?Although drawn so that it may not appear obvious at first glance, the circuit of Fig

Count the number of branches and nodes in the circuit in Fig. 3.4. If \(i_x = 3\ A\) and the 18 V source delivers 8 A of current, what is the value of \(R_A\)? (Hint: You need Ohm’s law as well as KCL.)

Determine \(i_x\) and \(v_x\) in the circuit of Fig. 3.7.

For the circuit of Fig. 3.9, determine (a) \(v_{R2}\) and (b) \(v_2\), if \(v_{R1} = 1\ V\).

Determine \(v_x\) in the circuit of Fig. 3.11.

In the circuit of Fig. 3.12b, \(v_{s1} = 120\ V\), \(v_{s2} = 30\ V\), \(R_1 = 30\ \Omega\) , and \(R_2 = 15\ \Omega\) . Compute the power absorbed by each element.

In the circuit of Fig. 3.14, find the power absorbed by each of the five elements in the circuit.

Determine v in the circuit of Fig. 3.16.

For the single-node-pair circuit of Fig. 3.18, find \(i_A\), \(i_B\) , and \(i_C\).

Determine the current i in the circuit of Fig. 3.21 after first replacing the four sources with a single equivalent source.

Determine the voltage v in the circuit of Fig. 3.23 after first replacing the three sources with a single equivalent source.

Determine whether the circuit of Fig. 3.25 violates either of Kirchhoff’s laws.

Determine i in the circuit of Fig. 3.28.

Determine v in the circuit of Fig. 3.30 by first combining the three current sources, and then the two \(10\ \Omega\) resistors.

For the circuit of Fig. 3.32, calculate the voltage \(v_x\).

Use voltage division to determine \(v_x\) in the circuit of Fig. 3.36.

In the circuit of Fig. 3.39, use resistance combination methods and current division to find \(i_1\), \(i_2\), and \(v_3\).

Referring to the circuit depicted in Fig. 3.44, count the number of (a) nodes; (b) elements; (c) branches.

Referring to the circuit depicted in Fig. 3.45, count the number of (a) nodes; (b) elements; (c) branches.

For the circuit of Fig. 3.46: (a) Count the number of nodes. (b) In moving from A to B, have we formed a path? Have we formed a loop? (c) In moving from C to F to G, have we formed a path? Have we formed a loop?

For the circuit of Fig. 3.46: (a) Count the number of circuit elements. (b) If we move from B to C to D, have we formed a path? Have we formed a loop? (c) If we move from E to D to C to B, have we formed a path? Have we formed a loop?

Refer to the circuit of Fig. 3.47, and answer the following: (a) How many distinct nodes are contained in the circuit? (b) How many elements are contained in the circuit? (c) How many branches does the circuit have? (d) Determine if each of the following represents a path, a loop, both, or neither: (i) A to B (ii) B to D to C to E (iii) C to E to D to B to A to C (iv) C to D to B to A to C to E

Referring to the single node diagram of Fig. 3.49, compute: (a) \(i_B\), if \(i_A = 1\ A\), \(i_D = -2\ A\), \(i_C = 3\ A\), and \(i_E = 0\); (b) \(i_E\), if \(i_A = -1\ A\), \(i_B = -1\ A\), \(i_C = -1\ A\), and \(i_D = -1\ A\).

Determine the current labeled I in each of the circuits of Fig. 3.50.

In the circuit shown in Fig. 3.51, the resistor values are unknown, but the 2 V source is known to be supplying a current of 7 A to the rest of the circuit. Calculate the current labeled \(i_2\).

The voltage source in the circuit of Fig. 3.52 has a current of 1 A flowing out of its positive terminal into resistor \(R_1\). Calculate the current labeled \(i_2\).

In the circuit depicted in Fig. 3.53, \(i_x\) is determined to be 1.5 A, and the 9 V source supplies a current of 7.6 A (that is, a current of 7.6 A leaves the positive reference terminal of the 9 V source). Determine the value of resistor \(R_A\).

Determine the current labeled \(I_3\) in the circuit of Fig. 3.55.

Study the circuit depicted in Fig. 3.56, and explain (in terms of KCL) why the voltage labeled \(V_x\) must be zero.

In many households, multiple electrical outlets within a given room are often all part of the same circuit. Draw the circuit for a four-walled room which has a single electrical outlet per wall, with a lamp (represented by a \(1\ \Omega\) resistor) connected to each outlet.

For the circuit of Fig. 3.57: (a) Determine the voltage \(v_1\) if \(v_2 = 0\) and \(v_3 = -17\ V\). (b) Determine the voltage \(v_1\) if \(v_2 = -2\ V\) and \(v_3 = +2\ V\). (c) Determine the voltage \(v_2\) if \(v_1 = 7\ V\) and \(v_3 = 9\ V\). (d) Determine the voltage \(v_3\) if \(v_1 = -2.33\ V\) and \(v_2 = -1.70\ V\).

For each of the circuits in Fig. 3.58, determine the voltage \(v_x\) and the current \(i_x\).

Use KVL to obtain a numerical value for the current labeled i in each circuit depicted in Fig. 3.59.

