Electric Circuits I Final Examination

Transcription

1 The University of Toledo s8fs_elci7.fm - EECS:300 Electric Circuits I Electric Circuits I Final Examination Problems Points.. 3. Total 34 Was the exam fair? yes no

2 The University of Toledo s8fs_elci7.fm - EECS:300 Electric Circuits I Problem points Given is the electrical circuit model shown in Figure.. I I G V I 3 G G 8 S G 9 S I I 0 A V V ρi G3 V V G 3 V 3 V V VV G3 + - G 4 G 3 6 S G 4 7 S ρ 5 Ω I G3 0 I V Figure. The electric circuit model with positive reference directions for currents and voltages that ought to be calculated. Problem Statement Demonstrate an ability to solve the electrical circuit model of Figure. applying the Nodal Voltage Method to determine: (a) voltage V G3 across resistor G 3 and current I G3 that flows through resistor G 3, (b) currnet I V through the voltage source V V, (c) power P V delivered/consumed to/from the electrical circuit by the voltage source V V. 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.. Select the reference node, and and indicate in Figure. the positive reference direction nodal volatages Then prepare the set of canonical form nodal-voltage equations. Show your work in the space reserved for equation (-). Canonical NVM equations only can be written for two nodes, since nodal voltage V 3 is equal to the electromotive force V V of the voltage source which is connected between the node numbered 3 and the reference node, as it is shown in Figure.. G V - G V - G 3 V 3 -I I -G V + G V - G 3 V 3 I I with V 3 V V ρi G3 ρg 3 V (-)

3 The University of Toledo s8fs_elci7.fm - 3 EECS:300 Electric Circuits I. Show the expressions for the self and mutual conductances of the nodes, and calculate their values. Show your work in the space reserved for equation (-). Self conductances of the two nodes for which the nodal voltage method equations (-) have been written are, G G G + G S 8 S G G + G S G G 4 S 9 S G 6 S G (-) G G 0S 3 6 S G 0S G G G 3 G 3 G 8 S 4 7 S 3 8 S G 3 9 S G 3 G 3 G 9 S.3 Show the solution method for the nodal voltages. Show your work in the space reserved for equations (-3). Substituting the known expression (-) for V 3 into the canonical form nodal voltage equations, and rearranging the terms we obtain a system of two equations in two unknown voltages, V and V, (G - G 3 ρg 3 )V - G V -I I G V - G V - G 3 ρg 3 V -I I -G V + G V - G 3 ρg 3 V I I -(G + G 3 ρg 3 )V + G V I I V 3 V V ρi G3 ρg 3 V (-3).4 Calculate the numerical values, for the nodal voltages. Show your work in the space reserved for equations (-4) The system determinant of equations (-3) and the two needed numerator determinants are, G G - G 3 ρg 8 S 3 -G (G - G 3 ρg 3 )G - G(G + G 3 ρg 3 ) G -G - G 3 ρg 9 S 3 G ( ) S G 3 6 S -I I -G I I G G 3 ρg I I G + I I G AS G 3 8 S G - G 3 ρg 3 -I I I I (G - G 3 ρg 3 ) - I I (-G - G 3 ρg 3 ) G 3 9 S -G - G 3 ρg 3 I I 0( ) -0( ) AS (-4). G 4 7 S G 4 S G 6 S G 0S

