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Unformatted text preview: ons a nd is
sustained by t his s ystem without t he a id of any external input. T he e xponential
c urrent w aveform is t herefore t he c haracteristic mode of this R C c ircuit.
M athematically we know t hat a ny c ombination o f c haracteristic modes can be
s ustained by t he s ystem alone w ithout r equiring a n e xternal i nput. This fact can
be readily verified for the series R L c ircuit shown in Fig. 2.2. T he loop equation
for this system i s (D + 2)y(t) = f (t) I t h as a single characteristic r oot>. = - 2, a nd t he c haracteristic mode is e - 2t. We
now verify t hat a loop current y (t) = c e- 2t c an be sustained t hrough t his circuit
w ithout a ny i nput voltage. T he i nput voltage f (t) required t o drive a loop current
y (t) = c e- 2t is g iven by f (t) = L dy
= + R y(t) ~(ce-2t) + 2 ce- 2t
dt = _ 2ce- 2t + 2 ce- 2t =0
Clearly, t he l oop c urrent y (t) = c e- 2t is s ustained by the R L c ircuit on its own,
w ithout t he n ecessity of a n e xternal input.
T he Resonance Phenomenon We have s een t hat a ny signal consisting of a system's characteristic modes is
s ustained by t he s ystem on its own; t he system offers no obstacle to such signals.
Imagine w hat w ould h appen if we a ctually drove the system with a n e xternal i nput
t hat is one of i ts c haracteristic modes. This would be like pouring gasoline on a fire
in a d ry forest o r hiring a n alcoholic t o t aste liquor. An alcoholic would gladly do t he
j ob w ithout pay. T hink w hat will happen if he were paid by the a mount o f liquor he
tasted! T he s ystem response to characteristic modes would n aturally b e very high.
\Ve call this b ehavior t he r esonance p henomenon. An intelligent discussion of T he i mpulse function 8(t) is also used in determining t he response of a linear
system t o a n a rbitrary i nput f (t). I n C hapter 1 we e xplained how a system response
t o a n i nput f (t) m ay be found by breaking this i nput i nto narrow rectangular
pulses, as illustrated in Fig. 1.27a, a nd t hen s umming t he s ystem response to all
t he c omponents. T he r ectangular pulses become impulses in t he l imit as their
widths approach zero. Therefore, t he s ystem response is t he s um of its responses
t o various impulse components. This discussion shows t hat if we know t he s ystem
response t o a n impulse i nput, we c an determine the system response to a n a rbitrary
i nput f (t). We now discuss a m ethod o f d etermining h (t), t he u nit impulse response
of a n LTIC s ystem described by t he n th-order differential equation Q (D)y(t) = P (D)f(t) (2.17a) where Q (D) a nd P (D) a re t he polynomials shown in Eq. (2.2). Recall t hat noise
considerations restrict practical systems t o m ::; n. U nder this constraint, t he m ost
general case is m = n. T herefore, Eq. (2.17a) can b e expressed as (Dn + a n_ID n- 1 + ... + a ID + ao)y(t) =
(bnDn + bn_ID n - 1 + ... + bID + b o)f(t) (2.17b) Before deriving t he general expression for t he u nit impulse response h(t), i t is
i lluminating t o u nderstand q ualitatively t he n ature o f h(t). T he im...
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