Abstract
In the present context, a transformation establishes a one-to-one relation between two sets of objects. Such a relation exists between sinusoidal functions of time and the corresponding phasor representations, provided a specification of the frequency of the sinusoid accompanies the phasor representation (to make the relation one-to-one).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The integral in (18.2) is an example of a line integral, so called because the path of integration is along a line defined by \(s = c\) that is not necessarily coincident with one of the axes. You need not be alarmed by this integral because we never need to use it.
- 2.
In this book, signal means current or voltage.
- 3.
You will see this proved if you take a course in complex variables.
- 4.
In physical problems, we are free to choose a time origin. We could generalize by treating right-sided functions that equal zero for \(t < t_0\), but that would clutter the development without providing any useful increase in generality.
- 5.
This requirement is somewhat stronger than necessary, but is easily tested and completely satisfactory in all practical applications.
- 6.
Please accept on faith that the result is consistent with that of a rigorous development, such as provided by the theory of generalized functions (or distribution theory).
- 7.
After Paul Dirac (1902–1984), a British physicist credited with introducing the delta function.
- 8.
In general, the dimension of a delta function is the reciprocal of the dimension of the argument.
- 9.
The left sides of the referenced relations can be defined in a self-consistent manner, but we do not need those more general relations in this book.
- 10.
Entries 11–13 are useful mainly for obtaining inverse Laplace transforms.
- 11.
Note that A is associated with p, which has a positive imaginary part and A * is associated with p *.
- 12.
Recorded data or signals can be advanced relative to the time at which they were recorded, but that is not a true advance because it must be preceded by a delay (the recording time) sufficient to allow the advance.
- 13.
After the British physicist Oliver Heaviside (1850–1925), who contributed much to electrical engineering.
- 14.
The ideal differentiator described in Chapter 8 is rarely (if ever) used in practice.
- 15.
In truth, frequency-domain analysis can be generalized through the Fourier transformation, and resulting properties and methods allow analysis of virtually any linear circuit and transformable excitation, not just sinusoids. You will learn about the Fourier transformation if you take a subsequent course in signal processing or communication systems, where it is favored over the Laplace transformation.
- 16.
There are several definitions of stability. The one given here is called the BIBO (bounded-input-bounded output) definition and is an appropriate definition for linear circuit models. Note that no physical circuit can actually produce an unbounded output. Mathematical studies of stability necessarily deal with circuit models and thus with mathematical models of currents and voltages.
- 17.
One can cook up excitations whose zeros cancel (mathematically) one or more poles of a transfer function, but in the real world, such cancellation is imperfect and all unforced terms are present, regardless of the form of the excitation.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Glisson, T.H. (2011). Laplace Transformation and s-Domain Circuit Analysis. In: Introduction to Circuit Analysis and Design. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9443-8_18
Download citation
DOI: https://doi.org/10.1007/978-90-481-9443-8_18
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-9442-1
Online ISBN: 978-90-481-9443-8
eBook Packages: EngineeringEngineering (R0)