First-Order Delay Transfer Function

The following circuits have first-order delay transfer functions.

(a) R-L circuit (Fig. 2.10)

First-order differential equation with constant coefficients in time domain reads i(t)R + L(di/dt) = vin(t) or i(t) + (L/R)(di/dt) = vin(t)/R, where t = L/R is a

time constant.

In Fig. 2.10 the voltage vin(t) is the input of the network, and the current i(t) is its response (or output). That is, the R-L circuit can be represented by a

i(s)(1 + st)

(vin(s)/R)

or

xout(s) _ i(s) _ 1 1 _ G3

xin(s) vin(s) R (1 + st) (1 + st) ’

with t = L/R and gain G3=(1/R) < 1.

Fig. 2.10 R-L circuit acting as a first-order delay network

i(t)

R L

Vin(t)

Fig. 2.11 Representation of transfer function of a first – order delay network in a block diagram

Fig. 2.12 Integrator with negative feedback generates a first-order delay network
Xin(s)	Xini(s)t	Xouti(s)^

Xouti(s) 0=1

(b) Integrator with negative feedback (Fig. 2.12).

xouti(s) xini(s) ; xin(s) xouti(s) — xini(s), xouti(s) — (xin(s) xouti (s))

xouti(s) 1 1

xin (s) St(1 + st) St + 1

with gain G — 1.