Calculator
Formula Used
The calculator uses per-unit-length parameters: series impedance Z′ = R′ + jωL′ and shunt admittance Y′ = G′ + jωC′.
Characteristic impedance: Z0 = √(Z′ / Y′) and propagation constant: γ = √(Z′Y′).
Input impedance for a line of length l with load ZL: Zin = Z0 · (ZL + Z0·tanh(γl)) / (Z0 + ZL·tanh(γl)). In lossless mode, it applies tanh(jβl)=j·tan(βl).
How to Use This Calculator
- Choose a model: lossless or lossy.
- Enter frequency and line length with units.
- Provide L′ and C′, then add R′ and G′ if lossy.
- Select the load type or enter custom R and X.
- Press Calculate to view Z0, γ, Zin, and matching metrics.
- Use CSV or PDF buttons to export the computed summary.
Example Data Table
Sample values below illustrate typical RF coax-like parameters.
| Mode | f (MHz) | l (m) | R′ (Ω/m) | L′ (µH/m) | G′ (S/m) | C′ (pF/m) | ZL (Ω) |
|---|---|---|---|---|---|---|---|
| Lossless | 100 | 1 | 0 | 0.25 | 0 | 100 | 50 + j0 |
| Lossy | 100 | 10 | 0.5 | 0.25 | 1e-5 | 100 | 75 + j25 |
| Lossy | 900 | 5 | 0.8 | 0.20 | 2e-5 | 90 | Open |
Transmission Line Impedance in Context
Transmission lines are not just long wires; their voltage and current vary with position and time. When the line length becomes a meaningful fraction of a wavelength, the load “seen” at the source depends on frequency and length. This tool computes that behavior using distributed parameters.
Characteristic Impedance from RLGC
The characteristic impedance Z0 is the ratio of traveling-wave voltage to current on an infinite line. With per‑unit‑length values R′, L′, G′, C′, the calculator forms Z′ = R′ + jωL′ and Y′ = G′ + jωC′, then evaluates Z0 = √(Z′/Y′). In low‑loss conditions, Z0 ≈ √(L′/C′).
Propagation Constant and Losses
The propagation constant γ = α + jβ controls both attenuation and phase shift. Here γ = √(Z′Y′), where α (nepers per meter) represents loss and β (radians per meter) sets electrical length. Skin effect typically increases R′ with frequency, while dielectric loss often appears as G′.
Input Impedance versus Length
For a finite line of length l, the load is transformed by the line into an input impedance Zin. The calculator applies the general expression with tanh(γl). In lossless mode, it simplifies to the familiar tangent form, which highlights quarter‑wave and half‑wave behaviors.
Reflection Coefficient and VSWR
Mismatch is quantified by the reflection coefficient Γ = (ZL − Z0)/(ZL + Z0). The magnitude |Γ| predicts standing waves and power delivery. The calculator reports VSWR as (1 + |Γ|)/(1 − |Γ|), along with return loss in dB, which is commonly used in RF test reports.
Typical Parameter Ranges
Real lines vary widely: coax often has Z0 near 50 Ω or 75 Ω, while twisted pair can be 100 Ω or higher. Microstrip impedance depends on trace width, substrate height, and permittivity; it is often specified within ±10%. Use the example table as a starting point, then refine RLGC to match your medium.
Using Results for Matching Decisions
If Zin is far from your source impedance, you can adjust line length, select a different Z0, or add a matching network. Quarter‑wave transformers, stub tuners, and series/shunt reactances are common options. The computed electrical length βl helps you choose practical, frequency‑specific solutions.
Validation and Common Pitfalls
Keep units consistent: L′ in H/m, C′ in F/m, and frequency in Hz. Extremely small losses can make results sensitive to rounding, so compare lossless and lossy modes as a sanity check. When available, validate Z0 and phase delay against a datasheet or VNA measurement.
Frequently Asked Questions
1) What is the difference between Z0 and Zin?
Z0 is a property of the line itself. Zin is what the source sees at the line input and depends on load impedance, frequency, and line length.
2) When should I use lossless mode?
Use lossless mode for short lines, low frequencies, or when attenuation is negligible compared with your accuracy needs. For long runs or high frequencies, include R′ and G′.
3) Why does Zin become purely resistive at certain lengths?
At half‑wave multiples, a lossless line repeats the load impedance. At quarter‑wave points, it inverts impedance. These special lengths can cancel reactive parts, making Zin appear resistive.
4) How do I interpret return loss?
Return loss expresses how much power is reflected due to mismatch. Larger positive values mean better matching. For example, 20 dB return loss corresponds to about 10% reflected voltage magnitude.
5) Can I model frequency‑dependent conductor loss?
Yes. If you have a model for R′(f) (often proportional to √f for skin effect), run the calculator at multiple frequencies and update R′ each time to observe trends.
6) What load should I enter for an open or short?
For an open circuit, enter a very large resistance (for example 1e9 Ω) with zero reactance. For a short, enter a very small resistance (for example 1e-6 Ω) with zero reactance.
7) Why can VSWR be extremely high?
If |Γ| approaches 1, nearly all energy reflects and the VSWR formula grows rapidly. This often happens with open/short loads or when the load impedance is far from Z0.