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RF & Microwave Engineering
1. Transmission lines
A distributed line is described by per-unit-length $R,L,G,C$:
\[Z_0=\sqrt{\frac{R+j\omega L}{G+j\omega C}},\qquad \gamma=\sqrt{(R+j\omega L)(G+j\omega C)}.\]For an ideal lossless line,
\[Z_0=\sqrt{L/C},\qquad \gamma=j\beta.\]2. Reflection and mismatch
At a load $Z_L$,
\[\Gamma_L=\frac{Z_L-Z_0}{Z_L+Z_0}.\]Return loss and VSWR are
\[RL=-20\log_{10}|\Gamma|,\] \[VSWR=\frac{1+|\Gamma|}{1-|\Gamma|}.\]Mismatch loss is
\[ML=-10\log_{10}(1-|\Gamma|^2).\]3. Input impedance of a line
For a lossless line of length $l$,
\[Z_{in}=Z_0\frac{Z_L+jZ_0\tan\beta l}{Z_0+jZ_L\tan\beta l}.\]At $l=\lambda/4$, a quarter-wave transformer can match two real impedances using
\[Z_t=\sqrt{Z_0Z_L}.\]4. Smith chart
The Smith chart maps normalized impedance/admittance to complex reflection coefficient. It is especially useful for:
- moving along a transmission line;
- adding series/shunt reactance;
- visualizing matching trajectories;
- reading VSWR and return loss;
- understanding narrowband matching sensitivity.
5. S-parameters
For a two-port network,
\[\mathbf b=\mathbf S\mathbf a.\]- $S_{11}$: input reflection;
- $S_{21}$: forward transmission;
- $S_{12}$: reverse transmission/isolation;
- $S_{22}$: output reflection.
At high frequency, S-parameters are practical because incident/reflected traveling waves are easier to measure than terminal open/short currents and voltages.
6. Rectangular waveguide
For a rectangular guide with broad dimension $a$ and height $b$, cutoff is
\[f_{c,mn}=\frac{c}{2\sqrt{\epsilon_r}}\sqrt{\left(\frac{m}{a}\right)^2+\left(\frac{n}{b}\right)^2}.\]The dominant mode is usually TE$_{10}$:
\[f_{c,10}=\frac{c}{2a\sqrt{\epsilon_r}}.\]Waveguides carry modes, not simple TEM waves.
7. Resonators and quality factor
A resonator stores electric and magnetic energy. A useful relation is
\[Q=2\pi\frac{\text{energy stored}}{\text{energy lost per cycle}} \approx\frac{f_0}{\Delta f_{3\text{dB}}}.\]Loaded $Q$ includes internal loss and external coupling.
8. Filters and coupled resonators
Microwave filters are implemented with transmission-line sections, cavities, microstrip resonators, dielectric resonators, waveguide irises and lumped/distributed hybrids. Key specifications include insertion loss, return loss, fractional bandwidth, group delay, rejection, power handling and temperature stability.
9. Couplers, dividers and hybrids
Directional couplers sample forward/reverse waves. Wilkinson dividers provide matched splitting and isolation. 90°/180° hybrids enable balanced amplifiers, mixers, beamforming and measurement networks.
Worked example — 50 Ω to 100 Ω quarter-wave match
\[Z_t=\sqrt{50\times100}=70.7\ \Omega.\]At 2 GHz in a medium with effective dielectric constant $\epsilon_{eff}=2.25$,
\[\lambda_g\approx\frac{c}{f\sqrt{\epsilon_{eff}}}\approx0.10\ \text{m},\]so a quarter-wave section is roughly 25 mm before discontinuity/end-effect corrections.
10. Measurement workflow
A VNA is the central RF network instrument. Good practice:
- calibrate at the actual reference plane;
- choose source power that does not compress the DUT;
- use appropriate IF bandwidth/averaging;
- control cable movement;
- de-embed fixtures only with a defensible model;
- inspect phase/group delay, not only magnitude.
See Measurements & Instruments.
11. Engineering reality
References
- D. M. Pozar, Microwave Engineering.
- R. E. Collin, Foundations for Microwave Engineering.
- S. Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in Communication Electronics.
- MIT OpenCourseWare, Electromagnetics and Applications.
Microwave and space component reference
For a hardware-oriented map of the components that make up real microwave and satellite RF chains—including LNAs, PAs/TWTAs, mixers, filters, switches, circulators, limiters, PLL/VCOs, beamformers, ADC/DACs, waveguide hardware, MMIC technologies, and space qualification—see Microwave & Space Electronics Components.