In the circuit of Fig. 3.60, it is determined that \(v_1 = 3\ V\) and \(v_3 = 1.5\ V\). Calculate \(v_R\) and \(v_2\).

In the circuit of Fig. 3.60, a voltmeter is used to measure the following: \(v_1 = 2\ V\) and \(v_3 = -1.5\ V\). Calculate \(v_x\).

Determine the value of \(v_x\) as labeled in the circuit of Fig. 3.61.

Consider the simple circuit shown in Fig. 3.62. Using KVL, derive the expressions \(v_{1}=v_{s} \frac{R_{1}}{R_{1}+R_{2}} \quad \text { and } \quad v_{2}=v_{s} \frac{R_{2}}{R_{1}+R_{2}}\)

(a) Determine a numerical value for each current and voltage (\(i_1\), \(v_1\), etc.) in the circuit of Fig. 3.63. (b) Calculate the power absorbed by each element and verify that they sum to zero.

The circuit shown in Fig. 3.64 includes a device known as an op amp. This device has two unusual properties in the circuit shown: (1) \(V_d = 0\ V\), and (2) no current can flow into either input terminal (marked “?” and “+” inside the symbol), but it can flow through the output terminal (marked “OUT”). This seemingly impossible situation—in direct conflict with KCL—is a result of power leads to the device that are not included in the symbol. Based on this information, calculate \(V_{out}\). (Hint: two KVL equations are required, both involving the 5 V source.)

The circuit of Fig. 3.12b is constructed with the following: \(v_{s1}=-8\mathrm{\ V}\), \(R_{1}=1 \Omega\), \(v_{s2}=16\mathrm{\ V}\), and \(R_{2}=4.7 \Omega\). Calculate the power absorbed by each element. Verify that the absorbed powers sum to zero.

Obtain a numerical value for the power absorbed by each element in the circuit shown in Fig. 3.65.

Compute the power absorbed by each element of the circuit of Fig. 3.66.

Compute the power absorbed by each element in the circuit of Fig. 3.67 if the mysterious element X is (a) a \(13\ \Omega\) resistor; (b) a dependent voltage source labeled \(4v_{1}\), “+” reference on top; (c) a dependent voltage source labeled \(4i_x\), “+” reference on top.

Kirchhoff’s laws apply whether or not Ohm’s law applies to a particular element. The I-V characteristic of a diode, for example, is given by \(I_{D}=I_{S}\left(e^{V_{D} / V_{T}}-1\right)\) where \(V_T = 27\ mV\) at room temperature and \(I_S\) can vary from \(10^{?12}\) to \(10^{?3}\ A\). In the circuit of Fig. 3.68, use KVL/KCL to obtain \(V_D\) if \(I_S = 29\ pA\). (Note: This problem results in a transcendental equation, requiring an iterative approach to obtaining a numerical solution. Most scientific calculators will perform such a function.)

Referring to the circuit of Fig. 3.69, (a) determine the two currents \(i_1\) and \(i_2\); (b) compute the power absorbed by each element.

Determine a value for the voltage v as labeled in the circuit of Fig. 3.70, and compute the power supplied by the two current sources.

Referring to the circuit depicted in Fig. 3.71, determine the value of the voltage v.

Determine the voltage v as labeled in Fig. 3.72, and calculate the power supplied by each current source.

Determine the numerical value for \(v_{eq}\) in Fig. 3.74a, if (a) \(v_1 = 0\), \(v_2 = -3\ V\), and \(v_3 = +3\ V\); (b) \(v_1 = v_2 = v_3 = 1\ V\); (c) \(v_1 = -9\ V\), \(v_2 = 4.5\ V\), \(v_3 = 1\ V\).

Determine the numerical value for \(i_{eq}\) in Fig. 3.74b, if (a) \(i_1 = 0\), \(i_2 = -3\ A\), and \(i_3 = +3\ A\); (b) \(i_1 = i_2 = i_3 = 1\ A\); (c) \(i_1 = -9\ A\), \(i_2 = 4.5\ A\), \(i_3 = 1\ A\).

For the circuit presented in Fig. 3.75, determine the current labeled i by first combining the four sources into a single equivalent source.

Determine the value of \(v_1\) required to obtain a zero value for the current labeled i in the circuit of Fig. 3.76.

(a) For the circuit of Fig. 3.77, determine the value for the voltage labeled v, after first simplifying the circuit to a single current source in parallel with two resistors. (b) Verify that the power supplied by your equivalent source is equal to the sum of the supplied powers of the individual sources in the original circuit.

(a) Determine the values for \(I_X\) and \(V_Y\) in the circuit shown in Fig. 3.79. (b) Are those values necessarily unique for that circuit? Explain. (c) Simplify the circuit of Fig. 3.79 as much as possible and still maintain the values for v and i. (Your circuit must contain the \(1\ \Omega\) resistor.)

Determine the equivalent resistance of each of the networks shown in Fig. 3.80.

For each network depicted in Fig. 3.81, determine a single equivalent resistance.

