Problems 5 circuit wiley

5/18/13 Problems

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Problems

Section 5.2 Source Transformations

P 5.2-1 The circuit shown in Figure P 5.2-1a has been divided into two parts. The circuit shown inFigure P 5.2-1b was obtained by simplifying the part to the right of the terminals using sourcetransformations. The part of the circuit to the left of the terminals was not changed.

(a) Determine the values of Rt and vt in Figure P 5.2-1b.

(b) Determine the values of the current i and the voltage v in Figure P 5.2-1b. The circuit inFigure P 5.2-1b is equivalent to the circuit in Figure P 5.2-1a. Consequently, the current iand the voltage v in Figure P 5.2-1a have the same values as do the current i and thevoltage v in Figure P 5.2-1b.

(c) Determine the value of the current ia in Figure P 5.2-1a.

FIGURE P 5.2-1

P 5.2-2 Consider the circuit of Figure P 5.2-2. Find ia by simplifying the circuit (using source

transformations) to a single-loop circuit so that you need to write only one KVL equation tofind ia.

FIGURE P 5.2-2

P 5.2-3 Find vo using source transformations if in the circuit shown in Figure P 5.2-3.

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FIGURE P 5.2-3

Hint: Reduce the circuit to a single mesh that contains the voltage source labeled vo.

Answer:

vo = 28 V

P 5.2-4 Determine the value of the current ia in the circuit shown in Figure P 5.2-4.

FIGURE P 5.2-4

P 5.2-5 Use source transformations to find the current ia in the circuit shown in Figure P 5.2-5.

FIGURE P 5.2-5

Answer:

ia = 1 A

P 5.2-6 Use source transformations to find the value of the voltage va in Figure P 5.2-6.

FIGURE P 5.2-6

Answer:

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va = 7 V

*P 5.2-7 Determine the power supplied by each of the sources in the circuit shown in Figure P 5.2-7.

FIGURE P 5.2-7

P 5.2-8 The circuit shown in Figure P 5.2-8 contains an unspecified resistance R.

(a) Determine the value of the current i when R = 4 Ω.

(b) Determine the value of the voltage v when R = 8 Ω.

(c) Determine the value of R that will cause i = 1 A.

(d) Determine the value of R that will cause v = 16 V.

FIGURE P 5.2-8

P 5.2-9 Determine the value of the power supplied by the current source in the circuit shown inFigure P 5.2-9.

FIGURE P 5.2-9

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Section 5.3 Superposition

P 5.3-1 The inputs to the circuit shown in Figure P 5.3-1 are the voltage source voltages v1 and v2.

The output of the circuit is the voltage vo. The output is related to the inputs by

where a and b are constants. Determine the values of a and b.

FIGURE P 5.3-1

P 5.3-2 A particular linear circuit has two inputs, v1 and v2, and one output, vo. Three measurements

are made. The first measurement shows that the output is vo = 4 V when the inputs are v1= 2

V and v2 = 0. The second measurement shows that the output is vo = 10 V when the inputs

are v1= 0 and v2 = −2.5 V. In the third measurement, the inputs are v1 = 3 V and v2 = 3 V.

What is the value of the output in the third measurement?

P 5.3-3 The circuit shown in Figure P 5.3-3 has two inputs, vs and is, and one output io. The output

is related to the inputs by the equation

Given the following two facts:

and

Determine the values of the constants a and b and the values of the resistances are R1 and R2.

FIGURE P 5.3-3

Answer:

a = 0.6 A/A, b = 0.02 A/V, R1 = 30 Ω, and R2 = 20 Ω.

P 5.3-4 Use superposition to find v for the circuit of Figure P 5.3-4.

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FIGURE P 5.3-4

P 5.3-5 Use superposition to find i for the circuit of Figure P 5.3-5.

FIGURE P 5.3-5

Answer:

i = −2 mA

P 5.3-6 Use superposition to find i for the circuit of Figure P 5.3-6.

FIGURE P 5.3-6

Answer:

i = 3.5 mA

P 5.3-7 Use superposition to find the value of the voltage va in Figure P 5.3-7.

