ECET 345 Enthusiastic Learning / snaptutorial.com

1. ECET 345 Week 1 Homework For more classes visit www.snaptutorial.com ECET 345 Week 1 Homework 1.Express the following numbers in Cartesian (rectangular) form. 2.Express the following numbers in polar form. Describe the quadrant of the complex plane, in which the complex number is located. 3.(a) A continuous-time sine wave has a frequency of 60 Hz, an amplitude of 117 V, and an initial phase of π/4 radians. Describe this signal in a mathematical form using the Sin function. 4. A sinusoidal signal described by 50 Cos (20πt + π/4) passes through a linear time invariant (LTI) system that applies a gain of 1.5 and a phase lag of π/2 radians to the signal. Write the mathematical expression that describes the signal that will come out of the LTI system. 5.A sinusoidal signal described by 20 Cos (2πt + π/4) passes through a linear time invariant (LTI) system that applies a gain of 2 and a time delay of 0.125 seconds to the signal. Write the mathematical expression that describes the signal that will come out of the LTI system. 6. Apply the principle of superposition to determine whether the following systems are linear. Sketch what the plot of the function looks like. 7. A continuous time system, described by y(t) = 5 Cos (2*π*20*t + π/2), is sampled at a rate 320 Hz. 8. Sketch the odd and even part of the following discrete signal. (See pages 13–14 of the text.) 9. Express the signal given in Problem 8 as the sum of the following

2. ****************************************** ECET 345 Week 1 iLab Observation of Wave- Shapes and Their Spectrum For more classes visit www.snaptutorial.com Objective of the lab experiment: The objective of this experiment is to observe the shapes of different kinds of signals such as sine waves, square waves, and so on and to study how the shape of a signal alters its spectrum. ****************************************** ECET 345 Week 1 iLab Signal Observation and Recreation

3. For more classes visit www.snaptutorial.com Objective: Using Multisim, create virtual circuits and experimentally observe the closest equivalent of four key signals (impulse, sinusoidal, exponential, and square wave) on the oscilloscope. ****************************************** ECET 345 Week 2 Homework For more classes visit www.snaptutorial.com ECET 345 Week 2 Homework 1.Redraw the following schematics with the impedance of each of the element shown in Laplace domain. Then determine the overall impedance of the entire circuit between the two ends of the shown circuit and express it in Laplace domain as a ratio of two polynomials in s, with the coefficients of the highest power if s in the numerator and denominator are made unity. (Follow the method outlined in the lecture

4. to determine the impedances of elements in Laplace domain and then use the formulas for combining impedances in series and parallel.) 2. (a) Apply Laplace transform to the following differential equation and express it as an algebraic equation in s. 3. An RC circuit with an initial condition is shown below. The toggle switch is closed at t = 0. Assuming that a current i(t) flows clockwise in the circuit, Write the integral equation that governs the behavior of the circuit current and solve it for the current in the circuit i(t) and voltage across the capacitor as a function of time using Laplace transforms. Note the polarity of the initial condition as marked in the figure. (Take help from the document “Solving RC, RLC, and RL Circuits Using Laplace Transforms” (located in Doc Sharing) and the Week 2 Lecture to see how initial conditions are entered in Laplace domain.) 4. The voltage in a circuit, expressed in Laplace domain, is given by the questions below. 5.An RLC circuit is shown below. There is an initial voltage of 5 V on the capacitor, with polarity as marked in the circuit. The switch is closed at t = 0 and a current i(t) is assumed to flow clockwise. Write the integral-differential equation of this circuit using Kirchoff’s method (sum of all voltages around a loop is zero). Apply Laplace transform as outlined in the lecture for Week 2 and in the document “Solving RC, RLC, and RL Circuits Using Laplace Transforms” (located in Doc Sharing) and write i(s) in Laplace transform notation. Express the denominator with the coefficient of the highest power of s unity. Then invert to obtain the current in time domain, i(t). ****************************************** ECET 345 Week 2 iLab Response of RC circuits

5. For more classes visit www.snaptutorial.com Objective of the lab experiment: The objective of this experiment is to experimentally measure the impulse and step response of an RC circuit and compare it to theoretical results using Laplace transform. ****************************************** ECET 345 Week 2 Lab Response OfRc Circuits (100% Score) For more classes visit www.snaptutorial.com Objective of the lab experiment: The objective of this experiment is to experimentally measure the step response of an RC circuit and compare it to response prediced using MATLAB

6. ****************************************** ECET 345 Week 3 Homework For more classes visit www.snaptutorial.com ECET 345 Week 3 Homework The transfer function of a circuit is given by Express the transfer function in a form in which the coefficients of the highest power ofs are unity in both numerator and denominator. What is the characteristic equation of the system? (Hint: see this week’s lecture for a definition of characteristic equation.) Determine the order of the transfer function. Determine where the poles and zeroes of the system are located. ____________ Using MATLAB, plot the pole zero map and the Bode plot of the two transfer functions and paste the graphs below. Identify and briefly discuss the differences between the Bode plot of the two transfer functions. ******************************************

