2.8 Applications

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2.8 Applications

Resistors are often used to model de vices that con vert electrical ener gy into heat or other forms of energy. Such devices include conducting wire, light bulbs, electric heaters, stoves, ovens, and loudspeakers. In this section, we will consider two real-life problems that apply the concepts developed in this chapter: electrical lighting systems and design of dc meters.

2.8.1 Lighting Systems

Lighting systems, such as in a house or on a Christmas tree, often consist of N lamps connected either in parallel or in series, as sho wn in Fig. 2.55. Each lamp is modeled as a resistor . Assuming that all the lamps are identical and Vo is the power-line voltage, the voltage across each lamp is Vo for the parallel connection and Vo /N for the series connection. The series connection is easy to manufa cture but is seldom used in practice, for at least two reasons. First, it is less reliable; when a lamp fa ils, all the lamps go out. Second, it is harder to maintain; when a lamp is bad, one must test all the lamps one by one to detect the faulty one.

Figure 2.54 For Practice Prob. 2.15.

So far, we have assumed that connecting wires are perfect conductors (i.e., conductors of zero resistance). In real physical systems, however, the resistance of the connecting wire may be appreciably large, and the modeling of the system must include that resistance.

Historical

Thomas Alva Edison (1847–1931) was perhaps the greatest American inventor. He patented 1093 inventions, including such history-making inventions as the incandescent electric bulb, the phonograph, and the first commercial motion pictures.

Born in Milan, Ohio, the youngest of seven children, Edison received only three months of formal education because he hated school. He was home-schooled by his mother and quickly began to read on his own. In 1868, Edison read one of Faraday’s books and found his calling. He moved to Menlo Park, New Jersey, in 1876, where he man aged a well-staffed research laboratory. Most of his inventions came out of this laboratory. His laboratory served as a model for modern research organ iza tions. Because of his diverse interests and the over whelming number of his inventions and patents, Edison began to estab lish manufacturing companies for making the devices he invented. He designed the first electric power station to supply electric light. Formal electrical engineering education began in the mid-1880s with Edison as a role model and leader.

Library of Congress

(a) Parallel connection of light bulbs, (b) series connection of light bulbs.

Example 2.16

Three light bulbs are connected to a 9-V battery as shown in Fig. 2.56(a). Calculate: (a) the total current supplied by the battery , (b) the current through each bulb, (c) the resistance of each bulb.

Solution:

(a) The total po wer supplied by the battery is equal to the total po wer absorbed by the bulbs; that is,

$$p = 15 + 10 + 20 = 45$$

W

Since p = V I, then the total current supplied by the battery is

$$I = \frac{p}{V} = \frac{45}{9} = 5 \text{ A}$$

(b) The bulbs can be modeled as resistors as shown in Fig. 2.56(b). Since R1 (20-W bulb) is in parallel with the battery as well as the series com bination of R2 and R3,

$$V_1 = V_2 + V_3 = 9 \text{ V}$$

The current through R1 is

$$I_1 = \frac{p_1}{V_1} = \frac{20}{9} = 2.222 \text{ A}$$

By KCL, the current through the series combination of R2 and R3 is

$$I_2 = I - I_1 = 5 - 2.222 = 2.778 \text{ A}$$

(c) Since p = I 2 R,

$$R_1 = \frac{p_1}{l_1^2} = \frac{20}{2.222^2} = 4.05 \ \Omega$$

\n

$$R_2 = \frac{p_2}{l_2^2} = \frac{15}{2.777^2} = 1.945 \ \Omega$$

\n

$$R_3 = \frac{p_3}{l_3^2} = \frac{10}{2.777^2} = 1.297 \ \Omega$$

Refer to Fig. 2.55 and assume there are six light b ulbs that can be con nected in parallel and six dif ferent light bulbs that can be connected in series. In either case, each light bulb is to operate at 40 W. If the voltage at the plug is 115 V for the parallel and series connections, calculate the current through and the voltage across each bulb for both cases.

Answer: 115 V and 347.8 mA (parallel), 19.167 V and 2.087 A (series).

