0195136047.pdf

5.4 APPLICATIONS OF OPERATIONAL AMPLIFIERS 255

Rf

0.707R

i

Rf

Ri

vo

f 0

BW

fl fh

vi

f

Figure 5.4.17Frequency response of voltage-

gain magnitude of a bandpass active filter.

BW=fh−fl, forfl<< fh

Analog Computers

Although not used as much as the digital computer (which nowadays forms the basic tool for
numerical analysis and the solution of algebraic as well as differential equations), the analog
computer still retains some significant advantages over the digital computer. A physical system
can be represented by a set of differential equations which can be modeled on an analog computer
that uses continuously varying voltages to represent system variables. The differential equation
is solved by the computer, whereas the modeled quantities are readily varied by adjusting passive
components on the computer. The mathematical functions (integration, addition, scaling, and
inversion) are provided by op amps.
The analog computer components are shown in Figure 5.4.18. Let the ordinary differential
equation to be solved be

d^2 y(t)

dt^2

+a 1

dy(t)

dt

+a 2 y(t)=f(t) (5.4.50)

which can be rearranged by isolating the highest derivative term as

y ̈=−a 1 y ̇−a 2 y+f (5.4.51)

subject to the initial conditions

y( 0 )=y 0 (5.4.52a)

and

dy

dt

∣

∣

∣

∣

t= 0

=y 1 (5.4.52b)

Thesimulationof the solution is accomplished by the connection diagram of Figure 5.4.19. Time
scaling is done by redefiningtast =ατ, in which case the differential equation [Equation
(5.4.50)] becomes

d^2 y(τ)

dτ^2

=−αa 1

dy(τ)

dτ

−α^2 a 2 y(τ)+α^2 f(τ) (5.4.53)