# Electric Circuits I Final Examination

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1 EECS:300 Electric Circuits I ffs_elci.fm - Electric Circuits I Final Examination Problems Points Total 38 Was the exam fair? yes no //3

2 EECS:300 Electric Circuits I ffs_elci.fm - Problem 4 points Given is the electric circuit model, shown in Figure.. R = 5Ω R = Ω R 3 = 4Ω R 4 = 3Ω V V = 0V I C = 5A R R 3 G 3 I V V V - + R R 4 G G V I C C I VN V V I C G 4 (a) Figure. The electric circuit model with positive reference directions for currents and voltages that ought to be calculated. (a)original drawing of the circuit model. (b)equivalent representation with equivalent energy sources selected to satisfy the requirement which makes possible writing the equations of the required analyses technique. 0 (b) For the electric circuit model of Figure., demonstrate an ability to use the nodal voltage method analysis technique to determine a partial solution that includes: (a) showing the active-convention coupled positive-reference-direction voltage V C across the current source I C, Hint # Solution (b) power delivered by the energy sources V V and I C to the circuit of Figure., (c) electrical energy converted to heat in the circuit of Figure. during a time interval t=4 minutes. For full credit: all equations, all answers to questions, all circuit models and other graphical representations are expected to be entered into the space designated for them; all shown numerical results must be preceded by the symbolic and numeric expressions whose evaluation produces these numerical results. For full credit, an explicit demonstration of understanding the following solution steps is expected.. When the solution process, or just a simplification of the electrical model, involves replacement of a part of the circuit by an equivalent circuit, show the new equivalent form of the electric circuit model in the space reserved for Figure.(b); also write in the space reserved for equation (-) any voltage-current relation that is needed to complete the electrical model shown in Figure,(b). In any case, indicate in Figure. the positive reference directions for the selected nodal voltages. As NVM is based on the application of the KCL, and a current-voltage relation does not exist for an ideal voltage source, the application of NVM requires that part of the circuit consisting of the series connection of the voltage source V V and resistor R be replaced by an equivalent current source circuit. After this replacement is applied, the circuit model of Figure.(b) is obtained, and the equation (-) explicitly defines the introduced current source parameter I VN //3

3 EECS:300 Electric Circuits I ffs_elci.fm - 3 as, I VN = G V V =0. 0 = 4 A (-) It should be noted and well understood that the two currents, I V and I VN, are two different currents: - I V is the actual current flowing through the voltage source V V in the circuit model of Figure.(a), - I VN, which appears only in the circuit model of Figure.(b), is the parameter of the current source I VN of the Norton s equivalent circuit for the series connection of the voltage source V V and resistor R of the circuit model in Figure.(a),. For the circuit model of Figure.(b), prepare the set of canonical form nodal-voltage equations. Show your work in the space reserved for equations (-3). Since the circuit model of Figure.(b) does not contain any voltage sources, the NVM is directly applicable to it without further modifications; the corresponding canonical form NVM system of equations is, G V - G V = I VN - G V + G V = -I C (-).3 Calculate the numerical values of the coefficients in equations (-) (the self and mutual conductances of the nodes); show the calculation in the space reserved for equations (-3). G = R = = 0. S 5 G 3 = R3 = = 0.5 S 4 G = R = = 0.5 S G 4 = R4 = = 0.33 S 3 G = G + G +G 3 = = 0.95 S G = G = G 3 = 0.5 S -3) G = G 3 + G 4 = = 0.58 S //3

