Solved: Derive Eqs. 3.443.49 from Eqs. 3.413.43. The following two hints should help you

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) Find the power dissipated in each resistor in the 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 dissipated.

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 resistors.

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.

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 for each of the circuits in Fig. P3.8.

Find the equivalent resistance \(R_{\mathrm{ab}}\) 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 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 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 \(5 \ \mathrm{k} \Omega\) 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 \(4 \ \mathrm{k} \Omega\) using a minimum number of identical resistors from Appendix H.

a) Calculate the no-load voltage \(v_{o}\) for the voltage divider circuit shown in Fig. P3.12. b) Calculate the power dissipated in \(R_{1}\) and \(R_{2}\) 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 \(R_{1}\) and \(R_{2}\).

In the voltage-divider circuit shown in Fig. P3.13, the no-load value of \(v_{o}\) is 4 V. When the load resistance \(R_{L}\) is attached across the terminals a and b, \(v_{o}\) drops to 3 V. Find \(R_{L}\).

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 \(3.6 \ \mathrm{k} \Omega\). When the divider is loaded, \(v_{o}\) is not to drop below 7.5 V. a) Design the divider circuit to meet the specifications just mentioned. Specify the numerical values of \(R_{1}\) and \(R_{2}\). 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 1 W resistors. What is the smallest resistor from Appendix H that can be used as \(R_{L}\) before one of the resistors in the divider is operating at its dissipation limit?

Find the power dissipated in the \(5 \ \Omega\) resistor in the current divider circuit in Fig. P3.16.

For the current divider circuit in Fig. P3.17 calculate a) \(i_{o}\) and \(v_{o}\). b) the power dissipated in the \(6 \ \Omega\) 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: \(i_{g}=50 \mathrm{~mA} ; v_{g}=25 \mathrm{~V} ; i_{1}=0.6 i_{2}\); \(i_{3}=2 i_{2} ; \text { and } i_{4}=4 i_{1}\).

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 \(R_{1}, R_{2} \text {, and } R_{3}\) 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 \(v_{1}=12 \mathrm{~V}\), \(v_{2}=5 \mathrm{~V}, \text { and } v_{3}=-12 \mathrm{~V}\).

a) 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 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 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 \(v_{o}=k v_{s}\) at no load ( \(R_{L}=\infty\) ) and \(v_{o}=\alpha v_{s}\) at full load ( \(R_{L}=R_{o}\) ). Note that by definition \(\alpha<k<1\). a) Show that \(R_{1}=\frac{k-\alpha}{\alpha k} R_{o}\) and \(R_{2}=\frac{k-\alpha}{\alpha(1-k)} R_{o}\) b) Specify the numerical values of \(R_{1}\) and \(R_{2}\) if \(k=0.85, \alpha=0.80\), and \(R_{o}=34 \ \mathrm{k} \Omega\). c) If \(v_{s}=60 \mathrm{~V}\), specify the maximum power that will be dissipated in \(R_{1}\) and \(R_{2}\). d) Assume the load resistor is accidentally short circuited. How much power is dissipated in \(R_{1}\) and \(R_{2}\)?

a) Show that the current in the kth branch of the circuit in Fig. P3.22(a) is equal to the source current \(i_{g}\) times the conductance of the kth branch divided by the sum of the conductances, that is, \(i_{k}=\frac{i_{g} G_{k}}{G_{1}+G_{2}+G_{3}+\cdots+G_{k}+\cdots+G_{N}}\) b) Use the result derived in (a) to calculate the current in the \(5 \ \Omega\) 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 \(6 \ \mathrm{k} \Omega\) resistor, positive at the top. b) Use the result from part (a) and voltage division to find the voltage across the \(5 \ \mathrm{k} \Omega\) resistor, positive on the left.

Look at the circuit in Fig. P3.3(d). a) Use current division to find the current in the \(50 \ \Omega\) resistor from left to right. b) Use the result from part (a) and current division to find the current in the \(70 \ \Omega\) 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 \(25 \ \Omega\) resistor, positive at the left. b) Using your result from (a), find the current flowing in the \(25 \ \Omega\) resistor from left to right. c) Starting with your result from (b), use current division to find the current in the \(50 \ \Omega\) resistor from top to bottom. d) Using your result from part (c), find the voltage drop across the \(50 \ \Omega\) resistor, positive at the top. e) Starting with your result from (d), use voltage division to find the voltage drop across the \(60 \ \Omega\) resistor, positive at the top.

