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Principles and Applications of Electrical Engineering

Giorgio Rizzoni, James Kearns

Chapter 2

Equivalent Networks - all with Video Answers

Educators

Chapter Questions

Problem 0

The resistance of the device $D$ in Eigure P2.90 is a nonlinear function of pressure $P$. The $i-v$ characteristics of $D$ are shown for various pressures. Assume:
a. Plot the DC load line.
b. Plot the voltage across $D$ as a function of pressure.
c. Determine the current through $D$ when $P=30$ psig.

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Problem 1

Apply voltage division to the circuit of Eigure P2.1. Assume that $v_S=9 \mathrm{~V}, R_1=8 \mathrm{k} \Omega, R_2=R_3=10 \mathrm{k} \Omega, R_4=12 \mathrm{k} \Omega$. Find $v_2$.

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00:57

Problem 2

Refer to Eigure P2.2, and assume that $v_S=12 \mathrm{~V}, R_1=5 \Omega, R_2=3 \Omega, R_3=4 \Omega$, and $R_4=5 \Omega$. Apply voltage division to each resistive branch and KVL to find
the voltage $v_{a b}$ -

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:40

Problem 3

Apply voltage division to each circuit in Eigure P2.3 to find the value of $R_o$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 4

Apply the concepts of equivalent parallel resistance, voltage division, and current division to determine the current through each of the resistors $R_4, R_5$, and $R_6$ in Figure P2, 4, $v_S=10 \mathrm{~V}, R_1=20 \Omega, R_2=40 \Omega, R_3=10 \Omega, R_4=R_5=R_6= 15 \Omega$.

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Problem 5

The voltage divider network of Figure P2.5 is designed to provide $v_{\text {out }}=v_S />2$. However, in practice, the resistors may not be perfectly matched; that is, their tolerances are such that the resistances are unlikely to be identical. Apply voltage division to relate $v_{\text {out }}$ to $v_S$ and take the derivative of $v_{\text {out }}$ to find an expression for $d v_{\text {out }}$ in terms of the tolerances $d R_1 / R_1$ and $d R_2 / R_2$. Assume $v_S=$ 10 V and nominal resistance values of $R_1=R_2=5 \mathrm{k} \Omega$.
a. If the resistors have $\pm 5$ percent tolerance, find the expected range of possible output voltages.
b. Find the expected output voltage range for a tolerance of $\pm 1$ percent.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 6

Apply voltage division to the circuit in Figure P2.6 to find an expression for the voltage across the variable resistor $R$. Use that expression to determine and plot the power absorbed by $R$, ranging from 0 to $30 \Omega$. Plot the power absorption as a function of $R$. Assume that $v_S=15 \mathrm{~V}, R_S=10 \Omega$.

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03:19

Problem 7

Apply voltage division to the circuit shown in Figure P2.7 to determine the terminal voltage $v_o$ of the voltage source and the power absorbed by $R_o$.

Thomas Thompson
Thomas Thompson
Numerade Educator

Problem 8

With no load $R_o$ attached to the terminals of the nonideal source in Figure P2.7, the voltage drop $v_o$ is 50.8 V . When a $R_o=10 \Omega$ load is attached, that voltage drop is 49 V . Apply voltage division to find an expression for $v_o$ in terms of $v_S$. $R_S$ and $R_o$.
a. Determine $v_S$ and $R_S$ for this nonideal source.
b. What voltage would be measured at the terminals in the presence of a 15 $\Omega$ load resistor?
c. How much current could be drawn from the nonideal source under shortcircuit conditions?

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03:03

Problem 9

Apply voltage division and KVL to determine the voltage $p_o$ across terminals $A$ and $B$ in Figure P2.9.

Thomas Thompson
Thomas Thompson
Numerade Educator
03:10

Problem 10

Refer to Figure P2.10 and assume $v_S=15 \mathrm{~V}, R_1=12 \Omega, R_2=5 \Omega, R_3=8 \Omega, R_4= 2 \Omega, R_5=4 \Omega, R_6=2 \Omega$, and $R_7=1 \Omega$. Apply voltage division to find:
a. The voltage $v_{a c}$ across nodes $a$ and $c$.
b. The voltage $v_{b d}$ across nodes $b$ and $d$.