FIGURE P 5.3-7

Answer:

va = 7 V

P 5.3-8 Use superposition to find the value of the current ix in Figure P 5.3-8.

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FIGURE P 5.3-8

Answer:

P 5.3-9 The input to the circuit shown in Figure P 5.3-9a is the voltage source voltage vs. The output

is the voltage vo. The current source current, ia, is used to adjust the relationship between the

input and output. The plot shown in Figure P 5.3-9b specifies a relationship between the inputand output of the circuit. Design the circuit shown in Figure P 5.3-9a to satisfy thespecification shown in Figure P 5.3-9b.

FIGURE P 5.3-9

Hint: Use superposition to express the output as vo = cvs + dia where c and d are constants

that depend on R1, R2, and A. Specify values of R1, R2, and A to cause the required value of

c.Finally, specify a value of ia to cause the required value of dia.

P 5.3-10 The input to the circuit shown in Figure P 5.3-10 is the voltage source voltage, vs. The output is the

voltage vo. The current source current, ia, is used to adjust the relationship between the input and

output. Design the circuit so that input and output are related by the equation vo = 2vs + 9.

FIGURE P 5.3-10

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Hint: Determine the required values of A and ia.

P 5.3-11 The circuit shown in Figure P 5.3-11 has three inputs: v1, v2, and i3. The output of the circuit is vo.

The output is related to the inputs by

where a, b, and c are constants. Determine the values of a, b, and c.

FIGURE P 5.3-11

P 5.3-12 Determine the voltage vo(t) for the circuit shown in Figure P 5.3-12.

FIGURE P 5.3-12

P 5.3-13 Determine the value of the voltage vo in the circuit shown in Figure P 5.3-13.

FIGURE P 5.3-13

*P 5.3-14 The circuit shown in Figure P 5.3-14 has two inputs, v1 and v2, and one output, vo. The output is

related to the input by the equation

where a and b are constants that depend on R1, R2, and R3.

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(a) Use superposition to show that when R3 = R1 || R2 and R2 = nR1,

(b) Design this circuit so that a = 4b.

FIGURE P 5.3-14

P 5.3-15 The input to the circuit shown in Figure P 5.3-15 is the current i1. The output is the voltage vo. The

current i2 is used to adjust the relationship between the input and output. Determine values of the

current i2 and the resistance, R, that cause the output to be related to the input by the equation

FIGURE P 5.3-15

P 5.3-16 Determine values of the current, ia, and the resistance, R, for the circuit shown in Figure P 5.3-16.

FIGURE P 5.3-16

P 5.3-17 The circuit shown in Figure P 5.3-17 has three inputs: v1, i2, and v3. The output of the circuit is the

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current io. The output of the circuit is related to the inputs by

where a, b, and c are constants. Determine the values of a, b, and c.

FIGURE P 5.3-17

P 5.3-18 Using the superposition principle, find the value of the current measured by the ammeter in Figure P5.3-18a.

FIGURE P 5.3-18 (a) A circuit containing two independent sources. (b) The circuit after theideal ammeter has been replaced by the equivalent short circuit and a labelhas been added to indicate the current measured by the ammeter, im.

Hint: Figure P 5.3-18b shows the circuit after the ideal ammeter has been replaced by the

equivalent short circuit and a label has been added to indicate the current measured by the ammeter,im.

Answer:

P 5.3-19 Using the superposition principle, find the value of the voltage measured by the voltmeter in Figure

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P 5.3-19a.

FIGURE P 5.3-19 (a) A circuit containing two independent sources. (b) The circuit after theideal voltmeter has been replaced by the equivalent open circuit and a labelhas been added to indicate the voltage measured by the voltmeter, vm.

Hint: Figure P 5.3-19b shows the circuit after the ideal voltmeter has been replaced by the

equivalent open circuit and a label has been added to indicate the voltage measured by the voltmeter,vm.

Answer:

Section 5.4 Thévenin's Theorem

P 5.4-1 Determine values of Rt and voc that cause the circuit shown in Figure P 5.4-1b to be the

Thévenin equivalent circuit of the circuit in Figure P 5.4-1a.