2.8.2 Design of DC Meters

By their nature, resistors are used to control the flow of current. We take advantage of this property in se veral applications, such as in a poten tiometer (Fig. 2.57). The word potentiometer, derived from the w ords potential and meter, implies that potential can be metered out. The potentiometer (or pot for short) is a three-terminal de vice that operates on the principle of v oltage division. It is essentially an adjustable v oltage divider. As a voltage regulator, it is used as a volume or level control on radios, TVs, and other devices. In Fig. 2.57,

$$V_{\text{out}} = V_{bc} = \frac{R_{bc}}{R_{ac}} V_{\text{in}}$$

(2.58)

where Rac = Rab + Rbc. Thus, Vout decreases or increases as the sliding contact of the pot moves toward c or a, respectively.

Another application where resistors are used to control current flow is in the analog dc meters—the ammeter, voltmeter, and ohmmeter, which measure current, voltage, and resistance, respectively. Each of these meters employs the d’Arsonval meter movement, shown in Fig. 2.58. The movement consists essentially of a movable iron-core coil mounted on a pivot between the poles of a permanent magnet. When current flows through the coil, it creates a torque which causes the pointer to deflect. The amount of current through the coil determines the deflection of the pointer, which is registered on a scale attached to the meter movement. For example, if the meter movement is rated 1 mA, 50 Ω, it w ould take 1 mA to cause a full-scale deflection of the meter movement. By introducing additional circuitry to the d’Arsonval meter mo vement, an am meter, voltmeter, or ohmmeter can be constructed.

Consider Fig. 2.59, where an analog v oltmeter and ammeter are con nected to an element. The voltmeter measures the voltage across a load and

Practice Problem 2.16

Figure 2.57 The potentiometer controlling potential levels.

An instrument capable of measuring voltage, current, and resistance is called a multimeter or a volt-ohm meter (VOM).

A load is a component that is receiving energy (an energy sink), as opposed to a generator supplying energy (an energy source). More about loading will be discussed in Section 4.9.1.

A d’Arsonval meter movement.

Connection of a voltmeter and an ammeter to an element.

is therefore connected in parallel with the element. As shown in Fig. 2.60(a), the voltmeter consists of a d’Arson val movement in series with a resistor whose resistance Rm is deliberately made very large (theoretically, infinite), to minimize the current drawn from the circuit. To extend the range of voltage that the meter can measure, series multiplier resistors are often connected with the voltmeters, as shown in Fig. 2.60(b). The multiple-range voltmeter in Fig. 2.60(b) can measure v oltage from 0 to 1 V, 0 to 10 V, or 0 to 100 V, depending on whether the switch is connected to R1, R2, or R3, respectively.

Let us calculate the multiplier resistor Rn for the single-range voltmeter in Fig. 2.60(a), or Rn = R1, R2, or R3 for the multiple-range voltmeter in Fig. 2.60(b). We need to determine the value of Rn to be connected in series with the internal resistance Rm of the voltmeter. In any design, we consider the worst-case condition. In this case, the worst case occurs when the fullscale current Ifs =Im flows through the meter. This should also correspond

Figure 2.60 Voltmeters: (a) single-range type, (b) multiple-range type.

to the maximum v oltage reading or the full-scale v oltage Vfs. Since the multiplier resistance Rn is in series with the internal resistance Rm,

$$V_{\text{fs}} = I_{\text{fs}} (R_n + R_m) \tag{2.59}$$

From this, we obtain

$$R_n = \frac{V_{\text{fs}}}{I_{\text{fs}}} - R_m \tag{2.60}$$

Similarly, the ammeter measures the current through the load and is connected in series with it. As shown in Fig. 2.61(a), the ammeter consists of a d’Arsonval movement in parallel with a resistor whose resistance Rm is deliberately made very small (theoretically, zero) to minimize the v oltage drop across it. To allow multiple ranges, shunt resistors are often connected in parallel with Rm as sho wn in Fig. 2.61(b). The shunt resistors allow the meter to measure in the range 0–10 mA, 0–100 mA, or 0–1 A, depending on whether the switch is connected to R1, R2, or R3, respectively.

