1. EGR 2201 Unit 11 Sinusoids and Phasors
Read Alexander & Sadiku, Chapter 9 and Appendix B.
Homework #11 and Lab #11 due next week.
Quiz next week.
-Handouts: Quiz 10, Unit 11 practice sheet.
-Set up scope & func gen to display waveforms

2. DC Versus AC
In a direct-current (DC) circuit, current flows in one direction only.
The textbook’s Chapters 1 through 8 cover DC circuits.
In an alternating-current (AC) circuit, current periodically reverses direction.
The book’s Chapters 9 through 11 cover AC circuits.

3. The Math Used in AC Circuits
Our study of AC circuits will rely heavily on two areas of math:
Sine and cosine functions
Complex numbers
We’ll review the math after introducing some terminology used in discussing AC voltages and currents.

4. Waveforms
The graph of a current or voltage versus time is called a waveform.
Example:
Note that this is an AC waveform: negative values of voltage mean the opposite polarity (and therefore opposite direction of current flow) from positive values.

5. Periodic Waveforms
Often the graph of a voltage or current versus time repeats itself. We call this a periodic waveform.
Common shapes for periodic waveforms include:
Sinusoid
Square
Triangle
Sawtooth
Image from
Sinusoids are the most important of these.
Show sine, square, & triangle waves on scope.

6. Cycle In a periodic signal, each repetition is called a cycle.
How many cycles are shown in the diagram below?
Do practice question 1a.

7. Waveform Parameters
Important parameters associated with periodic waveforms include:
Period T
Frequency f
Angular Frequency 
Amplitude Vm (or Peak Value Vp)
Peak-to-Peak Value
Instantaneous Values

8. Period
The time required for one cycle is called the waveform’s period.
The symbol for period is T.
Period is measured in seconds, abbreviated s.
Example: If a waveform repeats itself every 3 seconds, we’d write
T = 3 s
Do practice question 1b.

9. Frequency
A waveform’s frequency is the number of cycles that occur in one second.
The symbol for frequency is f.
Frequency is measured in hertz, abbreviated Hz.
Some old-timers say “cycles per second” instead of “hertz.”
Example: If a signal repeats itself 20 times every second, we’d write
f = 20 Hz

10. Period and Frequency
Period and frequency are the reciprocal of each other:
f = 1 / T
T = 1 / f
For practice with these relationships, play my Frequency-Period game.
Do practice question 1c.

11. Radians
Recall that the radian (rad) is the SI unit for measuring angle.
It is related to degrees by
 radians = 180
We’ll often need to convert between radians and degrees:
To convert radians to degrees, multiply by 180 𝜋 .
To convert degrees to radians, multiply by 𝜋 180 .

12. Angular Frequency
The quantity 2f, which appears in many equations, is called the angular frequency.
Its symbol is , and its unit is rad/s:
 = 2f
Do practice question 1d.

13. One Question, Three Answers
So we have three ways of answering the question, “How fast is the voltage (or current) changing?”
Period, T, unit = seconds (s)
Tells how many seconds for one cycle.
Frequency, f, unit = hertz (Hz)
Tells how many cycles per second.
Angular frequency, , unit = rad/s
Tells size of angle covered per second, where one complete cycle corresponds to an angle of 2 radians (which is 360).

14. Relating T, f, and 
If you know any one of these three (period, frequency, angular frequency), you can easily compute the other two.
The key equations that you must memorize are:
T = 1/f
 = 2f = 2/T

15. Amplitude (or Peak Value)
The maximum value reached by an ac waveform is called its amplitude or peak value.
Do practice question 1e.

16. Peak-to-Peak Value
A waveform’s peak-to-peak value is its total height from its lowest value to its highest value.
Many waveforms are symmetric about the horizontal axis. In such cases, the peak-to-peak value is equal to twice the amplitude.
Do practice question 1f.

17. Instantaneous Value
The waveform’s instantaneous value is its value at a specific time.
A waveform’s instantaneous value constantly changes, unlike the previous parameters (period, frequency, angular frequency, amplitude, peak-to-peak value), which usually remain constant.
Do practice question 1g.