Keshav Singh
Keshav Singh
Numerade Educator
02:41

Problem 11

The circuit of Eigure P2.11 is used to measure the internal resistance $r_B$ of a battery.
a. A fresh battery is being tested, and it is found that the voltage $v_{\text {out }}$ is 2.28 V with the switch open and 2.27 V with the switch closed. Apply voltage division to find the internal resistance of the battery.
b. The same battery is tested one year later. $v_{\text {out }}$ is found to be 2.28 V with the switch open but 0.31 V with the switch closed. Apply voltage division to find the one-year-old internal resistance of the battery.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 12

For the circuit shown in Figure P2.12, assume $i_S=5 \mathrm{~A}, R_1=10 \Omega, R_2=7 \Omega, R_3= 8 \Omega, R_4=4 \Omega$, and $R_5=2 \Omega$. How many nodes are in the circuit? Use series and parallel equivalent resistance concepts to simply the network to the left of the current source into a single equivalent resistance. Apply current division to find the magnitude of the current through the branch containing $R_4$ and $R_5$.

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06:07

Problem 13

Consider the practical ammeter, depicted in Figure P2.13, consisting of an ideal ammeter in series with a $1-\mathrm{k} \Omega$ resistor. (An ideal ammeter acts like a shortcircuit.) The meter sees a full-scale deflection when the current through it is 30 $\mu \mathrm{A}$. Depending on the setting of the rotary switch, the ammeter will read full scale when the current $I$ equals $10 \mathrm{~mA}, 100 \mathrm{~mA}$ and 1 A . Apply current division to determine the appropriate values of $R_1, R_2$, and $R_3$.

Vishal Gupta
Vishal Gupta
Numerade Educator

Problem 14

How many nodes are in the circuit shown in Figure P2.14? Use series and parallel equivalent resistance concepts to simplify the network to the right of node $V_1$ into a single equivalent resistance. Apply current division to find the magnitude of the current through the $3 \Omega$ resistor.

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Problem 15

How many nodes are in the circuit shown in Figure P2.15? Use series and parallel equivalent resistance concepts to simplify the network to the right of the current source into a single equivalent resistance. Apply current division to find the magnitude of the current through $R_1$. Assume $R_1=10 \Omega, R_2=9 \Omega, R_3=4 \Omega$, $R_4=4 \Omega, i_S=2 \mathrm{~A}$.

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Problem 16

How many nodes are in the circuit shown in Figure P2.16? Apply current division to find the current through each resistive branch. Apply KVL and Ohm's law to find the magnitude of the voltage across nodes $a$ and $b$. Assume $R_1=12 \Omega, R_2=10 \Omega, R_3=5 \Omega, R_4=2 \Omega, I_S=3 \mathrm{~A}$.

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00:53

Problem 17

Find the equivalent resistance seen by the voltage source in Eigure P2.17. Use that result and voltage division to find $v_2$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 18

Find the equivalent resistance seen by the voltage source and the current $i$ in the circuit of Eigure P2.18.

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15:38

Problem 19

In the circuit of Eigure P2.19, the power absorbed by the $15-\Omega$ resistor is 15 W . Find $R$.

Manish Haldankar
Manish Haldankar
Numerade Educator
03:16

Problem 20

Find the equivalent resistance between terminals $a$ and $b$ in the circuit of Figure P2.20.

Supratim Pal
Supratim Pal
Numerade Educator

Problem 21

For the circuit shown in Eigure P2.21, find the equivalent resistance seen by the voltage source. How much power is delivered by it?

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05:09

Problem 22

For the circuit shown in Eigure P2.22, find the equivalent resistance seen by the current source. How many nodes are in the circuit? Assume $R_1=2 \Omega, R_2=3 \Omega$, $R_3=85 \Omega, R_4=2 \Omega$, and $R_5=4 \Omega$.

Thomas Thompson
Thomas Thompson
Numerade Educator

Problem 23

Refer to Eigure P2.23. Assume $v_S=20 \mathrm{~V}, R_1=10 \Omega, R_2=5 \Omega, R_3=8 \Omega, R_4= 2 \Omega, R_5=4 \Omega, R_6=2 \Omega, R_7=1 \Omega$, and $R_8=10 \Omega$. How many nodes are in the circuit?
a. Determine the equivalent resistance seen by the voltage source $v_S$.
b. Apply voltage division to find the voltage across $R_7$ and $R_8$.

