# Electric Circuits I FINAL EXAMINATION

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1 EECS:300, Electric Circuits I s6fs_elci7.fm - Electric Circuits I FINAL EXAMINATION Problems Points Total 34 Was the exam fair? yes no 5//6

2 EECS:300, Electric Circuits I s6fs_elci7.fm - Problem points Given is the electric circuit model, shown in Figure.. I C βi R R Vv 30V I β 5 R 6Ω R 3 R I R R 4Ω R 3 8Ω VC I C R 5 I 3 I R 4 R 4 7Ω R 5 5Ω V v + - Figure. The electric circuit model with positive reference directions for currents and voltages that ought to be calculated. Problem Statement For the electric circuit model of Figure., demonstrate an ability to: (a) use the Mesh Current Method analysis technique on a DC circuit that contains both, the voltage and current sources, (b) assign the active-convention coupled positive-reference-direction voltage and current to an ideal electrical energy source, (c) apply the determinant method for solving systems of linear algebraic equations, (d) calculate the power delivered by the energy sources V V and I C, (e) calculate the amount of electrical energy converted to heat in the circuit of Figure. during a time interval t4 minutes. 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, an explicit demonstration of understanding the following solution steps is expected.. Select and indicate in Figure. the positive reference directions for a feasible set of meshcurrents. 5//6

3 EECS:300, Electric Circuits I s6fs_elci7.fm - 3. Using the set of mesh currents selected in section., prepare the set of canonical form Mesh Current equations for the electrical circuit shown in Figure.. Show your work in the space reserved for equations (-). Using the Mesh Current Method, we only need to write two normal form equations, since Mesh Current I 3 is equal to the current βi R of the dependent current source. R I - R I - R 3 I 3 0 -R I + R I - R 3 I 3 V V (-) expressing I R in terms of the mesh currents which pass through R we obtain mesh current I3 expressed in terms of the rest of mesh currents as, I 3 βi R β(i - I).3 Eliminate from equations (-) the one mesh current whose expression in terms of the other two mesh currents is determined by the electric circuit model in Figure.. Show your work in the space reserved for equations (-). Substituting the known expressions for I 3 into the mesh current equations (-), and rearranging them we obtain, (R- β R3) I - (R - β R3)I 0 - (R+ β R3) I + (R+ β R3)I Vs (-).4 Calculate the numerical values of the coefficients (the self and mutual resistances of the meshes) in equations (-). Show your calculation in the space reserved for equations (-3). Self resistances of the two meshes for which the mesh current method equations (-) and (-)h been written are, R R + R + R Ω(-4) R R + R 4 + R Ω(-5) (-3) and the mutual resistances of those meshes are R R R 4 Ω(-6) R 3 R 3 R 3 8 Ω.(-7) R 3 R 3 R 5 5 Ω. 5//6

4 EECS:300, Electric Circuits I s6fs_elci7.fm Calculate the values of determinants involved in the solution process of the system of equations (-). Show your calculation in the space reserved for equations (-4). R- β R3 -R + β R3 -R- β R3 R+ β R3 (R- β R3)(R+ βr3) + (- R +β R3)(R+ βr3) (8-5. 8). ( ) + ( )(4+5. 5) Ω 0 -R + β R3 V V R+ β R3 -V V (-R+β R3) 0. (4-5. 8) 30(- 86) -580VΩ (-4) R- β R3 0 -R- β R3 V V V V (R- β R3) 0(8-5. 8)30. (-6) -860VΩ.6 Calculate the values of mesh currents indicated in the electric circuit model of Figure.. Show your calculation in the space reserved for equations (-5). I A I A I 3 β(i - I) 5( ) 75A (-5).7 Applying the active convention for coupled positive reference directions of the current and voltage of a circuit element, show in Figure. the positive reference direction for the voltage V C of the current source I C ; then calculate the value of the voltage V C and the power P C delivered to the circuit by the current source I C. Show your work in the space reserved for equations (-6). V C (I 3 - I )R 3 + (I 3 - I )R 5 ( ). 8+(75-39) V P C I C V C I 3 V C kw (-6) 5//6

