chap3 - 80 Simple Resistive Circuits Current division is a...

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Unformatted text preview: 80 Simple Resistive Circuits Current division is a circuit analysis tool that is used to find the current through a single resistance from a col- lection of parallel-connected resistances when the cur- rent into the collection is known: R 1.: sq —i I R} ’ where if is the current through the resistance R]- and i is the current into the parallel-connected resistances whose equivalent resistance is Req- (See page 66.) A voltmeter measures voltage and must be placed in par- allel with the voltage being measured.An ideal voltmeter has infinite internal resistance and thus does not alter the voltage being measured. (See page 68.) An ammeter measures current and must be placed in series with the current being measured. An ideal amme- ter has zero internal resistance and thus does not alter the current being measured. (See page 68.) Problems Se Fig ZonlflkQ - Digital meters and analog meters have internal resist- ance, which influences the value of the circuit variable being measured. Meters based on the d’Arscnval meter movement deliberately include internal resistance as a way to limit the current in the movement’s coil. (See page 68.) - The Wheatstone bridge circuit is used to make precise measurements of a resistor’s value using four resistors, a dc voltage source, and a galvanometer. A Wheatstone bridge is balanced when the resistors obey Eq. 3.33, resulting in a galvanOmeter reading of 0 A. (See page 71.) - A circuit with three resistors connected in a A configu- ration (or a 11' configuration) can be transformed into at equivalent circuit in which the three resistors are Y con nected (or T connected). The A-to-Y transformation i: given by Eqs 3.44—3.46; the Y—to-A transformation i given by Eqs. 3.47—3.49. (See page 74.) cfions 3.14.2 3.1 For each of the circuits shown, a) identify the resistors connected in series. ure P3.1 3k0 8k!) 5k!) 61:0 (80 40.0 509 SOVQ 450 30.0 60.0 b) simplify the circuit by replacing the serie: connected resistors with equivalent resistors. (b) 3.2 For each of the circuits shown in Fig. P32, ' a) identify the resistors connected in parallel, b) simplify the circuit by replacing the parallel- connected resistors with equivalent resist0rs. 3.3 3) Find the power dissipated in each resistor in the _ mt! circuit shown in Fig. 3.9. b) Find the power delivered by the 120 V source. c) Show that the power delivered equals the power j dissipated. 1.3.4 a) Show that the solution of the circuit in Fig. 3.9 WK! (see Example 3.1) satisfies Kirchhoff’s current law at junctions x and y. 189 3.5 3.6 .II ." Problems 81 b) Show that the solution of the circuit in Fig. 3.9 satisfies Kirchhoff’s voltage law around every closed loop. Find the equivalent resistance seen by the source in each of the circuits of Problem 3.1. Find the equivalent resistance seen by the source in each of the circuits of Problem 3.2. 3.7 Find the equivalent resistance Rab for each of the PM! circuits in Fig. P33. 3.8 Find the equivalent resistance Rab for each of the 95"“ circuits in Fig. P3.8. “K 1:».- 20 k0 14o son so a mo no as b 20 129 82 3.9 PSPIEE Simple Resistive Circuits a) In the circuits in Fig. P3.9(a)—(c), find the equiv- alent resistance Rah. b) For each circuit find the power delivered by the source. Sections 3.3—3.4 3.10 Find the power dissipated in the 30 Q resistor in the PSFICE circuit in Fig. P3.10. Figure P3.10 30 A 0l.