Masters G.M. Renewable and Efficient Electric Power Systems

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118 THE ELECTRIC POWER INDUSTRY

pass through turbine blades, causing a shaft to spin. The function of the generator,

then, is to convert the rotational energy of the turbine shaft into electricity.

3.3.1 A Simple Generator

Electric generators are all based on the fundamental concepts of electromagnetic

induction developed by Michael Faraday in 1831. Faraday discovered that moving

a conductor through a magnetic ﬁeld induces an electromagnetic force (emf), or

voltage, across the wire, as suggested in Fig. 3.9a. A generator, very simply,

is an arrangement of components designed to cause relative motion between a

magnetic ﬁeld and the conductors in which the emf is to be induced. Those

conductors, out of which ﬂows electric power, form what is called the armature.

Most large generators have the armature windings ﬁxed in the stationary portion

of the machine (called the stator), and the necessary relative motion is caused

by rotating the magnetic ﬁeld, as shown in Fig. 3.9b.

To help illustrate generation of ac, consider the simple generator shown in

Fig. 3.10. The rotor in this case is just a 2-pole magnet (1 north pole and 1

south pole), which for now we can consider to be just an ordinary permanent

magnet. The stator consists of iron, shaped somewhat like a C (backwards, in

this case), with some copper wire (the armature) wrapped around the iron. The

purpose of the iron in the stator is to provide a low reluctance path for the

magnetic ﬂux lines, channeling as much ﬂux as possible through the copper

armature windings. You may recall from Chapter 1 that the low reluctance of

ferromagnetic materials (e.g., iron) causes ﬂux to prefer to stay in the iron,

which is exactly analogous to current-carrying electrons wanting to stay in copper

wires. And just as low-resistance copper wire allows more current to ﬂow, low

reluctance ferromagnetic materials allow more magnetic ﬂux (ﬂux doesn’t “ﬂow,”

however).

As the rotor turns, magnetic ﬂux passes through the stator and the arma-

ture windings, in one direction, then diminishes to zero, then increases in the

−

Voltage

Motion

Magnetic

field

Conductor

Armature

conductors

Rotor

(a) (b)

Stator

Load

Figure 3.9 Voltage and current can be created by (a) moving a conductor through a

magnetic ﬁeld, or (b) moving the magnetic ﬁeld past the conductors. The armature wind-

ings indicate current ﬂow into the page with an “x” and current out of the page with a

dot (the x is meant to resemble the feathers of an arrow moving away from you; the dot

is the point of the arrow coming toward you).

POLYPHASE SYNCHRONOUS GENERATORS 119

Flux

lines

Stator

Rotor

Figure 3.10 As the permanent-magnet rotor turns, it causes magnetic ﬂux within the

iron stator to vary (approximately) sinusoidally. The windings around the stator therefore

see a time-varying ﬂux, which creates a voltage across their terminals.

(a)

(b)

Figure 3.11 Changing ﬂux in the stator creates an emf voltage across the windings.

other direction. Ideally, the ﬂux φ would vary sinusoidally as suggested in

Fig. 3.11a. From Faraday’s law, whenever a winding links a time-varying amount

of magnetic ﬂux φ, there will be a voltage e (electromotive force) created across

the winding:

e = N

dφ

(3.1)

3.3.2 Single-Phase Synchronous Generators

Suppose we want to generate voltage at a frequency of 60 Hz so that it will

match the frequency of conventional (U.S.) power. How fast would the rotor of

the simple generator in Fig. 3.10 have to turn? Each revolution of the rotor gives

one voltage cycle, so

= shaft rotation rate =

1 revolution

cycle

60 cycles

sec

60 sec

min

= 3600 rpm

(3.2)

120 THE ELECTRIC POWER INDUSTRY

To generate 60 Hz using this 2-pole generator would therefore require that the

rotor turn at a ﬁxed rate of exactly 3600 rpm. Such a ﬁxed-speed machine is

called a synchronous generator since it is synchronized with the utility grid.

Most conventional electric power is generated using synchronous generators (this

is not the case for wind turbines, however).

While we could imagine the magnetic ﬁeld in the rotor of a generator to be

created using a permanent magnet, as suggested in Fig. 3.10, that would greatly

limit the amount of power that could be generated. Instead, the magnetic ﬁeld is

created by sending dc through brushes and slip rings into conductors afﬁxed to

the rotor. Field windings may be imbedded into slots that run along the rotor as

shown in Fig. 3.12a, or they may be wound around what are called salient poles,

as shown in Fig. 3.12b. Salient pole rotors are less expensive to fabricate and

are often used in slower-spinning hydroelectric generators. High-speed turbines

and generators use round rotors, which are better able to handle the centrifugal

forces and resulting stresses. Figure 3.13a shows a complete 2-pole generator

with a round rotor, while Fig. 3.13b is a 4-pole generator with a salient rotor.