Radio astronomy as an RF receiver problem
For cryogenic LNAs, SIS mixers, low-noise feeds/OMTs, phase-stable LO distribution, IF chains, ADC/channelizer backends, RFI linearity and interferometric correlation, see Radio Astronomy — RF Circuits & Receiver Systems.
RF transceiver architectures
For an interview-focused comparison of superheterodyne, zero-IF, low-IF, image-reject, IF-sampling and direct-RF-sampling receivers—plus direct-conversion/direct-RF transmitters, TDD/FDD front ends, mixers, synthesizers, I/Q impairments and calibration—see RF Transceiver Architectures — Interview Ready.
Modern RF signal generation
For the internal architecture of a modern RF/vector signal generator—including reference clocks, PLLs, baseband I/Q, interpolation, DDS/NCO, DACs, I/Q modulation, direct-RF generation, ALC, phase noise and multi-channel coherence—see Modern Signal Generator.
Spectrum analyzer signal processing
For the internal RF/DSP chain of a modern real-time analyzer—using the Signal Hound BB60D as a concrete example—see Spectrum Analyzer Signal Processing.
RF band quick reference
For a compact table of GSM/LTE/NR operating bands plus L/S/C/X/Ku/K/Ka/V/W microwave bands, see RF Bands — Compact Reference.
Cellular RF standards
For NR/LTE transmitter, receiver, RRM, conformance and test-environment specifications, see 3GPP Standards — What an RF & Wireless Engineer Should Know.
Related: Antennas · Signal Integrity & Power Integrity · Paul Traps · EMC · Calculators · Worked Examples
Advanced study — network stability, power waves and nonlinear RF design
Cascading linear networks correctly
| For a two-port network, (S)-parameters are referenced to port impedances. Power transfer depends on source/load match, so ( | S_{21} | ^2) is not generally the transducer power gain. When cascading stages, use an ABCD or chain-scattering representation with consistent port conventions, then convert the combined network back to (S)-parameters. Avoid multiplying individual (S_{21}) terms when substantial interstage reflections are present. |
Stability and matching
For a two-port transistor network define (\Delta=S_{11}S_{22}-S_{12}S_{21}) and
\[K=\frac{1-|S_{11}|^2-|S_{22}|^2+|\Delta|^2}{2|S_{12}S_{21}|}.\]| For the usual two-port small-signal stability test, (K>1) and ( | \Delta | <1) indicate unconditional stability for passive source and load terminations at the frequency examined. This is a linear frequency-domain statement; check bias-dependent stability, out-of-band modes, packaging resonances and large-signal behavior separately. |
Noise and nonlinear tradeoffs
The Friis noise-factor cascade is
\[F_{\mathrm{tot}}=F_1+\frac{F_2-1}{G_1}+\frac{F_3-1}{G_1G_2}+\cdots,\]with linear available gains under the assumptions of the cascade model. Early attenuation can dominate noise figure, while early high gain may compromise blocker tolerance. Power amplifiers require large-signal load-pull, AM-AM/AM-PM and efficiency characterization instead of small-signal (S)-parameters alone.
Advanced exercise: design an LNA chain against simultaneous NF, IIP3 and stability constraints. Document impedance reference planes and a sensitivity analysis to component tolerances.
Further reading: D. M. Pozar, Microwave Engineering; G. Gonzalez, Microwave Transistor Amplifiers; NIST VNA uncertainty analysis.
Advanced topic — stability and cascade noise
The usual two-port small-signal stability quantities are
\[\Delta=S_{11}S_{22}-S_{12}S_{21},\qquad K=\frac{1-|S_{11}|^2-|S_{22}|^2+|\Delta|^2}{2|S_{12}S_{21}|}.\]| For a linear two-port, (K>1) and ( | \Delta | <1) indicate unconditional stability for passive source and load terminations at the frequency considered. Inspect out-of-band frequencies, bias dependence and large-signal operating points as well. Forward gain ( | S_{21} | ^2) is not generally the transducer gain if the source and load are mismatched. |
For a noise-factor cascade with linear available gains,
\[F_{\mathrm{total}}=F_1+\frac{F_2-1}{G_1}+\frac{F_3-1}{G_1G_2}+\cdots.\]Loss preceding the first LNA often dominates receiver noise figure; excessive early gain may instead compromise blocker tolerance.
Advanced experiment: design a two-stage receiver to meet NF, compression and stability simultaneously, then propagate VNA calibration and bias uncertainty into the predicted gain.
Further reading: D. M. Pozar, Microwave Engineering; NIST covariance-based VNA uncertainty.
Advanced RF applications
For systems that turn RF theory into calibrated observables, see RF Direction Finding, Precision Timing & Synchronization, Advanced Antenna Metrology, Radio Science & Tracking, and the Advanced Engineering Learning Map.