# Electric Circuits I. Midterm #1

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1 The University of Toledo Section number s5ms_elci7.fm - Electric Circuits I Midterm # Problems Points Total 5 Was the exam fair? yes no

2 The University of Toledo Section number s5ms_elci7.fm - 2 Problem 3 points Given is the electric circuit model, shown in Figure.. V V + - R 2 R 3 I R R R 4 v A v B Figure 3. Electrical model of a resistive network and its parameters values. R 25Ω R 2 5Ω R 3 7Ω R 4 8Ω V V 25V T 5s Problem statement For the electric circuit model of Figure., demonstrate an ability to apply the voltage/current divider formula, the Ohm s Law and power calculation in a resistive circuit to determine: - indicated voltages V A and V B, - the current flow and power dissipation in resistor R, - energy E R converted to heat in resistor R during time interval T. Problem solution Hint # For full credit, give answers to all questions, prepare all required circuit diagrams, write all equations for which the space is left, and show all symbolic and numerical expressions whose evaluation produces shown numerical results. An explicit demonstration of understanding the following solution steps is expected.. Apply the passive coupled positive reference convention for voltage V A and current I R to determine the positive reference direction for current I R through resistor R. Show the determined positive reference direction for current I R in the circuit model of Figure...2 Determine the value of the indicated voltage drop V A in the circuit model of Figure.. Show your calculation in the space reserved for equation (-). V A V V 25V (-).3 Determine the value of the indicated voltage drop V B in the circuit model of Figure.. Show your calculation in the space reserved for equation (-2). V B V V R 3 + R 4 R 2 + R 3 + R V (-2)

3 The University of Toledo Section number s5ms_elci7.fm Determine the current I R that flows through resistor R in the circuit model of Figure.. Show your calculation in the space reserved for equation (-3). I R V A 25 R 25 A (-3).5 Determine the amount of power P R that is dissipated in resistor R in the circuit model of Figure.. Show your calculation in the space reserved for equation (-4). P R V A I R W (-4).6 Determine the amount of energy W R that is dissipated in resistor R in the circuit model of Figure. during the time interval of T seconds. Show your calculation in the space reserved for equation (-5). W R P R Τ J (-5)

4 The University of Toledo Section number s5ms_elci7.fm - 4 Problem 2 7 points Given is the electric circuit model shown in Figure 2.(a). I C g V R3 R 0Ω R 4 R 2 5Ω R 3 2Ω R 4 R I V V v +- R 2 V C I C R 3 V R3 R 4 4Ω V V 40V g 2S I N V R R 2 I C V 2 V R3 2 R 3 0 (a) (b) Figure 2. 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)representation in which the series connection of the voltage source V V and the resistor R has been replaced by the equivalent Norton s circuit. Problem statement Using the electric circuit model of Figure 2.(a), demonstrate an ability to: - use the Nodal Voltage Method for solving the distribution of voltages in a resistive circuit, - apply the equivalence of Norton s and Thevenin s equivalent circuits to substitute one of these circuits by the other when so needed in a process of solving an electric circuit, - apply the determinant method for solving sets of simultaneous linear algebraic equations, - determine current flow through circuit elements using the calculated nodal voltages. Problem solution Hint # For full credit, give answers to all questions, prepare all required circuit diagrams, write all equations for which the space is left, and show all symbolic and numerical expressions whose evaluation produces shown numerical results. An explicit demonstration of understanding the following solution steps is expected. Hint #2 If it appears that KCL equations can not be applied to all "independent" nodes in the circuit model of Figure 2.(a), consider using an equivalent circuit model in which the needed KCL equations can be written. 2. Select the reference node, and indicate in Figure 2.(a) the positive reference directions for the nodal-voltages of remaining nodes in the model. In case you decided to use an equivalent circuit model, show the prepared model in the space reserved for Figure 2.(b), and write in the space reserved for equation (2-) any voltage-current relation that completes the same model. As NVM is based on the application of the KCL, and a current-voltage relation does not exist for an ideal voltage source, the series connection of the voltage source V V and resistor R ought to be replaced by its Norton s equivalent circuit. After this transformation is applied, the circuit model of Figure 2.(b) is

