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Unit 3: Introduction to Alternating Current and Voltage


In EET 150 you learned how to analyze circuits containing DC (direct-current) sources. In this course you'll learn how to analyze circuits containing AC (alternating-current) sources. AC electricity is more complicated than DC, so this unit will introduce some concepts and terms that we'll use for the rest of the course.

Actually, much of the AC material in this unit should be a review of things that you studied in EET 114. But we will pick up some new ideas, such as angular frequency and a way to write mathematical expressions for sinusoidal waves. For instance, here's a mathematical expression that describes a particular sinusoidal voltage:

     v = 4.62  V sin(968t + 50°).

At first sight, this may look complicated, but with a bit of practice you'll be able to interpret and use expressions of this kind.

In this unit we'll get a start on analyzing AC circuits. Fortunately, resistors in AC circuits behave pretty much the same way they behave in DC circuits. Therefore, you can use circuit rules that you already know--such as Ohm's Law, Kirchhoff's Voltage Law, and Kirchoff's Current Law--in pretty much the same way that you used them in DC circuits. (But as we'll see in later units, inductors and capacitors behave one way in DC circuits, and they behave entirely differently in AC circuits.)

First, read the following material in Thomas Floyd's Principles of Electric Circuits (8th edition):

  • Introduction to Alternating Current and Voltage (Chapter 11)

Then work through the e-Lesson below.

After you finish the e-Lesson, you'll be ready to take Quiz #3, perform Lab #3, and do Homework #3.


Unit 2 Review
  • This unit will build on material that you studied in Unit 2. So let's begin by taking this self-test to review what you learned in that unit.
  • Self-test

DC and AC
  • Direct current (DC) is current that flows in one direction only.
  • A DC voltage source is a voltage source that produces direct current.
    • Examples: Batteries and dc power supplies (such as the power supply built into the trainer that you use in lab) are DC voltage sources.
  • Alternating current (AC) is current whose direction periodically reverses.
  • An AC voltage source is a voltage source that produces alternating current.
    • Examples: Electrical outlets in the walls of your home provide alternating current. The trainer that you use in lab also contains an AC voltage source called a function generator.
  • Wisconsin Online learning object
Waveform
  • In most DC circuits, current and voltage remain constant as time passes. But in AC circuits the voltage and current change as time passes.
  • In Unit 2 of this course we'll see how to write down mathematical expressions that describe how the values change in time. But a simpler and more common way is to draw diagrams showing how the voltage or current changes in time.
  • Such a graph of a current or voltage versus time is called a waveform. Below are several examples of voltage waveforms. Notice that each of these diagrams plots voltage (measured in volts or millivolts) on the vertical axis, and time (measured in microseconds or milliseconds) on the horizontal axis.
  • First, here is an example of a triangle waveform. In a triangle waveform, voltage increases gradually along a straight line to its maximum value, and then gradually decreases along a straight line to its lowest value, and then starts increasing again.
    Triangle wave
  • Next, here is an example of a square waveform. In a square waveform, voltage remains constant at a high value for a while, then suddenly drops to a low value, where it stays for a while before suddenly jumping back to its high value.
    Square wave
AC versus Pulsating DC
  • Both of the examples just shown are AC waveforms, because the voltage actually changes polarity. (In other words, the voltage is positive sometimes and negative sometimes.)
  • On the other hand, you could have a waveform whose polarity does not change, even though its value does change as time passes. This would be a pulsating DC waveform. Here is an example:
    Pulsating DC triangle wave
Sine Wave
  • The waveform studied most frequently in electrical circuit theory is the sine wave. Here's an example:
    Sine wave
  • It's called a sine wave because this is the same shape that you get if you make a plot of the mathematical function y = sin(x), with x plotted on the horizontal axis and y on the vertical axis.
Sinusoid
  • Strictly speaking, a sine wave must pass through the origin (the point where the x-axis crosses the y-axis).
  • The more general term sinusoid is used to describe any waveform that has the same shape as a sine wave but that may be shifted to the right or to the left along the x-axis. The waveform shown just above is an example of a sinusoid. So is the following waveform, which has the same shape but is shifted horizontally so that it does not pass through the origin:
    Sinusoid
Sinusoids In, Sinusoids Out
  • Here's a remarkable feature of sinusoidal waveforms that makes them particularly easy to work with. Suppose you connect a function generator to any circuit containing resistors, inductors, and capacitors. If the function generator is set to produce a sinusoidal waveform, then every voltage drop and every current in the circuit will also be a sinusoid.
  • The same thing is not true for waveforms of other shapes. For instance, if the function generator is set to produce a triangle waveform, and if the circuit contains any inductors or capacitors, then the voltage drops across the components will be complicated waveforms, not simple triangle waveforms.
  • Fortunately, it turns out that sinusoids are not only the easiest waveforms to work with, they're also the most useful. Therefore we'll concentrate primarily on analyzing AC circuits that have sinusoidal voltage sources, rather than triangle or square-wave or other voltage sources.
Periodic Waveform
  • A periodic waveform is a waveform whose values are repeated at regular intervals.
  • All of the waveforms shown above are periodic waveforms.

