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Unformatted text preview: Problem 27 (3) Linear. Consider the responses to two arbitrary inputs: ' m) = an?)
yztr) = x202) Multiply ﬁrst by a and the second by bland add to get
ayiU) + [73:20) = M102) + M202) That is, for the input ax1(t)+ lax/2(1), we replace 1 by :2 to get the new output which is the righthand side of the above equation.
(b) Time varying. Consider the response to the delayed input: 3’00) = x02  1:) Now consider the delayed output due to the undelayed input:
y(r — 1:) = x[(r  1?] Clearly the two are not the same. ((2 ) Noncausal. Consider t = 2 which gives y(2) = x(4); i.e., the output depends on a future value of
the input. (d) Not zero memory. This follows from (c J where it was found that the output does not depend
only on values of the input at the present time only. Problem 210 Using Kirchoff’s voltage equation and Ohm’s law, the appropriate equations are ﬂ0=Lﬁm+ﬂ0
d:
ﬂn=mm
£0; 2 mm
d! R d! Substitute the last equation in the ﬁrst and rearrange to obtain 6%) + E = .6.
dr Ly“) Lxm (b) The proof is similar to those of Problems 28 and 29.
(c ) Consider _ r r
dyU—Tl : where [I d: d,” d: ll
As
I
1" Thus L“ L r) + £311 1) = £x(t — 1:) dr L L which shows that the system is fixed.
(d) Note that the solution to the homogeneous equation is yHU) = Ae 'R'”: t > 0 Assume a complete solution of this form where A is time varying. Substitute into the differential
equation of part (a) to obtain I
A(:) = f—Exﬂkkmdl +140
0 Since the inductor current is assumed 0 at 1’ = 0, this gives A0: 0, so the solution to the differential
equation is ‘ R R
y(t) = —x(}L)exp[F—(r  1)}Ak
it L Problem 212 (a) Linear; ﬁrst order; causal; time invariant. (b) Linear; first order; causal; time varying.
(c ) Nonlinear; second order; causal; time invariant. ((1) Nonlinear; zero order; causal; time invariant. W (a) By KVL and Ohm's law: 51 (tho)
R dt + h(:) = 6(1) where the forcing function being a delta function means that the response is the impulse response.
For t < 0. the impulse resp0nse is zero because the input is 0 and the initial conditions are assumed
0. For t > 0, we solve the homogeneous differential equation to get the solutiou h(t) : Ae'Rm‘, r > 0 To ﬁnd the condition required to ﬁxA, integrate the ditfereutial equation (with impulse forcing
function) through I = 0: mg dh(t)
R a:
o— 0? o.
d: + h(r)d: = 60M: : 1
l l Fromthe form of Mr), we see that it has a step discontinuity at r = 0 and therefore its derivative has
an impulse at r = 0. Thus the second term on the lefthand side integrates to 0 (it only has a step
discontinuity). The ﬁrst term on the lefthand side integrates to (UR)[h(0+)  h(0)]. Thus, the above
equation becomes h(0+) = RIL = A. and the impulse response becomes h(t) = {—e “Lam (b) Let the voltage across the resistor be visa) and the voltage across the inductor be vL(t). Thus
vL(r) + vR(r) = 6(t) But the voltage across the resistor cannot be proportional to an impulse because then the current
around the loop would be proportional to an impulse and this means the inductor voltage (L times
the derivative of the current) w0uld be proportional to the derivative of an impulse. Since there is
no derivative of an impulse on the righthand side to balance it, this cannot be the case. Therefore,
it must be true at time 0 that vL(t) = 50) and the Current possesses a step at time 0. In particular, 0v
. _ 1 F 1
40+) u Elana: _ L For t > 0, the current around the resistorinductor loop must satisfy
dim
dt L +Rt’(r) = 0 or £0) = Ae'R‘IL, t > 0 The constant A can be ﬁxed by setting t'(0) = i(0+) = 1/L. Thus, the same result is obtained for the
impulse response as obtained in part (a). W By KVL around the loop, I x(r) = Rita) + % fi(l)d}. +R2i(r) —w But i0) = 3’0)er Substitute this into the integrodifferential equation and differentiate once to get R1+R2dy(t) + 3’“) = E
R2 at: ch d: One way to ﬁnd the impulse response is to ﬁnd the step response and differentiate it. The solution
to the homogeneous equation is v R390
I a(t) = Ae'm‘ I > 0 With a step input, the righthand side of the differential equation is an impulse. To get the required
initial condition, we integrate the differential equation through I = 0: 0+
1131+]?2 da(t)dr+ 1 R2 0_ d: RZC Dr 0e
a(r)dr = 6(r)dr : 1
l i To match the righthand side, the integrand of the ﬁrst term on the lefthand side must contain a unit
impulse and, therefore, the second term on the lefthand side is proportional to a unit step. Hence
the integral on the secmrd term through I = 0 is 0 (a step discontinuity). The ﬁrst term is a perfect
differential. Thus, we obtain a(0+) = Rim?!1 + R,) as the required condition, and the step
response is R _ i R  +
2 6 "(RI Wan) and h(t) : La“) = 2 an) — —1—6 “‘R' MW)
R1 +R2 d‘ R1+R2 (R1+R2)C a(t) : W (a) The step response is 030‘) = I MAMA = exp(Rt/L)u(t) (b) The ramp response is a,(:) = fa,(l)dl = %[1~ cxp(—Rr/L)]u(r) .1 WM
Write the input as x0) = r0) — r(r — 1) — u(t # 2) Thus. by superposition, the output is
y(t) = a,(:)  a,(z — 1) — (1,0  2)
= in — exp(—R:/L)]u(r)  ﬁ {1 — cxp[R(t  1)/L)]u(r  1) — exp[—R(t _ 2)/L]u(t — 2) Sketches of the input and output are given below: 1.5 ‘u'
a...’
K 0.5 Fromm (a) The frequency response function is given in terms of the impulse response by How) 2 f h(t)e 'h'dr From the impulse response given in Problem 229, this gives jw HUto) = f[6(t) ~ £exp(erlL)u(t)]exp(jmt)dt = L (b) In terms off = mﬂn, the frequency l‘CSpOnSC function is Jff/f3 where f = R
3 1 + mf3 Err—L H0): Taking the magnitude, the amplitude response function is
[fl/fa «1 + (ﬂfg)2 AU): (c ) The phase response function is the argument of the frequency response function. It is given by 60‘) = g—m'v/fg) Plots of the amplitude and phase msp0nsc functions are given below: phase resp.; radians ...
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 Spring '08
 ATKIN

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