1.5 The significance of reactive components equations
15
Figure 1.5-4 Interpolation of slope points shows the function representing the derivative of ( )
From Figure
1.5-4 it is apparent that, in the case of a sinusoidal function, its derivative is an
identical curve which is however offset along the x-axis. In the case of ( ), this offset is 90⁰ and
hence its derivative is ( ).
In lookup tables however, this derivative is often stated as ( ) which is also correct by
virtue of the well know trigonometric equations. The author however prefers to think of the
derivative of a sinusoidal function in the latter from since it gives a better idea of what such a
derivative represents.
Let us now consider a capacitor and see how these mathematical concepts apply to the
physical world. Let us assume that the voltage across the capacitor may be described by a sinusoidal
function
( )
( )
Where represent the angular frequency
ͷ
of the stimulus. According to equation (1.4-1)(b)
( )
( )
( )
This means that, at any instant in time, the current
( )
will always be ahead
of the
voltage. This may be seen in Figure 1.5-4 where the derivative of the ( ) function, reaches a
peak
before the ( ) function does. This is what is meant by the current leading the voltage
in a capacitor.
A dual argument may be derived for inductors where the opposite applies i.e. the voltage
leads the current by
. This latter demonstration is left to the reader.
This is discussed in more details in section 4.2.
5
, where is the frequency in Hertz
( )
( )
(1.5-3)
Conquer Radio Frequency
15 www.cadence.com/go/awr