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

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

a) Find the voltage \(v_{x}\) in the circuit in Fig. P3.28 using voltage and/or current division. b) Replace the 18 V source with a general voltage source equal to \(V_{s}\). Assume \(V_{s}\) is positive at the upper terminal. Find \(v_{x}\) as a function of \(V_{s}\).

Find \(v_{o}\) in the circuit in Fig. P3.29 using voltage and/or current division.

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

For the circuit in Fig. P3.31, find \(i_{g}\) and then use current division to find \(i_{o}\).

For the circuit in Fig. P3.32, calculate \(i_{1}\) and \(i_{2}\) using current division.

A d’Arsonval ammeter is shown in Fig. P3.33. a) Calculate the value of the shunt resistor, \(R_{A}\), 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, \(R_{A}\), 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 d’Arsonval movement are used to build a 5 A ammeter. A resistance of \(20 \mathrm{~m} \Omega\) is placed across the terminals of the ammeter. What is the new full-scale range of the ammeter?

A d’Arsonval 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-scale reading 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 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?

A d’Arsonval voltmeter is shown in Fig. P3.37. Find the value of \(R_{v}\) for each of the following full-scale readings: (a) 50 V, (b) 5 V, (c) 250 mV, and (d) 25 mV.

Suppose the d’Arsonval voltmeter described in Problem 3.37 is used to measure the voltage across the \(45 \ \Omega\) resistor in Fig. P3.38. a) What will the voltmeter read? b) Find the percentage of error in the voltmeter reading if \(\% \text { error }=\left(\frac{\text { measured value }}{\text { true value }}-1\right) \times 100\).

The ammeter in the circuit in Fig. P3.39 has a resistance of \(0.1 \ \Omega\). 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 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: \(R_{1}=20 \ \mathrm{k} \Omega, R_{2}=80 \ \mathrm{k} \Omega, R_{C}=0.82 \ \mathrm{k} \Omega\), \(R_{E}=0.2 \ \mathrm{k} \Omega, V_{C C}=7.5 \mathrm{~V}, V_{0}=0.6 \mathrm{~V} \text {, and } \beta=39\). a) Calculate the value of \(i_{B}\) in microamperes. b) Assume that a digital multimeter, when used as a dc ammeter, has a resistance of \(1 \ \mathrm{k} \Omega\). If the meter is inserted between terminals b and 2 to measure the current \(i_{B}\), what will the meter read? c) Using the calculated value of \(i_{B}\) 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 \(900 \ \Omega / \mathrm{V}\). The other voltmeter is rated 150 V full scale and has a sensitivity of \(1200 \ \Omega / \mathrm{V}\). (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 \(50 \ \mathrm{k} \Omega\) precision resistor is also available.The \(50 \ \mathrm{k} \Omega\) 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 full scale 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 v in the circuit of Fig. P3.44(b)?

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 d’Arsonval voltmeter having a sensitivity of \(100 \ \Omega / \mathrm{V}\) 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 \(70 \ \mathrm{k} \Omega\) resistor? c) What will the voltmeter read if it is placed across the \(20 \ \mathrm{k} \Omega\) 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.

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 \(R_{1}, R_{2} \text {, and } R_{3}\). b) For each of the three ranges, calculate the percentage of error that this design strategy produces.

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 \(15 \ \mathrm{M} \Omega\) 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 \(10 \ \mathrm{M} \Omega\). a) What is the internal voltage of the source \(\left(v_{s}\right)\) in millivolts? b) What is the internal resistance of the source \(\left(R_{s}\right)\) in kilo-ohms?

Design a d’Arsonval voltmeter that will have the three voltage ranges shown in Fig. P3.48. a) Specify the values of \(R_{1}, R_{2} \text {, and } R_{3}\). b) Assume that a \(750 \ \mathrm{k} \Omega\) 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?

A \(600 \ \mathrm{k} \Omega\) 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 \(360 \ \mathrm{k} \Omega\) 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 \(R_{1}=500 \ \Omega, R_{2}=1000 \ \Omega \text {, and } R_{3}=750 \ \Omega\). a) What is the value of \(R_{x}\)? 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 \(3 \ \mathrm{k} \Omega\) resistor in the circuit in Fig. P3.52.

Find the detector current \(i_{d}\) in the unbalanced bridge in Fig. P3.53 if the voltage drop across the detector is negligible.

