Gibilisco S. Teach Yourself Electricity and Electronics

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Antennas

Many types of antennas exhibit resonant properties. The simplest type of resonant antenna, and the

only kind that will be mentioned here, is the center-fed, half-wavelength dipole antenna (Fig. 17-14).

The length L

, in feet, for a dipole antenna at a frequency of f

, in megahertz, is given by the

following formula:

= 468/f

This takes into account the fact that electromagnetic fields travel along a wire at about 95 percent

of the speed of light. A straight, thin wire in free space has a velocity factor of approximately 0.95.

If the length of the half-wave dipole is specified in meters as L

, then:

= 143/f

A half-wave dipole has a purely resistive impedance of about 73 Ω at its fundamental frequency f

But this type of antenna is also resonant at all harmonics of f

. The dipole is a full wavelength long at

; it is

⁄2 wavelength long at 3f

; it is two full wavelengths long at 4f

, and so on.

Radiation Resistance

At f

and all of the odd harmonics, the antenna behaves like a series-resonant RLC circuit with a

fairly low resistance. At all even harmonics, the antenna acts like a parallel-resonant RLC circuit with

a high resistance. Does this confuse you? There’s no resistor in Fig. 17-14! Where, you ask, does the

resistance come from in the half-wave dipole? The answer to this is rather esoteric, and it brings to

light an interesting property that all antennas have. It is called radiation resistance, and is a crucial

factor in the design and construction of all RF antenna systems.

When electromagnetic energy is fed into an antenna, power is radiated into space in the form

of radio waves. This is a manifestation of true power, just as the dissipation of power in a pure re-

sistance is a manifestation of true power. Although there is no physical resistor in Fig. 17-14, the ra-

diation of radio waves is like power dissipation in a pure resistance. In fact, if a half-wave dipole

antenna were replaced with a 73-Ω nonreactive resistor that could dissipate enough power without

burning out, a radio transmitter connected to the opposite end of the line wouldn’t know the dif-

ference. (But a receiver would!)

Problem 17-14

How many feet long is a quarter-wave section of transmission line at 7.05 MHz, if the velocity fac-

tor is 0.800?

Resonant Devices 281

17-14 The half-wave,

center-fed dipole is a

simple and efficient

antenna.

Just use the formula:

= 246v/f

= (246 × 0.800)/7.05

= 197/7.05

= 27.9 ft

Quiz

Refer to the text in this chapter if necessary. A good source is 18 or more correct. Answers are in the

back of the book.

1. The power in a pure reactance is

(a) radiated.

(b) true.

(d) apparent.

2. Which of the following is not an example of true power?

(a) Power in the form of heat, produced by dc flowing through a resistor

(b) Power in the form of electromagnetic fields, radiated from a radio antenna

(d) Power in the form of heat, produced by losses in an RF transmission line

3. Suppose the apparent power in a circuit is 100 W, and the imaginary power is 40 W. What is

the true power?

(a) 92 W

(b) 100 W

(d) It is impossible to determine from this information.

4. Power factor is equal to

(a) apparent power divided by true power.

(b) imaginary power divided by apparent power.

(d) true power divided by apparent power.

5. Suppose a circuit has a resistance of 300 Ω and an inductance of 13.5 µH in series, and is

operated at 10.0 MHz. What is the power factor?

(a) 0.334

(b) 0.999

(d) It cannot be determined from the information given.

282 Power and Resonance in Alternating-Current Circuits

6. Suppose a series circuit has Z = 88.4 Ω, with R = 50.0 Ω. What is the power factor, expressed

as a percentage?

(a) 99.9 percent

(b) 56.6 percent

(d) 29.5 percent

7. Suppose a series circuit has R = 53.5 Ω, with X = 75.5 Ω. What is the power factor, expressed

as a percentage?

(a) 70.9 percent

(b) 81.6 percent

(d) 63.2 percent

8. The phase angle in an ac circuit is equal to

(a) arctan (Z/R).

(b) arctan (R/Z).

