?(a) Using superposition, determine the voltage labeled \(v_x\) in the circuit

For the circuit of Fig. 5.4, use superposition to compute the current \(i_x\).

For the circuit of Fig. 5.7, use superposition to obtain the voltage across each current source.

For the circuit of Fig. 5.18, compute the current \(I_X\) through the \(47\ k \Omega\) resistor after performing a source transformation on the voltage source.

Use Thévenin’s theorem to find the current through the \(2\ \Omega\) resistor in the circuit of Fig. 5.28. (Hint: Designate the \(2\ \Omega\) resistor as network B.)

Determine the Thévenin and Norton equivalents of the circuit of Fig. 5.30.

Find the Thévenin equivalent for the network of Fig. 5.32. (Hint: a quick source transformation on the dependent source might help.)

Find the Thévenin equivalent for the network of Fig. 5.39. (Hint: Try a 1 V test source.)

Consider the circuit of Fig. 5.43. (a) If \(R_{out} = 3\ k \Omega\), find the power delivered to it. (b) What is the maximum power that can be delivered to any \(R_{out}\)? (c) What two different values of \(R_{out}\) will have exactly 20 mW delivered to them?

Linear systems are so easy to work with that engineers often construct linear models of real (nonlinear) systems to assist in analysis and design. Such models are often surprisingly accurate over a limited range. For example, consider the simple exponential function \(e^x\). The Taylor series representation of this function is \(e^{x} \approx 1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\cdots\) (a) Construct a linear model for this function by truncating the Taylor series expansion after the linear term. (b) Evaluate your model function at x = 0.000001, 0.0001, 0.01, 0.1, and 1.0. (c) For which values of x does your model yield a “reasonable” approximation to \(e^x\)? Explain your reasoning.

Construct a linear approximation to the function y(t) = 4 sin 2t . (a) Evaluate your approximation at t = 0, 0.001, 0.01, 0.1, and 1.0. (b) For which values of t does your model provide a “reasonable” approximation to the actual (nonlinear) function y(t)? Explain your reasoning.

Considering the circuit of Fig. 5.48, employ superposition to determine the two components of \(i_8\) arising from the action of the two independent sources, respectively.

(a) Employ superposition to determine the current labeled i in the circuit of Fig. 5.49. (b) Express the contribution the 1 V source makes to the total current i in terms of a percentage. (c) Changing only the value of the 10 A source, adjust the circuit of Fig. 5.49 so that the two sources contribute equally to the current i.

(a) Employ superposition to obtain the individual contributions each of the two sources in Fig. 5.50 makes to the current labeled \(i_x\). (b) Adjusting only the value of the rightmost current source, alter the circuit so that the two sources contribute equally to \(i_x\).

(a) Determine the individual contributions of each of the two current sources in the circuit of Fig. 5.51 to the nodal voltage \(v_1\). (b) Determine the percentage contribution of each of the two sources to the power dissipated by the \(2\ \Omega\) resistor.

(a) Determine the individual contributions of each of the two current sources shown in Fig. 5.52 to the nodal voltage labeled \(v_2\). (b) Instead of performing two separate PSpice simulations, verify your answer by using a single dc sweep. Submit a labeled schematic, relevant Probe output, and a short description of the results.

After studying the circuit of Fig. 5.53, change both voltage source values such that (a) \(i_1\) doubles; (b) the direction of \(i_1\) reverses, but its magnitude is unchanged; (c) both sources contribute equally to the power dissipated by the \(6\ \Omega\) resistor.

Consider the three circuits shown in Fig. 5.54. Analyze each circuit, and demonstrate that \(V_{x}=V_{x}^{\prime}+V_{x}^{\prime \prime}\) (i.e., superposition is most useful when sources are set to zero, but the principle is in fact much more general than that).

Employ superposition principles to obtain a value for the current \(I_x\) as labeled in Fig. 5.56.

(a) Employ superposition to determine the individual contribution from each independent source to the voltage v as labeled in the circuit shown in Fig. 5.57. (b) Compute the power absorbed by the \(2\ \Omega\) resistor.

For the circuit of Fig. 5.59, plot \(i_L\) versus \(v_L\) corresponding to the range of \(0\ \leq\ R\ \leq\ \infty\).

Determine the current labeled I in the circuit of Fig. 5.60 by first performing source transformations and parallel-series combinations as required to reduce the circuit to only two elements.

