Foundations of Analog and Digital Circuits Mas - Wordpress ...

32 CHAPTER ONE the circuit abstraction
i
0
Slope =
v
1
---
R
FIGURE 1.28 Plot of the v–i
relationship for a resistor.
i
0
+
V
-
(a)
i
(b)
FIGURE 1.29 (a) Independent
voltage source with assigned
terminal variables, (b) v–i relationship
for the voltage source.
+
v
-
V
v
Most often, resistances and conductances are thought of as time-invariant
parameters. But if the temperature of a resistor changes, then so too can
its resistance and conductance. Thus, a linear resistor can be a time-varying
element.
1.6.2 ELEMENT LAWS
From the viewpoint of circuit analysis, the most important characteristic of a
two-terminal element is the relation between the voltage across and the current
through its terminals, or the v i relationship for short. This relation, called
the element law, represents the lumped-parameter summary of the electronic
behavior of the element. for example, as seen in Equation 1.15,
v = iR
is the element law for the resistor. The element law is also referred to as the
constituent relation, or the element relation. In order to standardize the manner
in which element laws are expressed, the current and voltage for all two-terminal
elements are defined according to the associated variables convention shown in
Figure 1.19. Figure 1.28 shows a plot of the v i relationship for a resistor when
v and i are defined according to the associated variables convention.
The constituent relation for the independent voltage source in Figure 1.26b
supplying a voltage V is given by
v = V (1.18)
when its terminal variables are defined as in Figure 1.29a. A plot of the v i
relationship is shown in Figure 1.29b. Observe the clear distinction between
the element parameter V and its terminal variables v and i.
Similarly, the element law for the ideal wire (or a short circuit) is given by
v = 0. (1.19)
Figure 1.30a shows the assignment of terminal variables and Figure 1.30b
plots the v i relationship.
Finally, the element law for an open circuit is given by
i = 0. (1.20)
Figure 1.31a shows the assignment of terminal variables and Figure 1.31b
plots the v − i relationship.
Comparing the v i relationship for the resistor in Figure 1.28 to those for a
short circuit in Figure 1.30 and an open circuit in Figure 1.31, it is evident that
the short circuit and open circuit are limiting cases for a resistor. The resistor
approaches the short circuit case as its resistance approaches zero. The resistor
approaches the open circuit case as its resistance approaches infinity.