Electric Circuits I. Midterm #1 Examination

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1 EECS:2300, Electric Circuits I s8ms_elci7.fm - Electric Circuits I Midterm # Examination Problems Points Total 5 Was the exam fair? yes no

2 EECS:2300, Electric Circuits I s8ms_elci7.fm - 2 Problem 4 points Given is the electric circuit model shown of Figure., for which it is known that the element E delivers power P E to the rest of electric circuit. One-port Network A V AB I A I E E V E I E2 E 2 V E2 P E = 80 W I E = 30 A I E2 = 2 A B Figure. The electric circuit model with positive reference directions for currents and voltages that are known. Problem Statement On the example of the electric circuit model of Figure., demonstrate an ability to: (a) apply the concepts of active-convention and passive-convention coupled positivereference-directions for the voltage drop V X across a circuit element X, and the current I X flowing through the circuit element X; (b) determine the power P X that the circuit element X delivers/receives to/from the rest of the electric circuit. 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. Problem Solution For full credit, explicit demonstration of understanding the following solution steps is expected.. Being it known, that circuit element E in the electric circuit model of Figure.: (a) delivers power P E to the electric circuit, and (b) that the positive reference direction of its current I E is shown in the circuit model, determine, and show in the circuit model of Figure. the active-convention coupled positive reference direction for the voltage V E across element E.

3 EECS:2300, Electric Circuits I s8ms_elci7.fm Calculate the magnitude of voltage V E. Show your work in the space reserved for equation (-). V E = P E. I E. = = 6V (-).3 For the circuit element E 2, calculate the amount of power P E2 that E 2 delivers/receives to/from the circuit whose model is shown in Figure.. Show in the space reserved for equation (-2): your work, and your answers regarding the power P E2 being received or delivered. by E2. P E2 = I E2 V E2 = I E2 ( V E ) = 2 ( 6) = - 72W Since: (-2) (a) calculated power P E2 has a negative value, and (b) positive directions of reference current I E2 and voltage V E2 correspond to active convention the nature of the element E 2 does not agree with the active convention, so E 2 must be of different nature, i.e. a passive circuit element, that therefore only can receive power. yes no not applicable x element E 2 delivers power P E2 to the circuit of Figure., x element E 2 receives power P E2 from the circuit. of Figure...4 Based on your answers in parts. through.3, do a calculation that will make it possible to determine whether the One-port network in the circuit of Figure. delivers or receives power. Show in the space reserved for equation (-3): your work, and your answers regarding the power P AB being received or delivered. by the One-port network. By KCL: I E2 - I E +I A = 0 I A = I E - I E2 = 30-2 = 8A P AB = I A V AB = I A V E = 8 6 = 08 W Accordingly to: (-3) (a) positive directions of reference current I A and voltage V AB correspond to passive convention, (b) calculated power P AB has a positive value, and therefore, the one-port network receives power. yes no not applicable x One-port Network delivers power P AB to the circuit of Figure., x One-port Network receives power P AB from the circuit. of Figure..

4 EECS:2300, Electric Circuits I s8ms_elci7.fm - 4 Problem 2 6 points Given is the electric circuit model of Figure 2.. I D I I V I V R R I 2 I R R 2 I R2 I 3 V R2 R 3 I I R4 R = 4Ω R 2 = 5Ω R 3 = 2Ω = 8Ω I I = 0 A I D = βi R4 β = 0 Fig.2. An electric circuit model. Problem Statement For the electric circuit model of Figure 2., demonstrate an ability to: (a) use the Mesh Current Method (MCM) analysis technique to determine a partial solution to the circuit model of Figure 2., (b) show in the circuit model of Figure 2. the active-convention coupled positive-referencedirection voltage V I across the current source I I, (c) apply the determinant method for solving systems of linear algebraic equations, 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 shown results. Problem Solution For full credit, explicit demonstration of understanding the following solution steps is expected. 2. Indicate in Figure 2. the positive reference directions for nodal voltages that you will be using for writing the MCM mathematical model for the electric circuit model of Figure 2.. Since the circuit model of Figure 2. contains two current sources I I and I D that do not have a resistor connected in parallel with them: (a) the set of MCM equations for this circuit model may not contain equations which are written for either one of the meshes that contain I I and I D ; therefore, (b) one has to select two of the independent meshes in such a way that, these two meshes contain each just one of the sources I I and I D, (c) consequently, the mesh selection shown in Figure 2. has the mesh M2 containing the current source I I, and the mesh M3 containing the current source I D ; (d) eventually, only one mesh current equation can be written for mesh M, while the mesh currents I 2 and I 3 can be expressed, and their values calculated, in terms of I I and I respectively.

