Exercise 3 (Resistive Network Analysis)

Transcription

1 Circuit Analysis Exercise 0/0/08 Problem. (Hambley.49) Exercise (Resistive Network Analysis) Problem. (Hambley.5)

2 Circuit Analysis Exercise 0/0/08 Problem. (Hambley.59) Problem 4. (Hambley.68)

3 Circuit Analysis Exercise 0/0/08 Problem 5. (Hambley.75)

4 Circuit Analysis Exercise 0/0/08 Problem 6. (Hambley.8) 4

5 Circuit Analysis Exercise 0/0/08 Problem 7. (Hambley.84) Problem 8. (Hambley.88) 5

6 Circuit Analysis Exercise 0/0/08 Problem 9. (Hambley.94) 6

7 Circuit Analysis Exercise 0/0/08 Problem 0. (Rizzoni.0) Using node voltage analysis in the circuit of Figure, find the three indicated node voltages. Let I 0.A; R 00 ; R 75 ; R 5 ; R 50 ; R 00 ; Known quantities: Figure The current source value, the voltage source value and the resistance values for the circuit shown in Figure P.0. The three node voltages indicated in Figure P.0 using node voltage analysis. At node : v 00 v v A At node : v v 75 v 5 v v 50 At node : v v v i i 0 For the voltage source we have: v 0 v olving the system, we obtain: v 4.4, v 4.58, v 5.4 and, finally, i 54 ma. Problem. (Rizzoni.0) For the circuit of Figure P.0, use mesh current analysis to find the matrices required to solve the circuit, and solve for the unknown currents. Hint:You may find source transformations useful. 7

8 Circuit Analysis Exercise 0/0/08 Circuit shown in Figure P.0. Mesh equation in matrix form and solve for currents. after source transformation, we can have the equivalent circuit shown in the right hand side. We can write down the following matrix I, 0 I 4 I4 5 olve the equation, we can have I, A.66 I I A I A I A Problem. (Rizzoni.0) Using mesh current analysis, find the current i in the circuit of FigureP.0. 8

9 Circuit Analysis Exercise 0/0/08 Known quantities: The values of the resistors in the circuit of Figure P.0. The current in the circuit of Figure P.0 using mesh current analysis. ince I is unknown, the problem will be solved in terms of this current. For mesh #, it is obvious that: i I For mesh #: For mesh #: olving, i 0.645I i 0.48I i i 5 i 5 0 i 4 i 5 i Then, i i i and i0.48i0.645i0.6i Problem. (Rizzoni.56) Find the Thevenin equivalet resistance seen by the load resistor R L in the circuit of Figure P.56. Problem.56 Known quantities: Circuit shown in Figure P.56. 9

10 Circuit Analysis Exercise 0/0/08 Thevenin equivalent circuit To find R T, we need to make the current source an open circuit and voltage sources short circuits, as follows: Note that this circuit has only three nodes. Thus, we can re-draw circuit as shown: and combine the two parallel resistors to obtain: the the Thus, R T 50 (50.) 00.8 Problem 4. (Rizzoni.58) Find the Thevenin equivalent of the circuit connected to R L in the Figure, where R 0 ; R 0 ; R 0., and R ; g p Figure Known quantities: Circuit shown in Figure P.58. Thevenin equivalent circuit To find RT, we short circuit the source tarting from the left side, ( 0.) 00.99, ( ) 0.89 Therefore, we have R T Ω.. 0

11 Circuit Analysis Exercise 0/0/08 To find, we apply mesh analysis: v oc Two resistors are omitted because no current flows through them and they, therefore, do not affect voc i -0i i -0i 0 = + olving for i, i 06. A we obtain, vt voc 0i 4. Problem 5. (Rizzoni.59) The Wheatstone bridge circuit shown in Figure is used in a number of practical applications. One traditional use is in determining the value of an unknown resistor R x. Find the value of the voltage ab = a - b in terms of R, R x, and s. If R k, s and ab =m, what is the value of R x? Figure Known quantities: Circuit shown in Figure P.59. alue of resistance R x R R+R R R+Rx a) We have x ab a-b - ab = R - x R + R x b) For kw R, s, ab m, R x R x 996 Ω 000 +R x