Notice the 4-pole generator has four poles in both the rotor and stator.

Adding more poles allows the generator to spin more slowly while still pro-

ducing a desired frequency for its output power. For example, when the rotor in

a 4-pole machine makes one revolution, it generates two cycles on the output

lines. This means that it only needs to rotate half as fast as the 2-pole machine,

namely 1800 rpm, in order to generate 60-Hz ac. In general, rotor speed N

as a

function of number of poles p and output frequency required f is given by

= shaft rotation rate (rpm) =

1 revolution

(p/2) cycles

f cycles

60 s

min

(3.3)

120f

(3.4)

Magnetic flux

Field windings

Magnetic flux

(a) (b)

Figure 3.12 Field windings on (a) 2-pole, round rotor and (b) 4-pole, salient rotor.

CARNOT EFFICIENCY FOR HEAT ENGINES 121

(a) (b)

Flux

Output

Rotor

Stator

Output

Figure 3.13 (a) A 2-pole machine has one N and one S pole on the rotor and on the

stator. (b) A 4-pole machine has 4 poles on the rotor and 4 on the stator.

TA BLE 3.2 Shaft Rotation (rpm) as a Function of

Desired Output Frequency and Number of Poles

Poles

p 50 Hz rpm 60 Hz rpm

2 3000 3600

4 1500 1800

6 1000 1200

8 750 900

10 600 720

12 500 600

While the United States uses 60 Hz exclusively for power, Europe and parts

of Japan use 50 Hz. Table 3.2 provides a convenient summary of rotor speeds

required for a synchronous generator to deliver power at 50 Hz and at 60 Hz.

3.3.3 Three-Phase Synchronous Generators

The machines shown thus far have been single-phase generators. In Chapter 2,

we saw the value of 3-phase power, especially when large amounts of power are

needed. To provide 3-phase power, we can keep the simple rotor with a single

pair of north and south poles, but we need to add another winding to the stator

as has been done in Fig. 3.14. Now during each revolution of the shaft, the rotor

sweeps by each of the three stator windings, thereby inducing a voltage in each

stator that is 120

◦

out of phase with the adjacent windings.

Figure 3.15 shows a 4-pole, 3-phase generator with a salient-pole rotor, which

means it spins at only half the rotor speed compared to a 2-pole machine.

122 THE ELECTRIC POWER INDUSTRY

(a) (b)

2p/3 2p4p/3

Figure 3.14 (a) A 2-pole, 3-phase synchronous generator. (b) Three-phase stator output

voltage.

B′

A′

C′

A′

B′

C′

Figure 3.15 A 4-pole, 3-phase, wye-connected, synchronous generator with a 4-pole

rotor. The dc rotor current needs to be delivered to the rotor through brushes and slip rings.

3.4 CARNOT EFFICIENCY FOR HEAT ENGINES

Over 90% of U.S. electricity is generated in power plants that convert heat into

mechanical work. The heat may be the result of nuclear reactions, fossil-fuel

combustion, or even concentrated sunlight focused onto a boiler. Almost all of

this 90% is based on a heat source boiling water to make steam that spins a turbine

and generator, but there is a rapidly growing fraction that is generated using gas

turbines. The best new fossil-fuel power plants use a combination of both steam

turbines and gas turbines to generate electricity with very high efﬁciency.

Steam engines, gas turbines, and internal-combustion engines are examples

of machines that convert heat into useful work. What we are interested in here

is, How efﬁciently can they do so? This same question will be asked when we

describe fuel cells, photovoltaics, and wind turbines in future chapters, and in each

case we will encounter quite interesting, fundamental limits to their maximum

possible energy-conversion efﬁciencies.

CARNOT EFFICIENCY FOR HEAT ENGINES 123

3.4.1 Heat Engines

Very simply, a heat engine extracts heat Q

from a high-temperature source,

such as a boiler, converts part of that heat into work W , usually in the form of

a rotating shaft, and rejects the remaining heat Q

into a low-temperature sink

such as the atmosphere or a local body of water. Figure 3.16 provides a general

model describing such engines.

