Unformatted text preview: EECS 314 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above Winter 2011 Homework set 4 Problem 1 (30 points) Power transfer from the source to the load resistance This circuit diagram shows a voltage source VS with the source resistance RS to the left of the terminals a, b, and the load resistor RL to the right of the terminals a, b. The source parameters VS and RS are fixed (constant); the load resistance RL can be varied. Your goal is to study and explain what happens in the circuit as RL is varied. Show your work for all parts on additional pages; write the results in the space below. Part 1 (10 points) Derive the algebraic expressions for the current I in the circuit, the voltage V across the load resistor (the same as the voltage across the terminals a and b), and the power PL absorbed by the load resistor in terms of VS , RS , and RL. I = Calculate the maximal values of the current IMAX , voltage VMAX , and power PL,MAX in terms of VS and RS (Hint: The maximal power is absorbed by the load resistance that equals the source resistance). IMAX = VMAX = PL.MAX = V = PL = © 2011 Alexander Ganago Last printed 1/27/11 5:13 PM Page 1 of 2 File: 2011 W 314 HW 04 p1.doc EECS 314 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above Winter 2011 Homework set 4 Problem 1 Part 2 (10 points) Assume VS = 120 V, RS = 100 Ω, and RL varied from 1 Ω to 10,000 Ω = 10 kΩ. Use software, with which you are comfortable (MATLAB, Excel, etc.), to plot 3 curves: 1. Normalized current I/ IMAX 2. Normalized voltage V/VMAX 3. Normalized power P/PMAX as functions of the load resistance RL . Use a logarithmic axis for the load resistance and clearly label which curve shows what on your plot (hand‐written labels are OK). Your plot should show at least 5 points per decade (any interval, over which varies by a factor of 10, such as from 3 to 30 Ω). Attach your plot with your name, discussion section #, and the date of work (all in ink). If you had to write computer code (in MATLAB, etc.) include it along with your plot, with your name and date of work printed as part of the code. If you use EXCEL, attach a one‐page printout of the spreadsheet, with your name and date of work printed on top. Part 3 (10 points) Use your calculations (Part 1 of this problem) and computer‐generated plot (Part 2 of this problem) to explain why the power P absorbed by the load (or transferred to the load) decreases at low load resistances, and why it decreases at high load resistances. Write a brief discussion on whether your plot agrees with your algebraic results. From your plot, determine the ranges of load resistances, within which the power transferred to the load remains: above 90% of the max power above 70% of the max power above 50% of the max power. © 2011 Alexander Ganago Last printed 1/27/11 5:13 PM Page 2 of 2 File: 2011 W 314 HW 04 p1.doc In ohms In % of RS EECS 314 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above Winter 2011 Homework set 4 Problem 2 (30 points) Volt‐Amp characteristics of sources represented as their Thevenin equivalents Part 1 (15 points) Assume that the unknown source (the box on the diagram below) can be represented with its Thevenin equivalent circuit (the voltage source VT in series with the source resistance RT). This table gives values of the current I through the load resistor RL: Load resistance RL Current I 1 Ω 5 Ω 1 2 A 6 A Calculate the following parameters and write your answers in the table below: open‐circuit voltage VT in volts; short‐circuit current ISC in amps; the source resistance RT in ohms. Also calculate the load resistance connected to the terminals a, b, to which the source transfers the maximal power, and the maximal power that can be dissipated in the load. VT (volts) ISC (amps) RT (ohms) RL, max power !(ohms ) PMAX (watts ) Show your work on an additional page. Sketch the volt‐amp characteristic of the source in the format shown here: Clearly indicate the numerical values of the open‐circuit voltage and short‐circuit current, and clearly show the two data points listed in the table above. Your sketch (on an additional page) can be either hand‐written or computer‐generated. © 2011 Alexander Ganago Last printed 1/27/11 10:07 PM Page 1 of 2 File: 2011 W 314 HW 04 p2.doc EECS 314 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above Winter 2011 Homework set 4 Problem 2 Part 2 (15 points) As in Part 1, assume that the unknown source can be represented with its Thevenin equivalent circuit. Do not assume that the source is the same as in Part 1. The table below gives values of the power transferred to the load resistor RL: Load resistance RL Power PL 2 Ω 8 Ω 1600 W 1152 W Calculate the following parameters and write your answers in the table below: open‐circuit voltage VT in volts; short‐circuit current ISC in amps; the source resistance RT in ohms. Also calculate the load resistance connected to the terminals a, b, to which the source transfers the maximal power, and the maximal power that can be dissipated in the load. VT (volts) ISC (amps) RT (ohms) RL, max power !