5 EECS:300, Electric Circuits I s6fs_elci7.fm Based on the calculation result (-6), determine whether the current source I C delivers, or absorbs power in the circuit of Figure.. Check the correct answer on both lines below, yes no not applicable x current source I C delivers power to the circuit of Figure., x current source I C absorbs 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 source I C, the positive sign of the numerical value of the power P C implies that the current source I C delivers power to the circuit of Figure.. 5//6

6 EECS:300, Electric Circuits I s6fs_elci7.fm - 6 Problem points Given is the electric circuit model shown in Figure.(a). R L a R 0 Ω Z T a v v + - C V oc L5mH C 5µF V T + - V ab (a) b v v 40cos30πt V (b) b Figure.(a) An electric circuit model. (b)phasor domain Thévenin's equivalent model for the circuit shown in Figure.(a). Problem Statement For the electric circuit model of Figure.(a): - prepare the Thévenin s equivalent circuit with respect to the pair of terminals a and b; - determine maximum real power that the circuit of Figure.(a) can deliver to a load connected to the pair of terminals a and b. 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 calculated numerical results. Problem Solution Explicit demonstration of understanding the following solution steps is expected.. For the circuit shown in Figure.(a), prepare the electrical model of the phasor domain Thévenin's equivalent circuit with respect to terminals a and b. Show the prepared model in the space reserved for Figure.(b). Determine the symbolic expressions (in terms of the circuit parameters R, L, C and V vm ), and determine the numerical values of the real and imaginary parts of the Thévenin's equivalent circuit voltage V T. Show the work and results in the space designated for equation (-). Electromotive force V T is equal to the voltage V oc between terminals a and b when they are left "open circuited". When there are no external elements connected between terminals a and b, no current flows through the inductor L and, therefore, no voltage drop appears across the impedance Z L, which makes the voltage V oc equal the voltage V c across the capacitor s impedance Z C, which voltage, in turn, is 5//6

7 EECS:300, Electric Circuits I s6fs_elci7.fm - 7 determined by the voltage divider formula, V V Z C V V jωc V T V ab V oc V C Z R +Z C R + jωc - j j j9.54 V V V +jωcr (- jωcr)v m +ω C R 0 o (-) ωcr 30π (ωcr) Determine the numerical values of the modulus V T, and argument φ VT, of the Thévenin's equivalent voltage V T ; show your calculation in the space reserved for equation (-). modv T V T V T V (-) argv T φ VT arctg rad 5.6 o Determine the symbolic expressions in terms of circuit parameters (R, L, and C), for the Thévenin's equivalent impedance Z T ; show your calculation in the space designated for equation (-3). With the voltage source deenergized, the impedance seen from the terminal pair a and b is that of Z L connected in series with the parallel connection of Z R and Z C. Z T Z L + R Z C Z R Z C Z R +Z C + Z L R R + R ωl(+ω C R ) ωcr +(ωcr) + j +(ωcr) jωc jωc + jωl R(- jωcr) +(ωcr) + jωl (-3).5 Determine the numerical values of the real and imaginary parts, R T and X T, of the Thévenin s equivalent impedance (Z T R T +jx T ); show your calculation in the space designated for equation 5//6

