RF Journey

01 / 10

What actually is RF?

What is travelling from a transmitter to a receiver?

RF stands for radio frequency. Radio waves belong to the same family of electromagnetic waves as visible light, but repeat much more slowly. In a circuit, a signal is a changing quantity we can measure, such as a voltage. By controlling its changes, we can use it to carry information.

A transmitter creates a changing voltage and current.
A cable or a PCB trace—a metal track on a circuit board—guides the signal to an antenna.
The antenna launches a travelling pattern of electric and magnetic fields into space.
A receiving antenna responds to those fields, producing an electrical signal that a receiver can interpret.

A field describes the electrical or magnetic influence at each position in space. Fields also carry energy along cables: in a coaxial cable, an inner conductor and an outer metal shield guide them. Voltage and current describe what we measure at the conductors; fields describe what is happening in the surrounding space.

An electrical change takes time to travel. RF engineering pays attention to the path’s dimensions and to how that travel time compares with changes in the signal.

From an electrical signal to a radio wave

Slow-motion visualization

At a fixed point, the fields keep oscillating. Farther away, their peaks are smaller as the wave spreads over a larger area.

Electric field ↕Magnetic field ⊙ out of screen / ⊗ into screen

About this illustration

The goal is to connect the whole radio link: a transmitter drives an antenna through a cable or PCB trace, a wave travels through space, and a receiving antenna produces an electrical signal. Notice that the fields change at fixed locations while the wave pattern moves outward.

Motion is slowed, and sizes and signal strengths are illustrative. The curve shows field values, not a wavy flight path. Cable delay and losses, reflections, and the complex fields close to the antennas are omitted. This is not a prediction of a real radio’s range or received voltage.

These ideas will be explained in more detail in later modules, especially Fields & Waves, and in the Transmission Lines & Matching (in preparation) and Antennas & Propagation (in preparation) courses.

Think about itCould a signal that never leaves a coax cable still be RF?

Answer

Yes. A radio-frequency signal can remain guided inside a cable. It does not have to be sent wirelessly to be RF.

02 / 10

A signal that repeats

When we say that a 2.45 GHz signal repeats, what exactly is repeating?

Start with a voltage at one point on a cable. It rises, crosses its middle value, falls, and eventually returns to the same value while moving in the same direction. That full repeat is one cycle.

A sinusoid, or sine wave, is a smooth repeating pattern. Imagine keeping a voltage probe at one point and recording its reading as time passes. Place time on the horizontal axis and voltage on the vertical axis: the collected readings draw the familiar curve. An oscilloscope is an instrument that displays this view.

We can also measure an electric field instead of voltage. Now imagine many fixed field sensors along the route, all read at the same instant. Their readings give a spatial snapshot: field value vertically, position horizontally. One view follows a point through time; the other compares positions at one time.

Explore the animation

What do the two graphs tell us about the same wave?

A Wave in Space Becomes a Waveform in Time

Slow-motion visualizationElectric field · same vertical scale

A stays in one place as the wave passes. Recording the field there over time draws the waveform.

Both dots show the same reading now.

About this illustration

The goal is to connect a travelling wave with a measurement at one fixed place. The left view compares positions right now; the right keeps the latest two cycles measured at A. New readings enter at “Now” and older readings scroll left. One period is the time for one complete cycle.

This is an ideal electric field with constant strength, shown in slow motion. Both views use the same relative field scale. The curve’s height represents the field, not a wavy flight path; the dot at A represents its reading, not a moving sensor. Spreading, losses and instrument effects are omitted.

Period and wavelength are explained next in this module. Travelling fields are explored further in Fields & Waves.

Common misconceptionThe sine-wave curve is the path travelled by the wave.

It is not. The wave can propagate along a straight cable or through space while the plotted voltage or field value rises and falls. The curve shows value versus time or position; it does not show a particle or a packet of energy snaking along that route.

After the picture is clear, a sinusoid can be written as $x(t)=A\sin(2\pi ft+\phi)$ . The symbols name ideas we will build one at a time: amplitude $A$ , frequency $f$ , time $t$ , and phase $\phi$ , with the angle written in radians.

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03 / 10

Frequency & period

How can we describe the repetition without saying “very fast”?

Frequency, $f$ , counts how many complete cycles occur each second. The unit is hertz (Hz): 1 Hz means one cycle per second. A 100 MHz signal repeats 100 million times per second. A 2.45 GHz signal repeats 2.45 billion times per second.

Kilohertz (kHz) means thousands of cycles per second; megahertz (MHz) means millions; gigahertz (GHz) means billions.
A nanosecond (ns) is one billionth of a second. A picosecond (ps) is one trillionth: 1 ns = 1,000 ps.

Period, $T$ , asks the inverse question: how long does one cycle take? Frequency is cycles per second; period is seconds per cycle.

Predict, then explore

If frequency doubles, what happens to the period?

Try 1, 2, and 4 Hz and watch the marked period.

Interactive period model

One repeat takes time.

This scale is always one second long. At 1 Hz, that second contains one cycle.

Frequency

One period $T$ = 1 s

The blue dimension line marks one complete cycle, from one crest to the next.

Reveal the explanation

It halves. Twice as many cycles must fit into the same second, so each cycle gets half as much time.

