Transmission Line Impedance Characteristics

Characteristic Impedance (Z₀)

The ratio of voltage to current for a wave propagating in one direction on an infinitely long transmission line. It is determined by the line's physical construction and materials.

• Formula: Z₀ = √(L/C), where L is inductance per unit length and C is capacitance per unit length.
• Typical Values: Common values include 50 Ω, 75 Ω, and 300 Ω, depending on the application.
• Importance: Critical for impedance matching to prevent reflections and maximize power transfer.

Voltage Standing Wave Ratio (VSWR)

A measure of the impedance mismatch between the transmission line and the load. It represents the ratio of the maximum voltage to the minimum voltage along the line.

• Definition: VSWR = V_max / V_min
• Relationship to Reflection Coefficient (Γ): VSWR = (1 + |Γ|) / (1 - |Γ|)
• Ideal Value: A VSWR of 1:1 indicates a perfect match (no reflections). Higher VSWR values indicate greater mismatch.

Reflection Coefficient (Γ)

The ratio of the reflected voltage wave to the incident voltage wave. It indicates the magnitude and phase of the signal reflected from the load.

• Formula: Γ = (Z_L - Z₀) / (Z_L + Z₀), where Z_L is the load impedance and Z₀ is the characteristic impedance.
• Values: Ranges from -1 to +1 for real impedances; complex values are possible for complex impedances.
• Interpretation: Γ = 0 indicates a perfect match. Γ = 1 indicates a complete reflection with the same phase. Γ = -1 indicates a complete reflection with a 180-degree phase shift.

Load Impedance (Z_L)

The impedance of the device connected at the end of the transmission line.

• Impact on Matching: A load impedance equal to the characteristic impedance (Z_L = Z₀) provides a matched condition, minimizing reflections and maximizing power transfer.
• Mismatch Consequences: A mismatched load impedance (Z_L ≠ Z₀) causes reflections, leading to standing waves, reduced power transfer, and potential damage to the source.

Transmission Line Length (l) and Electrical Length

The physical length of the transmission line and its equivalent electrical length are crucial factors.

• Electrical Length: Measured in wavelengths (λ) or radians (βl), where β is the phase constant (2π/λ).
• Impact on Impedance: The observed impedance varies periodically with the electrical length. Specific lengths can transform the load impedance.
• Quarter-Wave Transformer: A λ/4 transmission line transforms a load impedance Z_L to an impedance Z₀²/Z_L.

Formulas for Impedance Transformation

Equations describing how the load impedance is transformed by the transmission line.

• General Formula: Z_in = Z₀ (Z_L + j Z₀ tan(βl)) / (Z₀ + j Z_L tan(βl)), where Z_in is the impedance, Z₀ is the characteristic impedance, Z_L is the load impedance, β is the phase constant, and l is the length of the line.