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Unformatted text preview: l by which a signal is expressed as a sum o f everlasting
exponentials est. T he relati:ve a mount o f a c omponent est is F (s). Therefore F (s),
t he Laplace transform of f (t), r epresents t he s pectrum of exponential components
of f (t). Moreover, H (s) is t he s ystem response (or gain) t o a s pectral component
est, a nd t he o utput signal s pectrum is t he i nput s pectrum F (s) t imes t he s pectral
response (gain) H (s) [yes) = F(s)H(s)J.
T he Laplace transform changes integro-differential equations of L TIC systems
into algebraic equations. Therefore, solving these integro-differential equations reduces t o solving algebraic equations. T he Laplace transform method cannot generally be used for time-varying p arameter s ystems or for nonlinear systems.
T he t ransfer function of a system may also be defined as a ratio of t he Laplace
transform of t he o utput t o t he Laplace transform of t he i nput when all initial
conditions are zero (system in zero state). I f F (s) is t he Laplace transform of t he
i nput f (t) a nd Y (s) is t he Laplace transform of t he corresponding o utput yet) (when
all initial conditions are zero), t hen y es) = F (s)H(s), where H (s) is t he system
transfer function. T he s ystem transfer function H (s) is t he Laplace transform of the
system impulse response h (t). Like t he impulse response h (t), t he t ransfer function
H (s) is also a n e xternal description of t he s ystem.
Electrical circuit analysis can also be carried o ut by using a transformed circuit
method, in which all signals (voltages a nd c urrents) are represented by their Laplace
transforms, all elements by their impedances (or admittances), a nd initial conditions
by their equivalent sources (initial condition generators). In this method, a network
can be analyzed as if it were a resistive circuit.
Large systems can b e considered as suitably interconnected subsystems represented by blocks. Each subsystem, being a smaller system, can b e readily analyzed
a nd r epresented by its i nput-output relationship, such as its transfer function. Analysis o f large systems can be carried o ut w ith t he knowledge of i nput-output relationships of its subsystems a nd t he n ature o f i nterconnection o f various subsystems.
LTIC s ystems can be realized by scalar multipliers, summers, a nd i ntegrators.
A given transfer function can be synthesized in many different ways. Canonical,
cascade, a nd parallel forms of realization are discussed. I n p ractice, all t he building
blocks (scalar multipliers, summers, a nd i ntegrators) can be obtained from operational amplifiers.
Feedback systems are closed-loop systems mainly used t o c ounteract t he effects
of unpredictable variations in t he s ystem parameters, t he load, a nd t he environment.
Such systems are designed for a specified speed a nd t he s teady-state error. For
speed, t he useful transient parameters are rise time, peak time, a nd s ettling time.
Percent overshoot indicates how smoothly t he system o utput rises to its final value.
We can relate t he s tea...
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