a) Find an expression for the equivalent resistance of two

a) Show that the solution of the circuit in Fig. 3.9 (see Example 3.1) satisfies Kirchhoffs current law at junctions x and y. b) Show that the solution of the circuit in Fig. 3.9 satisfies Kirchhoffs voltage law around every closed loop.

a) Show that the solution of the circuit in Fig. 3.9 (see Example 3.1) satisfies Kirchhoffs current law at junctions x and y. b) Show that the solution of the circuit in Fig. 3.9 satisfies Kirchhoffs voltage law around every closed loop.

For each of the circuits shown in Fig. P3.3, a) identify the resistors connected in series, b) simplify the circuit by replacing the seriesconnected resistors with equivalent resistor

For each of the circuits shown in Fig. P3.4, a) identify the resistors connected in parallel, b) simplify the circuit by replacing the parallelconnected resistors with equivalent resistors.

For each of the circuits shown in Fig. P3.3, a) find the equivalent resistance seen by the source, b) find the power developed by the source.

6 For each of the circuits shown in Fig. P3.4, a) find the equivalent resistance seen by the source, b) find the power developed by the source.

a) In the circuits in Fig. P3.7(a)(d), find the equivalent resistance seen by the source. b) For each circuit find the power delivered by the source.

Find the equivalent resistance Rb for each of the circuits in Fig. P3.8.

Find the equivalent resistance Rb for each of the circuits in Fig. P3.9.

a) Find an expression for the equivalent resistance of two resistors of value R in series. b) Find an expression for the equivalent resistance of n resistors of value R in series. c) Using the results of (a), design a resistive network with an equivalent resistance 3 k of using two resistors with the same value from Appendix H. d) Using the results of (b), design a resistive network with an equivalent resistance of 4 k using a minimum number of identical resistors from Appendix H.

a) Find an expression for the equivalent resistance of two resistors of value R in parallel. b) Find an expression for the equivalent resistance of n resistors of value R in parallel. c) Using the results of (a), design a resistive network with an equivalent resistance of 5 k using two resistors with the same value from Appendix H. d) Using the results of (b), design a resistive network with an equivalent resistance of 4 k using a minimum number of identical resistors from Appendix H.

a) Calculate the no-load voltage v for the voltagedivider circuit shown in Fig. P3.12. b) Calculate the power dissipated in R and R c) Assume that only 0.5 W resistors are available. The no-load voltage is to be the same as in (a). Specify the smallest ohmic values of R and R .

In the voltage-divider circuit shown in Fig. P3.13, the no-load value of v is 4 V. When the load resistance RL is attached across the terminals a and b, drops to 3 V. Find RL

The no-load voltage in the voltage-divider circuit shown in Fig. P3.14 is 8 V. The smallest load resistor that is ever connected to the divider is When the divider is loaded, v is not to drop below 7.5 V. a) Design the divider circuit to meet the specifications just mentioned. Specify the numerical values of R and R 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

Assume the voltage divider in Fig. P3.14 has been constructed from 1W resistors. What is the smallest resistor from Appendix H that can be used as RL before one of the resistors in the divider is operating at its dissipation limit?

Find the power dissipated in the 5 resistor in the current divider circuit in Fig. P3.16.

For the current divider circuit in Fig. P3.17 calculate a) i and v . b) the power dissipated in the 6 resistor. c) the power developed by the current source

Specify the resistors in the current divider circuit in Fig. P3.18 to meet the following design criteria:ig = 50 mA; vg = 25 V; i = 0.6i; i = 2i2; and i = 4i

There is often a need to produce more than one voltage using 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 R and R in the circuit in Fig. P3.19 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 , v = 12 V, v = 5 V and v = 12 V

The voltage divider in Fig. P3.20(a) is loaded with the voltage divider shown in Fig. P3.20(b); that is, a is connected to , and b is connected to Find v b) Now assume the voltage divider in Fig. P3.20(b) is connected to the voltage divider in Fig. P3.20(a) by means of a current-controlled voltage source as shown in Fig. P3.20(c). Find v c) What effect does adding the dependent-voltage source have on the operation of the voltage divider that is connected to the 380 V source?

A voltage divider like that in Fig. 3.13 is to be designed so that v = kv at no load (RL = ) and v = kvs at full load (RL=Rv ). Note that by definition a < k < l. Show that R1 = k - a ak Ro and b) Specify the numerical values of R and R if k = 0.85, a = 0.80 and R = 34 .c) If v = 60 V , specify the maximum power that will be dissipated in R and R d) Assume the load resistor is accidentally short circuited. How much power is dissipated in Rand R?

a) Show that the current in the kth branch of the circuit in Fig. P3.22(a) is equal to the source current times the conductance of the kth branch divided by the sum of the conductances, that is, b) Use the result derived in (a) to calculate the current in the 5 resistor in the circuit in Fig. P3.22(b).

