1.5 The significance of reactive components equations
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1.5 The significance of reactive components equations
1.5.1 Introduction
In the author's experience, far too often mathematical expressions and operators are used
without their actual meaning in a physical or, for that matter, mathematical context being explained
adequately. In this section the author aims to give a more 'real' and intuitive explanation of the
meaning and the application of derivatives and complex numbers. A good understanding of these
mathematical concepts forms the perfect platform onto which one can build knowledge of circuits
and systems at radio frequency.
1.5.2 Derivatives
The derivative of a function is too often seen as a mathematical entity which may be found
in a lookup table. One should not forget however that the derivate of a function at a specific point
represents the slope of the tangent to that function at that very point. Below are some derivatives
for simple and commonly encountered rational functions.
These may be found in mathematical tables however one should be able to work them out
quite easily just by finding the slopes of the tangents at various points for each curve and joining the
dots to find the expression of the derivative.
For , this process is illustrated in Figure 1.5-1 which shows that this graphical method yields
the same result as that shown by equation (1.5-1). This is also shown as an animation in video 1.2
For , this process is illustrated in (1.5-2), which yet again confirms the result shown by
equation (1.5-2). This is also shown as an animation in video 1.3
What is obvious from both the mathematical and graphical approach, is that the derivative
of a simple power function will always be an order of power lower than the function and hence
increase more slowly. This result may be generalised as
So it all seems quite straight forward for simple rational functions in that the derivate simply
increases at a lower rate than the function from which it was derived. However, often enough in
electronic circuits, we are interested in periodic functions and stimuli which vary in a sinusoidal
fashion, so let us derive graphically the derivative of the sine function. This is shown in Figure 1.5-3
and Figure 1.5-4 and also demonstrated with an animation i n video 1.4
(1.5-1)
(1.5-2)
Conquer Radio Frequency
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