FIGURE P 5.4-1

Hint: Use source transformations and equivalent resistances to reduce the circuit in Figure P

5.4-1a until it is the circuit in Figure P 5.4-1b.

Answer:

Rt = 5 Ω and voc = 2 V

P 5.4-2 The circuit shown in Figure P 5.4-2b is the Thévenin equivalent circuit of the circuit shownin Figure P 5.4-2a. Find the value of the open-circuit voltage, voc, and Thévenin resistance,

Rt.

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FIGURE P 5.4-2

Answer:

voc = −12 V and Rt = 16 Ω

P 5.4-3 The circuit shown in Figure P 5.4-3b is the Thévenin equivalent circuit of the circuit shownin Figure P 5.4-3a. Find the value of the open-circuit voltage, voc, and Thévenin resistance,

Rt.

FIGURE P 5.4-3

Answer:

voc = 2 V and Rt = 4 Ω

P 5.4-4 Find the Thévenin equivalent circuit for the circuit shown in Figure P 5.4-4.

FIGURE P 5.4-4


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FIGURE P 5.4-5

Answer:

voc = −2 V and Rt = −8/3 Ω


FIGURE P 5.4-6

P 5.4-7 The circuit shown in Figure P 5.4-7 has four unspecified circuit parameters: vs, R1, R2, and

d, where d is the gain of the CCCS.

(a) Show that the open-circuit voltage, voc, the short-circuit current, isc, and the Thévenin

resistance, Rt, of this circuit are given by

and

(b) Let R1 = R2 = 1 kΩ. Determine the values of vs and d required to cause voc = 5 V and Rt

= 625 Ω.

FIGURE P 5.4-7

P 5.4-8 A resistor, R, was connected to a circuit box as shown in Figure P 5.4-8. The voltage, v, wasmeasured. The resistance was changed, and the voltage was measured again. The results areshown in the table. Determine the Thévenin equivalent of the circuit within the box andpredict the voltage, v, when R = 8 kΩ.

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FIGURE P 5.4-8

P 5.4-9 A resistor, R, was connected to a circuit box as shown in Figure P 5.4-9. The current, i, wasmeasured. The resistance was changed, and the current was measured again. The results areshown in the table.

(a) Specify the value of R required to cause i = 2 mA.

(b) Given that R > 0, determine the maximum possible value of the current i.

FIGURE P 5.4-9

Hint: Use the data in the table to represent the circuit by a Thévenin equivalent.

P 5.4-10 Measurements made on terminals a–b of a linear circuit, Figure P 5.4-10a, which is known to bemade up only of independent and dependent voltage sources and current sources and resistors, yieldthe current–voltage characteristics shown in Figure P 5.4-10b. Find the Thévenin equivalent circuit.

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FIGURE P 5.4-10

P 5.4-11 For the circuit of Figure P 5.4-11, specify the resistance R that will cause current ib to be 2 mA.

The current ia has units of amps.

FIGURE P 5.4-11

Hint: Find the Thévenin equivalent circuit of the circuit connected to R.

P 5.4-12 For the circuit of Figure P 5.4-12, specify the value of the resistance RL that will cause current iLto be −2 A.

FIGURE P 5.4-12

Answer:

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RL = 12 Ω

P 5.4-13 The circuit shown in Figure P 5.4-13 contains an adjustable resistor. The resistance R can be set toany value in the range 0 ≤ R ≤ 100 kΩ.

(a) Determine the maximum value of the current ia that can be obtained by adjusting R. Determine

the corresponding value of R.

(b) Determine the maximum value of the voltage va that can be obtained by adjusting R. Determine

the corresponding value of R.

(c) Determine the maximum value of the power supplied to the adjustable resistor that can beobtained by adjusting R. Determine the corresponding value of R.