18. Lead and Lag
When two waveforms have the same frequency but are not “in phase” with each other, we say that the one shifted to the left leads the other one.
And we say that the one shifted to the right lags the other one.
Do practice question 2 a,b.

19. Phase Angle
To quantify the idea of how far a waveform is shifted left or right relative to a reference point, we assign each waveform a phase angle .
Recall that one complete cycle corresponds to an angle of 2 radians, or 360.
A positive phase angle causes the waveform to shift left along the x-axis.
A negative phase angle causes it to shift right.
Do practice question 2c.

20. Sinusoids
A sinusoid is a sine wave or a cosine wave or any wave with the same shape, shifted to the left or right.
Sinusoids arise in many areas of engineering and science. They are the waveform used most frequently in electrical circuit theory.
The waveform we’ve been looking at is a sinusoid.

21. Amplitude, Frequency, Phase Angle
Any two sinusoids must have the same shape, but can vary in three ways:
Amplitude (maximum value)
Angular frequency (how fast the values change)
Phase angle (how far shifted to the left or right)
We’ll use mathematical expressions for sinusoids that specify these three things….
Draw three pairs of sinusoids, each differing in one of these quantities.

22. Mathematical Expression For a Sinusoid
The mathematical expression for a sinusoidal voltage looks like this: v(t) = Vmcos(t + ) where Vm is the amplitude,  is the angular frequency, and  is the phase angle (relative to some reference).
Example:
v(t) = 20 cos(180t + 30) V
Do practice questions 3a - g.

23. Calculator’s Radian Mode and Degree Mode
Recall that when using your calculator’s trig buttons (such as cos), you must pay attention to whether the calculator is in radian mode or degree mode.
Example: If the calculator is in radian mode, then cos(90) returns 0.448, which is the cosine of 90 radians.
But if the calculator is in degree mode, then cos(90) returns 0, which is the cosine of 90 degrees.

24. Caution: Radians and Degrees
In the expression for a sinusoid, v(t) = Vmcos(t + )  is usually given in degrees, but  is always given in radians per second.
Recommendation: To compute a sinusoid’s instantaneous value, leave your calculator in radian mode, and convert  to radians.
Do practice question 3h on instantaneous value, plus question 4.

25. Schematic Symbols for Independent Voltage Sources
Several different symbols are commonly used for voltage sources:
Type of Voltage Source
Symbol Used in Our Textbook
Symbol Used in Multisim Software
Generic voltage source (may be DC or AC)
DC voltage source
AC sinusoidal voltage source
Note that Multisim’s symbol for AC voltage source specifies amplitude, frequency, and phase angle.

26. Function Generator
We use a function generator to produce periodic waveforms.

27. Trainer’s Function Generator
Regular Output,
controlled by all four knobs. In this course we’ll always use this one.
No matter which one
of these you use, you
must also use the
GROUND connection.
Demo measuring a waveform on the scope, explaining how the RANGE switch and FREQUENCY knob determine the frequency. Also demo Fluke 45 as freq counter.
TTL Mode Output, controlled only by the FREQUENCY and RANGE knobs. Used in Digital Electronics courses.

28. Oscilloscope We use an oscilloscope to display waveforms.
Using it, we can measure amplitude, period, and phase angle of ac waveforms, as well as dc values, transients, and more.

29. Oscilloscope Challenge Game
The oscilloscope is a complex instrument that you must learn to use.
To learn the basics, play my Oscilloscope Challenge game at

30. Sine or Cosine?
Any sinusoidal waveform can be expressed mathematically using either the sine function or the cosine function.
Example: these two expressions describe the same waveform: v(t) = 20 sin(300t + 30) v(t) = 20 cos(300t  60)
In a problem where you’re given a mixture of sines and cosines, your first step should be to convert all of the sines to cosines.
Note that the amplitude and angular freq are the same---only the phase angle differs.

31. Trigonometric Identities Relating Sine and Cosine
You can convert from sine to cosine (or vice versa) using the trig identities sin(x + 90) = cos(x) sin(x  90) = cos(x) cos(x + 90) = sin(x) cos (x  90) = sin(x)
These identities reflect the fact that the cosine function leads the sine function by 90.