Verify that the power absorbed by the \(7\ \Omega\) resistor in Fig. 5.22a remains the same after the source transformation illustrated in Fig. 5.22c.

(a) Determine the current labeled i in the circuit of Fig. 5.61 after first transforming the circuit such that it contains only resistors and voltage sources. (b) Simulate each circuit to verify the same current flows in both cases.

(a) Using repeated source transformations, reduce the circuit of Fig. 5.62 to a voltage source in series with a resistor, both of which are in series with the \(6\ M \Omega\) resistor. (b) Calculate the power dissipated by the \(6\ M \Omega\) resistor using your simplified circuit.

(a) Using as many source transformations and element combination techniques as required, simplify the circuit of Fig. 5.63 so that it contains only the 7 V source, a single resistor, and one other voltage source. (b) Verify that the 7 V source delivers the same amount of power in both circuits.

(a) Making use of repeated source transformations, reduce the circuit of Fig. 5.64 such that it contains a single voltage source, the \(17\ \Omega\) resistor, and one other resistor. (b) Calculate the power dissipated by the \(17\ \Omega\) resistor. (c) Verify your results by simulating both circuits with PSpice or another suitable CAD tool.

Make use of source transformations to first convert all three sources in Fig. 5.65 to voltage sources, then simplify the circuit as much as possible and calculate the voltage \(V_x\) which appears across the \(4\ \Omega\) resistor. Be sure to draw and label your simplified circuit.

(a) With the assistance of source transformations, alter the circuit of Fig. 5.66 such that it contains only current sources. (b) Simplify your new circuit as much as possible, and calculate the power dissipated in the \(7\ \Omega\) resistor. (c) Verify your solution by simulating both circuits with PSpice or another appropriate CAD tool.

Transform the dependent source in Fig. 5.67 to a voltage source, then calculate \(V_0\).

With regard to the circuit represented in Fig. 5.68, first transform both voltage sources to current sources, reduce the number of elements as much as possible, and determine the voltage \(v_3\).

Referring to Fig. 5.69, determine the Thévenin equivalent of the network connected to \(R_L\). (b) Determine \(v_L\) for \(R_L = 1\ \Omega,\ 3.5\ \Omega,\ 6.257\ \Omega,\ \text{and}\ 9.8\ \Omega\)..

(a) With respect to the circuit depicted in Fig. 5.69, obtain the Norton equivalent of the network connected to \(R_L\). (b) Plot the power dissipated in resistor \(R_L\) as a function of \(i_L\) corresponding to the range of \(0\ <\ R_L\ <\ 5\ \Omega\). (c) Using your graph, estimate at what value of \(R_L\) does the dissipated power reach its maximum value.

(a) Obtain the Norton equivalent of the network connected to \(R_L\) in Fig. 5.70. (b) Obtain the Thévenin equivalent of the same network. (c) Use either to calculate \(i_L\) for \(R_L = 0\ \Omega,\ 1\ \Omega,\ 4.923\ \Omega\), and \(8.107\ \Omega\).

(a) Determine the Thévenin equivalent of the circuit depicted in Fig. 5.71 by first finding \(V_{oc}\) and \(I_{sc}\) (defined as flowing into the positive reference terminal of \(V_{oc}\)). (b) Connect a \(4.7\ k \Omega\) resistor to the open terminals of your new network and calculate the power it dissipates.

Referring to the circuit of Fig. 5.71: (a) Determine the Norton equivalent of the circuit by first finding \(V_{oc}\) and \(I_{sc}\) (defined as flowing into the positive reference terminal of \(V_{oc}\)). (b) Connect a \(1.7\ k \Omega\) resistor to the open terminals of your new network and calculate the power supplied to that resistor.

(a) Employ Thévenin’s theorem to obtain a two-component equivalent for the network shown in Fig. 5.73. (b) Determine the power supplied to a \(1\ M \Omega\) resistor connected to the network if \(i_1 = 19\ \mu A\), \(R_1 = R_2 = 1.6\ M \Omega\), \(R_2 = 3\ M \Omega\), and \(R_4 = R_5 = 1.2\ M \Omega\). (c) Verify your solution by simulating both circuits with PSpice or another appropriate CAD tool.

Determine the Thévenin equivalent of the network shown in Fig. 5.74 as seen looking into the two open terminals.

