### Complex variables and the Laplace transform for engineers

This text is designed to remedy that need by supplying graduate engineering students especially electrical engineering with a course in the basic theory of complex variables, which in turn is essential to the understanding of transform theory. Presupposing a good knowledge of calculus, the book deals lucidly and rigorously with important mathematical concepts, striking an ideal balance between purely mathematical treatments that are too general for the engineer, and books of applied engineering which may fail to stress significant mathematical ideas.

The text is divided into two basic parts: The first part Chapters 1—7 is devoted to the theory of complex variables and begins with an outline of the structure of system analysis and an explanation of basic mathematical and engineering terms. Chapter 2 treats the foundation of the theory of a complex variable, centered around the Cauchy-Riemann equations.

The next three chapters — conformal mapping, complex integration, and infinite series — lead up to a particularly important chapter on multivalued functions, explaining the concepts of stability, branch points, and riemann surfaces. Numerous diagrams illustrate the physical applications of the mathematical concepts involved.

### COMPLEX VARIABLES AND INTEGRAL TRANSFORMS

Time-domain functions are written in lower-case, and the resultant s-domain functions are written in upper-case. We use this notation, because we can convert F s back into f t using the inverse Laplace transform. The inverse laplace transform converts a function in the complex S-domain to its counterpart in the time-domain.

Laplace Transforms: Partial Fractions (Imaginary Roots)

Its mathematical definition is as follows:. The inverse transform is more difficult mathematically than the transform itself is. However, luckily for us, extensive tables of laplace transforms and their inverses have been computed, and are available for easy browsing.

## Complex Variables And The Laplace Transform For Engineers - Wilbur R. LePage

In plain English, the laplace transform converts differentiation into polynomials. The only important thing to remember is that we must add in the initial conditions of the time domain function, but for most circuits, the initial condition is 0, leaving us with nothing to add. This is useful for finding the initial conditions of a function needed when we perform the transform of a differentiation operation see above.

Similar to the Initial Value Theorem, the Final Value Theorem states that we can find the value of a function f, as t approaches infinity, in the laplace domain, as such:. This is useful for finding the steady state response of a circuit. The final value theorem may only be applied to stable systems. If we have a circuit with impulse-response h t in the time domain, with input x t and output y t , we can find the Transfer Function of the circuit, in the laplace domain, by transforming all three elements:. In this situation, H s is known as the "Transfer Function" of the circuit.

It can be defined as both the transform of the impulse response, or the ratio of the circuit output to its input in the Laplace domain:. Transfer functions are powerful tools for analyzing circuits. If we know the transfer function of a circuit, we have all the information we need to understand the circuit, and we have it in a form that is easy to work with. When we have obtained the transfer function, we can say that the circuit has been "solved" completely.

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Earlier it was mentioned that we could compute the output of a system from the input and the impulse response by using the convolution operation. As a reminder, given the following system:. Where the asterisk denotes convolution, not multiplication. However, in the S domain, this operation becomes much easier, because of a property of the laplace transform:.

Where the asterisk operator denotes the convolution operation. The same is valid for Young or others. It is a wrong approach to reject past for present or future even in life since past is equal to experience and future can be more economically build up if already made errors are not repeated. I remember a very funny situation, visiting a conference on advanced hydraulics, a team of young engineers presented a "new" and "revolutionary" pump control totally mechanical with less sensitivity than the,, at that time already used, electronic electrohydraulic systems.

This was the consequence of not looking back at evolution and also neglecting the bases of tribology or servo-control. By the way most of differential equations used in dynamic analysis of control loops are not from the last year! Hi, guy, you owned me a GA at the front issues. The Laplace transform just changes the description of a signal or the relationship between output and input transfer function into a form that is easier to manipulate mathematically. The "b" denotes the frequency that it is oscillating. A larger "a" and the signal decays rapidly. If "a" is negative, the signal grows.

## Complex Variables and the Laplace Transform for Engineers by Lepage Wilbur

A larger "b" and the signal oscillates at a higher frequency. Every body over complicates the description of the LaPlace Transform, especially the prof's in college! First, the FFT is a "fast" way to compute the spectrum of any time series waveform.

At Tektronix we used it on the now obsolete Dynamic Analyzer to determine the poles and zeros of structures e. The poles are complex; the real part is the Frequency of resonance, the imaginary is the damping of the resonance describes how quickly the pole will damp to zero. This is often ploted in the "s" plane with F on one axis, damping on the other. Two other parameters are associated with each pole; Magnitude and Phase. The phase is usually ZERO or degrees for uncoupled poles. So, each pole of a system has these properties; Frequency, Damping, Magnitude, and Phase.

For some special applications e. Great explanation! New Post. Comments Format:. Subscribe to Discussion :.