Page 150 - Maxwell House

130 Chapter 3

we can calculate the parasitic inductance as

2
ℒ = = (3.36)
0
2 8
It turns out that the “ideal” capacitor is a fiction and does not exist at any frequency except DC
when = 0. So it is natural to ask how to distinguish the capacitor from inductance and
⁄

vise versa. Evidently, the natural way is to compare the stored energies. If | ⁄ | ≪ 1 the

circuit element must behave mostly as a capacitor. The element with contrariwise combination
of energy behaves primarily as an inductance. So let us check this ratio for the disk plate
capacitor using (3.25) and (3.34)

2 2
1 1
| ⁄ | = � � � � (3.37)

8
1
Here = is the speed of light in dielectric filling up the gap between plates and =
� 0
√ . In case of sinusoidal voltage with the frequency = and from (3.37)
⁄

we have
2 2
1 2 1
| ⁄ | = � � = � � ≪ 1 (3.38)

8 8
where = 2 is the arc length of capacitor plate and = is the wavelength in the
⁄
medium with the refractive index n. Therefore, the parallel plate design behaves as a capacitor
until the perimeter of its plate is short compared to the wavelength of electromagnetic field.
Now we can repeat all the above analysis replacing the term “capacitor” on “inductor” and
come to a conclusion that, in principle, the “ideal” inductor is a fiction at any frequency too. It
follows from the fact that the alternating magnetic field is the source of the alternative electrical
fields. If so, it must be some concentration of electric energy around the coils creating a parasitic
capacitance. Figure 3.1.11c illustrates a slightly idealized structure of electric fields exerted by
a solenoid. Omitting all math transformations, we obtain

4 2

= ( ) 2 � � (3.39)

0
16ℎ
2 ℎ
= = (3.40)

2 8
In case of sinusoidal voltage with the frequency = and the electric energy
⁄

storage is negligible compare to the magnetic storage if
2 2
1 2 1
| | = � � = � � ≪ 1 (3.41)
⁄

8 8
Therefore, the solenoid behaves almost as an “ideal” inductor until the ratio of the perimeter
of its turn divided by the number or turns is short compared to the wavelength of the