If the flux is constant for t < о and differs from this constant value by 0(t) for t> o; i. e.,

then the expected value of the signal from an a-c system is, from Equations (3-38) and (3-47),

/• со

<S(t> = АкФ0 I v2(x) dx •’o

At the input to the integrating circuit the expected value of voltage is, from Equations (3-39) and (3-47), ■ ■

ф (t-x)v2(x)dx

(3-49)

The application of Equation (3-48) to an actual case is usually quite difficult, leading to a large quantity of algebraic manipulation during which mistakes are very probable. An alternative method, that lends itself to some approximation and considerable simplification, is as follows.

Equation (3-49) is used to obtain the expected value of voltage at the input to the integrating circuit. This expression is replaced by a simpler one that should give essentially the same output; a knowledge of the properties of integrating circuits is useful here. This simpler expres­sion is then vised as the driving function in the differential Equation (3-2) of the integrating circuit to obtain the expected value of the signal, <S(t)> . Examples of this procedure are included in Subsection 3. 2. If an exact answer is desired, use either Equation (3-48), or Equations (3-49) and (3-2) with no approximations.