32. A Graphical Method Instead of Trig Identities
Remembering and applying trig identities may be difficult.
The book describes a graphical method that relies on the following diagram:
To use it, recall that we measure positive angles counterclockwise, and negative angles clockwise.
Do practice question 5.

33. Mathematical Review: Complex Numbers
The system of complex numbers is based on the so-called imaginary unit, which is equal to the square root of 1.
Mathematicians use the symbol i for this number, but electrical engineers use j: or
Check whether your calculator recognizes sqrt of -1.

34. Rectangular versus Polar Form
Any complex number can be expressed in three forms:
Rectangular form
Example: 3 + j 4
Polar form
Example: 5  53.1
Exponential form
Example: 5e j 53.1 or 5e j 0.927

35. Rectangular Form
In rectangular form, a complex number z is written as the sum of a real part x and an imaginary part y:
z = x + jy
Ex: z = 5 + j2

36. The Complex Plane
We often represent complex numbers as points in the complex plane, with the real part plotted along the horizontal axis (or “real axis”) and the imaginary part plotted along the vertical axis (or “imaginary axis”).
Do practice question 6.

37. Polar Form
In polar form, a complex number z is written as a magnitude r at an angle :
z = r
The angle  is measured from the positive real axis.

38. Converting Between Rectangular and Polar Forms
We will very often have to convert from rectangular form to polar form, or vice versa.
This is easy to do if you remember a bit of right-angle trigonometry.

39. Converting from Rectangular Form to Polar Form
Given a complex number z with real part x and imaginary part y, its magnitude is given by and its angle is given by

40. Inverse Tangent Button on Your Calculator
When using your calculator’s tan1 (inverse tangent) button, pay attention to whether the calculator is in degree mode or radian mode.
Also recall that the calculator’s answer may be in the wrong quadrant, and that you may need to adjust the answer by 180.
The tan1 button always returns an angle in Quadrants I or IV, even if you want an answer in Quadrants II or III.
Do practice question 7.

41. Converting from Polar Form to Rectangular Form
Given a complex number z with magnitude r and angle , its real part is given by and its imaginary part is given by
You don’t have to worry about possibly having to adjust the calculator’s answer, as we did above with rect-to-polar conversions.
Do practice question 8.

42. Exponential Form r  rej
Complex numbers may also be written in exponential form. Think of this as a mathematically respectable version of polar form.
In exponential form,  should be in radians.
Polar form
Exponential Form
r 
rej
One reason this is important: to enter numbers in polar form in MATLAB, you actually have to enter them in exponential form.
Do practice question 9.
Example:
330 
3ej/6

43. Euler’s Identity
The exponential form is based on Euler’s identity, which says that, for any ,

44. Mathematical Operations
You must be able to perform the following operations on complex numbers:
Addition
Subtraction
Multiplication
Division
Complex Conjugate

45. Addition
Adding complex numbers is easiest if the numbers are in rectangular form.
Suppose z1 = x1+jy1 and z2 = x2+jy2 Then z1 + z2 = (x1+x2) + j(y1+y2)
In words: to add two complex numbers in rectangular form, add their real parts to get the real part of the sum, and add their imaginary parts to get the imaginary part of the sum.
Do practice question 10 a.

46. Subtraction
Subtracting complex numbers is also easiest if the numbers are in rectangular form.
Suppose z1 = x1+jy1 and z2 = x2+jy2 Then z1  z2 = (x1x2) + j(y1y2)
In words: to subtract two complex numbers in rectangular form, subtract their real parts to get the real part of the result, and subtract their imaginary parts to get the imaginary part of the result.

47. Multiplication
Multiplying complex numbers is easiest if the numbers are in polar form.
Suppose z1 = r1 1 and z2 = r2 2 Then z1  z2 = (r1r2)  (1+ 2)
In words: to multiply two complex numbers in polar form, multiply their magnitudes to get the magnitude of the result, and add their angles to get the angle of the result.

48. Division
Dividing complex numbers is also easiest if the numbers are in polar form.
Suppose z1 = r1 1 and z2 = r2 2 Then z1 ÷ z2 = (r1 ÷ r2)  (1  2)
In words: to divide two complex numbers in polar form, divide their magnitudes to get the magnitude of the result, and subtract their angles to get the angle of the result.
Do practice question 10 b - d.