$T=\frac1f\qquad\text{and}\qquad f=\frac1T$

T

is period [s];

f

is frequency [Hz].

To calculate the period of our 2.45 GHz signal, first write the frequency in hertz, then take its reciprocal. Convert the result from seconds into a smaller unit:

$T=\frac{1}{2.45\times10^9\,\mathrm{Hz}}\approx4.08\times10^{-10}\,\mathrm{s}$ $T\approx0.408\,\mathrm{ns}=408\,\mathrm{ps}$ Multiply seconds by 10⁹ to get nanoseconds, or by 10¹² to get picoseconds.

At 100 MHz, the same method gives 10 ns per cycle. Frequency and period are two ways to describe the same repetition; the unit conversion changes how we write it.

One metre of cable

Imagine a cable in which a signal travels at about 2 × 10⁸ m/s. One metre adds about 5 ns of delay. That is half of a 100 MHz cycle, but more than twelve complete 2.45 GHz cycles. An ideal cable may simply delay a sine wave without changing its shape. Later, we’ll explore why this delay matters when signals combine.

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04 / 10

Wavelength

Period is one cycle viewed in time. What does that same cycle look like across distance?

Freeze a travelling sinusoid for an instant. The distance from one point in the cycle to the next identical point is its wavelength, written $\lambda$ . Crest to crest is convenient, but any matching pair of points works.

The source sets the rhythm. If it produces the same number of cycles each second but those cycles travel more slowly, successive crests sit closer together. Frequency stays the same while wavelength decreases. Velocity factor (VF) compares the wave's speed with the speed of light: VF = 0.66 means 66% of that speed.

Same rhythm. Different spacing.

Slow-motion visualizationBoth: 100 MHz · 10 ns per cycle

Both signals repeat at the same rate. The slower wave in the cable travels less far each cycle, so its crests are closer together.

Dots follow crests · same scales in both plots

About this illustration

The goal is to show why slower travel means a shorter wavelength at the same frequency. Both sources complete each cycle together. The dots follow crests, and each bracket spans one wavelength.

These are ideal, steady sine waves with equal relative field strengths. Both plots share the same distance scale and slowed clock. The curves show field values along each path, not a wavy cable or moving particles. Spreading, losses and reflections are omitted; 66% of light speed is an example cable speed.

Guided waves and real cable behaviour are explored further in Transmission Lines & Matching (in preparation).

Predict, then explore

If frequency doubles while propagation velocity stays constant, what happens to wavelength?

Compare frequency first, then compare free space with the 0.66-VF cable. The optional PCB example models a slower wave on a circuit board.

Interactive wavelength model

One repeat takes up distance.

This scale is always 3 metres long. At 100 MHz in free space, those 3 metres contain 1 wavelength.

Frequency

Propagation medium

One wavelength $\lambda$ = 3 mPhase velocity: $c$

The blue line marks one wavelength. Here $c$ is the speed of light in free space. Cable velocity factor (VF) 0.66 means 66% of that speed. The example PCB uses effective permittivity $\varepsilon_{\mathrm{eff}} = 3$ , a material-and-geometry value that gives a speed of $\frac{c}{\sqrt{3}}$ , about 58% of $c$ . These are illustrative values: actual speeds depend on the construction and operating frequency.

Reveal the explanation

It halves. The wave completes twice as many cycles while travelling the same distance in one second, so each spatial cycle occupies half the distance.

$v_{p} = f\lambda \qquad \lambda = \frac{v_{p}}{f}$

v_p

is phase velocity—the speed of a crest [m/s];

f

is frequency [Hz];

\lambda

is wavelength [m].

In free space, use $v_p=c$ , where $c$ = 299,792,458 m/s. A 100 MHz wave is therefore about 3 m long. At 2.45 GHz, it is only about 122 mm long.

In the cable with VF = 0.66, multiply the free-space wavelength by 0.66. At 2.45 GHz, that gives about 80.8 mm instead of 122.4 mm. The period is still 408 ps: each cycle takes the same time but travels less far.

Go deeperWhat does the PCB setting mean?

The explorer uses effective relative permittivity, $\varepsilon_{\mathrm{eff}}$ , to describe how a PCB line's materials and geometry slow its wave. In this simplified model, $v_p=\frac{c}{\sqrt{\varepsilon_{\mathrm{eff}}}}$ . A value of 3 gives about 58% of the speed of light. It describes the whole line, including where its fields travel, rather than just the board material.

Common misconceptionHigher frequency means the signal travels faster.

Frequency says how often the signal repeats, not how fast it propagates. In vacuum, electromagnetic waves of every frequency travel at $c$ . When propagation speed stays constant, higher frequency means a shorter wavelength. In some materials, propagation speed also changes with frequency. This effect is called dispersion.

Common misconceptionWavelength is always the free-space value, so 2.45 GHz is always about 12.2 cm.

$c/f$ is the free-space wavelength. In a uniform, non-magnetic dielectric with low loss, relative permittivity $\varepsilon_r$ reduces the propagation speed and wavelength:

\lambda \approx \frac{c}{f\sqrt{\varepsilon_r}} = \frac{\lambda_0}{\sqrt{\varepsilon_r}}

Here $\lambda_0 = c/f$ is the free-space wavelength. At the same frequency, a larger $\varepsilon_r$ gives a shorter wavelength. For example, $\varepsilon_r = 4$ halves it.

