Lossless Transmission Lines [Part-I]

Lossless Transmission Line

A transmission line for which the resistance per unit length R’, and conductance G’ are negligible compared to ωL’ and ωC’, respectively; approximated by R’ and G’ are approximately zero.

γ = jβ = jω((L’C’)^1/2)

Lossless-Line Propagation Constant (The propagation constant of a lossless line)

α = 0

Attenuation Constant of a Lossless Line

β = ω((L’C’)^1/2)

Phase Constant of a Lossless Line

Z0 = (L’/C’)^1/2

Characteristic Impedance of a Lossless Line:
The real-valued characteristic impedance of a lossless transmission line

λ = (2π)/β

Guide Wavelength: The wavelength of a wave propagating on a transmission line

u_p = ω/β

Phase Velocity: The speed at which a constant phase point propagates along the line

β = ω((με)^1/2)

Phase Constant in Terms of Material Parameters: Expression relating phase constant to permeability and permittivity

u_p = 1/((με)^1/2)

Phase Velocity in Terms of Material Parameters

ε_r = ε/ε_0

Relative Permittivity: Ratio of a material’s permittivity to free-space permittivity

c = 1/(((μ0)(ε0))^1/2)

Speed of Light in Free Space — Electromagnetic wave velocity in vacuum

u_p = c/(ε_r^1/2)

Phase Velocity in Terms of Relative Permittivity (Phase velocity for common TEM lines)

λ = λ0/(ε_r^1/2)

Guide Wavelength in Terms of Relative Permittivity (Wavelength on a transmission line)

λ0 = c/f

Free-Space Wavelength
Wavelength in air or vacuum

Nondispersive Transmission Line

A transmission line in which all frequency components propagate with the same phase velocity

Incident Wave

The wave traveling from the generator toward the load, represented by the factor of e^-jβz

Reflected Wave

The wave traveling from the load toward the generator, represented by the factor of e^jβz

Lossless-Line Voltage Expression

Lossless-Line Current Expression

Z_L = (Ṽ_L)/(Ĩ_L)

Load Impedance

Load Voltage

• Total voltage evaluated at the load position z = 0

• Ṽ_L = ((V_0)^+) + ((V_0)^-)

Load Current

• Total current evaluated at the load position z = 0

• Ĩ_L = ((V_0)^+)/Z0 - ((V_0)^-)/Z0

Γ = ((V_0)^-)/((V_0)^+)

Voltage Reflection Coefficient: Ratio of reflected voltage amplitude to incident voltage amplitude

Γ = (Z_L - Z0)/(Z_L + Z0)

Reflection Coefficient in Terms of Impedances: Expression relating reflection coefficient to load and characteristic impedance

z_L = (Z_L)/Z0

Normalized Load Impedance: Load impedance normalized to the characteristic impedance

Γ = (z_L - 1)/(z_L )

Reflection Coefficient Using Normalized Impedance

-Γ = ((I_0)^-)/((I_0)^+)

Current Reflection Coefficient: Ratio of reflected current amplitude to incident current amplitude

Γ = |Γ|(e^j(θ_r))

Polar Form of Reflection Coefficient: Complex representation of the reflection coefficient

Matched Load

Condition in which the load impedance equals the characteristic impedance: Z_L = Z0 resulting in Γ = 0

For Z_L = ∞: Γ = 1

Open-Circuit Load Reflection Coefficient

For Z_L = 0: Γ = -1

Short-Circuit Load Reflection Coefficient

Standing Wave Voltage Expression

Voltage on a line containing incident and reflected waves

Standing Wave Current Expression

Current on a line containing incident and reflected waves

Magnitude of Standing-Wave Voltage

e^jx + e^-jx = 2(cos(x))

Euler Cosine Identity