Waveform Parameters
  • Important parameters associated with periodic waveforms include:
    • Period
    • Frequency
    • Instantaneous Value
    • Peak Value
    • Peak-to-Peak Value
    • RMS Value (also called effective value)
    • Average Value
  • Each of these terms is explained below.
Cycle
  • The plot of a periodic waveform shows a regularly repeating pattern of values, each of which is called a cycle.
  • Example: In the picture of the sine wave shown below, we see a little less than two full cycles. The first cycle extends from 0 ms to 50 ms, and the second (incomplete) cycle extends from 50 ms to the edge of the chart, where it is cut off:
    Sine wave
  • Self-test
Period
  • The time required for the values to rise and fall through one complete cycle is called the period of the waveform.
  • The symbol for period is T.
  • Period is measured in units of seconds, abbreviated s.
  • Example: The sine wave shown above has a period of 50 ms.
  • Self-test
Frequency
  • The frequency of an ac waveform is the number of cycles that occur in one second.
  • The symbol for frequency is f.
  • Frequency is measured in units of cycles per second, or Hertz, abbreviated Hz.
How Period and Frequency Are Related
  • Period and frequency are the reciprocal of each other:
    f = 1 ÷ T

    and
    T = 1 ÷ f
  • Example: The sine wave shown earlier has a period of 50 ms. Therefore, its frequency is 20 Hz.
  • Self-test
Instantaneous Value
  • The instantaneous value of an ac waveform is its value at a specific instant of time.
  • We'll see below how you can use the mathematical expression for a waveform to find the waveform's instantaneous values at specific times.
  • You can read off approximate instantaneous values from the graph of a waveform.
  • Example: At 20 ms, the sine wave shown earlier has an instantaneous value of about 300 mV.
  • Self-test
Peak Value
  • The maximum value reached by an ac waveform is called its peak value.
    • If the waveform is a voltage waveform, then its peak value is also called its peak voltage, abbreviated Vp.
    • If the waveform is a current waveform, then its peak value is also called its peak current , abbreviated Ip.
  • The peak value of a waveform is sometimes also called its amplitude, but the term “peak value” is more descriptive.
  • Example:The sine wave shown earlier has a peak value of 500 mVp. Notice that I write a "p" after the unit to show that I'm talking about a peak value.
  • Self-test
Peak-to-Peak Value
  • The peak-to-peak value is the difference between a waveform's positive peak value and its negative peak value.
    • If the waveform is a voltage waveform, then its peak-to-peak value is also called its peak-to-peak voltage, abbreviated Vpp.
    • If the waveform is a current waveform, then its peak-to-peak value is also called its peak-to-peak current , abbreviated Ipp.
  • If the waveform is symmetrical about the time axis, then the peak-to-peak value equals twice the peak value.
  • Example: The sine wave shown earlier has a peak-to-peak value of 1 Vpp. That's the difference between the maximum positive value (which is 500 mV) and the maximum negative value (which is -500 mV). Notice that I wrote a "pp" after the unit to show that I'm talking about a peak-to-peak value.
  • Self-test
p and pp
  • As mentioned in the two examples above, we write p as the subscript of a quantity or unit to show that we're talking about a peak value, and we write pp as the subscript of a quantity or unit when we're talking about a peak-to-peak value.
  • Example: For the sine wave above, we could write

    Vp = 500 mVp

    or

    Vpp = 1 Vpp.