In the Wheatstone bridge circuit shown in Fig. 3.26, the ratio \(R_{2} / R_{1}\) can be set to the following values: 0.001, 0.01, 0.1, 1, 10, 100, and 1000. The resistor \(R_{3}\) can be varied from \(1 \text { to } 11,110 \ \Omega\), in increments of \(1 \ \Omega\). An unknown resistor is known to lie between \(4 \text { and } 5 \ \Omega\). What should be the setting of the \(R_{2} / R_{1}\) 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 \(\Delta\) (\(25,37.5 \text {, and } 40 \ \Omega\)) 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 \(100 \ \Omega\) ) with its equivalent \(\Delta\).

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 \(125 \ \Omega\) ) with its equivalent \(\Delta\).

a) Find the equivalent resistance \(R_{\mathrm{ab}}\) in the circuit in Fig. P3.58 by using a \(\text { Y-to- } \Delta\) transformation involving resistors \(R_{2}, R_{3}, \text { and } R_{5}\). b) Repeat (a) using a \(\Delta \text {-to-Y }\) transformation involving resistors \(R_{3}, R_{4}, \text { and } R_{5}\). c) Give two additional \(\Delta \text {-to-Y }\) or \(\text { Y-to- } \Delta\) transformations that could be used to find \(R_{\mathrm{ab}}\).

Use a \(\Delta \text {-to-Y }\) transformation to find the voltages \(v_{1}\) and \(v_{2}\) in the circuit in Fig. P3.59.

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

Use a \(\text { Y-to- } \Delta\) transformation to find (a) \(i_{o}\); (b) \(i_{1}\); (c) \(i_{2}\); and (d) the power delivered by the ideal current source in the circuit in Fig. P3.61.

Find \(i_{o}\) and the power dissipated in the \(140 \ \Omega\) resistor in the circuit in Fig. P3.62

For the circuit shown in Fig. P3.63, find (a) \(i_{1}\) (b) v, (c) \(i_{2}\), and (d) the power supplied by the voltage source.

Show that the expressions for \(\Delta\) conductances as functions of the three Y conductances are \(\begin{aligned}G_{a} & =\frac{G_{2} G_{3}}{G_{1}+G_{2}+G_{3}}, \\ G_{b} & =\frac{G_{1} G_{3}}{G_{1}+G_{2}+G_{3}}, \\ G_{c} & =\frac{G_{1} G_{2}}{G_{1}+G_{2}+G_{3}},\end{aligned}\) where \(G_{a}=\frac{1}{R_{a}}, \quad G_{1}=\frac{1}{R_{1}}, \quad \text { etc. }\)

Derive Eqs. 3.44–3.49 from Eqs. 3.41–3.43. The following two hints should help you get started in the right direction: 1) To find \(R_{1}\) as a function of \(R_{a}, R_{b} \text {, and } R_{c}\), first subtract Eq. 3.42 from Eq. 3.43 and then add this result to Eq. 3.41. Use similar manipulations to find \(R_{2}\) and \(R_{3}\) as functions of \(R_{a}, R_{b}\), and \(R_{c}\). 2) To find \(R_{b}\) as a function of \(R_{1}, R_{2}\), and \(R_{3}\), take advantage of the derivations obtained by hint (1), namely, Eqs. 3.44–3.46. Note that these equations can be divided to obtain \(\frac{R_{2}}{R_{3}}=\frac{R_{c}}{R_{b}}\), or \(R_{c}=\frac{R_{2}}{R_{3}} R_{b}\), and \(\frac{R_{1}}{R_{2}}=\frac{R_{b}}{R_{a}}\), or \(R_{a}=\frac{R_{2}}{R_{1}} R_{b}\). Now use these ratios in Eq. 3.43 to eliminate \(R_{a}\) and \(R_{c}\). Use similar manipulations to find \(R_{a}\) and \(R_{c}\) as functions of \(R_{1}, R_{2}\), and \(R_{3}\).

