The relationshipt between Wavelength and Frequency is determined by simple equation that you learned in high school physics. The equation is simple, but RF engineers use it all the time. The wavelength sets the size of an antenna, the spacing of an antenna array and the length of a matching stub, so a quick feel for it saves a lot of calculation.
- How are wavelength and frequency related ?
- What is the wavelength of typical applications ?
- How does the medium change the wavelength ?
How are wavelength and frequency related ?
A wave moves forward by one wavelength during one period. The number of periods per second is the frequency, so the distance covered per second is wavelength x frequency. That distance is the velocity, and the equation below follows from it.
Wavelength = Velocity / Frequency
This equation says
i) When frequency is same, Wavelength gets longer as Velocity gets higher in vice versa.
ii) When Velocity is same, Wavelength gets shorter as Frequency gets higher in vice versa
In free space the velocity is the speed of light, c = 299 792 458 m/s. Rounding it to 3 x 108 m/s changes the result by only 0.07 %. With that rounding, the equation gives a handy shortcut: wavelength in meters = 300 / frequency in MHz. For example, 300 MHz gives 1 m and 3 GHz gives 10 cm. You can check any row of the table below with it.
The same equation also turns a length into a phase. A wave takes one period to cross one wavelength, and one period is 360 deg of phase. So a line of length L delays the signal by 360 x L / λ deg. For example, at 3 GHz the wavelength is 10 cm, so every 1 cm of path adds 36 deg of phase. This is why cable lengths matter in a phased array or in a balanced circuit, even when the loss of the cable is negligible.
Wavelength is inversely proportional to frequency : ten times the frequency gives one tenth of the wavelength.Wavelength in meters is 300 / f in MHz : this free space shortcut is accurate to 0.07 %.Length is phase at RF : a path of L adds 360 x L / λ deg, which is 36 deg per cm at 3 GHz.
What is the wavelength of typical applications ?
The table below lists the free space wavelength for frequencies from 30 MHz up to car radar. Read it with antenna size in mind. A quarter wave antenna is a quarter of the value in the table, so the table also tells you roughly how large the antenna for each application is.
If we assume that the velocity of electromagnetic wave is same as the speed of light which is 3 x 10^8 m/s. Wavelength for some typical application can be calculated as follows.
|
Frequency (Hz) |
Wavelength (m) |
Wavelength (cm) |
Comments |
|
30 Mhz |
10 |
1000 |
|
|
100 Mhz |
3 |
300 |
FM Radio |
|
300 Mhz |
1 |
100 |
|
|
900 Mhz |
0.333333333 |
33.33333333 |
Analog Celluar Phone |
|
1000 Mhz(1Ghz) |
0.3 |
30 |
|
|
2000 Mhz (2Ghz) |
0.15 |
15 |
WCDMA |
|
2400 Mhz (2.4 Ghz) |
0.125 |
12.5 |
Bluetooth, WLAN |
|
5800 Mhz (5.8 Ghz) |
0.051724138 |
5.172413793 |
WLAN |
|
15000 Mhz (15 Ghz) |
0.02 |
2 |
Possible 5G Candidate (mmWave) |
|
26000 Mhz (26 Ghz) |
0.011538462 |
1.153846154 |
Possible 5G Candidate (mmWave) |
|
76000 Mhz (76 Ghz) |
0.003947368 |
0.394736842 |
Car Radar |
Let's turn two rows into antenna sizes. At 900 MHz the wavelength is 33.3 cm, so a quarter wave antenna is about 8.3 cm long. At 2.4 GHz the wavelength is 12.5 cm, and a quarter wave is about 3.1 cm. The same scaling explains the mmWave rows. NR band n257, 26500 MHz to 29500 MHz in 38.101-2, has a wavelength of about 1.07 cm at 28 GHz. A half wavelength element spacing is then only about 5.4 mm, so a phone can hold a whole antenna array in a small module. For comparison, NR band n78, 3300 MHz to 3800 MHz in 38.101-1, has a wavelength of about 8.6 cm at 3.5 GHz.
Antenna size follows the wavelength : a quarter wave antenna is 8.3 cm at 900 MHz and 3.1 cm at 2.4 GHz.mmWave wavelengths are about 1 cm : at 28 GHz, half a wavelength is about 5.4 mm, small enough for an array in a phone.
How does the medium change the wavelength ?
The table assumes free space. Inside a cable, on a PCB or in a dielectric antenna, the wave travels slower, and the frequency does not change. So the wavelength gets shorter, and a component designed with the free space value comes out too long.
If the electronic wave travel through special type of media (not free space), the velocity would change. According to the equation shown above, the wavelength would change as the velocity changes.
A common example is the case when the wave travel through a dielectrics. (See Dielectric page if you are not familiar with what it is and how it influence the velocity of electromagnetic wave)
For a non magnetic dielectric with relative permittivity εr, the velocity is c / √εr. So the wavelength is λ = λ0 / √εr, where λ0 is the free space value. For example, take FR-4 with εr = 4.4. At 2.4 GHz, λ0 = 12.5 cm becomes 12.5 / √4.4 = 6.0 cm inside the material. In a coax cable with solid polyethylene, εr is about 2.25, so the velocity is 1 / √2.25 = 0.67 of c. This ratio is the velocity factor on cable datasheets.
One more detail matters on a PCB. A microstrip line has the dielectric below it and air above it, so the wave sees an effective permittivity between 1 and εr. The wavelength on the line therefore lies between the two values above. The Microstrip page covers how to calculate it.
The frequency stays the same in every medium : only the velocity changes, and the wavelength follows it.A dielectric shortens the wavelength by √εr : FR-4 with εr = 4.4 cuts it to 48 % of the free space value.The velocity factor of a cable is 1 / √εr : a solid polyethylene coax has a velocity factor of about 0.67.