  • Similarly, if we were dealing with a current waveform whose peak value is 20 mA and whose peak-to-peak value is 40 mA, we could write

    Ip = 20 mAp

    or

    Ipp = 40 mApp.

  • Some other textbooks use pk (instead of p) as the abbreviation for peak values, and p-p (instead of pp) as the abbreviation for peak-to-peak values. So you may see these other abbreviations used from time to time.

Oscilloscope
  • The oscilloscope is an instrument designed to display waveforms. Using it, you can measure period, frequency, peak values, peak-to-peak values, and other important quantities.
  • Shown here is a Tektronix 2213, one of the types of oscilloscopes that we have in Sinclair's electronics labs. At the left is the screen on which waveforms are displayed. To the right are the knobs and switches that you can adjust to control the waveform's appearance.
    Oscilloscope--click for larger photo
  • Below are three separate photos showing how a triangle wave, a square wave, and a sine wave look on the oscilloscope screen.
    Triangle wave displayed on the oscilloscope

    Square wave displayed on the oscilloscope

    Sine wave displayed on the oscilloscope
Using the Oscilloscope to Measure Voltage
  • The oscilloscope displays a graph of voltage versus time, with voltage plotted on the vertical axis and time plotted on the horizontal axis.
  • To measure a waveform's peak-to-peak voltage, you count how many vertical divisions (squares) the waveform covers on the oscilloscope's screen, and then you multiply this number times the setting of the oscilloscope VOLTS-PER-DIVISION knob.
  • This learning object will show you how to do it:
  • Wisconsin Online learning object
Using the Oscilloscope to Measure Period and Frequency
  • Remember, the oscilloscope displays a graph of voltage versus time, with voltage plotted on the vertical axis and time plotted on the horizontal axis.
  • To measure a waveform's period, you count how many horizontal divisions (squares) the waveform covers on the oscilloscope's screen, and then you multiply this number times the setting of the oscilloscope SECONDS-PER-DIVISION knob.
  • Once you know the waveform's period, you can use the formula f = 1 ÷ T to find its frequency.
  • This learning object will show you how to do it:
  • Wisconsin Online learning object
Oscilloscope Challenge Game
  • The oscilloscope is a complicated piece of equipment. You'll need plenty of practice to learn how to use it correctly.
  • To brush up on your oscilloscope skills, take some time right now to play Oscilloscope Challenge. In particular, work through the game's "Study" section, which is a tutorial containing several pages of notes to help you identify and use the oscilloscope's controls. This will be a good preparation for Lab #2, in which you'll use a real oscilloscope to make measurements.

RMS Value (or Effective Value)
  • The root-mean-square (rms) value or effective value of an ac waveform is a measure of how effective the waveform is in producing heat in a resistance.
  • Example: If you connect a 5 Vrms source across a resistor, it will produce the same amount of heat as you would get if you connected a 5 V dc source across that same resistor. On the other hand, if you connect a 5 V peak source or a 5 V peak-to-peak source across that resistor, it will not produce the same amount of heat as a 5 V dc source.
  • That's why rms (or effective) values are useful: they give us a way to compare ac voltages to dc voltages.
  • To show that a voltage or current is an rms value, we write rms after the unit: for example, Vrms = 25 V rms.
  • A multimeter set to AC mode measures rms values.
Relationship Between Peak Values & RMS Values
  • You need to be able to convert peak values to rms values, and vice versa. There are some standard conversion factors that let you do this.
  • For a sine wave, to convert from peak values to rms values, use these equations:

    For voltages, Vrms ≈ 0.707 × Vp

    For currents, Irms ≈ 0.707 × Ip

    To convert in the other direction (from rms values to peak values), use these equations:

    For voltages, Vp ≈ 1.414 × Vrms

    For currents, Ip ≈ 1.414 × Irms

  • Note: the "squiggly" equals sign means approximately equal. These approximate conversion factors are close enough for our purposes.
  • These equations are valid only for sinusoidal waveforms. For other wave shapes, there are other numbers (which we won't need in this course) that you would use to convert between peak values and rms values.
  • Self-test
True rms Meter
  • Some inexpensive multimeters measure peak values of AC waves, and then use the equations given above to compute rms values.  Since these equations hold only for sine waves, these meters give incorrect rms values for non-sinusoidal waves.
  • A true rms meter gives correct rms value for any AC wave. The multimeters in Sinclair's EET labs are true rms meters.

Average Value
  • The average value of a waveform is the average of its values over a time period.
  • Any waveform that is symmetrical about the time axis has zero average value over a complete cycle.
    • For example, consider the sine wave shown below. Over one cycle, this waveform's positive values exactly cancel out its negative values, so its average value over a complete cycle is zero.
    • Sometimes, though, it's useful to refer to the waveform's average value over a half cycle. So, just looking at the positive "hump" of the sine wave below, what is its average value? Using calculus, it can be shown that a sine wave's average value over a half ccyle is equal to 0.636 times its peak value.
      Sine wave
  • The average value of a waveform is also called its DC value.
  • When a waveform is measured with a multimeter set to DC mode, the meter indicates the waveform's average value over a complete cycle, which would be zero for the sine wave pictured above.
  • Wisconsin Online learning object

Phase of a Sine Wave
  • The pictures of sinusoidal waveforms shown above had voltage on the vertical axis and time on the horizontal axis. That's a useful picture when you're interested in knowing how the voltage changes as time goes by.
  • Recalling that one complete cycle of a sine wave is generated when a generator's rotor rotates through an angle of 360° (a full rotation), here's another way of plotting a sine wave:
    Sine wave showing degrees of phase
  • Here we're plotting voltage on the vertical axis, just as before, but now we're plotting degrees of the rotor's rotation on the horizontal axis. Notice that one complete cycle of the sine wave corresponds to 360°.
  • The quantity on the horizontal axis in this sort of picture is called the phase of the sine wave. For instance, we can see that this particular sine wave has a voltage of 0 V when its phase is 0°, and a voltage of 300 V when its phase is 90°, and so on.
  • Wisconsin Online learning object
Radians
  • Often we measure angles in degrees, but the radian (rad) is another unit for measuring angles. Since both units are widely used, it's important to be comfortable with both units and to be able to convert from degrees to radians (or vice versa).
  • A full circle is equal to 360°, and it's also equal to 2p radians. And since p is approximately equal to 3.14, this means that 360° is approximately equal to 6.28 radians. As an equation:

    360° = 2p rad ≈ 6.28 rad

  • Dividing both sides of that equation by 2, we can also see that

    180° = p rad ≈ 3.14 rad

  • Dividing both sides again by 2, we can also see that

    90° = p/2 rad ≈ 1.57 rad

  • Using these equalities, we can say that the sine wave pictured above has a voltage of 300 V when its phase is p/2 rad, and a voltage of 0 V when its phase is p rad, and so on.
Converting Between Degrees and Radians
  • To convert any angle from radians to degrees, multiply by 180 ÷ p.
  • In the other direction, to convert from degrees to radians, multiply by p ÷ 180.
  • Self-test
Phase Shift
  • Often we'll find ourselves dealing with two or more sinusoids that have the same frequency, one of which is shifted to the right or the left of the other one. For example:
    Two sinusoids with a phase shift
  • In the case pictured here, notice that the pink sinusoid is rising through zero (the horizontal axis) at the same time that the blue sinusoid is reaching its maximum value. Therefore the pink sinusoid has a phase of 0° at the same time that the blue sinusoid has a phase of 90°.
  • We express this by saying that there's a 90° phase shift (also called a 90° phase angle) between the two waveforms.
Lead and Lag
  • In cases like the one shown above, we're interested not only in how far apart the two waveforms are shifted, but which one "comes before" the other one.
  • The waveform shifted farther to the left is said to lead the other waveform.
    • So in the picture shown above, the blue waveform leads the pink waveform by 90°.
  • As another way of saying the same thing, the waveform shifted farther to the right is said to lag the other waveform.
    • So in the picture shown above, the pink waveform lags the blue waveform by 90°.
  • Self-test
Using the Oscilloscope to Measure Phase Shifts
  • By displaying two waveforms simultaneously and then measuring the time interval between corresponding points on the two waveforms, we can determine the phase shift between them.
  • In particular, if T is the period of the waveforms, and t is the time interval between corresponding points on the two waveforms, then the phase shift φ is given by the equation:

