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:
  • $\tilde{V}(z) = V<em>0^+ e^{-\gamma z} + V</em>0^- e^{\gamma z}$
  • $\tilde{I}(z) = I<em>0^+ e^{-\gamma z} + I</em>0^- e^{\gamma z}$
• The characteristic impedance of the line is given by:
  • $Z<em>0 = \frac{V</em>0^+}{I_0^+} = \sqrt{\frac{R' + j\omega L'}{G' + j\omega C'}}$

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) $\mu<em>c$ and $\\sigma</em>c$ 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: $\gamma = \sqrt{(R' + j\omega L')(G' + j\omega C')}$; $u<em>p = \frac{\omega}{\beta}$; $Z</em>0=\sqrt{\frac{(R'+j\omega L')}{(G'+j\omega C')}}$.
• Lossless: $\alpha = 0, \beta = \omega \sqrt{\varepsilon<em>r}/ c$; $u</em>p = c/\sqrt{\varepsilon<em>r}$; $Z</em>0=\sqrt{L'/C'}$.
• Lossless coaxial: $\alpha = 0, \beta = \omega \sqrt{\varepsilon<em>r}/ c$; $u</em>p = c/\sqrt{\varepsilon<em>r}$; $Z</em>0 = (60/\sqrt{\varepsilon_r}) ln(b/a)$.
• Lossless two-wire: $\alpha = 0, \beta = \omega \sqrt{\varepsilon<em>r}/ c$; $u</em>p = c/\sqrt{\varepsilon<em>r}$; $Z</em>0 = (120/\sqrt{\varepsilon_r}) ln[(D/d) + \sqrt{(D/d)^2 -1}]$.
• Lossless parallel-plate: $u<em>p = c/\sqrt{\varepsilon</em>r}$; $Z<em>0 = (120\pi/\sqrt{\varepsilon</em>r}) (h/w)$.

Lossless Microstrip Line

• Phase velocity in dielectric: $u<em>p = \frac{c}{\sqrt{\varepsilon</em>{eff}}}$.
• $Z<em>0 = \frac{60}{\sqrt{\varepsilon</em>{eff}}}ln\left[\frac{6 + (2\pi - 6)e^{-t}}{s} + \sqrt{1 + \left(\frac{4}{s}\right)^2}\right]$

Microstrip

• The characteristic impedance of the microstrip line is given by
  • $Z<em>0 = \frac{60}{\sqrt{\varepsilon</em>{eff}}}ln\left[\frac{6 + (2\pi - 6)e^{-t}}{s} + \sqrt{1 + \left(\frac{4}{s}\right)^2}\right]$
  • 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:
  • $\varepsilon<em>{eff} = \frac{\varepsilon</em>r + 1}{2} + \frac{\varepsilon_r - 1}{2} \left(1 + \frac{10}{s}\right)^{-0.555} = 1.7686$
  • $Z<em>0 = \frac{60}{\sqrt{\varepsilon</em>{eff}}}ln\left[\frac{6 + (2\pi - 6)e^{-t}}{s} + \sqrt{1 + \left(\frac{4}{s}\right)^2}\right] = 148.5372$
  • $\alpha = 0$ Np/m