Page 55 - Biosystems Engineering

36 Chapter Two

harvester driving line (Fig. 2.2) is further detailed to include blocks
for the individual subsystems.

2.2.2 Mathematical Representation
Although a qualitative block diagram representation, such as the one
shown in Fig. 2.2, provides a structured insight in the connections
between the different subprocesses of the system, for design purposes
we usually need to describe its dynamic behavior quantitatively. For
this purpose, we need a mathematical representation of each block.
Let us represent each signal by a time function. Then a system can
be defined as a function G, which transfers the input signal u(t) enter-
ing the system into an output signal y(t) exiting the system:
yt() = G u t()) (2.1)
(
Function G is referred to as an input–output transfer function, or sim-
ply transfer function.
For some simple linear systems, like basic electronic components,
the dynamic behavior is well known and can be accurately described
using mathematical models. A few examples of such simple systems
and the corresponding models are given here.
The potential v(t) over a resistor is equal to current i(t) multiplied by
the resistance R. When we consider the current i(t) through the resistor
to be the input u(t) and the potential v(t) over the resistor to be the out-
put y(t), the resistor transfer function can be written as follows:
vt() = Ri t() or y t() = Ku t() (2.2)

where the constant gain K is equal to the resistance R. The transfer
function in Eq. (2.2) is called a constant gain multiplier, which is the
simplest type of system. Other examples of constant gain multipliers
are a linear mechanical spring, a tooth wheel connection, and conduc-
tive heat transfer through a material.
The voltage v(t) over a solenoid is not proportional to the current
i(t) through it, but to its time derivative:

vt() = L di t() (2.3)
dt
where L is the inductance of the solenoid. If we again consider the
current i(t) to be the input u(t) and the voltage v(t) to be the output,
Eq. (2.3) can be rewritten as follows to obtain the transfer function:

du t ()
yt () = K (2.4)
dt
This is a differentiator where the gain K equals the inductance L.
Other examples of this differentiator action can be found: between