    φ = (t ÷ T) × 360°

  • Self-test
  • The following animated learning object also shows how to measure phase angles. Their explanation is a little different from mine, but you should be able to see it's the same idea. They first count the number of "spaces" (by which they mean hashmarks) in a complete cycle, then divide that by 360° to find the number of degrees per "space," and then multiply that times the number of "spaces" between the two waveforms. The end result will be the same that you'd get using the formula φ = (t ÷ T) × 360°.
  • Wisconsin Online learning object

Lowercase and Uppercase
  • From EET 150 you know that uppercase letters are used to represent most DC quantities.
    • For example, V represents DC voltage, and I represents DC current.
  • On the other hand, most AC quantities are represented by lowercase letters.
    • For example, v represents AC voltage, and i represents AC current.
  • Within AC, uppercase letters are also used to represent constants that don't change with time.
    • For example, Vp represents peak voltage, and Ip represents peak current. Peak voltage and peak current don't continually change as time passes, so that's why they're represented by uppercase letters.
Mathematical Expression for a Sine Wave
  • The mathematical expression for a voltage sine wave with peak value Vp is

    v = Vp sin(θ)

  • Here the Greek letter θ (theta) stands for the phase of the sine wave. So in this expression we're multiplying the peak value times the sine of the wave's phase.
  • For example, in the sine wave shown below, the peak value is 300 V, so the expression for this sine wave is

    v = 300 V sin(θ)

    Sine wave showing degrees of phase
  • In this graph we're plotting θ on the horizontal axis and v on the vertical axis.
  • With this expression, you can use your calculator to compute instantaneous values of a sine wave at a particular phase. For example, the wave shown above has an instantaneous value of 282 V at a phase of 110°, since

    300 V × sin(110°) = 282 V

  • Similarly, the expression for a current sine wave with peak value Ip is

    i = Ip sin(θ)

  • Self-test
Mathematical Expression for a Phase-Shifted Wave
  • Above you learned that a phase-shifted sinusoid is one that is shifted to the right or to the left along the horizontal axis, so that the wave does not pass through the origin.
  • Mathematically, this is shown by having a fixed angle added to θ in the expression for the waveform. In other words, instead of having an expression of the form

    v = Vp sin(θ)

    we'll have an expression that looks like this:

    v = Vp sin(θ + φ)

    where φ is a constant angle called (not surprisingly) the phase shift.
  • Note: φ is the Greek letter phi, pronounced "fie."
  • For example, here is the graph of v = 300 V sin(θ + 90°):
    Sine wave with a phase shift of +90 degrees
  • Notice that this graph has the same shape as the sine wave shown just above, but this one is shifted 90° (one-quarter of a cycle) to the left.
  • A positive phase shift causes the waveform to shift left along the x-axis, and a negative phase shift causes it to shift right.
  • As another example, here is the graph of v = 300 V sin(θ − 90°):
    Sine wave with a phase shift of -90 degrees
  • Notice that this graph has the same shape as the sinusoids shown above, but this one is shifted 90° (one-quarter of a cycle) to the right of the origin.
  • In each of these diagrams, we're looking at only one cycle of the waveform. You should imagine the wave extending indefinitely to the left and to the right.
  • Self-test

