Resonant circuits often consist of coils and capacitors in series or parallel, but there are other kinds of hardware that exhibit resonance. Some of these are as follows.

#### Piezoelectric Crystals

Pieces of quartz, when cut into thin wafers and subjected to voltages, will vibrate at high frequencies. Because of the physical dimensions of such a piezoelectric crystal, these vibrations occur at a precise frequency fo, and also at whole-number multiples of fo. These multiples, 2fo, 3fo, 4fo, and so on, are called harmonic frequencies or simply harmonics. The frequency fo is called the fundamental frequency or simply the fundamental. The fundamental, fo, is defined as the lowest frequency at which resonance occurs. Quartz crystals can be made to act like LC circuits in electronic devices. A crystal exhibits an impedance that varies with frequency. The reactance is zero at fo and the harmonic frequencies.

#### Cavities

Lengths of metal tubing, cut to specific dimensions, exhibit resonance at very high, ultrahigh, and microwave radio frequencies. They work in much the same way as musical instruments resonate with sound waves. But the waves are electromagnetic, rather than acoustic. Such cavities, also called

cavity resonators, have reasonable physical dimensions at frequencies above about 150 MHz. Below this frequency, a cavity can be made to work, but it is long and unwieldy. Like crystals, cavities resonate at a fundamental frequency fo, and also at harmonic frequencies.

#### Sections of Transmission Line

When a transmission line is cut to 1⁄ 4 wavelength, or to any whole-number multiple of this, it behaves as a resonant circuit. The most common length for a transmission-line resonator is a 1⁄ 4 wavelength. Such a piece of transmission line is called a quarter-wave section.

When a quarter-wave section is short-circuited at the far end, it acts like a parallel-resonant LC circuit, and has a high resistive impedance at the resonant frequency fo. When it is open at the far end, it acts as a series-resonant LC circuit, and has a low resistive impedance at fo. In effect, a quarterwave section converts an ac short circuit into an ac open circuit and vice versa, at a specific frequency fo.

The length of a quarter-wave section depends on the desired fo. It also depends on how fast the electromagnetic energy travels along the line. This speed is specified in terms of a velocity factor, abbreviated v. The value of v is given as a fraction of the speed of light. Typical transmission lines have velocity factors ranging from about 0.66 to 0.95 (or 66 percent to 95 percent). This factor is provided by the manufacturers of prefabricated lines such as coaxial cable.

If the frequency in megahertz is fo and the velocity factor of a line is v, then the length Lft of a quarter-wave section of transmission line, in feet, is given by this formula:

**Lft = 246v/fo**

The length Lm in meters is given by this:

**Lm = 75.0v/fo**

We use L here to stand for “length,” not “inductance”!

#### Antennas

**The half-wave, center-fed dipole is a simple and efficient antenna.**

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 (above figure). The length Lft, in feet, for a dipole antenna at a frequency of fo, in megahertz, is given by the following formula:

**Lft = 468/fo**

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 Lm, then:

**Lm = 143/fo**

A half-wave dipole has a purely resistive impedance of about 73 Ω at its fundamental frequency fo. But this type of antenna is also resonant at all harmonics of fo. The dipole is a full wavelength long at 2fo; it is 3⁄ 2 wavelength long at 3fo; it is two full wavelengths long at 4fo, and so on.

#### Radiation Resistance

At fo 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 resistance is a manifestation of true power. Although there is no physical resistor in above figure, the radiation 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 difference. (But a receiver would!)

#### Problem 1

**How many feet long is a quarter-wave section of transmission line at 7.05 MHz, if the velocity factor is 0.800?**

Just use the formula:

**Lft = 246v/fo
= (246 × 0.800)/7.05
= 197/7.05
= 27.9 ft **