Therefore, the room air capacity
transfer
function
is:
1
s
K
s
q
s
r
r
total
r
(4.12)
and
3600
3600
0
pa
a
r
v
i
pa
a
r
pa
a
r
v
f
i
st
pa
a
r
d
c
V
n
AU
s
c
V
s
c
V
n
s
AU
c
V
s
(4.13)
can be considered as the disturbance.

117
Equation (4.9) can be solved by using
inverse
Laplace transformation
if the suppled
total
q
is
known
.
This is pretty much all there is to deriving a linear transfer function. Naturally, some are more
complex than this case, but the procedural principles do not differ: write down the governing
differential equation (s) in a convenient linear form; express the variables as deviations such that
all other non-deviant terms become constants and disappear; finally, express in 'time constant'
form. Of course few problems present us with a convenient set of linear differential equations
and we shall see how to deal with these cases later. Meanwhile, we will pay further consideration
to the linear case.
4.9.
Analysis of P, PI, and PID Control Systems
4.9.1.
PID Control Systems for HVAC
The simple control system shown includes an air temperature sensor, a controller that compares
the sensed temperature to the set point, a steam valve controlled by the controller, and the coil
itself. (Two-position control applies to an actuator that is either fully open or fully closed.)
Figure 4-4-16 Block Diagram of a Control System
Equivalent control diagram for heating coil:
In Figure 4-4-16 the
G
(s) represent functions relating the input to the output of each module (the
expressions in parentheses are examples of the
G
(s) discussed in the last section of this chapter
on
Laplace transforms
).
Voltages V represent both temperatures (set point and coil outlet) and
the controller output to the valve in electronic control systems.
Proportional control
corrects the controlled variable in proportion to the difference between the
controlled variable and the set point.
For example, a proportional controller would make a 10%
increase in the coil heat output rate if a 10% decrease in the coil outlet air temperature were
sensed.
The proportionality constant between the error and the controller output is called the
gain
K
p
.
Following equation shows the characteristic of a proportional controller: