Transmission-line power flow

In the time domain the instantaneous line power associated with the chosen transverse section is p(z,t)=v(z,t)i(z,t). Because chapter 01 fixed current positive toward +z and voltage from the lower negative reference to the upper positive reference, the sign of p is not arbitrary.

A positive value means that, under this bookkeeping convention, electromagnetic energy is crossing the section toward +z. A negative value means net transfer toward −z. The arrows remain definitions even when the solved power is negative.

This is the same passive-sign discipline used in circuit theory, now applied locally along a distributed line. We do not flip arrows to make a result positive.

Let the real waveforms be the real parts of peak-amplitude phasors multiplied by exp(+jωt). Averaging their product over one period gives the complex-power quantity S(z)=(1/2)v̲(z)i̲*(z). Its real part P is average transported power and its imaginary part Q is the local reactive-power bookkeeping quantity.

The factor 1/2 is mandatory because AoN uses PEAK phasors. If root-mean-square phasors are used instead, that factor is absorbed into the phasor amplitudes. Mixing the two conventions creates the familiar factor-of-two error.

P is measured in watts. Q is measured in volt-amperes reactive, and S in volt-amperes. Q is not dissipated energy per second; it tracks quadrature exchange associated with electric and magnetic storage.

For a pure forward wave, i̲⁺=v̲⁺/Z₀. Substituting into complex power gives S⁺=(|v̲⁺|²/2)/Z₀*. Therefore P⁺=(|v̲⁺|²/2)Re(1/Z₀), which is positive on the selected passive branch.

For a pure backward wave the SAME +z current reference gives i̲⁻=−v̲⁻/Z₀. Its real power in the +z bookkeeping direction is therefore negative: P⁻=−(|v̲⁻|²/2)Re(1/Z₀). This is the energy-flow reason the backward-current minus sign cannot be discarded.

If the forward voltage carries exp(−γz), its magnitude carries exp(−αz), so its power carries exp(−2αz). Amplitude attenuation and power attenuation therefore differ by a factor two in the exponent, exactly as chapter 04's Np/dB audit predicted.

Write a(z)=V⁺exp(−γz) and b(z)=V⁻exp(+γz). Then v̲=a+b and i̲=(a−b)/Z₀. Substitution produces a directional difference plus an interference term b a*−a b*. The latter is purely imaginary before multiplication by 1/Z₀*.

On a LOSSLESS line Z₀ is real. The interference term therefore contributes only to Q, while the average real power is exactly P=(|a|²−|b|²)/(2Z₀). Relative phase can move voltage/current maxima and reactive exchange without changing the net transported real power.

On a lossy line Z₀ is generally complex, so multiplying the purely imaginary cross term by 1/Z₀* can create a real contribution. For an arbitrary two-wave state on a lossy line, calculate P from the TOTAL v̲ and i̲; do not blindly subtract two isolated-wave formulas.

Differentiate S=(1/2)v̲i̲* with respect to z and use d v̲/dz=−Z′i̲ and d i̲/dz=−Y′v̲. Because the derivative of i̲* contains Y′*, no heuristic loss argument is needed.

The real part is especially revealing: dP/dz=−(R′|i̲|²+G′|v̲|²)/2. Series resistance removes conductor power and shunt conductance removes dielectric/leakage power, each per unit length. For passive R′,G′≥0 the forward bookkeeping power cannot increase merely through distributed dissipation.

The imaginary part is dQ/dz=(ω/2)(C′|v̲|²−L′|i̲|²). It measures the local imbalance between electric and magnetic reactive storage. On a pure lossless traveling wave those two terms are equal and Q is spatially constant.

For an exact transverse electromagnetic mode, the average electromagnetic power through a cross-section A is the integral of the time-average Poynting vector: P=(1/2)Re∫(E×H*)·ẑ dA. The line variables are field integrals, so the circuit-looking expression and the field expression must agree.

