Reflection Coefficient Simulator — Complex Gamma and Z_in Transform
Compute Gamma(d) along the line in real time from the load reflection coefficient and electrical distance d/lambda. Watch the phase rotation on the complex plane while VSWR and input impedance update live.
Parameters
Characteristic Z_0
Ω
Load Re(Z_L)
Ω
Load Im(Z_L)
Ω
Electrical distance d/λ
λ
d/λ = 0 is the load end; 0.5 corresponds to a full 360° rotation of the phase (lossless line assumed).
Results
—
|Γ|
—
∠Γ (phase)
—
VSWR = (1+|Γ|)/(1−|Γ|)
—
|Z_in| at current point
Reflection coefficient on the complex plane
Red dot = load Γ_L / Blue dot = Γ(d) at current point / Green dashed circle = constant |Γ| / Blue dashed arc = locus going from load toward source (clockwise)
Theory & Key Formulas
The strength of reflection on a transmission line is described by a complex reflection coefficient Γ, fixed at the load end by:
$$\Gamma_L = \frac{Z_L - Z_0}{Z_L + Z_0}$$
At a distance d from the load toward the source (β = 2π/λ is the phase constant):
On a lossless line |Γ| stays constant while only the phase rotates, so Γ traces a circle centered at the origin of the complex plane.
What is the Reflection Coefficient Simulator
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In RF work you hear "reflection coefficient" and "VSWR" all the time. What do they actually represent?
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Roughly speaking, they describe how much of a wave traveling down a transmission line bounces back. When the characteristic impedance Z_0 of the line and the load Z_L attached at the end disagree, part of the wave reflects. The ratio of reflected to incident voltage is the reflection coefficient Γ, given by $\Gamma_L = (Z_L - Z_0) / (Z_L + Z_0)$. Set Z_L to 50 Ω in the tool above — Γ snaps to the origin, the perfect match.
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A complex number is hard to picture. What do the real and imaginary parts mean physically?
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The magnitude |Γ| says "how strongly it reflects" and the angle ∠Γ says "how much the phase is shifted when it does". If the load is a pure resistor equal to Z_0, Γ=0. If it is a pure resistor not equal to Z_0, Γ is real (phase 0° or 180°). Add reactance and Γ becomes complex. With the default Z_L = 100 + j50 you should see |Γ| ≈ 0.447 and phase 26.6°. On the complex plane it sits up and to the right of the origin.
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When I press "Sweep distance", the blue dot moves clockwise around a circle. What is that?
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That is Γ(d), the reflection coefficient as you move from the load toward the source along the line. On a lossless line |Γ| does not change, so on the complex plane the point just rotates along the green circle. One wavelength of travel rotates the phase by 4π, which is two full turns, so it returns to the same point. The fact that Γ traces a circle on a Smith chart is the formula $\Gamma(d) = \Gamma_L e^{-j2\beta d}$ shown visually.
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The VSWR card shows 2.62 with the defaults. What does that mean?
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It is the Voltage Standing Wave Ratio — how wildly the voltage swings along the line. A perfect match gives 1; total reflection gives infinity. Antenna data sheets often say "VSWR ≤ 2.0", and engineers tune matching networks to meet that. The neat thing is that VSWR depends only on |Γ|, so it reads the same anywhere on the line — that is why "the antenna VSWR is bad" is even a meaningful sentence.
Frequently Asked Questions
At an electrical distance of 0.25 the phase rotates by π and the reflection coefficient flips sign (Γ(λ/4) = −Γ_L). Converting to impedance gives Z_in = Z_0² / Z_L: the load impedance is transformed to "the characteristic impedance squared divided by the load". This is the basis of the well-known quarter-wave transformer used for impedance matching. For Z_0 = 50 Ω and Z_L = 100 Ω, a quarter wave line presents Z_in = 25 Ω at its input.
For ordinary passive loads, |Γ| ≤ 1. Only when the load is an active element (a negative-resistance device, an oscillating transistor, and so on) can |Γ| > 1 occur, which means the load is injecting energy into the line. Designers of oscillators and reflection amplifiers deliberately work in the |Γ| > 1 region, but this tool assumes a passive load and so usually shows |Γ| < 1.
