ECE4122 Chap.2 Uniform Plane Waves and Polarisation
Uniform Plane Waves and Polarisation
A uniform plane wave has transverse electric and magnetic fields that are constant across a plane normal to propagation. Its exponential factor describes travel and attenuation; its vector amplitude describes orientation and polarisation. Intrinsic impedance links E and H, while the propagation constant separates attenuation α from phase β.
Polarisation comes from two orthogonal transverse components.
Relative amplitude and phase determine whether the field tip traces a line, circle or ellipse. Average power requires a declared peak or RMS convention and, in phasor form, the complex conjugate of H.
What this chapter covers
- 01
Travelling-wave phasors and propagation direction
- 02
Attenuation and phase constants
- 03
Intrinsic impedance and field recovery
- 04
Average Poynting vector
- 05
Linear, circular and elliptical polarisation
- 06
Peak and RMS field conventions
Worked example · free
Recover a magnetic field and power density
- +1Read +z propagation from the spatial factor.
- +2Use H = (1/η)(a_z × E) to obtain 0.200aᵧe^{-j6z} A/m.
- +1Compute λ = 2π/6 = 1.047 m.
- +2Use peak-phasor power |E|²/(2η) = 2.40 W/m² in +z.
Key terms
- Plane Wave
- A wave whose fields have no transverse variation over planes normal to the propagation direction.
- Intrinsic Impedance
- The complex ratio of electric to magnetic field for a travelling wave in a medium.
- Propagation Constant
- The quantity α + jβ describing exponential attenuation and spatial phase change.
- Polarisation Ellipse
- The path traced by the electric-field tip at a fixed point as time advances.
- Axial Ratio
- The major-axis to minor-axis ratio of a polarisation ellipse.
Uniform Plane Waves and Polarisation FAQ
How can I tell which way a phasor wave travels?
Under the e^{jωt} convention, e^{-jβz} travels toward +z and e^{+jβz} toward -z. State the time convention first, because changing it changes the sign rule while leaving the physical wave unchanged.
Does a ninety-degree phase shift always mean circular polarisation?
No. Quadrature components make an ellipse whose axes align with the component directions. Circular polarisation additionally requires equal component magnitudes. Unequal quadrature components produce elliptical polarisation.
Why is a complex conjugate used in average power?
The conjugate combines phasors so the real part represents time-average power and preserves the phase relationship between E and H. Omitting it can reverse or distort reactive and real power terms.
What changes in a lossy medium?
Amplitude decays through α, phase advances through β and intrinsic impedance can become complex. E and H need not be in phase, so lossless shortcuts for power and wavelength should be replaced by the general phasor relations.
Exam move
For each wave, write the time convention, propagation direction and E/H orientation before inserting numbers. Practise classifying polarisation from amplitude ratio and phase difference separately. Recompute average power once with peak phasors and once with RMS values so the convention difference is explicit. Use E/H and E × H as independent checks. Use a field-triad worksheet for each wave.
State the time convention, read the sign of the spatial exponential, and draw the propagation unit vector. Place the supplied electric-field components on transverse axes, then obtain magnetic-field direction with the cross product rather than a memorised sign. Divide by the appropriate intrinsic impedance and check that E cross H carries real power in the declared travel direction.
Choose the material model before simplifying. Compare conductivity with omega times permittivity to decide whether the medium is lossless, low-loss, a general lossy dielectric or a good conductor. From the exact propagation constant, identify attenuation and phase separately.
Check that alpha has inverse-length units, beta produces wavelength through two pi divided by beta, and complex intrinsic impedance permits a phase difference between E and H. Use skin depth only after the good-conductor assumptions are justified. Classify polarisation in a fixed order.
Extract the two orthogonal transverse magnitudes, calculate their phase difference and decide whether the tip path is linear, circular or elliptical. Equal magnitudes are required for a circle; quadrature alone is insufficient. State propagation direction and viewing convention before assigning handedness.
Sketch the field at two or four times separated by a quarter period to verify the rotation rather than relying on a remembered label. Alternate peak and RMS power drills. With peak phasors use one half of the real part of E cross H conjugate. With RMS phasors omit that extra factor. Check the result using both electric- and magnetic-field forms when the medium is lossless.
Predict how power changes if field amplitude doubles or intrinsic impedance changes, then compare the calculation with that scaling. End by stating attenuation direction, wavelength, polarisation and average-power direction as separate conclusions. Keeping those layers separate prevents a correct scalar magnitude from masking a reversed wave or an inconsistent phasor convention.
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