Transmission Lines and Wave Propagation

Traveling Waves

• Transient waves: caused by a sudden disturbance.
• Continuous periodic waves: generated by a repetitive source.
• Waves transport energy.
• Waves possess velocity.
• Many waves exhibit linearity, meaning they do not interfere with the propagation of other waves and can pass through each other.

Sinusoidal Waves in Lossless Media

• A medium is considered lossless if it does not diminish the amplitude of a wave traveling within it or on its surface.
• y(x, t) = A \cos \left( \frac{2\pi t}{T} - \frac{2\pi x}{\lambda} + \phi_0 \right), where:
  • A is the amplitude of the wave.
  • T is the time period.
  • \lambda is the spatial wavelength.
  • \phi_0 is a reference phase.
• Phase velocity is given by: u_p = \frac{\lambda}{T}
• Wave direction:
  • +\textit{x} direction: coefficients of t and x have opposite signs.
  • -\textit{x} direction: coefficients of t and x have the same sign (both positive or both negative).

Wave Frequency and Period

• The frequency of a sinusoidal wave, f, is the reciprocal of its time period T: f = \frac{1}{T} (Hz).
• The relationship between phase velocity, frequency, and wavelength is given by: u_p = f \lambda
• The general form of a sinusoidal wave can be expressed as: y(x, t) = A \cos(\omega t - \beta x), where:
  • \omega = 2\pi f is the angular frequency (rad/s).
  • \beta = \frac{2\pi}{\lambda} is the phase constant or wavenumber (rad/m).

Phase Lead and Lag

• y(x, t) = A \cos(\omega t - \beta x + \phi_0)
• When \phi_0 is positive, it indicates a phase lead.
• When \phi_0 is negative, it indicates a phase lag.

Wave Travel in Lossy Media

• In lossy media, the wave amplitude attenuates as it propagates, described by an attenuation factor.
• The wave equation in a lossy medium is given by: y(x,t) = Ae^{-\alpha x} \cos(\omega t - \beta x + \phi_0), where \alpha is the attenuation constant.

Circuit Theory vs. Transmission Line

• Circuit theory deals with lumped elements, while transmission line theory deals with distributed parameters.

Transmission Lines

• A transmission line connects a generator to a load.
• Examples of transmission lines:
  • Two parallel wires
  • Coaxial cable
  • Microstrip line
  • Optical fiber
  • Waveguide

Types of Transmission Modes

• TEM (Transverse Electromagnetic) mode: Electric and magnetic fields are orthogonal to each other, and both are orthogonal to the direction of propagation.

Transmission Line Model

• Transmission lines can be modeled using distributed circuit parameters:
  • R': The combined resistance of both conductors per unit length, in \Omega/m.
  • L': The combined inductance of both conductors per unit length, in H/m.
  • G': The conductance of the insulation medium between the two conductors per unit length, in S/m.
  • C': The capacitance of the two conductors per unit length, in F/m.

Transmission-Line Equations

• Differential form of the transmission-line equations:
  • \frac{\partial v(z, t)}{\partial z} = -R' i(z, t) - L' \frac{\partial i(z, t)}{\partial t}
  • \frac{\partial i(z, t)}{\partial z} = -G' v(z, t) - C' \frac{\partial v(z, t)}{\partial t}
• Phasor representation for AC signals:
  • v(z, t) = Re[\tilde{V}(z) e^{j\omega t}]
  • i(z, t) = Re[\tilde{I}(z) e^{j\omega t}]
• Second-order differential equation:
  • \frac{\partial i(z, t)}{\partial z} = G' C' \frac{\partial v(z, t)}{\partial t}

Derivation of Wave Equations

• Combining the transmission-line equations leads to second-order differential equations known as the Telegrapher's equations.
• These equations involve a complex propagation constant and describe attenuation and phase constants.

Solution of Wave Equations

• The general solutions for voltage and current waves on a transmission line are:
  • \tilde{V}(z) = V0^+ e^{-\gamma z} + V0^- e^{\gamma z}
  • \tilde{I}(z) = I0^+ e^{-\gamma z} + I0^- e^{\gamma z}
• The characteristic impedance of the line is given by:
  • Z0 = \frac{V0^+}{I_0^+} = \sqrt{\frac{R' + j\omega L'}{G' + j\omega C'}}

Solution of Wave Equations

• Wave along +z: coefficients of t and z have opposite signs.
• Wave along -z: coefficients of t and z have the same sign.

