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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.
Related: Antennas · EMC · Calculators · Worked Examples