For a PCB surface trace, use effective relative permittivity $\varepsilon_{\mathrm{eff}}$ in place of $\varepsilon_r$ , because its fields extend through both the board and the air. With $\varepsilon_{\mathrm{eff}} = 3$ , the wavelength at 2.45 GHz is about 70.6 mm instead of 122.4 mm in free space.

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05 / 10

Why wavelength matters in RF

When does a trace stop behaving like an ideal connection?

Electrical length compares a structure's physical length with the wavelength travelling through it. It answers a practical question: how much of one cycle fits along this structure?

Think about itWhich is electrically longer: the same 30 mm trace at 10 MHz or at 2.45 GHz?

Answer

The 2.45 GHz trace. Its physical length is unchanged, but the wavelength is far shorter, so 30 mm occupies a much larger fraction of a cycle. In a typical guided PCB structure, the wavelength is shorter still.

Compare its physical length with the signal's wavelength in that structure. A 30 mm feature is tiny beside the roughly 30 m free-space wavelength at 10 MHz. At 2.45 GHz, 30 mm is almost one quarter of a free-space wavelength, and it can be an even larger fraction of the shorter guided wavelength on a cable or PCB.

Wavelength and physical dimensions at 2.45 GHz

The free-space wavelength is about 122.4 mm. One quarter is about 30.6 mm. That does not make every 30.6 mm object an antenna, but it tells an RF engineer to stop treating that size as automatically negligible. In fact, dimensions well below a quarter-wavelength can already affect a circuit. We will explore when and why in When an Interconnect Becomes a Transmission Line (in preparation).

Same distance. Different phase.

Slow-motion visualizationBoth: 30 mm in free space · ≈ 100 ps travel time

At 100 MHz, A and B rise and fall almost together. At 2.45 GHz, B lags by nearly a quarter-cycle. The same travel time is a larger part of the shorter cycle.

Electric field · travels →

About this illustration

The goal is to connect distance with phase. These curves show field values along the path at one instant. A and B stay in place. Only a short part of each wave is visible, so the 100 MHz curve is nearly flat.

Both ideal sine waves travel at light speed, using equal field scales and one slowed clock; 2.45 GHz oscillates 24.5 times faster. Spreading, losses and reflections are omitted. Cables and PCB traces have different speeds. Delay alone does not damage the wave.

The later Phase section explains how differences in arrival time affect signals that combine. Guided waves are explored further in Transmission Lines & Matching (in preparation).

To name a position within a repeating cycle, imagine one turn around a circle. One complete cycle is 360°, a half-cycle is 180°, and a quarter-cycle is 90°. That position is called phase. The degrees describe progress through a cycle; the cable itself does not need to bend.

One turn. One cycle.

Slow-motion visualizationOne complete turn = 360°

The point’s angle is the phase; its height gives the sine wave’s value. Each turn adds another cycle as earlier values scroll left.

Linked dots · same height

About this illustration

The circle connects an angle with a position in a sine-wave cycle. We start at the right-hand point, call that 0°, and turn counterclockwise. New values appear at “Now” while earlier values scroll left. The graph keeps the latest one and a half cycles visible, and the bracket marks one full turn.

This is a mathematical picture of an ideal sine wave, not a particle travelling in a circle or a cable bending. Both views use the same relative height scale. The later Phase section explains how differences in phase affect signals that combine.

Now follow one crest from A to B in the earlier animation. The same crest reaches B about 100 ps after it reaches A. In this ideal propagation model, that travel time is the wave's phase delay. We write it as $\tau$ and measure it in seconds.

The corresponding phase lag is the difference in cycle position between A and B at the same instant, measured in degrees. A delay of one quarter of a period produces a phase lag of 90°. This phase change along the path is its electrical length:

$\theta = \frac{360^{\circ} L}{\lambda } = 360^{\circ}f\tau$

L

is physical length;

\tau=\frac L{v_p}

is phase delay [s] in this model;

\theta

is electrical length [degrees].

To use this relationship, first determine the propagation velocity along the actual path. Divide physical length by that velocity to get the delay, then compare the delay with one period. Using free-space wavelength for a cable would underestimate this phase change when the wave travels more slowly in the cable.

The lumped approximation neglects travel time within a component and treats ideal connecting wires as having negligible delay. A resistor can still have a voltage across its terminals. When travel time must be included, different positions can have different voltage, current, and phase. The structure has distributed behavior.

A delay does not automatically damage a signal: an ideal cable can deliver the same waveform later. It matters when we compare arrival times or combine signals that have taken different paths. In those cases, assuming that both ends change together gives the wrong answer.

A preview of the next lessons

Impedance describes the relationship between voltage and current. A sudden change in that relationship along a path can send part of a wave back as a reflection. The circuit connected at the end is the termination. We will study these in later modules; for now, keep the central idea that a connection takes time to carry a change.

Go deeperThere is no universal length cutoff

Rules such as “shorter than one tenth of a wavelength is lumped” are rough screening guidelines, not laws. Acceptable error, impedance discontinuities, loss, bandwidth, and measurement goals matter. Fast digital edges also contain frequency components far above the clock rate, so rise time can make a low-clock-rate interconnect an RF problem.