Phasors
  • A phasor is a vector that represents an AC electrical quantity, such as a voltage waveform or a current waveform.
  • The phasor's length represents the voltage's or current's peak value.
  • The phasor's angle represents the voltage's or current's phase.
  • For example, in the following diagram, think of the phasor as representing a particular voltage waveform that has, let's say, a peak voltage of 4 V p. If you wanted to draw on this diagram another phasor representing a voltage waveform with a peak voltage of 8 V p, then you would draw a new phasor that's twice as long as the original one.
Rotating Phasors
  • Phasors have two main uses in studying AC circuits. In the first use, we imagine a phasor rotating counterclockwise around the origin at a speed that depends on the waveform's frequency. (Higher-frequency waveforms rotate more quickly than lower-frequency waveforms.)
  • As the phasor rotates, the waveform's instantaneous value at any time is equal to the phasor's y-component.
  • For example, looking at our sample diagram again, imagine the phasor to be rotating counterclockwise around the origin, and think of this diagram as a "snapshot" of the phasor at one instant in time. As we know from above, the phasor's length represents the waveform's peak voltage, Vp. The phasor's y-component, which is equal to Vp sin(θ), represents the waveform's instantaneous voltage.
Phasor Diagrams
  • The second, and more important, use of phasors is to represent the relationship between two or more waveforms with the same frequency.
  • For example, consider the following diagram, which shows two phasors labeled v1 and v2.
    • Phasor v1 is drawn at an angle of 0°, and it has a length of 10 units.
    • Phasor v2 is drawn at an angle of 45°, and it's half as long as v1. (Measure it with a ruler if you don't believe me.)

  • Such a diagram might represent two voltage waveforms in a circuit. From the diagram we can see that v1's peak voltage is twice as great as v2's peak voltage. (Assuming that the units are V, then v1 has a peak voltage of 10 V p, which means that v2 must have a peak voltage of 5 V p.) We can also see that v2 leads v1 by a phase shift of 45°.
  • In terms of the equations for sinusoidal waveforms that you studied in the previous unit, this diagram would then be a pictorial representation of the equations

    v1 = 10 V sin(θ)

    v2 = 5 V sin(θ + 45°)

  • The equations and the phasor diagram convey the same information, but the diagram can be easier to understand and interpret, especially in cases where you're dealing with half a dozen waveforms instead of just two.
  • The same information can also be conveyed using a sinusoidal diagram such as the following:
  • Carefully compare the phasor diagram, the equations, and the sinusoidal diagram given above, until you're convinced that they all describe the same pair of waveforms.
  • So we can use sinusoidal diagrams, mathematical equations, or phasor diagrams to describe the relationships among waveforms in a circuit. Of these three representations, phasor diagrams are the probably the easiest to use and to understand, so they are widely used in AC circuit analysis.
  • Self-test
Angular Frequency
  • As mentioned above, when you imagine a phasor rotating about the origin, the waveform's frequency determines the speed of the phasor's rotation. In particular, if the waveform's frequency is f, then the phasor will rotate with an angular speed of 2pf.
  • This quantity 2pf, which appears in many equations, is called the waveform's angular frequency.
  • Its symbol is ω, and its unit is radians per second (rad/s):

    ω = 2pf

  • Of course, since a waveform's frequency is equal to the reciprocal of its period (f = 1 ÷ T), we can also write

    ω = 2p ÷ T

  • So if you know a waveform's frequency or period, you can easily compute its angular frequency (or vice versa).
  • Note that ω is the Greek letter omega; it's not a w.
  • Self-test
Vp sin(ωt)
  • Until now, we've been writing the mathematical expressions for voltage waveforms as

    v = Vp sin(θ)

  • Recall that here Vp is a constant equal to the waveform's peak voltage. On the other hand, θ is a variable that changes continually as time passes. It can also be shown that θ=ωt, where ω is the waveform's angular frequency and t is time (measured in seconds).
  • Making this subsitution for θ, the expression for a voltage sine wave with peak value Vp and angular frequency ω becomes

    v = Vp sin(ωt)

  • Similarly, for a current sine wave with peak value Ip and angular frequency ω,

    i = Ip sin(ωt)

