response by applying Duhamel’s theorem.2’” By this theorem the time in g(t) is replaced 
by a fixed parameter ¢’ and the response becomes a function of ¢and¢*% Then f(¢) is ob- 
tained from the integral 
t 
fo 2 | f(t—t,0%) dt’ [101] 
0 
If the response of the system to a sinusoidal disturbance of circular frequency w is required, 
g(t’) may be written as 
g(t’) =g* cos wt’ [102] 
If the values of g(t’) and f(¢,¢’) from Equations [102] and [100] are used in Equation [101], 
the frequency response of the system becomes 
f(t) = g*A(@) cos (wt - &) + cos? 5 et /M [103] 
where the phase angle 6 is 
tan 6 =Mo [104] 
and the amplitude is 
A(@) = cos 6 {105] 
After an appreciable time has passed the transient term becomes very small and the frequency 
response is given by the first term of Equation [103]. When the circular frequency w = 1/M, 
the amplitude has diminished to 0.707 and the response is distorted by a phase angle 6 = 7/4. 
CONSTANT-CURRENT COATED WIRE 
For the case of acoated hot wire an equivalent time constant may also be defined as 
the time in which the wire response attains 1 - e~! of its final value. Thus the time con- 
stant M, of a constant-current coated wire responding to a step-like change in convective 
cooling is, from Equation [42], 
Dy, A, Gag Bn Me xig~1 [106] 
n=1 Qa, 
yB,2M, =1+log A, Ge 
ea [107] 
vlog [1+ 37 Anfant St oe 
n=2 A,G,a,. 
24 
