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CLAMPING
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The continuous functionf(t) is said to be clamped to produce the function fc(t) as shown in Fig. 22.3. The period of sampling is T and the frequency is ws = 2dT radians/time. The clamping can be described mathematically by the combination of impulse modulation and the application of a zero-order hold as will soon be shown.
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T 2T 3T 4T 5T
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FIGURE 22-3 Clamping.
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SAMPLEDDATACONTROLSYSTEMS
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FIGURE 22-4 Impulse modulation.
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Impulse Modulation
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As shown in Fig. 22.4, an impulse-modulated function consists of a sequence of impulse functions, the magnitudes of which equal the values of the continuous function at sampling instants. The impulse-modulated function is given the symbol f*(t). A convenient symbol for the impulse modulation sampler is shown as a switch (J ), which closes momentarily every T sec. It should be noted that the impulse modulation switch is purely symbolic, for there is no switch of this type present in the hardware used in sampling signals.
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Zero-order Hold
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The zero-order hold is defined by the transfer function 1 - e-Ts G/t,&) = s
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(22.1)
Combining impulse modulation with the zero-order hold, as shown in Fig. 22.5, provides the clamping. To illustrate how the zero-order hold shapes the sequence of impulses into a clamped signal, a block diagram of the hold is shown in Fig. 22.6 in which an integrator and a transport lag are connected to implement the zero-order hold. The operation of a zero-order hold can be understood by expressing F,(s) as follows (see Fig. 22.5):
Fc(s) = F+(s)1 - e-Ts F*(s)e-Ts _ F*(s) s S s
(22.2)
where F*(s) is the Laplace transform of the impulse-modulated function, f*(t). This expression shows that the clamped function fc(t) is obtained by a combination of integration, transport lag, and subtraction. Recall that integration in Eq. (22.2) is represented by l/s and transport lag by emTs. To understand the signals at the output of the integrator, one must recall that the integration of an impulse
f(t)
F(s)
f*(t)
F*(s)
1 -@
fc(t)
F,(s)
F(z)
FIGURE 22-5 Clamping.
SAMPLING
Z-TRANSFORMS
f(c)
2T 3T 4T
T2T3T
FIGURE 22-6 Construction of a clamped signal.
function is a step function. The impulse occurring at t = 0 results in a pulse* of width T. (See Fig. 22.6.) The impulse occurring at t = T results in a pulse of width T starting at t = T. The remaining impulses each contribute a pulse of width T starting at successive values of T. The height of each pulse equals the magnitude of the impulse that produced it. The combination of these pulse functions provide the stair-step function associated with clamping. This rather mechanical description of clamping through implementation of the zero-order hold transfer function may give the reader an intuitive feel for the abstract mathematical expression involved.
LAPLACE TRANSFORM OF THE IMPULSE-MODULATED FUNCTION
Let i(t) be a sequence, or train, of unit impulses that are separated by period T. This may be expressed as follows: co
i(t) = s(t) + @t - T) + s(t - 2T) + ... = c i (t - nT) n=O (22.3)
The starred function f*(t) may be written as the product of f(t) and i(t): f*(t) = fW(t) Introducing the expression for i(t) from EQ. (22.3) into this equation gives
f*(t) = f(t) g*(t - nT) n=O (22.4)
*The pulse occurs because the integration of the impulse at I = 0 is combined with the integration of a delayed impulse of the same magnitude, but opposite sign, at t = 2 .
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Since i(t) has a non-zero value only at sampling instants (t = nT), f(t) may be replaced by f(nT) and then placed within the summation to produce the result f*(t) = &T)s(r n=O - nT) (22.5)
Applying the Laplace transform of a unit. impulse, which is unity, and applying the theorem on translation of a function (See Chap. 4) to this expression for f*(t) give the Laplace transform of the impulse-modulated function:
F*(s) = LCf*(t)} = -&T)e- T n=O
(22.6)
The function F*(s) is referred to as the starred transform of f(t). An alternate form for F*(s) which is useful in proofs and derivations is given in the appendix of this chapter. This alternate form is based on the Fourier series representation of a periodic function.
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