7. ECET 345 Week 3 Lab Transfer Function Analysis Of Continuous Systems For more classes visit www.snaptutorial.com ECET 345 Week 3 Lab Transfer Function Analysis of Continuous Systems Objective of the lab experiment: The objective of this experiment is to create continuous (s domain) transfer functions in MATLAB and explore how they can be manipulated to extract relevant data. We shall first present an example of how MATLAB is used for s (Laplace) domain analysis, and then the student shall be required to perform specified analysis on a given circuit. ****************************************** ECET 345 Week 4 Homework For more classes visit

8. www.snaptutorial.com ECET 345 Week 4 Homework 1. A shiny metal disk with a dark spot on it, as shown in figure below, is rotating clockwise at 100 revolutions/second in a dark room. A human observer uses a strobe that flashes 99 times/second to observe the spot on the metal disk (a strobe is a flashing light whose rate of flashing can be varied). The spot appears to the human observer as if it is rotating slowly 2. (a) A system samples a sinusoid of frequency 480 Hz at a rate of 100 Hz and writes the sampled signal to its output without further modification. Determine the frequency that the sampling system will generate in its output. 3. The spectrum of an analog signal is shown below, containing . Such a signal is sampled by an ideal impulse sampler at a 100 Hz rate. List the first 10 positive frequencies that will be produced by the replication. (Hint: Follow the method outlined in the lecture for spectrum replication of sampled signals.) 4. The spectrum of an analog signal is shown below. It is sampled, with an ideal impulse sampler, at a rate of 200 Hz 5. Determine the Z transform of the signal,, shown below using the basic definition of Z transform . All values not shown can be assumed to be zero. 6. a) A simulation diagram is shown below. Determine the difference equation associated with the diagram. 7. An analog signal is given by f(t) = t (i.e., it increases linearly with time and is thus is a unit ramp.) It is convolved with a second signal, g(t), which is of the form g(t) = 1 (i.e., it has a constant value of 1 or is a unit step function). The two signals are shown below.

9. ****************************************** ECET 345 Week 4 iLab Part 1 RC Circuit Frequency Response For more classes visit www.snaptutorial.com Objective of the lab experiment: The objective of this experiment is to experimentally measure the frequency response of a simple RC circuit using Multisim and observe how changing R and C will affect the outcome. ****************************************** ECET 345 Week 4 iLab Part 2 Experimental Observation of Aliasing For more classes visit www.snaptutorial.com

10. Objective of the lab experiment: The objective of this experiment is to observe the effect of aliasing in a discrete sampling system and to measure how aliasing alters the frequency of an input signal that is beyond the Nyquist limit. This lab can also be used to quantitatively and qualitatively observe the effect of an antialiasing filter, even though we do not do so in this exercise. ****************************************** ECET 345 Week 4 Lab Experimental Observation Of Aliasing (100% Score) For more classes visit www.snaptutorial.com ECET 345 Week 4 Lab Objective of the lab experiment: The objective of this experiment is to observe the effect of aliasing in a discrete sampling system and to measure how aliasing alters the frequency of an input signal that is beyond the Nyquist limit. This lab can also be used to quantitatively and qualitatively observe the effect of an antialiasing filter, even though we do not do so in this exercise.

11. ****************************************** ECET 345 Week 5 Homework For more classes visit www.snaptutorial.com 1.Using z-transform tables (page 776 of text or equivalent), find the z- transform of 2.Find the inverse z-transform, x(n), of the following functions by bringing them into a form such that you can look up the inverse z- transform from the tables. This will require some algebraic and /or trigonometric manipulation/calculation. You will also need a table of z- transforms (page 776 of text or equivalent). When computing the value of trigonometric functions, keep in mind that the arguments are always in radians and not in degrees. 3.Find the first seven values (i.e., x(n) for n = 0 to 6) of the function given below. Hint: Manually calculate the three parts separately for various values of n and add or subtract them point by point for various values of n. For example, for n = 2 equals 2 * 2 * 1 (or 4); for n = 5 equals 2 * u(2) or 2 * 1 = 2; and so on. Also keep in mind that u(n - k) is a unit step function delayed byksamples, and hence it will be zero for all values of (n - k), which are negative and 1 otherwise.

12. 4.The simulation diagram of a discrete time system is shown below. Find the first six output (y(0) to y(6)) of the system when an input x(n) , as computed in problem 3, is applied to the discrete time system. ****************************************** ECET 345 Week 5 Ilab Convolution Of Signals Solution (100% Score) For more classes visit www.snaptutorial.com Objective of the lab experiment: The objective of this experiment is to demonstrate how the convolution is used to process signals entering a system. 1. Convolution in the time domain is equivalent to what mathematical operation in the frequency domain? 2. When we convolve the triangular 10 Hz input with the impulse response of the 50 Hz low-pass filter, why is it that the peaks of output become rounded and not a sharp point as in the input triangular function? 3. Why is it that we get no (or very little) output when we convolve the 60 Hz sinusoid with the impulse response of the filter? 4. When we apply the 10 Hz output, which is within the pass band of the filter, we see that we get nearly the same sinusoid in the output except