49. Complex Conjugate
Given a complex number in rectangular form, z = x + jy its complex conjugate is simply z* = x  jy
Given a complex number in polar form, z = r  its complex conjugate is simply z* = r 

50. Performing Complicated Operations on Complex Numbers
Solving a problem may require us to perform many operations on complex numbers.
Example: 6∠30°+(5−𝑗3) (2+𝑗4)
With a powerful calculator such as the TI-89, you can do this quickly and easily. With most calculators it’s more tedious, since you must repeatedly convert between rectangular and polar forms.
Another option is to use MATLAB.

51. Entering Complex Numbers in MATLAB
Entering a number in rectangular form: >> z1 = 3 + j  4
Entering a number in polar (actually, exponential) form: >> z5 = 3  exp(j  −30  pi / 180)
You must give the angle in radians, not degrees.
MATLAB always displays complex numbers in rectangular form, no matter how you enter them.
Do practice question 11.

52. Operating on Complex Numbers in MATLAB
Use the usual mathematical operators for addition, subtraction, multiplication, division: >> z8 = z1 + z5
>> z9 = z5 / z6 and so on.

53. Built-In Complex Functions in MATLAB
Useful MATLAB functions:
real(z1) gives z1’s real part
imag(z1) gives z1’s imaginary part
abs(z1) gives z1’s magnitude
angle(z1) gives z1’s angle in radians
conj(z1) gives z1’s complex conjugate
Since angle(z1) gives you the angle in radians, you must multiply this by 180/pi if you want the angle in degrees.
Do practice question 12.

54. Online Alternative to MATLAB
Recall that wolframalpha.com is a free online math tool.
An earlier example in MATLAB:
Same example in WolframAlpha:
>> z1 = 3 + j  4 >> z5 = 3  exp(j  −30  pi / 180)
>> z8 = z1 + z5 >> abs(z8)
>> angle(z8)  180 / pi
Note that you must use i rather than j for the imaginary unit.
Next slide shows results of this command.

55. Online Alternative to MATLAB (cont'd.)
Here is part of the result of the previous command in wolframalpha.com:
Rect. form
Polar form

56. Reminder About Calculators
In this course I’ll let you use any calculator or MATLAB on the exams.
But the Fundamentals of Engineering exam and the Principles and Practice of Engineering exam have a restrictive calculator policy.
To succeed on those exams, you must be able to do complex-number math with a “bare-bones” calculator.
Do practice questions 13 and 14.

57. Useful Properties of j
j is the only number whose reciprocal is equal to its negation:
Therefore, for example,
Also, 𝑗=190 = 𝑒 𝑗 𝜋 2
Therefore multiplication by j is equivalent to a counterclockwise rotation of 90 in the complex plane.
BEFORE CLASS, SET UP SCOPE FOR NEXT SLIDE.

58. Sinusoids Everywhere
If you connect a sinusoidal voltage source or current source to a circuit made up of resistors, capacitors, and inductors, then:
Every voltage and every current in the circuit will also be a sinusoid.
All of these sinusoids will have the same frequency.
Statement #1 is not true for sources of other waveshapes, such as triangle waves, square waves, or sawtooth waves.
Demo using R=10k, C=0.1uF, and source f=400 Hz. Display source voltage and cap voltage on scope with different waveshapes.

59. Kirchhoff’s Laws in AC Circuits
KCL and KVL hold in AC circuits.
But to apply these laws, we must add (or subtract) sinusoids instead of adding (or subtracting) numbers.
Example: In the circuit shown, KVL tells us that v = v1 + v2. Suppose that v1 = 10 cos(200t + 30) V and v2 = 4 cos(200t + 90) V How can we add those to find v?

60. Adding Sinusoids
We often need to find the sum of two or more sinusoids.
A unique mathematical property of sinusoids: the sum of sinusoids of the same frequency is always another sinusoid of that frequency.
You can’t make the same statement for triangle waves, square waves, sawtooth waves, or other waveshapes.
-Show that it’s not true for square waves.

61. Adding Sinusoids (Continued)
For example, if we add v1 = 10 cos(200t + 30) V + v2 = 4 cos(200t + 90) V we’ll get another sinusoid of the same angular frequency, 200 rad/s:
v1 + v2 = Vm cos(200t + ) V
But how do we figure out the resulting sinusoid’s amplitude Vm and phase angle ?

