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Singular Perturbation Methods in Control

Singular perturbations and time-scale innovations have been brought to regulate engineering within the overdue Nineteen Sixties and feature considering develop into universal instruments for the modeling,analysis and layout of keep watch over platforms. The 1986 version of this ebook, reprinted the following in its unique shape, offers the theoretical beginning for consultant regulate purposes.

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The amplitude of this complex exponential is I = CV j 2 f . di , assuming the current to be a complex Finally, for the inductor, where v = L dt exponential results in the voltage having the form v = LIj 2 fe j 2 f t, making its complex amplitude V = LIj 2 f . The major consequence of assuming complex exponential voltage and currents is that the ratio Z = V=I for each element does not depend on time. This quantity is known as the element’s impedance. The impedance is, in general, a complexvalued, frequency-dependent quantity.

Taking the derivative of the second equation and plugging the first equation into the result yields the equation governing the voltage. d2V (x) = (Ge + j 2 f Ce)(Re + j 2 f Le)V (x) dx2 This equation’s solution is V (x) = V+e; x + V;e+ x : Calculating its second derivative and comparing the result with our equation for the voltage can check this solution. 4) Thus, depends on frequency, and we express it in terms of real and imaginary parts as indicated. The quantities V + and V; are constants determined by the source 58 Analog Signal Processing and physical considerations.

We express this dependence as v (x t) and i(x t). When we place a sinusoidal source at one end of the transmission line, these voltages and currents will also be sinusoidal (the transmission line model consists of linear circuit elements). As is customary in Most conductors are cylinders, with current flowing down the cylinder’s axis. This static view does not carry over to sinusoidal fields, however. As frequency increases, Maxwell’s equations indicate that current flows axially only within a shell of depth from the conductor’s surface.