Input impedance along a transmission line

At distance d from the load toward the generator, define the input impedance by the total voltage/current ratio at that plane. This is not the impedance of a new physical element; it is the impedance presented by the terminated line to anything connected at that reference plane.

Using the same local reflected-to-incident voltage ratio Γ(d), factor the forward-wave contribution from both voltage and current. Their ratio immediately gives Z_in/Z₀=(1+Γ)/(1−Γ).

The inverse relation Γ=(Z_in−Z₀)/(Z_in+Z₀) is algebraically identical to the load formula because a reference plane can itself be regarded as a new 'port' looking into the remaining line plus load.

Chapter 01 established Γ(d)=Γ_L exp(−2γd), with d measured from the load toward the generator. Substitute the load reflection coefficient Γ_L=(Z_L−Z₀)/(Z_L+Z₀) into the local impedance relation.

After multiplying numerator and denominator by exp(+γd) and collecting hyperbolic functions, the result is the exact uniform-line transformation shown below.

No low-loss approximation has been used. Z₀ and γ may both be complex, provided they belong to the same uniform line and physical branch conventions established in RF-00.

For a lossless line γ=jβ and Z₀ is real positive. The identity tanh(jx)=j tan x converts the exact hyperbolic expression into the familiar tangent form.

This formula is not a separate microwave rule; it is simply the same reflection translation written in impedance coordinates.

Because tan(βd) is real on the ideal lossless line, even a purely resistive load generally transforms into a complex input impedance except at special electrical lengths.

If Z_L=Z₀, the tangent-form numerator and denominator contain the same factor, so Z_in=Z₀ for every d. A matched uniform line therefore looks matched at every reference plane.

For a short circuit Z_L=0, the lossless expression becomes Z_in=jZ₀tan(βd). At d=0 it is zero, as required. For an open circuit, taking the Z_L→∞ limit gives Z_in=−jZ₀cot(βd), which diverges at the load plane.

The short and open expressions are purely reactive on an ideal lossless line: neither termination absorbs average power.

Since tan[β(d+λ_g/2)]=tan(βd+π)=tan(βd), the complete lossless input impedance repeats after every half guided wavelength.

At d=λ_g/4, tan(βd) tends to ±∞. For a finite nonzero load, the dominant terms give Z_in=Z₀²/Z_L. Thus normalized impedance is inverted at this special electrical length.

An ideal short therefore appears as an open a quarter wavelength away, and an ideal open appears as a short. Chapter 04 will turn this special point into a systematic quarter-wave matching method; here it is only a consequence and consistency check of the general transformation.

Define normalized impedance z_n=Z_in/Z₀. Then z_n=(1+Γ)/(1−Γ) and Γ=(z_n−1)/(z_n+1). These are fractional-linear (Möbius) transforms.

On a lossless line, moving the reference plane rotates Γ around a circle of constant |Γ| while z_n follows the corresponding impedance locus. This algebra is the foundation of the Smith chart, but no graphical construction is needed yet.

The key conceptual point is that changing reference plane does not change the physical load; it changes the local phase between forward and backward waves, and therefore the local voltage/current ratio.

When α>0, γ=α+jβ and tanh(γd) is not periodic in d. The exact half-wave repetition of the lossless line is therefore broken.

For a sufficiently long passive line with αd≫1, tanh(γd)→1. The input impedance tends to Z₀ independently of a finite load at the far end. Physically, the round-trip information about the termination is exponentially suppressed before returning to the observation plane.

The same conclusion follows from Γ(d)=Γ_Lexp(−2γd): its magnitude decays as exp(−2αd), so the local state approaches Γ=0 and hence Z_in=Z₀.

Take a lossless 50 Ω line terminated in 100 Ω. At d=λ_g/8, tan(βd)=tan(π/4)=1. Substitution gives Z_in=40−j30 Ω. The physical load is purely resistive, yet a plane one eighth wavelength away sees a capacitive input reactance.

If the same 50 Ω line is terminated in 25 Ω at d=λ_g/8, the result is 40+j30 Ω. At d=λ_g/4, the 100 Ω load becomes 25 Ω, illustrating the inversion Z₀²/Z_L without yet using it as a design recipe.

For a lossy example with Z₀=50 Ω, Γ_L=1/3, αd=0.20 and βd=π/4, the local reflection coefficient is −j(1/3)e^(−0.40)≈−j0.2234. Transforming it gives Z_in≈45.24−j21.28 Ω: attenuation pulls the local impedance trajectory toward Z₀.

1. Starting from the total two-wave voltage and current at a plane, derive Z_in/Z₀=(1+Γ)/(1−Γ).
2. Invert the previous relation to recover Γ=(z_n−1)/(z_n+1).
3. Starting from Γ(d)=Γ_Lexp(−2γd) and Γ_L=(Z_L−Z₀)/(Z_L+Z₀), derive the tanh form for Z_in(d).
4. Use γ=jβ to derive the lossless tangent form.
5. A 50 Ω lossless line is short-circuited. Find Z_in at d=λ_g/8 and d=λ_g/4.
6. A 50 Ω lossless line is open-circuited. Find Z_in at d=λ_g/8 and d=λ_g/4.
7. Compute Z_in for Z₀=50 Ω, Z_L=100 Ω and d=λ_g/8.
8. Prove that any lossless terminated line repeats its input impedance after λ_g/2.
9. For Z₀=50 Ω and Z_L=100 Ω, evaluate the quarter-wave input impedance and explain why systematic matching use is deferred to the next chapter.
10. Explain, using both tanh(γd) and Γ(d), why a sufficiently long passive lossy line appears terminated in Z₀.

Chapter 04 will isolate the quarter-wave case and turn it into a design tool: choose the section impedance, derive the geometric-mean transformer for real loads, quantify bandwidth and loss limits, and distinguish single-section matching from more general matching networks.