62. Adding Sinusoids: A Common Mistake
Here’s a common mistake that is almost guaranteed to give you the wrong answer: v1 = 10 cos(200t + 30) V v2 = 4 cos(200t + 90) V v1 + v2 = 14 cos(400t + 120) V

63. Using MATLAB to Plot the Sinusoids We’re Adding
We have v1 = 10 cos(200t + 30) V v2 = 4 cos(200t + 90) V and we’re trying to find v1 + v2.
In MATLAB:
>> fplot('10 * cos(200 * t + pi/6)', [0, 0.1]) >> hold on >> fplot(‘4 * cos(200 * t + pi/2)', [0, 0.1], 'r')
>> fplot('10*cos(200*t + pi/6) + 4*cos(200*t + pi/2)', [0, 0.1], 'g')
Turn on the grid in MATLAB (Edit > Axes Properties and then check boxes). From the MATLAB graph you can see that amplitude is approx and phase angle is approx. 50 degrees.

64. Complex Numbers to the Rescue!
One method for adding sinusoids relies on trig identities.
You learned this method if you took the course named Introductory Math for Engineering Applications (EGR 1101).
But we’ll use a simpler method, which relies on complex numbers.
In fact, the only reason we’re interested in complex numbers (in this course) is that they give us a simple way to add and subtract sinusoids.

65. Phasors
A phasor is a complex number that represents the amplitude and phase angle of a sinusoidal voltage or current.
The phasor’s magnitude r is equal to the sinusoid’s amplitude.
The phasor’s angle  is equal to the sinusoid’s phase angle.
Example: To represent the sinusoid v(t) = 10 cos(200t + 30) V we’ll use the phasor V = 1030 V

66. Time Domain and Phasor Domain
Some fancy terms:
We call an expression like cos(200t + 30) V the time-domain representation of a sinusoid.
We call 30 V the phasor-domain representation of the same sinusoid. (It’s also called the frequency-domain representation.)
Do practice problems 15 and 16.

67. Using Phasors to Add Sinusoids
To add sinusoids of the same frequency:
If any of your sinusoids are expressed using sine, convert them all to cosine.
Write the phasor-domain version of each sinusoid.
Add the phasors (which are just complex numbers).
Write the time-domain version of the resulting phasor.

68. Example of Using Phasors to Add Sinusoids
To add v1 = 10 cos(200t + 30) V v2 = 4 cos(200t + 90) V
Transform from time domain to phasor domain: V1 = 1030 V and V2 = 490 V
Add the phasors: V1 + V2 = 1030 + 490 = 12.4946.1 V
Transform from phasor domain back to time domain: v1 + v2 = cos(200t ) V
Do practice problem 17.

69. Phasor Relationships for Circuit Elements
We’ve seen how we can use phasors to add sinusoids.
Next we’ll revisit the voltage-current relationships for resistors, inductors, and capacitors, assuming that their voltages and currents are sinusoids.

70. Phasor Relationship for Resistors
For resistors we have, in the time domain:
v = iR
Example: If i = 2 cos(200t + 30) A and R = 5 , then v = 10 cos(200t + 30) V
For this same example, in the phasor domain we have: If I = 230 A and R = 5 , then V = 1030 V
So we can write V = IR.

71. What This Means
For resistors, if i is a sinusoid, then v will be a sinusoid with the same frequency and phase angle as i.
Therefore i and v reach their peak values at the same instant.
We say that a resistor’s voltage and current are in phase.

72. Summary for Resistors In the time domain: In the phasor domain:
Do practice problem 18.
In the phasor domain:

73. Phasor Relationship for Inductors
For inductors we have, in the time domain:
𝑣=𝐿 𝑑𝑖 𝑑𝑡
Example: If i = 2 cos(200t + 30) A and L = 5 H, then v = 2000 cos(200t + 120) V
For this same example, in the phasor domain we have: If I = 230 A and L = 5 H, then V = 2000120 V
So we can write V = jLI.
Question: Why is V = jLI nicer than 𝑣=𝐿 𝑑𝑖 𝑑𝑡 ?