(ohms ) PMAX (watts ) Show your work on an additional page. Sketch the volt‐amp characteristic of the source in the format shown here: Clearly indicate the numerical values of the open‐circuit voltage and short‐circuit current, and clearly show the two data points listed in the table above. Your sketch (on an additional page) can be either hand‐written or computer‐generated. © 2011 Alexander Ganago Last printed 1/27/11 10:07 PM Page 2 of 2 File: 2011 W 314 HW 04 p2.doc EECS 314 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above Winter 2011 Homework set 4 Problem 3 (30 points) Practical Perspective: The Load Line Determines the Operating Point of a Non‐Ohmic Device Background: If a non‐ohmic electronic device such as a semiconductor diode, whose Volt‐Amp characteristic is known (for example, as a file of lab data), is connected to a linear circuit that includes sources and resistors, we can apply the concept of the Thevenin equivalent circuit in order to determine the operating voltage and current, as well as the power absorbed by the non‐ohmic device. The strategy is straightforward: (1) Replace the linear circuit with its Thevenin equivalent; (2) Obtain its Volt‐Amp characteristic and plot it as a straight line, which is also known as the load line; (3) On the same graph, plot the non‐linear Volt‐Amp characteristic of the electronic device; (4) Determine the crossing point of the two Volt‐Amp characteristics; (5) The coordinates of the crossing point are the operating voltage and current. This graph shows lab data for a semiconductor diode obtained with the same equipment and the same VI that you used in DC Lab, along with the Volt‐Amp characteristic of the Thevenin equivalent circuit with VT = VS = 4 V and RT = RS = 4 kΩ. The operating point is highlighted with a circle; the arrows show the operating voltage of about 1.62 V and the operating current of about 0.6 mA; their product equals about 0.972 mW and is the power absorbed by the semiconductor diode. © 2011 Alexander Ganago Last printed 1/27/11 8:09 PM Page 1 of 2 File: 2011 W 314 HW 04 p3.doc EECS 314 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above Winter 2011 Homework set 4 Problem 3 (continued) Use file 2007‐07‐24 IV diode a4_.xls available on CTools web site (HW 04 folder) and determine the operating voltage in volts, the operating current in mA, as well as the power in mW absorbed by the semiconductor diode connected to the following circuit: Note that for this circuit the Thevenin equivalent resistance equals: RT = RS + RX Assume the following parameters: VS = 5 V; RS = 200 Ω, RP = 1 kΩ. Write your results in the table below. Operating voltage, V Operating current, mA Power in mW absorbed by the diode Attach the printout of the volt‐amp characteristic of the diode. Remember to write on it your name, date of work, and discussion section number (in ink!). On your printout show the load lines for all resistance values (either draw them by hand using a ruler or plot them with a computer program, with which you are comfortable), as well as arrows that indicate the operating current and voltage for each resistance value. Show your calculations for RX = 200 Ω. RX = 1 kΩ RX = 500 Ω RT = 200 Ω RX = 0 © 2011 Alexander Ganago Last printed 1/27/11 8:09 PM Page 2 of 2 File: 2011 W 314 HW 04 p3.doc EECS 314 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above Winter 2011 Homework set 4 Problem 4 (10 points) Saw‐Tooth, or Ramp Waveform and Its Applications The Saw‐Tooth waveform is used to move, or sweep material objects whose position depends on the applied voltage. For example, due to piezoelectric effect, voltage causes mechanical displacement, which can be used for making positioning tools with extreme accuracy. In old televisions and oscilloscopes with Cathode Ray Tubes, or CRTs, the applied voltage displaces the beam of electrons (or cathode rays, according to their old name). Behind each television screen (old, not a flat one) sits a CRT, where beams of electrons are produced by electron guns (one for each color) and deflected by the electric fields, which are generated by high‐voltage power supplies connected to the pairs of vertical and horizontal plates so that each beam sweeps the entire surface of the screen. To each pair of plates, a high‐voltage Saw‐Tooth (ramp) signal is applied, which makes the beam sweep the whole the length (or height) of the screen at a constant speed, and ensures nearly instantaneous return of the beam to the original position (during the return motion the beam is blanked, its brightness made “blacker than black”). A simplified sequence of scans is shown on the sketch (the return motion is shown with dashed lines). Note that the beam moves slowly from top to bottom, and moves fast from left to right, because the frequency of the saw‐tooth waveform for vertical deflection is low, while the frequency of the saw‐tooth waveform for horizontal deflection is high. © 2011 