5 The University of Toledo Section number s5ms_elci7.fm - 5 obtained, and equation (2-) explicitly defines the introduced current source parameter I N. I N G V V (2-) 2.2 For the selected circuit model in Figure 2., prepare the set of general form nodal-voltage equations. Show your work in the space reserved for equations (2-2). Based on the nodal voltages indicated in Figure 2.(b), normal form of the NVM system of equations is, G V - G 2 V 2 I C - I N (2-2).5 -G 2 V + G 22 V 2 -I C Since the current parameter I C of the dependent current source depends on the voltage of resistor R 3, I C g V R3 voltage V R3 ought to be expressed in terms of nodal voltages, so that yet another unknown voltage V R3 is not introduced into the system of equations (2-2), V R3 V 2 I C g V Rearrange the equations (2-2) so that known terms appear at the right hand side, and the unknown terms at the left hand side of the equality sign; show the resulting equations in the space reserved for equations (2-3). G V - G 2 V 2 g V 2 - G V V -G 2 V + G 22 V 2 -g V 2 (2-3) G V - (G 2 + g)v 2 - G V V -G 2 V + (G 22 +g)v Calculate the numerical values of the coefficients in equations (2-3) (the self and mutual resistances of the nodes); show the calculation in the space reserved for equations (2-4). G R G 2 R S 0.2 S G G + G 2 + G S G 2 G 2 G S G 3 R3 2 S G 22 G 3 + G S (2-4) G 4 R S

6 The University of Toledo Section number s5ms_elci7.fm Prepare expressions (in terms of the nodal-voltage equation coefficients), and calculate the values, of determinants involved in the solution of equations (2-3); show the calculation in the space reserved for equations (2-5). G -G 2 - g -G 2 G 22 +g -G (G 22 +g) -G 2 (G 2 +g) 5. (0.75+2) (0.25+2) S 2 - G V V -G 2 - g 0 G 22 +g -G V V (G 22 +g) (0.75+2) SA (2-5) 2 G -G V V -G 2 0 -G V V G SA 2.6 Calculate the numerical values of the nodal-voltages/mesh-currents; show the calculation in the space reserved for equations (2-6). V V V V (2-6) 2.7 Indicate in the circuit of Figure 2.(a) the active convention positive reference directions for the: - current I V of the voltage source V V, and - the e.m.f. V C of the current source I C, then calculate the values of current I V and voltage V C. Show the calculation in the space reserved for equations (2-7). Since voltage source V V and resistor R are connected in series, their currents are equal. OL: I V I R (V V + V ) G [40 + (-.58)] A KVL: V C V - V (-.05) -0.43V (2-7)

7 The University of Toledo Section number s5ms_elci7.fm - 7 Problem 3 5 points Given is a resistive network whose electrical circuit model is shown in Figure 3.. a R 2 R 36Ω V V + - R ab b R R 3 R 5 R 4 R 2 9Ω R 3 3Ω R 4 5Ω R 5 2Ω Figure 3. Electrical model of a resistive network and its parameters values. Problem statement For the electric circuit model of Figure 3., demonstrate an ability to apply the series/parallel reduction method to determine the value of the equivalent resistance R ab seen by the voltage source V V between the terminals a and b Problem solution Hint # For full credit, give answers to all questions, prepare all required circuit diagrams, write all equations for which the space is left, and show all symbolic and numerical expressions whose evaluation produces shown numerical results. An explicit demonstration of understanding the following solution steps is expected. 3. For the resistive part of the network of Figure 3., prepare the graphical representations of three equivalent networks of gradually decreasing complexity, which result when resistors connected in series/parallel are replaced by an equivalent resistor. Show the three graphical representations, in the order of their creation, in the space reserved for Figure 3.2. Label the equivalent resistances at each reduction step by R xy, where x and y are indices of your choice

8 The University of Toledo Section number s5ms_elci7.fm For the resistive part of the network of Figure 3., prepare the graphical representations of three equivalent networks of gradually decreasing complexity, which result when resistors connected in series/parallel are replaced by an equivalent resistor. Show the three graphical representations, in the order of their creation, in the space reserved for Figure 3.2. Label the equivalent resistances at each reduction step by R xy, where x and y are indices of your choice. R 34 a a R 234 a R 2 R ab R R 5 R ab R R 5 R ab R R 2345 b (a) (b) (c) Figure 3.2 Reduction steps for the resistive network of Figure 3.. (a)equivalent network after reducing the series connections of resistors R and R 2. (b)equivalent network after reducing the series connection of resistors R 2 and R 3 to the equivalent resistor R 23. (c)equivalent network after reducing the series connection of resistors R 5 and R 23 to the equivalent resistor R Calculate all equivalent resistances indicated in Figure 3.2; show the work in the space reserved for equations (3-). R 34 R 3 + R Ω R R 234 R 34 R 2 34 R R34 + R Ω (3-) R 2345 R R Ω R 2345 R 8 36 R ab R 2345 R R R Ω