12 Circuit Analysis Exercise 0/0/08 Problem 6. (Rizzoni.4) With reference to Figure 4, using superposition, determine the component of the current through R that is due to s. 450 ; R 7; R 5; R 0 R 4 R 5 Figure 4 Known quantities: The values of the voltage sources and of the resistors in the circuit of Figure P.4: 450 R 7 R 5 R 0 R 4 R 5 The component of the current through R that is due to, using superposition. uppress by replacing it with a short circuit. Redraw the circuit. A solution using equivalent resistances looks reasonable. R and R 4 are in parallel: R 4 R R 4 7 R R R4is in series with R : R 4 R 4 R R eq R 5 R R 4 R 5 R R 4 5 R R OL: CD: I R eq A 5 I R I R R R A

13 Circuit Analysis Exercise 0/0/08 Problem 7. (Rizzoni.) In the circuit the Figure 5, assume the source voltage and source current and all resistances are known. a. Write the node equations required to determine the node voltages. b. Write the matrix solution for each node voltage in terms of the known parameters. Known quantities: Circuit of Figure P. with voltage source, Figure 5, current source, I, and all resistances. a. The node equations required to determine the node voltages. b. The matrix solution for each node voltage in terms of the known parameters. a) pecify the nodes (e.g., A on the upper left corner of the circuit in Figure P.0, and B on the right corner). Choose one node as the reference or ground node. If possible, ground one of the sources in the circuit. Note that this is possible here. When using KCL, assume all unknown current flow out of the node. The direction of the current supplied by the current source is specified and must flow into node A. KCL: I a R a R R a b R 0 b I R R KCL: b a R a R b R b 0 R 4 0 b R R R 4 R b) Matrix solution:

14 Circuit Analysis Exercise 0/0/08 a I R R R R R R 4 R R R R R R R 4 I R R R R 4 R R R R R R R 4 R R b R R I R R R R R R R R R R 4 R R R R I R R R R R R 4 R R Notes:. The denominators are the same for both solutions.. The main diagonal of a matrix is the one that goes to the right and down.. The denominator matrix is the "conductance" matrix and has certain properties: a) The elements on the main diagonal [i(row) = j(column)] include all the conductance connected to node i=j. b) The off-diagonal elements are all negative. c) The off-diagonal elements are all symmetric, i.e., the i j-th element = j i-th element. This is true only because there are no controlled (dependent) sources in this circuit. d) The off-diagonal elements include all the conductance connected between node i [row] and node j [column]. Problem 8. (Rizzoni.4) Using KCL, perform node analysis on the circuit shown in Figure 6, and determine the voltage across R 4. Note that one source is a controlled voltage source! Let 5 ; A 70; R.k; R.8k; R 6.8k; 0 R4 Figure 6 4

15 Circuit Analysis Exercise 0/0/08 Known quantities: Circuit shown in Figure P.4 5 A 70 R. k R.8 k R 6.8 k R 4 0 The voltage across R 4 using KCL and node voltage analysis. Node analysis is not a method of choice because the dependent source is [] a voltage source and [] a floating source. Both factors cause difficulties in a node analysis. A ground is specified. There are three unknown node voltages, one of which is the voltage across R 4. The dependent source will introduce two additional unknowns, the current through the source and the controlling voltage (across R ) that is not a node voltage. Therefore 5 equations are required: KCL 0 KCL IC 0 R R R R KCL I C 04KL R 0R R R4 5KL A 0 A A R R ubstitute using Equation [5] into Equations [], [] and [] and eliminate (because it only appears twice in these equations). Collect terms: A R R R R R R I C 0 A R R A I C A R R R R I C 0 R R R 4 R R R R R R A 70 R R A R R R R A (5)70 R mA A 5 R.0 (5) mA olving, we have: Notes: R R4 5. m. This solution was not difficult in terms of theory, but was terribly long and arithmetically cumbersome. This was because the wrong method was used. There are only mesh currents in the circuit; the sources were voltage sources; therefore, a mesh analysis is the method of choice. 5

16 Circuit Analysis Exercise 0/0/08. In general, a node analysis will have fewer unknowns (because one node is the ground or reference node) and will, in such cases, be preferable. 6