74. What This Means
For inductors, if i is a sinusoid, then v will be a sinusoid with the same frequency as i, but i will lag v by 90.

75. Summary for Inductors In the time domain: In the phasor domain:
Do practice problem 19.
In the phasor domain:

76. Phasor Relationship for Capacitors
For capacitors we have, in the time domain:
𝑖=𝐶 𝑑𝑣 𝑑𝑡
Example: If v = 2 cos(200t + 30) V and C = 5 F, then i = 2000 cos(200t + 120) A
For this same example, in the phasor domain we have: If V = 230 V and C = 5 F, then I = 2000120 A
So we can write I = jCV.

77. What This Means
For capacitors, if i is a sinusoid, then v will be a sinusoid with the same frequency as i, but i will lead v by 90.

78. Summary for Capacitors
In the time domain:
Do practice problem 20.
In the phasor domain:

79. Summary: Textbook’s Table 9.2
Working in the frequency domain (phasor domain) is nicer than working in the time domain because we can use algebra instead of calculus and differential equations.

80. A Memory Aid
To remember whether current leads or lags voltage in a capacitor or inductor, remember the phrase
“ELI the ICEman”
(For this to make sense, you must know that E is sometimes used as the abbreviation for voltage.)

81. Impedance
The impedance Z of an element or a circuit is the ratio of its phasor voltage V to its phasor current I:
𝐙= 𝐕 𝐈
Impedance is measured in ohms.
Like resistance, impedance represents opposition to current: for a fixed voltage, greater impedance results in less current.

82. A Resistor’s Impedance
For resistors, V = IR, so a resistor’s impedance is:
𝐙= 𝐕 𝐈 =𝑅
So a resistor’s impedance is a pure real number (no imaginary part), and is simply equal to its resistance.
To emphasize this, we could write
𝐙= 𝑅+𝑗0 or
𝐙= 𝑅∠0°
Do practice problem 21a.

83. Resistors and Frequency
A resistor’s impedance does not depend on frequency, since Z=R for a resistor.
Therefore, a resistor doesn’t oppose high-frequency current any more or less than it opposes low-frequency current.

84. An Inductor’s Impedance
For inductors, V = jLI, so an inductor’s impedance is:
𝐙= 𝐕 𝐈 =𝑗𝜔𝐿
So an inductor’s impedance is a pure imaginary number (no real part).
To emphasize this, we could write
𝐙=0+𝑗𝜔𝐿 or
𝐙= 𝜔𝐿∠90°
Do practice problem 21b.

85. Inductors and Frequency
The magnitude of an inductor’s impedance is directly proportional to frequency, since Z = jL for an inductor.
Therefore, an inductor opposes high-frequency current more than it opposes low-frequency current.
Also, as 0, Z0, which is why inductors act like short circuits in dc circuits.

86. A Capacitor’s Impedance
For capacitors, I = jCV, so an inductor’s impedance is:
𝐙= 𝐕 𝐈 = 1 𝑗𝜔𝐶 =− 𝑗 𝜔𝐶
So a capacitor’s impedance is a pure imaginary number (no real part).
To emphasize this, we could write
𝐙=0− 𝑗 𝜔𝐶 or
𝐙= 1 𝜔𝐶 ∠−90°
Do practice problem 21c.

87. Capacitors and Frequency
The magnitude of a capacitor’s impedance is inversely proportional to frequency, since 𝐙= − 𝑗 𝜔𝐶 for a capacitor.
Therefore, a capacitor opposes low-frequency current more than it opposes high-frequency current.
Also, as 0, Z, which is why capacitors act like open circuits in dc circuits.

88. Summary: Impedances of the Basic Elements
Ignore this column for now.

89. Ohm’s Law Generalized
We know that in DC circuits, Ohm’s law applies only to resistors, and says: 𝑣=𝑖𝑅
In AC circuits, a generalized form of Ohm’s law replaces resistance R with impedance Z, and applies to all elements: 𝐕=𝐈𝐙
In this generalized form of Ohm’s law, V, I, and Z are complex numbers.