Resistor networks are sometimes used as volume control 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, the circuit designer will select the values of \(R_{1}\) and \(R_{2}\) so that the ratio of \(v_{o} / v_{i}\) and the resistance seen by the input voltage source \(R_{\mathrm{ab}}\) both have a specified value. a) Show that if \(R_{\mathrm{ab}}=R_{\mathrm{L}}\), then \(R_{\mathrm{L}}^{2}=4 R_{1}\left(R_{1}+R_{2}\right)\), \(\frac{v_{o}}{v_{i}}=\frac{R_{2}}{2 R_{1}+R_{2}+R_{\mathrm{L}}}\). b) Select the values of \(R_{1}\) and \(R_{2}\) so that \(R_{\mathrm{ab}}=R_{\mathrm{L}}=300 \ \Omega\) and \(v_{o} / v_{i}=0.5\). c) Choose values from Appendix H that are closest to \(R_{1}\) and \(R_{2}\) from part (b). Calculate the percent error in the resulting values for \(R_{\mathrm{ab}}\) and \(v_{0} / v_{1}\) if these new resistor values are used.

a) The fixed-attenuator pad shown in Fig. P3.67 is called a bridged tee. Use a \(\text { Y-to- } \Delta\) transformation to show that \(R_{\mathrm{ab}}=R_{\mathrm{L}}\) if \(R=R_{\mathrm{L}}\). b) Show that when \(R=R_{\mathrm{L}}\), the voltage ratio \(v_{o} / v_{i}\) equals 0.50

The design equations for the bridged-tee attenuator circuit in Fig. P3.68 are \(R_{2}=\frac{2 R R_{\mathrm{L}}^{2}}{3 R^{2}-R_{\mathrm{L}}^{2}}\), \(\frac{v_{o}}{v_{i}}=\frac{3 R-R_{\mathrm{L}}}{3 R+R_{\mathrm{L}}}\), when \(R_{2}\) has the value just given. a) Design a fixed attenuator so that \(v_{i}=3.5 v_{o}\) when \(R_{\mathrm{L}}=300 \ \Omega\). 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 \(\Delta R=0\). Show that if \(\Delta R \ll R_{o}\) the bridge output voltage is approximately \(v_{o} \approx \frac{-\Delta R R_{4}}{\left(R_{o}+R_{4}\right)^{2}} v_{\mathrm{in}}\) b) Given \(R_{2}=1 \ \mathrm{k} \Omega, R_{3}=500 \ \Omega, R_{4}=5 \ \mathrm{k} \Omega\), and \(v_{\text {in }}=6 \mathrm{~V}\), what is the approximate bridge output voltage if \(\Delta R\) is 3% of \(R_{o}\)? c) Find the actual value of \(v_{o}\) in part (b).

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

Assume the error in \(v_{o}\) in the bridge circuit in Fig. P3.69 is not to exceed 0.5%. What is the largest percent change in \(R_{o}\) that can be tolerated?

a) Using Fig. 3.38 derive the expression for the voltage \(V_{y}\). b) Assuming that there are \(p_{y}\) 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 \(\alpha\) and \(\beta\). a) 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 \(V_{x}\) and \(V_{y}\).

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?

Derive the s-domain equivalent circuit shown in Fig. 13.4 by expressing the inductor current i as a function of the terminal voltage and then finding the Laplace transform of this time-domain integral equation.

Find the Thvenin equivalent of the circuit shown in Fig. 13.7

Find the Norton equivalent of the circuit shown in Fig. 13.3.

A resistor, a 12.5 mH inductor, and a 0.5 F capacitor are in series. a) Express the s-domain impedance of this series combination as a rational function. b) Give the numerical value of the poles and zeros of the impedance.

An resistor, a 25 mH inductor, and a 62.5 pF capacitor are in parallel. a) Express the s-domain impedance of this parallel combination as a rational function. b) Give the numerical values of the poles and zeros of the impedance.

A resistor is in series with a 62.5 F capacitor. This series combination is in parallel with a 400 mH inductor. a) Express the equivalent s-domain impedance of these parallel branches as a rational function. b) Determine the numerical values of the poles and zeros.

Find the poles and zeros of the impedance seen looking into the terminals a,b of the circuit shown in Fig. P13.7.

Find the poles and zeros of the impedance seen looking into the terminals a,b of the circuit shown in Fig. P13.8.

Find and in the circuit shown in Fig. P13.9 if the initial energy is zero and the switch is closed at t = 0.

Repeat Problem 13.9 if the initial voltage on the capacitor is 150 V positive at the upper terminal.

The switch in the circuit shown in Fig. P13.11 has been in position x for a long time. At the switch moves instantaneously to position y. a) Construct an s-domain circuit for b) Find c) Find io. Io

The switch in the circuit in Fig. P13.12 has been closed for a long time. At the switch is opened. a) Find for b) Find for v t 0. o i t 0.

