Quarter-wave transformations

Consider a lossless transformer section with characteristic impedance Z_t, physical length ℓ and electrical length θ=βℓ, terminated by Z_L. The chapter-03 input-impedance law applies directly, with Z₀ replaced by Z_t because the middle section is a different uniform line.

At θ=π/2, tan θ diverges. For a finite nonzero load, the dominant numerator and denominator terms leave the exact quarter-wave inversion Z_in=Z_t²/Z_L.

This is an impedance TRANSFORMATION. It becomes a MATCH only after we specify what impedance the upstream line or source wants to see.

Let the upstream line/reference impedance be a positive real R_S and let the load be a positive real R_L. Exact matching at the design plane means Z_in(f₀)=R_S.

At the quarter-wave frequency, Z_in=Z_t²/R_L. Therefore Z_t²=R_SR_L and the positive physical characteristic impedance is Z_t=√(R_SR_L).

The geometric mean lies between the two positive resistances. If R_L>R_S, then R_S<Z_t<R_L; if R_L<R_S, the inequalities reverse. The section does not dissipate the mismatch in the ideal model: it redistributes voltage/current through wave interference so the upstream port sees the desired ratio.

The design condition is β(f₀)ℓ=π/2. Therefore ℓ=π/[2β(f₀)]=λ_g(f₀)/4, where λ_g is the guided wavelength of the transformer section.

For an ideal nondispersive TEM line with phase velocity v_p, ℓ=v_p/(4f₀). In a homogeneous dielectric v_p=1/√(με). For a quasi-TEM microstrip, an effective permittivity may be used only within the accuracy of the chosen model.

If β is dispersive, do not replace the design condition by c/[4f₀√ε_eff] without checking the model. The exact condition is always the phase condition β(f₀)ℓ=π/2.

Keep the ideal transformer designed with Z_t²=R_SR_L, but let θ move away from π/2. Evaluate the input impedance with the general tangent formula and then compare it with R_S through Γ_in=(Z_in−R_S)/(Z_in+R_S).

The geometric-mean choice causes the imaginary term in the numerator of Γ_in to cancel exactly. A convenient finite expression is Γ_in=(R_L−R_S)cosθ / [(R_L+R_S)cosθ+j2√(R_SR_L)sinθ].

At θ=π/2 the numerator vanishes. Away from that point the mismatch grows continuously. A single quarter-wave section is therefore intrinsically narrowband unless the impedance ratio is already close to one.

Assume a nondispersive transformer so θ(f)=(π/2)(f/f₀). Let the permitted reflection magnitude be ρ_m<1, equivalently a minimum return loss RL_min=−20log₁₀ρ_m.

Solving |Γ_in|≤ρ_m gives |cosθ|≤C, where C=2√(R_SR_L)ρ_m/[|R_L−R_S|√(1−ρ_m²)]. If C≥1, this particular reflection requirement is satisfied for all θ in the first 0-to-π interval; otherwise the matched band around π/2 is finite.

For 0<C<1 the ideal fractional bandwidth is Δf/f₀=(4/π)asin(C). This closed form depends on the impedance ratio and the chosen reflection tolerance, not on f₀ itself under the nondispersive assumptions.

At θ=π/2 the ideal inversion is Z_in=Z_t²/Z_L. If Z_L has reactance and Z_t is real, Z_in generally remains complex. One positive real degree of freedom Z_t cannot force both the real and imaginary parts of Z_in to equal a prescribed real R_S.

A single ideal quarter-wave section therefore gives an exact real-to-real match, not a universal complex-load matcher. A reactive load can first be moved along a line, resonated, or combined with stubs/lumped/reactive sections before a quarter-wave real-to-real transformation is applied.

This distinction matters later on the Smith chart: moving along a line rotates Γ; adding matching elements changes the load itself. The present chapter keeps those operations conceptually separate.

For a lossy transformer section use the exact chapter-03 formula with γ=α+jβ. At the nominal phase condition βℓ=π/2, tanh(αℓ+jπ/2)=coth(αℓ), so the ideal infinite tangent is replaced by a finite hyperbolic factor.

The exact input impedance becomes Z_in=Z_t[Z_L+Z_t coth(αℓ)]/[Z_t+Z_L coth(αℓ)]. The simple geometric-mean choice no longer produces an exact match in general.

Conductor and dielectric loss also reduce delivered power. In practical design the section impedance and length may therefore be optimized with the frequency-dependent complex γ(f) and Z_t(f), rather than by applying the lossless square-root formula unchanged.

Design an ideal nondispersive transformer between R_S=50 Ω and R_L=100 Ω at f₀=2.00 GHz. The required characteristic impedance is Z_t=√5000≈70.71 Ω.

If the section phase velocity is 2.00×10⁸ m/s, λ_g=0.100 m and ℓ=25.0 mm. At f₀, Z_in=50 Ω and Γ_in=0.

At 1.80 GHz, θ=0.45π=81°. The exact detuned expression gives |Γ_in|≈0.0552, corresponding to return loss ≈25.2 dB. At 1.50 GHz, |Γ_in|≈0.134 and return loss ≈17.5 dB.

For a 20 dB return-loss requirement, ρ_m=0.1. Here C≈0.284, giving an ideal fractional bandwidth ≈36.7%, approximately 1.63–2.37 GHz. Real dispersion and loss break the exact symmetry around f₀.

1. Starting from the lossless input-impedance formula, derive Z_in=Z_t²/Z_L at θ=π/2.
2. Derive Z_t=√(R_SR_L) for an exact match between two positive real impedances.
3. Match 50 Ω to 200 Ω. Find Z_t and state whether it lies between the endpoint impedances.
4. A transformer has v_p=1.8×10⁸ m/s and f₀=3 GHz. Find its quarter-wave physical length.
5. For a 50 Ω to 100 Ω transformer, derive the finite detuned Γ_in(θ) expression after imposing Z_t²=R_SR_L.
6. For the same 50-to-100 Ω transformer, compute |Γ_in| at θ=π/3 and the corresponding return loss.
7. Given a 20 dB minimum return loss for the 50-to-100 Ω case, calculate the ideal nondispersive fractional bandwidth.
8. Explain why Z_L=50+j30 Ω cannot generally be matched to 50 Ω by choosing only a real Z_t at exactly λ_g/4.
9. For a lossy λ_g/4 section with αℓ=0.10, evaluate coth(αℓ) approximately and explain qualitatively why the ideal inversion is weakened.
10. A 50 Ω line must match a 75 Ω load at 1 GHz. Find Z_t. Then state which additional information is still required before a physical width and length can be fabricated.

Chapter 05 will build the Smith chart from the bilinear map Γ=(z−1)/(z+1): constant-resistance and constant-reactance circles, movement along a line, impedance/admittance views and matching trajectories. The chart will be derived as geometry of equations already established, not taught as a graphical shortcut.