FIGURE P 5.4-13

P 5.4-14 The circuit shown in Figure P 5.4-14 consists of two parts, the source (to the left of the terminals)and the load. The load consists of a single adjustable resistor having resistance 0 ≤ RL ≤ 20 Ω. The

resistance R is fixed but unspecified. When RL = 4 Ω, the load current is measured to be io = 0.375

A. When RL = 8 Ω, the value of the load current is io = 0.300 A.

(a) Determine the value of the load current when RL = 10 Ω.

(b) Determine the value of R.

FIGURE P 5.4-14

P 5.4-15 The circuit shown in Figure P 5.4-15 contains an unspecified resistance, R. Determine the value ofR in each of the following two ways.

(a) Write and solve mesh equations.

(b) Replace the part of the circuit connected to the resistor R by a Thévenin equivalent circuit.Analyze the resulting circuit.

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FIGURE P 5.4-15

P 5.4-16 Consider the circuit shown in Figure P 5.4-16. Replace the part of the circuit to the left of terminalsa–b by its Thévenin equivalent circuit. Determine the value of the current io.

FIGURE P 5.4-16

P 5.4-17 An ideal voltmeter is modeled as an open circuit. A more realistic model of a voltmeter is a largeresistance. Figure P 5.4-17a shows a circuit with a voltmeter that measures the voltage vm. In

Figure P 5.4-17b, the voltmeter is replaced by the model of an ideal voltmeter, an open circuit. Thevoltmeter measures vmi, the ideal value of vm.

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FIGURE P 5.4-17

As Rm → ∞, the voltmeter becomes an ideal voltmeter, and vm → vmi. When Rm < ∞, the voltmeter

is not ideal, and vm > vmi. The difference between vm and vmi is a measurement error caused by the

fact that the voltmeter is not ideal.

(a) Determine the value of vmi.

(b) Express the measurement error that occurs when Rm = 1000 Ω as a percentage of vmi.

(c) Determine the minimum value of Rm required to ensure that the measurement error is smaller

than 2 percent of vmi.

P 5.4-18 Determine the Thévenin equivalent circuit for the circuit shown in Figure P 5.4-18.

FIGURE P 5.4-18

P 5.4-19 Given that in the circuit shown in Figure P 5.4-19, consider these two observations:

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Determine the following:

(a) The maximum value of iR and the value of R that causes iR to be maximal

(b) The maximum value of vR and the value of R that causes vR to be maximal

(c) The maximum value of and the value of R that causes pR to be maximal

FIGURE P 5.4-19

P 5.4-20 Consider the circuit shown in Figure P 5.4-20. Determine

(a) The value of vR that occurs when R = 9 Ω.

(b) The value of R that causes vR = 5.4 V.

(c) The value of R that causes iR = 300 mA.

FIGURE P 5.4-20

Section 5.5 Norton's Equivalent Circuit

P 5.5-1 The part of the circuit shown in Figure P 5.5-1a to the left of the terminals can be reduced toits Norton equivalent circuit, using source transformations and equivalent resistance. Theresulting Norton equivalent circuit, shown in Figure P 5.5-1b, will be characterized by theparameters:

(a) Determine the values of and R1.

(b) Given that , determine the maximum values of the voltage, v, and of thepower, p = vi.

Answer:

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, and .

FIGURE P 5.5-1

P 5.5-2 Two black boxes are shown in Figure P 5.5-2. Box A contains the Thévenin equivalent ofsome linear circuit, and box B contains the Norton equivalent of the same circuit. With accessto just the outsides of the boxes and their terminals, how can you determine which is which,using only one shorting wire?

FIGURE P 5.5-2 Black boxes problem.

P 5.5-3 Find the Norton equivalent circuit for the circuit shown in Figure P 5.5-3.

FIGURE P 5.5-3

Answer:

Rt = 2 Ω and isc = −7.5 A


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FIGURE P 5.5-4

P 5.5-5 The circuit shown in Figure P 5.5-5b is the Norton equivalent circuit of the circuit shown inFigure P 5.5-5a. Find the value of the short-circuit current, isc, and Thévenin resistance, Rt.