The Smith chart is the complex Γ plane that this tool draws (inside the unit circle) with constant-resistance and constant-reactance circles of the normalized impedance z = Z/Z_0 overlaid on it. This tool plots constant-|Γ| circles; the Smith chart adds the impedance grid so that the impedance can be read off directly. In practice matching designs plot Γ on a Smith chart and alternate straight-line travel along the line with discrete jumps from added components.
This tool assumes a lossless line, so |Γ| stays constant. Real coaxial cables or microstrips have conductor and dielectric loss; then Γ(d) = Γ_L * exp(−2αd) * exp(−j2βd), with an attenuation constant α. The magnitude shrinks as you move away from the load, and the trajectory on the complex plane spirals inward rather than tracing a circle. VSWR measured at the source end looks better than at the load end, so cable measurements need care.
Real-World Applications
Antenna and radio matching design: Almost every RF device — mobile phones, Wi-Fi, broadcast transmitters — uses reflection coefficients to match antennas to the radio circuit. A poor VSWR sends transmit power back into the amplifier and destroys it, or kills receive sensitivity, so engineers spend their days nudging Γ toward the origin with matching networks (LC tuning, quarter-wave transformers, stubs). The S-parameter S11 measured by a network analyzer is exactly the input reflection coefficient.
Cable testing (TDR): Time-Domain Reflectometry sends a step pulse down a cable and watches the reflections returning from discontinuities, locating breaks and bad connectors by time of arrival. The polarity and size of the reflection tell you whether the fault looks like an open (Γ=+1) or a short (Γ=−1), and the round-trip time plus the propagation velocity gives the distance. It is widely used for LAN cabling and high-frequency coaxial maintenance.
Radar and feed lines: In the waveguide system of a radar transmitter, a poor match at a rotary joint or antenna feed sends reflected power back into the high-power amplifier and burns it out. Isolators and circulators are essential to absorb reflected energy in a termination, and the allowable reflection is specified as a |Γ| limit. Optical fibre links care about return loss (typically −40 dB or better) for the same reason.
Education and electromagnetic theory: The reflection coefficient concentrates many core ideas of wave engineering into one quantity — superposition of incident and reflected waves, complex impedance, phase rotation, standing waves. It appears in third- and fourth-year undergraduate EM and communication courses, and an interactive complex-plane view like this tool can suddenly make all that abstract algebra click.
Common Misconceptions and Cautions
The most common misconception is to think that "if Z_L is real, there is no reflection". Reflection vanishes only when Z_L = Z_0 (as complex numbers); a purely resistive load with the wrong value still reflects. For instance, plugging a 75 Ω resistor into a 50 Ω system gives Γ = (75−50)/(75+50) = 0.2 and VSWR = 1.5. Drag the Im(Z_L) slider in this tool down to zero with Re(Z_L) = 100 Ω and you will see Γ = 1/3, phase 0°, VSWR = 2.0. "Real, therefore matched" is wrong.
The next most common error is to assume that increasing d/λ "spreads Γ outward" or "gradually matches" the load. On a lossless line |Γ| is exactly constant; Γ only rotates clockwise on a circle around the origin. It never moves closer to the matched state |Γ|=0. The trick of "matching by line length" simply chooses a phase where Z_in happens to be convenient, and you still need an LC element or a stub to drag Γ to the origin. Watch the |Γ| stat card while sweeping d/λ in the tool — it does not budge.
Finally, remember that this tool reports the voltage reflection coefficient, not the power reflection ratio. Reflected power is |Γ|² of the incident power, so |Γ| = 0.5 reflects 25% of the power and only 75% reaches the load. The return loss in dB is RL = −20 log₁₀|Γ|, so |Γ| = 0.5 is about 6 dB, and |Γ| = 0.1 is 20 dB. A datasheet specifying "return loss ≥ 20 dB" means |Γ| < 0.1 and VSWR < 1.22. Keeping the voltage form, the power form, and the dB form separate is where serious RF design starts.