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Problems Section 5.2 Source Transformations 5.2-1a has been divided into two parts. The circuit shown in P 5.2-1 The circuit shown in Figure P 5.2-1a Figure P 5.2-1b 5.2-1b was obtained by simplifying the part to the right of the terminals using source transformations. The part of the circuit to the left of the terminals was not changed.
of Rt and v t in Figure P 5.2-1b 5.2-1b. (a) Determine the values of R 5.2-1b. The circuit in (b) Determine the values of the current i and the voltage v in Figure P 5.2-1b Figure P 5.2-1b 5.2-1b is equivalent to the circuit in Figure P 5.2-1 a. Consequently, the current i and the voltage v in Figure P 5.2-1a 5.2-1a have the same values as do the current i and the voltage v in Figure P 5.2-1b 5.2-1b. 5.2-1a. (c) Determine the value of the current ia in Figure P 5.2-1a
FIGURE P 5.2-1
s impli mplifying the ci c ircui rc uitt (using ( using sourc e P 5.2-2 Consider the circuit of Figure P 5.2-2. Find ia by si transformations) to a single-loop circuit so that you need to write only one KVL equation to find ia.
FIGURE P 5.2-2
transf ormations ons if P 5.2-3 Find v o using sourc e transformati
in the circuit shown show n in Figure 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 v o.
An swe r :
v o = 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 An swe r :
ia = 1 A P 5.2-6 Use source transformations to find the value of the voltage v a in Figure P 5.2-6.
FIGURE P 5.2-6
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Problems
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 in
Figure P 5.2-9.
FIGURE P 5.2-9
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Problems
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 v 1 and v 2.
The output of the circuit is the voltage v o. 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, v 1 and v 2 , and one output, v o. Three measurements
are made. The first measurement shows that the output is v o = 4 V when the inputs are v 1= 2 V and v 2 = 0. The second measurement shows that the output is v o = 10 V when the inputs are v 1= 0 and v 2 = −2.5 V. In the third measurement, the inputs are v 1 = 3 V and v 2 = 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, v s 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 An swe r :
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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Problems
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 An swe r :
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 An swe r :
i = 3.5 mA P 5.3-7 Use superposition to find the value of the voltage v a in Figure P 5.3-7.
FIGURE P 5.3-7 An swe r :
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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Problems
FIGURE P 5.3-8 An swe r :
P 5.3-9 The input to the circuit shown in Figure P 5.3-9a is the voltage source voltage v s . The output
is the voltage v o. 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 input and output of the circuit. Design the circuit shown in Figure P 5.3-9a to satisfy the specification shown in Figure P 5.3-9b.
FIGURE P 5.3-9
Use superposition to express the output as v o = cv s + 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
Hint:
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, v s . The output is the voltage v o. 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 v o = 2v s + 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: v 1, v 2, and i3. The output of the circuit is v o. 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 v o(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 v o 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, v 1 and v 2, and one output, v o. 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 v o. 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: v 1, i2, and v 3. 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 P 5.3-18a.
FIGURE P 5.3-18 (a) A circuit containing two indepen dent s ources. ( b) The circuit after the ideal ammeter has been replaced by the equivalent s hort circuit and a label has been ad ded to indicate the current measu red by the ammeter, im .
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.
Hint:
An swe r :
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 indepen dent s ources. ( b) The circuit after the ideal voltmeter has been replaced by the equivalent open circuit and a label has b een added t o indicate the vo ltage measu red by the voltmeter, vm.
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, v m.
Hint:
An swe r :
Section 5.4 Thévenin's Theorem P 5.4-1 Determine values of Rt and v oc that cause the circuit shown in Figure P 5.4-1b to be the
Thévenin equivalent circuit of the circuit in Figure P 5.4-1 a.
FIGURE P 5.4-1
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.
Hint:
An swe r :
Rt = 5 Ω and v oc = 2 V P 5.4-2 The circuit shown in Figure P 5.4-2b is the Thévenin equivalent circuit of the circuit shown
in Figure P 5.4-2a. Find the value of the open-circuit voltage, v oc, and Thévenin resistance, Rt .
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FIGURE P 5.4-2 An swe r :
v oc = −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 shown
in Figure P 5.4-3a. Find the value of the open-circuit voltage, v oc, and Thévenin resistance, Rt .
FIGURE P 5.4-3 An swe r :
v oc = 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 P 5.4-5 Find the Thévenin equivalent circuit for the circuit shown in Figure P 5.4-5.
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FIGURE P 5.4-5 An swe r :
v oc = −2 V and Rt = −8/3 Ω P 5.4-6 Find the Thévenin equivalent circuit for the circuit shown in Figure P 5.4-6.
FIGURE P 5.4-6 P 5.4-7 The circuit shown in Figure P 5.4-7 has four unspecified circuit parameters: v s, R 1, R2, and
d , where d is the gain of the CCCS.
(a) Show that the open-circuit voltage, v oc, the short-circuit c urrent, isc, and the Thévenin
resistance, Rt , of this circuit are given by
and
(b) Let R1 = R2 = 1 kΩ. Determine the values of v s and d required to cause v oc = 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, was
measured. The resistance was changed, and the voltage was measured again. The results are shown in the table. Determine the Thévenin equivalent of the circuit within the box and predict 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, was
measured. The resistance was changed, and the current was measured again. The results are shown 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 be made up only of independent and dependent voltage sources and current sources and resistors, yield the 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 i b 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 iL to be −2 A.