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Problem 24

Find the equivalent resistance seen by the voltage source in Eigure P2.24. How many nodes are in the circuit? Assume $R_1=12 \Omega, R_2=5 \Omega, R_3=8 \Omega, R_4=2 \Omega$, $R_5=4 \Omega, R_6=2 \Omega, R_7=1 \Omega, R_8=10 \Omega$, and $R_9=10 \Omega$.

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01:39

Problem 25

For the circuit shown in Figure P2.25, assume $v_S=10 \mathrm{~V}, R_1=9 \Omega, R_2=4 \Omega, R_3 =4 \Omega, R_4=5 \Omega$, and $R_5=4 \Omega$. Find:
a. The number of nodes in the circuit.
b. The equivalent resistance seen by the voltage source $v_{S^{-}}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:34

Problem 26

Determine the equivalent resistance of the infinite network of resistors in the circuit of Eigure P2.26.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:45

Problem 27

In the circuit of Eigure P2.27, find the equivalent resistance between terminals $a$ and $b$ if terminals $c$ and $d$ are open and again if terminals $c$ and $d$ are shorted together. Also, find the equivalent resistance between terminals $c$ and $d$ if terminals $a$ and $b$ are open and again if terminals $a$ and $b$ are shorted together.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:45

Problem 28

Refer to Eigure P2.27 and determine the equivalent resistance between terminals $a$ and $b$ if terminal $c$ is wired (shorted) to terminal $a$ and terminal $d$ is wired (shorted) to terminal $b$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:40

Problem 29

Apply the node voltage method to find the magnitude of the current through the voltage source. Use it and the definition of equivalent resistance between two nodes to find the equivalent resistance seen by the voltage source in Figure P2.29. How many nodes are in the circuit? Assume: $R_1=12 \Omega, R_2=5 \Omega, R_3= 8 \Omega, R_4=2 \Omega$, and $R_5=4 \Omega$.

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator

Problem 30

Refer to Figure P2.30 and assume $v_S=15 \mathrm{~V}, R_1=12 \Omega, R_2=5 \Omega, R_3=8 \Omega, R_4= 2 \Omega, R_5=4 \Omega, R_6=2 \Omega, R_7=1 \Omega$, and $R_8=R_9=10 \Omega$. Find:
a. The number of nodes in the circuit.
b. The equivalent resistance seen by the voltage source $v_S$.

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Problem 31

Refer to Figure P2.31, and assume that $v_S=7 \mathrm{~V}, i_S=3 \mathrm{~A}, R_1=20 \Omega, R_2=12 \Omega$, and $R_3=10 \Omega$. Apply the principle of superposition to find:
a. The component of $i_1$ due to $v_S$.
b. The component of $i_2$ due to $i_S$.

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02:53

Problem 32

With reference to Figure P2.32, determine the current $i$ through $R_1$ due only to the source $V_{S 2}$.

Thomas Thompson
Thomas Thompson
Numerade Educator
03:10

Problem 32

Refer to Eigure P2.33 and use the principle of superposition to find the voltages at nodes $A, B$, and $C$. Assume $V^1=12 \mathrm{~V}, V_2=10 \mathrm{~V}, R_1=2 \Omega, R_2=8 \Omega, R_3= 12 \Omega, R_4=8 \Omega$.

Keshav Singh
Keshav Singh
Numerade Educator
00:53

Problem 34

Use the principle of superposition to determine the voltage $v$ across $R_2$ in Figure P2.34.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:34

Problem 35

Refer to Eigure P2. 35 and use the principle of superposition to determine the component of the current $i$ through $R_3$ that is due to $V_{S 2}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:25

Problem 35

Apply source transformations to find the mesh current $I_3$ for the circuit shown in Eigure P2.45.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator

Problem 36

The circuit shown in Eigure P2.63 is one form of a differential amplifier. Find an expression for the voltage drop $v_{b a}$ from terminal $b$ to terminal $a$ in terms of $v_1$ and $v_2$ using Thévenin's or Norton's theorem. Notice that the figure implies zero current through sources $v_1$ and $v_2$.