(d) arctan (X/R).

9. Suppose an ac ammeter and an ac voltmeter indicate that there are 220 W of VA power in a

circuit that consists of a resistance of 50 Ω in series with a capacitive reactance of −20 Ω. What is

the true power?

(a) 237 W

(b) 204 W

(d) 81.6 W

10. Suppose an ac ammeter and an ac voltmeter indicate that there are 57 W of VA power in a

circuit. The resistance is known to be 50 Ω, and the true power is known to be 40 W. What is the

absolute-value impedance?

(a) 50 Ω

(b) 57 Ω

(d) It is impossible to determine on the basis of this data.

11. Which of the following should be minimized in an RF transmission line?

(a) The load impedance

(b) The load resistance

(d) The transmitter power

Quiz 283

12. Which of the following does not increase the loss in a transmission line?

(a) Reducing the power output of the source

(b) Increasing the degree of mismatch between the line and the load

(d) Raising the frequency

13. Which of the following is a significant problem that standing waves can cause in an RF

transmission line?

(a) Line overheating

(b) Excessive power loss

(d) All of the above

14. Suppose a coil and capacitor are in series. The inductance is 88 mH and the capacitance is

1000 pF. What is the resonant frequency?

(a) 17 kHz

(b) 540 Hz

(d) 540 kHz

15. Suppose a coil and capacitor are in parallel, with L = 10.0 µH and C = 10 pF. What is f

(a) 15.9 kHz

(b) 5.04 MHz

(d) 50.4 MHz

16. Suppose you want to build a series-resonant circuit with f

= 14.1 MHz. A coil of 13.5 µH is

available. How much capacitance is needed?

(a) 0.945 µF

(b) 9.45 pF

(d) 945 pF

17. Suppose you want to build a parallel-resonant circuit with f

= 21.3 MHz. A capacitor of

22.0 pF is available. How much inductance is needed?

(a) 2.54 mH

(b) 254 µH

(d) 2.54 µH

18. A

⁄4-wave section of transmission line is cut for use at 21.1 MHz. The line has a velocity

factor of 0.800. What is its physical length in meters?

(a) 11.1 m

(b) 3.55 m

284 Power and Resonance in Alternating-Current Circuits

(d) 2.84 m

19. What is the fourth harmonic of 800 kHz?

(a) 200 kHz

(b) 400 kHz

(d) 4.00 MHz

20. Suppose you want to build a

⁄2-wave dipole antenna designed to have a fundamental

resonant frequency of 3.60 MHz. How long should you make it, as measured from end to end in

feet?

(a) 130 ft

(b) 1680 ft

(d) 515 ft

Quiz 285

TRANSFORMERS ARE USED TO OBTAIN THE OPTIMUM VOLTAGE FOR THE OPERATION OF A CIRCUIT OR

system. Transformers can also match impedances between a circuit and a load, or between two

different circuits. Transformers can be used to provide dc isolation between electronic circuits

while letting ac pass. Another application is to mate balanced and unbalanced circuits, feed systems,

and loads.

Principle of the Transformer

When two wires are near each other and one of them carries a fluctuating current, a fluctuating cur-

rent is induced in the other wire. This effect is known as electromagnetic induction. All ac transformers

work according to the principle of electromagnetic induction. If the first wire carries sine-wave ac of a

certain frequency, then the induced current is sine-wave ac of the same frequency in the second wire.

The closer the two wires are to each other, the greater is the induced current, for a given current

in the first wire. If the wires are wound into coils and placed along a common axis (Fig. 18-1), the

induced current will be greater than if the wires are straight and parallel. Even more coupling, or ef-

ficiency of induced-current transfer, is obtained if the two coils are wound one atop the other.

Primary and Secondary

The two windings, along with the core on which they are wound, constitute a transformer. The first

coil is called the primary winding, and the second coil is known as the secondary winding. These are

often spoken of simply as the primary and the secondary. The induced current in the secondary cre-

ates a voltage between its end terminals. In a step-down transformer, the secondary voltage is less

than the primary voltage. In a step-up transformer, the secondary voltage is greater than the primary

voltage. The primary voltage is abbreviated E

pri

, and the secondary voltage is abbreviated E

sec

. Un-

less otherwise stated, effective (rms) voltages are always specified.

