📻 Frequency Conversion Calculator

Frequency is one of the most fundamental concepts in physics, engineering, electronics, and everyday life. It describes how often a repeating event occurs per unit of time. Whether you are talking about the oscillation of a sound wave, the rotation of an electric motor, the clock cycles of a microprocessor, or the electromagnetic waves carrying your Wi-Fi signal — all are described mathematically by frequency.

The SI (International System of Units) unit of frequency is the Hertz (Hz), defined as one cycle per second. Named in honor of the German physicist Heinrich Rudolf Hertz (1857–1894), who first conclusively demonstrated the existence of electromagnetic waves in 1887, the hertz is the universal reference point from which all other frequency units derive.

The metric (SI) prefix system scales the hertz across an enormous range — from the ultra-slow oscillations of gravitational waves to the rapid cycles of visible light. The key prefixes are:

• Millihertz (mHz): \( 1\,\text{mHz} = 10^{-3}\,\text{Hz} \) — Very slow oscillations; ELF signals, seismic waves, ocean tides.
• Hertz (Hz): \( 1\,\text{Hz} = 1 \) cycle/s — Power grid (50/60 Hz), audio bass notes, EEG brain waves.
• Kilohertz (kHz): \( 1\,\text{kHz} = 10^3\,\text{Hz} \) — AM radio, human speech frequencies (300 Hz–3.4 kHz), sonar.
• Megahertz (MHz): \( 1\,\text{MHz} = 10^6\,\text{Hz} \) — FM radio (88–108 MHz), older CPUs, VHF TV.
• Gigahertz (GHz): \( 1\,\text{GHz} = 10^9\,\text{Hz} \) — Wi-Fi (2.4/5 GHz), modern CPUs, 5G micro-bands.
• Terahertz (THz): \( 1\,\text{THz} = 10^{12}\,\text{Hz} \) — T-ray imaging, molecular spectroscopy, future 6G research.

Every periodic signal has two complementary descriptors: its frequency \( f \) and its period \( T \). They are precise mathematical inverses of each other:

This inverse relationship has profound practical implications. A signal at 60 Hz (North American mains power) completes one cycle every \( T = 1/60 \approx 16.67\,\text{ms} \). A CPU running at 3.5 GHz completes one clock cycle every \( T = 1/(3.5 \times 10^9) \approx 286\,\text{ps} \) (picoseconds). The faster the oscillation, the shorter its period.

In physics and engineering, we often work with angular frequency \( \omega \) (the Greek letter omega), measured in radians per second (rad/s). This is especially common in:

• Wave equations: \( y(t) = A\sin(\omega t + \phi) \)
• AC circuit analysis: impedance \( Z_C = \frac{1}{j\omega C} \), \( Z_L = j\omega L \)
• Rotational mechanics: \( \omega \) relates to torque and angular momentum
• Control systems: Bode plots use \( \omega \) on the frequency axis

Revolutions Per Minute (RPM) is the non-SI unit most commonly used in mechanical engineering for motors, turbines, engines, compressors, fans, and drills. The conversion to and from Hz is straightforward because there are 60 seconds in one minute:

For traveling waves — electromagnetic, acoustic, or water waves — frequency and wavelength \( \lambda \) are linked by the wave's propagation speed \( v \):

This equation governs all wave phenomena. For electromagnetic waves in free space, \( v = c \approx 3 \times 10^8\,\text{m/s} \):

• AM radio at 1 MHz: \( \lambda = 3 \times 10^8 / 10^6 = 300\,\text{m} \) — very long wavelength
• FM radio at 100 MHz: \( \lambda = 3\,\text{m} \)
• Wi-Fi at 2.4 GHz: \( \lambda = 0.125\,\text{m} = 12.5\,\text{cm} \)
• Visible light at 500 THz: \( \lambda = 600\,\text{nm} \) — green light

For sound in air at 20°C, \( v \approx 343\,\text{m/s} \). Middle C (261.63 Hz) has a wavelength of \( \lambda = 343/261.63 \approx 1.31\,\text{m} \).

In digital signal processing, the Nyquist–Shannon sampling theorem states that a continuous signal with maximum frequency \( f_{\max} \) must be sampled at a rate \( f_s \) satisfying: \[ f_s \geq 2 f_{\max} \] This minimum rate is called the Nyquist rate. Sampling below it causes aliasing — high-frequency components appear as false low-frequency signals. Audio CDs sample at 44,100 Hz (44.1 kHz), safely above twice the 20 kHz human hearing limit.

Every physical system has a natural resonant frequency \( f_0 \) at which it oscillates most readily when disturbed. For an LC electrical circuit: \[ f_0 = \frac{1}{2\pi\sqrt{LC}} \] where \( L \) is inductance (in henrys) and \( C \) is capacitance (in farads). This formula underlies the design of radio tuners, oscillator circuits, and bandpass filters. Mechanical systems (bridges, buildings, pendulums) have analogous resonant frequencies determined by stiffness and mass.

When a wave source or observer moves, the observed frequency differs from the emitted frequency. The Doppler formula for sound is: \[ f_{\text{obs}} = f_{\text{src}} \times \frac{v + v_{\text{obs}}}{v - v_{\text{src}}} \] where \( v \) is the wave speed, \( v_{\text{obs}} \) is the observer's speed (positive toward source), and \( v_{\text{src}} \) is the source's speed (positive away from observer). This effect explains why an ambulance siren sounds higher as it approaches and lower as it recedes.