- 20 n 3.13 M516“ new PSPIEE c) Using the results of (b), design a resistive 11 work with an equivalent resistance of usi 1 k0 resistors. d) Using the results of (b), design a resistive 11 work with an equivalent resistance of 5.51 using 2 kfl resistors. a) Calculate the no-load voltage so for the volta; divider circuit shown in Fig. P313. b) Calculate the power dissipated in R1 and R2. c) Assume that only 0.5 W resistors are availat The no-load voltage is to be the same as in { Specify the smallest ohmic values of R1 and I Figure P3.13 3.11 For the circuit in Fig. P311 calculate m” a) no and to. b) the power dissipated in the 12 Q resistor. c) the power developed by the current source. Figure P3.11 100 12 I} 3.14 In the voltage-divider circuit shown in Fig. P3 "SPICE the no-load value of no is 4 V. When the load res ance R L is attached across the terminals a and b 40 o o 20 c 180 c drops to 3 v. Find RL. Figure 93.14 3.12 a) Find an expression for the equivalent resistance of two resistors of value R in parallel. RL b) Find an expression for the equivalent resistance of n resistors of value R in parallel. Figure P3.9 a 4 Q 14 Q 12 0 O 144 V 16 .0 10 Q 18 (I a 18 [l b 12 .0 b 20 V 15 n 48 Q 10 o ( ) 14 .0 2.5 (1 4 .0 o w» b 6 Q (a) 101} 5.6 o (C) 1211 The no-load voltage in the voltage»divider circuit shown in Fig. P3.15 is 20 V. The smallest load resistor that is ever connected to the divider is 48 k0. When the divider is loaded, 1),, is not to drop below 16 V. a) Design the divider circuit to meet the specifica- tions just mentioned. Specify the numerical value of R1 and R2. b) Assume the power ratings of commercially available resistors are 1/16, 1/8, 1/4, 1, and 2 W. What power rating would you specify? Figure P3.15 Assume the voltage divider in Fig. P315 has been constructed from 0.15 W resistors. How small can RL be before one of the resistors in the divider is operating at its dissipation limit? a) The voltage divider in Fig. P3.1?(a) is loaded with the voltage divider shown in Fig. P3.17(b); that is, a is connected to a’, and b is connected to b“. Find on. b) Now assume the voltage divider in Fig. P3.17(b) is connected to the voltage divider in Fig. P3.17(a) by means of a current-controlled voltage source as shown in Fig. P3.17(c). Find v0. c) What effect does adding the dependent-voltage source have on the operation of the voltage divider that is connected to the 480 V source? 93.11 60 k0 SUkQ s=_. \ Problems 83 3.18 There is often a need to produce more than one 9335515?" voltage usrng a voltage divider. For example, the memory components of many personal computers require voltages of —12 V, 5 V, and +12 V, all with respect to a common reference terminal. Select the values of R1, R2, and R3 in the circuit in Fig. P118 to meet the following design requirements: a) The total power supplied to the divider circuit by the 24 V source is 80 W when the divider is unloaded. b) The three voltages, all measured with respect to the common reference terminal, are 1.11 = 12 V, v2 = 5V,and v3 = —12 V. Figure P3.18 24V 3.19 A voltage divider like that in Fig. 3.13 is to be P3553153" designed so that '00 a kt), at no load (R L = 00) and v0 = as, at full load (RL = R0). Note that by defi- nitiona < k < 1. a) Show that k — or R1 _ ak R0 and k — a 32 ' a(1 — k)R”‘ b) Specify the numerical values of R1 and R2 if k = (185,0: = 0.80, and R0 = 34 [(0. c) If v, = 60 V, specify the maximum power that will be dissipated in R1 and R2. d) Assume the load resistor is accidentally short circuited. How much power is dissipated in R1 and R2? 84 Simple Resistive Circuits 3.20 a) Show that the current in the kth branch of the P571“ circuit in Fig. P3.20(a) is equal to the source current i3 times the conductance of the kth branch divided by the sum of the conductances, that is, . EsGk 1" G1+Gg+Ga+-~+Gk+~-+GN' b) Use the result derived in (a) to calculate the cur- rent in the 6.25 {1 resistor in the circuit in Fig. P3.20(b). Figure P3.20 1142mA 0.25.0 2.5.0 10‘I 6.25.0 100 200 (b) 3.21 Specify the resistors in the circuit in Fig. P321 to DESIGN meet the following design criteria: PROBLEM ig 2 SInAwg = 1V;i] = 4&2; i2 = 83:3; and £3 = 5.14. Figure P3.21 + '8 ER Iyl‘. R1 {2‘ R2 33; R3 1:4' R4 3.22 Look at the circuit in Fig. P3.1(a). a) Use current division to find the current flowing from t0p to bottom in the 10 k9. resistor. b) Using your result from (a), find the voltage drop across the 10 k0 resistor, positive at the top. c) Using your result from (b), use voltage division to find the voltage drop across the 6 k0 resistor, positive at the top. d) Using your result from (c), use voltage division to find the voltage drop across the 5 k!) resistor, positive at the left. 3.23 Look at the circuit in Fig. P3.1(b). a) Use voltage division to find the voltage drop across the 240 Q resistor, positive at the left. b) Using your result from (a), find the current flow- ing in the 240 Q resistor from left to right. c) Using your result form (b), use current division to find the current in the 140 Q. resistor. 