(a) Simplify the circuit of Fig. 3.82 as much as possible by using source and resistor combinations. (b) Calculate i, using your simplified circuit. (c) To what voltage should the 1 V source be changed to reduce i to zero? (d) Calculate the power absorbed by the \(5\ \Omega\) resistor.

(a) Simplify the circuit of Fig. 3.83, using appropriate source and resistor combinations. (b) Determine the voltage labeled v, using your simplified circuit. (c) Calculate the power provided by the 2 A source to the rest of the circuit.

Making appropriate use of resistor combination techniques, calculate \(i_3\) in the circuit of Fig. 3.84 and the power provided to the circuit by the single current source.

Calculate the voltage labeled \(v_x\) in the circuit of Fig. 3.85 after first simplifying, using appropriate source and resistor combinations.

Determine the power absorbed by the \(15 \Omega\) resistor in the circuit of Fig. 3.86.

Calculate the equivalent resistance \(R_{eq}\) of the network shown in Fig. 3.87 if \(R_1 = 2R_2 = 3R_3 = 4R_4\) etc. and \(R_{11} = 3 \Omega\).

Show how to combine four \(100 \Omega\) resistors to obtain an equivalent resistance of (a) \(25 \Omega\); (b) \(60 \Omega\); (c) \(40 \Omega\).

In the voltage divider network of Fig. 3.88, calculate (a) \(v_2\) if v = 9.2 V and \(v_1 = 3\ V\); (b) \(v_1\) if \(v_2 = 1\ V\) and v = 2 V; (c) v if \(v_1 = 3\ V\) and \(v_2 = 6\ V\); (d) \(R_1/R_2\) if \(v_1 = v_2\); (e) \(v_2\) if v = 3.5 V and \(R_1 = 2R_2\); (f ) \(v_1\) if v = 1.8 V, \(R_1 = 1\ k \Omega\), and \(R_2 = 4.7\ k \Omega\).

Choose a voltage v < 2.5 V and values for the resistors \(R_1\), \(R_2\), \(R_3\), and \(R_4\) in the circuit of Fig. 3.90 so that \(i_1 =1\ A\), \(i_2 =1.2\ A\), \(i_3 = 8\ A\), and \(i_4 = 3.1\ A\).

Employ voltage division to assist in the calculation of the voltage labeled \(v_x\) in the circuit of Fig. 3.91.

A network is constructed from a series connection of five resistors having values \(1 \Omega\), \(3 \Omega\), \(5 \Omega\), \(7 \Omega\), and \(9 \Omega\). If 9 V is connected across the terminals of the network, employ voltage division to calculate the voltage across the \(3 \Omega\) resistor, and the voltage across the \(7 \Omega\) resistor.

Employing resistance combination and current division as appropriate, determine values for \(i_1\), \(i_2\), and \(v_3\) in the circuit of Fig. 3.92.

In the circuit of Fig. 3.93, only the voltage \(v_x\) is of interest. Simplify the circuit using appropriate resistor combinations and iteratively employ voltage division to determine \(v_x\).

A common midfrequency model for a field effect–based amplifier circuit is shown in Fig. 3.95. If the controlling parameter \(g_m\) (known as the transconductance) is equal to 1.2 mS, employ current division to obtain the current through the \(1\ k \Omega\) resistor, and then calculate the amplifier output voltage \(v_{out}\).

The circuit depicted in Fig. 3.96 is routinely employed to model the midfrequency operation of a bipolar junction transistor–based amplifier. Calculate the amplifier output \(v_{out}\) if the transconductance \(g_m\) is equal to 322 mS.

With regard to the circuit shown in Fig. 3.97, compute (a) the voltage across the two \(10 \Omega\) resistors, assuming the top terminal is the positive reference; (b) the power dissipated by the \(4 \Omega\) resistor.

Delete the leftmost \(10 \Omega\) resistor in the circuit of Fig. 3.97, and compute (a) the current flowing into the left-hand terminal of the \(40 \Omega\) resistor; (b) the power supplied by the 2 V source; (c) the power dissipated by the \(4 \Omega\) resistor.

Although drawn so that it may not appear obvious at first glance, the circuit of Fig. 3.73 is in fact a single-node-pair circuit. (a) Determine the power absorbed by each resistor. (b) Determine the power supplied by each current source. (c) Show that the sum of the absorbed power calculated in (a) is equal to the sum of the supplied power calculated in (c).

Consider the seven-element circuit depicted in Fig. 3.98. (a) How many nodes, loops, and branches does it contain? (b) Calculate the current flowing through each resistor. (c) Determine the voltage across the current source, assuming the top terminal is the positive reference terminal.

Although drawn so that it may not appear obvious at first glance, the circuit of Fig. 3.73 is in fact a single-node-pair circuit. (a) Determine the power absorbed by each resistor. (b) Determine the power supplied by each current source. (c) Show that the sum of the absorbed power calculated in (a) is equal to the sum of the supplied power calculated in (c).

Consider the seven-element circuit depicted in Fig. 3.98. (a) How many nodes, loops, and branches does it contain? (b) Calculate the current flowing through each resistor. (c) Determine the voltage across the current source, assuming the top terminal is the positive reference terminal.