(a) Determine the Norton equivalent of the circuit depicted in Fig. 5.74 as seen looking into the two open terminals. (b) Compute power dissipated in a \(5\ \Omega\) resistor connected in parallel with the existing \(5\ \Omega\) resistor. (c) Compute the current flowing through a short circuit connecting the two terminals.

For the circuit of Fig. 5.75: (a) Employ Norton’s theorem to reduce the network connected to \(R_L\) to only two components. (b) Calculate the downward directed current flowing through \(R_L\) if it is a \(3.3\ k \Omega\) resistor. (c) Verify your answer by simulating both circuits with PSpice or a comparable CAD tool.

(a) Obtain a value for the Thévenin equivalent resistance seen looking into the open terminals of the circuit in Fig. 5.76 by first finding \(V_{oc}\) and \(I_{sc}\). (b) Connect a 1 A test source to the open terminals of the original circuit after shorting the voltage source, and use this to obtain \(R_{TH}\). (c) Connect a 1 V test source to the open terminals of the original circuit after again zeroing the 2 V source, and use this now to obtain \(R_{TH}\).

Refer to the circuit depicted in Fig. 5.77. (a) Obtain a value for the Thévenin equivalent resistance seen looking into the open terminals by first finding \(V_{oc}\) and \(I_{sc}\). (b) Connect a 1 A test source to the open terminals of the original circuit after deactivating the other current source, and use this to obtain \(R_{TH}\). (c) Connect a 1 V test source to the open terminals of the original circuit, once again zeroing out the original source, and use this now to obtain \(R_{TH}\).

Obtain a value for the Thévenin equivalent resistance seen looking into the open terminals of the circuit in Fig. 5.78 by (a) finding \(V_{oc}\) and \(I_{sc}\), and then taking their ratio; (b) setting all independent sources to zero and using resistor combination techniques; (c) connecting an unknown current source to the terminals, deactivating (zero out) all other sources, finding an algebraic expression for the voltage that develops across the source, and taking the ratio of the two quantities.

With regard to the network depicted in Fig. 5.79, determine the Thévenin equivalent as seen by an element connected to terminals (a) a and b; (b) a and c; (c) b and c. (d) Verify your answers using PSpice or other suitable CAD tool. (Hint: Connect a test source to the terminals of interest.)

Determine the Thévenin and Norton equivalents of the circuit represented in Fig. 5.80 from the perspective of the open terminals. (There should be no dependent sources in your answer.)

Determine the Norton equivalent of the circuit drawn in Fig. 5.81 as seen by terminals a and b. (There should be no dependent sources in your answer.)

With regard to the circuit of Fig. 5.82, determine the power dissipated by (a) a \(1\ k \Omega\) resistor connected between a and b; (b) a \(4.7\ k \Omega\) resistor connected between a and b; (c) a \(10.54\ k \Omega\) resistor connected between a and b.

Determine the Thévenin and Norton equivalents of the circuit shown in Fig. 5.83, as seen by an unspecified element connected between terminals a and b.

Referring to the circuit of Fig. 5.84, determine the Thévenin equivalent resistance of the circuit to the right of the dashed line. This circuit is a common source transistor amplifier, and you are calculating its input resistance.

Referring to the circuit of Fig. 5.85, determine the Thévenin equivalent resistance of the circuit to the right of the dashed line. This circuit is a common-collector transistor amplifier, and you are calculating its input resistance.

The circuit shown in Fig. 5.86 is a reasonably accurate model of an operational amplifier. In cases where \(R_i\) and A are very large and \(R_{o} \sim 0\), a resistive load (such as a speaker) connected between ground and the terminal labeled \(v_{\mathrm{out}}\) will see a voltage \(?R_f /R_1\) times larger than the input signal \(v_{\mathrm{in}}\). Find the Thévenin equivalent of the circuit, taking care to label \(v_{\mathrm{out}}\).

(a) For the simple circuit of Fig. 5.87, graph the power dissipated by the resistor R as a function of \(R/R_S\), if \(0\ \leq\ R\ \leq\ 3000\ \Omega\). (b) Graph the first derivative of the power versus \(R/R_S\), and verify that maximum power is transferred to R when it is equal to \(R_S\).

For the circuit drawn in Fig. 5.88, (a) determine the Thévenin equivalent connected to \(R_{out}\). (b) Choose \(R_{out}\) such that maximum power is delivered to it.

