Power flow, R/T & wave impedance

The instantaneous macroscopic Poynting vector is S(t)=E(t)×H(t). For harmonic fields AoN writes E(t)=Re{Ẽe^{jωt}} and H(t)=Re{H̃e^{jωt}}, where Ẽ and H̃ are PEAK complex amplitudes. Multiplying the two real parts produces a constant term plus a term oscillating at 2ω; the latter averages to zero over one period.

Therefore the cycle-averaged real power-flux density is one half the real part of Ẽ×H̃*. Here * denotes complex conjugation. The factor 1/2 belongs to peak phasors. If a root-mean-square (RMS) phasor convention were used instead, that factor would not appear.

The complex vector (1/2)Ẽ×H̃* is useful because its real part is average active power flow. Its imaginary part participates in the complex Poynting theorem and should not be casually called transported real power.

In a homogeneous isotropic lossless medium, η=√(μ/ε) is real and H̃=(1/η)k̂×Ẽ. Because Ẽ is transverse, Ẽ×(k̂×Ẽ*)=|Ẽ|²k̂. Substituting into the average Poynting expression gives a flux parallel to propagation.

The same result can be written using H. Thus the field-amplitude ratio E/H is not merely a unit conversion: it sets how much power density corresponds to a given electric or magnetic amplitude.

The accepted AoN wave diagram below remains the signed geometry reference: E, H/B and propagation form a right-handed triad, and the average power points with k for a passive lossless plane wave.

At an interface, a plane wave crossing a surface with normal n transports only the normal component of its Poynting vector through that surface. For a propagating lossless wave making angle θ from the normal, the normal flux is |E|² cosθ/(2η).

Let r and t denote the electric FIELD-amplitude Fresnel coefficients of chapter 04, in either s or p basis. The reflected wave stays in medium 1 with the same η₁ and |cosθ_i|, so the reflected-power fraction is R=|r|². The transmitted wave generally has a different impedance and angle, so T requires an extra flux factor.

This is the precise reason that |t|² is usually NOT the transmitted-power fraction.

For two passive lossless media with no absorbing sheet and no other source at the interface, the normal time-averaged flux entering the interface must equal the sum leaving it. Therefore R+T=1 for each uncoupled s or p polarization.

This identity is a powerful audit of both algebra and basis conventions. A Fresnel amplitude set that violates R+T=1 in a lossless propagating case has a missing impedance/cosine factor, an inconsistent t definition, or a sign/basis error.

At the Brewster angle r_p=0, the ideal lossless p-polarized interface has R_p=0 and T_p=1. Under total internal reflection, |r|=1 and the propagating normal transmitted power is zero even though a nonzero evanescent field exists in medium 2.

Reuse chapter 04: n₁=1, n₂=1.5, θ_i=45°, θ_t≈28.13°, with nonmagnetic media. The common transmitted flux factor is n₂cosθ_t/(n₁cosθ_i)≈1.87083.

For s polarization, r_s≈−0.303337 and t_s≈0.696663, giving R_s≈0.092013 and T_s≈0.907987. For p, r_p≈+0.0920134 and t_p≈0.728009, giving R_p≈0.00846646 and T_p≈0.991534.

Both sums equal one within rounding. Notice again that t_p²≈0.530 does NOT equal T_p≈0.992; the medium/angle factor is essential.

Now keep AoN's exp(+jωt) convention and model a homogeneous isotropic medium with real ε, μ and bulk conductivity σ. Ampère–Maxwell becomes ∇×H=(σ+jωε)E. Combining with Faraday gives a complex propagation constant and complex intrinsic impedance.

With the spatial convention exp(−jkz), a passive decaying wave uses k=β−jα with α>0, because exp(−jkz)=exp(−jβz)exp(−αz). The physical square-root branch must therefore be selected by passivity/decay, not by an arbitrary algebraic sign.

When η is complex, E and H are not exactly in phase. The average Poynting vector must be calculated from (1/2)Re(E×H*), not by replacing η with |η| in the lossless formula.

A useful limiting check is the good-conductor regime σ≫ωε. The passive branch then gives equal phase and attenuation constants, β≈α≈√(ωμσ/2); the skin depth is δ_s=1/α, and the intrinsic impedance approaches (1+j)√[ωμ/(2σ)] under AoN’s +jωt convention.

For a real Ohmic bulk conductivity σ, the cycle-averaged power transferred from the field to matter is (1/2)σ|E|² per unit volume under the peak-phasor convention. This is the harmonic counterpart of instantaneous E·J.

For an infinitesimally thin isotropic conductive sheet with surface current K=σ_sE_t, the average absorbed power per unit AREA is (1/2)Re(σ_s)|E_t|². The tangential field must be the field actually acting on the sheet.

For a passive planar interface or sheet problem with only the reflected and transmitted plane-wave channels, define the dimensionless absorptance A=1−R−T. This equality is an energy balance, not an independent optical law. If diffraction or other outgoing modes exist, their normalized powers must also be subtracted; if the model contains gain or an active sheet, A need not be positive.

A vacuum plane wave with peak E=120 V/m carries average power density E²/(2η₀)≈19.11 W/m². Frequency does not appear explicitly because for a specified vacuum field amplitude the intrinsic impedance is frequency-independent in the ideal model.

A conductive sheet with Re(σ_s)=1.00 mS exposed to a tangential peak field of 10.0 V/m dissipates (1/2)(10⁻³)(10²)=0.050 W/m². This is a local sheet-loss density, not automatically the absorptance; absorptance requires division by incident normal power and a self-consistent field solution.

EM-04 now provides the bridge from Maxwell fields to RF quantities: wave impedance, reflected/transmitted power and field normalization. Transmission-line characteristic impedance and scattering parameters will reuse this logic, but they are port/network quantities rather than synonyms for intrinsic wave impedance.

1. Derive ⟨S⟩=(1/2)Re(Ẽ×H̃*) by expanding the product of the two real harmonic fields and averaging one period.
2. A lossless medium has η=250 Ω and peak |E|=50 V/m. Compute average plane-wave power density.
3. Derive the transmitted-power coefficient T from the incident and transmitted normal Poynting fluxes.
4. Explain why R=|r|² needs no extra η or cosθ factor when reflection occurs back into the same lossless medium.
5. Using the chapter-04 air→glass 45° amplitudes, reproduce R_s,T_s and verify their sum.
6. Repeat for p polarization and explain quantitatively why |t_p|² is far from T_p.
7. At an ideal Brewster angle, what are R_p and T_p? State the assumptions needed.
8. For ideal total internal reflection, explain how a nonzero evanescent transmitted field can coexist with zero net normal transmitted power.
9. With exp(+jωt−jkz), show why k=β−jα with α>0 gives decay toward +z.
10. A sheet has Re(σ_s)=2.5 mS and tangential peak |E_t|=8 V/m. Compute its local average absorbed power density. What additional information is needed to convert that into absorptance?

EM-04 is complete. The next RF course can now derive distributed transmission-line equations, characteristic impedance and reflection from Maxwell-compatible field/power concepts without conflating line impedance with intrinsic wave impedance.