A line ten wavelengths long is not automatically worse than one three wavelengths long. At that frequency, an ideal line with no losses or reflections delays a sine wave by ten or three complete cycles. Both outputs line up with the input once settled, but the ten-wavelength line delivers a new change later. Real cables generally lose more signal as length or frequency increases; reflections also depend on the connections. Electrical length alone does not tell us whether a connection will work well.

Frequency and wavelength tell us when geometry matters. Next we need a different question: how large is the signal we are sending through that geometry?

06 / 10

Amplitude

Two signals can repeat at the same rate but have very different magnitudes. How do we say exactly how large one is?

Think about itWhich has more power: a 1 V signal or a 0.5 V signal?

Answer

There is not enough information. Ask: 1 V peak, peak-to-peak, or RMS? Across what impedance? What waveform? Only then can the powers be compared.

Amplitude is the magnitude of a stated quantity relative to a stated reference. We will begin with voltage relative to 0 V, but RF engineers also measure current, electric field, magnetic field, and power. “Amplitude” is incomplete until the quantity and unit are clear.

For a sine wave centered on 0 V, peak voltage, $V_{\mathrm{pk}}$ , runs from 0 V to one crest. Peak-to-peak voltage, $V_{\mathrm{pp}}$ , runs from the negative peak to the positive peak. RMS voltage, $V_{\mathrm{rms}}$ , is the equivalent DC voltage that would produce the same average power in a resistor.

Interactive amplitude model

Same timing. Different height.

This view always shows a zero-offset 1 kHz sine wave. Changing the peak voltage changes how far the signal rises above and below 0 V; it does not change the timing.

Peak voltage,

V_{\mathrm{pk}}

Peak voltage · $V_{\mathrm{pk}}$: 1 V
Peak-to-peak · $V_{\mathrm{pp}}$: 2 V
RMS voltage · $V_{\mathrm{rms}}$: 0.707 V

The blue line measures peak voltage from the 0 V reference to the crest. Waveform limit: these readouts assume a zero-offset sine. Other waveform shapes have different RMS-to-peak relationships, and a DC offset changes the total RMS value.

DC means a steady voltage here. Picture a resistor warming up: a voltage that changes sign can heat it just as a steady voltage can. For a resistor, instantaneous power is $p(t)=\frac{v(t)^2}{R}$ . Squaring makes both positive and negative voltages contribute positive power, so a zero average voltage does not imply zero heating.

RMS stands for root mean square: square the voltage as it changes over time, average those squared values over the measurement interval, then take the square root. For a sine wave centered on 0 V, the average squared voltage over complete cycles is $V_{\mathrm{pk}}^2/2$ .

$V_{\mathrm{pp}} = 2V_{\mathrm{pk}}$ $V_{\mathrm{rms}} = \sqrt{\frac{V_{\mathrm{pk}}^2}{2}} = \frac{V_{\mathrm{pk}}}{\sqrt{2}}$ These conversions apply to a sine wave centered on 0 V.

A load is the circuit receiving the signal. Let ours be a 50 Ω resistor, with a sine-wave voltage measured directly across it. For a 1 Vpp sine wave, the positive peak is 0.5 V, so the RMS voltage is about 0.354 V:

$V_{\mathrm{rms}}=\frac{0.5\,\mathrm V}{\sqrt2}\approx0.354\,\mathrm V$ $P_{\mathrm{avg}}=\frac{V_{\mathrm{rms}}^2}{R}=\frac{(0.5/\sqrt2)^2}{50}\,\mathrm W$ $P_{\mathrm{avg}}=0.0025\,\mathrm W=2.5\,\mathrm{mW}$ Power is energy delivered per second. A milliwatt (mW) is one thousandth of a watt.

By comparison, 1 Vrms across the same resistor gives 20 mW. Both may be casually called “one volt,” yet their powers differ by a factor of eight. The next module, RF Power, develops this relationship. For now, always include peak, peak-to-peak, or RMS with a voltage value.

Common misconceptionAmplitude directly equals power.

A voltage amplitude alone is not power. For a resistive load, $P=\frac{V_{\mathrm{rms}}^2}{R}$ , so waveform, impedance, and where the voltage is measured all matter.

Common misconception1 V always means the same thing.

It might mean peak, peak-to-peak, RMS, or even a source setting specified for a particular load. A trustworthy RF measurement states the qualifier, waveform, impedance, and measurement location: for example, “1 Vpp sine measured across the 50 Ω resistor at the cable output.”

Amplitude tells us how large a signal is. To understand what happens when signals meet, we also need to know where they are within their cycles.

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07 / 10

Phase

If two sine waves have the same frequency and amplitude, can they still behave differently?

Yes. They can be at different positions within their cycles. Phase describes that relative position as an angle: one complete cycle is 360°. To make a phase statement, choose the signal you are comparing against and the location where you compare them. That location, such as a cable connector, is the reference plane.

A signal leads another when it reaches the same part of its cycle earlier; it lags when it reaches that part later. Compare their marked positive peaks. Signals are coherent here when they have the same frequency and maintain a fixed phase relationship, so the comparison stays consistent from one cycle to the next.