13. for a time delay. How is the time delay a signal experiences as it passes through a system related to the phase characteristic of the system response? ****************************************** ECET 345 Week 5 iLab Convolution of Signals For more classes visit www.snaptutorial.com Objective of the lab experiment: The objective of this experiment is to demonstrate how the convolution is used to process signals entering a system ****************************************** ECET 345 Week 6 Homework For more classes visit www.snaptutorial.com

14. ECET 345 Week 6 Homework 1.Find the z-transform x(z) of x(n) = . Hint: Follow the method used in the lecture for Week 6. Also, when evaluating the numerical value of a trig function, keep in mind that the arguments of trig functions are always in radians and not in degrees. 2. Find the system transfer function of a causal LSI system whose impulse response is given by and express the result in positive powers of z. Hint: The transfer function is just the z-transform of impulse response. However, we must first convert the power of -0.5 from (n - 1) to (n - 2) by suitable algebraic manipulation. 3. Express the following signal, x(n), in a form such that z-transform tables can be applied directly. In other words, write it in a form such that the power of 0.25 is (n-1) and the argument of sin is also expressed with a (n-1) multiplier. 4. The transfer function of a system is given below. Find its impulse response in n-domain. Hint: First expand using partial fraction expansion and then perform its inversion using z-transform tables 5. The transfer function of a system is given by 6. A simulation diagram is shown below. We apply a unit impulse to such a system. Determine the numerical values of the first three outputs. You are free to use MATLAB where appropriate or do it entirely by hand. ****************************************** ECET 345 Week 6 iLab Z-Domain Analysis of Discrete Systems

15. For more classes visit www.snaptutorial.com Objective of the lab experiment: The objective of this experiment is to perform z domain analysis of discrete (sampled) signals and systems and extract useful information (such as impulse and step response, pole zero constellation, frequency response, etc.) from a z domain description of the system, such as its transfer function. We shall also study conversion of analog transfer functions (in s domain) into equivalent z domain transfer functions using bilinear transform. ****************************************** ECET 345 Week 6 Lab Z-Domain Analysis Of Discrete Systems (100% Score) For more classes visit www.snaptutorial.com ECET 345 Week 6 Lab Z-Domain Analysis of Discrete Systems Objective of the lab experiment:

16. The objective of this experiment is to perform z domain analysis of discrete (sampled) signals and systems and extract useful information (such as impulse and step response, pole zero constellation, frequency response, etc.) from a z domain description of the system, such as its transfer function. Equipment list: • MATLAB ****************************************** ECET 345 Week 7 Homework For more classes visit www.snaptutorial.com 1.A sine wave of 60 Hz, amplitude of 117 V, and initial phase of zero (or 117 sin(2π*60t) is full wave rectified and sampled at 2,048 samples per second after full wave rectification. Research the Fourier series for a full wave rectified sine wave (on the Internet or in circuit theory books, such as Linear Circuits by Ronald E. Scott) and write it below. Then write a MATLAB program that samples and stores 4,096 points of full wave rectified sine wave and performs Fourier analysis (FFT) of the full wave rectified sine wave on the stored points. Plot the results in both linear and log scale (in two separate figures) and extract the amplitude of the DC component and the first four harmonics (first , second, third, and fourth multiple of the

17. fundamental frequency) of the Fourier analysis, then enter them in the table given below. The DC component is given by the first number in the Fourier analysis. Hint: Full wave rectification can be achieved in MATLAB simply by taking the absolute value (abs command) of the sine wave. ****************************************** ECET 345 Week 7 ilab Fourier Analysis Of Time Domain Signals Solution (100% Score) For more classes visit www.snaptutorial.com Objective of the lab experiment: The objective of this experiment is to perform Fourier analysis to obtain frequency domain signature of signals and systems that are measured or whose characteristics are known in time domain. Towards this end, we shall learn how to use Fourier transform to obtain Bode plots of systems from time domain data passing through the system. We shall also learn the equivalence of convolution operation in time domain with multiplication operation in frequency domain. (a) Application of Fourier transform to time domain signals (a) What are the frequencies of the first seven peaks in the FFT?

18. (b) Does the spectrum contain only even, only odd, or both even and odd harmonic peaks? (c) Research the Fourier series expansion of a triangular wave using the Internet. From the formula you come up with, compare the amplitudes and frequencies of the harmonics that you found with what the theory says they should be. Explain any differences. ECET 345 Week 7 iLab Fourier Analysis of Time Domain Signals For more classes visit www.snaptutorial.com Objective of the lab experiment: The objective of this experiment is to perform Fourier analysis to obtain frequency domain signature of signals and systems that are measured or whose characteristics are known in time domain. Towards this end, we shall learn how to use Fourier transform to obtain Bode plots of systems from time domain data passing through the system. We shall also learn the equivalence of convolution operation in time domain with multiplication operation in frequency domain.

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