Unit 3A – Resistors in Series and in Parallel; Voltage ...



Chapter 2 – Circuit Elements

Study Guide

Objectives:

1. Understand the symbols for and the behavior of the following ideal basic circuit elements: independent voltage and current sources, dependent voltage and current sources, and resistors.

2. Be able to state Ohm’s law, Kirchhoff’s current law, and Kirchhoff’s voltage law, and be able to use these laws to analyze simple circuits.

3. Know how to calculate the power for each element in a simple circuit and be able to determine whether or not the power balances for the whole circuit.

Mastering the Objectives:

1. Read the Introduction and Section 2.1.

a) Plot the voltage as a function of current in the independent voltage source shown:

The plot shows that it is not possible to determine the current through an independent voltage source if all you know is the value of the voltage.

b) Plot the current as a function of the voltage drop for the independent current source shown:

The plot shows that you cannot determine the voltage drop across an independent current source if all you know is the value of the current.

c) Plot the voltage vs as a function of the controlling current, ix, for the current-controlled voltage source shown:

Now plot the voltage vs as a function of the current is through the dependent voltage source if the controlling current ix = 2 V.

Compare these two plots.

d) Give the units for the following variables in Fig. 2.2:

( ______ ( ______ ( ______ ( ______

e) What change could be made in Figs. 2.3(b) and (c) to make the interconnection valid?

f) Show that the power generated equals the power absorbed in the circuit of Fig. 2.3(e).

g) What change could be made in Figs. 2.4(a) and (d) to make the interconnection valid?

h) Show that the power generated equals the power absorbed in the circuits of Figs. 2.3(b) and (c).

i) Solve Assessment Problems 2.1 and 2.2.

2. Read Section 2.2.

a) Write the Ohm’s law equation for the resistors shown below. Remember that the current arrow points to the sign to use in the equation.

b) Find the conductance of the two resistors in part (a).

c) If i1 = 5 mA in the resistor in part (a), find v1 and the power.

If v2 = 25 V in the resistor in part (a), find i2 and the power.

d) Use the results of Example 2.3 to show that the power balances for each circuit in Fig. 2.6.

e) Solve Assessment Problem 2.3 and Chapter Problem 2.11.

3. Read Section 2.3

a) The concepts of short circuit and open circuit, introduced in Example 2.4 and Fig. 2.10, are important for understanding the operation of a switch and for understanding concepts introduced in later chapters. Summarize your understanding of short circuit and open circuit as follows:

i. Draw a short circuit and label the voltage drop across it and the current flowing through it.

ii. What is the resistance of the short circuit?

iii. What is the voltage drop across the short circuit?

iv. How much current can flow through the short circuit? (Hint – use Ohm’s law with the resistance and voltage you just determined.)

v. Draw an open circuit and label the voltage drop across it and the current flowing through it.

vi. What is the resistance of the open circuit?

vii. What is the current flowing through the open circuit?

viii. How much voltage can drop across the open circuit?

b) Add a column to the table in Fig. 2.13(b) and fill in the power absorbed by the resistor for each value of vT. Plot the power versus vT below:

If you were only given the values in the voltage column and the power column, could you find the resistor value? How?

c) Solve Chapter Problem 2.4.

4. Read Section 2.4.

a) There are three different ways to state KCL:

• The sum of all the currents entering a node is zero.

• The sum of all the currents leaving a node is zero.

• The sum of the currents entering a node equals the sum of the currents leaving that node.

Consider the following circuit fragment:

i. Write the KCL equation at the node by summing all of the currents entering. If a current is leaving, switch the direction of the arrow and re-label its current.

ii. Write the KCL equation at the node by summing all of the currents leaving. If a current is entering, switch the direction of the arrow and re-label its current.

iii. Write the KCL equation at the node by equating the sum of the currents entering and the sum of the currents leaving. You will not need to change any of the current arrows or their labels.

iv. Prove that the three KCL equations you wrote are the same.

b) When writing a KVL equation you can traverse the closed loop in either the clockwise direction or the counter-clockwise direction. Always pick a starting point in your closed loop and a direction. An easy way to determine which sign to use for a voltage term is to write down the first sign you come to as you traverse the loop. For example, look at the “a” loop in Fig. 2.17. Start to the left of the 1 ( resistor and traverse the loop in the clockwise direction:

• The first sign is “–” and the voltage is v1, so write “–v1.”

• The next sign is “+” and the voltage is v2, so write “–v1 + v2.”

• The next sign is “+” and the voltage is v4, so write “–v1 + v2 + v4.”

• The next sign is “–” and the voltage is vb, so write “–v1 + v2 + v4 – vb.”

• The next sign is “–” and the voltage is v3, so write “–v1 + v2 + v4 – vb – v3.”

• We have returned to the starting point, so complete the equation:

– v1 + v2 + v4 – vb – v3 = 0

Use this technique to write a KVL equation for the circuit below. Start to the left of the 5 ( resistor and traverse the loop in the clockwise direction.

Now write a KVL equation for the same circuit, but start below the dependent source and traverse the loop in the counter-clockwise direction. Prove that the two KVL equations you wrote are the same.

c) Use the circuit in Fig. 2.19 to find v1 using Kirchhoff’s laws and Ohm’s law. Do not use io and i1 in your equations. Instead, write a KVL equation around the left loop, summing the three voltages. Then use Ohm’s law and KCL to sum the currents leaving node “b” in terms of the voltages v1 and vo. Solve the two equations to find vo and v1.

d) Use the data in the table of Fig. 2.20(b) to plot the current as a function of the voltage. Write the equation of the resulting straight line. Then, follow the steps in Example 2.9 to construct a circuit model for the device of Fig. 2.20(a) that consists of a current source and a resistor, connected as shown below:

Find the values for i and R. Finally, connect a 10 ( resistor between terminals “a” and “b” and calculate the power the resistor absorbs. Why is the answer the same as in Example 2.9?

e) Solve Assessment Problem 2.6 and Chapter Problems 2.16 and 2.21.

5. Read Section 2.5.

a) Using the values of i(, io, and vo on p. 50, show that the power generated equals the power absorbed for the circuit in Fig. 2.22.

b) Construct a different set of six independent equations for the circuit of Fig. 2.24 as follows:

i. Write KCL equations at nodes “b,” “c,” and “d” by summing the currents entering each node.

ii. Write the constraint equation that equates ic with the dependent source current.

iii. Write two KVL equations for the paths “dcbd” and “abcda,” traversing these loops in a counter-clockwise direction.

If you are really brave, use this new set of equations to find the current iB in terms of the known circuit variables. You can follow the steps enumerated on pages 52 and 53, with slight modifications. You should get the result shown in Eq. 2.25.

c) Solve Assessment Problem 2.9.

Assessing Your Mastery:

Review the Objectives for this unit. Once you are satisfied that you have achieved these Objectives, take the Chapter 2 Test.

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