4 EECS:300 Electric Circuits I ffs_elci.fm Prepare expressions (in terms of the nodal voltage equation coefficients), and calculate the values, of determinants involved in the solution of equations (-); show the calculation in the space reserved for equations (-4). = G -G -G G = G G -G G = = = 0.49 S = I VN -G - I C G = I VN G - G I C = = =.07 AS (-4) = G I VN -G - I C = - I C G + G I VN = = = AS.5 Based on the values of determinants obtained in the step.4, calculate the numerical values of the nodal voltages; show the calculation in the space reserved for equations (-5). V = = =. V (-5) V = = 3.75 = V Indicate in the circuit of Figure.(a) the active convention positive reference directions for the: - current I V flowing through the voltage source V V, and - the voltage difference V C across the current source I C, then determine the values of current I V and voltage V C. Show the calculation in the space reserved for equations (-6). Since there exists no current-voltage relation for an ideal voltage source, the current I V can not be determined directly; instead, as the voltage source V V is connected in series with the resistor R, the current I V through voltage source V V is equal to the current I R that flows through the resistor R. I V = I R = (V V -V )G = (0 -.) 0. = 3.56A V C = - V = - (- 7.65) = 7.65 V (-6) //3

5 EECS:300 Electric Circuits I ffs_elci.fm Calculate the power which the two energy sources deliver/consume to/from the circuit of Figure.; show the calculation in the space reserved for equations (-7). The power delivered by the energy source V V is calculated as, P V = V V I V = = 7. W The power delivered by the energy source I C is calculated as, P C = V C I C = = 38.5 W (-7).8 Calculate the amount of electrical energy W converted to heat in the circuit of Figure. during the time interval t; show the calculation in the space reserved for equations (-8). The power converted to heat in the circuit of Figure. is equal to the power delivered to the circuit by the two energy sources, V V and I C, P V + P C = = 09.37W and the energy converted to heat in the circuit during the time interval t is equal to the energy delivered to the circuit by energy sources, W = ( P V + P C ) t = = 6.5 kj (-8) //3

6 EECS:300 Electric Circuits I ffs_elci.fm - 6 Problem points Given is the electric circuit model and the circuit element parameter values as shown in Figure 3.(a). i I C i L L R R = 0Ω C = 00 µf L = 0 mh I I Z C I L Z L V L Z R i I = I m cosωt A I m = 0A (a) f = 0Hz (b) igure. An electric circuit specification. (a)tme domain electrical model. (b)phasor domain representation of (a). For the time domain electric circuit model of Figure.(a), demonstrate an ability to:. derive the corresponding phasor domain circuit model and determine its parameter values;. apply equivalents of resistive circuits analysis methods to the phasor domain circuit analysis; 3. apply specifically the voltage/current divider formula in the phasor domain analysis of electric circuits; 4. determine the complex power delivered/consumed by circuit elements. Prepare a partial solution of the electric circuit whose electrical model and circuit element parameter values are shown in Figure.(a). For full credit, the partial solution ought to include: (a) the phasor domain representation of the circuit as specified under. below, (b) the phasor domain representation V R of voltage v R across resistor R, as specified under.,.3 and.4 below, (c) the phasor domain representation I L of current i L through inductor L, and the phasor domain representation V L of the voltage across impedance Z L, as specified under.5 below, (d) complex power S L delivered to the impedance Z L, as specified under.6 below. Hint # For full credit: all equations, all answers to questions, all circuit models and other graphical representations are expected to be entered into the space designated for them; all shown numerical results must be preceded by the symbolic and numeric expressions whose evaluation produces the numerical results. Solution An explicit demonstration of understanding the following solution steps is expected.. For the time domain circuit model shown in Figure.(a), prepare the phasor domain representation and show it in the space reserved for Figure.(b). Hint# Denote the impedances of the resistor R by Z R, denote the impedance of the capacitor //3

