1.5 The significance of reactive components equations
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1.5.4 Exponential Functions
1.5.4.1 Tips and Tricks
Lastly let us now look at some mathematical properties which are peculiar to exponential functions.
It may be useful in some instances to represent the imaginary constants and as exponentials in
order to make the 90ι and -90ι phase shifts which they represent more explicit and simplify
calculations.
This may be easily achieved by means of Euler's formulae as shown in (1.5-8).
( ) ( )
It is also sometimes useful to represent a multiplicative factor of '-1', in terms of complex
exponentials.
( ) ( )
In order to represent a polarity inversion as a 180ι phase shift.
The derivatives of exponential functions are also a special case. From lookup tables, we
know that
So
the derivative of a basic exponential function is actually the exponential function itself! However,
if the exponent is multiplied by a constant , then this constant appears as a multiplicative factor in
the derivative
This
also applies if the constant is complex hence
( )
So
the derivative of a complex exponential is shifted in phase by
⁄
with respect to the original
exponential.
(1.5-10)
(1.5-8)
(1.5-9)
Conquer Radio Frequency
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