Calculation result
Detailed values
| Free-space wavelength | 2.9979246 m |
|---|---|
| Propagation velocity | 1.978630e+8 m/s |
| Phase constant β | 3.1755228 rad/m |
| Electrical length | 90° |
| Normalized impedance | 1 − j4.3298e-17 |
| Length error | 0% |
| Actual section length | 494.657556 mm |
| Ideal transformer Z | 70.710678 Ω |
Microstrip result
Coaxial result
- Required D/d ratio
- 5.857758
- Required outer ID
- 5.857758 mm
Saved scenarios
Frequency-response analysis
Multi-section design table
| Section | Characteristic impedance | Physical length | Electrical length | Profile note |
|---|---|---|---|---|
| 1 | 59.460356 Ω | 494.657556 mm | 90° | geometric profile |
| 2 | 70.710678 Ω | 494.657556 mm | 90° | geometric profile |
| 3 | 84.089642 Ω | 494.657556 mm | 90° | geometric profile |
Sweep data table
| Frequency | Zin real | Zin imag | |Γ| | VSWR | Return loss | Delivered power |
|---|
Formula used
Zt = √(Z0 × ZL)l = vp / (4f) = VF × c / (4f)Zin = Zt × (ZL + jZt tan βl) / (Zt + jZL tan βl)Zin = Zt × (ZL + Zt tanh γl) / (Zt + ZL tanh γl)Γ = (Zin − Z0) / (Zin + Z0)VSWR = (1 + |Γ|)/(1 − |Γ|), RL = −20log10|Γ|Z0 ≈ 60/√εr × ln(D/d)Preflected = |Γ|² × 100%How to use this calculator
- Choose a mode, then enter the system and load impedances.
- Set the design frequency and propagation method.
- Leave physical length at zero for an ideal quarter-wave section.
- Enable the lossy model when cable attenuation is known.
- Run the sweep to inspect bandwidth and off-frequency performance.
- Use PCB or coaxial synthesis for practical dimensions.
- Export results, save a scenario, or print a calculation report.
Engineering guidance and limitations
Resistive loads
A single quarter-wave section gives an exact match between two positive real impedances at its design frequency. The required characteristic impedance is their geometric mean.
Reactive loads
Complex loads generally require an additional tuning element or a shifted line location. Use the complex mode as an analyzer rather than assuming an exact match.
Bandwidth
A single section is narrowband. Higher impedance ratios usually reduce useful bandwidth, while multi-section transformers can broaden the response.
Velocity factor
Physical length depends on propagation velocity. Use manufacturer data for cable and an effective dielectric model for printed transmission lines.
PCB fabrication
Board dielectric tolerance, copper thickness, solder mask, etching, roughness, and connector discontinuities can shift impedance and electrical length.
Measurement
Verify critical designs using a calibrated vector network analyzer. Trim length carefully because frequency response shifts with every dimensional change.
Frequently asked questions
Why is the transformer one-quarter wavelength long?
At ninety electrical degrees, the transmission-line impedance relationship becomes an impedance inverter.
Can it match a complex load directly?
Not usually. The reactive part commonly needs cancellation or line-position adjustment.
Does the physical length equal free-space wavelength divided by four?
Only in free space. Real lines use guided wavelength determined by propagation velocity.
What happens away from the design frequency?
Electrical length changes, so the impedance transformation and match deteriorate.
Should connector length be included?
Include every section contributing meaningful electrical phase, especially at microwave frequencies.
Why does the PCB result differ from another field solver?
Closed-form microstrip equations are approximations. Stackup details and solver models can differ.
How can bandwidth be increased?
Use multiple transformer sections, a tapered line, or another matching-network topology.
Can a commercial 75-ohm cable replace a 70.71-ohm design?
Often, but the residual mismatch should be evaluated across the required bandwidth.
How accurate is the lossy-line model?
It uses constant attenuation and phase velocity. Real cable properties may vary with frequency.
What is a practical trimming method?
Start slightly long, measure, then shorten in small controlled steps.