3.24 a) Find the voltage 1.1; in the circuit in Fig. P324. PSPICE b) Replace the 30 V source with a general voltage source equal to V}. Assume V; is positive at the upper terminal. Find 1),, as a function of V}. Figure P334 60 kfl 15 kfl 3.25 Find '01 and 1); in the circuit in Fig. P325. “’1“ Figure 8.25 12 Q 50 0 25A 25.!) 3.26 Find ’00 in the circuit in Fig. P326. “5"” Figure P3.26 3k!) 12 k0 3.27 Find i0 and is in the circuit in Fig. P327. ”m Figure P12? J.— 100 15.0 35.0 3.28 For the circuit in Fig. P328, calculate (a) 1'0 and .- mics (b) the power dissipated in the 15 0 resistor. Figure P3.28 4 fl 2 Q 120V0100 I 3.29 The current in the 12 Q resistor in the circuit in 1’5"“ Fig. P329 is 1 A, as shown. a) Find ”3- b) Find the power dissipated in the 20 0 resistor. Figure P129 20.0 ’W’» NW 39 6Q action 3.5 -"3.30 a) Show for the ammeter circuit in Fig. P330 that the current in the d’Arsonval movement is always 1 / 25th of the current being measured. b) What would the fraction be if the 100 mV, 2 mA movement were used in a 5 A ammeter? c) Would you expect a uniform scale on a dc d’Arsonval ammeter? Figure P330 100 mV, 2 mA (25/12) (1 ”-3.31 The ammeter in the circuit in Fig. P331 has a resist- ' ance of 0.5 It. What is the percentage of error in the reading of this ammeter if measured value % error = true value —1) X 100? Problems 85 Figure P3.3 1 3.32 The ammeter described in Problem 3.31 is used to measure the current to in the circuit in Fig P332. What is the percentage of error in the measured value? Figure P332 3.33 A d’Arsonva] voltmeter is shown in Fig. P333. Find the value of Rt, for each of the following full-scale readings: (a) 100 V, (b) 5 V, and (c) 100 mV. Figure P3.33 Voltmeter 3.34 Suppose the d’Arsonval voltmeter described in Problem 3.33 is used to measure the voltage across the 24 Q resistor in Fig. P332. 3) What will the voltmeter read? b) Using the definition of the percentage of error in a meter reading found in Problem 3.31, what is the percentage of error in the voltmeter reading? 3.35 A shunt resistor and a 50 mV, 1 mA d’Arsonva] movement are used to build a 10 A ammeter. A resistance of 0.015 I) is placed across the terminals of the ammeter. What is the new full-scale range of the annneter? 86 3.36 DESIGN PROBLEM 3.37 DESIGN PROBLEM 3.38 PSPIEE 3.39 Simple Resistive Circuits A d’Arsonval movement is rated at 1 mA and 50 mV. Assume 0.5 W precision resistors are avail- able to use as shunts. What is the largest full-scale- reading ammeter that can be designed? Explain. A d’Arsonval arnmeter is shown in Fig. P337. Design a set of d’Arsonval ammeters to read the fol- lowing full-scale current readings: (a) 5 A, (b) 2 A, (c) 1 A, and (d) 50 mA. Specify the shunt resistor R A for each ammeter. Figure P137 The elements in the circuit in Fig. 2.24 have the follow- ing values: R1 = 20 k0, R2 = 80 k0, RC = 0.82 k0, RE 2 0.2 k”, VCC = 7.5 V, VG = 0.6 V, and B = 39. 