Same signals. Different sums.

Slow-motion visualizationSame frequency · each input has peak 1

ABSum ·

\mathrm{A}+\mathrm{B}

In phase0° · no delay

Quarter-cycle lag90° · ¼ period

Half-cycle lag180° · ½ period

In phase, the sum doubles. Half a cycle apart, the inputs cancel but remain present.

Relative voltage · same scale throughout

About this illustration

This compares how three fixed phase differences change the sum of two equal sine waves. B lags A by zero, a quarter, or half a cycle. Every graph uses the same voltage scale and two-cycle time window, with new values appearing at “Now”.

The sum adds two ideal voltage contributions at one observation point. It is not a wiring diagram or a model of a specific combiner. Cancellation here does not destroy energy; a real system determines where that energy goes. Later lessons on transmission lines (in preparation) and antenna arrays (in preparation) explore how path delays and phase affect real systems.

In the optional experiment below, each sine wave is represented by an arrow, called a phasor. Its length represents amplitude, and its angle represents phase. The reference arrow points right. A second arrow a quarter-turn from it means 90° of phase difference; two opposite arrows mean 180°. These are cycle angles, not directions in which the signals travel.

Optional experiment

Explore phase and time delay

Compare the waveforms and their phasors, then keep the phase offset fixed and change frequency. Watch how the equivalent time shift changes.

Reference · 0°Signal · 90°At 0°, the arrows overlap.

Phase offset

Frequency

Phase difference90°

Equivalent time shift−102 ps

Signal 2leads the reference

Combined peak amplitude1.41 $A$

2A\left|\cos\left(\frac{\Delta\phi}{2}\right)\right|

The addition is partly constructive. Here, $A$ is the peak amplitude of either input signal. At 180°, lead and lag are equivalent modulo one cycle. A fixed phase relationship requires equal-frequency, coherent signals.

For two equal sine waves, let $A$ be the peak amplitude of either input:

At 0°, equal signals rise and fall together. Their sum has amplitude $2A$ .
At 90°, one signal is one quarter-cycle ahead or behind. Their sum has amplitude $\sqrt{2}A$ .
At 180°, one signal is at its positive peak when the other is at its negative peak. Ideal equal signals cancel.

Cancellation describes the sum at the observation point. It does not say that energy disappears. In a physical system, energy may travel elsewhere, return toward a source, or be absorbed by other parts of the circuit. The simple graph models only the summed value.

We can now convert a known phase delay into the phase lag it produces at a particular frequency:

$|\Delta \phi| = 360^{\circ}f\tau$ This gives the total phase-lag magnitude for a delay

\tau

at frequency

f

. A display that shows only one turn can show 450° as 90°, hiding the extra whole cycle.

A fixed 100 ps delay is only 3.6° at 100 MHz, 36° at 1 GHz, and about 88° at 2.45 GHz. The delay did not change. Its size relative to the cycle did.

Why engineers care

Cable length, PCB routing, filters, antenna spacing, and measurement fixtures all create phase differences. Those differences can produce reinforcement, cancellation, beam steering, or a misleading measurement, depending on where and how signals combine.

Common misconceptionPhase is an absolute property of one signal.

Phase is meaningful relative to another coherent signal, a defined time origin, and a stated reference plane. Move the reference plane along a cable and the measured phase changes.

Common misconceptionA fixed phase shift always represents the same time delay.

It does not. A 90° lag is 2.5 ns at 100 MHz, 250 ps at 1 GHz, and about 102 ps at 2.45 GHz. Always state the frequency with a phase-to-delay conversion.

Go deeperThe sign of delay

For a sine wave written as $\sin(2\pi f t + \phi)$ , delaying the signal by $\tau$ seconds gives $\sin(2\pi f(t - \tau) + \phi)$ : each peak arrives $\tau$ seconds later. At frequency $f$ in hertz, the phase change is $\Delta\phi = -360^\circ f\tau$ . The minus sign means the delayed signal lags behind the reference.

A phase display showing only one turn reports the angle modulo $360^\circ$ , meaning whole turns are removed. For example, $-450^\circ$ can appear as $-90^\circ$ . A single reading at one frequency therefore cannot distinguish a quarter-cycle delay from one full cycle plus a quarter-cycle.

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08 / 10

Why do we keep using sine waves?

Bluetooth and Wi-Fi carry data. Their real waveforms are not one endless, perfect sine wave, so why do RF engineers keep drawing sinusoids?

Because a complicated signal can be represented by sinusoidal components with different frequencies, amplitudes, and phases. Add the right components together and they reproduce the original waveform. This is the intuition behind Fourier analysis; we do not need the transform mathematics yet to use the idea.

The waveform's repetition frequency is its fundamental. Components at whole-number multiples of that frequency are called harmonics: for a 1 Hz fundamental, the third harmonic is 3 Hz and the fifth is 5 Hz. The builder uses these small numbers so we can read the graph easily.

Predict, then explore

What happens when we add smaller sinusoids at three and five times a starting frequency?

Build the approximation one component at a time.

Interactive harmonic builder

Can smooth sine waves build a square-like shape?

Start with a 1 Hz fundamental. Then add smaller odd harmonics. The light traces show the individual sinusoids; the blue trace is their exact point-by-point sum.

