Frequency and Period: The Relationship Every Engineer Should Know
Frequency and period are inversely related properties of any repeating signal. Learn how to convert between them, understand Hertz, and apply these concepts to oscillators, timers, and signal analysis.
The Fundamental Relationship
Every periodic signal — a sine wave, a square wave, a clock pulse — repeats itself over and over. Two properties describe this repetition:
- Period (T): The time it takes for one complete cycle to occur, measured in seconds (s).
- Frequency (f): The number of complete cycles that occur per second, measured in hertz (Hz).
They are exactly reciprocal:
f = 1 / T
T = 1 / f
This simple relationship is one of the most fundamental in all of electronics and physics.
Units and Prefixes
Frequency spans an enormous range across different applications. SI prefixes make the numbers manageable:
| Unit | Symbol | Value | Example application |
|---|---|---|---|
| Hertz | Hz | 1 cycle/s | Mains power frequency (50Hz/60Hz) |
| Kilohertz | kHz | 10³ Hz | Audio, AM radio |
| Megahertz | MHz | 10⁶ Hz | FM radio, CPUs (older), Wi-Fi |
| Gigahertz | GHz | 10⁹ Hz | Modern CPUs, 5G, microwave |
| Terahertz | THz | 10¹² Hz | THz imaging, light waves |
Period follows the inverse pattern — higher frequency means shorter period:
| Frequency | Period |
|---|---|
| 1 Hz | 1 s |
| 50 Hz | 20 ms |
| 1 kHz | 1 ms |
| 1 MHz | 1 µs |
| 1 GHz | 1 ns |
| 2.4 GHz (Wi-Fi) | ≈ 416 ps |
Worked Examples
Example 1: UK mains power
UK mains electricity runs at 50Hz. What is the period?
T = 1 / f = 1 / 50 = 0.02 s = 20 ms
This is why fluorescent lighting powered by mains can flicker at 100Hz (twice per cycle) — visible to high-speed cameras and sometimes to people sensitive to flicker.
Example 2: Arduino timer frequency
You set an Arduino Timer1 to generate an interrupt every 250 µs. What frequency is this?
f = 1 / T = 1 / (250 × 10⁻⁶) = 4000 Hz = 4 kHz
Example 3: 555 timer oscillator
A 555 timer circuit is configured in astable mode with R1 = 1kΩ, R2 = 10kΩ, C = 100nF. The output frequency is approximately:
f ≈ 1.44 / ((R1 + 2×R2) × C)
= 1.44 / ((1000 + 20000) × 100×10⁻⁹)
= 1.44 / (21000 × 10⁻⁷)
= 1.44 / 0.0021
≈ 686 Hz
Period ≈ 1/686 ≈ 1.46 ms.
Angular Frequency (ω)
In mathematics and signal processing, frequency is often expressed as angular frequency (ω):
ω = 2π × f (radians per second)
One complete cycle is 2π radians. Angular frequency appears in:
- Impedance of capacitors: Z_C = 1/(jωC)
- Impedance of inductors: Z_L = jωL
- Fourier transforms and Laplace transforms
- Control systems (Bode plots use ω on the x-axis)
Conversion:
ω = 2π / T
T = 2π / ω
Resonant Frequency
When a circuit contains both inductance and capacitance, they can resonate at a natural frequency where energy oscillates between the two components. The resonant frequency of an LC circuit is:
f₀ = 1 / (2π × √(L × C))
Where:
- L = inductance in henries (H)
- C = capacitance in farads (F)
Example: L = 100µH, C = 470pF:
f₀ = 1 / (2π × √(100×10⁻⁶ × 470×10⁻¹²))
= 1 / (2π × √(4.7×10⁻¹⁴))
= 1 / (2π × 2.168×10⁻⁷)
≈ 734 kHz
This is the principle behind:
- Tuned RF filters (radio receiver front-ends)
- Crystal oscillators (a quartz crystal acts as a very high-Q LC resonator)
- Switch-mode power supply (SMPS) resonant converters
Duty Cycle
For square waves and PWM signals, the duty cycle describes what fraction of the period the signal is in the high state:
Duty cycle = t_high / T × 100%
A 50% duty cycle means the signal is high for half the period. A 1kHz signal with 25% duty cycle is high for 250µs and low for 750µs.
Example: PWM for LED brightness control — 1kHz PWM with 30% duty cycle:
- Period = 1ms
- High time = 0.3ms
- Low time = 0.7ms
- Perceived brightness ≈ 30% of maximum
Practical Applications by Frequency Range
| Range | Application |
|---|---|
| < 1 Hz | Geological sensors, tidal models, biological rhythms |
| 1 Hz – 20 Hz | Mechanical vibration, industrial process control |
| 20 Hz – 20 kHz | Audio (full human hearing range) |
| 20 kHz – 200 kHz | Ultrasound, ultrasonic cleaners, some switching supplies |
| 100 kHz – 1 MHz | AM radio, inductive charging, RFID |
| 1 MHz – 100 MHz | FM radio, short-wave, oscilloscope bandwidth |
| 100 MHz – 3 GHz | Wi-Fi, Bluetooth, 4G LTE, GPS |
| 3 GHz – 30 GHz | 5G, radar, microwave links |
| > 300 GHz | THz imaging, infrared light |
Use the Frequency/Period Converter
The Frequency and Period Converter on DevGizmo converts between frequency (Hz, kHz, MHz, GHz) and period (s, ms, µs, ns) and also calculates angular frequency. Enter either value to instantly compute the others.