How does nonlinearity create harmonics and distortion?

From RF Design Wiki · By the RF Design Wiki editorial team · Published October 21, 2026 · Last reviewed October 21, 2026 · How we research

A weakly nonlinear amplifier can be modeled by the power series y = a₁x + a₂x² + a₃x³. Driving it with one tone produces harmonics at 2f and 3f and, when a₃ opposes a₁, gain compression. Two tones produce intermodulation products at 2f₁ − f₂ and 2f₂ − f₁ that fall close to the wanted signal.^[1]

What happens with a single tone?

For x = A·cos ωt: the fundamental is (a₁A + ¾a₃A³), the second harmonic ½a₂A² and the third harmonic ¼a₃A³. Second-order products grow 2 dB per dB of input and third-order products 3 dB per dB.^[1]

What are desensitization and cross-modulation?

A strong blocker reduces the small-signal gain seen by a weak wanted signal (desensitization or blocking) through the a₃ term. If the blocker is amplitude-modulated, that modulation transfers to the wanted signal (cross-modulation), which matters in full-duplex systems where the transmitter leaks into the receiver.^[1]

Which distortion products matter most?

Third-order intermodulation (IM3): in-band and cannot be filtered. See IP3.
Second-order products (f₁ − f₂): critical in direct-conversion receivers; specified by IIP2.
Harmonics: regulated as spurious emissions; removed with low-pass filtering after the PA.^[2]
Compression: see P1dB.

Frequently asked questions

Do differential circuits cancel even harmonics?

Ideally yes; balance limits the cancellation in practice.

Is a mixer a nonlinear device?

Yes, intentionally: it relies on multiplication, which is a nonlinear (time-varying) operation.

Does the power-series model apply to PAs?

Only approximately; PAs near saturation and with memory effects need more complete behavioral models.

References

Online sources accessed October 21, 2026; printed works cited by edition. See all sources.

“RF Microelectronics, 2nd ed.”. B. Razavi, Prentice Hall. 2012.
“Recommendation ITU-R SM.329: Unwanted emissions in the spurious domain”. ITU-R. current version.

Categories: Fundamentals

Articles on RF Design Wiki

Start here	Main page What is RF design Learning path Design workflow
Fundamentals	Decibels and dBm RF power Wavelength and frequency Thermal noise Noise figure Noise temperature SNR and Eb/N0 Skin effect Q factor Nonlinearity and harmonics Intermodulation and IP3 P1dB compression Dynamic range and SFDR
Transmission lines	Transmission line theory Characteristic impedance Γ, VSWR and return loss Input impedance of a line Quarter-wave transformer Microstrip Stripline Coplanar waveguide Coaxial cable Waveguides RF connectors
Matching and S-parameters	Impedance matching Maximum power transfer Smith chart L-network matching Pi and T networks Stub matching Bode–Fano limit S-parameters Touchstone files Z, Y, ABCD parameters De-embedding Mixed-mode S-parameters
Filters and passives	RF filter basics Filter approximations LC filter design Microstrip filters Cavity and dielectric filters SAW and BAW filters Diplexers and duplexers Wilkinson divider Couplers and hybrids Baluns Circulators and isolators RF attenuators Passive components
Amplifiers	Gain definitions Amplifier stability LNA design Power amplifier basics PA classes Efficiency and PAE Doherty amplifier Linearization and DPD Load-pull Transistor technologies Transistor biasing ACPR and EVM
Mixers and frequency generation	RF mixers Mixer specifications Image frequency I/Q modulators Oscillator basics Crystal oscillators VCO design Phase noise Jitter from phase noise PLL basics PLL loop filter Fractional-N synthesis Direct digital synthesis Multipliers and dividers
Antennas and propagation	Antenna fundamentals Gain and directivity Dipole antenna Monopole antenna Patch antenna Arrays and beamforming Small antenna limits Polarization Friis equation Free-space path loss Link budget Multipath and fading Atmospheric and rain loss Path loss models Fresnel zones
Receivers and systems	Receiver architectures Superheterodyne Direct conversion Low-IF receiver Software-defined radio Receiver sensitivity Blocking and selectivity Transmitter architectures Modulation ADCs for RF Radar basics
PCB layout and materials	RF PCB layout RF PCB materials Grounding and vias Controlled-impedance stackup Decoupling RF shielding Thermal design
EMC and EMI	EMC and EMI basics FCC Part 15 CISPR standards Immunity and ESD EMI filters and ferrites Precompliance testing RF exposure limits
Test and measurement	Vector network analyzer VNA calibration Spectrum analyzer Noise figure measurement Phase noise measurement RF power measurement Signal generators TDR Antenna measurement Oscilloscopes for RF
Simulation	Simulation overview Harmonic balance EM simulation EM solver methods SPICE for RF
Standards and bands	Frequency bands Radar band letters ISM bands 5G NR bands Wi-Fi and Bluetooth RF GNSS frequencies Amateur radio bands Spectrum regulation
Reference	Formula cheat sheet Textbooks and references Glossary FAQ All sources
About	About Editorial policy