How is the reflection coefficient computed?

The reflection coefficient is Γ = (Z_L − Z_0)/(Z_L + Z_0), where Z_0 is the characteristic impedance and Z_L the load. When Z_L is complex (has reactance), Γ is also complex with magnitude |Γ| and phase angle. |Γ| = 0 is a perfect match and |Γ| = 1 is total reflection. With a transformer of ratio N on the primary side, replace Z_L by Z_in = N² · Z_L.

What does VSWR measure?

VSWR is the ratio of voltage maxima to minima on the line: SWR = (1 + |Γ|)/(1 − |Γ|). SWR = 1 is a perfect match (only forward waves, no standing pattern). Larger SWR means stronger reflection. Practical targets are SWR < 1.5; at SWR ≥ 3 most transmitters fold back output to protect the final stage. The job of an antenna tuner is to drive SWR down toward 1.

How do you obtain the power transfer efficiency?

The fraction of incident power that reaches the load is η = 1 − |Γ|². For example |Γ| = 0.2 gives 4 % reflected power and 96 % efficiency. |Γ| = 0.5 (about SWR = 3) drops efficiency to 75 %, showing why matching matters. The mismatch loss in dB is 10·log10(1/(1−|Γ|²)).

Why does a transformer ratio N change the apparent impedance?

An ideal transformer with turns ratio N = N1/N2 preserves power but scales voltage by N and current by 1/N, so the impedance seen from the primary is Z_in = N² · Z_L. To match an 8 Ω speaker to a 50 Ω line you need N² = 50/8 = 6.25, i.e. N ≈ 2.5. Real transformers also have leakage inductance and winding loss that limit bandwidth, so they are used together with LC or λ/4 networks.

Impedance Matching Simulator — Reflection Coefficient and SWR

Q: How do you obtain the power transfer efficiency?

The fraction of incident power that reaches the load is η = 1 − |Γ|². For example |Γ| = 0.2 gives 4 % reflected power and 96 % efficiency. |Γ| = 0.5 (about SWR = 3) drops efficiency to 75 %, showing why matching matters. The mismatch loss in dB is 10·log10(1/(1−|Γ|²)).

Q: Why does a transformer ratio N change the apparent impedance?

An ideal transformer with turns ratio N = N1/N2 preserves power but scales voltage by N and current by 1/N, so the impedance seen from the primary is Z_in = N² · Z_L. To match an 8 Ω speaker to a 50 Ω line you need N² = 50/8 = 6.25, i.e. N ≈ 2.5. Real transformers also have leakage inductance and winding loss that limit bandwidth, so they are used together with LC or λ/4 networks.

Parameters

Characteristic impedance Z_0

Load resistance R_L

Load reactance X_L

Transformer ratio N = N1/N2

The sweep moves the load resistance from 1 Ω to 1000 Ω and crosses the matching point R_L = Z_0/N². A transformer ratio N transforms the impedance by N².

Results

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Reflection |Γ|

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VSWR

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Reflected |Γ|²

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Power efficiency

Matching Schematic

Source -> transmission line (Z_0) -> transformer (N:1) -> load Z_L. Incident (yellow) and reflected (orange) waves are shown, with the equivalent input impedance Z_in = N²·Z_L labeled.

Smith Chart

Maps the normalized impedance z = Z_in/Z_0 to the Γ plane. Center = matching point (z=1), yellow dot = current Γ, dashed = SWR circle. Right end = open, left end = short, upper half = inductive, lower half = capacitive.

Theory & Key Formulas

Reflection coefficient (complex):

$$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$$

Voltage standing wave ratio:

$$\mathrm{SWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}$$

Reflected power and transfer efficiency:

$$\eta = 1 - |\Gamma|^2$$

Impedance transformation by ratio N:

$$Z_{in} = N^2 \, Z_L$$

$|\Gamma| = 0$ and $\mathrm{SWR} = 1$ mean a perfect match (no reflection). $\mathrm{SWR} < 1.5$ is the practical engineering target.

What is the Impedance Matching Simulator?

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My radio's manual says "tune the antenna SWR below 1.5". What is SWR, and is the difference between 1.5 and 3 really significant?

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Good question. SWR stands for Voltage Standing Wave Ratio, the ratio of the voltage maximum to the minimum along the line. SWR = 1 is a perfect match — no standing wave, only forward power. A larger SWR means more reflection and a stronger standing-wave pattern. SWR = 1.5 reflects about 4 % of the power; SWR = 3 reflects 25 %, so a quarter of the transmitter output never reaches the antenna. Try the defaults (Z_0=50, R_L=75, X_L=0, N=1) — you should see SWR = 1.50.

