CONTINUOUS-TIME SIGNALS AND SYSTEMS - Signals, Systems, and Transforms - page 61

and

v L (t) = L di(t)
dt

. (2.53)

Hence, the transformation notation of (2.52) represents the explicit equations of (2.53). This
model, (2.53), is two equations, with the first a first-order linear differential equation with
constant coefficients. ( L and R are constants.)

We now discuss further the difference between a physical system and a model
of that system. Suppose that we have a physical circuit with an inductor, a resistor,
and some type of voltage source connected in series. Equation (2.53) may or may
not be an accurate model for this physical system. Developing accurate models for
physical systems can be one of the most difficult and time-consuming tasks for engi-
neers. For example, the model that related the thrust from the engines to the atti-
tude (pitch angle) of the Saturn V booster stage was a 27th-order differential
equation.
As a final point, consider again the circuit diagram of Figure 2.33. Engineers
can be very careless in diagrams of this type. The diagram may represent either

1. the physical interconnections of a power supply, a coil, and a resistor or

2. a circuit model of a physical system that contains any number of physical

devices (not necessarily three).

Hence, often, we do not differentiate between drawing a wiring diagram for
physical elements and drawing a circuit model. This carelessness can lead to confu-
sion, even for experienced engineers.

Interconnecting Systems

In this section, the system-transformation notation of (2.51) will be used to speci-
fy the interconnection of systems. First, we define three block-diagram elements.
The first element is a block as shown in Figure 2.34(a); this block is a graphical
representation of a system described by (2.51). The second element is a circle that
represents a summing junction as shown in Figure 2.34(b). The output signal of
the junction is defined to be the sum of the input signals. The third element is a
circle that represents a product junction, as shown in Figure 2.34(c). The output
signal of the junction is defined to be the product of the input signals.
We next define two basic connections for systems. The first is the parallel con-
nection and is illustrated in Figure 2.35(a). Let the output of System 1 be

y 1 (t) and

that of System 2 be y 2 (t). The output signal of the total system is then given by

y(t) = y 1 (t) + y 2 (t) = T 1 [x(t)] + T 2 [x(t)] = T[x(t)]. (2.54)

The notation for the total system is y(t) = T[x(t)].