Study the circuit of Fig. 5.89. (a) Determine the Norton equivalent connected to resistor \(R_{out}\). (b) Select a value for \(R_{out}\) such that maximum power will be delivered to it.

Assuming that we can determine the Thévenin equivalent resistance of our wall socket, why don’t toaster, microwave oven, and TV manufacturers match each appliance’s Thévenin equivalent resistance to this value? Wouldn’t it permit maximum power transfer from the utility company to our household appliances?

For the circuit of Fig. 5.90, what value of \(R_L\) will ensure it absorbs the maximum possible amount of power?

With reference to the circuit of Fig. 5.91, (a) calculate the power absorbed by the \(9\ \Omega\) resistor; (b) adjust the size of the \(5\ \Omega\) resistor so that the new network delivers maximum power to the \(9\ \Omega\) resistor.

Select a value for \(R_L\) in Fig. 5.93 such that it is ensured to absorb maximum power from the circuit.

Determine what value of resistance would absorb maximum power from the circuit of Fig. 5.94 when connected across terminals a and b.

Derive the equations required to convert from a Y-connected network to a \(\Delta\)-connected network.

Convert the \(\Delta-\) (or “\(\text {?}-\)”) connected networks in Fig. 5.95 to Y-connected networks.

Convert the Y- (or “T-”) connected networks in Fig. 5.96 to \(\Delta\)-connected networks.

For the network of Fig. 5.97, select a value of R such that the network has an equivalent resistance of \(9\ \Omega\). Round your answer to two significant figures.

For the network of Fig. 5.98, select a value of R such that the network has an equivalent resistance of \(70.6\ \Omega\).

Determine the effective resistance \(R_{\mathrm {in}}\) of the network exhibited in Fig. 5.99.

Calculate \(R_{in}\) as indicated in Fig. 5.100.

Employ \(\Delta\)/Y conversion techniques as appropriate to determine \(R_{\mathrm {in}}\) as labeled in Fig. 5.101.

(a) Determine the two-component Thévenin equivalent of the network in Fig. 5.102. (b) Calculate the power dissipated by a \(1\ \Omega\) resistor connected between the open terminals.

(a) Use appropriate techniques to obtain both the Thévenin and Norton equivalents of the network drawn in Fig. 5.103. (b) Verify your answers by simulating each of the three circuits connected to a \(1\ \Omega\) resistor.

(a) Replace the network in Fig. 5.104 with an equivalent three-resistor \(\Delta\) network. (b) Perform a PSpice analysis to verify that your answer is in fact equivalent. (Hint: Try adding a load resistor.)

Determine the power absorbed by a resistor connected between the open terminal of the circuit shown in Fig. 5.105 if it has a value of (a) \(1\ \Omega\); (b) \(100\ \Omega\) ; (c) \(2.65\ k \Omega\); (d) \(1.13\ M \Omega\).

It is known that a load resistor of some type will be connected between terminals a and b of the network of Fig. 5.106. (a) Change the value of the 25 V source such that both voltage sources contribute equally to the power delivered to the load resistor, assuming its value is chosen such that it absorbs maximum power. (b) Calculate the value of the load resistor.

A \(2.57\ \Omega\) load is connected between terminals a and b of the network drawn in Fig. 5.106. Unfortunately, the power delivered to the load is only 50% of the required amount. Altering only voltage sources, modify the circuit so that the required power is delivered and both sources contribute equally.

A load resistor is connected across the open terminals of the circuit shown in Fig. 5.107, and its value was chosen carefully to ensure maximum power transfer from the rest of the circuit. (a) What is the value of the resistor? (b) If the power absorbed by the load resistor is three times as large as required, modify the circuit so that it performs as desired, without losing the maximum power transfer condition already enjoyed.

A backup is required for the circuit depicted in Fig. 5.107. It is unknown what will be connected to the open terminals, or whether it will be purely linear. If a simple battery is to be used, what no-load (“open circuit”) voltage should it have, and what is the maximum tolerable internal resistance?

Three 45 W light bulbs originally wired in a Y network configuration with a 120 V ac source connected across each port are rewired as a \(\Delta\) network. The neutral, or center, connection is not used. If the intensity of each light is proportional to the power it draws, design a new 120 V ac power circuit so that the three lights have the same intensity in the \(\Delta\) configuration as they did when connected in a Y configuration. Verify your design using PSpice by comparing the power drawn by each light in your circuit (modeled as an appropriately chosen resistor value) with the power each would draw in the original Y-connected circuit.