The thermal efﬁciency of a heat engine is the ratio of work done to input

energy provided by the high-temperature source:

Thermal efﬁciency =

Net work output

Total heat input

(3.5)

Since energy is conserved,

= W + Q

(3.6)

which leads to another expression for efﬁciency:

Thermal efﬁciency =

− Q

= 1 −

(3.7)

3.4.2 Entropy and the Carnot Heat Engine

The most efﬁcient heat engine that could possibly operate between a hot and cold

thermal reservoir was ﬁrst described back in the 1820s by the French engineer

Sadi Carnot. To sketch out the basis for his famous equation, which links the

High-temperature

SOURCE,

Low-temperature

SINK,

HEAT

ENGINE

Figure 3.16 A heat engine converts some of the heat extracted from a high-temperature

reservoir into work, rejecting the rest into a low-temperature sink.

124 THE ELECTRIC POWER INDUSTRY

maximum possible efﬁciency of a heat engine to the temperatures of the hot and

cold reservoirs, we need to introduce the concept of entropy.

As is often the case in thermodynamics, the deﬁnition of this extremely impor-

tant quantity is not very intuitive. It can be described as a measure of molecular

disorder, or molecular randomness. At one end of the entropy scale is a pure

crystalline substance at absolute zero temperature. Since every atom is locked

into a predictable place, in perfect order, its entropy is deﬁned to be zero. In

general, substances in the solid phase have more ordered molecules and hence

lower entropy than liquid or gaseous substances. When we burn some coal,

there is more entropy in the gaseous end products than in the solid lumps we

burned. That is, unlike energy, entropy is not conserved in a process. In fact,

for every real process that occurs, disorder increases and the total entropy of the

universe increases.

The concept of ever-increasing entropy is enormously important. It tells us

that in any isolated system (e.g., the universe) in which the total energy cannot

change, the only processes that can occur spontaneously are ones that result in an

increase in the entropy of the system. One implication is that heat ﬂows naturally

from warm objects to cold ones, and not the other way around. It also dictates the

direction of certain chemical reactions, as we’ll see in Chapter 4 where entropy

will be used to determine the maximum possible efﬁciency of fuel cells.

When we started the analysis of the heat engine in Fig. 3.16, we began by

tabulating energy ﬂows. The ﬁrst law of thermodynamics treats energy in the

form of heat transfer on an equal basis with energy that shows up as work done

by the engine. For an entropy analysis, that is not the case. Work is considered to

be an idealized process in which no increase in disorder occurs, and hence it has

no accompanying entropy transfer. This is a key distinction. Processes involve

heat transfer and work. Heat transfer is accompanied by entropy transfer, but

work is entropy-free.

Obviously, to be a useful analysis tool, entropy must be described with

equations as well as mental images. Going back to the heat engine, if an amount

of heat Q is removed from a “large” thermal reservoir at temperature T (large

enough that the temperature of the reservoir doesn’t change as a result of this

heat loss), the loss of entropy S from the reservoir is deﬁned as

S =

(3.8)

where T is an absolute temperature measured using either the Kelvin or Rankine

scale. Conversions from Celsius to Kelvin and from Fahrenheit to Rankine are

K =

◦

C + 273.15 (3.9)

R =

◦

F + 459.67 (3.10)

Equation (3.8) suggests that entropy goes down as temperature goes up. We

know that high-temperature heat is more useful than the same amount at lower

CARNOT EFFICIENCY FOR HEAT ENGINES 125

temperature, which reminds us that entropy is not such a good thing. Less

is better!

If we apply (3.8) to a heat engine, along with the requirement that entropy

must increase during its operation, we can easily determine the maximum possible

efﬁciency of such a machine. Since there is no entropy change associated with

the work done, the requirement that entropy must increase (or, at best break

even) tells us that the entropy added to the low-temperature sink must exceed

the entropy removed from the high-temperature reservoir:

≥

(3.11)

Rearranging (3.11)

≥

(3.12)

and substituting into (3.7) gives us the following constraint on the efﬁciency of

a heat engine:

Thermal efﬁciency = 1 −

≤ 1 −

(3.13)

That is, the maximum possible efﬁciency of a heat engine is given by

max

= 1 −

(3.14)

This is the classical result described by Carnot. One immediate observation that

can be made from (3.14) is that the maximum possible heat engine efﬁciency

increases as the temperature of the hot reservoir increases or the temperature of

the cold reservoir decreases. In fact, since neither inﬁnitely high temperatures nor

absolute zero temperatures are possible, we must conclude that no real engine

can convert thermal energy into mechanical energy with 100% efﬁciency—there

will always be waste heat rejected to the environment.

The following examples illustrate how (3.8) and (3.14) can be used in the

entropy analysis of heat engines.