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Problem 36

Refer to Eigure P2.36 and use the principle of superposition to determine the current $i$ through $R_4$ due to the current source $i_S$. Assume: $R_1=12 \Omega, R_2=8 \Omega$, $R_3=5 \Omega, R_4=3 \Omega, v_S=3 \mathrm{~V}$, and $i_S=2 \mathrm{~A}$.

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Problem 37

Refer to Eigure P2.36 and use the principle of superposition to determine the current $i$ through $R_4$ due to the voltage source $v_S$. Assume: $R_1=12 \Omega, R_2=8 \Omega$, $R_3=5 \Omega, R_4=3 \Omega, v_S=3 \mathrm{~V}$, and $i_S=2 \mathrm{~A}$.

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Problem 38

Use the principle of superposition node to determine the voltages $V_a$ and $V_b$ in Eigure P2.38. Let $R_1=10 \Omega, R_2=4 \Omega, R_3=6 \Omega, R_4=6 \Omega, V_1=2 \mathrm{~V}, V_2=4 \mathrm{~V}, I 1 =2 \mathrm{~A}$.

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Problem 39

Use the principle of superposition to determine the current $i$ through $R_3$ in Figure P2 39. Let $R_1=10 \Omega, R_2=4 \Omega, R_3=2 \Omega, R_4=2 \Omega, R_5=2 \Omega, V S=10 \mathrm{~V}$, $i_S =2 \mathrm{~A}$.

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Problem 40

Figure P2.40 represents a temperature measurement system, where temperature $T$ is linearly related to the voltage source $V_{S 2}$ by a transduction constant $k$. Use the principle of superposition to determine the components of $V_{a b}$ due to $V_{S 1}$ and $V_{S 2}$ and then to determine the temperature.
In practice, the voltage across $R_3$ is used as the measure of temperature, which is introduced to the circuit through a temperature sensor modeled by the voltage source $V_{S 2}$ in series with $R_s$.

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Problem 41

In Figure P2.41, use the principle of superposition to determine the components of the current through the voltage source $v_S$ due to $v_S$ and $i_S$, respectively. Use those results to determine the total current through $v_S$ and the power supplied by it. Let $R_1=12 \Omega, R_2=10 \Omega, R_3=5 \Omega, R_4=5 \Omega, v_S=10 \mathrm{~V}, i_S=5 \mathrm{~A}$. (Note: Power is not a linear function of voltage or current and so power cannot be computed using the component currents separately.)

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05:12

Problem 42

Use the principle of superposition to determine components of the current $i_o$ through $R_1$ due to each independent source in Eigure P2.42. Let $R_1=8 \Omega, R_2= 2 \Omega, R_3=3 \Omega, R_4=4 \Omega, R_5=2 \Omega, V_1=15 \mathrm{~V}, I 1=2 \mathrm{~A}, I 2=3 \mathrm{~A}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 43

Apply two source transformations and current division in the circuit of Figure $\underline{\mathrm{P} 2,43}$ to find $I_2$. Let $R_1=12 \Omega, R_2=6 \Omega, R_3=10 \Omega, V_1=4 \mathrm{~V}, V_2=1 \mathrm{~V}$.

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Problem 44

Apply source transformations to find the voltage $V_o$ across $R_0$ for the circuit of Figure P2.44. Assume that $R_1=2 \Omega, R V=R_2=R_0=4 \Omega, V S=4 V$, and $i_S=0.5$ A.

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Problem 46

Apply source transformations to find the voltage $V$ across the current source in Figure P2.46.

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Problem 47

Apply a single source transformation and then voltage division to find the magnitude of the voltage across $R_1$ in Figure P2.47. Let $R_1=10 \Omega, R_2=5 \Omega, V_1 =2 \mathrm{~V}, V_2=1 \mathrm{~V}, i_S=2 \mathrm{~A}$.

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Problem 48

Transform each of the three Thévenin sources to Norton sources and apply current division to find the current through $R_1$ in Eigure P2.48. Let $R_1=6 \Omega, R_2 =3 \Omega, R_3=3 \Omega, V_1=4 \mathrm{~V}, V_2=1 \mathrm{~V}, V 3=2 \mathrm{~V}$.