7 EECS:300 Electric Circuits I ffs_elci.fm - 7 C by Z C, and denote the impedance of the inductor L by Z L. 3. Applying the current divider formula to the circuit of Figure.(b), express the real and imaginary parts of the current I L of the impedance Z L in terms of circuit element parameters R, L, C and I m. Show your calculation in the space reserved for equation (-). Z I L = I R R R(R - jωl) R - jωlr I = I Z m R +Z L R + jωl = I m R + (ωl) = I m R + (ωl) (-) R - ωlr Re{I L } = I m Im{I L } = I m R + (ωl) R + (ωl).3 Using the derived expression (-), determine numerical values of the real and imaginary parts of the current phasor I L. Show your calculation in the space reserved for equation (-) ωl = 40π = 7.54 Ω ωlr = = 5Ω (ωl ) = 7.54 = 56.7Ω R - jωlr I L = I m = 0 R + (ωl) 0 - j = 0( j0.33) = j6.6)a (-).4 Calculate the values of the module and argument of the phasor I L in the circuit of Figure.(b). Use the space reserved for equation (-3) to show your calculation, and the numerical values of the polar representation of the of the phasor I L. modi L = I L = = 8.8 A argi L = arctg -6.6 = arctg(-0.375) = -0.6 o = rad 7.6 (-3) I L = I L argi L = o A.5 Determine the values of the module and argument of the phasor domain representation of the voltage V L across the impedance Z L. Show the passive coupled positive reference direction of the voltage V L in the circuit of Figure.(b) and show your calculation in the space reserved for equation (-4). V L = Z L I L = jωl I L argi L = o o = o V (-4).6 Determine the values of the real and imaginary parts of the phasor domain representation of the complex power S L delivered to impedance Z L. Show your calculation in the space reserved for equation (-5). V L I L * o o o S L = = = = o = = (0 + j333) VAR (-5) //3

8 EECS:300 Electric Circuits I ffs_elci.fm - 8 Problem 3 points R = 3kΩ R = kω R 3 = kω V V = V I C = βi V β = 3 R R A R R A R 3 I V V V - + v R R 3 I C I V V V - + v R - + V CT (a) B (b) B Figure 3. An electric circuit specification. (a)electrical model of the circuit. (b)alternate simplified model of the For the electric circuit model of Figure 3.(a), demonstrate an ability to: (a) apply the Thevenin s/norton s theorem to simplify a circuit model, (b) use the obtained simplified circuit model to find a solution to the original circuit model. (c) determine solutions to circuits which contain dependent energy sources, (d) correctly apply the KVL to a closed path (loop) in the circuit model. Hint # For full credit: all equations, all answers to questions, all circuit models and other graphical representations are expected to be entered into the space designated for them; all shown numerical results must be preceded by the symbolic and numeric expressions whose evaluation produces these numerical results. Solution For full credit, an explicit demonstration of understanding the following solution steps is expected. 3. Show on the circuit model in Figure 3.(a): - the active convention coupled positive reference direction for the current I V of the voltage source V V, - the passive convention coupled positive reference direction for the voltage V R with respect to the current I V of the resistor R. 3. Apply the Thevenin s/norton s theorem to the circuit model in Figure 3.(a) to replace the part of the circuit to the right of terminals A and B with an equivalent circuit that does not contain a current source. Show the new circuit model in the space reserved for Figure 3.(b), and show in the space reserved for equation (3-) a symbolic expression for the electromotive force V CT of the new current dependant voltage source. I C = βi V V CT = R 3 I C (3-) V CT = βr 3 I V //3

9 EECS:300 Electric Circuits I ffs_elci.fm Apply the KVL to the loop (closed path) in the circuit model of Figure 3.(b), and write the obtained KVL equation in the space reserved for equation (3-), using the shown positive reference directions for the voltages and currents of circuit elements. V V - (R + R + R 3 ) I V + V CT = 0 V V - (R + R + R 3 ) I V + βr 3 I V = 0 (3-) 3.4 Combine equations (3-) and (3-) to obtain a symbolic form solution for the current IV that flows through the voltage source V V. Show your calculation in the space reserved for equation (3-3). V V - (R + R + R 3 ) I V + βr 3 I V = 0 V V - [R + R - (β-)r 3 ] I V = 0 I V = V V R + R - (β-)r 3 (3-3) 3.5 Calculate the numerical value for the current I V in the circuit model of Figure 3.. Show your calculation in the space reserved for equation (3-4). I V = V = V [3 + - (3 - )] 0 3 = R + R - (β-)r = A = 4 ma (3-4) 3.6 Determine the expression and calculate the value of the voltage drop V R across the resistor R in the circuit model of Figure 3.. Show your calculation in the space reserved for equation (3-5). V R = R I V = = 8 V (3-5) //3

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