®(t) = Ф + 0 (t) ,

Ak*Qe2Zi22

where Ф is the time-averaged value and <j>{t) has a power spectral density of G(w), then the time-averaged value of the signal from an a-c system, using Equation (3-51) for v(t) and Equation (3-46) for S(t), is. •

Since the first term on the right is the output due to the time-averaged flux, the second term represents the error due to the fluctuations of the flux. The fractional error is the ratio of these two terms: .

3,2.5 Value of the Signal for Transient in the Flux

In this section the expected value of the signal will be calculated for the case in which v(t) is given by Equation (3-51) and for three different transients in the flux.

The procedure that will be followed is:

a. Substitution of v(t) f Equation (3-51) J and the appropriate expression for <b(t) into Equation (3-49) to obtain < AV2 (t)> ;

b. Simplification of this expression for <AV2(t)>; .

2

c. Substitution of the simplified expression for <AV (t)>and the proper initial condition into the differential Equation (3-2) of the integrating circuit to obtain the expected value

. of the signal. <S(t)>; and

d. Simplification of <S(t) >. .

Case 1. The flux has one constant value for t<o and a second constant value for t>o; i. e

for t < o, and

®(t) = Ф0 + Д Ф for t > o.

So Equation (3-61) becomes

А(кДФ Qe Z12r ш

A k *o °e2 Z122

(шн + u’i).
yields the expected value of the signal
<S(t)>
Шн + WL
А кдф Qe2 Z122
WH + a, L
А (кДФОе Z12)^ 2t
-t/r
Since the correct value is
Шн + a, L
the error is E(t)
А к ДФ Qe2Z’i22 ‘-h2
-t/r e
WH+WL
	-t/т
wi>,H + a, L>
2t
and the fractional error is
Ф(0 = Фг	for t < o, and
ФМ = Ф0 + St for t >o,
then <Ht) = St , d(t-x) = St-Sx. and the expected value of Av (t) is

А к(Ф0 + St) Qe2 Z122
<AV2(t)> =	H
— AkSQe2Z122
WH-WL

-2ш1}

This expression approaches, with a time-constant of about l/w^, the simpler one

2

H

/kSQeZj2

Ш.

and, if t >l/wL, this simpler expression may be substituted into Equation (3-2) along with the initial condition

to obtain the expected value of the signal

* (i.. -,л’

Since the correct value is

il* Ш-±)— ( 1 — e-t/T ) .

and the fractional error is

Since we have assumed that т » l/w.^, this may be simplified to

Case 3. The flux is constant for t < о and increases exponentially (constant period, P) for t >o; i. e., ‘

®(t) = Ф0	for t<o, and
®'(t) = Ф0 et/p	for t>o,

then <(>(t) = Ф0(е1/,р-1), 0(t-x) = (e^’x^P-l), and the expected value of AV2(t) is

JH 2Р(ШН + Ші) (2P WHWL + РшН + PwL + t/P

шн + wL . (2Po>H + 1)(2Pwl + l)(PwH + Po>L + 1)

This is really the simplified expression for <AV^(t)>. The actual expression approaches this simpler one with a time constant of about 1/cuSo, if | P| l/o^ and т 3> 1/w^, this simpler Equation (3-74) may be substituted into the differential Equation (3-2) along with the initial condition

.. 2

<S(o)> =

to obtain the expected value of the a-c system signal

where

2P2(wh+wl)(2P2whcjl + Pwh+Pojl + 1)

(P + t) (2PwH+l) (2Pwl+1) (Ршн+Ршь+1)’

and

Since the correct value is

<S(t)>c = Get/P,

the error is.

. E(t) = G(H-l)(et/P-e’t/T ) + J (e2t/P-e’t//T ),’

and the fractional error is

( е‘/р-е-*(т+І)