain why the following i–v relation is not valid for inductors:
v(t)  L i(t)
(8.33)
Explain the following magnitude relationship between the voltage across and the current through an inductor:
XL 
VP VRMS

I P I RMS
(8.34)
where XL = _______________________________________________________________________________
Does this ratio represent resistance (note the units of ohms)? ________ Why or why not?
If the frequency of the AC signal changes in a circuit with an inductor, what happens to the voltage-to-current
ratio? Why?
Explain the following equations:
XL  f
(8.35)
X L  2 fL  L
(8.36)
V (I  90) V
Z L 
   90  X L   90  0  jX L  j 2fL
I I
I
(8.37)
V
Z L  L  X L   90   jX L   j 2fL   jL
IL
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(8.38)
Class Notes Ch. 8
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What is the name of Z L ?___________________________________________________________________
What is the magnitude of the impedance of an inductor?___________________________________________
What is the real part of the impedance of an inductor? __________________________________________
What is the imaginary part of the impedance of an inductor? _______________________________________
What does an imaginary impedance represent in terms of energy or power?____________________________
Why is the positive sign required for the angle? _________________________________________________
What does the difference in angle suggest as a fundamental difference between a resistance and a reactance?
Explain.
Example 8.3.1
Determine the (a) reactance and (b) impedance of an inductor if the effective current
is 25 mA when the peak-to-peak voltage is 18 V. (c) Compare and explain the
similarities and the differences between Examples 8.2.1 and 8.3.1.
Given:
Desired:
Strategy:
Solution:
Answer: Z L  255  90    j 255 
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Class Notes Ch. 8
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Explain the relationships demonstrated in Figure 8.14 below:
Explain the quantities and relationships demonstrated in Figure 8.15 below:
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Class Notes Ch. 8
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8.4 RESPONSE OF RL AND RC SERIES CIRCUITS TO AC
In DC circuits, resistances in series ________________.
Similarly, in AC circuits, impedances in series ___________________.
Consider the circuit in Figure 8.16:
What is the total impedance of the circuit? Explain it
using Eq. (8.39).
ZT  Z R  Z L  ( R  j 0)  (0  jX L )  R  jX L
(8.39)
Impedances must be added as _________________________________ numbers .
Example 8.4.1
Determine (a) the total impedance, (b) all phasor voltages, (c) the phasor current, and
(d) the time-domain expressions for the voltages and current of a 2 k resistor in
series with a 0.3 H inductor (as per Figure 8.16) at a frequency of 40 krads/s.
Given:
Desired:
Strategy:
a.
b.
c.
d.
Solution:
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Class Notes Ch. 8
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Answers:
b. c.d.
a.
ZT  (2  j12) k
Assume the source is the reference sine wave for phase shifts.
VS =100 V (peak)
 vS(t) = VS sin(t + S) = 10.0 sin(40,000t) V
VR = 1.64  80.5 V
 vR(t) = VR sin(t + R) = 1.64 sin(40,000t – 80.5°) V
VL = 9.86+9.5 V
 vL(t) = VL sin(t + L) = 9.86 sin(40,000t + 9.5°) V
I  0.822 80.5 mA
 i(t) = I sin(t + I) = 0.822 sin(40,000t – 80.5°) mA
Analysis:
1. Does vL(t) lead i(t) for the inductor by 90°?____________________________________
2.
Are vR(t) and i(t) in-phase for the resistor?____________________________________
3.
The sum of the peak voltages for the inductor and the resistor, 11.5 V, does not equal the
source voltage (10 V): why not?____________________________________________
4.
Example 8.4.2
Verify KVL with actual values:
VS  VR  VL  0
Determine the total impedance of a 20  resistor in series with a 3 F capacitor at a
frequency of 4000 Hz.
Given:
Desired:
Strategy:
Solution:
Answer: ZT  20.0  j13.3 
Contemporary Electric Circuits, 2nd ed., ©Prentice-Hall, 2008
Strangeway, Petersen, Gassert, and Lokken
Class Notes Ch. 8
Page 22
Learning Objectives
Discussion: Can you perform each learning objective for this chapter? (Examine each one.)
As a result of successfully completing this chapter, you should be able to:
1.
Explain why complex numbers are used in AC circuit analysis.
2.
Explain what a complex number is, what j is, and what the polar and rectangular forms of complex numbers
are.
3.
Perform complex number arithmetic calculations, including polar–rectangular form conversions.
4.
Describe what a phasor is.
5.
Convert back and forth between time domain and phasor form for AC voltages and currents.
6.
Describe and determine the reactances of inductors and capacitors.
7.
Describe and determine the impedances of resistances, inductors, and capacitors.
8.
Use phasors to determine AC voltages, currents, and impedances of components and the entire circuit for
RC and RL series circuits.
9.
Describe magnitude and phase relationships for voltages, currents, and impedances in RC and RL series
circuits.
Suggestion:
Now would be a good time to read the Chapter Review near the end of Chapter 8.
Contemporary Electric Circuits, 2nd ed., ©Prentice-Hall, 2008
Strangeway, Petersen, Gassert, and Lokken
Class Notes Ch. 8
Page 23