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Problem 49

Simplify the circuit in Figure P2.49 by applying source transformations to the right half of the circuit. Solve for the node voltage $v_1$.

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Problem 50

The circuit shown in Figure P2.50 is a simplified DC version of an AC threephase wye-wye (Y-Y) electrical distribution system commonly used to supply industrial loads, particularly rotating machines.
a. Determine the number of nonreference nodes.
b. Determine the number of unknown node voltages.
c. Apply source transformations to find $\sigma_e$.

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01:39

Problem 51

Apply source transformations to simplify the circuit in Eigure P2.21. Solve for the magnitude of the current through the $1 \Omega$ resistor.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator

Problem 52

Apply source transformations to reduce the one-port network on the left side of Eigure P2.82 to a Thévenin source. Apply voltage division to solve for the magnitude of the measured voltage when the voltmeter is attached.

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Problem 53

Find the Thévenin equivalent of the network seen by the $3-\Omega$ resistor in Eigure P2.53.

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Problem 54

Find the Thévenin equivalent of the network seen by the $3-\Omega$ resistor in Eigure P2,54. Use it and voltage division to find the voltage $v$ across the 3 - $\Omega$ resistor.

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Problem 55

Find the Norton equivalent of the network seen by $R_2$ in Figure P2.55. Use it and current division to compute the current $i$ through $R_2$. Assume $I_1=10 \mathrm{~A}, I_2= 2 \mathrm{~A}, V_1=6 \mathrm{~V}, R_1=3 \Omega$, and $R_2=4 \Omega$.

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Problem 56

Find the Norton equivalent of the network between nodes $a$ and $b$ in Figure P2,56

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05:39

Problem 57

Find the Thévenin equivalent of the network seen by $R$ in Eigure P2.57, and use the result to compute the current $i_R$. Assume $V_o=10 \mathrm{~V}, I_o=5 \mathrm{~A}, R_1=2 \Omega, R_2= 2 \Omega, R_3=4 \Omega$, and $R=3 \Omega$.

Narayan Hari
Narayan Hari
Numerade Educator
05:09

Problem 58

Find the Thévenin equivalent resistance seen by the load $R_o$ in Eigure P2.58

Thomas Thompson
Thomas Thompson
Numerade Educator
05:39

Problem 59

Find the Thévenin equivalent of the network seen by the load $R_o$ in Eigure P2.59.

Narayan Hari
Narayan Hari
Numerade Educator
05:09

Problem 60

Find the Thévenin equivalent network seen by the load $R_o$ in Eigure P2.60, where $R_1=10 \Omega, R_2=20 \Omega, R_g=0.1 \Omega$, and $R_p=1 \Omega$.

Thomas Thompson
Thomas Thompson
Numerade Educator
03:44

Problem 61

A Wheatstone bridge such as that shown in Figure P2.61 is used in numerous practical applications, such as determining the value of an unknown resistor $R_X$.
Find the Thévenin equivalent network seen by terminals $a$ and $b$ in terms of $R$, $R_X$, and $V_S$. Use it to find the value of $R_x$ when $R=1 \mathrm{k} \Omega, V_S=12 \mathrm{~V}$, and $V_{a b}=$ 12 mV .

Abhishek Jana
Abhishek Jana
Numerade Educator
03:44

Problem 62

Thévenin's theorem can be useful when dealing with a Wheatstone bridge. For the circuit of Eigure P2.62:
a. Express the Thévenin equivalent resistance seen by the load resistor $R_o$ in terms of $R_1, R_2, R_3$, and $R_X$.
b. Determine the Thévenin equivalent network seen by $R_o$. Apply voltage division and use the result to compute the power dissipated by $R_o$. Assume $R_o=500 \Omega, V_S=12 \mathrm{~V}, R_1=R_2=R_3=1 \mathrm{k} \Omega$, and $R_X=996 \Omega$.
c. When $R_o$ is replaced by an open-circuit, the Thévenin equivalent network supplies no power. What is the net power supplied by the entire Wheatstone bridge circuit when $R_o$ is replaced by an open-circuit? Are the results the same? What do you conclude?