Lossless Transmission Line

• Conditions for a lossless line: R' << \omega L' and G' << \omega C'.
• Propagation constant: \gamma = \alpha + j\beta = j\omega \sqrt{L'C'}, which implies \alpha = 0
• Phase constant: \beta = \omega \sqrt{L'C'}.
• Phase velocity: u_p = \frac{1}{\sqrt{L'C'}}.
• Characteristic impedance: Z_0 = \sqrt{\frac{L'}{C'}}.
• Nondispersive line: If sinusoidal waves of different frequencies travel on a transmission line with the same phase velocity, the line is called nondispersive.

Dispersion

• Dispersionless line: Does not distort signals passing through it regardless of its length.
• Dispersive line: Distorts the shape of the input pulses because the different frequency components propagate at different velocities. The degree of distortion is proportional to the length of the dispersive line.

Example 2-1: Air Line

• An air line is a transmission line in which air separates the two conductors, which renders G' = 0 because \sigma = 0. In addition, assume that the conductors are made of a material with high conductivity so that R' \approx 0. For an air line with a characteristic impedance of 50 \Omega and a phase constant of 20 rad/m at 700 MHz, find the line inductance L' and the line capacitance C'.

Transmission-line parameters

• The pertinent constitutive parameters apply to all three lines and consist of two groups: (1) \muc and \\sigmac are the magnetic permeability and electrical conductivity of the conductors, and (2) \varepsilon, \mu, and \sigma are the electrical permittivity, magnetic permeability, and electrical conductivity of the insulation material separating them.

Characteristic parameters of transmission lines

• General case: \gamma = \sqrt{(R' + j\omega L')(G' + j\omega C')}; up = \frac{\omega}{\beta}; Z0=\sqrt{\frac{(R'+j\omega L')}{(G'+j\omega C')}}.
• Lossless: \alpha = 0, \beta = \omega \sqrt{\varepsilonr}/ c; up = c/\sqrt{\varepsilonr}; Z0=\sqrt{L'/C'}.
• Lossless coaxial: \alpha = 0, \beta = \omega \sqrt{\varepsilonr}/ c; up = c/\sqrt{\varepsilonr}; Z0 = (60/\sqrt{\varepsilon_r}) ln(b/a).
• Lossless two-wire: \alpha = 0, \beta = \omega \sqrt{\varepsilonr}/ c; up = c/\sqrt{\varepsilonr}; Z0 = (120/\sqrt{\varepsilon_r}) ln[(D/d) + \sqrt{(D/d)^2 -1}].
• Lossless parallel-plate: up = c/\sqrt{\varepsilonr}; Z0 = (120\pi/\sqrt{\varepsilonr}) (h/w).

Lossless Microstrip Line

• Phase velocity in dielectric: up = \frac{c}{\sqrt{\varepsilon{eff}}}.
• Z0 = \frac{60}{\sqrt{\varepsilon{eff}}}ln\left[\frac{6 + (2\pi - 6)e^{-t}}{s} + \sqrt{1 + \left(\frac{4}{s}\right)^2}\right]

Microstrip

• The characteristic impedance of the microstrip line is given by
  • Z0 = \frac{60}{\sqrt{\varepsilon{eff}}}ln\left[\frac{6 + (2\pi - 6)e^{-t}}{s} + \sqrt{1 + \left(\frac{4}{s}\right)^2}\right]
  • where
    • t = (30.67)^{\frac{0.75}{\varepsilon_r}}
• \beta = \omega \sqrt{\varepsilon_{eff}}

2016 Test 1 Q1

• A 200 mm long lossless microstrip transmission line uses a w=0.6 mm wide conducting strip on a substrate h=1.8 mm thick, with \varepsilon_r = 2.3 at an operating frequency of f = 10 GHz.
• Determine:
  • a) the characteristic impedance of the transmission line,
  • b) the attenuation constant.
• Solution:
  • \varepsilon{eff} = \frac{\varepsilonr + 1}{2} + \frac{\varepsilon_r - 1}{2} \left(1 + \frac{10}{s}\right)^{-0.555} = 1.7686
  • Z0 = \frac{60}{\sqrt{\varepsilon{eff}}}ln\left[\frac{6 + (2\pi - 6)e^{-t}}{s} + \sqrt{1 + \left(\frac{4}{s}\right)^2}\right] = 148.5372
  • \alpha = 0 Np/m