90. A Typical AC Circuit Problem
Suppose we want to find i(t) in this circuit.
Here are the steps:
Transform to the phasor domain (i.e., write the voltage source in phasor form Vs and find the resistor’s impedance ZR and the capacitor’s impedance ZC).
Combine ZR and ZC to find the circuit’s total impedance Zeq. (See next slide.)
Apply Ohm’s law: I = Vs ÷ Zeq.
Transform back to the time domain by converting I to i(t).

91. Combining Impedances in Series
The equivalent impedance of series-connected impedances is the sum of the individual impedances:
𝐙 𝑒𝑞 = 𝐙 1 + 𝐙 2 +…+ 𝐙 𝑁
Thus, series-connected impedances combine like series-connected resistors.
Do practice problem 23.

92. Combining Impedances in Parallel
The equivalent impedance of parallel-connected impedances is given by the reciprocal formula:
𝐙 𝑒𝑞 = 𝐙 𝐙 2 +…+ 1 𝐙 𝑁
For two impedances in parallel we can also use the product-over-sum formula: 𝐙 𝑒𝑞 = 𝐙 1 𝐙 2 𝐙 1 + 𝐙 2
Thus, parallel-connected impedances combine like parallel-connected resistors.
Do practice problems 24, 25, 26, 27.

93. Voltage-Divider Rule
As in dc circuits, the voltage-divider rule lets us find the voltage across an element in a series combination if we know the voltage across the entire series combination.
Example: In the circuit shown,
𝐕 1 = 𝐙 1 𝐙 1 + 𝐙 2 𝐕 and 𝐕 2 = 𝐙 2 𝐙 1 + 𝐙 2 𝐕

94. Current-Divider Rule
As in dc circuits, the current-divider rule lets us find the current through an element in a parallel combination if we know the current through the entire parallel combination.
Example: In the circuit shown,
𝐈 1 = 𝐙 2 𝐙 1 + 𝐙 2 𝐈 and 𝐈 2 = 𝐙 1 𝐙 1 + 𝐙 2 𝐈
Do practice problem 28.

95. Summary of Chapter 9
We’ve seen that we can apply these familiar techniques to sinusoidal ac circuits in the phasor domain:
Ohm’s law (𝐕=𝐈𝐙)
Kirchhoff’s laws (KVL and KCL)
Series and parallel combinations
Voltage-divider rule
Current-divider rule
In each case, we must use complex numbers (phasors) instead of real numbers.

96. What’s Next?
In Chapter 10 we’ll see that we can also apply these other familiar techniques in the phasor domain:
Nodal analysis
Mesh analysis
Superposition
Source transformation
Thevenin’s theorem
Norton’s theorem

97. General Procedure for Analyzing AC Circuits with Sinusoidal Sources
Transform the circuit from the time domain to the phasor domain.
Solve the problem using circuit techniques (Ohm’s law, Kirchhoff’s laws, voltage-divider rule, mesh analysis, etc.). This will involve doing math with complex numbers instead of real numbers.
Transform the resulting phasor to the time domain.

98. Terminology: Impedance, Resistance, and Reactance
Since impedance Z is a complex number, we can write it in rectangular form:
𝐙=𝑅+𝑗𝑋
We call the real part (R) the resistance.
We call the imaginary part (X) the reactance.
Impedance, resistance, and reactance are measured in ohms.

99. Impedance, Resistance, and Reactance of Single Elements
Resistor R 
Inductor
jL L
Capacitor
−𝑗 𝜔𝐶
−1 𝜔𝐶
Inductors and capacitors are called reactive elements because they have reactance but no resistance.

100. Admittance
Recall that the reciprocal of resistance is called conductance, abbreviated G:
G = 1 / R
The reciprocal of impedance is called admittance, abbreviated Y:
Y = 1 / Z
Conductance and admittance are measured in siemens (S).
Do practice problem 22.

101. More Terminology: Admittance, Conductance, and Susceptance
Since admittance Y is a complex number, we can write it in rectangular form:
𝐘=𝐺+𝑗𝐵
We call the real part (G) the conductance.
We call the imaginary part (B) the susceptance.
Admittance, conductance, and susceptance are measured in siemens.

102. Admittance, Conductance, and Susceptance of Single Elements
Resistor G 
Inductor
−𝑗 𝜔𝐿
−1 𝜔𝐿
Capacitor
jC C