The switch in the circuit in Fig. P13.13 has been in position a for a long time. At it moves instantaneously from a to b. a) Construct the s-domain circuit for b) Find c) Find v t 0.

The switch in the circuit in Fig. P13.14 has been in position a for a long time. At the switch moves instantaneously to position b. a) Construct the s-domain circuit for b) Find c) Find d) Find for e) Find for t 7 0.

The switch in the circuit in Fig. P13.15 has been closed for a long time before opening at a) Construct the s-domain equivalent circuit for b) Find c) Find for i t 0.

The make-before-break switch in the circuit in Fig. P13.16 has been in position a for a long time.At it moves instantaneously to position b. Find for t 0.

a) Find the s-domain expression for in the circuit in Fig. P13.17. b) Use the s-domain expression derived in (a) to predict the initial- and final-values of c) Find the time-domain expression for a) Find the s-domain expression for in the circuit in Fig. P13.17. b) Use the s-domain expression derived in (a) to predict the initial- and final-values of c) Find the time-domain expression for

Find the time-domain expression for the current in the capacitor in Fig. P13.17. Assume the reference direction for is down.

There is no energy stored in the circuit in Fig. P13.19 at a) Use the mesh current method to find b) Find the time domain expression for c) Do your answers in (a) and (b) make sense in terms of known circuit behavior? Explain.

There is no energy stored in the circuit in Fig. P13.20 at the time the voltage source is turned on, and a) Find and b) Find and c) Do the solutions for and make sense in terms of known circuit behavior? Explain.

There is no energy stored in the circuit in Fig. P13.21 at the time the sources are energized. a) Find and b) Use the initial- and final-value theorems to check the initial- and final-values of and c) Find and for t 0.

There is no energy stored in the circuit in Fig. P13.22 at a) Find b) Find c) Does your solution for make sense in terms of known circuit behavior? Explain.

Find in the circuit shown in Fig. P13.23 if . There is no energy stored in the circuit at t = 0.

The switch in the circuit in Fig. P13.24 has been closed for a long time before opening at Find for t 0.

There is no energy stored in the circuit in Fig. P13.25 at the time the switch is closed. a) Find for b) Does your solution make sense in terms of known circuit behavior? Explain.

The initial energy in the circuit in Fig. P13.26 is zero. The ideal voltage source is a) Find b) Use the initial- and final-value theorems to find and vo v (q). c) Do the values obtained in (b) agree with known circuit behavior? Explain. d) Find vo(t).

There is no energy stored in the circuit in Fig. P13.27 at the time the current source turns on. Given that a) Find b) Use the initial- and final-value theorems to find and c) Determine if the results obtained in (b) agree with known circuit behavior.

The switch in the circuit seen in Fig. P13.28 has been in position a for a long time. At it moves instantaneously to position b. a) Find b) Find Figure P13.28 a b 5.625 H if 3  24 V 5  vo 0.1 F 20  25if t  0 vo. Vo. t = 0

The switch in the circuit seen in Fig. P13.29 has been in position a for a long time before moving instantaneously to position b at a) Construct the s-domain equivalent circuit for b) Find and c) Find and v2 V . 2 v1

There is no energy stored in the capacitors in the circuit in Fig. P13.30 at the time the switch is closed. a) Construct the s-domain circuit for b) Find and c) Find and d) Do your answers for and make sense in terms of known circuit behavior? Explain.

There is no energy stored in the circuit in Fig. P13.31 at the time the current source is energized. a) Find and b) Find and c) Find and d) Find and e) Assume a capacitor will break down whenever its terminal voltage is 1000 V. How long after the current source turns on will one of the capacitors break down?

The switch in the circuit shown in Fig. P13.32 has been open for a long time. The voltage of the sinusoidal source is The switch closes at Note that the angle in the voltage expression determines the value of the voltage at the moment when the switch closes, that is, a) Use the Laplace transform method to find i for b) Using the expression derived in (a), write the expression for the current after the switch has been closed for a long time. c) Using the expression derived in (a), write the expression for the transient component of i. d) Find the steady-state expression for i using the phasor method. Verify that your expression is equivalent to that obtained in (b). e) Specify the value of so that the circuit passes directly into steady-state operation when the switch is closed.

Beginning with Eq. 13.65, show that the capacitor current in the circuit in Fig. 13.19 is positive for and negative for Also show that at the current is zero and that this corresponds to when is zero

There is no energy stored in the circuit in Fig. P13.34 at the time the voltage source is energized. a) Find , , and b) Find , , and for i i L t 0.