Now our objective is to obtain the multiplier shunt Rn for the singlerange ammeter in Fig. 2.61(a), or Rn =R1, R2, or R3 for the multiple-range ammeter in Fig. 2.61(b). We notice that Rm and Rn are in parallel and that at full-scale reading I = Ifs = Im + In, where In is the current through the shunt resistor Rn. Applying the current division principle yields

$$I_m = \frac{R_n}{R_n + R_m} I_{\text{fs}}$$ $$\sum_{i=1}^{n} x_i$$ $$R_n = \frac{I_m}{I_{\text{fs}} - I_m} R_m \tag{2.61}$$

The resistance Rx of a linear resistor can be measured in tw o ways. An indirect way is to measure the current I that flows through it by connecting an ammeter in series with it and the v oltage V across it by con necting a voltmeter in parallel with it, as shown in Fig. 2.62(a). Then

$$R_x = \frac{V}{I} \tag{2.62}$$

The direct method of measuring resistance is to use an ohmmeter . An ohmmeter consists basically of a d’Arson val movement, a v ariable resistor or potentiometer, and a battery, as shown in Fig. 2.62(b). Applying KVL to the circuit in Fig. 2.62(b) gives

$$E = (R + R_m + R_x)I_m$$

or

$$R_x = \frac{E}{I_m} - (R + R_m)$$

(2.63)

The resistor R is selected such that the meter gives a full-scale deflection; that is, Im = Ifs when Rx = 0. This implies that

$$E = (R + R_m) I_{\text{fs}} \tag{2.64}$$

Substituting Eq. (2.64) into Eq. (2.63) leads to

$$R_x = \left(\frac{I_{\text{fs}}}{I_m} - 1\right)(R + R_m) \tag{2.65}$$

As mentioned, the types of meters we ha ve discussed are kno wn as analog meters and are based on the d’Arsonval meter movement. Another type of meter , called a digital meter, is based on acti ve circuit elements

Figure 2.61

Ammeters: (a) single-range type, (b) multiple-range type.

Figure 2.62 Two ways of measuring resistance: (a) using an ammeter and a voltmeter, (b) using an ohmmeter.

Historical

Library of Congress

Samuel F. B. Morse (1791–1872), an American painter, invented the telegraph, the first practical, commercialized application of electricity.

Morse was born in Charlestown, Massachusetts, and studied at Yale and the Royal Academy of Arts in London to become an artist. In the 1830s, he became intrigued with developing a telegraph. He had a working model by 1836 and applied for a patent in 1838. The U.S. Senate appro priated funds for Morse to construct a telegraph line between Baltimore and Washington, D.C. On May 24, 1844, he sent the famous first message: “What hath God wrought!” Morse also developed a code of dots and dashes for letters and numbers, for sending messages on the telegraph. The development of the telegraph led to the invention of the telephone.

such as op amps. For example, a digital multimeter displays measurements of dc or ac voltage, current, and resistance as discrete numbers, instead of using a pointer deflection on a continuous scale as in an analog multimeter. Digital meters are what you would most likely use in a modern lab. However, the design of digital meters is beyond the scope of this book.

Example 2.17 Following the v oltmeter setup of Fig. 2.60, design a v oltmeter for the following multiple ranges:

(a) 0–1 V (b) 0–5 V (c) 0–50 V (d) 0–100 V Assume that the internal resistance Rm = 2 kΩ and the full-scale current Ifs =100μA.

Solution:

We apply Eq. (2.60) and assume that R1, R2, R3, and R4 correspond with ranges 0–1 V, 0–5 V, 0–50 V, and 0–100 V, respectively. (a) For range 0–1 V,

$$R_1 = \frac{1}{100 \times 10^{-6}} - 2000 = 10,000 - 2000 = 8 \text{ k}\Omega$$

(b) For range 0–5 V,

$$R_2 = \frac{5}{100 \times 10^{-6}} - 2000 = 50,000 - 2000 = 48 \text{ k}\Omega$$

(c) For range 0–50 V,

$$R_3 = \frac{50}{100 \times 10^{-6}} - 2000 = 500,000 - 2000 = 498 \text{ k}\Omega$$

(d) For range 0–100 V,

$$R_4 = \frac{100 \text{ V}}{100 \times 10^{-6}} - 2000 = 1,000,000 - 2000 = 998 \text{ k}\Omega$$

Note that the ratio of the total resistance (Rn + Rm) to the full-scale voltage Vfs is constant and equal to 1/Ifs for the four ranges. This ratio (given in ohms per v olt, or Ω/ V) is known as the sensitivity of the v oltmeter. The larger the sensitivity, the better the voltmeter.

Following the ammeter setup of Fig. 2.61, design an ammeter for the following multiple ranges: (a) 0–1 A (b) 0–100 mA (c) 0–10 mA Take the full-scale meter current as Im = 1 mA and the internal resistance of the ammeter as Rm = 50 Ω.

Answer: Shunt resistors: 50 mΩ, 505 mΩ, 5.556 Ω.