Visual
Key formulas
| Name | Formula | Variables / Notes |
|---|---|---|
| Cutoff Frequency of Rectangular Waveguide (TE_mn) | f_c = (c / 2) x sqrt((m/a)^2 + (n/b)^2) | c = speed of light (3 x 10^8 m/s); a = broad wall width (m); b = narrow wall height (m); m, n = mode indices; dominant mode TE10: f_c = c / (2a) |
| VSWR and Reflection Coefficient | VSWR = (1 + |Gamma|) / (1 - |Gamma|) | Gamma = (Z_L - Z_0) / (Z_L + Z_0) = reflection coefficient; Z_L = load impedance; Z_0 = characteristic impedance; VSWR = 1 means perfect match; VSWR = infinity means total reflection |
| Return Loss | RL = -20 log10(|Gamma|) dB | |Gamma| = magnitude of reflection coefficient; higher return loss is better; RL = 0 dB means full reflection; RL = infinity means perfect match; RL > 10 dB is acceptable in most designs |
| Noise Figure | NF = 10 log10(F) dB, F = SNR_in / SNR_out | F = noise factor (linear); SNR_in = input signal-to-noise ratio; SNR_out = output signal-to-noise ratio; NF = 0 dB means noiseless device |
| Friis Noise Formula (Cascaded Stages) | F_total = F1 + (F2 - 1)/G1 + (F3 - 1)/(G1 x G2) | F1, F2, F3 = noise factors of stages 1, 2, 3; G1, G2 = available power gains of stages 1, 2; first stage noise factor dominates; use low-noise amplifier (LNA) as first stage |
| Guide Wavelength | lambda_g = lambda_0 / sqrt(1 - (f_c/f)^2) | lambda_g = wavelength inside waveguide; lambda_0 = free-space wavelength at operating frequency f; f_c = cutoff frequency; lambda_g is always greater than lambda_0 |
Key concepts
S-Parameters
S11 = input reflection coefficient (return loss). S21 = forward transmission (insertion gain or loss). S12 = reverse transmission (isolation). S22 = output reflection coefficient. For a lossless reciprocal two-port network, the S-matrix is unitary and S12 = S21. Measured at matched source and load impedances (50 ohm in RF work).
Waveguide Modes
TE (Transverse Electric) modes have no E-field component in the propagation direction. TM (Transverse Magnetic) modes have no H-field in the propagation direction. TEM mode (both transverse) cannot propagate in a single-conductor waveguide. The dominant mode TE10 has the lowest cutoff frequency in rectangular waveguides and is used in most practical systems.
Microstrip Line
A microstrip has a conducting strip on a dielectric substrate over a ground plane. Characteristic impedance Z_0 depends on strip width W and substrate height h. Wider strips give lower impedance. Effective permittivity epsilon_eff is between the substrate permittivity and 1 (air above). Wave speed on microstrip is c / sqrt(epsilon_eff).
Impedance Matching
Quarter-wave transformer: insert a lambda/4 section of impedance Z_t = sqrt(Z_0 x Z_L) between source and load. Valid only at the design frequency. Single-stub matching: place an open or short-circuited stub at a specific distance from the load to cancel the reactive part of the input admittance.
PIN Diode and Schottky Diode in Microwaves
PIN diode acts as a current-controlled variable resistor at microwave frequencies (I-region charge modulation). Used for switches and attenuators. Schottky diode has no minority carrier storage, giving very fast switching and low noise. Used as a detector or mixer at millimetre wave frequencies.
Tables
Rectangular Waveguide Mode Summary
| Mode | Cutoff Formula | Notes |
|---|---|---|
| TE10 (dominant) | f_c = c/(2a) | Lowest f_c, standard mode |
| TE20 | f_c = c/a | Second TE mode, avoid |
| TE01 | f_c = c/(2b) | If b < a/2, above TE10 |
| TM11 (lowest TM) | f_c = c/2 sqrt(1/a^2+1/b^2) | No TM modes with m or n=0 |
S-Parameter Interpretation
| Parameter | Description | Ideal Value |
|---|---|---|
| S11 | Input reflection | 0 (matched) |
| S22 | Output reflection | 0 (matched) |
| S21 | Forward gain | High for amplifier |
| S12 | Reverse isolation | 0 for amplifier |
VSWR to Return Loss Conversion
| VSWR | |Gamma| | Return Loss (dB) |
|---|---|---|
| 1.0 (perfect) | 0.00 | Infinite |
| 1.5 | 0.20 | 14.0 dB |
| 2.0 | 0.33 | 9.5 dB |
| 3.0 | 0.50 | 6.0 dB |
| infinity | 1.00 | 0 dB |
Quick facts
- Standard WR-90 rectangular waveguide operates in X-band (8.2 to 12.4 GHz) with a = 22.86 mm, b = 10.16 mm and TE10 cutoff at 6.56 GHz.
- At 10 GHz, free-space wavelength lambda_0 = 30 mm.
- A perfect impedance match corresponds to VSWR = 1, |Gamma| = 0, and return loss = infinity dB.
- Phase velocity in a waveguide (v_p = c / sqrt(1-(f_c/f)^2)) is always greater than the speed of light; group velocity (v_g) is always less than c, and v_p x v_g = c^2.
- A quarter-wave transformer works only at its design frequency; its bandwidth is inversely proportional to the impedance ratio Z_L / Z_0.
- The noise temperature T_e of an amplifier with noise factor F is T_e = (F - 1) x 290 K.
- Klystron amplifies using velocity modulation; magnetron oscillates using crossed electric and magnetic fields; TWT (traveling wave tube) achieves broadband amplification via wave-particle interaction.
- Attenuation in a rectangular waveguide due to finite conductivity increases with frequency, approximately proportional to f^0.5 at high frequencies.
Exam shortcuts
- For TE10 dominant mode cutoff: f_c = c/(2a). If a = 2.3 cm, f_c = 3x10^10 / (2 x 2.3) = 6.52 GHz. Operating frequency must be above this and below the next mode f_c = c/a = 13.04 GHz.
- VSWR to |Gamma|: |Gamma| = (VSWR - 1) / (VSWR + 1). To find VSWR from |Gamma|: VSWR = (1 + |Gamma|) / (1 - |Gamma|). Memorise this pair; it appears in almost every microwave exam.
- For Friis noise figure, always keep F in linear (not dB). Convert: F = 10^(NF_dB/10). Then apply formula. Common mistake is adding dB values directly, which gives wrong results.
- Quarter-wave transformer impedance: Z_t = sqrt(Z_0 x Z_L). For Z_0 = 50 ohm and Z_L = 200 ohm: Z_t = sqrt(50 x 200) = sqrt(10000) = 100 ohm. This single step formula is sufficient for short questions.
- Guide wavelength is always longer than free-space wavelength. If a question gives lambda_g and lambda_0 and asks for f_c: f_c = f x sqrt(1 - (lambda_g/lambda_0)^-2). Rearrange from the guide wavelength formula.