FIGURE P 5.4-12
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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 to any 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 v a 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 be obtained 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 of R 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 terminals a–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 large resistance. Figure P 5.4-17a shows a circuit with a voltmeter that measures the voltage v m. In Figure P 5.4-17b, the voltmeter is replaced by the model of an ideal voltmeter, an open circuit. The voltmeter measures v mi, the ideal value of v m.
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Problems
FIGURE P 5.4-17
As Rm → ∞, the voltmeter becomes an ideal voltmeter, and v m → v mi. When Rm < ∞, the voltmeter is not ideal, and v m > v mi. The difference between v m and v mi is a measurement error caused by the fact that the voltmeter is not ideal.
(a) Determine the value of v mi. (b) Express the measurement error that occurs when Rm = 1000 Ω as a percentage of v mi. (c) Determine the minimum value of Rm required to ensure that the measurement error is smaller than 2 percent of v mi. 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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Problems
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 v R and the value of R that causes v R 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 v R that occurs when R = 9 Ω. (b) The value of R that causes v R = 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 to
its Norton equivalent circuit, using source transformations and equivalent resistance. The resulting Norton equivalent circuit, shown in Figure P 5.5-1b, will be characterized by the parameters:
(a) Determine the values of (b) Given that power, p = vi.
An swe r :
and R1. , determine the maximum values of the voltage, v, and of the
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Problems
,
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 of
some linear circuit, and box B contains the Norton equivalent of the same circuit. With access to 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 An swe r :
Rt = 2 Ω and isc = −7.5 A P 5.5-4 Find the Norton equivalent circuit for the circuit shown in Figure P 5.5-4.
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Problems
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 in
Figure P 5.5-5a. Find the value of the short-circuit current, isc, and Thévenin resistance, Rt .
FIGURE P 5.5-5 An swe r :
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 in
Figure P 5.5-6a. Find the value of the short-circuit current, isc, and Thévenin resistance, Rt .
FIGURE P 5.5-6 An swe r :
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 the
following methods: (a) Replace the part of the circuit to the left of terminals a–b by its the Norton equivalent circuit. Use current division to determine the value of R. (b) Analyze the circuit shown Figure P 5.5-7 using mesh equations. Solve the mesh equations to determine the value of R.
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Problems
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 characterized
by
Determine the values of i and v.
FIGURE P 5.5-8 P 5.5-9 Find the Norton equivalent circuit for the circuit shown in Figure P 5.5-9.
FIGURE P 5.5-9 P 5.5-10
Find the Norton equivalent circuit for the circuit shown in Figure P 5.5-10.
FIGURE P 5.5-10 P 5.5-11
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Problems
An ideal ammeter is modeled as a short circuit. A more realistic model of an ammeter is a small resistance. 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. The ammeter 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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Problems
FIGURE P 5.5-12 An swe r :
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 variable resistance R shown in Figure P 5.5-13.
FIGURE P 5.5-13 An swe r :
i = 20/(8 + R) A
Section 5.6 M aximum Power Transfe r 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 voltage source voltage, V s, 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/m 2).
(a) Find the load resistance, RL, for maximum power transfer. (b) Find the maximum power transferred when is = 1 A.
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Problems
FIGURE P 5.6-2 Circuit model of pho tovoltaic 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 An swe r :
R = 60 Ω and P max = 54 mW P 5.6-4 For the circuit in Figure P 5.6-4, prove that for R s 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 An swe r :
max pL = 0.75 W P 5.6-6 Determine the maximum power that can be absorbed by a resistor, R, connected to terminals
a–b of the circuit shown in Figure P 5.6-6. Specify the required value of R.
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Problems
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 safely
receive 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 a
potentiometer. T he 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 power received 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 R t
may be varied between 1 Ω and 5 Ω. Select Rt so that maximum power is delivered to RL.
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Problems
An swe r :
13.9 W P 5.6-10
A resistive circuit was connected to a variable resistor, and the power delivered to the resistor was measured as shown in Figure P 5.6-10. Determine the Thévenin equivalent circuit.
FIGURE P 5.6-10 An swe r :
Rt = 20 Ω and v oc = 20 V
Section 5.8 Using PSpice to De te rmine the Thév en in Equivale nt 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 the
part of the circuit to the left of the terminals circuit A and the part of the circuit to the right of the 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 the modified circuit. (d) Compare the node voltages of circuit B before and after replacing circuit A by its Thévenin 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 of
R and is listed in the table. Are the data consistent?
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Problems
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 and
voltage v corresponding to several values of the resistance R. The results are shown in the table 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 the open-c ircuit voltage is
and
. In lab, you built the circ uit using R = 110 Ω and RL
= 40 Ω and measured that i = 54.5 mA. Is this measurement consistent with the prelab calculations? 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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Problems
FIGURE P 5.9-4 P 5.9-5 Figure P 5.9-5 shows a circuit and some corresponding data. Two resistances, R 1 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 Thévenin equivalent circuit. Use the data in rows 2 and 3 of the table to find the values of Rt and
v oc, 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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