Power and Energy

1.5 Power and Energy

Although current and voltage are the two basic variables in an electric circuit, they are not sufficient by themselves. For practical purposes, we need to know how much power an electric device can handle. We all know from experience that a 100-watt bulb gives more light than a 60-watt bulb. We also know that when we pay our bills to the electric utility companies, we are paying for the electric energy consumed over

a certain period of time. Thus, power and energy calculations are

1.5 Power and Energy

To relate power and energy to voltage and current, we recall from physics that:

Power is the time rate of expending or absorbing energy, measured in watts (W).

We write this relationship as

where p is power in watts (W), w is energy in joules (J), and t is time

in seconds (s). From Eqs. (1.1), (1.3), and (1.5), it follows that

or Reference polarities for power using the passive sign convention: (a) absorbing power, (b) supplying power.

p ⫽ vi

When the voltage and current directions The power p in Eq. (1.7) is a time-varying quantity and is called the

conform to Fig. 1.8 (b), we have the ac- instantaneous power . Thus, the power absorbed or supplied by an ele-

tive sign convention and p ⫽ ⫹ vi. ment is the product of the voltage across the element and the current

through it. If the power has a ⫹ sign, power is being delivered to or absorbed by the element. If, on the other hand, the power has a ⫺ sign,

3A 3A 3A

power is being supplied by the element. But how do we know when the power has a negative or a positive sign?

Current direction and voltage polarity play a major role in deter- mining the sign of power. It is therefore important that we pay atten-

4V 4V

tion to the relationship between current i and voltage v in Fig. 1.8(a). The voltage polarity and current direction must conform with those + −

shown in Fig. 1.8(a) in order for the power to have a positive sign.

This is known as the passive sign convention. By the passive sign con- (b)

(a) (a)

vention, current enters through the positive polarity of the voltage. In Figure 1.9

this case, p ⫽ ⫹vi or v i 70 implies that the element is absorbing Two cases of an element with an absorbing power of 12 W: (a) p⫽ 4 ⫻ 3 ⫽ 12 W, power. However, if p ⫽ ⫺vi or v i 60 , as in Fig. 1.8(b), the element (b) W. p⫽ 4 ⫻ 3 ⫽ 12 is releasing or supplying power.

Passive sign convention is satisfied when the current enters through 3A the positive terminal of an element and p 3A ⫽ ⫹ vi. If the current enters

3A 3A through the negative terminal, p ⫽ ⫺ vi.

Unless otherwise stated, we will follow the passive sign conven-

4V 4V

tion throughout this text. For example, the element in both circuits of

Fig. 1.9 has an absorbing power of ⫹ 12 W because a positive current

enters the positive terminal in both cases. In Fig. 1.10, however, the

element is supplying power of ⫹ 12 W because a positive current enters

(a) (a)

(b)

the negative terminal. Of course, an absorbing power of ⫺ 12 W is Figure 1.10

equivalent to a supplying power of ⫹ 12 W. In general,

Two cases of an element with a supplying power of 12 W: (a) p⫽⫺ 4⫻3⫽

12 Chapter 1

Basic Concepts

In fact, the law of conservation of energy must be obeyed in any electric circuit. For this reason, the algebraic sum of power in a cir- cuit, at any instant of time, must be zero:

ap⫽0

This again confirms the fact that the total power supplied to the circuit must balance the total power absorbed.

From Eq. (1.6), the energy absorbed or supplied by an element

from time t 0 to time t is

w⫽ p dt ⫽ v i dt

Energy is the capacity to do work, measured in joules (J).

The electric power utility companies measure energy in watt-hours (Wh), where

1 Wh ⫽ 3,600 J

Example 1.4

An energy source forces a constant current of 2 A for 10 s to flow through a light bulb. If 2.3 kJ is given off in the form of light and heat energy, calculate the voltage drop across the bulb.

Solution:

The total charge is

¢q ⫽ i ¢t ⫽ 2 ⫻ 10 ⫽ 20 C

The voltage drop is

Practice Problem 1.4

To move charge q from point a to point b requires ⫺ 30 J. Find the voltage drop v ab if: (a) q⫽ 6 C, (b) q⫽⫺ 3 C.

Answer: (a) ⫺ 5 V, (b) 10 V.

Example 1.5

Find the power delivered to an element at t⫽ 3 ms if the current enter-

ing its positive terminal is i⫽ 5 cos 60 p t A

and the voltage is: (a) v⫽ 3i , (b) v⫽ 3 di 兾dt .

1.5 Power and Energy

Solution:

(a) The voltage is v⫽ 3i ⫽ 15 cos 60 p t ; hence, the power is p ⫽ vi ⫽ 75 cos 2 60 p t W

At ms, t⫽ 3

p⫽ 75 cos (60 p⫻ 3 ⫻ 10 3 ) ⫽ 75 cos 2 0.18 p⫽ 53.48 W

(b) We find the voltage and the power as di

v⫽3 ⫽ 3(⫺60 p )5 sin 60 p t⫽⫺ 900 p sin 60 p t V

dt

p ⫽ vi ⫽ ⫺4500 p sin 60 p t cos 60 p t W

At ms, t⫽ 3

p ⫽ ⫺4500 p sin 0.18 p cos 0.18 p W

⫽ ⫺14137.167 sin 32.4⬚ cos 32.4⬚ ⫽ ⫺6.396 kW

Find the power delivered to the element in Example 1.5 at t⫽ 5 ms

Practice Problem 1.5

if the current remains the same but the voltage is: (a) v⫽ 2i V,

(b) V. v⫽ a10 ⫹ 5 i dt 冮 b

Answer: (a) 17.27 W, (b) 29.7 W.

How much energy does a 100-W electric bulb consume in two hours?

Example 1.6

Solution:

w ⫽ pt ⫽ 100 (W) ⫻ 2 (h) ⫻ 60 (min/h) ⫻ 60 (s/min) ⫽ 720,000 J ⫽ 720 kJ

This is the same as w⫽ pt ⫽ 100 W ⫻ 2 h ⫽ 200 Wh

A stove element draws 15 A when connected to a 240-V line. How

Practice Problem 1.6

long does it take to consume 180 kJ?

Answer:

50 s.

14 Chapter 1

Basic Concepts

Historical

1884 Exhibition In the United States, nothing promoted the future of electricity like the 1884 International Electrical Exhibition. Just imagine a world without electricity, a world illuminated by candles and gaslights, a world where the most common transportation was by walk- ing and riding on horseback or by horse-drawn carriage. Into this world an exhibition was created that highlighted Thomas Edison and reflected his highly developed ability to promote his inventions and products. His exhibit featured spectacular lighting displays powered by an impres- sive 100-kW “Jumbo” generator.

Edward Weston’s dynamos and lamps were featured in the United States Electric Lighting Company’s display. Weston’s well known col- lection of scientific instruments was also shown.

Other prominent exhibitors included Frank Sprague, Elihu Thompson, and the Brush Electric Company of Cleveland. The American Institute of Electrical Engineers (AIEE) held its first technical meeting on Octo- ber 7–8 at the Franklin Institute during the exhibit. AIEE merged with the Institute of Radio Engineers (IRE) in 1964 to form the Institute of Electrical and Electronics Engineers (IEEE).

Smithsonian Institution.

1.6 Circuit Elements