  • Self-test
Instantaneous Value
  • The instantaneous value of an ac waveform is its value at a specific instant of time. You can use the mathematical expression for a waveform to find the waveform's instantaneous values at specific times.
  • Example: The instantaneous value of 10 V sin(377t) at time 3 s is equal to 266 mV, since
    10 x sin(377 x 3) = 266 mV
  • Since ω is in rad/s, your calculator must be in Radians mode when you do this calculation.
  • Self-test
General Form of a Sinusoid
  • We've just seen that the mathematical expression for a voltage or current sine wave contains two important pieces of information: the wave's peak value (Vp or Ip) and its angular frequency (ω).
  • If we now add in the possibility that the wave may also have a phase shift (φ), we find that the general expression for a sinusoidal voltage or current is

    v = Vp sin (ωt + φ)

    or

    i = Ip sin (ωt + φ)

  • Often the phase angle φ is given in degrees, but the angular frequency ω is almost always given in radians per second. So, before you can use the calculator to do a computation involving these quantities, you must convert φ from degrees to radians. If you don't do this, you'll be mixing degrees with radians, and you'll get the wrong answer. (That would be like trying to add 2 inches plus 10 centimeters without first converting one of those quantities so that they both have the same unit.)
  • Of course, since we know that ω = 2pf, we could also write these two equations as

    v = Vp sin (2pft + φ)

    i = Ip sin (2pft + φ)

  • Self-test

Analyzing AC Circuits
  • In later units we'll find that analyzing an AC circuit can get pretty tricky if the circuit contains capacitors or inductors. But if the circuit contains only resistors, then you can analyze it using the same techniques you've learned for DC circuits, with a couple of slight changes mentioned below. Once you understand how to apply Ohm's law, Kirchhoff's Laws, and the power laws to DC circuits, you already know almost everything you need to analyze resistive AC circuits.
Sinusoids In, Sinusoids Out
  • Here's a remarkable feature of sinusoidal waveforms that makes them particularly easy to work with. Suppose you connect a function generator to any circuit containing resistors, inductors, and capacitors. If the function generator is set to produce a sinusoidal waveform, then every voltage drop and every current in the circuit will also be a sinusoid.
  • The same thing is not true for waveforms of other shapes. For instance, if the function generator is set to produce a triangle waveform, and if the circuit contains any inductors or capacitors, then the voltage drops across the components will be complicated waveforms, not simple triangle waveforms.
  • Fortunately, it turns out that sinusoids are not only the easiest waveforms to work with, they're also the most useful. Therefore we'll concentrate primarily on analyzing AC circuits that have sinusoidal voltage sources, rather than triangle or square-wave or other voltage sources.
Phase in Resistors
  • The voltage across any resistor and the current through that resistor have the same phase angle.  They reach their peak values at the same instant.
  • We say that the resistor's voltage and current are in phase with each other.
  • For instance, in the diagram below, suppose the tall blue waveform represents the voltage across a resistor, and the short purple waveform represents the current through that resistor. These two waveforms are in phase with each other, since they reach their peak values at the same instant.
    Phase relation of current and voltage in a resistor
  • This means that in a phasor diagram, the phasor for a resistor's voltage must point in the same direction as the phasor for that resistor's current.
  • Self-test
Ohm's Law for Resistors
  • Ohm's law can be applied to resistors in AC circuits:

    Ip = Vp ÷ R
  • In words, this says that a resistor's peak current is equal to the resistor's peak voltage drop divided by the resistor's resistance.
  • The same law also applies to peak-to-peak values and rms values:
    Ipp = Vpp ÷ R
    and
    Irms = Vrms ÷ R
  • Of course, you can also rearrange any of these equations algebraically if you wish to solve for voltage or resistance. For example, if you wish to calculate peak voltage based on known values of resistance and peak current, you would rearrange the first equation to the form Vp = Ip × R.
  • Self-test
KVL and KCL for AC Circuits
  • Recall from your studies of DC circuits that Kirchhoff's Voltage Law (KVL) says that the sum of the voltage drops around any closed loop in a circuit equals the sum of the voltage rises around that loop.
  • Recall also that Kirchhoff's Current Law (KCL) says that the sum of all currents entering a point is equal to the sum of all currents leaving that point.
  • You can also apply KVL and KCL to AC circuits that contain just resistors, as long as you're careful to use all peak values, or all rms values, or all peak-to-peak values.
Power in a Resistor
  • Recall from EET 150 that, in DC circuits, you can use any one of the following equations to find the power dissipated in a resistor
    P = I2 × R