3) Calculate the value of :53 in microamperes. b) Assume that a digital multimeter, when used as a dc ammeter, has a resistance of 1 k0. If the meter is inserted between terminals b and 2 to measure the current :53, what will the meter read? c) Using the calculated value of is in (a) as the cor- rect value, what is the percentage of error in the measurement? The voltage-divider circuit shown in Fig. P339 is designed so that the no-load output voltage is 7/9ths of the input voltage. A d’Arsonval voltmeter having a sensitivity of 100 Q / V and a full-scale rat- ing of 200 V is used to check the operation of the circuit. a) What will the voltmeter read if it is placed across the 180 V source? b) What will the voltmeter read if it is placed across the 70 k0. resistor? c) What will the voltmeter read if it is placed across the 20 k0 resistor? d) Will the voltmeter readings obtained in parts (b) and (c) add to the reading recorded in part (a)? Explain why or why not. Figure P339 3.40 You have been told that the dc voltage of a power supply is about 500V. When you go to the instrumen' room to get a dc voltmeter to measure the power supply voltage, you find that there are only two dc voltmeters available. The voltmeters are rated 400 V full scale and have a sensitivity of 1000 Q/V. a) How can you use the two voltmeters to checl the power supply voltage? b) What is the maximum voltage that can be measured? c) If the power supply voltage is 504 V, what wil each voltmeter read? 3.41 Assume that in addition to the two voltmeter described in Problem 3.40, a 50 k0 precision resis tor is also available. The 50 k0 resistor is connecter in series with the series-connected voltmeters. Thi circuit is then connected across the terminals of thl power supply. The reading on the voltmeters i 328 V. What is the voltage of the power supply? 3.42 The voltmeter shown in Fig. P3.42(a) has a full scale reading of 800 V. The meter movement i rated 100 mV and 1 mA. What is the percentage 0 error in the meter reading if it is used to measur the voltage 1) in the circuit of Fig. P3.42(b)? ure P3442 — — — — w 800 V '. I -' Rm I + ' l 3.5 mA 0 600 kn o I mV : ' 1 mA lCommon _ (a) (b) 6.43 A 600 k!) resistor is connected from the 200 V ter- minal to the common terminal of a dual-scale volt- meter, as shown in Fig. P3.43(a). This modified voltmeter is then used to measure the voltage across the 360 k0 resistor in the circuit in Fig. P3.43(b). a) What is the reading on the 500 V scale of the meter? b) What is the percentage of error in the measured voltage? Figure P3.43 600 k!)- l 40 k0 I | I Modified | : voltmeter I em 0 i 1 ' | lCommon ll Problems 87 3.44 Assume in designing the multirange voltmeter P2353" shown in Fig. P3.44 that you ignore the resistance of the meter movement. a) Specify the values of R1, R2, and R3. b) For each of the three ranges, calculate the percent» age of error that this design strategy produces. Figure P3.“ R1 50 V R2 20 v R3 50 mV 2 V 1 mA Common 3.45 Design a d’Arsonval voltmeter that will have the DESIGN three voltage ranges shown in Fig. P3.45. PROBLEM a) Specify the values of R1, R2, and R3. b) Assume that a 500 k0 resistor is connected between the 100 V terminal and the common terminal. The voltmeter is then connected to an unknown voltage using the common terminal and the 200 V terminal. The voltmeter reads 188 V. What is the unknown voltage? c) What is the maximum voltage the voltmeter in (b) can measure? Figure P3.45 200 V R3 100 V R2 50 V R1 10 mV 2 mA Common 88 3.46 Simple Resistive Circuits The circuit model of a dc voltage source is shown in Fig. P3.46. The following voltage measurements are made at the terminals of the source: (1) With the terminals of the source open, the voltage is meas- ured at 80 mV, and (2) with a 10 M0 resistor con- nected to the terminals the voltage is measured at 72 mV.A]1 measurements are made with a digital voltmeter that has a meter resistance of 10 M0. a) What is the internal voltage of the source (’03) in millivolts? b) What is the internal resistance of the source (R5) in kilo-ohms? Figure P3 .45 I Rx I I | I I v I Terminals of I 5 l the source I | I | _________ I Sections 3.6—3.7 3.4.? 