4 The University of Toledo s8fs_elci7.fm - 4 EECS:300 Electric Circuits I, and the solutions for the nodal voltages are V V V V V 3 V V ρg 3 V V (-4).5 Using the shown positive reference direction for the current I G3 of the resistor G 3, and applying the passive convention for coupled positive reference directions of the current and voltage of G 3, indicate in Figure. the positive reference direction for the voltage V G3 across the resistor G 3 ; and calculate the value of the voltage V G3 and the current I G3. Sshow your work in the space reserved for equation (-5). V G3 V 0.08 V I G3 V G3 G A (-5).6 Applying the active convention for coupled positive reference directions of the current and voltage of circuit elements, show in Figure. the positive reference direction for the current I V of the voltage source V V ; then calculate the value of the current I V and the power P V delivered to the circuit by the voltage source V V. Show your work in the space reserved for equation (-6). KCL3: I V (V 3 - V )G + (V 3 - V )G ( )8 + ( ) A P V I V V V W (-6).7 Based on the result of calculation in part.6, determine whether the voltage source V V delivers, or receives power in the circuit of Figure.. Check the correct answer on both lines below, yes no not applicable x voltage source V V delivers power to the circuit of Figure., x voltage source V V receives power from the circuit of Figure.. Since the active convention has been selected for the coupled positive reference directions of the current and voltage of the voltage source V V, the positive sign of the numerical value of the power P V implies that the voltage source V V delivers power to the circuit of Figure.

5 The University of Toledo s8fs_elci7.fm - 5 EECS:300 Electric Circuits I Problem points v S + - C (a) i L L v R R R Ω C 0 mf L 5 mh v S 0cosωt V f 60 Hz Figure. An electric circuit specification. (a)electrical model. (b)phasor representation. Z C V il S + - Z L V R (b) Z R Problem Statement For the electrical circuit model of Figure.(a), demonstrate an ability to: (a) prepare its phasor domain representation, (b) determinethe phasor domain representation V R of voltage v R across resistor R, as specified under.,.3 and.4 below, (c) determinethe phasor domain representation of current I L through inductor L, as specified under.5 below, (d) calculate the reactive power component Q L of the complex power S L delivered to the phasor domain impedance Z L of the inductor 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 shown results. Problem Solution For full credit, explicit demonstration of understanding the following solution steps is expected.. For the electrical circuit model of Figure.(a), prepare the phasor domain representation in which the parameters of passive elements R, C, and L are denoted respectively by impedances Z R, Z C, and Z L. Show your work in the space reserved for Figure.(b).. Applying the voltage divider formula to the circuit of Figure.(b), express the real and imaginary parts of the voltage V R across the impedance Z R in terms of circuit element parameters R, L, and C. Show your calculation in the space reserved for equation (-). V R V S Z R Z L Z RL Z R +Z L Z R Z L V Z S V S RL + Z C Z R Z L Z + Z R Z L +(Z R +Z L )Z C (-) Z C R +Z L jωlrv sm -ω LCRV sm R + jωl jωlr + -ω LCR + R + jω L jωc ω L CR V sm [R(ω LC -) + jωl] R (ω LC -) + ω L

6 The University of Toledo s8fs_elci7.fm - 6 EECS:300 Electric Circuits I.3 Using the derived expression (-), determine numerical values of the real and imaginary parts of the voltage V R. Show your calculation in the space reserved for equation (-) It does make the computations less prone to errors, and faster, if the following five expression evaluated in advance, ω L C (0π) ω L C ω L 0π Ω ω L Ω R (ω L C - ) Ω ω L CR V sm (ω L C -) Re{V R } R (ω L C-) + ω L (-) 0.6 V Im{V R } ω L CR V sm ωl R (ω L C-) + ω L V.4 Calculate the numerical values of the module and argument of the voltage V R in the circuit of Figure.(b). Show your calculation, and the numerical value of the phasor V R in the space reserved for equation (-3). modv R V R argv R arctg V R. _ / 0.3 V. V 7.3 o 0.3 rad (-3).5 Determine the expression for the phasor representation of the current through inductor L. Show your calculation in the space reserved for equation (-4) I L V R Z L V R. /_ 0.3 jω L.885 /_ π/ 5.84 _ / -.7 A 5.84 _/ -7.8 o (.73 - j5.57)a.6 Determine the expression for the reactive power component Q L of the complex power S L delivered to the phasor domain impedance Z L of the inductor L. Show your calculation in the space reserved for equation (-5). (-4) Q L X L I Lef ω L I Lef ω L I L VAR (-5)