5 EECS:2300, Electric Circuits I s8ms_elci7.fm Write the set of canonical form MCM equations for the circuit model of Figure 2,. Show your work in the space reserved for equations (2-). The canonical form of MCM equations, for unknown mesh current I in the circuit model of Figure 2. is M: R I - R 2 I 2 - R 3 I 3 = 0 (2-) as the other two mesh currents are determined by their relations to the currents of current sources, I 2 = I I = 0A I 3 = I D = βi R4 = βi = 2.3 Show the solution method, and calculate the numerical values, for the mesh currents. Show your work in the space reserved for equations (2-2). the two obtained expression for I 2 and I 3 can be used to eliminate I 2 and I 3 from the canonical MCM equation for mesh M, R I - R 2 I I - R 3 βi = 0 which, after a rearrangement results in one equation with one unknown mesh current I, (R - R 3 β)i = R 2 I I with the coefficients from equation (2-) determined as, R = R 2 +R 3 + = = 5 Ω (2-2) R 2 = R 2 = 5 Ω R 3 = R 3 = 2Ω so that the values of the three mesh currents in the circuit model of Figure 2. are determined as I = R 2 I I = R - R 3 β I 2 = I I = 0A I 3 = I D = βi R4 = βi = 0(-0) = -00A 50 = = -0A - 5 Testing the correctness by KCL: I D - I R3 - I R4 = I 3 - (I 3 - I ) - I =I 3 - I 3 + I - I = 0A

6 EECS:2300, Electric Circuits I s8ms_elci7.fm Based on the positive reference direction for the current I R which is shown in Figure 2., and using the passive-convention coupled reference-directions of current and voltage, show in Fig.2. the positive reference direction for the voltage V R across the resistor R. 2.5 Using the active-convention coupled positive reference directions, show in Fig.2. the positive reference direction for the voltage V I across the current source I I. 2.6 Calculate the numerical value of the voltage V I with respect to its indicated positive reference direction. Show your work in the space reserved for equations (2-3). V I = - V R + V R2 = -R I R + R 2 I R2 = -R (I 3 I 2 ) +R 2 (I 2 I ) = =-4(-00-0) + 5[0 - (-0)] = -4(-0) + 5(20) = = 540V (2-3)

7 EECS:2300, Electric Circuits I s8ms_elci7.fm - 7 Problem 3 5 points Given is a resistive network whose electric circuit model is shown in Figure 3.. a R 3 R = 36Ω = 5Ω V V R 2 = 9Ω R 5 = 2Ω R R b R 5 R 3 = 3Ω V V = 60V t = 5 minutes Figure 3. Electyrical Circuit Model of a resistive network. Problem Statement For the electric circuit model of Figure 3., demonstrate an ability to: (a) apply the series/parallel resistance reduction method to determine the value of the equivalent resistance between the terminals of an electrical energy source in the circuit; (b) calculate the amount of energy that an independent DC voltage source delivers to a resistive electric circuit within a given period of time t. 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 shown results. Problem Solution Problem Solution For full credit, explicit demonstration of understanding the following solution steps is expected. 3. For 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

8 EECS:2300, Electric Circuits I s8ms_elci7.fm step by R xy, where x and y are two indices of your choice. a a V V + - R ab b R 3 R 2 R 5 R R 25 ab R 3 V V + - b (a) a (b) R R 235 ab V V + - b (c) Figure 3.2 Reduction step networks of the resistive network of Figure 3.. (a)reduced network after replacing the paralel connection of resistors R and R 2 by the equivalent resistor R 2. (b)reduced network after replacing the series connection of resistors R 2 and R 5 by the equivalent resistor R 25. (c)reduced network after replacing the paralel connection of resistors R 3 and R 25 by the equivalent resistor R Calculate the equivalent resistor resistances shown in Figure 3.2. Show your work in the space reserved for equations (3-). R R 2 = R R 2 = R = = 7.2Ω R + R R 25 = R 2 + R 5 = =9.2Ω R R 235 = R 3 R 25 = 3 R = = 2.6 Ω R3 + R (3-) R ab = + R 235 = = 7.6Ω 3.3 Calculate the amount of energy that voltage source V V delivers within a given time period t to the resistors in electric circuit model of Figure 3.. Show your work in the space reserved for equations (3-2). Denoting by P V and W V the power and energy that are delivered by the voltage source V V, W V = P V t = V V t = 5 60 = 84. kj 7.6 R ab (3-2)

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