FIGURE P 5.5-5

Answer:

isc = 1.13 A and Rt = 7.57 Ω

P 5.5-6 The circuit shown in Figure P 5.5-6b is the Norton equivalent circuit of the circuit shown inFigure P 5.5-6a. Find the value of the short-circuit current, isc, and Thévenin resistance, Rt.

FIGURE P 5.5-6

Answer:

isc = −24 A and Rt = −3 Ω

P 5.5-7 Determine the value of the resistance R in the circuit shown in Figure P 5.5-7 by each of thefollowing methods:

(a) Replace the part of the circuit to the left of terminals a–b by its the Norton equivalentcircuit. Use current division to determine the value of R.

(b) Analyze the circuit shown Figure P 5.5-7 using mesh equations. Solve the meshequations to determine the value of R.

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FIGURE P 5.5-7

P 5.5-8 The device to the right of terminals a−b in Figure P 5.5-8 is a nonlinear resistor characterizedby

Determine the values of i and v.

FIGURE P 5.5-8


FIGURE P 5.5-9


FIGURE P 5.5-10

P 5.5-11

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An ideal ammeter is modeled as a short circuit. A more realistic model of an ammeter is a smallresistance. Figure P 5.5-11a shows a circuit with an ammeter that measures the current im. In

Figure P 5.5-10b, the ammeter is replaced by the model of an ideal ammeter, a short circuit. Theammeter measures imi, the ideal value of im.

FIGURE P 5.5-11

As Rm → 0, the ammeter becomes an ideal ammeter and im → imi. When Rm > 0, the ammeter is

not ideal and im < imi. The difference between im and imi is a measurement error caused by the fact

that the ammeter is not ideal.

(a) Determine the value of imi.

(b) Express the measurement error that occurs when Rm = 20 Ω as a percentage of imi.

(c) Determine the maximum value of Rm required to ensure that the measurement error is smaller

than 2 percent of imi.

P 5.5-12 Determine values of Rt and isc that cause the circuit shown in Figure P 5.5-12b to be the Norton

equivalent circuit of the circuit in Figure P 5.5-12a.

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FIGURE P 5.5-12

Answer:

Rt = 3 Ω and isc = −2 A

P 5.5-13 Use Norton's theorem to formulate a general expression for the current i in terms of the variableresistance R shown in Figure P 5.5-13.

FIGURE P 5.5-13

Answer:

i = 20/(8 + R) A

Section 5.6 Maximum Power Transfer

P 5.6-1 The circuit shown in Figure P 5.6-1 consists of two parts separated by a pair of terminals.Consider the part of the circuit to the left of the terminals. The open circuit voltage is

, and short-circuit current is . Determine the values of (a) the voltagesource voltage, Vs, and the resistance R2 and (b) the resistance R that maximizes the power

delivered to the resistor to the right of the terminals, and the corresponding maximum power.

FIGURE P 5.6-1

P 5.6-2 The circuit model for a photovoltaic cell is given in Figure P 5.6-2 (Edelson, 1992). The

current is is proportional to the solar insolation (kW/m2).

(a) Find the load resistance, RL, for maximum power transfer.

(b) Find the maximum power transferred when is = 1 A.

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FIGURE P 5.6-2 Circuit model of photovoltaic cell.

P 5.6-3 For the circuit in Figure P 5.6-3, (a) find R such that maximum power is dissipated in R and(b) calculate the value of maximum power.

FIGURE P 5.6-3

Answer:

R = 60 Ω and Pmax = 54 mW

P 5.6-4 For the circuit in Figure P 5.6-4, prove that for Rs variable and RL fixed, the power dissipated

in RL is maximum when Rs = 0.

FIGURE P 5.6-4

P 5.6-5 Find the maximum power to the load RL if the maximum power transfer condition is met for

the circuit of Figure P 5.6-5.

FIGURE P 5.6-5

Answer:

max pL = 0.75 W

P 5.6-6 Determine the maximum power that can be absorbed by a resistor, R, connected to terminalsa–b of the circuit shown in Figure P 5.6-6. Specify the required value of R.

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FIGURE P 5.6-6 Bridge circuit.

