For estimating and education only. Results come from the simplified models described below, are provided as is with no warranty, and are not a substitute for engineering analysis, simulation or measurement. Check them before relying on them in a design, purchase or safety decision. See the terms of use.
Equations
Single section
Z_1 = \sqrt{Z_0 R_L}Ω\ell = \frac{\lambda}{4} = \frac{v_f\,c}{4 f_0}mBinomial
\ln\frac{Z_{n+1}}{Z_n} \approx 2^{-N}\binom{N}{n}\ln\frac{R_L}{Z_0}\Gamma_m = \frac{\mathrm{VSWR}_\mathrm{max} - 1}{\mathrm{VSWR}_\mathrm{max} + 1}Bandwidth, N = 1
\frac{\Delta f}{f_0} = 2 - \frac{4}{\pi}\cos^{-1}\left[\frac{\Gamma_m}{\sqrt{1 - \Gamma_m^2}}\,\frac{2\sqrt{Z_0 R_L}}{|R_L - Z_0|}\right]Bandwidth, binomial
\frac{\Delta f}{f_0} = 2 - \frac{4}{\pi}\cos^{-1}\left[\frac{1}{2}\left(\frac{\Gamma_m}{|A|}\right)^{1/N}\right], \qquad A = 2^{-N}\,\frac{R_L - Z_0}{R_L + Z_0}Sweep
Z_\mathrm{in} = Z_n\,\frac{Z + jZ_n\tan\theta}{Z_n + jZ\tan\theta}, \qquad \theta = \frac{\pi}{2}\,\frac{f}{f_0}section by section, from the loadVariables
| Symbol | Meaning | Unit |
|---|---|---|
Z_0 | Source line impedance | Ω |
R_L | Load resistance | Ω |
Z_n | Impedance of section n | Ω |
N | Number of sections | — |
\Gamma_m | Largest acceptable reflection coefficient | — |
v_f | Velocity factor of the section line | — |
Worked example
| Input | Value |
|---|---|
| Line | 100 Ω |
| Load | 50 Ω |
| Sections | 3, binomial |
| Limit | |Γ| ≤ 0.05 (VSWR 1.105) |
| Result | Value |
|---|---|
| Section impedances | 91.7 / 70.71 / 54.53 Ω |
| Section length (λ/4 at f0) | 18.74 mm |
| Load mismatch without matching | 2 : 1 |
| Bandwidth at VSWR ≤ 1.105 | 71.3 % |
| Bandwidth at VSWR ≤ 1.105 | 2850 MHz |
Pozar’s Example 5.6: sections of 91.7, 70.7 and 54.5 Ω, and about 70 % bandwidth for |Γ| ≤ 0.05.
Load this example in the calculatorValidity
- Real load, lossless TEM sections.
- Binomial impedances use the small-reflection approximation. For ratios above about 4:1, exact tables differ slightly. The plotted sweep is exact for the impedances shown.
- Step discontinuities between sections are not modeled. In microstrip they shift the center frequency.
Sources
- D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012. §5.4–5.6 (quarter-wave and binomial multisection transformers).
- G. L. Matthaei, L. Young and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, Artech House, 1980.