Abhishek Jana
Abhishek Jana
Numerade Educator

Problem 63

Use source transformations to find the Thévenin equivalent network seen by resistor $R_3$ in the circuit of Figure P2.38. Assume $R_1=10 \Omega, R_2=4 \Omega, R_3=R_4= 6 \Omega, V_1=2 \mathrm{~V}, V_2=4 \mathrm{~V}$ and $I 1=2 \mathrm{~A}$.

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Problem 65

Find the Thévenin equivalent network seen by resistor $R_4$ in the circuit of Figure P2.33. Assume $R_1=2 \Omega, R_2=8 \Omega, R_3=12 \Omega, R_4=8 \Omega, V_1=12 \mathrm{~V}$ and $V_2=10 \mathrm{~V}$.

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01:39

Problem 66

Find the Thévenin equivalent network seen from node $a$ to node $b$ in Figure P2.66. Let $R_1=10 \Omega, R_2=8 \Omega, R_3=5 \Omega, R_4=4 \Omega, R_5=1 \Omega, V S=10 \mathrm{~V}, i_S=2 \mathrm{~A}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 67

Find the Thévenin equivalent network seen by $R_3$ in Figure P2,40. Compute the Thévenin (open-circuit) voltage $V_T$ in terms of the temperature $T$. Use that result to determine the temperature when $R_3$ is attached to that network.

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Problem 68

Find the Norton equivalent network seen by $R_5$ in Eigure P2.68. Use it and current division to compute the current through $R_5$. Assume $R_1=15 \Omega, R_2=8 \Omega$, $R_3=4 \Omega, R_4=4 \Omega, R_5=2 \Omega, I 1=2 \mathrm{~A}, I 2=3 \mathrm{~A}$.

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05:09

Problem 69

Find the Thévenin equivalent network seen by $R$ in Eigure P2.69. Use it and voltage division to compute the magnitude of the voltage across $R$. Assume:

Thomas Thompson
Thomas Thompson
Numerade Educator
04:26

Problem 70

Find the Norton equivalent network between terminals $a$ and $b$ in Figure P2,70 Let $R_1=6 \Omega, R_2=3 \Omega, R_3=2 \Omega, R_4=2 \Omega, V s=10 \mathrm{~V}, i_S=3 \mathrm{~A}$.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:06

Problem 71

Find the Norton equivalent network seen by $R_4$ in Eigure P2.71. Use it and current division to determine the current through $R_4$. Assume $R_1=8 \Omega, R_2=5 \Omega$, $R_3=4 \Omega, R_4=3 \Omega, V o=10 \mathrm{~V}$, and $I o=2 \mathrm{~A}$.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator

Problem 72

The Thévenin equivalent network seen by a load $R_o$ is depicted in Figure P2.72. Assume $V_T=10 \mathrm{~V}, R_T=2 \Omega$, and that the value of $R_o$ is such that maximum power is transferred to it. Determine:
a. The value of $R_o$.
b. The power $P_o$ dissipated by $R_o$.

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Problem 73

The Thévenin equivalent network seen by a load $R_o$ is depicted in Figure P2.72. Assume $V_T=25 \mathrm{~V}, R_T=100 \Omega$, and that the value of $R_o$ is such that maximum power is transferred to it. Determine:
a. The value of $R_o$.
b. The power $P_o$ dissipated by $R_o$.

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Problem 74

A practical voltage source is modeled in Figure P2.74 as an ideal source $V_S$ in series with a resistance $R_S$. This model accounts for internal power losses found in a real voltage source. The following data characterizes the real (nonideal) source:

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Problem 75

A practical voltage source is modeled in Figure P2.74 as an ideal source $V_S$ in series with a resistance $R_S$. This model accounts for internal power losses found in a real voltage source. A load $R$ is connected across the terminals of the model. Assume: $$V_s=12 \mathrm{~V} \quad R_s=0.3 \Omega$$ Plot the power dissipated in the load as a function of the load resistance. What can you conclude?

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08:05

Problem 76

Consider NiMH hobbyist batteries depicted in Figure P2.76.
a. If $V_1=12.0 \mathrm{~V}, R_1=0.15 \Omega$, and $R_o=2.55 \Omega$, find the load current $I_o$ and the power dissipated by the load.
b. If battery 2 with $V_2=12 \mathrm{~V}$ and $R_2=0.28 \Omega$ is placed in parallel with battery 1 , apply source transformations to determine whether the load current $I_o$ will increase or decrease. Will the power dissipated by the load increase or decrease? By how much?