The windings of a transformer have inductance, because they are coils. The required induc-

tances of the primary and secondary depend on the frequency of operation, and also on the resistive

part of the impedance in the circuit. As the frequency increases, the needed inductance decreases. At

high resistive impedances, more inductance is generally needed than at low resistive impedances.

286

CHAPTER

Transformers and

Impedance Matching

Turns Ratio

The primary-to-secondary turns ratio in a transformer is the ratio of the number of turns in the pri-

mary, T

pri

, to the number of turns in the secondary, T

sec

. This ratio is written T

pri

sec

or T

pri

sec

. In

a transformer with excellent primary-to-secondary coupling, the following relationship always

holds:

pri

sec

= T

pri

sec

That is, the primary-to-secondary voltage ratio is always equal to the primary-to-secondary turns

ratio (Fig. 18-2).

Problem 18-1

Suppose a transformer has a primary-to-secondary turns ratio of exactly 9:1. The ac voltage at the

primary is 117 V rms. Is this a step-up transformer or a step-down transformer? What is the voltage

across the secondary?

This is a step-down transformer. Simply plug in the numbers in the preceding equation and

solve for E

sec

, as follows:

pri

sec

= T

pri

sec

117/E

sec

= 9.00

1/E

sec

= 9.00/117

sec

= 117/9.00

= 13.0 V rms

Principle of the Transformer 287

18-1 Magnetic lines of flux

between two aligned

coils of wire when one

of the coils carries

fluctuating or

alternating current.

18-2 The primary voltage (E

pri

) and

secondary voltage (E

sec

) in a

transformer depend on the number

of turns in the primary winding

pri

) versus the number of turns in

the secondary winding (T

sec

Problem 18-2

Consider a transformer with a primary-to-secondary turns ratio of exactly 1:9. The voltage at the

primary is 121.4 V rms. Is this a step-up transformer or a step-down transformer? What is the volt-

age at the secondary?

This is a step-up transformer. Plug in numbers and solve for E

sec

, as follows:

121.4/E

sec

= 1/9.000

sec

/121.4 = 9.000

sec

= 9.000 × 121.4

= 1093 V rms

Sometimes the secondary-to-primary turns ratio is given, rather than the primary-to-secondary

turns ratio. This is written T

sec

pri

. In a step-down unit, T

sec

pri

is less than 1. In a step-up unit,

sec

pri

is greater than 1. When you hear someone say that such-and-such a transformer has a cer-

tain “turns ratio,” say 10:1, be sure of which ratio is meant, T

pri

sec

or T

sec

pri

! If you get it wrong,

you’ll have the secondary voltage wrong by a factor of the square of the turns ratio.

Ferromagnetic Cores

If a ferromagnetic substance such as laminated iron or powdered iron is placed within the pair of

coils, the extent of coupling is increased far above that possible with an air core. But this improve-

ment in coupling is obtained at a price. Some energy is invariably lost as heat in the core. Also, fer-

romagnetic cores limit the maximum frequency at which a transformer will work well.

The schematic symbol for an air-core transformer consists of two inductor symbols back-to-

back (Fig. 18-3A). If a laminated iron core is used, two parallel lines are added to the schematic sym-

bol (Fig. 18-3B). If the core is made of powdered iron, the two parallel lines are broken or dashed

(Fig. 18-3C).

In transformers for 60-Hz utility ac, and also for low audio-frequency (AF) use, sheets of an

alloy called silicon steel, glued together in layers, are often employed as transformer cores. The sili-

con steel is sometimes called transformer iron. The reason layering is used, rather than making the

core from a single mass of metal, is that the magnetic fields from the coils cause currents to flow in

a solid core. These eddy currents go in circles, heating up the core and wasting energy that would oth-

erwise be transferred from the primary to the secondary. Eddy currents are choked off by breaking

up the core into layers, so that currents cannot flow very well in circles.

