Transmission Line Impedance Calculator
Created by: Ethan Brooks
Last updated:
Compute the complex input impedance and SWR of a coaxial feedline using the exact transmission line formula, with charts showing impedance versus electrical length for stub and matching design.
Transmission Line Impedance Calculator
Amateur RadioCalculate complex input impedance, electrical length, and SWR of a coaxial feedline at any load and frequency.
What is a Transmission Line Impedance Calculator?
The transmission line input impedance calculator computes the complex impedance seen at the transmitter end of a feedline given the antenna load impedance, characteristic impedance of the coax, the physical length of the line, and the velocity factor of the dielectric. This is a fundamental RF engineering calculation that explains why changing the length of a coaxial run can dramatically alter what your transmitter sees — even when the antenna itself is unchanged.
For amateur radio operators, this calculation is most important when building stub tuners, designing quarter-wave matching sections, or troubleshooting antenna systems where the feed-point impedance is well away from 50 ohms. A quarter-wave section of carefully chosen impedance cable can transform one real impedance to another. A half-wave section reproduces the load impedance at the input — a useful property for remote switching or for feeding the same antenna from different locations.
The calculator also shows how a line terminated in an open or short circuit produces a pure reactance at its input, which varies from highly inductive to highly capacitive as line length changes. This behaviour is the basis for stub matching, where a shunt stub of calculated length is connected at a specific point on the main feedline to cancel an unwanted reactive component and present 50 ohms to the transmitter.
Velocity factor is essential in these calculations. A cable with VF of 0.66 carries signals at 66 percent of free-space speed, making a physical metre of cable equivalent to 1 / 0.66 = 1.52 metres of electrical length. Ignoring velocity factor when cutting stubs or quarter-wave transformers leads to resonance errors that can dramatically raise SWR even when the lengths look correct on paper.
How the Transmission Line Impedance Calculator Works
The calculator implements the exact lossless transmission line input impedance formula: Z_in = Z₀ × (Z_L + j·Z₀·tan βℓ) / (Z₀ + j·Z_L·tan βℓ), where β = 2πf / (c × VF) is the propagation constant in radians per metre, ℓ is the physical line length, Z₀ is the characteristic impedance, and Z_L is the complex load impedance. For general complex loads the formula is evaluated in full using complex arithmetic to produce the real and imaginary parts of Z_in.
Special cases are flagged when the line length is close to a quarter-wave (βℓ = 90°) or half-wave (βℓ = 180°) multiple. At a quarter-wave, Z_in approaches Z₀² / Z_L, enabling impedance transformation. At a half-wave, Z_in reproduces Z_L. These special lengths are highlighted in the results table. The chart displays impedance magnitude from 0 to two full wavelengths of electrical length so you can see how Z_in cycles with line length.
Transmission line impedance formula
Z_in = Z₀ × (Z_L + j·Z₀·tan βℓ) / (Z₀ + j·Z_L·tan βℓ)
β = 2π × f / (c × VF) where c = 299,792,458 m/s
Electrical length (°) = β × ℓ × (180/π)
Quarter-wave transformer: Z_in = Z₀² / Z_L (when βℓ = 90°)
Half-wave line: Z_in = Z_L (when βℓ = 180°)
SWR = (|Z_in| + Z₀) / (Z₀ − Z₀²/|Z_in|) ... derived from (Z_in − 50) / (Z_in + 50)
Example Calculations
Example 1: 40 metre dipole (73 Ω) with 66 ft of RG-213
A 40 metre resonant dipole presents about 73 Ω at the feed point. Into 50 Ω RG-213, the SWR is 1.46:1. At 66 feet, the electrical length at 7.15 MHz is about 33°. The input impedance stays close to 73 Ω for short runs because the mismatch is modest; a tuner or direct coax connection both work well.
Example 2: Quarter-wave transformer for 75 Ω to 50 Ω
To match a 75 Ω antenna to a 50 Ω feedline, the quarter-wave transformer cable should have Z₀ = √(75 × 50) = 61.2 Ω. Standard 75 Ω cable is not far off, and the input impedance at the transmitter end of a quarter-wave section will be 75²/50 = 112.5 Ω — which shows that the transformer impedance must be chosen correctly for the application.
Example 3: Open-circuit stub for VHF matching
A 12-inch open stub of RG-58 (VF 0.66) at 144 MHz has an electrical length of about 62°. The input impedance is −j·Z₀·cot(62°) = −j·50·cot(62°) ≈ −j27 Ω — a capacitive reactance. Connecting this stub in shunt at the right point on the main feedline can cancel an inductive mismatch and present 50 Ω to the transmitter.