Fundamental · 1 Hz · amplitude 1
Sum

Included harmonics

What changed

\sin(2\pi t)

The 1 Hz fundamental sets the repetition rate. By itself, it remains smoothly rounded everywhere.

Adding these three smooth components makes the sum more square-like. It is still an approximation. The heights are relative to the first component, whose amplitude is 1; they are not voltages measured in a circuit.

Why these harmonics and amplitudes?

The target square wave spends equal time above and below zero, with no average offset. This symmetry means it needs only odd harmonics (1, 3, 5, and so on). Their amplitudes decrease as $1,\ \frac{1}{3},\ \frac{1}{5}$ . Three terms cannot reproduce a perfect square wave.

Equal high and low times are called a 50% duty cycle. A different duty cycle can also need even harmonics; a nonzero average adds a constant, or DC, component.

An ideal square wave with levels −1 and +1 requires an infinite sum of these odd harmonics, multiplied by $4/\pi \approx 1.27$ . Our example leaves out this factor so the first sine has amplitude 1. Including it would make every component and their sum about 27% taller, without changing their proportions or the shape of the waveform.

Reveal the explanation

Their sum develops flatter tops and steeper transitions. It begins to resemble a square wave, even though every ingredient is a smooth sinusoid.

How does this help a radio send information? Start with a steady RF sine wave called a carrier. The transmitter deliberately varies its amplitude, frequency, or phase according to a message. This is modulation. For example, changing the carrier's amplitude in step with an audio tone lets a receiver recover that tone from the amplitude changes.

That changing signal needs additional frequency components to describe it. In the simple amplitude-modulation example below, a carrier and two nearby components combine into a wave whose height rises and falls. These nearby components are sidebands. Bandwidth describes the width of the frequency range used to describe a signal; it is different from the carrier frequency at the centre of that range. A signal may still have a small amount of power outside this range. We therefore need to state how the bandwidth is defined, for example as the frequency range containing 99% of the signal's total power.

How a message changes a carrier

Slow-motion visualizationAmplitude modulation · one repeating message

Signal over time

The same signal in frequency

Frequency →

In amplitude modulation (AM), the carrier’s amplitude varies with the message signal, while its frequency stays constant.

VoltageEnvelope

About this illustration

The frequency graph is another view of the same signal. Each line marks a frequency present in it: the carrier in the middle and two extra components called sidebands. For example, an 8 MHz carrier varied by a 1 MHz message tone has sidebands at 7 MHz and 9 MHz. These three sine waves add together to produce the changing waveform.

This example uses one continuously repeating tone, slowed down for viewing. The spectrum shows the relative amplitude of the carrier and each sideband. These amplitudes remain constant in this example. Signals that start and stop spread across nearby frequencies; that effect is not shown. Later Signals & Modulation (in preparation) lessons explain these ideas in more detail.

Go deeperWhy AM has two sidebands

The upper sideband is above the carrier frequency; the lower sideband is below it. These names describe frequency, not the positive and negative parts of the voltage waveform.

For an 8 MHz carrier modulated by a 1 MHz tone, the lower sideband is at $8 - 1 = 7\,\mathrm{MHz}$ and the upper sideband is at $8 + 1 = 9\,\mathrm{MHz}$ . The carrier remains at 8 MHz.

Why two? Conventional AM multiplies the carrier by an amplitude that varies with the message. For a sinusoidal message, that multiplication produces components at both the sum and the difference of the carrier and message frequencies. A message such as speech contains many frequencies, so each sideband occupies a range rather than a single line.

In conventional AM, both sidebands carry the same message information. We can suppress either one and still recover the complete message with a suitable receiver. Transmitting only one sideband is called single-sideband modulation (SSB). The carrier is usually suppressed too. For a message such as speech, SSB uses approximately half the bandwidth of conventional AM and avoids transmitting power in the carrier and the duplicate sideband.

Removing a sideband also changes the waveform. Even if we keep the carrier, the envelope generally no longer follows the message exactly. A simple AM receiver that recovers the message by following this envelope can therefore introduce distortion. An SSB receiver instead combines the received signal with a locally generated carrier reference to recover the message.

We explore these tradeoffs further in Why Modulation Exists (in preparation).

One signal, two useful views
Time-domain signal	Frequency-domain view
An infinite-duration ideal sine wave	One ideal frequency component
A repeating non-sinusoidal waveform	Harmonics at whole-number multiples of its repetition frequency; a steady DC component if its average is not zero
A modulated RF packet	A band of nearby components around its operating frequency
A short pulse or fast edge	A broad range of frequency components

Representing a signal as sine-wave components also helps us predict how a filter or amplifier will affect it. We can analyze each component separately and add the results, provided the circuit responds approximately linearly. In this range, doubling the input signal's voltage amplitude approximately doubles the output signal's voltage amplitude. Many circuits behave this way only when the signal amplitude is low enough; this is called small-signal operation.

In a filter, cable, or amplifier with fixed operating conditions, a steady sine wave then keeps its frequency, although its amplitude and phase may change. What counts as small depends on the circuit. At higher signal levels, this proportional response can break down, introducing distortion and additional frequency components.