Example 3.1 Entropy Analysis of a Heat Engine. Consider a 40% efﬁcient

heat engine operating between a large, high-temperature reservoir at 1000 K

(727

◦

C) and a large, cold reservoir at 300 K (27

◦

C).

a. If it withdraws 10

J/s from the high-temperature reservoir, what would be

the rate of loss of entropy from that reservoir and what would be the rate

of gain by the low-temperature reservoir?

b. Express the work done by the engine in watts.

c. What would be the total entropy gain of the system?

126 THE ELECTRIC POWER INDUSTRY

Solution

a. The loss of entropy from the high-temperature source would be

S

loss

J/s

1000 K

= 1000 J/s · K

Since 40% of the heat removed from the source is converted into work,

the remaining 60% is heat transfer into the cold-temperature sink. The rate

of entropy gain by the sink would be

S

gain

0.60 × 10

J/s

300 K

= 2000 J/s · K

c. The heat engine converts 40% of its input energy into work, which is

Work = 0.40 × 10

J/s×

J/s

= 400 kW

d. Since there is no entropy associated with the work done by the heat engine,

the total change in entropy of the entire system is the loss from the source

plus the gain to the sink:

S

total

=−1000 + 2000 +0 =+1000 J / s · K

(a) The example (b) A Carnot engine

= 10

J/s

W = 0.4 × 10

J/s

= 0.6 × 10

J/sK

=−1000 J/sK

∆

= 2000 J/sK

∆

= 0

∆

TOTAL

=+1000 J/sK

SOURCE

= 1000 K

= 10

J/s

= 0.7 × 10

J/s

= 0.3 × 10

J/sK

∆

=−1000 J/sK

∆

= 1000 J/sK

∆

= 0

∆S

TOTAL

= 0 J/sK

SINK

= 300 K

SINK

= 300 K

SOURCE

= 1000 K

70% eff

HEAT

ENGINE

40% eff

HEAT

ENGINE

Figure 3.17 Energy and entropy analysis of two heat engines. The example heat engine

(a) shows a net increase in entropy, while the Carnot engine (b) does not.

STEAM-CYCLE POWER PLANTS 127

The fact that there was a net increase in entropy in Example 3.1 tells us the

engine hasn’t violated the Carnot efﬁciency limit, which from (3.14) we know

would be 70% for this 1000 K source and 300 K sink. Figure 3.17 summarizes

the energy and entropy analysis for the example heat engine as well as for a

perfect Carnot engine.

3.5 STEAM-CYCLE POWER PLANTS

Conventional thermal power plants can be categorized by the thermodynamic

cycles they utilize when converting heat into work. Utility-scale thermal power

plants are based on either (a) the Rankine cycle, in which a working ﬂuid is alter-

nately vaporized and condensed, or (b) the Brayton cycle, in which the working

ﬂuid remains a gas throughout the cycle. Most baseload thermal power plants,

which operate more or less continuously, are Rankine cycle plants in which steam

is the working ﬂuid. Most peaking plants, which are brought on line as needed

to cover the daily rise and fall of demand, are gas turbines based on the Brayton

cycle. The newest generation of thermal power plants use both cycles and are

called combined-cycle plants.

3.5.1 Basic Steam Power Plants

The basic steam cycle can be used with any source of heat, including combustion

of fossil fuels, nuclear ﬁssion reactions, or concentrated sunlight onto a boiler.

The essence of a fossil-fuel-ﬁred steam power plant is diagrammed in Fig. 3.18.

In the steam generator, fuel is burned in a ﬁring chamber surrounded by a boiler

that transfers heat through metal tubing to the working ﬂuid. Water circulating

through the boiler is converted to high-pressure, high-temperature steam. During

this conversion of chemical to thermal energy, losses on the order of 10% occur

due to incomplete combustion and loss of heat up the stack.

High-pressure steam is allowed to expand through a set of turbine wheels

that spin the turbine and generator shaft. For simplicity, the turbine in Fig. 3.18

is shown as a single unit, but for increased efﬁciency it may actually consist

of two or sometimes three turbines in which the exhaust steam from a higher-

pressure turbine is reheated and sent to a lower-pressure turbine, and so forth.

The generator and turbine share the same shaft allowing the generator to convert

the rotational energy of the shaft into electrical power that goes out onto trans-

mission lines for distribution. A well-designed turbine may have an efﬁciency

approaching 90%, while the generator may have a conversion efﬁciency even

higher than that.

The spent steam is drawn out of the last turbine stage by the partial vacuum

created in the condenser as the cooled steam undergoes a phase change back to

the liquid state. The condensed steam is then pumped back to the boiler to be

reheated, completing the cycle.

The heat released when the steam condenses is transferred to cooling water,

which circulates through the condenser. Usually, cooling water is drawn from a