3.48 PSPICE 3.49 PSPICE Assume the ideal voltage source in Fig. 3.26 is replaced by an ideal current source. Show that Eq. 3.33 is still valid. The bridge circuit shown in Fig. 3.26 is energized from a 21 V dc source. The bridge is balanced when R, = 800 (LR? = 1200 (Land R3 = 600 .Q. a) What is the value of RI? b) How much current (in milliamperes) does the dc source supply? c) Which resistor in the circuit absorbs the most power? How much power does it absorb? (1) Which resistor absorbs the least power? How much power does it absorb? Find the detector current id in the unbalanced bridge in Fig. P349 if the voltage drop across the detector is negligible. Figure P3439 A“ ‘9' 45 k!) 3.50 PSPICE 3.51 PSPILE 3.52 PSPIEE 3.53 PSPICE Find the power dissipated in the 13 Q resistor in the circuit in Fig. P350. Figure P150 3 I} [email protected] 300 'l 18!! In the Wheatstone bridge circuit shown in Fig. 3.26 the ratio RZ/Rl can be set to the following values 0.001, 0.01, 0.1, 1, 10, 100, and 1000. The resistor R; can be varied from 1 to 11,110 [2, in increments o: 1 0. An unknown resistor is known to lie betweer 4 and 5 Q. What should be the setting of the Rz/R- ratio so that the unknown resistor can be measuret to four significant figures? Use a A-to-Y transformation to find the voltages e and a: in the circuit in Fig. P352. Figure P3.52 280 100 3) Find the equivalent resistance Rab in the circui in Fig. P353 by using a A-to-Y transformatio involving the resistors R2, R3, and R4. b) Repeat (a) using a Y—to-A transformatio: involving resistors R2, R4, and R5. c) Give two additional A-to-Y or Y-to-A transfor mations that could be used to find Rab. Figure P153 20 fl Problems 89 3.54 Find the equivalent resistance Rab in the circuit in 3.58 a) Find the resistance seen by the ideal voltage 55m Fig. P3.54. PSPICE source in the circuit in Fig. P358. b) If vat, equals 600 V, how much power is dissi- F'i P354 . . gure pated 111 the 15 0. resrstor? 15 Q 10 Q a Figure P353 3 20 600 17 Q l. 30 o as O 38 11 “ 15 o 9 14 fl .. h ‘ 60 0 85 Q :3 5 In the circuit in Fig P3. 55(a) the device labeled D '=' " t! represents a component that has the equivalent cir- b cuit shown' 111 Fig. P3 55(b) The labels on the termi- nals of D show how the device is connected to the circuit.Find ex and the power absorbed by the device. 3.59 Use a Y—to-A transformation to find (a) to; (b) i1; PSPICE (c) 1'2; and (d) the power delivered by the ideal cur- "Qm P355 rent source in the circuit in Fig. P359. b Figure P359 320 I) 6.25 I} a 15 0 C (b) Fin? I'o andfihépf’wef diSSiPa‘ed in the 14“ 9 reSiS' 3.60 For the circuit shown in Fig. P360, find (a) :1, (b) v, tor 1n the c1rcu1tm F1g.P3.56. PSPICE (c) i2, and (d) the power supplied by the voltage Figure P3.56 source- 22 Q 20 0 Figure P3.60 240 V _ 8 Q 10 o 12 n 27 .0 Find Rab in the circuit in Fig. P357. “9“" ”3°57 3.61 Derive Eqs. 3.4mm from Eqs. 3.41—3.43. The fol- 1.8 kn 13 k0 lowing two hints should help you get started in the right direction: 1.8 kfl 1.8 k!) a) To find R1 as a function of Ra, Rb, and Re, first . subtract Eq. 3.42 from Eq. 3.43 and then add th1s h result to Eq. 3.39. Use similar manipulations to 1.8 k0 1.8 k0 find R2 and R3 as functions of Ra, Rb, and RC. 90 Simple Resistive Circuits b) To find Rb as a function of R1, R2, and R3, take advantage of the derivations obtained by hint (1), namely, Eqs. 3.44—3.46. Note that these equa- tions can be divided to obtain R2 Re R2 #— = — R = —R and a1 Rb R2 — = —— R = —R . R2 R; or " R1 5 Now use these ratios in Eq. 3.43 to eliminate RH and RC. Use similar manipulations to find RE and RC as functions of R1, R2, and R3. 3.62 Show that the expressions for A conductances as functions of the three Y conductances are G _ 0263 “ 61+ 62+ Gg’ G103 Gs = —~, G1 + 62 + 63 G. = —G‘GZ , GI + 62 + 63 where 1 1 Ga = F“, G1: 1:, etc. Sections 3.1-3.7 3.63 Resistor networks are sometimes used as volume- 92505359» control circuits. In this application, they are referred to as resistance attenuators or pads. A typi- cal fixed-attenuator pad is shown in Fig. P363. In designing an attenuation pad, the circuit designer will select the values of R1 and R2 so that the ratio of ’00 /vi and the resistance seen by the input voltage source Rab both have a specified value. a) Show that ifRab = RL, then Ri = 4am}:1 + R2), 1’ R2 E 2R1+R2+RL' 13) Select the ...
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