Look at the circuit in Fig. P3.3(a). a) Use voltage division to find the voltage across the 6 k resistor, positive at the top. b) Use the result from part (a) and voltage division to find the voltage across the 5 k resistor, positive on the left

Look at the circuit in Fig. P3.3(d). a) Use current division to find the current in the 50 k resistor from left to right. b) Use the result from part (a) and current division to find the current in the 70 k resistor from top to bottom.

Look at the circuit in Fig. P3.7(a). a) Use voltage division to find the voltage drop across the 25 k resistor, positive at the left. b) Using your result from (a), find the current flowing in the 25 k resistor from left to right. c) Starting with your result from (b), use current division to find the current in the 50 k resistor from top to bottom. d) Using your result from part (c), find the voltage drop across the 60 k resistor, positive at the top. e) Starting with your result from (d), use voltage division to find the voltage drop across the resistor, positive at the top.

Attach a 450 mA current source between the terminals ab in Fig. P3.9(a), with the current arrow pointing up. a) Use current division to find the current in the 36 resistor from top to bottom. b) Use the result from part (a) to find the voltage across the 36 resistor, positive at the top. c) Use the result from part (b) and voltage division to find the voltage across the 18 resistor, positive at the top. d) Use the result from part (c) and voltage division to find the voltage across the 10 resistor, positive at the top.

Attach a 6 V voltage source between the terminals ab in Fig. P3.9(b), with the positive terminal at the top. a) Use voltage division to find the voltage across the 4 resistor, positive at the top. b) Use the result from part (a) to find the current in the 4 resistor from left to right. c) Use the result from part (b) and current division to find the current in the 16 resistor from left to right. d) Use the result from part (c) and current division to find the current in the 10 resistor from top to bottom. e) Use the result from part (d) to find the voltage across the 10 resistor, positive at the top. f) Use the result from part (e) and voltage division to find the voltage across the 18 resistor, positive at the top

Find the voltage in the circuit in Fig. P3.28 using voltage and/or current division. Replace the 18 V source with a general voltage source equal Vs to Assume VS is positive at the upper terminal.Find vx ab a function of Vs.

Find in the circuit v in Fig. P3.29 using voltage and/or current division.

Find v and v in the circuit in Fig. P3.30 using voltage and/or current division.

For the circuit in Fig. P3.31, find and then use current division to find i.

For the circuit in Fig. P3.32, calculate i and i using current division.

A dArsonval ammeter is shown in Fig. P3.33. a) Calculate the value of the shunt resistor, RA, to give a full-scale current reading of 5 A. b) How much resistance is added to a circuit when the 5 A ammeter in part (a) is inserted to measure current? c) Calculate the value of the shunt resistor, RA, to give a full-scale current reading of 100 mA. d) How much resistance is added to a circuit when the 100 mA ammeter in part (c) is inserted to measure current?

A shunt resistor and a 50 mV, 1 mA dArsonval movement are used to build a 5 A ammeter. A resistance of 20 mis placed across the terminals of the ammeter. What is the new full-scale range of the ammeter?

A dArsonval movement is rated at 2 mA and 200 mV. Assume 1 W precision resistors are available to use as shunts. What is the largest full-scalereading ammeter that can be designed using a single resistor? Explain.

a) Show for the ammeter circuit in Fig. P3.36 that the current in the dArsonval 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 dArsonval ammeter?

A dArsonval voltmeter is shown in Fig. P3.37. Find the value Rv of for each of the following full-scale readings: (a) 50 V, (b) 5 V, (c) 250 mV, and (d) 25 mV.

Suppose the dArsonval voltmeter described in Problem 3.37 is used to measure the voltage across the 45 resistor in Fig. P3.38. a) What will the voltmeter read? b) Find the percentage of error in the voltmeter reading if % error = measured valuetrue value -1 * 100.

The ammeter in the circuit in Fig. P3.39 has a resistance of 01.1 Using the definition of the percentage error in a meter reading found in Problem 3.38, what is the percentage of error in the reading of this ammeter?

The ammeter described in Problem 3.39 is used to measure the current 1 in the circuit in Fig. P3.38.What is the percentage of error in the measured value?

The elements in the circuit in Fig.2.24 have the following values: R 20 k, R = 80 k, Rc = 0.82 k, RE = 0.2 K Vcc = 7.5 v, V = 0.6v and = 39 a) Calculate the value of iB in microamperes. b) Assume that a digital multimeter, when used as a dc ammeter, has a resistance 1 kof If the meter is inserted between terminals b and 2 to measure the current iB, what will the meter read? c) Using the calculated value of iB in (a) as the correct value, what is the percentage of error in the measurement?

