The variance of the period indication (the voltage proportional to growth rate) can be obtained by using Equation (2-14) with the expression for an individual pulse at the period-meter output,, or by integrating the power spectral density of the period indication. The latter method is used here because the power spectral density is also needed to calculate the false trip probability.

For a signal composed of the linear superposition of randomly-arriving pulses, the distributed portion of the power spectral density is given^ by

• so

G (W) = — Г!/■• (б) COS w 6 d б.’ (2-26)

77 Jo

where

‘ <jC. .

Ф(б) = 0. 5 г I v(t) v(t + 6)dt. (2-27)

‘ ‘ о,

In this last expression. 0. 5 r is the average arrival rate of the pulses and v(t) is the shape of one pulse that starts at t = o. But v(t) at the output of the log count-raie meter is given by Equation (2-13). Substitution of Equation (2-13) into Equations (2-27) and (2-26) gives the power spectral density there as

The transfer function of the rest of the system (period meter) is

where Rj, Rg, Cj. and Cg are identified in the schematic (Figure 2-2) of the period meter.

(2-30)

2 -1 in (volts) /(radian sec ) .

To normalize this expression, note that at a period of 1 second per neper the flux changes by a factor of 10 in 2. 303 seconds. This corresponds to a change of unity in к or. from Equation (2-17). a change of AV volts at the output of the log count-rale meter. Hence, at 1-second period.

dS = AV dt 2.303

At the input to the period meter, the rate of change of voltage is

AV H — — "A *

2. 303 _

and this produces a voltage at the output of the period meter of

— 1 “ — I

The power spectral density at the output of the period meter, in (nepers sec ) / (radian sec ), is obtained by dividing the right-hand side of Equation (2-30) by the square of Equation (2-31). The result is

(Equation (2-3a) was also used in obtaining this expression.)

The variance of the period indication is obtained by integrating Equation (2-32) over all frequencies; i. e.., ‘