A rather esoteric form of loss, called hysteresis loss, occurs in all ferromagnetic transformer cores,

but especially laminated iron. Hysteresis is the tendency for a core material to be sluggish in accept-

288 Transformers and Impedance Matching

18-3 Schematic symbols for transformers. At A, air core. At B, laminated

iron core. At C, ferrite or powdered iron core.

ing a fluctuating magnetic field. Laminated cores exhibit high hysteresis loss above the AF range,

and are therefore not good above a few kilohertz.

At frequencies up to several tens of megahertz, powdered iron works well for RF transformers.

This material has high magnetic permeability and concentrates the flux efficiently. High permeabil-

ity cores minimize the number of turns needed in the coils, and this minimizes the loss that occurs

in the wires.

Geometries

The properties of a transformer depend on the shape of its core, and on the way in which the wires

are wound on it. There are several different geometries used with transformers.

E Core

A common core for a power transformer is the E core, so named because it is shaped like the capital

letter E. A bar, placed at the open end of the E, completes the core assembly after the coils have been

wound on the E-shaped section (Fig. 18-4A).

The primary and secondary windings can be placed on an E core in either of two ways. The

simpler winding method is to put both the primary and the secondary around the middle bar of the

E (Fig. 18-4B). This is called the shell method of transformer winding. It provides maximum cou-

pling between the windings. However, this scheme results in considerable capacitance between the

primary and the secondary. Such interwinding capacitance can sometimes be tolerated, but often it

cannot. Another disadvantage of the shell geometry is that, when windings are placed one on top of

the other, the transformer cannot handle very much voltage. High voltages cause arcing between the

windings, which can destroy the insulation on the wires and lead to permanent short circuits.

Another winding method is the core method. In this scheme, one winding is placed at the bot-

tom of the E section, and the other winding is placed at the top (Fig. 18-4C). The coupling occurs

Geometries 289

18-4 At A, a utility transformer E core, showing both sections. At B, the shell

winding method. At C, the core winding method.

by means of magnetic flux in the core. The interwinding capacitance is lower than it is in a shell-

wound transformer because the windings are physically farther apart. Also, a core-wound trans-

former can handle higher voltages than a shell-wound transformer of the same physical size.

Sometimes the center part of the E is left out of the core when the core winding scheme is used.

Shell-wound and core-wound transformers are almost universally employed at 60 Hz. These

configurations are also common at AF.

Solenoidal Core

A pair of cylindrical coils, wound around a rod-shaped piece of powdered iron or ferrite, was once

a common configuration for RF transformers. Sometimes this type of transformer is still seen, al-

though it is most often used as a loopstick antenna in portable radio receivers and in radio direction-

finding equipment. The coil windings can be placed one atop the other, or they can be separated

(Fig. 18-5) to reduce the capacitance between the primary and secondary.

In a loopstick antenna, the primary serves to pick up the radio signals. The secondary winding

provides an optimum impedance match to the first amplifier stage, or front end, of the radio re-

ceiver. The use of transformers for impedance matching is discussed later in this chapter.

Toroidal Core

The toroidal core (or toroid ) has become common for winding RF transformers. The core is a

donut-shaped ring of powdered iron. The coils are wound around the donut. The complete assem-

bly is called a toroidal transformer. The primary and secondary can be wound one over the other, or

they can be wound over different parts of the core (Fig. 18-6). As with other transformers, when the

windings are one on top of the other, there is more interwinding capacitance than when they are

separated.

Toroids confine practically all the magnetic flux within the core material. This allows toroidal

coils and transformers to be placed near other components without inductive interaction. Also, a

toroidal coil or transformer can be mounted directly on a metal chassis, and the operation is not

affected (assuming the wire is insulated or enameled).

290 Transformers and Impedance Matching

18-5 A solenoidal-core

transformer.