ECE4122 Chap.3 Boundaries, Reflection and Transmission Lines
Boundaries, Reflection and Transmission Lines
Reflection occurs because incident and transmitted fields must satisfy boundary conditions across an impedance change. At a material interface, intrinsic impedances determine the coefficient; at a line termination, characteristic and load impedances play the same role.
The coefficient's magnitude gives reflected amplitude and its angle gives phase.
On a lossless line, reflection magnitude stays constant while phase rotates with position. Incident and reflected voltage form a standing-wave envelope described by VSWR. Input impedance depends on electrical length, so reference plane and direction must be declared.
What this chapter covers
- 01
Normal-incidence reflection and transmission
- 02
Oblique-incidence geometry and polarisation
- 03
Load reflection coefficient
- 04
Standing waves and VSWR
- 05
Input impedance along a lossless line
- 06
Transient reflections and reference planes
Worked example · free
Measure mismatch on a complex load
- +1Substitute Γ_L = (Z_L - Z₀)/(Z_L + Z₀).
- +2Evaluate (-20 + j40)/(80 + j40) as j0.50.
- +1Read magnitude 0.50 and phase +90°.
- +2Compute VSWR = (1 + 0.50)/(1 - 0.50) = 3.00.
Key terms
- Reflection Coefficient
- The complex ratio of reflected to incident amplitude at a specified reference plane.
- Transmission Coefficient
- The transmitted-to-incident field amplitude ratio across a stated boundary.
- Standing Wave
- The spatial envelope produced when incident and reflected waves of the same frequency superpose.
- Voltage Standing-Wave Ratio
- The maximum-to-minimum voltage ratio on a lossless line, determined by reflection magnitude.
- Electrical Length
- The phase length βl of a structure, which changes with frequency even when physical length is fixed.
Boundaries, Reflection and Transmission Lines FAQ
What does a negative reflection coefficient mean?
For a real normal-incidence coefficient, it means the reflected electric-field amplitude reverses relative to the incident field at the boundary. It does not mean negative power; reflected power follows the squared magnitude.
Why does input impedance change along a line?
The local voltage and current are sums of incident and reflected waves whose relative phase changes with position. Their ratio therefore varies even though the physical load at the end of the line is fixed.
How are VSWR and reflected power related?
VSWR first gives |Γ| through (s - 1)/(s + 1). Reflected power fraction is then |Γ|². Squaring is essential because Γ is an amplitude ratio. This separates the measured standing-wave envelope from the energy fraction returned toward the source.
Why should I state a reference plane?
Reflection-coefficient phase and input impedance change with distance from the load. Without a reference plane, two correct calculations at different positions can appear contradictory. Always attach the plane to the complex coefficient before comparing phase or transformed impedance.
Exam move
Draw the incident side, transmitted side and positive travel direction for every boundary problem. Predict the sign or limiting value of Γ before computing it. For lines, label the load plane and generator direction, then connect Γ, VSWR and input impedance as three views of one mismatch. Always convert amplitude reflection to power with a squared magnitude. Use one reference-plane diagram for every problem.
On a material boundary, label medium one, medium two, the unit normal and the three wave vectors. On a line, label generator direction, load plane, characteristic impedance and electrical distance. Predict the sign or quadrant of the reflection coefficient from the impedance step before performing complex division.
This prediction is especially useful for real loads: a load above the reference impedance gives positive real voltage reflection, while a lower real load gives a phase reversal. Separate amplitude and power throughout. Electric-field or voltage reflection uses the complex coefficient itself; reflected power uses its squared magnitude.
Transmission across dissimilar media needs the appropriate impedance factor, so a field transmission amplitude above one does not by itself violate conservation. Close a lossless boundary calculation by checking incident power against reflected plus transmitted power. For a passive load, confirm that reflection magnitude does not exceed one and that standing-wave ratio is at least one.
Build a transmission-line method around three equivalent descriptions. First calculate the load coefficient. Second obtain standing-wave ratio from magnitude and explain what phase information it discards. Third use electrical length to find an upstream coefficient or input impedance. Practise open, short, matched, quarter-wave and half-wave cases as anchors.
A half wavelength repeats the load, a quarter wavelength performs an impedance inversion, and a matched line remains matched at every plane. If a general tangent calculation contradicts one of those limits, inspect the sign and direction convention. Mix forward and inverse drills. Given a complex load, calculate reflection and standing-wave ratio.
Given standing-wave ratio and a measured minimum position, reconstruct magnitude and phase at a specified reference plane. Given input impedance and length, work backward to the load. Keep beta in radians per metre and convert physical distance to electrical phase before using a formula.
Finish by naming what is invariant and what is not: on an ideal lossless line reflection magnitude is constant, but phase and local impedance rotate with position. That statement prevents a correct load result from being attached to the wrong plane.
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