The two switches in the circuit shown in Fig. P13.35 operate simultaneously. There is no energy stored in the circuit at the instant the switches close. Find for by first finding the s-domain Thvenin equivalent of the circuit to the left of the terminals a, b.

There is no energy stored in the circuit in Fig. P13.36 at the time the switch is closed. a) Find b) Use the initial- and final-value theorems to find and c) Find i1.

a) Find the current in the resistor in the circuit in Fig. P13.36. The reference direction for the current is down through the resistor. b) Repeat part (a) if the dot on the 250 mH coil is reversed.

The magnetically coupled coils in the circuit seen in Fig. P13.38 carry initial currents of 300 and 200 A, as shown. a) Find the initial energy stored in the circuit. b) Find and I2 I . c) Find and d) Find the total energy dissipated in the 240 and resistors. e) Repeat (a)(d), with the dot on the 720 mH inductor at the lower terminal.

The make-before-break switch in the circuit seen in Fig. P13.39 has been in position a for a long time.At it moves instantaneously to position b. Find i t 0.

The switch in the circuit seen in Fig. P13.40 has been closed for a long time before opening at Use the Laplace transform method of analysis to find vo.

In the circuit in Fig. P13.41, switch 1 closes at and the make-before-break switch moves instantaneously from position a to position b. a) Construct the s-domain equivalent circuit for b) Find I1. c) Use the initial- and final-value theorems to check the initial and final values of d) Find for t 0+ i . 1

Verify that the solution of Eqs. 13.91 and 13.92 for yields the same expression as that given by Eq. 13.90.

There is no energy stored in the circuit seen in Fig. P13.43 at the time the two sources are energized. a) Use the principle of superposition to find b) Find for t 7 0.

The op amp in the circuit shown in Fig. P13.44 is ideal. There is no energy stored in the circuit at the time it is energized. If find (a) (b) (c) how long it takes to saturate the operational amplifier, and (d) how small the rate of increase in must be to prevent saturation.

Find in the circuit shown in Fig. P13.45 if the ideal op amp operates within its linear range and vg = 16u(t) mV.

The op amp in the circuit shown in Fig. P13.46 is ideal. There is no energy stored in the capacitors at the instant the circuit is energized. a) Find if and b) How many milliseconds after the two voltage sources are turned on does the op amp saturate?

The op amp in the circuit seen in Fig. P13.47 is ideal. There is no energy stored in the capacitors at the time the circuit is energized. Determine (a) (b) and (c) how long it takes to saturate the operational amplifier.

a) Find the transfer function for the circuit shown in Fig. P13.48(a). b) Find the transfer function for the circuit shown in Fig. P13.48(b). c) Create two different circuits that have the transfer function Use components selected from Appendix H and Figs. P13.48(a) and (b).

a) Find the transfer function for the circuit shown in Fig. P13.49(a). b) Find the transfer function for the circuit shown in Fig. P13.49(b). c) Create two different circuits that have the transfer function Use components selected from Appendix H and Figs. P13.49(a) and (b).

a) Find the transfer function for the circuit shown in Fig. P13.50. Identify the poles and zeros for this transfer function. b) Find three components from Appendix H which when used in the circuit of Fig. P13.50 will result in a transfer function with two poles that are distinct real numbers. Calculate the values of the poles. c) Find three components from Appendix H which when used in the circuit of Fig. P13.50 will result in a transfer function with two poles, both with the same value. Calculate the value of the poles. d) Find three components from Appendix H which when used in the circuit of Fig. P13.50 will result in a transfer function with two poles that are complex conjugate complex numbers. Calculate the values of the poles.

a) Find the numerical expression for the transfer function for the circuit in Fig. P13.51. b) Give the numerical value of each pole and zero of

Find the numerical expression for the transfer function of each circuit in Fig. P13.52 and give the numerical value of the poles and zeros of each transfer function.

The operational amplifier in the circuit in Fig. P13.53 is ideal. a) Find the numerical expression for the transfer function b) Give the numerical value of each zero and pole of Figure P13.53 VCC VCC 125 k 400 pF 250 k 1.6 nF vo v

The operational amplifier in the circuit in Fig. P13.54 is ideal. a) Find the numerical expression for the transfer function b) Give the numerical value of each zero and pole of Figure P13.54

The operational amplifier in the circuit in Fig. P13.55 is ideal. a) Derive the numerical expression of the transfer function for the circuit in Fig. P13.55. b) Give the numerical value of each pole and zero of H(s).

