Modern Control Engineering

Example Problems and Solutions 151

or

Noting that

we obtain

or

Taking the Laplace transforms of both sides of this last equation and substituting the initial

condition that u 0 (0)=0, we obtain

The block diagram of the system corresponding to this equation is shown in Figure 4–39.

A–4–11. Consider the thin, glass-wall, mercury thermometer system shown in Figure 4–40. Assume that the

thermometer is at a uniform temperature (ambient temperature) and that at t=0it is

immersed in a bath of temperature where ubis the bath temperature (which may be con-

stant or changing) measured from the ambient temperature Define the instantaneous ther-

mometer temperature by so that uis the change in the thermometer temperature satisfying

the condition that u(0)=0. Obtain a mathematical model for the system. Also obtain an electri-

cal analog of the thermometer system.

Solution.A mathematical model for the system can be derived by considering heat balance as fol-

lows: The heat entering the thermometer during dtsec is qdt, where qis the heat flow rate to the

thermometer. This heat is stored in the thermal capacitance Cof the thermometer, thereby rais-

ing its temperature by du. Thus the heat-balance equation is

Cdu=qdt (4–48)

Q–+u,

Q–.

Q–+ub ,

Q–

Qo(s)=

R

RCs+ 1

H(s)+

1

RCs+ 1

Qi(s)

RC

duo

dt

+uo=Rh+ui

C

duo

dt

=h+

1

R

Aui-uoB

Gc=

1

R

C

duo

dt

=h+GcAui-uoB

H(s)

1

RCs+ 1

R

RCs+ 1

Qi(s)

Qo(s)

++

Figure 4–39
Block diagram of the
air heating system
shown in
Figure 4–38.

Thermometer

Bath

Q +u

Q +ub

Figure 4–40
Thin, glass-wall,
mercury thermo-
meter system.