9 The University of Toledo Section number s5ms_elci7.fm - 9 Appendix Problem 2: Solution using the Mesh Current Method I V V v + - V C I C R 4 R 3 R 0Ω R 2 5Ω R 3 2Ω R 4 4Ω V v + - V V 40V g 2S + - I C g V R3 V C R 4 R 3 R R 2 V R3 R I R I 2 2 V R3 (a) Figure 2. 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)representation in which the parallel connection of the current source I C and the resistor R 4 has been replaced by the equivalent Thevenin s circuit. (b) 2. Indicate in Figure 2. the positive reference directions for the nodal-voltages/mesh-currents that you have selected for preparing the mathematical model; also write in the space reserved for equation (2- ) any voltage-current relation needed to define the outcome of having applied the Hint #2. As MCM is based on the application of the KVL, and a current-voltage relation does not exist for an ideal current source, the application of MCM requires that part of the circuit consisting of the parallel connection of the current source I C and resistor R 4 be replaced by its Thevenin s equivalent circuit. After this transformation is applied, the circuit model of Figure 2.(b) is obtained, and equation (2-) explicitly defines the introduced voltage source parameter V C. V C R 4 I C R 4 g V R3 (2-) 2.2 For the circuit model of Figure 2., prepare the set of general form nodal-voltage/mesh-current equations. Show your work in the space reserved for equations (2-2). Based on the mesh currents indicated in Figure 2.(b), normal form of the MCM system of equations is,

10 The University of Toledo Section number s5ms_elci7.fm - 0. R I - R 2 I 2 V V -R 2 I + R 22 I 2 V C (2-2) Since the voltage parameter V C of the dependent voltage source depends on the resistor voltage V R3, V C R 4 g V R3 voltage V R3 must be expressed in terms of the mesh currents, so that yet another unknown variable is not introduced into the system of equations (2-2), V R3 -R 3 I 2 V C R 4 g V R3 R 4 g (-R 3 I 2 ) -R 3 R 4 gi Rearrange the equations (2-2) so that known terms appear at the right hand side, and the unknown terms at the left hand side of the equality sign; show the resulting equations in the space reserved for equations (2-3). R I - R 2 I 2 V V -R 2 I + R 22 I 2 -R 3 R 4 gi 2 (2-3) R I - R 2 I 2 V V -R 2 I + (R 22 +R 3 R 4 g)i Calculate the numerical values of the coefficients in equations (2-3) (the self and mutual conductances/resistances of the nodes/meshes); show the calculation in the space reserved for equations (2-4). R R + R Ω R 2 R 2 R 2 5 Ω (2-4) R 22 R 2 + R 3 + R Ω 2.5 Prepare expressions (in terms of the nodal-voltage/mesh-current equation coefficients), and calculate the values, of determinants involved in the solution of equations (2-3); show the calculation

11 The University of Toledo Section number s5ms_elci7.fm - in the space reserved for equations (2-5). R -R 2 -R 2 R 22 +R 3 R 4 g R (R 22 +R 3 R 4 g) -R 2 R 2 5. ( ) Ω 2 V V -R 2 0 R 22 +R 3 R 4 g V V (R 22 +R 3 R 4 g) 40. ( ) VΩ (2-5) 2 R V V -R 2 0 V V R VΩ 2.6 Calculate the numerical values of the nodal-voltages/mesh-currents; show the calculation in the space reserved for equations (2-6). I A (2-6) I A Indicate in the circuit of Figure 2. the active convention positive reference directions for the: - current I VS of the voltage source V V, and - the e.m.f. V CS of the current source I C, then use OL to calculate the value of current I VS and KVL to calculate the value of voltage V CS. Show the calculation in the space reserved for equations (2-7). Since only one mesh current flows through the voltage source V V, current I VS is equal to the mesh current I. I V I A (2-7) KVL: V C (I 2 - I )R 2 + I 2 R 3 ( ) V 2.8 Calculate the power which the two energy sources deliver/consume to/from the circuit of Figure 2.;

12 The University of Toledo Section number s5ms_elci7.fm - 2 show the calculation in the space reserved for equations (2-8). The power delivered by the energy source V V is determined by P V V V I V W The power delivered by the energy source I C is determined by P C V C I C V C g V R3 V C g ( I 2 R 3 ) (-26 2) 2.9 W (2-8) 2.9 Calculate the amount of electrical energy W F. converted to heat in the circuit of Figure 2. during a three minutes time interval; show the calculation in the space reserved for equations (2-9). The power converted to heat in the circuit of Figure 2. is equal to the power delivered to the circuit by the two energy sources, V V and I C, therefore, P V + P C W then the energy converted to heat in the circuit during three minutes is (2-9) W F. ( P V + P C ) t kj

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