By William K. Roots (Auth.)
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9). t\T FIG. 10. function. 10 displays the transient response of an exponential lag to a unit impulse function. The time constant T again has two definitions: 1. 2. In Fig. 10 the initial slope of the response at t = 0 is —l/T2. Alternatively, in Fig. 10 tan φ = T 2 and cot ψ = 1/Γ2. In Fig. 10 θ = (i/or ~ (1/3)Γ| For practical purposes, the transient response of an exponential lag has decayed by time t > 4T, and the response time within 5% is 3T the response time within 2 % is AT 28 2 . 4. Harmonic Response The previous derivations of the responses of an exponential lag to unit inputs of the ramp, step, and impulse functions are by no means rigorous.
The Bode diagram of the frequency response of a transit delay •W i(i) l ,«"M h(t) I · 0 t m 1 . 0 f — T -»"■ (a) Λ0 1 . \ 0 1 j ^ί- -"-" / y (b) FIG. 3. Examples of the response of a transit delay to a unit step function (a) and a unit damped sinusoid (b); (a) i(t) = h(t) and 6(t) = h(t — τ). (b) t(t) = exp(£i) sin(oii) h(t) and d(t) = exp[£(i - τ)] sin|>(i - τ)] h(t - r). Qh(t) 0 ί A(«) 0 exp(ft) sin(a>i) Λ(ί) Ramp Damped sinusoid »(I) Time domain β(*) s plane 0 [Q exp( - « ) ] / [ ( * - ξ)(*2 [Q β ρ ( - « ) ] / * » [0exp(—TÎ)]/Î Response (Θ) Q«Ü/[(* - ζ)(ί 2 + ω>)] Q e x p [ « i - τ)] 8in[
A) The general case. 1 0 f FIG. 24. Examples of the use of the unit step function h(t) to denote f(t) = 0 when t < 0 and f(t) Φ 0 when t > 0. (a) f(t) = h(t) Qt. (b) /(f) = /*(*) sin(aji). (c)/(i) = *i + A(f)(0, - 0J = ^ + Ä(i)(-*i) + A(f )(*,). 46 2 . THERMAL-PROCESS RESPONSE In Table IV the unit impulse function has been given the symbol 8(t) and the unit step function, the symbol h(t). This latter symbol is also used with any other function f(t) to illustrate the following (see Fig. 24): f(t) = 0 | t < 0 f(t) Φ 0 \t>0 In using transform pairs the basic principle is (output) = (transfer function) χ (input), θ(ί) = G(s) x i(i) or 6(s)li(s) = G(s) As an example consider the time-domain equation for the response of an exponential lag with a unit impulse input: m = [1/(1 + TD)] 8(t) In the s plane this becomes T i (l/T) + s 1 ^Tls-i-i/T)} Using line ten of Table IV to transfer back into the time domain, β(ί) = ( 1 / Γ ) β χ ρ ( - / / Γ ) which confirms Eq.
