# 'XNH8QLYHUVLW\ (GPXQG73UDWW-U6FKRRORI(QJLQHHULQJ. ECE 110 Fall Test II. Michael R. Gustafson II

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2 Problem I: [20 pts.] The (mostly) Basics (1) Write the equations that relate the current to the voltage for the following three elements: L v 1 i 1 R v 2 v 3 C i 2 i 3 (2) For the reactive elements above, write the equation for the energy stored in each element: (3) Circle the appropriate entries in the sentence below: The ( voltage drop across / current through ) an inductor and the ( voltage drop across / current through ) a capacitor must be continuous. (4) Clearly using phasors, simplify the following signal into a single cosine: v a (t) = 5cos(10t 14 o ) + 6sin(10t + 22 o ) (5) Fill in the following table: Quantity Symbol Name of real part Symbol Name of imaginary part Symbol Impedance Admittance (6) (This one is not as basic...) A resistor is connected to an inductor inside a box, but you do not know if they are connected in series or parallel. Two terminals are sticking out of the box. You conduct two experiments by applying a sinusoidal voltage across the terminals and then measuring the steady state current going into the box. The inputs and outputs you measured are below: Input Voltage (V) Output Current (ma) 5cos(5000t) 3.11cos(5000t 43.1 o ) 5cos(20000t) 2.33cos(20000t 13.1 o ) How are the resistor and inductor connected? What are their values? Hint: look at the formulas for and values of the impedance and admittance.

3 Problem II: [20 pts.] Switched Circuit Given the following circuit: v x t=2 ms R 1 R 2 v a L R 3 i b where the switch has been closed for a very long time before t = 2 ms. The switch opens at 2 ms. Assuming v a = 8 V i b = 30 ma R 1 = 500 Ω R 2 = 4 kω R 3 = 1 kω L = 750 mh Determine and accurately sketch an expression for the voltage drop across resistor R 1, v x, for t > 0 ms. You domain should be at least three time constants past the switch time. On the sketch, be sure to clearly indicate time constants and how you approximated slopes at various locations. Note: if you get a constant value of v x for all time, you are doing it wrong.

4 Problem III: [22 pts.] Frequency and Steady-State Analysis Given the following circuit: i a R 2 C R 1 v x v b (1) Determine a transfer function H a (jω) between source I a and output V x. You should assume v b is 0 V for this. (2) Determine a transfer function H b (jω) between source V b and output V x. You should assume i a is 0 A for this. (Spoiler Alert - you are going to use this in parts (4) and (5) below). (3) Assuming i a (t) = 4cos(2t) + 7cos(35t) ma v b (t) = 6 + 4cos(35t) V R 1 = 2 kω R 2 = 8 kω C = 5 µf = F and further assuming the circuit has been place for a very long time prior to t = t 0 sec, determine an expression in the time domain for the steady state voltage v x (t) for t > t 0 sec. Note for the next two parts you are being asked about H b (jω) The transfer function related to the voltage source The one related to the source on the right 1 (4) Sketch a straight line approximation for the Bode magnitude plot of H b (jω). Be sure to label corners, slopes, and magnitudes. (5) Given the information above, what kind of filter does H b (jω) represent? Why do you think that? 1 In other words, not the a one.

5 Problem IV: [14 pts.] Thévenin-Norton Equivalents Given the following circuit: R 1 R 3 v 1 a b R 2 R 4 (1) Clearly determine the values of and draw the Thévenin equivalent for the circuit as seen from terminals a and b. Note that you must explicitly solve for any variables - you cannot leave a coupled system of expressions. Once you have fully solved for an unknown, however, you may use it later without needing to write in the substitution. Also, please do not spend time cleaning up expressions. Make sure a and b are clearly drawn on your equivalent circuit. (2) Clearly determine the values of and draw the Norton equivalent for the circuit as seen from terminals a and b. Same rules as above. (3) If a load resistor R L were placed between the terminals a and b, what value of this resistor would maximize the power transferred to that load? What is the value of the power transferred to that load?

6 Problem V: [24 pts.] Bode Plots and Filters (1) You are given a circuit to analyze and have properly come up with a transfer function for it of: H(jω) = Y(jω) X(jω) = (jω) (jω + 40)(jω ) = 4(jω) (1 + j ω 40 )(1 + j ω 6000 ) (a) Sketch a straight-line approximation for the magnitude portion of the Bode plot. Be sure to label the axes, including numerical values, along with all slopes and critical frequencies. (b) Sketch a straight-line approximation for the phase portion of the Bode plot. Be sure to label the axes, including numerical values, along with all slopes and critical frequencies. (c) Based on this and any other information at your disposal, what kind of filter do you believe this to be? Why do you believe that? You must provide some reasonable explanation in order to receive credit for this part. (d) Approximately what is/are the cutoff frequency/ies for this filter? How did you come to that conclusion? Describe the process you used or reference the Bode diagram. (e) What is the passband gain for this filter? (2) Design a voltage-to-voltage low pass filter with a maximum gain of 1 and a cutoff frequency of 4000 rad/s using a single 2.2 kω resistor, a single reactive element of your choice, and wires. (3) Design a voltage-to-voltage high pass filter with a maximum gain of 1 and a cutoff frequency of 4000 rad/s using a single 2.2 kω resistor, a single reactive element of the kind you did not use in (2), and wires. Which is to say, if you used a capacitor in (2) you must use an inductor here; if an inductor there then a capacitor here. (4) Given the straight-line approximation to the Bode magnitude plot below, and also given that any double corners below are represented by critically damped (i.e. repeated) roots, determine a reasonable expression for the transfer function G(jω) from which it was generated. Carefully note where the corners are. Also state what kind of filter you believe this is, why you believe that, approximately what the cutoff frequency/frequencies is/are, and what the passband gain is/is log 10 G(jω), db Approximation Actual ω, rad/s

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