7 The University of Toledo s8fs_elci7.fm - 7 EECS:300 Electric Circuits I Problem 3 points Figure 3. shows the electrical circuit model with two capacitors C and C. V V + - t 0 - t 0 + Q C 60 ff S C v C C v C R C 360 ff V C (0 - ) 5V V V 5V Figure 3. Electrical model of an electric cirucuit. Problem Statement Capacitors C and C have been independently precharged: - capacitor C has been precharged in the configuration of the circuit which existed before the moment t 0; - capacitor C has been precharged to the voltage V C (0 - ) in different circuit (not shown in Figure 3.). Considering that switch S is switched at time t0s from the position shown in Figure 3. to its other position, demonstrate an ability to determine the amounts of: - quantity of charge transfer Q (with respect to the indicated positive reference direction of charge flow) which will occur in the circuit between the time t0, and the time t, when transient current completely stops flowing in the circuit; - voltages across capacitors C and C, V C ( ) and V C ( ) with respect to positive reference directions indicated in Figure 3., at the time when transient current completely stops flowing in the circuit; - the energy stored in the capacitor C at 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 For full credit, explicit demonstration of understanding the following solution steps is expected. 3. Determine the voltage V C (0 + ) across capacitor C at t 0 +. Show your calculation in the space reserved for equation (3-). Since the voltage across a capacitor can not change instantaneously, and V C (0 ) V V 5V, V C (0 + ) V C (0 ) V V 5V (3-)

8 The University of Toledo s8fs_elci7.fm - 8 EECS:300 Electric Circuits I 3. Determine the charge Q C (0 + ) stored in the capacitance C at t 0 +. Show your calculation in the space reserved for equation (3-). Q C (0 + ) C V C (0 + ) fc (3-) 3.3 Determine the charge Q C (0 + ) stored in the capacitance C at t 0 +. Show your calculation in the space reserved for equation (3-3). Since V C (0 ) is known, Q C (0 + ) C V C (0 + ) C V C (0 ) fc (3-3) 3.4 Determine the total charge stored in capacitors C and C at t 0 +. Show your calculation in the space reserved for equation (3-4). Q C (0 + ) + Q C (0 + ) fc (3-4) 3.5 Prepare the relation between voltages V C ( ) and V C ( ) across capacitors C and C at t. Show the prepared relation in the space reserved for equation (3-5). V C ( ) V C ( ) (3-5) 3.6 Prepare the expression for total charge stored on capacitors C and C at t as a function of the voltages V C ( ) and V C ( ), and capacitances C and C. Show your calculation in the space reserved for equation (3-6). Q C ( ) + Q C ( ) C V C ( ) + C V C ( ) (3-6) 3.7 Determine the total quantity of charge stored in capacitors C and C at t. Show your

9 The University of Toledo s8fs_elci7.fm - 9 EECS:300 Electric Circuits I calculation in the space reserved for equation (3-7). Any charge flow (the current in the circuit) after t0 + only causes a redistribution of charge between C and C ; i.e. the charge only flows from one capacitor to the other, but the total quantity of charge stored on C and C does not change. (3-7) Q C ( ) + Q C ( ) Q C (0 + ) + Q C (0 + ) 50 fc 3.8 Combine equations (3-5) and (3-6) to obtain an equation in which V C ( ) is the only unknown, then solve the equation for V C ( ) and calculate its numerical value. Show your calculation in the space reserved for equation (3-8). Q C ( ) + Q C ( ) C V C ( ) + C V C ( ) (3-8) Q C ( ) + Q C ( ) 50 V C ( ) 6V C + C ( ) Determine the charge Q C ( ) stored on the capacitor C at t. Show your calculation in the space reserved for equation (3-9). Q C ( ) C V C ( ) fc (3-9) 3.0 Determine the change in charge Q C in the capacitance C between t0 + and t. Show your calculation in the space reserved for equation (3-0). Q C Q C ( ) - Q C (0 + ) fc (3-0) 3. Determine the energy stored in the capacitor C at t. Show your calculation in the space reserved for equation (3-).. W C ( ) C V C ( ) fj (3-)