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



Chapter 3 – Simple Resistive Circuits

Study Guide

Objectives:

1. Be able to recognize resistors connected in series and in parallel and use the rules for combining series-connected resistors and parallel-connected resistors to yield equivalent resistance.

2. Know how to design simple voltage-divider and current-divider circuits.

3. Be able to use voltage division and current division appropriately to solve simple circuits.

4. Be able to determine the reading of an ammeter when added to a circuit to measure current; be able to determine the reading of a voltmeter when added to a circuit to measure voltage.

5. Understand how a Wheatstone bridge is used to measure resistance.

6. Know when and how to use delta-to-wye equivalent circuits to solve simple circuits.

Mastering the Objectives:

1. Read the Introduction and Section 3.1.

a) What two methods can you use to determine whether two resistors are in series?

b) The equivalent resistance of a collection of series-connected resistors is

smaller than / larger than / the same as (circle one)

the value of the largest single resistor.

c) Define the term “black box.”

d) If 100 V is applied to the black box containing the seven resistors in Fig. 3.4, the current into the box is 25 A. What resistor can be placed in another black box so that it is impossible two tell the two black boxes apart?

e) Solve Chapter Problem 3.1

2. Read Section 3.2.

a) What characteristics of parallel-connected resistors are missing in Fig. 3.6?

b) Four parallel-connected resistors have the values 1.5 k(, 3 k(, 4 k(, and 6 k(. A friend tells you that the equivalent resistance of these four resistors is 2 k(. Without doing any computations, you tell the friend that 2 k( cannot possibly be the correct answer. How did you know that?

c) Determine the equivalent resistance of the four resistors discussed in part (b).

d) State in words the equivalent resistance of two resistors in parallel.

e) Show that the solution in Example 3.1 satisfies

i) KCL at each node;

ii) KVL around each loop (there are three loops);

iii) The requirement for power balance.

f) Solve Assessment Problem 3.1 and Chapter Problems 3.2 and 3.6.

3. Read Section 3.3

a) Describe the type of circuit that is best analyzed using voltage division.

b) In words, what is the relationship between the voltage drop across a single resistor in a collection of series-connected resistors to the source voltage?

c) In the voltage divider circuit of Fig. 3.12, if v2 > v1 is R2 > R1 or is R1 > R2? Why?

d) In the current divider circuit of Fig. 3.15, if i2 > i1 is R2 > R1 or is R1 > R2? Why?

e) Define the term “load.”

f) What is the relationship, in words, of the output voltage of an unloaded voltage divider to the output voltage of a loaded voltage divider?

g) Suppose the tolerance on the resistors in Fig. 3.14 of Example 3.2 is decreased to 5%. Now what are the maximum and minimum values of vo?

h) Suppose you want vo in Fig. 3.14 of Example 3.2 to vary no more than 1% from its nominal value. What is the largest tolerance allowed for the 25 k( and 100 k( resistors?

i) Calculate the current in the rest of the resistors in Fig. 3.17 of Example 3.3.

j) Solve Assessment Problems 3.2 and 3.3.

4. Read Section 3.4.

a) In Fig. 3.18, if the voltage drop across Rj is larger than the voltage drop across any other resistor, is Rj larger or smaller than all the other resistors? Use Eq. 3.30 to prove your answer.

b) In Fig. 3.19, if the current through Rj is larger than the current through any other resistor, is Rj larger or smaller than all the other resistors? Use Eq. 3.32 to prove your answer.

c) Use voltage division, current division, and Ohm’s law to find the current and voltage for all of the resistors in Fig. 3.20 of Example 3.4.

d) Solve Assessment Problem 3.4

e) Describe the type of circuit that is best analyzed using voltage division.

f) In words, what is the relationship between the voltage drop across a single resistor in a collection of series-connected resistors to the source voltage?

g) Define the term “load.”

h) What is the relationship, in words, of the output voltage of an unloaded voltage divider to the output voltage of a loaded voltage divider?

i) Recalculate the maximum and minimum values of the output voltage in Example 3.2 if the resistors have a tolerance of (2%.

5. Read Sections 3.5 and 3.6

a) Fill in the blanks:

An ammeter is used to measure _________________. An ammeter is placed in ____________ with the component it is measuring. An ideal ammeter behaves like a(n) __________________. A real analog ammeter consists of a meter movement in _______________ with a resistor. The purpose of the resistor is to _____________________________________.

A voltmeter is used to measure _________________. A voltmeter is placed in ____________ with the component it is measuring. An ideal voltmeter behaves like a(n) __________________. A real analog voltmeter consists of a meter movement in _______________ with a resistor. The purpose of the resistor is to _____________________________________.

b) From Example 3.5, what are the two methods that can be used to determine the effective resistance of an ammeter, Rm?

c) From Example 3.6, what are the two methods that can be used to determine the effective resistance of a voltmeter, Rm?

d) Solve Assessment Problems 3.5 and 3.6.

e) What is a galvanometer?

f) When Eq. 3.33 is satisfied, we say that the bridge is balanced. The easiest way to remember the condition for a balanced bridge is to note that if Eq. 3.33 is satisfied, the product of one set of opposite resistors equals the product of the other set. In Fig. 3.25, resistors R1 and Rx are opposite, as are resistors R2 and R3, so if the bridge is balanced,

R1Rx = R2R2

Show that this condition is equivalent to Eq. 3.33.

g) Solve Assessment Problem 3.7 and Chapter Problem 3.49.

6. Read Section 3.7.

a) Redraw the delta-connected resistors and the wye-connected resistors in Fig. 3.31 by superimposing them. The result should look like Fig. 3.33. Now use the superimposed resistors to find the pattern used in transforming from one type of interconnection to the other. For example, from Eqs. 3.44 – 3.46 we see that a wye-connected resistor equals the product of the two delta-connected resistors on either side, divided by the sum of the three delta-connected resistors. What is the pattern used to calculate a delta-connected resistor from the wye-connected resistors? (Use Eqs. 3.47 – 3.49.)

b) If R1 = R2 = R3 in the wye-connection of Fig. 3.31, what are the values of the delta-connected resistors? If Ra = Rb = Rc in the delta-connection of Fig. 3.31, what are the values of the wye-connected resistors?

c) Notice that in Fig. 3.32 of Example 3.7, there are two wye-connected sets of resistors: the 100 ( – 25 ( – 40 ( resistors on the left and the 125 ( – 25 ( – 37.5 ( resistors on the right. Repeat the problem stated in Example 3.7, but now replace the wye-connected resistor on the left with an equivalent set of delta-connected resistors, and then simplify with series and parallel combinations of resistors.

d) Solve Assessment Problem 3.8 and Chapter Problem 3.53.

Assessing Your Mastery:

Review the Objectives for this unit. Once you are satisfied that you have achieved these Objectives, take the Chapter 3 Test.

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