You have been told that the dc voltage of a power supply is about 350 V.When you go to the instrument room to get a dc voltmeter to measure the power supply voltage, you find that there are only two dc voltmeters available. One voltmeter is rated 300 V full scale and has a sensitivity of 900 /V The other voltmeter is rated 150 V full scale and has a sensitivity of 1200 /V (Hint: you can find the effective resistance of a voltmeter by multiplying its rated full-scale voltage and its sensitivity.) a) How can you use the two voltmeters to check the power supply voltage? b) What is the maximum voltage that can be measured? c) If the power supply voltage is 320 V, what will each voltmeter read

Assume that in addition to the two voltmeters described in Problem 3.42, a 50 k precision resistor is also available.The 50 k resistor is connected in series with the series-connected voltmeters. This circuit is then connected across the terminals of the power supply. The reading on the 300 V meter is 205.2 V and the reading on the 150 V meter is 136.8 V. What is the voltage of the power supply?

The voltmeter shown in Fig. P3.44(a) has a fullscale reading of 500 V. The meter movement is rated 100 mV and 0.5 mA. What is the percentage of error in the meter reading if it is used to measure the voltage in the circuit of Fig. P3.44(b)? Figure P3.44 100 mV 0.5 mA Common

The voltage-divider circuit shown in Fig. P3.45 is designed so that the no-load output voltage is 7/9ths of the input voltage. A dArsonval voltmeter having a sensitivity of 100 /V and a full-scale rating 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 k resistor? c) What will the voltmeter read if it is placed across the 20 k 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 P3.4

Assume in designing the multirange voltmeter shown in Fig. P3.46 that you ignore the resistance of the meter movement. a) Specify the values of R, R and R b) For each of the three ranges, calculate the percentage of error that this design strategy produces. Figure P3.46

The circuit model of a dc voltage source is shown in Fig. P3.47. The following voltage measurements are made at the terminals of the source: (1) With the terminals of the source open, the voltage is measured at 50 mV, and (2) with a 15 M resistor connected to the terminals, the voltage is measured at 48.75 mV. All measurements are made with a digital voltmeter that has a meter resistance of 10 M. a) What is the internal voltage of the source ( vs) in millivolts? b) What is the internal resistance of the source (Rs )in kilo-ohms?

Design a dArsonval voltmeter that will have the three voltage ranges shown in Fig. P3.48. a) Specify the values of R, R and R b) Assume that a 750 k resistor is connected between the 150 V terminal and the common terminal. The voltmeter is then connected to an unknown voltage using the common terminal and the 300 V terminal. The voltmeter reads 288 V. What is the unknown voltage? c) What is the maximum voltage the voltmeter in (b) can measure? Figure P3.48

A 300 k resistor is connected from the 200 V terminal to the common terminal of a dual-scale voltmeter, as shown in Fig. P3.49(a). This modified voltmeter is then used to measure the voltage across the 360 k resistor in the circuit in Fig. P3.49(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?

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 24 V dc source. The bridge is balanced when R = 500 , R = 1000 and R = 750 a) What is the value Rx of 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? d) Which resistor absorbs the least power? How much power does it absorb?

Find the power dissipated in the resistor in the circuit in Fig. P3.52. Figure P3.52 750  5 k 15 k 25 k 3 k 5 k

Find the detector current id in the unbalanced bridge in Fig. P3.53 if the voltage drop across the detector is negligible. Figure P3.53

In the Wheatstone bridge circuit shown in Fig. 3.26, the ratio R/R 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 , in increments of 1 . An unknown resistor is known to lie between 4 AND 5 . What should be the setting of the R/R ratio so that the unknown resistor can be measured to four significant figures?

Find the current and power supplied by the 40 V source in the circuit for Example 3.7 (Fig. 3.32) by replacing the lower (25, 37.5, and 40 ) with its equivalent Y.

Find the current and power supplied by the 40 V source in the circuit for Example 3.7 (Fig. 3.32) by replacing the Y on the left (25, 40, and 100 ) with its equivalent .

Find the current and power supplied by the 40 V source in the circuit for Example 3.7 (Fig. 3.32) by replacing the Y on the right (25, 37.5, and 125 ) with its equivalent .

a) Find the equivalent resistance Rab in the circuit in Fig. P3.58 by using a Y-to- transformation involving resistors R, R and R b) Repeat (a) using a -to-Y transformation involving resistors R,R and R c) Give two additional -to-Y or Y-to- transformations that could be used to find Rab. Figure P3.58 R4 R5 13  20  30

Use a -to-Y transformation to find the voltages v and v in the circuit in Fig. P3.59. Figure P3.59

a) Find the resistance seen by the ideal voltage source in the circuit in Fig. P3.60. b) If vab equals 400 V, how much power is dissipated in the 31 resistor?