A spectrum analyzer uses this second view. The horizontal axis is frequency, and the vertical axis shows signal level at those frequencies. An oscilloscope helps us see when the waveform changes; a spectrum analyzer helps us see which frequencies are present. They describe the same signal in different ways.

Common misconceptionA real transmitter radiates at exactly one infinitely narrow frequency.

An ideal unchanging sine wave that exists forever has one frequency. A real packet carries a finite group of data: it starts, stops, and changes to represent the message. Those changes spread its frequency content over a range. Real hardware can add unwanted components too.

Go deeperA compact Fourier idea

A periodic waveform can be represented schematically as:

x(t)=x_{\mathrm{DC}}+\sum_{n=1}^{\infty}A_n\sin(2\pi n f_0 t+\phi_n)

Here, $f_0$ is the repetition frequency. Each integer $n=1,2,3,\ldots$ selects a component at frequency $n f_0$ . Some component amplitudes may be zero. This is a representation of one waveform, not a claim that tiny physical sine waves are independently travelling inside it.

The formal network property used above is called linear and time-invariant: scaled inputs produce scaled outputs, sums can be analyzed component by component, and the behavior does not change merely because the experiment starts later. Real RF hardware only approximates this model over a stated signal range and bandwidth.

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RF spectrum

Where do kHz, MHz, and GHz sit, and where do familiar radio systems fit?

The electromagnetic spectrum is the full range of electromagnetic frequencies, including radio, infrared, visible light, and X-rays. The spectrum of one signal instead describes which frequencies that particular signal contains. A Wi-Fi signal occupies a small part of the much larger electromagnetic spectrum.

The frequency prefixes introduced earlier give us the scale: 1 MHz is 1,000 kHz, and 1 GHz is 1,000 MHz. Convert to the same unit before comparing two systems.

Think about itWhich is higher in frequency: 100 MHz FM or the 1.575 GHz GPS L1 signal?

Answer

GPS L1. Convert to the same unit first: 1.575 GHz is 1,575 MHz, about 15.75 times 100 MHz.

Illustrative locations of familiar RF systems
Technology	Illustrative frequency	Unit	Engineering relevance
AM broadcast	roughly 500-1,700 kHz	kHz	Compact receiving antennas are often electrically short
FM broadcast	around 100 MHz	MHz	About 3 m free-space wavelength at 100 MHz
Cellular	hundreds of MHz to several GHz	MHz / GHz	Many bands; the exact range depends on system and region
GPS L1	1.57542 GHz	GHz	A precise satellite-navigation carrier frequency
Bluetooth and 2.4 GHz Wi-Fi	around 2.4 GHz	GHz	The frequency range used for this lesson’s 2.45 GHz example
5 and 6 GHz Wi-Fi	several GHz	GHz	The same path delay creates a larger carrier-phase shift

These are orientation examples, not an allocation chart. Actual bands and channel rules depend on the technology, standard, and jurisdiction. When compliance or measurement matters, state the exact frequency range rather than relying only on a familiar label.

Common misconceptionMicrowave and RF are completely separate things.

Microwave is a name commonly used for the higher-frequency part of the broader RF domain; the exact lower boundary varies by convention. At 2.45 GHz, a signal is in UHF under the usual 300 MHz-3 GHz band nomenclature and is also commonly described as microwave.

Standard RF band designators

These decade-wide names help engineers communicate scale. Their wavelengths below are approximate free-space values. They become shorter in ordinary dielectric transmission lines such as the cable and PCB examples used here.

Common RF band designators
Band	Frequency	Approx. free-space wavelength
VLF	3-30 kHz	100-10 km
LF	30-300 kHz	10-1 km
MF	300 kHz-3 MHz	1 km-100 m
HF	3-30 MHz	100-10 m
VHF	30-300 MHz	10-1 m
UHF	300 MHz-3 GHz	1 m-10 cm
SHF	3-30 GHz	10-1 cm
EHF	30-300 GHz	10-1 mm

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Putting it all together — a 2.45 GHz signal

Delay and phase over 30 mm: free space versus a cable

Suppose two observation points are 30 mm apart along the direction a 2.45 GHz wave travels. First use free space as a reference case. Then we will replace it with a 30 mm cable whose velocity factor is 0.66. Frequency and distance stay the same; propagation speed changes.

$T=\frac{1}{2.45\times10^9\,\mathrm{Hz}}\approx408\,\mathrm{ps}$ $\tau=\frac{0.030\,\mathrm m}{299\,792\,458\,\mathrm{m/s}}\approx100\,\mathrm{ps}$ $\theta=360^\circ\frac{\tau}{T}\approx88.3^\circ$ Rounded results are shown; the phase calculation uses unrounded period and delay.

The signal takes nearly a quarter of a cycle to cross this distance. A model that treats both points as having the same phase would therefore miss a large phase difference. Now calculate the cable's speed, wavelength, and delay:

$v_p=0.66c\approx1.979\times10^8\,\mathrm{m/s}$ $\lambda=\frac{v_p}{f}=0.66\lambda_0\approx80.8\,\mathrm{mm}$ $\tau=\frac{0.030\,\mathrm m}{0.66c}\approx152\,\mathrm{ps}$ $\theta=360^\circ\frac{\tau}{T}\approx133.7^\circ$ The free-space wavelength is

\lambda_0

. The cable calculation uses the same 408 ps period and unrounded values throughout.