    P = V2 ÷ R

    P = V  × I
    In these equations, V is the DC voltage drop across the resistor, and I is the DC current through the resistor.
  • In AC circuits, similar equations apply, but you must be sure to use the resistor's rms voltage and rms current.:

    P = Irms2 × R

    P = Vrms2 ÷ R

    P = Vrms  × Irms

    In these equations, Vrms is the rms voltage drop across the resistor, and Irms is the rms current through the resistor. If you use peak values or peak-to-peak values instead of rms values, you'll get the wrong answer for the power.
  • Self-test

Superimposed DC and AC Voltages
  • Many circuits that you'll encounter in the field contain not just AC voltages sources or just DC voltage sources, but a combination of AC and DC voltage sources.
  • As a very simple example of this, the following diagram shows a series circuit containing a DC voltage source, and AC voltage source, and a single resistor.
  • In this case, the voltage across the resistor is a 10 V pp sine wave superimposed on 8 V DC. What this means is that we'll have a 10 V pp sine wave that is "moved up" 8 V from where it would normally be, as in the following diagram.
  • Notice the scale on the vertical axis. We have a 10 V pp sine wave, but instead of ranging from −5 V to +5 V, the sine wave's voltage ranges from +3 V to +13 V.
  • When you use a multimeter or oscilloscope to make measurements in such a circuit, you must pay close attention to whether your equipment is set to measure AC values or DC values (or both). Unit 9 of EET 114 was devoted to this topic, so you should go back and review the e-Lesson for that unit, along with its Self-Test questions, before continuing with this lesson.

Review of Electrical Quantities
  • The table below summarizes the electrical quantities that we've studied so far. The table shows the abbreviation for each quantity, along with the standard unit for measuring the quantity and the abbreviation for the unit.
  • You studied the first eleven quantities (through inductance) in EET 150. The others have been added in this course.
    Quantity
    Abbreviation
    Unit
    Abbreviation for the Unit
    charge
    Q
    coulomb
    C
    current
    I
    ampere
    A
    voltage (or emf)
    V (or E)
    volt
    V
    resistance
    R
    ohm
    Ω
    conductance
    G
    siemens
    S
    energy (or work)
    W
    joule
    J
    power
    P
    watt
    W
    efficiency
    η
     
     
    capacitance
    C
    farad
    F
    time constant
    τ
    second
    s
    inductance
    L
    henry
    H
    period
    T
    second
    s
    frequency
    f
    hertz
    Hz
    peak voltage
    Vp
    volts peak
    V p
    peak current
    Ip
    amps peak
    A p
    peak-to-peak voltage
    Vpp
    volts peak-to-peak
    V pp
    peak-to-peak current
    Ipp
    amps peak-to-peak
    A pp
    rms voltage
    Vrms
    volts rms
    V rms
    rms current
    Irms
    amps rms
    A rms
    time
    t
    second
    s
    varying voltage
    v
    volt
    V
    varying current
    i
    ampere
    A
    angle
    θ
    radian or degree
    rad or °
    phase angle
    φ
    radian or degree
    rad or °
    angular frequency
    ω
    radians per second
    rad/s
  • Notice that the abbreviations for quantities are usually written with italicized letters, while the abbreviations for units are usually written with plain, non-italicized letters.
  • We'll continue to learn about more new quantities throughout this course. Unit 4 will include an expanded version of the table.

Unit 3 Review
  • This e-Lesson has covered several important topics, including:
    • waveforms
    • waveform parameters, including:
      • period
      • frequency
      • instantaneous value
      • peak value
      • peak-to-peak value
      • rms value (also called effective value)
      • average value
    • the oscilloscope
    • mathematical expressions for waveforms
    • phasors
    • mathematical expressions for waveforms
    • resistors in AC circuits
    • average power
    • measuring phase shifts.
  • To finish the e-Lesson, take this self-test to check your understanding of these topics.
  • Self-test

Congratulations! You've completed the e-Lesson for this unit.


Nick Reeder | Electronics Engineering Technology | Sinclair Community College

Send comments to nick.reeder@sinclair.edu