 ###### Find the total steady-state error due to a unit step input and a unit step disturbance in the system of Figure 4.
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###### Consider the signal x(t) with Fourier transform as shown 1) What is the minimum sampling rate for sampling this signal without causing aliasing? 2) Suppose the sampling rate is fs = 1.5 MHz . Plot the spectrum of the sampled signal. Give a precise plot. 3) Suppose the sampling rate is now fs = 4 MHz . Plot the spectrum of the sampled signal. Give a precise plot. Show how to recover the original continuous-time analog signal x(t). 4) Suppose we quantize the samples amplitudes (obtained with the sampling rate 4 MHz) to 128 levels. Find the resulting bit rate for the digital signal.
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###### What is the fundamental period of the below signal? You need to find the fundamental frequency of e and e 5 separately. The fundamental period of [n] is the least common multiple of the se two numbers.
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###### A second-order control system has the closed-loop transfer function T(s)=Y(s)/R(s). The system specifications for a step input follow: (1) (15 pts) Percent overshoot P.O.≤5% (angle information is required) (2) (15 pts) Settling time Ts<4s (3) (15 pts) Peak time Tp<1s. Show the permissible area for the poles of T(s) corresponding to each of the desired response requirements. Use a 2% settling criterion to determine settling time.
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###### 3. (30) A feedback system is shown by the figure below a) (10 pts) Determine the feedback system transfer function Y(s)/R(s). b) (12 pts) Determine the steady-state error for a unit step when K=0.4 and Gp(s)=3. c) (8 pts) Select an appropriate value for Gp(s) so that the steady-state error is equal to zero for the unit step input.
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###### 4. (30+10 bonus) The block diagram for a wheelchair control system is shown below. Typical values for the time constants are:a) (10 pts) Determine the characteristic equation of the feedback system b) (10 pts) Generate the Routh array for the wheelchair control system c) (10 pts) Determine the limiting gain K=K1K2K3 for a stable system d) (10 bonus pts) When the gain K is set equal to half of the maximum limiting value, determine and explain whether the settling time (to within 2% of the final value of the system) is less than 4s.
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###### 2. (30) A feedback system with negative unity feedback has a loop transfer function 2( 8) ( ) ( ) ( ) ( 4) C s L s G s G s s s     (a) (10 pts) Determine the closed-loop transfer function T(s)=Y(s)/R(s). (b) (10 pts) Using the figure on the right, determine the overshoot for a step input response. (c) (10 pts) Determine the steady-state error of the closed-loop system for a ramp input r(t)=t.
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###### 1. (30) A unity negative feedback control system has the loop transfer function: L(S)=Gc(s)G(s)= K/s(s+2k))(20 pts) Determine the closed-loop transfer function and percent overshoot due to a unit step input. (2) (5 pts) Determine the settling time (using a 2% settling criterion) due to a unit step input. (3) (5 pts) For what value of K is the settling time less than 1 second?
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###### Exa) Calculate the output 𝒚𝒌of LTID system with the following input, 𝒙𝒌, and impulse response, 𝒉𝒌 𝑥𝑛=(0.4)^𝑛*𝑢[𝑛] ℎ𝑛=(0.8)^𝑛*𝑢[𝑛]
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###### Determine the DTFT of the following signal 𝑥𝑛=4(1/2)^𝑛*𝑢(𝑛)
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###### 1. Find the most compact form of Fourier Transform X(f) of the time-domain signals x(t) below (both part (a) and (b)) using Fourier Integral. [Should show the result by the calculation of Integral !]
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###### 2. (a) For the time domain signal x(t)= 4Cos(2π200)t +8 Sin2(2π200)t, Find All Complex (Exponential) Fourier Series Coefficients Cn . (5) (b) For the given signal x(t) in Part (a), Find X(f), then Sketch its Magnitude Spectrum in frequency domain.[Sketch both positive and negative frequencies too] (3) (c) Determine the required Minimum Bandwidth [in Hz] of the ideal LPF to pass the signal x(t) without losing any power. (2)
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###### . This problem is on the AM-DSB modulation using single tone message m(t) using carrier c(t), where, m(t) = 2Cos(20πt) and C(t)= 10Cos(100πt). The Modulated Signal is generally written as S(t)=Ac[1+kam(t)] Cos2πfct. Hence Ac =10, and fc=50Hz from C(t) given above. (a) For Ka =0.1, decide the value of modulation index. (2) (b) Decide the maximum value of Ka to avoid envelope distortion. (2) (c) In case of (a), decide the maximum and minimum absolute values of the envelope of the signal S(t). (2) 4 (d) Among the total power of S(t), Decide the Portion of the Power (in %) due to Carrier frequency fc for ka=0.1 assuming 1Ω load. (4)
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###### 1.LetF⃗=xˆz+yˆx+zˆyandevaluatetheintegral􏰈 F⃗·d⃗lwhereCisthecirclex2+y2=a2in C the z = 0 plane. (Hint: Use Stokes’ theorem.) 2. A cylinder is filled with a uniform volume charge described by 􏰇K, ρ≤a, ρv= 0, ρ>a, where K and a are given constants. In addition, a line charge having uniform density λ0 resides along the axis of the cylinder. What is the value of λ0 if E⃗ vanishes for all ρ > a?
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