Meghan Miholics
Meghan Miholics
Numerade Educator
02:02

Problem 77

A thermistor is a nonlinear device that changes its terminal resistance value as its surrounding temperature changes. The resistance and temperature generally have a relation in the form of:
a. If $R_0=300 \Omega$ and $\beta=-0.01 \mathrm{~K}-1$, plot $R_{\mathrm{th}}(T)$ as a function of the surrounding temperature $T$ for $350 \leq T \leq 750$.
b. If the thermistor is in parallel with a $250-\Omega$ resistor, find the expression for the equivalent resistance and plot $R_{\mathrm{th}}(T)$ on the same graph for part a.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:44

Problem 78

A moving-coil meter movement has a meter resistance $r_M=200 \Omega$, and fullscale deflection is caused by a meter current $i_m=10 \mu \mathrm{~A}$. The meter is to be used to display pressure, as measured by a sensor, up to a maximum of 100 kPa . Models of the meter and pressure sensor are shown in Eigure P2.78 Page 158 along with the relationship between measured pressure and the sensor output $v_o$ -
a. Devise a circuit that will produce the desired behavior of the meter, showing all appropriate connections between the terminals of the sensor and the meter.
b. Determine the value of each component in the circuit.
c. What is the linear range, that is, the minimum and maximum pressure that can accurately be measured?

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
01:55

Problem 79

A circuit that measures the internal resistance of a practical ammeter is shown in Eigure P2.79, where $R_S=50,000 \Omega, v_S=12 \mathrm{~V}$, and $R_p$ is a variable resistor that can be adjusted at will.
a. Assume that $r_a \ll 50,000 \Omega$. Estimate the current $i$.
b. If the meter displays a current of $150 \mu \mathrm{~A}$ when $R_p=15 \Omega$, find the internal resistance of the meter $r_{a^{-}}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
07:06

Problem 80

A practical voltmeter has an internal resistance $r_m$. What is the value of $r_m$ if the meter reads 11.81 V when connected as shown in Eigure P2.80? Assume $V_S=$ 12 V and $R_S=25 \mathrm{k} \Omega$.

Bruce Edelman
Bruce Edelman
Numerade Educator

Problem 81

Using the circuit of Eigure P2.80, find the voltage that the meter reads if $V_S=24$ V and $R_S$ has the following values: $R_S=0.2 r_m, 0.4 r_m, 0.6 r_m, 1.2 r_m, 4 r_m, 6 r_m$, and $10 r_m$. How large (or small) should the internal resistance of the meter be relative to $R_S$ ?

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05:41

Problem 82

A voltmeter is used to determine the voltage across a resistive element in the circuit of Figure P2.82. The instrument is modeled by an ideal voltmeter in parallel with a $120-\mathrm{k} \Omega$ resistor, as shown. The meter is placed to measure the voltage across $R_4$. Assume $R_1=8 \mathrm{k} \Omega, R_2=22 \mathrm{k} \Omega, R_3=50 \mathrm{k} \Omega, R_S=125 \mathrm{k} \Omega$, and $i_S=120 \mathrm{~mA}$. Find the voltage across $R_4$ with and without the voltmeter in the circuit for the following values:
a. $R_4=100 \Omega$
b. $R_4=1 \mathrm{k} \Omega$
c. $R_4=10 \mathrm{k} \Omega$
d. $R_4=100 \mathrm{k} \Omega$

Km Neeraj
Km Neeraj
Numerade Educator

Problem 83

An ammeter is used as shown in Eigure P2.83. The ammeter model consists of an ideal ammeter in series with a resistance. The ammeter model is placed in the branch as shown in the figure. Find the current through $R_5$ both with and without the ammeter in the circuit for the following values, assuming that Page $159 R_S=20 \Omega, R_1=800 \Omega, R_2=600 \Omega, R_3=1.2 \mathrm{k} \Omega, R_4=150 \Omega$, and $v_S=24 \mathrm{~V}$.
a. $R_5=1 \mathrm{k} \Omega$
b. $R_5=100 \Omega$
c. $R_5=10 \Omega$
c. $R_5=1 \Omega$