There is no energy stored in the circuit in Fig. P13.56 at the time the switch is opened.The sinusoidal current source is generating the signal The response signal is the current a) Find the transfer function b) Find c) Describe the nature of the transient component of without solving for io i (t). d) Describe the nature of the steady-state component of without solving for e) Verify the observations made in (c) and (d) by finding io(t).

a) Find the transfer function as a function of for the circuit seen in Fig. P13.57. b) Find the largest value of that will produce a bounded output signal for a bounded input signal. c) Find for 3, 0, 4, 5, and i i m = g = 5u(t) A.

In the circuit of Fig. P13.58 is the output signal and is the input signal. Find the poles and zeros of the transfer function, assuming there is no initial energy stored in the linear transformer or in the capacitor.

A rectangular voltage pulse is applied to the circuit in Fig. P13.59. Use the convolution integral to find vo.

Interchange the inductor and resistor in Problem 13.59 and again use the convolution integral to find vo.

a) Use the convolution integral to find the output voltage of the circuit in Fig. P13.52(a) if the input voltage is the rectangular pulse shown in Fig. P13.61. b) Sketch versus t for the time interval Figure P13.61

a) Repeat Problem 13.61, given that the resistor in the circuit in Fig. P13.52(a) is decreased to b) Does decreasing the resistor increase or decrease the memory of the circuit? c) Which circuit comes closer to transmitting a replica of the input voltage?

a) Given find when and are the rectangular pulses shown in Fig. P13.63(a). b) Repeat (a) when changes to the rectangular pulse shown in Fig. P13.63(b). c) Repeat (a) when changes to the rectangular pulse shown in Fig. P13.63(c). d) Sketch versus t for (a)(c) on a single graph. e) Do the sketches in (d) make sense? Explain.

a) Find when and are the rectangular pulses shown in Fig. P13.64(a). b) Repeat (a) when changes to the rectangular pulse shown in Fig. P13.64(b). c) Repeat (a) when changes to the rectangular pulse shown in Fig. P13.64(c).

The voltage impulse response of a circuit is shown in Fig. P13.65(a). The input signal to the circuit is the rectangular voltage pulse shown in Fig. P13.65(b). a) Derive the equations for the output voltage. Note the range of time for which each equation is applicable. b) Sketch for v -1 t 34 s.

Assume the voltage impulse response of a circuit can be modeled by the triangular waveform shown in Fig. P13.66.The voltage input signal to this circuit is the step function a) Use the convolution integral to derive the expressions for the output voltage. b) Sketch the output voltage over the interval 0 to 15 s. c) Repeat parts (a) and (b) if the area under the voltage impulse response stays the same but the width of the impulse response narrows to 4 s. d) Which output waveform is closer to replicating the input waveform: (b) or (c)? Explain.

a) Assume the voltage impulse response of a circuit is h(t) = b 0, t 6 0; 10e-4t , t 0

Use the convolution integral to find the output voltage if the input signal is h(t) = c 0, t 6 0; 10(1 - 2t), 0 t 0.5 s; 0, t 0.5 s. b) Repeat (a) if the voltage impulse response is c) Plot the output voltage versus time for (a) and

Use the convolution integral to find in the circuit seen in Fig. P13.68 if Figure P13.68

a) Use the convolution integral to find in the circuit in Fig. P13.69(a) if is the pulse shown in Fig. P13.69(b). b) Use the convolution integral to find c) Show that your solutions for and are consistent by calculating and at and Figure P13.69 ig 100 k (a) vo io ig (mA) 50 50 (b) 0 200 100 t (ms) 0.2 mF 200 200 + ms. 100

a) Find the impulse response of the circuit shown in Fig. P13.71(a) if is the input signal and is the output signal. b) Given that has the waveform shown in Fig. P13.71(b), use the convolution integral to find c) Does have the same waveform as Why or why not?

a) Find the impulse response of the circuit seen in Fig. P13.72 if is the input signal and is the output signal. b) Assume that the voltage source has the waveform shown in Fig. P13.71(b). Use the convolution integral to find c) Sketch for d) Does have the same waveform as ? Why or why not?

The current source in the circuit shown in Fig. P13.73(a) is generating the waveform shown in Fig. P13.73(b). Use the convolution integral to find at t = 5 ms.

