Nonlinear Control Sy.. - Free

202 CHAPTER 8. PASSIVITY
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i
Figure 8.1: Passive network.
In Figure 8.1 the voltage across the terminals of the box is denoted by v, and the
current in the circuit element is denoted by i. The assignment of the reference polarity for
voltage, and reference direction for current is completely arbitrary. We have
thus, the energy absorbed by the circuit at time "t" is
w(t) = J v(t)i(t) dt = J
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p(t) = v(t)i(t) (8.3)
v(t)i(t) dt + J v(t)i(t) dt. (8.4)
The first term on the right hand side of equation (8.4) represents the effect of initial
conditions different from zero in the circuit elements. With the indicated sign convention,
we have
(i) If w(t) > 0, the box absorbs energy (this is the case, for example, for a resistor).
(ii) If w(t) < 0, the box delivers energy (this is the case, for example, for a battery, with
negative voltage with respect to the polarity indicated in Figure 8.1).
In circuit theory, elements that do not generate their own energy are called passive,
i.e., a circuit element is passive if
f t v(t)i(t) dt > 0. (8.5)
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Resistors, capacitors and inductors indeed satisfy this condition, and are therefore called
passive elements. Passive networks, in general, are well behaved, in an admittedly ambiguous
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