# FREQUENCY RESPONSE OF FIRST-ORDER SYSTEM

The first-order transfer function, written in terms of the time constant T is

1

7ТТ/Г

whence

К = lim G(s) = T

s^O

ш/шп

(a)

Figure 7.13 Frequency-response curves—second-order system.

10 2 10“1 10° 101 to (rad/sl to) |

№)

Figure 7.14 Frequency-response functions, elevator angle input. Jet transport cruising at high altitudes, (a) Speed amplitude, (h) Speed phase.

The frequency response is determined by the vector G(ico)

T

G(ico) = KMe’v = ———

1 + itoT

whence

1 — icoT

From (7.5,9), M and <p are found to be

1

(1 + co2T2)m

ip = — tan-1 шТ

A vector plot of Mei<p is shown in Fig. 7.9. This kind of diagram is sometimes called the transfer-function locus. Plots of M and <p are given in Figs. 7.10a and b. The abscissa is fT or log wT where / = ш/ітг, the input frequency. This is the only parame-

Figure 7.16 Frequency-response functions, elevator angle input. Jet transport cruising at high altitude, (a) Pitch-rate amplitude. (b) Pitch-rate phase. |

ter of the equations, and so the curves are applicable to all first-order systems. It should be noted that at w = 0, M = 1 and <p = 0. This is always true because of the definitions of К and G(s)—it can be seen from (7.4,5) that G(0) = K.

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