Use a Y-to- transformation to find (a) i(b) i (c) i and (d) the power delivered by the ideal current source in the circuit in Fig. P3.61.

Find i and the power dissipated in the 140 resistor in the circuit in Fig. P3.62.

For the circuit shown in Fig. P3.63, find (a)i (b) v, (c) i , and (d) the power supplied by the voltage source. Figure P3.63

Show that the expressions for conductances as functions of the three Y conductances are

Derive Eqs. 3.443.49 from Eqs. 3.413.43. The following two hints should help you get started in the right direction: 1) To find R as a function of Ra ,Rb and Rc, first subtract Eq. 3.42 from Eq. 3.43 and then add this result to Eq. 3.41. Use similar manipulations to find R and R and as functions of Ra ,Rb and Rc 2) To find Rb as a function of R, R and R, take advantage of the derivations obtained by hint (1), namely, Eqs. 3.443.46. Note that these equations can be divided to obtain. R2 R3=RcRb, or Rc = R2R3Rb and R1 R2=RbRa,orRa = R2R1Rb Now use these ratios in Eq. 3.43 to eliminate Ra and Rc. Use similar manipulations to find Ra and Rc as function of R, R and R.

Resistor networks are sometimes used as volumecontrol circuits. In this application, they are referred to as resistance attenuators or pads.A typical fixed-attenuator pad is shown in Fig. P3.66. In designing an attenuation pad, thecircuit designer will select the values of and so that the ratio of and the resistance seenby the input voltage source both have aspecified value.a) Show that if , then R2 L = 4R1(R1 + R2),vo vi = R2 2R1 + R2 + RL.Select the values of and so that and c) Choose values from Appendix H that are closest to and from part (b). Calculate the percent error in the resulting values for and if these new resistor values are used.

a) The fixed-attenuator pad shown in Fig. P3.67 is called a bridged tee. Use a Y-to- transformation to show that Rab = RL if R = RL. b) Show that when R = RL , the voltage ratio v/v equals 0.50. Figure P3.67

The design equations for the bridged-tee attenuator circuit in Fig. P3.68 are R2 = 2RR2 L 3R2 - R2 L, vo vi = 3R - RL 3R + RL when R has the value just given. a) Design a fixed attenuator so that vi = 3.5 v when RL = 30 b) Assume the voltage applied to the input of the pad designed in (a) is 42 V. Which resistor in the pad dissipates the most power? c) How much power is dissipated in the resistor in part (b)? d) Which resistor in the pad dissipates the least power? e) How much power is dissipated in the resistor in part (d)?

a) For the circuit shown in Fig. P3.69 the bridge is balanced when R = 0. Show that if R << R the bridge output voltage is approximately b) Given R = 1 k, R = 500 , R = 5,and vin = 6v, what is the approximate bridge output voltage if R is 3% of R? c) Find the actual value of v in part (b). Figure P3.69

a) If percent error is defined as % error = B approximate value true value -1R * 100,show that the percent error in the approximation of v in Problem 3.69 is % error = -(R)R3 (R2 + R3)R4* 100.b) Calculate the percent error in v, using the valuesin Problem 3.69(b).

Assume the error in v in the bridge circuit in Fig. P3.69 is not to exceed 0.5%. What is the largest percent change in R that can be tolerated?

a) Using Fig. 3.38 derive the expression for the voltage Vy. b) Assuming that there are py pixels in the y-direction, derive the expression for the y-coordinate of the touch point, using the result from part (a).

A resistive touch screen has 5 V applied to the grid in the x-direction and in the y-direction. The screen has 480 pixels in the x-direction and 800 pixels in the y-direction. When the screen is touched, the voltage in the x-grid is 1 V and the voltage in the y-grid is 3.75 V.) a) Calculate the values of and ) Calculate the x- and y-coordinates of the pixel at the point where the screen was touched.

A resistive touch screen has 640 pixels in the x-direction and 1024 pixels in the y-direction. The resistive grid has 8 V applied in both the x- and y-directions. The pixel coordinates at the touch point are (480, 192). Calculate the voltages Vx and Vy.

Suppose the resistive touch screen described in Problem 3.74 is simultaneously touched at two points, one with coordinates (480, 192) and the other with coordinates (240, 384). a) Calculate the voltage measured in the x- and y-grids. b) Which touch point has your calculation in (a) identified?