Common Amateur Radio Uses
- Design quarter-wave matching transformers to bridge antenna feed-point impedances to 50 Ω feedlines without a lumped-component tuner.
- Calculate the input impedance of an open or short stub for VHF and UHF matching, phasing arrays, or band-rejection traps.
- Predict what impedance a transmitter or ATU will see after a long feedline run to a high-SWR antenna.
- Determine the electrical length of a feedline for NVIS or phased array applications where line length affects pattern and feed current balance.
- Understand why moving the ATU location (antenna end versus transmitter end) changes which impedances the tuner must match.
- Verify that a half-wave feedline presents the same impedance as the antenna feed point, which can be exploited for remote tuner placement or switching.
Tips for Better Ham Radio Planning
The calculator assumes a lossless line, which is an excellent approximation for short amateur feedlines but diverges slightly on long runs with high-loss cable. For lossy lines, the input impedance at extreme SWR values (like open or short terminations) approaches Z₀ rather than the extreme reactive values predicted by the lossless model. For most practical amateur applications — matching at VHF and UHF, designing HF traps, or planning phasing cables — the lossless model is accurate enough to guide construction.
When using the quarter-wave transformer property, remember that it only transforms real impedances into real impedances at exactly the quarter-wave frequency. Off-frequency, the transformed impedance acquires a reactive component and the SWR rises. For broadband applications, lumped-component matching or a two-section transformer designed to present an acceptable SWR over a range is more appropriate than a simple quarter-wave section.
Frequently Asked Questions
What is transmission line input impedance and why does it matter?
The input impedance of a transmission line is the complex impedance seen looking into the feedline from the transmitter side. It depends on the characteristic impedance Z₀, the load impedance at the antenna end, the electrical length of the line, and the velocity factor. Matching this impedance to 50 ohms at the transmitter is key to minimising reflected power and maximising power transfer, which is exactly what an antenna tuner does.
Why does a quarter-wave feedline transform impedance?
A quarter-wave section has the special property that its input impedance equals Z₀ squared divided by the load impedance. This makes a quarter-wave transformer useful for impedance matching — a 75 Ω coax quarter-wave section can transform a 112 Ω antenna feed point to exactly 50 Ω, or a section of calculated characteristic impedance can bridge any two real impedances. At exactly λ/2, the line simply reproduces the load impedance, making half-wave lines impedance-transparent.
What happens when the load impedance is purely reactive?
A reactive load (capacitive or inductive) changes both the magnitude and phase of the input impedance continuously as line length increases. An open-circuit termination (infinite load impedance) produces a capacitive input impedance for lines shorter than a quarter-wave and an inductive input for lines between λ/4 and λ/2. A short-circuit termination does the opposite. These properties are exploited in stub tuners and reactive matching networks.
How does velocity factor affect transmission line electrical length?
Velocity factor (VF) is the ratio of the signal propagation speed in the cable to the speed of light. A cable with VF of 0.66 carries signals at 66 percent of light speed, so a physical length corresponds to a longer electrical wavelength. A physical quarter-wave section at 0.66 VF is shorter than the free-space quarter-wave by a factor of 0.66. The calculator applies VF when converting your entered physical length to the electrical length used in the impedance formula.
What is the difference between series and shunt stubs?
A series stub is inserted in-line with the feedline and presents its input impedance in series with the transmission path. A shunt stub (open or short) is connected in parallel with the main line at a specific point and adds a parallel reactance at that point to cancel unwanted reactive components. Both techniques use the transmission line input impedance equations this calculator computes, applied to the specific stub geometry needed for the matching problem.
When would I use an open-circuit versus a short-circuit stub?
Short-circuit stubs are easier to construct in practice — a simple fold or connector short at the far end — and are commonly used in VHF and UHF matching applications. Open-circuit stubs are slightly simpler to analyse theoretically. In coax, short-circuit stubs are preferred because an open end on a coax stub can radiate and couple to nearby conductors. In open-wire or ladder-line stubs, either termination is practical and the choice depends on the required reactance at the feed point.
Sources and References
- ARRL Antenna Book, 24th edition — Transmission Lines chapter, input impedance derivation and stub matching techniques.
- ARRL Handbook for Radio Communications — RF transmission line theory and practical matching applications.
- Pozar, D.M., Microwave Engineering, 4th edition — Transmission line theory and Smith chart analysis.
- ITU-R Recommendations on transmission line parameters for VHF and UHF amateur bands.