Transmission Lines — Characteristic Impedance, Reflection & VSWR¶

Apparatus layout diagram for this experiment

Labelled apparatus layout — original illustration prepared for this guide (not a photograph). See the drag-to-rotate 3-D schematic above for a spatial view of the same setup.

3-D apparatus schematic · drag to rotate, scroll to zoom · A transmission line of characteristic impedance Z₀ terminated in a load Z_L; any mismatch produces a reflected wave characterised by reflection coefficient Γ, which sets up a standing-wave pattern measured by the VSWR.

Aim¶

To compute the reflection coefficient and voltage standing-wave ratio (VSWR) on a transmission line of characteristic impedance Z₀ terminated in a mismatched load impedance Z_L, and to see how impedance matching minimises reflected power.

Theoretical Background¶

A transmission line — a coaxial cable, a twin-lead, or a printed microstrip trace — has an intrinsic characteristic impedance Z₀ determined by its geometry and the dielectric between its conductors. If the line is terminated in a load whose impedance Z_L differs from Z₀, not all of the incident wave's power can be absorbed by the load; a fraction is reflected back towards the source, described by the (generally complex) reflection coefficient Γ = (Z_L−Z₀)/(Z_L+Z₀). The superposition of the incident and reflected waves sets up a standing-wave pattern along the line, with the ratio of maximum to minimum voltage amplitude called the voltage standing-wave ratio (VSWR). A perfectly matched load (Z_L = Z₀) gives Γ = 0 and VSWR = 1 (no standing wave, all power delivered to the load), while a short-circuit or open-circuit load gives |Γ| = 1 and an infinite VSWR. Minimising VSWR through careful impedance matching is a central concern in RF and microwave engineering, since a high VSWR both wastes transmitted power and can stress the source.

Governing Formula¶

\[ \Gamma = \dfrac{Z_L - Z_0}{Z_L + Z_0}, \qquad VSWR = \dfrac{1+|\Gamma|}{1-|\Gamma|} \]

Symbol	Meaning
Γ	voltage reflection coefficient (dimensionless)
Z₀	line characteristic impedance (Ω)
Z_L	load impedance (Ω, assumed resistive here)
VSWR	voltage standing-wave ratio, always ≥ 1

Interactive Controls¶

Launch the simulator and use the following controls on the Transmission Lines — Characteristic Impedance, Reflection & VSWR panel:

Control	Symbol	Range	Default
Line characteristic impedance	Z₀	25–150 Ω	50 Ω
Load impedance (resistive)	Z_L	5–500 Ω	75 Ω

Procedure¶

Fix Z₀ = 50 Ω (a standard RF line impedance) and vary Z_L from 5 Ω to 500 Ω, recording Γ and VSWR at each step.
Plot VSWR versus Z_L and identify the value of Z_L giving the minimum VSWR (it should be Z_L = Z₀).
Verify that Z_L = 0 (short circuit) and Z_L → ∞ (open circuit) both give |Γ| = 1 and a very large VSWR.
Repeat for a different line impedance Z₀ (e.g. 75 Ω, as used in cable-TV systems) and note how the optimal load shifts accordingly.
Discuss what fraction of incident power is reflected for a given Γ (reflected power fraction = |Γ|²), and compute it for one representative mismatch.

Observation Table¶

| S. No. | Z₀ (Ω) | Z_L (Ω) | Γ | VSWR | Reflected power fraction |Γ|² | |---|---|---|---|---|---|| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

Graph¶

VSWR vs load impedance Z_L

Calculations¶

Use the governing formula above with your tabulated observations to compute the required result for each row. Show at least one complete sample calculation with units.

Result¶

| Quantity | Z_L giving minimum VSWR | Expected value | % deviation | |---|---|---|---|| | | | | |

Precautions¶

Treat the load as purely resistive in this simplified model; a reactive load would give a complex Γ with an additional phase angle.
Remember VSWR is always ≥ 1, with 1 representing a perfectly matched, reflection-free termination.
Vary Z_L over a wide enough range to clearly locate the VSWR minimum.
Do not confuse the characteristic impedance Z₀ (a property of the line itself) with the load impedance Z_L (a property of the termination).

Maximum Permissible Error¶

\[ \text{Evaluated numerically from } Z_0 \text{ and } Z_L \text{ via } \Gamma \]

VSWR is a nonlinear function of the reflection coefficient Γ, so its propagated uncertainty is best evaluated numerically by perturbing Z₀ and Z_L in turn rather than via one closed-form expression.

Viva-Voce Questions¶

Define the reflection coefficient Γ and the voltage standing-wave ratio (VSWR).
Why does a matched load (Z_L = Z₀) give zero reflection?
What are the VSWR values for a short-circuit and an open-circuit termination, and why?
Explain the practical importance of minimising VSWR in an RF transmitter feed line.
What technique(s) can be used to match a mismatched load to a transmission line?