(a) Explain in general terms how source transformation can be used to simplify a circuit prior to analysis. (b) Even if source transformations can greatly simplify a particular circuit, when might it not be worth the effort? (c) Multiplying all the independent sources in a circuit by the same scaling factor results in all other voltages and currents being scaled by the same amount. Explain why we don’t scale the dependent sources as well. (d) In a general circuit, if we set an independent voltage source to zero, what current can flow through it? (e) In a general circuit, if we set an independent current source to zero, what voltage can be sustained across its terminals?

The load resistor in Fig. 5.108 can safely dissipate up to 1 W before overheating and bursting into flame. The lamp can be treated as a \(10.6\ \Omega\) resistor if less than 1 A flows through it and a \(15\ \Omega\) resistor if more than 1 A flows through it. What is the maximum permissible value of \(I_s\)? Verify your answer with PSpice.

A certain red LED has a maximum current rating of 35 mA, and if this value is exceeded, overheating and catastrophic failure will result. The resistance of the LED is a nonlinear function of its current, but the manufacturer warrants a minimum resistance of \(47\ \Omega\) and a maximum resistance of \(117\ \Omega\). Only 9 V batteries are available to power the LED. Design a suitable circuit to deliver the maximum power possible to the LED without damaging it. Use only combinations of the standard resistor values given in the inside front cover.

As part of a security system, a very thin \(100\ \Omega\) wire is attached to a window using nonconducting epoxy. Given only a box of 12 rechargeable 1.5 V AAA batteries, one thousand \(1\ \Omega\) resistors, and a 2900 Hz piezo buzzer that draws 15 mA at 6 V, design a circuit with no moving parts that will set off the buzzer if the window is broken (and hence the thin wire as well). Note that the buzzer requires a dc voltage of at least 6 V (maximum 28 V) to operate.

Use the technique of \(Y- \Delta\) conversion to find the Thévenin equivalent resistance of the circuit of Fig. 5.47.

(a) Using superposition, determine the voltage labeled \(v_x\) in the circuit represented in Fig. 5.55. (b) To what value should the 2 A source be changed to reduce \(v_x\) by 10%? (c) Verify your answers by performing three dc sweeps in PSpice (one for each source). Submit a labeled schematic, relevant Probe output, and a short description of the results.

Perform an appropriate source transformation on each of the circuits depicted in Fig. 5.58, taking care to retain the \(4\ \Omega\) resistor in each final circuit.

(a) Employ Thévenin’s theorem to obtain a simple two-component equivalent of the circuit shown in Fig. 5.72. (b) Use your equivalent circuit to determine the power delivered to a \(100\ \Omega\) resistor connected to the open terminals. (c) Verify your solution by analyzing the original circuit with the same \(100\ \Omega\) resistor connected across the open terminals.

Referring to the circuit of Fig. 5.92, (a) determine the power absorbed by the \(3.3\ \Omega\) resistor; (b) replace the \(3.3\ \Omega\) resistor with another resistor such that it absorbs maximum power from the rest of the circuit.

Use the technique of \(Y- \Delta\) conversion to find the Thévenin equivalent resistance of the circuit of Fig. 5.47.

(a) Using superposition, determine the voltage labeled \(v_x\) in the circuit represented in Fig. 5.55. (b) To what value should the 2 A source be changed to reduce \(v_x\) by 10%? (c) Verify your answers by performing three dc sweeps in PSpice (one for each source). Submit a labeled schematic, relevant Probe output, and a short description of the results.

Perform an appropriate source transformation on each of the circuits depicted in Fig. 5.58, taking care to retain the \(4\ \Omega\) resistor in each final circuit.

(a) Employ Thévenin’s theorem to obtain a simple two-component equivalent of the circuit shown in Fig. 5.72. (b) Use your equivalent circuit to determine the power delivered to a \(100\ \Omega\) resistor connected to the open terminals. (c) Verify your solution by analyzing the original circuit with the same \(100\ \Omega\) resistor connected across the open terminals.

Referring to the circuit of Fig. 5.92, (a) determine the power absorbed by the \(3.3\ \Omega\) resistor; (b) replace the \(3.3\ \Omega\) resistor with another resistor such that it absorbs maximum power from the rest of the circuit.