Recalling that the convolution operator obeys the distributive property and that any
function of t convolved with 8(t) is simply the function itself yields
I t 1 -oAplate 8oAplate
Vout(t) = -Re R-cu(t) CAp Eor, (t) + OApaEnorm(t) (3-44)
The convolution is computed via the convolution integral, as shown in Equations
3-36 and 3-37, yielding
OAplate oAplate ,O O
Vout(t) = Apate orm(t) Aplate Enorm(l)e uR (t- l)dl (3-45)
The effect of the unit-step function in the second term of Equation 3-45 can be
incorporated into the limits of integration. Furthermore, the term e RC can be factored
out of the integral. This yields
Vou(t AplateEnorm) Aplatee t Enorm(l)edl (3-46)
Vot (t)C2 e RC- n Rmomt
The above expression is valid for an arbitrary electric-field, Enorm(t), with no
frequency constraints. The antenna output voltage consists of two terms, the first of
which is the electric field scaled by the quantity (eoA) /C. This is the output of an
ideal flat-plate electric-field antenna. The second term is the effect of the non-ideal
low-frequency response of the antenna. As R approaches infinity, the second term
approaches zero and the output is that of an ideal flat-plate electric-field antenna. This
can also be seen by considering the frequency-domain expression for the antenna
output voltage and allowing R to approach infinity. The same argument can be applied
to allowing the quantities RC and RC2 to approach infinity. If C alone is allowed to
approach infinity then in the limit the output voltage will be zero. However, L'Hopital's
Rule can be invoked, and it can be seen that the second term will decrease to zero
before the first. Therefore, it can be said that for very large values of C, the antenna
output voltage will be approximately that of an ideal antenna, with very low gain. Of