Radio-Frequency Transformers

In radio receivers and transmitters, transformers can be categorized generally by the method of construction used. Some have primary and secondary windings, just like utility and audio units. Others employ transmission-line sections. These are the two most common types of transformer found at radio frequencies.

Wire-Wound Types

In wire-wound RF transformers, powdered-iron cores can be used up to quite high frequencies. Toroidal cores are common, because they are self-shielding (all of the magnetic flux is confined within the core material). The number of turns depends on the frequency, and also on the permeability of the core.

In high-power applications, air-core coils are often preferred. Although air has low permeability, it has negligible hysteresis loss, and will not heat up or fracture as powdered-iron cores sometimes do. The disadvantage of air-core coils is that some of the magnetic flux extends outside of the coil. This affects the performance of the transformer when it must be placed in a cramped space, such as in a transmitter final-amplifier compartment.

A major advantage of coil-type transformers, especially when they are wound on toroidal cores, is that they can be made to work over a wide band of frequencies, such as from 3.5 MHz to 30 MHz. These are called broadband transformers.

Transmission-Line Types

A quarter-wave matching section of transmission line. The input impedance is Rin, the output impedance is Rout, and the characteristic impedance of the line is Zo.
As you recall, any transmission line has a characteristic impedance, or Zo, that depends on the line construction. This property is sometimes used to make impedance transformers out of coaxial or parallel-wire line.

Transmission-line transformers are always made from quarter-wave sections. From the previous section, remember the formula for the length of a quarter-wave section:
Lft = 246v/fo
where Lft is the length of the section in feet, v is the velocity factor expressed as a fraction, and fo is the frequency of operation in megahertz. If the length Lm is specified in meters, then:
Lm = 75v/fo
Suppose that a quarter-wave section of line, with characteristic impedance Zo, is terminated in a purely resistive impedance Rout. Then the impedance that appears at the input end of the line, Rin, is also a pure resistance, and the following relations hold:
Zo2 = RinRout
Zo = (RinRout)1/2

This is illustrated in above figure. The first of the preceding formulas can be rearranged to solve for Rin in terms of Rout, or vice versa:
Rin = Zo2/Rout
Rout = Zo2/Rin

These equations are valid at the frequency fo for which the line length measures 1⁄ 4 wavelength. Sometimes, the word “wavelength” is replaced by the lowercase Greek letter lambda (λ), so you will occasionally see the length of a quarter-wave section denoted as (1⁄ 4)λ or 0.25λ.

Neglecting line losses, the preceding relations hold at all odd harmonics of fo, that is, at 3fo, 5fo, 7fo, and so on. At other frequencies, a quarter wave section of line does not act as a transformer. Instead, it behaves in a complex manner that is beyond the scope of this discussion.

Quarter-wave transmission-line transformers are most often used in antenna systems, especially at the higher frequencies, where their dimensions become practical. A quarter-wave matching section should be made using unbalanced line if the load is unbalanced, and balanced line if the load is balanced.

A disadvantage of quarter-wave sections is the fact that they work only at specific frequencies. But this is often offset by the ease with which they are constructed, if radio equipment is to be used at only one frequency, or at odd-harmonic frequencies.

Problem 1

Suppose an antenna has a purely resistive impedance of 100 Ω. It is connected to a 1⁄ 4-wave section of 75-Ω coaxial cable. What is the impedance at the input end of the section?
Use the formula from above:
Rin = Zo2/Rout
= 752/100
= 5625/100
= 56 Ω

Problem 2

Consider an antenna known to have a purely resistive impedance of 600 Ω. You want to match it to the output of a radio transmitter designed to work into a 50.0-Ω pure resistance. What is the characteristic impedance needed for a quarter-wave matching section?
Use this formula:
Z2 = RinRout
= 600 × 50
= 30,000

Zo = (30,000)1/2
= 173 Ω

It may be difficult to find a commercially manufactured transmission line that has this particular characteristic impedance. Prefabricated lines come in standard Zo values, and a perfect match might not be obtainable. In that case, the closest obtainable Zo should be used. In this case, it would probably be 150 Ω. If nothing is available anywhere near the characteristic impedance needed for a quarter wave matching section, then a coil-type transformer can be used instead.

What about Reactance?

Things are simple when there is no reactance in an ac circuit using transformers. But often, especially in RF antenna systems, pure resistance doesn’t occur naturally. It has to be obtained by using inductors and/or capacitors to cancel the reactance out. The presence of reactance in a load makes a perfect match impossible with an impedance-matching transformer alone.

Recall that inductive and capacitive reactances are opposite in effect, and that their magnitudes can vary. If a load presents a complex impedance R + jX, it is possible to cancel the reactance X by deliberately introducing an equal and opposite reactance −X. This can be, and often is, done by connecting an inductor or capacitor in series with a load that contains reactance as well as resistance. The result is a pure resistance with a value equal to (R + jX ) − jX, or simply R.

When wireless communications is contemplated over a wide band of frequencies, adjustable impedance-matching and reactance-canceling networks can be placed between the transmitter and the antenna system. Such a device is called a transmatch or an antenna tuner. These devices not only match the resistive portions of the transmitter and load impedances, but they can tune out reactances in the load. Transmatches are popular among amateur radio operators, who use equipment apable of operation from less than 2 MHz up to the highest known radio frequencies.