Q: How does a quarter-wave transformer (λ/4 transformer) achieve impedance matching?

The input impedance of a λ/4 line is Z_in = Z₀²/Z_L (for a real load). Therefore, inserting a λ/4 line with characteristic impedance Z₀' = √(Z_S × Z_L) between two impedances Z_S and Z_L achieves matching. For example, to match 50Ω and 100Ω, use a λ/4 line with Z₀' = √(50×100) = 70.7Ω. This method provides matching at a single frequency; multi-section designs are needed for broadband matching.

Q: How does line loss (attenuation) affect impedance?

For a lossy transmission line, the input reflection coefficient becomes Γ_in = Γ_L × e^(−2αd) × e^(−j2βd) (α: attenuation constant [Np/m], d: line length), so |Γ_in| decreases as line length increases. This appears as improved VSWR at the input, but the signal is also attenuated equally. For example, with 3 dB/m loss over 1 m, |Γ_in| is approximately halved. In highly lossy systems, insertion loss management may take priority over impedance matching in the design.

Question 1

What is the reflection coefficient Γ and how is it calculated?

Accepted Answer

The reflection coefficient Γ (gamma) is a complex number representing the fraction of voltage reflected at the termination of a transmission line. It is calculated as Γ = (Z_L − Z₀)/(Z_L + Z₀). When Z_L = Z₀ (perfect match), Γ = 0 (no reflection); Z_L = ∞ (open circuit), Γ = +1; Z_L = 0 (short circuit), Γ = −1. |Γ| is related to return loss RL = −20log₁₀|Γ| [dB], and VSWR = (1+|Γ|)/(1−|Γ|).

Question 2

How do I read a Smith chart?

Accepted Answer

A Smith chart overlays constant-resistance circles and constant-reactance arcs on the complex reflection coefficient plane (|Γ|≤1). The center point (Γ=0) represents the characteristic impedance Z₀ (perfect match). Points to the right indicate higher resistance; the upper semicircle is inductive (XL>0); the lower is capacitive (XC<0). Adding λ/4 of line length rotates the point 90° counter-clockwise (half revolution), and λ/2 returns to the original point. Impedance matching designs use stubs or λ/4 transformers to move to the center (Γ=0).

Question 3

How does a quarter-wave transformer (λ/4 transformer) achieve impedance matching?

Accepted Answer

The input impedance of a λ/4 line is Z_in = Z₀²/Z_L (for a real load). Therefore, inserting a λ/4 line with characteristic impedance Z₀' = √(Z_S × Z_L) between two impedances Z_S and Z_L achieves matching. For example, to match 50Ω and 100Ω, use a λ/4 line with Z₀' = √(50×100) = 70.7Ω. This method provides matching at a single frequency; multi-section designs are needed for broadband matching.

Question 4

How does line loss (attenuation) affect impedance?

Accepted Answer

For a lossy transmission line, the input reflection coefficient becomes Γ_in = Γ_L × e^(−2αd) × e^(−j2βd) (α: attenuation constant [Np/m], d: line length), so |Γ_in| decreases as line length increases. This appears as improved VSWR at the input, but the signal is also attenuated equally. For example, with 3 dB/m loss over 1 m, |Γ_in| is approximately halved. In highly lossy systems, insertion loss management may take priority over impedance matching in the design.

Transmission Line · Reflection Coefficient · Impedance Matching

Voltage Standing Wave Pattern

|Z_in| vs Line Length

Formulas