Abstract
This chapter describes how real sinusoids are represented using complex quantities called phasors; how circuits containing resistors, capacitors, and inductors are represented using complex quantities called impedances and admittances; and how phasors, impedances, and admittances are used for analyzing sinusoidally excited circuits. This is a long chapter because it extends methods treated in several previous chapters to sinusoidally excited circuits.
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Notes
- 1.
Chapter 16 introduces Fourier analysis.
- 2.
Angular frequency is called radian frequency in some books.
- 3.
In Chapter 14, we express angles in degrees, as is conventional in the electric power industry.
- 4.
Determining frequency and initial phase from such a graph requires calculations, albeit simple ones.
- 5.
More generally, within any whole number of periods.
- 6.
If two sinusoids have different frequencies, then one alternately leads and lags the other. The duration of an interval during which one leads (or lags) the other depends upon the frequencies.
- 7.
In this book, a tilde (~) denotes a complex representation of a real quantity; thus, \(\tilde x(t)\) denotes the complex representation of a real signal \(x(t)\).
- 8.
Recall that the positive direction for an angle is counter-clockwise.
- 9.
It is meaningless to add the phasors for sinusoids having different frequencies.
- 10.
Fourier analysis is introduced in Chapter 16.
- 11.
Section 12.16 treats superposition in the context of sinusoidal excitation.
- 12.
Indeed, this is why the cosine, rather than the sine, is chosen as the standard form for a sinusoidal current or voltage.
- 13.
If there is more than one source, we may use superposition and consider one at a time.
- 14.
Of course, we can also use admittances. But conventionally, elements are represented by their impedances in circuit diagrams, even if circuit equations are subsequently written in terms of admittances.
- 15.
The effective resistances of physical inductors and capacitors are not zero and their reactances are not exactly those given in (12.48). These and other departures of physical components from ideal behavior are discussed in Section 12.22.
- 16.
Note that for any particular load at any particular frequency, reactance and susceptance have opposite signs (if non-zero).
- 17.
These remarks pertain to passive circuits.
- 18.
We define the admittance triangle for sake of completeness; however, admittance triangles are rarely seen in practice.
- 19.
Because the standard form for a sinusoid is a cosine, a dc component can be treated as a sinusoid whose frequency is zero.
- 20.
For example, at sufficiently high frequencies, an inductor exhibits shunt capacitance. Section 12.22 is an introductory treatment of such parasitic effects.
- 21.
As we show in a subsequent chapter, it is possible to emulate inductance using an active device and capacitance. Thus it is possible for an active circuit containing only active devices, resistors, and capacitors to exhibit resonance. Also, as discussed below, a physical component (such as a capacitor) is not ideal and can exhibit all three of resistance, capacitance and inductance and can be self-resonant at a sufficiently high frequency.
- 22.
Remember that the voltage across an inductor is proportional to both the inductance and the rate of change of the voltage.
- 23.
Actually, the circuit considered here only approximates a true gyrator, but provides a useful approximation at minimal cost.
- 24.
The impedance at resonance is of interest because it determines the maximum current required of a circuit driving the impedance.
- 25.
Adapted from Bogatin, Eric. Signal and Power Integrity – Simplified. Prentice-Hall, Englewood Cliffs, NJ. 2010.
- 26.
Leslie Green, RF Inductor modeling for the 21 st Century, EDN, September 27, 2001.
- 27.
- 28.
- 29.
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Glisson, T.H. (2011). Sinusoids, Phasors, and Impedance. In: Introduction to Circuit Analysis and Design. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9443-8_12
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DOI: https://doi.org/10.1007/978-90-481-9443-8_12
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