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Problem 84

Figure P2.84 shows an aluminum cantilevered beam loaded by the force $F$. Strain gauges $R_1, R_2, R_3$, and $R_4$ are attached to the beam as shown in Figure P2.84 and connected into the circuit shown. The force causes a tension stress on the top of the beam that causes the length (and therefore the resistance) of $R_1$ and $R_4$ to increase and a compression stress on the bottom of the beam that causes the length (and therefore the resistance) of $R_2$ and $R_3$ to decrease. The result is a voltage of 50 mV at node $B$ with respect to node $A$. Determine the force if

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Problem 85

Refer to Eigure P2.84 but assume that the cantilevered beam loaded by a force $F$ is made of steel. Strain gauges $R_1, R_2, R_3$, and $R_4$ are attached to the beam and connected in the circuit shown. The force causes a tension stress on the top of the beam that causes the length (and therefore the resistance) of $R_1$ and $R_4$ to increase and a compression stress on the bottom of the beam that causes the length (and therefore the resistance) of $R_2$ and $R_3$ to decrease. The result is a voltage $v_{B A}$ across nodes $B$ and $A$. Determine this voltage if $F=1.3 \mathrm{MN}$ and

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01:53

Problem 86

Apply nodal analysis to find two equations in terms of the node voltages $v_1$ and $v_2$ shown in Eigure P2.86 The two nonlinear resistors $R_a$ and $R_b$ are characterized by:
The resulting nonlinear (but not transcendental) equations cannot be solved by the methods used for simultaneous linear equations. While the equations can be solved analytically, consult your instructor before attempting to solve these equations.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 87

Many practical circuit elements are nonlinear; however, it is usually possible to linearize the V-I relationship near any specific point on the nonlinear V-I curve. Such a point is often referred to as an operating point. In other words, in the vicinity of an operating point [ $V_0, I_0$ ] the $V-I$ relationship can be linearly approximated by:
The inverse of the slope $m$ at the operating point is defined as incremental resistance $R_{\text {inc }}$ :
a. Refer to Figure P2.87 and find the operating point of the nonlinear element.
b. Find the incremental resistance of the nonlinear element at the operating point.
c. If $V_T$ is increased to 20 V , what is the new operating point and the new incremental resistance?

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Problem 88

The device $D$ in the circuit in Eigure P2.88 is an induction motor with a nonlinear $i-v$ characteristic. Determine the current through and the voltage across the motor.

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Problem 89

The nonlinear diode in Figure P2,89 has the $i-v$ characteristic shown. Assume:
Determine the voltage across and the current through the diode.

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Problem 91

The nonlinear device $D$ in Figure P2.91 has the following transcendental $i-v$ characteristic:
Assume that $V_S=2 \mathrm{~V}$ and $R=40 \Omega$. Determine an expression for the DC load line. Then use an iterative technique to determine the voltage across and current through $D$.

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Problem 92

The resistance of the device $D$ in Eigure P2.90 is a nonlinear function of pressure $P$. The $i-v$ characteristics of $D$ are shown for various pressures. Assume: $$V_g=3.0 \mathrm{~V} \quad R=100 \Omega$$ Construct the DC load line and determine the current through $D$ when $P=40$ psig.

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05:15

Problem 93

The so-called forward-bias $i-v$ relationship for a silicon diode is: $$i_0=I_{\mathrm{sar}}\left|e^{\left(t_{\mathrm{s}} / v_{n+2}\right)}-1\right|$$ where $I_{\text {SAT }}$ and $V_{\text {thermal }}$ are known as the saturation current and thermal voltage, respectively. At room temperature ( $20^{\circ} \mathrm{C}$ ): $$I_{\mathrm{SAT}}=10^{-12} \mathrm{~A} \text { and } V_{\mathrm{lam} \mathrm{~m}}=\frac{k T}{q}=253 \mathrm{mV}$$ where $k$ is Boltzmann's constant, $T$ is absolute temperature in kelvins, and $q$ is the charge of an electron.
Consider the circuit shown in Figure P2.93. KVL applied around the loop results in a transcendental equation for the loop current $i=i_D$. Such equations cannot be solved in terms of a closed-form expression $i=\ldots$. Instead, graphical or iterative procedures must be used.

Keshav Singh
Keshav Singh
Numerade Educator