The cable takes about 52 ps longer to carry a change over the same 30 mm. That is why its electrical length is about 134° rather than 88°. The table brings the time and distance views together:

A connected engineering view of 2.45 GHz
Concept	At 2.45 GHz	Engineering meaning
Frequency	2.45 GHz	2.45 billion cycles each second
Period	408 ps	Only 0.408 ns for one complete cycle
Free-space wavelength	122.4 mm	One cycle viewed across distance
Free-space quarter-wave	30.6 mm	A useful first physical scale, not a universal antenna length
30 mm in free space	100 ps / 88.3°	Almost one quarter of a cycle
30 mm in 0.66-VF cable	152 ps / 133.7°	The slower wave makes the same length electrically longer
Spectrum location	UHF; commonly microwave	In the 2.4 GHz region used by Bluetooth and Wi-Fi

For an actual connection, you need its physical length, operating frequency, and propagation speed. Use the cable's specified velocity factor or an appropriate model for the PCB line; the free-space result alone does not give the guided-wave delay.

Use the frequency, wavelength, and period calculator to check these scales at another frequency or propagation velocity.

Frequency still tells us nothing about amplitude or power. The designer must separately state voltage or power, its qualifier, the impedance, and the reference plane. And a real Bluetooth or Wi-Fi transmission is not an infinitely thin line at 2.45 GHz: modulation distributes its energy across a range of frequency components.

The connected mental model

Frequency sets the cycle rate. Period gives the time per cycle. Propagation velocity turns that time into wavelength. Wavelength makes physical dimensions electrically meaningful. Delay appears as phase, phase controls how coherent signals add, and Fourier components explain how a real modulated signal occupies spectrum.

Think about itIf the frequency doubles to 4.90 GHz while the structure and velocity stay unchanged, what changes?

Answer

Period and wavelength halve. The physical delay stays the same, but it spans twice as many cycles, so the electrical length and total phase lag double. A layout that was manageable at 2.45 GHz may be much less forgiving at 4.90 GHz.

Ungraded review

Check your understanding

Answer each question in your own words, then reveal the model answer.

01A fixed sensor records one cycle every 10 ns. Can you determine the wavelength from this measurement alone? Why?
Model answer
No. This tells us the period. To find the wavelength, we also need the propagation speed: the wavelength is the distance the wave travels during those 10 ns. In free space, that is about 3 m. In a slower cable, it is shorter even though the period is still 10 ns.
02Why can a voltage that averages to zero still heat a resistor, and which voltage measure predicts that heating?
Model answer
Positive and negative voltage both produce positive power in a resistor because instantaneous power is voltage squared divided by resistance. RMS voltage is the steady DC voltage that would produce the same average power. A zero-offset sine has RMS voltage equal to its peak voltage divided by the square root of two.
03Two equal sine waves at the same frequency stay half a cycle apart. Explain their sum without using a formula.
Model answer
At every instant, one contributes the opposite value of the other: when one is +1 V, the other is −1 V. Their sum is zero at the observation point. This does not mean energy disappears; its destination depends on the physical system combining the signals.
04A 2.45 GHz sinusoid travels in free space. What are its approximate period, wavelength, and phase change across 30 mm?
Model answer
About 408 ps, 122.4 mm, and 88.3°. The 30 mm path is already close to one quarter of a cycle.
05The same 2.45 GHz signal enters a cable with velocity factor 0.66. What stays fixed, and what changes?
Model answer
Frequency and period stay fixed. Phase velocity and wavelength fall; the wavelength becomes about 80.8 mm, so the same 30 mm path grows to about 133.7° electrically.
06Why does a real 2.45 GHz data packet occupy a band rather than one infinitely narrow spectral line?
Model answer
It starts, stops, and is modulated. Those time-domain changes require a range of sinusoidal frequency components around the operating frequency.

References and further reading

Standards and primary sources

Recommended texts

Michael Steer, Microwave and RF Design II — Transmission Lines: electrical length and the one-twentieth-wavelength criterion
Michael Steer, Fundamentals of Microwave and RF Design: wavelength in a dielectric and effective permittivity in microstrip
David M. Pozar, Microwave Engineering
Ulaby and Ravaioli, Fundamentals of Applied Electromagnetics
Ramo, Whinnery, and Van Duzer, Fields and Waves in Communication Electronics

Something unclear or incorrect? Suggest a correction

RF Signals

What actually is RF?

From an electrical signal to a radio wave

A signal that repeats

A Wave in Space Becomes a Waveform in Time

Frequency & period

One repeat takes time.

Wavelength

Same rhythm. Different spacing.

One repeat takes up distance.

Why wavelength matters in RF

Same distance. Different phase.

One turn. One cycle.

Amplitude

Same timing. Different height.

Phase

Same signals. Different sums.

Explore phase and time delay

Why do we keep using sine waves?

Can smooth sine waves build a square-like shape?

How a message changes a carrier

Signal over time

The same signal in frequency

RF spectrum

Standard RF band designators

Putting it all together — a 2.45 GHz signal

Check your understanding

Try 5 quick questions.

References and further reading

Standards and primary sources

Recommended texts