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Yes, I see SWR = 1.50, |Γ| = 0.200, reflected 4.00 %, efficiency 96.0 %. Where do these numbers come from?

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The formula is Γ = (Z_L − Z_0)/(Z_L + Z_0). With Z_L = 75 Ω and Z_0 = 50 Ω, Γ = 25/125 = 0.2 (purely real). |Γ| = 0.2, reflected power |Γ|² = 0.04, transfer efficiency η = 1 − |Γ|² = 0.96 = 96 %. SWR = (1+0.2)/(1−0.2) = 1.5. On the Smith chart the yellow dot is 0.2 to the right of the origin (matching point). The closer to the centre, the better the match.

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There is also a slider for the transformer ratio N. What does it do?

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It lets a transformer transform the impedance: the primary side sees Z_in = N² · Z_L. Here you have a 75 Ω antenna on a 50 Ω line and the match is off. Set N to √(50/75) ≈ 0.816; then Z_in = 0.816² × 75 ≈ 49.9 Ω, essentially a perfect match. Try N ≈ 0.82 in the slider — |Γ| should drop close to 0.000. This is the core idea of matching networks used between transmitters, feedlines and antennas.

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What happens when I change the load reactance X_L?

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X_L is the reactive part of the load: positive is inductive, negative is capacitive. Real antennas become reactive when driven off resonance, since they are no longer pure resistors. Set X_L to +50 — |Γ| jumps to about 0.318 and the yellow dot moves into the upper half of the Smith chart (inductive). With X_L = −50 the dot moves to the lower half (capacitive). Antenna tuners cancel the reactive part with adjustable series/parallel reactances to bring the spot back toward the centre.

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What exactly is a Smith chart? Why all those nested circles?

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It is a plot of the complex reflection coefficient Γ on the unit disk, invented in 1939 by Philip H. Smith. The outer circle is |Γ| = 1 (total reflection) and the centre is Γ = 0 (perfect match). Vertical groups of circles are loci of constant normalised resistance r = R/Z_0; the upper and lower arcs are loci of constant normalised reactance x = X/Z_0. Every impedance maps to a unique point. Moving R_L and X_L shows the geometric meaning of these grids and makes matching feel intuitive.

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What does the "Sweep R_L" button reveal?

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It animates R_L from 1 Ω to 1000 Ω and crosses the matching point R_L = Z_0/N². With N = 1 the match is at R_L = 50 Ω, where |Γ| drops to 0. Below 50 Ω, Γ is negative (toward short); above 50 Ω, Γ is positive (toward open). The yellow dot slides along the real axis on the Smith chart. Set N = 2 and the matching point moves to R_L = Z_0/N² = 12.5 Ω, showing how the transformer "shifts" the match position.

Physical Model and Key Equations

When a transmission line with characteristic impedance $Z_0$ terminates in a load $Z_L = R_L + jX_L$, part of the incident wave reflects at the boundary. The voltage reflection coefficient is $\Gamma = (Z_L - Z_0)/(Z_L + Z_0)$, derived from voltage and current continuity. If $Z_L$ is complex, $\Gamma$ is complex too with magnitude $|\Gamma| \in [0, 1]$ and a phase angle. $\Gamma = 0$ is a perfect match (no reflection); $|\Gamma| = 1$ is total reflection (open, short, or purely reactive load).

The interference of incident and reflected waves forms a standing pattern with voltage maxima $|V^+|(1 + |\Gamma|)$ and minima $|V^+|(1 - |\Gamma|)$, so $\mathrm{SWR} = (1 + |\Gamma|)/(1 - |\Gamma|)$. In power terms the fraction reflected back to the source is $|\Gamma|^2$ and the power transfer efficiency to the load is $\eta = 1 - |\Gamma|^2$. An ideal transformer with primary-to-secondary ratio $N:1$ transforms the secondary load to $Z_{in} = N^2 \cdot Z_L$ as seen from the primary, so the turn ratio is a direct knob on matching.

With the default values $Z_0 = 50$ Ω, $R_L = 75$ Ω, $X_L = 0$ Ω, $N = 1.0$ the simulator outputs $Z_{in} = 75$ Ω, $\Gamma = 0.200$, $\mathrm{SWR} = 1.50$, reflected power $4.00 \%$ and efficiency $96.0 \%$. This is exactly the textbook case of a 50 Ω coax driving a 75 Ω TV cable — even a 25 Ω gap costs 4 % of the power.