The sinusoidal voltage pulse shown in Fig. P13.74(a) is applied to the circuit shown in Fig. P13.74(b). Use the convolution integral to find the value of at t = 75 ms.

a) Show that if then b) Use the result given in (a) to find if F(s) = a s(s + a) 2 .

The operational amplifier in the circuit seen in Fig. P13.76 is ideal and is operating within its linear region. a) Calculate the transfer function b) If what is the steady-state expression for ?

The op amp in the circuit seen in Fig. P13.77 is ideal. a) Find the transfer function b) Find if c) Find the steady-state expression for if vg = 2 cos 10,000t V.

he transfer function for a linear time-invariant circuit is If what is the steady-state expression for vo

The transfer function for a linear time-invariant circuit is If what is the steady-state expression for ?

When an input voltage of is applied to a circuit, the response is known to be What will the steady-state response be if vg = 120 cos 6000 t V

Show that after coulombs are transferred from to in the circuit shown in Fig. 13.47, the voltage across each capacitor is (Hint: Use the conservation-of-charge principle.)

The voltage source in the circuit in Example 13.1 is changed to a unit impulse; that is, a) How much energy does the impulsive voltage source store in the capacitor? b) How much energy does it store in the inductor? c) Use the transfer function to find d) Show that the response found in (c) is identical to the response generated by first charging the capacitor to 1000 V and then releasing the charge to the circuit, as shown in Fig. P13.82.

There is no energy stored in the circuit in Fig. P13.83 at the time the impulsive voltage is applied. a) Find for b) Does your solution make sense in terms of known circuit behavior? Explain.

The inductor in the circuit shown in Fig. P13.84 is carrying an initial current of at the instant the switch opens. Find (a) (b) (c) and (d) where is the total flux linkage in the circuit.

a) Let in the circuit shown in Fig. P13.84, and use the solutions derived in Problem 13.84 to find and b) Let in the circuit shown in Fig. P13.84 and use the Laplace transform method to find and i2 i (t).

The parallel combination of and in the circuit shown in Fig. P13.86 represents the input circuit to a cathode-ray oscilloscope (CRO). The parallel combination of and is a circuit model of a compensating lead that is used to connect the CRO to the source. There is no energy stored in or at the time when the 10 V source is connected to the CRO via the compensating lead. The circuit values are and a) Find b) Find c) Repeat (a) and (b) given is changed to 64 pF

Show that if in the circuit shown in Fig. P13.86, will be a scaled replica of the source voltage.

The switch in the circuit in Fig. P13.88 has been closed for a long time. The switch opens at Compute (a) (b) (c) (d) (e) (f) and (g) v(t).

There is no energy stored in the circuit in Fig. P13.89 at the time the impulse voltage is applied. a) Find for b) Find for c) Find for d) Do your solutions for and make sense in terms of known circuit behavior? Explain.

The switch in the circuit in Fig. P13.90 has been in position a for a long time. At the switch moves to position b. Compute (a) (b) (c) (d) (e) (f) and (g) Figure P13.90 a b 20 k 100 V t  0 1.6 mF v3 2.0 mF v2 0.5 mF v1 i(t)

There is no energy stored in the circuit in Fig. P13.91 at the time the impulsive current is applied. a) Find for b) Does your solution make sense in terms of known circuit behavior? Explain.

Assume the line-to-neutral voltage in the 60 Hz circuit of Fig. 13.59 is Load is absorbing 1200 W; load is absorbing 1800 W; and load is absorbing 350 magnetizing VAR. The inductive reactance of the line is Assume does not change after the switch opens. a) Calculate the initial value of and b) Find and using the s-domain circuit of Fig. 13.60. c) Test the steady-state component of using phasor domain analysis. d) Using a computer program of your choice, plot v vs. t for 0 t 20 ms.

Assume the switch in the circuit in Fig. 13.59 opens at the instant the sinusoidal steady-state voltage is zero and going positive, i.e., a) Find for b) Using a computer program of your choice, plot vs. t for c) Compare the disturbance in the voltage in part (a) with that obtained in part (c) of Problem 13.92.

The purpose of this problem is to show that the line-to-neutral voltage in the circuit in Fig. 13.59 can go directly into steady state if the load is disconnected from the circuit at precisely the right time. Let where Assume does not change after is disconnected. a) Find the value of (in degrees) so that goes directly into steady-state operation when the load is disconnected. b) For the value of found in part (a), find for c) Using a computer program of your choice, plot on a single graph, for vo before and after load is disconnected.