P 5.6-7 Figure P 5.6-7 shows a source connected to a load through an amplifier. The load can safelyreceive up to 15 W of power. Consider three cases:

(a) A = 20 V/V and Ro = 10 Ω. Determine the value of RL that maximizes the power

delivered to the load and the corresponding maximum load power.

(b) A = 20 V/V and RL = 8 Ω. Determine the value of Ro that maximizes the power delivered

to the load and the corresponding maximum load power.

(c) Ro = 10 Ω and RL = 8 Ω. Determine the value of A that maximizes the power delivered to

the load and the corresponding maximum load power.

FIGURE P 5.6-7

P 5.6-8 The circuit in Figure P 5.6-8 contains a variable resistance, R, implemented using apotentiometer. The resistance of the variable resistor varies over the range 0 ≤ R ≤ 1000 Ω.The variable resistor can safely receive 1/4 W power. Determine the maximum powerreceived by the variable resistor. Is the circuit safe?

FIGURE P 5.6-8

P 5.6-9 For the circuit of Figure P 5.6-9, find the power delivered to the load when RL is fixed and Rt

may be varied between 1 Ω and 5 Ω. Select Rt so that maximum power is delivered to RL.

FIGURE P 5.6-9

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Answer:

13.9 W

P 5.6-10 A resistive circuit was connected to a variable resistor, and the power delivered to the resistor wasmeasured as shown in Figure P 5.6-10. Determine the Thévenin equivalent circuit.

FIGURE P 5.6-10

Answer:

Rt = 20 Ω and voc = 20 V

Section 5.8 Using PSpice to Determine the Thévenin Equivalent Circuit

P 5.8-1 The circuit shown in Figure P 5.8-1 is separated into two parts by a pair of terminals. Call thepart of the circuit to the left of the terminals circuit A and the part of the circuit to the right ofthe terminal circuit B. Use PSpice to do the following:

(a) Determine the node voltages for the entire circuit.

(b) Determine the Thévenin equivalent circuit of circuit A.

(c) Replace circuit A by its Thévenin equivalent and determine the node voltages of themodified circuit.

(d) Compare the node voltages of circuit B before and after replacing circuit A by itsThévenin equivalent.

FIGURE P 5.8-1

Section 5.9 How Can We Check …?

P 5.9-1 For the circuit of Figure P 5.9-1, the current i has been measured for three different values ofR and is listed in the table. Are the data consistent?

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FIGURE P 5.9-1

P 5.9-2 Your lab partner built the circuit shown in Figure P 5.9-2 and measured the current i andvoltage v corresponding to several values of the resistance R. The results are shown in thetable in Figure P 5.9-2. Your lab partner says that RL = 8000 Ω is required to cause i = 1 mA.

Do you agree? Justify your answer.

FIGURE P 5.9-2

P 5.9-3 In preparation for lab, your lab partner determined the Thévenin equivalent of the circuit

connected to RL in Figure P 5.9-3. She says that the Thévenin resistance is and

the open-circuit voltage is . In lab, you built the circuit using R = 110 Ω and RL

= 40 Ω and measured that i = 54.5 mA. Is this measurement consistent with the prelabcalculations? Justify your answers.

FIGURE P 5.9-3

P 5.9-4 Your lab partner claims that the current i in Figure P 5.9-4 will be no greater than 12.0 mA,regardless of the value of the resistance R. Do you agree?

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FIGURE P 5.9-4

P 5.9-5 Figure P 5.9-5 shows a circuit and some corresponding data. Two resistances, R1 and R, and

the current source current are unspecified. The tabulated data provide values of the current, i,and voltage, v, corresponding to several values of the resistance R.

(a) Consider replacing the part of the circuit connected to the resistor R by a Théveninequivalent circuit. Use the data in rows 2 and 3 of the table to find the values of Rt and

voc, the Thévenin resistance, and the open-circuit voltage.

(b) Use the results of part (a) to verify that the tabulated data are consistent.

(c) Fill in the blanks in the table.

(d) Determine the values of R1 and is.

FIGURE P 5.9-5

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