A homogeneous coaxial line makes the equality explicit. With E_r=V/[r ln(b/a)] and H_φ=I/(2πr), integrate E_r H_φ* over r from a to b and φ from 0 to 2π. The logarithm from the area integral cancels the voltage-field normalization and leaves V I*.

Thus P=(1/2)Re(VI*) exactly for the TEM coax mode. For quasi-TEM lines this remains the controlled modal approximation. Once higher modes become important, power must be evaluated from the full modal/field content rather than from one scalar voltage/current pair.

For peak phasors on a lossless line, the time-average electric and magnetic energies per unit length are C′|V|²/4 and L′|I|²/4. A pure forward wave has I=V/Z₀ with Z₀²=L′/C′, so the electric and magnetic contributions are equal.

The total average stored energy per unit length is therefore w′=C′|V|²/2. Dividing P=|V|²/(2Z₀) by w′ gives 1/(Z₀C′)=1/√(L′C′)=v_p. In this nondispersive lossless TEM model the energy-transport speed equals the phase velocity.

This equality must not be exported blindly to dispersive or strongly lossy structures, where electromagnetic energy density itself requires the appropriate dispersive material expression and energy/group velocities need separate care.

For a real positive reference impedance Z₀ and AoN PEAK phasors, define normalized forward/backward amplitudes a_p=V⁺/√(2Z₀) and b_p=V⁻/√(2Z₀). They have units √W, and the lossless real power becomes P=|a_p|²−|b_p|².

This normalization is useful because squared amplitude directly represents directional power. It is a bridge to later scattering-parameter work, but RF-00 stops here: reflection coefficients, standing-wave ratio and Smith-chart transformations belong to RF-01.

Do not extend this simple normalization unchanged to an arbitrary COMPLEX reference impedance. Power-wave definitions for complex references require an explicit convention; Kurokawa's power-wave formalism is one important example and will be treated only when scattering variables are introduced.

Numerical audit: on a 50 Ω lossless line let V⁺=2.00∠0° V peak and V⁻=0.600∠70° V peak. Then P⁺=40.0 mW, P⁻=−3.60 mW and P=36.4 mW. Directly evaluating (1/2)v̲i̲* at z=0 gives S≈36.4 mW+j22.55 mvar: the relative phase changes Q but not the lossless net real power.

1. Starting from v(t)=V_m cos(ωt+φ_v) and i(t)=I_m cos(ωt+φ_i), derive P=(1/2)V_m I_m cos(φ_v−φ_i).
2. A pure forward wave has peak voltage 1.50 V on a lossless 75 Ω line. Compute its average power.
3. A pure backward wave has local peak voltage magnitude 1.20 V on the same line. What is its signed +z average power?
4. For a lossless real-Z₀ line, prove from the two-wave solution that relative phase cannot change the NET average real power.
5. For Z₀=50 Ω, V⁺=2∠0° V peak and V⁻=0.6∠70° V peak at z=0, compute P and Q from the total phasors.
6. Differentiate S=(1/2)v̲i̲* and use the telegrapher equations to derive dP/dz.
7. For a weakly lossy real-Z₀ approximation, show that the pure-forward distributed loss density divided by P is R′/Z₀+G′Z₀=2α.
8. Carry out the coaxial Poynting-flux integral using E_r=V/[r ln(b/a)] and H_φ=I/(2πr) and recover (1/2)Re(VI*).
9. For a pure lossless wave, derive P/⟨w′⟩=v_p from Z₀=√(L′/C′).
10. On a 50 Ω real reference, convert V⁺=1.0 V peak and V⁻=0.25 V peak to a_p,b_p and compute net power. Explain why the same square-root normalization is not automatically valid for complex reference impedance.

RF-00 is now complete: the distributed RLGC model, telegrapher equations, characteristic impedance, propagation/loss and power flow form one continuous Maxwell-to-line chain. RF-01 will add boundary mismatch, reflection coefficient, standing waves, input impedance, quarter-wave transformations and the Smith chart without changing the reference conventions established here.