AC Power Analysis

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AC Power Analysis

Four things come not back: the spoken word; the sped arrow; time past; the neglected opportunity.

—Al Halif Omar Ibn

Enhancing Your Career

Career in Power Systems

The discovery of the principle of an ac generator by Michael Faraday in 1831 was a major breakthrough in engineering; it provided a convenient way of generating the electric po wer that is needed in e very electronic, electrical, or electromechanical device we use now.

Electric power is obtained by converting energy from sources such as fossil fuels (gas, oil, and coal), nuclear fuel (uranium), h ydro energy (water falling through a head), geothermal energy (hot water, steam), wind energy, tidal ener gy, and biomass ener gy (wastes). These various ways of generating electric po wer are studied in detail in the field of power engineering, which has become an indispensable subdiscipline of electrical engineering. An electrical engineer should be f amiliar with the analysis, generation, transmission, distrib ution, and cost of electric power.

The electric po wer industry is a v ery large employer of electrical engineers. The industry includes thousands of electric utility systems ranging from large, interconnected systems serving large regional areas to small power companies serving indi vidual communities or f actories. Due to the comple xity of the po wer industry, there are numerous elec trical engineering jobs in dif ferent areas of the industry: po wer plant (generation), transmission and distribution, maintenance, research, data acquisition and flow control, and management. Since electric po wer is used e verywhere, electric utility companies are e verywhere, of fering exciting training and steady emplo yment for men and w omen in thou sands of communities throughout the world.

A pole-type transformer with a lowvoltage, three-wire distribution system. © Dennis Wise/Getty Images RF

Learning Objectives

By using the information and exercises in this chapter you will be able to:

• 1. Fully understand instantaneous and average power.
• 2. Understand the basics of maximum average power.
• 3. Understand effective or rms values and how to calculate them and to understand their importance.
• 4. Understand apparent power (complex power), power, and reactive power and power factor.
• 5. Understand power factor correction and the importance of its use.

11.1 Introduction

Our effort in ac circuit analysis so f ar has been focused mainly on cal culating voltage and current. Our major concern in this chapter is power analysis.

Power analysis is of paramount importance. Po wer is the most important quantity in electric utilities, electronic, and communication systems, because such systems in volve transmission of po wer from one point to another . Also, e very industrial and household electrical device—every fan, motor, lamp, pressing iron, TV, personal computer has a po wer rating that indicates ho w much po wer the equipment re quires; e xceeding the po wer rating can do permanent damage to an appliance. The most common form of electric po wer is 50- or 60-Hz ac power. The choice of ac o ver dc allowed high-voltage power transmission from the power generating plant to the consumer.

We will be gin by defining and deriving instantaneous power and average power. We will then introduce other power concepts. As practical applications of these concepts, we will discuss how power is measured and reconsider how electric utility companies charge their customers.

11.2 Instantaneous and Average Power

As mentioned in Chapter 2, the instantaneous power p(t) absorbed by an element is the product of the instantaneous v oltage v(t) across the ele ment and the instantaneous current i(t) through it. Assuming the passive sign convention,

$$p(t) = v(t)i(t)$$

(11.1)

The instantaneous power (in watts) is the power at any instant of time.

It is the rate at which an element absorbs energy.

Consider the general case of instantaneous po wer absorbed by an arbitrary combination of circuit elements under sinusoidal excitation, as

We can also think of the instantaneous power as the power absorbed by the element at a specific instant of time. Instantaneous quantities are denoted by lowercase letters.

shown in Fig. 11.1. Let the v oltage and current at the terminals of the circuit be

$$v(t) = V_m \cos(\omega t + \theta_v)$$

(11.2a)

$$i(t) = I_m \cos(\omega t + \theta_i)$$

(11.2b)

where Vm and Im are the amplitudes (or peak values), and θv and θi are the phase angles of the voltage and current, respectively. The instantaneous power absorbed by the circuit is

$$p(t) = v(t)i(t) = V_m I_m \cos(\omega t + \theta_v) \cos(\omega t + \theta_i)$$

(11.3)