Sine Waves: Peak, RMS and Average

Everything up to here has been direct current: a steady push in one direction. Radio is not made of that. A radio signal is a current that reverses direction millions of times a second, and the moment you start describing such a current you discover that the simple question "what is the voltage?" has four different correct answers. This lesson sorts those four out, fixes the vocabulary of cycles and frequency, and gives you the one division sum — 300 divided by the frequency — that turns a band in megahertz into a length of wire.

What alternating current is, and why radio needs it

In a direct current the electrons drift steadily one way. In an alternating current they slosh: the driving voltage rises from zero to a maximum in one direction, falls back through zero, rises to a maximum the other way, and returns. That whole out-and-back journey is one cycle, and it repeats.

Three things about radio make alternating current not merely convenient but compulsory:

• Only a changing current radiates. A steady current in a wire produces a steady magnetic field that stays wrapped around the wire. A changing current produces a changing field, and a changing magnetic field produces an electric field, which produces a magnetic field, and the pair walk away from the antenna at the speed of light. Feed a dipole from a battery and nothing leaves it.
• Only a changing current works a transformer. As the transformers lesson showed, mutual inductance responds to change alone, which is why the mains is distributed as AC and why a power supply begins with a transformer.
• Frequency is what lets stations share the spectrum. If every transmission were DC there would be one channel in the world. Because each station alternates at its own rate, a tuned circuit can pick one out and ignore the rest.

Why the sine wave is the shape

An alternating quantity could in principle have any repeating shape — a square wave, a triangle, a sawtooth. The sine wave matters more than the others for two reasons.

First, it is what rotation naturally produces. Spin a coil steadily in a magnetic field and the voltage induced in it follows the sine of the angle it has turned through: maximum when the coil is cutting the field squarely, zero when it is moving along the field. Every alternator in every power station is doing exactly this, which is why the mains is a sine wave and not something else.

Second, the sine wave is the only shape that contains a single frequency. Any other repeating waveform can be built up by adding sine waves of higher frequencies — its harmonics — and that is not just mathematics but a licensing problem: a transmitter whose output is a distorted, non-sinusoidal wave is radiating on frequencies you are not authorised to use. A clean carrier is a clean sine.

Cycle, period, frequency

Three words for the same repetition, seen from different angles.

The hertz is named for Heinrich Hertz, who first produced and detected radio waves in the laboratory. One hertz is one cycle per second; a kilohertz is a thousand, a megahertz a million, a gigahertz a thousand million. Period and frequency are simply two ways of stating the same fact, so each is the reciprocal of the other:

Worked example 1 — the period of the mains

Indian mains runs at 50 Hz. T = 1 ÷ 50 = 0.02 s = 20 ms. Each cycle lasts twenty milliseconds, so each half cycle lasts ten. Remember that ten: it is why a full-wave rectifier delivers a pulse every 10 ms, a fact the power supplies lesson leans on.

Worked example 2 — the period of a 40 m signal

A carrier on 7.1 MHz has a period of 1 ÷ 7 100 000 = 0.000000141 s, that is 141 nanoseconds. In the time one mains cycle takes, that carrier completes about 142 000 cycles.

The four numbers people mean by "the voltage"

Because the waveform is moving, no single number describes it completely. Four different ones are in daily use, and the exam expects you to convert between them.

Those two factors are √2 and 1 ÷ √2, and they are worth recognising as such, because they turn up again in bandwidth and in the half-power point in the resonance lesson.

Set the explorer below to peak value and type 100. Then press the Indian mains button and watch the same four lines rearrange themselves around 230 V. The point of the widget is that these are not four different waves — they are four measurements of one wave.

The average trap

The average of a complete sine cycle is zero. It must be: the waveform spends exactly as long below the axis as above it, and the two halves cancel precisely. The figure of 0.637 is the average of a half cycle, or equally of a full-wave rectified sine, where every half cycle has been folded up above the axis. For a half-wave rectified sine — one hump, then a gap — the average over the whole cycle is half of that, 0.318 × peak.

What RMS actually means

RMS stands for root mean square, and the name is the recipe: square every instantaneous value, take the mean of those squares, then take the square root. Squaring is what makes it work, because power depends on the square of the current and a squared value is never negative. The result is the number the exam states in words: the RMS value of an alternating current is the value of direct current that would produce the same heating effect in the same resistance. One ampere RMS heats a 50 Ω dummy load exactly as much as one ampere from a battery.

That is why RMS is the value everything is rated in. Fuses, transformer windings, resistor dissipation, transmitter output and the mains itself are all quoted RMS, and an ordinary AC voltmeter is calibrated to indicate RMS whatever it internally measures.

Form factor and crest factor

Form factor is not an academic curiosity. A moving-coil meter with a rectifier in front of it physically responds to the average of the rectified wave, not to the RMS. Manufacturers simply print the scale 1.11 times higher so that it reads RMS on a sine wave — which is also why such a meter misreads a distorted or non-sinusoidal waveform. Crest factor tells you how far above the RMS figure the insulation and the semiconductors have to survive.

Indian mains: 230 V, 50 Hz, and a 325 V peak

Many electronics textbooks are written for American mains and quote 120 V at 60 Hz. For this exam and for your own safety the numbers are 230 V RMS at 50 Hz.

Worked example 3 — the peak of the mains

peak = 1.414 × 230 = 325 V, and peak-to-peak is 2 × 325 = 650 V.

A capacitor connected across the mains sees 325 V, not 230 V, so a 250 V part will fail. A rectifier diode fed from the mains must block a reverse peak of 325 V. The reservoir capacitor in a transformer supply charges to the peak of the secondary, which is why a 12 V transformer gives roughly 17 V of unregulated DC.

Angles and the radian

Because a sine wave comes from rotation, positions within a cycle are described as angles. One complete cycle is 360°, the peak occurs at 90°, the zero crossing at 180° and the negative peak at 270°. Engineers also use the radian, the angle subtended by an arc equal to the radius: one full turn is 2π radians, so one radian is about 57.3°. Angular frequency, written ω, is 2πf radians per second — the same 2πf that appears in every reactance formula in the next lesson.

Wavelength

A radio wave travels at very nearly 300 000 000 metres per second. In the time of one cycle it therefore advances a certain distance, and that distance is the wavelength, λ. Divide the speed by the frequency and, if you work in megahertz, the awkward zeros cancel:

Worked example 4

What is the wavelength at 14.2 MHz? λ = 300 ÷ 14.2 = 21.1 m — which is why 14 MHz is called the 20 metre band. At 145 MHz, λ = 300 ÷ 145 = 2.07 m, the 2 metre band. Running it the other way, a station on the 40 metre band is near 300 ÷ 40 = 7.5 MHz, and indeed the band sits at 7.0–7.2 MHz.

Antenna lengths come straight out of this sum, so it is worth being fluent with it. The antenna calculator does the arithmetic including the end-effect correction, but do a few by hand first.

Audio frequencies and radio frequencies

The dividing line is human hearing. Audio frequencies run roughly from 20 Hz to 20 kHz — the range a healthy young ear can detect, and the range a microphone and loudspeaker handle. Speech intelligibility needs only about 300 Hz to 3 kHz, which is why an SSB transmitter is filtered to about that width.

Anything above the audio range is a radio frequency. The ITU's numbered bands begin at 3 kHz with VLF, and in practice a frequency is treated as radio if it can be radiated usefully from an antenna. The distinction is a practical one in the shack: audio-frequency currents are handled with ordinary wire and electrolytic capacitors, radio-frequency currents need coaxial cable, screening and components that behave themselves at megahertz.