Alexander Ganago Last printed 1/27/11 8:20 PM Page 1 of 2 File: 2011 W 314 HW 04 p4.doc EECS 314 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above Winter 2011 Homework set 4 Problem 4 (continued) The US standard for television involves 30 frames (called fields) per second; each frame consists of 525 lines (out of which only 483 are visible; others – at the edges of the screen – are blanked). In other words, the frequency in Hz for the vertical sweep equals 30 (the number of frames); the frequency in Hz for the horizontal sweep equals the product of 30 and 525 For the ramp signals used in this TV standard, calculate and write your answers below: Frequency, in Hz Period, in µsec Rise Time (from 10% level to 90% level), in µsec Fall Time (from 10% level to 90% level), in µsec Vertical sweep Horizontal sweep Show your work below and/or on additional pages. © 2011 Alexander Ganago Last printed 1/31/11 10:34 AM Page 2 of 2 File: 2011 W 314 HW 04 p4.doc EECS 314 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above Winter 2011 Homework set 4 Problem 5 (50 points) Build a Saw‐Tooth Waveform as a Sum of Its Harmonics Part 1 (10 points) Consider a saw‐tooth waveform at 30 Hz and 5 kVppk = 5,000 Vppk, which can be used for the vertical deflection of electron beams in a TV set. Calculate the peak amplitude in volts of each sinusoid corresponding to the fundamental and the harmonics up to the 5th of this waveform. Write your results in the table below. Component Fundamental 2nd harmonic 3rd harmonic 4th harmonic 5th harmonic Frequency (Hz) 30 Vpk (volts) Show your work = a sample step‐by‐step calculation for the 5th harmonic. Use additional pages as needed. Part 2 (20 points) Make a computer‐generated plot of the sinusoidal components that sum up to this waveform. Your plot should show 3 curves: (1) The fundamental only (2) The sum of the fundamental and harmonics up to the 3rd (3) The sum of the fundamental and harmonics up to the 5th. Use equation 6 of the file Saw‐tooth harmonics. On the plot show 2 to 5 periods of the waveforms. Use distinct line styles to show different curves and clearly explain what is what. Make sure that the peak‐to‐peak amplitude of the waveforms on your plot is reasonably close to 5,000 Vppk, and that the period is close to 33 msec. © 2011 Alexander Ganago Last printed 1/27/11 8:24 PM Page 1 of 2 File: 2011 W 314 HW 04 p5.doc EECS 314 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above Winter 2011 Homework set 4 Problem 5 Part 3 (20 points) Appreciate the importance of the phase shifts in the summation of harmonic components. The same sinusoidal components that comprise the saw‐tooth wave above can also sum up to an alternative waveform if you neglect the phase shifts of even‐numbered harmonics: Make a computer‐generated plot of the alternative waveform according to equation 7 of the file Saw‐tooth harmonics. Your plot should show 3 curves: (1) The fundamental only (2) The sum of the fundamental and harmonics up to the 3rd (3) The sum of the fundamental and harmonics up to the 5th. Notice that the shapes of waveforms on your plots in Parts 2 and 3 of this problem are distinct (mirror images of each other). If you use MATLAB, attach the printouts of your computer code and plot for each part of this problem. If you use EXCEL or another software package that does not generate the code, write by hand the formulas for each of the sums (with numerical values for the amplitudes and the frequencies). On each page with your work, including printouts of the plots, include the following information: Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking this information clearly written. © 2011 Alexander Ganago Last printed 1/27/11 8:24 PM Page 2 of 2 File: 2011 W 314 HW 04 p5.doc EECS 314 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above Winter 2011 Homework set 4 Problem 6 (30 points) Equivalent Resistance Determine the equivalent resistance between terminals C and D in each circuit. In Circuits 2, 3, and 4, wires (thick curved lines) have zero resistance. Hint: Redrawing the circuit may be among the key tools for successful solution. Part 1 (20 points) For each circuit, write the equivalent resistance RCD in the simplified algebraic form, such as given in the following example RAB = [R1  (R5+R4+R3)]+R2 for Circuit 1. Write your answers below. Show your work below and/or on additional pages. Circuit 1 RCD = Circuit 2 RCD = © 2010 Alexander Ganago Last printed 1/27/11 10:05 PM Page 1 of 2 File: 2011 W 314 HW 04 p6.doc EECS 314 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above Winter 2011 Homework set 4 Problem 6 Part 1 Continued Circuit 3 RCD = Circuit 4 RCD = Part 2 (10 points) Assume the resistance values: R1= 10 Ω, R2 = 20 Ω, R3 = 30 Ω, R4 = 40 Ω, R5 = 50 Ω, and calculate RCD in circuit 4. Write your answer: RCD = _______________ Ω. Show your work. © 2010 Alexander Ganago Last printed 1/27/11 10:05 PM Page 2 of 2 File: 2011 W 314 HW 04 p6.doc ...
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