10.6: Summary
- Page ID
- 98499
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The most simple of AC waveforms is the sine wave. It can be thought of as the vertical displacement of a vector rotating at a constant speed, like the second hand of a clock. The length of the second hand represents the height or amplitude of the sine wave and the speed of rotation represents its period. As the speed tends to be very fast, it is more convenient for us to use frequency, which is just the reciprocal of the period. Sines can be displaced vertically, which is also called a DC offset, as well as having a horizontal or time shift. When expressed relative to a single cycle, this change is referred to as the phase shift. Finally, waveforms such as square waves, triangle waves and even more complex waveforms such as voice or music can be described in terms of combinations of sounds. Indeed, Fourier analysis tells us that any repeating waveform can be described as a series of sines each with the appropriate frequency, amplitude and phase shift. In order to determine the “effective DC value” of a sine wave, that is, the value that produces the same power dissipation, RMS values are used. The RMS value of a sine wave is its peak value divided by the square root of two (approximately equal to 0.707 of peak).
Complex numbers are used to describe AC voltages and currents, among other things. They consist of two parts: a real part and an imaginary part that is plotted perpendicular to the real part. All mathematical operations on complex numbers must follow vector rules. This includes basic trigonometric operations.
Reactance can be thought of as the imaginary axis version of resistance. That is, it restricts current flow. The difference is that there is a 90 degree phase shift between the current and voltage through a reactive element while the two are in phase for resistive elements. Capacitive reactance is inversely proportional to frequency while inductive reactance is directly proportional to frequency. The combination of resistance and reactance is known as impedance. Phasor diagrams may be used to plot the components of a complex impedance as well as show the relations between voltages or currents in a circuit.
Review Questions
1. What is a sine wave? Describe its constituent parameters (amplitude, frequency, offset, etc.).
2. What is the relationship between the frequency, period and phase of a sine wave?
3. What is the mathematical relationship between a sine wave and a cosine wave?
4. What is meant by the term RMS (root-mean-square) and what does it have to do with AC versus DC voltages, currents and powers?
5. Describe the difference between a scalar and a vector.
6. What is a complex number?
7. Detail the relationships between resistance, reactance, impedance, conductance, susceptance and admittance.
8. Describe how capacitive reactance varies with frequency.
9. Describe how inductive reactance varies with frequency.
10. Describe a phasor diagram.