Real-world Applications

Radio transmitters and antennas: A typical chain is a 50 Ω PA stage, 50 Ω coax and an antenna ideally near 50 Ω. If the antenna drifts off resonance and acquires reactance, SWR climbs, the reflected wave returns to the transmitter and can stress the final transistors. Modern transceivers fold output power back when SWR exceeds about 3, and antenna tuners are used in the field to push SWR back toward 1.

TV and CATV systems: Domestic terrestrial TV uses 75 Ω coax, while instrumentation is mostly 50 Ω. Adapting between the two requires a 75/50 matching balun. The default case (Z_0 = 50, R_L = 75, SWR = 1.5, reflected 4 %) is harmless for analog TV but can cause bit errors and ISI on digital links, so purpose-built baluns are used.

Audio power amplifiers and speakers: Vacuum-tube amplifiers with output impedances of several kΩ drive 8 Ω loudspeakers through output transformers. The matching ratio is $N^2 = Z_{plate}/Z_{spk}$; for 5 kΩ : 8 Ω the turns ratio is $N \approx 25$. Mismatched transformers reduce peak power and introduce distortion. Transistor amplifiers, with near-zero output impedance, do not need this transformer because they drive the speaker as a voltage source.

Inter-stage matching in RF amplifiers: Microwave ICs rarely present 50 Ω at every transistor port, so designers insert LC matching networks (L-, π- or T-section) between stages. On a Smith chart this is a sequence of moves (series L, shunt C and so on) from the device impedance to the centre. This tool is an entry point to transformer-based matching; LC networks and λ/4 transformers complement it for narrow- and broad-band cases.

Common Misconceptions and Pitfalls

First, it is wrong to assume that reflected power simply turns into heat in the line. The reflected wave travels back to the source where it is either re-absorbed by the source impedance or dissipated by the transmitter's protection circuit. On a long line the wave attenuates round-trip, so SWR meters at the transmitter and at the antenna can give different readings. Always measure SWR close to the antenna for trustworthy matching.

Second, beware of thinking that a low SWR means an efficient antenna. SWR measures only the line/load match, not the radiation. A 50 Ω dummy load at the end of a coax has SWR = 1 but radiates nothing — all the power becomes heat in the resistor. A real antenna must be confirmed with field-strength measurements, not by SWR alone.

Third, do not assume that transformer matching is frequency-independent. An ideal transformer's $N^2$ scaling is independent of frequency, but real transformers have leakage inductance, winding capacitance and core losses, so they have a finite bandwidth and a high-frequency roll-off. Wideband matches use transmission-line transformers (ferrite cores plus PTFE coax); narrowband matches use LC networks. This simulator assumes an ideal transformer, so it does not show leakage or saturation effects.

Frequently Asked Questions

The Series RLC tool (ac-impedance-rlc) studies the frequency response of a lumped circuit where R, L and C are connected at a single node — the voltage has a single phase at any instant. The present tool deals with the terminal of a transmission line, where the wave's voltage varies with position (a standing pattern can form). Use lumped approximations when the line length is much smaller than λ/10, and transmission-line theory otherwise.

Memorize the centre (matching point z = 1), the right end (open, z = ∞), the left end (short, z = 0), the upper half (inductive) and the lower half (capacitive). Constant-resistance circles arrange themselves along the real axis; constant-reactance arcs run above and below. Moving R_L and X_L on this tool makes the grid pattern intuitive. In design work, every matching component (series L, shunt C, etc.) corresponds to a specific motion on the chart — once those motions are memorized, an entire matching network can be designed from a single printed Smith chart.

It depends on the equipment specification, but typical thresholds are: SWR < 1.5 (reflected 4 %) is excellent and the design target for commercial transmitters; SWR < 2.0 (11 %) is acceptable for continuous use in consumer gear; SWR < 3.0 (25 %) is the limit where most transmitters fold back output. Above SWR ≈ 3 the final stage is at risk and the protection circuit will shut down. Also remember that low SWR does not eliminate ohmic and dielectric losses in the cable itself, which add several dB on long runs.

This page models an ideal transformer only. Out of scope are: (1) quarter-wave transformer matching, where a λ/4 line of $Z_T = \sqrt{Z_0 \cdot Z_L}$ matches Z_0 to Z_L (narrowband, strongly frequency-dependent); (2) LC matching networks (L, π or T sections) that move arbitrary loads to Z_0 with lumped elements; (3) stub matching with shorted or open transmission-line stubs, common on microstrip boards. Combine this tool with transmission-line and microwave-transmission simulators for a complete view of matching theory.