Characteristic impedance

For the +z contribution, chapter 02 gave i⁺=(γ/Z′)v⁺. Therefore the forward traveling-wave voltage/current ratio is Z′/γ. Because γ²=Z′Y′, the same quantity is γ/Y′ and also √(Z′/Y′).

AoN defines this modal ratio as the characteristic impedance Z₀. Its unit is ohms because Z′/Y′ has units Ω². This definition belongs to a UNIFORM line mode; it does not require a load to be attached.

For the reverse wave, the total line current is still referenced toward +z, so i⁻=−v⁻/Z₀. Consequently V⁻/I⁻=−Z₀ under the fixed current reference. Dropping this sign corrupts later power and reflection formulas.

The algebraic square root has two signs. Do not choose the sign of Z₀ independently from γ. AoN first chooses the forward passive propagation branch γ with Re(γ)=α≥0, then defines Z₀=Z′/γ.

For an ordinary passive line with L′>0, C′>0 and nonnegative R′,G′, this choice gives the physically useful branch with Re(Z₀)>0. Flipping both γ and Z₀ merely swaps which exponential is called forward; flipping only one breaks the first-order telegrapher equations.

This linked branch choice is more reliable than memorizing 'take the positive square root', especially when Z₀ is complex.

Set R′=G′=0. Then Z′=jωL′ and Y′=jωC′, so their ratio is L′/C′. The characteristic impedance is therefore real and positive: Z₀=√(L′/C′).

Combine this with v_p=1/√(L′C′). The pair {Z₀,v_p} can be inverted to recover the storage parameters: L′=Z₀/v_p and C′=1/(Z₀v_p).

In this ideal nondispersive line model neither Z₀ nor v_p depends on frequency. Real interconnects can depart from that through conductor loss, dielectric dispersion, frequency-dependent field confinement and higher modes.

Characteristic impedance is not a material constant. The per-unit-length L′ and C′ depend on the transverse field geometry as well as ε and μ. A clean exact transverse electromagnetic (TEM) example is a homogeneous coaxial line with inner radius a and outer-conductor inner radius b.

For that geometry, L′=(μ/2π)ln(b/a) and C′=2πε/ln(b/a). Their ratio gives Z₀=(1/2π)√(μ/ε)ln(b/a)=η ln(b/a)/(2π), where η=√(μ/ε) is the intrinsic wave impedance of the filling medium.

The formula displays the distinction clearly: η is a FIELD ratio of the medium, while Z₀ multiplies it by a geometry factor. Different line geometries in the same dielectric therefore have different characteristic impedances.

With loss, Z₀ is generally complex: Z₀=√[(R′+jωL′)/(G′+jωC′)]. A complex voltage/current ratio means the local traveling-wave voltage and current phasors are not exactly in phase.

If R′≪ωL′ and G′≪ωC′, expand the square root to first order. Under AoN's exp(+jωt) convention, the correction is −(j/2)[R′/(ωL′)−G′/(ωC′)] multiplying the lossless value.

This approximation is useful only when both dimensionless loss ratios are small. Near DC, or on a strongly resistive/leaky line, use the exact expression.

For peak phasors, the time-averaged line power crossing a transverse section in the +z bookkeeping direction is P=(1/2)Re{v i*}. Apply this to a single forward wave i⁺=v⁺/Z₀.

The forward power is (1/2)|V⁺|²Re(1/Z₀), positive for a passive line branch. For a reverse wave the fixed +z current reference contributes the minus sign, so the +z-directed power is negative: P⁻=−(1/2)|V⁻|²Re(1/Z₀).

On a lossless line Z₀ is real, reducing these to ±|V|²/(2Z₀). The reverse-current sign from chapter 02 is therefore exactly what makes a reverse wave transport energy toward −z.

Z₀ is a property of a uniform line mode. A load impedance Z_L is a boundary condition connected at a termination. The input impedance Z_in is the voltage/current ratio seen at a chosen position after forward and backward waves have combined; it therefore depends on Z_L, line length and γ.

Intrinsic wave impedance η=E/H belongs to an electromagnetic mode in a medium. In a TEM line, Z₀ is derived by integrating fields into a line voltage and current, so geometry enters. The coax example made this distinction explicit.

A line can have Z₀=50 Ω while being terminated in 75 Ω, shorted, open-circuited or connected to a complex device. Z₀ does not become the load; the mismatch determines reflected-wave amplitude, which belongs to RF-01.

For L′=250 nH/m and C′=100 pF/m, Z₀=√(L′/C′)=50.0 Ω and v_p=2.00×10^8 m/s. These are the same storage values used in chapter 02.

Now add R′=10 Ω/m and G′=2.0 mS/m at 100 MHz. The exact expressions give Z₀≈50.0316−j0.7945 Ω and γ≈0.14998+j3.14199 m⁻¹. The line remains close to 50 Ω but voltage and current acquire a small relative phase.

If the forward peak voltage amplitude is 2.00 V, the average forward power is about 39.96 mW using (1/2)|V⁺|²Re(1/Z₀). Using 50 Ω as if the lossy impedance were exactly real would be a small approximation here, not an identity.

1. Starting from d v̲/dz=−Z′i̲ and a pure forward exponential, derive Z₀=Z′/γ.
2. Use γ²=Z′Y′ to prove Z′/γ=γ/Y′=√(Z′/Y′), including a unit check.
3. Explain why the sign of Z₀ cannot be chosen independently from the branch of γ.
4. For a lossless line with L′=360 nH/m and C′=90 pF/m, compute Z₀ and v_p.
5. Invert Z₀=√(L′/C′) and v_p=1/√(L′C′) to derive L′ and C′ in terms of Z₀ and v_p.
6. A homogeneous nonmagnetic coax has ε_r=2.25 and b/a=3. Calculate its ideal TEM Z₀ using 60 ln(b/a)/√ε_r.
7. Derive the first-order low-loss approximation for Z₀ by expanding the exact ratio.
8. For a pure backward wave, use i⁻=−v⁻/Z₀ to show that the +z-directed average power is negative on a passive line.
9. Explain the difference among Z₀, Z_L, Z_in and η without using the word 'impedance' alone as the definition.
10. Using R′=10 Ω/m, L′=250 nH/m, G′=2 mS/m, C′=100 pF/m at 100 MHz, compare the loss ratios R′/(ωL′) and G′/(ωC′) and judge whether the low-loss approximation is justified.

Chapter 04 will return to γ and study propagation loss and distortion in detail: low-loss α and β, conductor versus dielectric contributions, the Heaviside distortionless condition, group/phase delay and the limits of a frequency-independent RLGC model.