8 EECS:300, Electric Circuits I s6fs_elci7.fm - 8 (-4). ωcr 30π (ωcr).6.58 ωl 30π Ω Z T R ωl(+ω C R ) ωcr +(ωcr) + j +(ωcr) j 5(+.58) (-4) ( j0.)ω.6 Prepare the expression which shows the relation between the impedance Z T of the circuit in Figure.(b) and the impedance Z LD which would draw maximum real power from the circuit in Figure.(a) when connected between terminals a and b. Show your calculation in the space designated for equation (-5). Z LD is simply the complex conjugate of Z T. Z LD Z T * R T - jx T ( j0.)ω (-5).7 Determine the numerical value of the current I LD that would be delivered to the load impedance Z LD defined by equation (-5), when, connected to terminals a and b in the circuit of Fig.(a); Show your calculation in the space designated for equation (-6). Using the Thevenin's equivalent circuit of Figure.(b), the current through impedance Z L when connected to terminals a and b can be expressed as V I LD T ZT +Z LD V T V T R T +jx T +R T -jx o T R o A T 3.88 (-6).8 Determine the numerical value of the maximum real power that a load could draw when connected between terminals a and b of the circuit in Figure.(a). Show your calculation in the space designated for equation (-7). Using the derived current through Z L, the real power that could be delivered to Z L can be expressed as, I LD R T 3. maxp LD (-7) 0W 5//6

9 EECS:300, Electric Circuits I s6fs_elci7.fm - 9 Problem 3 0 points Figure 3. shows the electrical model of a single capacitor RC-circuit. R I R II t0 - t0 + S i R R R N0 Figure 3. Electrical mode of a single capacitor RC-circuit. i C N C v C v R3 i R3 R 3 C 0mF V C (0- ) 0V I I 0A R I 50Ω R 50Ω R 50Ω R 3 50Ω Problem Statement Considering that switch S (which models a transistor) is closed at time t 0s, demonstrate an ability to: - determine the total quantity of the charge transfer to the capacitor, Q C, that will take place between times t0 and t, - determine voltage across resistor R 3, V R3 ( ), at the time when transient currents completely stop flowing in the circuit. 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 calculated numerical results. Explicit demonstration of understanding the following solution steps is expected. 3. Given the indicated positive reference directions for the voltages v C and v R3, show in the circuit model of Figure 3. the corresponding passive-convention positive-reference directions for currents through resistor R 3 and capacitor C. 3. To establish a base for clear thinking, write in the space designated for equation (3-) two KCL equations for the node N in Figure 3.: (a) one equation valid for the moment t0 -, and (b) the other one equation valid for the time t. Notice which of the currents values is equal to zero in both equations, and how that influences the strategy for solving the problem. Show your work in the space designated for equations (3-). KCL(0 - ): i R (0 - ) - i C (0 - ) - i R3 (0 - ) 0, where i R (0 - ) i C (0 - ) i R3 (0 - ) 0 (3-) KCL( ): i R ( ) - i C ( ) - i R3 ( ) 0, where i C ( ) 0 and i R ( ) i R3 ( ) 0 5//6

10 EECS:300, Electric Circuits I s6fs_elci7.fm Determine the common value of the currents i R ( ) and i R3 ( ) at the time when transient currents completely stops flowing in the circuit. Show your work in the space reserved for equations (3-). Since by definition, at t i C ( ) 0, the current i R3 ( ) is determined by the DC current divider formula, (3-) R I 50 i R3 ( ) I I 0 4A R I + R + R + R Determine the voltage v C ( ) across the capacitor C at the time when transient currents completely stop flowing in the circuit. Show your work in the space reserved for equations (3-3). v C ( ) i R3 ( )(R + R 3 ) 4(50+50) 400V (3-3) 3.5 Determine the charge Q C ( ) stored in the capacitor C at the time t. Show your work in the space reserved for equations (3-4). Q C ( ) C. v C ( ) C (3-4) 3.6 Determine the determine the total quantity of the charge transfer to the capacitor, Q C, between the times t0 -, and t. Show your work in the space reserved for equations (3-5). Q C Q C ( ) - Q C (0 - ) 4-0 4C (3-5) 3.7 Determine the energy W C stored in the capacitor C at the time when transient currents completely stopped flowing in the circuit. Show your work in the space reserved for equations (3-4). W C C. v C ( ) J (3-4) 3.8 Bonus Question: How much electrical energy has been converted to heat in the circuit of Figure 3. during the process which charged the capacitor C to the voltages v C ( ) and v C ( ) respectively. For full credit, mark your answers yes, no, or not applicable in all the given choices! yes no not applicable x W C x W C W C x. 5//6

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