Transmission Lines Overview

Examples of Transmission Lines: controlled geometry lines
Circuit component size compared to operating wavelength, λ
TL Analysis is needed if size is not too small compared to wavelength

Circuit Theory Analysis of Controlled Geometry Lines

RLCG Physically based Model (linear) - Distributed parameters
Use KVL and KCL to develop voltage-current relationships
Analysis in both Time and Frequency Domains
Second order differential equation for voltage as function of position and time

Solution in Frequency Domain

$$V\left(z\right)=V^+e^{-\gamma{}z}+V^-e^{+\gamma{}z}$$ $I\left( z \right)=~\left( 1/{{Z}_{o}} \right)\cdot \left\{ {{V}^{+}}{{e}^{-\gamma z}}-{{V}^{-}}{{e}^{+\gamma z}} \right\}$ $$Z_o=\sqrt{\left\{R_z+j\omega{}L_z\right\}/\left\{G_z+j\omega{}C_z\right\}}$$ $\gamma =\alpha +j\beta =\sqrt{\left\{ {{R}_{z}}+j\omega {{L}_{z}} \right\}\cdot \left\{ {{G}_{z}}+j\omega {{C}_{z}} \right\}}$

Harmonic solution in Time Domain

Using Laplace Transform
$$v\left(z,t\right)=\left\vert{}V^+\right\vert{}e^{-\alpha{}z}\cos{\left(\omega{}t-\beta{}z+{\varphi{}}_v^+\right)} +\left\vert{}V^-\right\vert{}e^{+\alpha{}z}\cos{\left(\omega{}t+\beta{}z+{\varphi{}}_v^-\right)}$$ $$i\left(z,t\right)=\left\vert{}V^+/Z_o\right\vert{}e^{-\alpha{}z}\cos{\left(\omega{}t-\beta{}z+{\varphi{}}_i^+\right)} -\left\vert{}V^-/Z_o\right\vert{}e^{+\alpha{}z}cos⁡(\omega{}t+\beta{}z+{\varphi{}}_i^-)$$

Transient solution in Time Domain

Only for Lossless Line $R_{z}$=$G_{z}$=0
$$v\left(z,t\right)=v^+⁡(t-[\sqrt{L_z \cdot C_z}] \cdot z)+v^-⁡(t+[\sqrt{L_z \cdot C_z}] \cdot z))$$ $$i\left(z,t\right)=\left\{{1/\sqrt{L_z \cdot C_z}}\right\}\left\{v^+{\left(t-[\sqrt{L_z \cdot C_z}] \cdot z\right)}-v^-⁡(t+[\sqrt{L_z \cdot C_z}] \cdot z)\right\}$$

Physical Insight

Two traveling waves: ''$+z$'' wave and ''$-z$'' wave both have same $Z_{o}$ and $\gamma{}$

$Z_{o}$: Characteristic impedance, $Ohms$ -- $\gamma{}$=$\alpha{}$+j$\beta{}$: Complex propagation constant, m$^{-1 }$

$\alpha{}$: Attenuation coefficient, Nepers/m -- $\beta{}$: Phase constant, rad/m

Phase velocity $c_{ph}$=$\omega{}$/$\beta{}$, m/s -- Wavelength λ$=c_{ph}/f$=2$\pi{}$/$\beta{}$, m

$V^+$ and $V^-$ are determined by boundary conditions at TL source and load ends

Reflection Coefficient

Define Reflection coefficient $\Gamma{}(z)$=$\ \left(V^-e^{+\gamma{}z}\right)/\left(V^+e^{-\gamma{}z}\right)$ =''$-z$ wave'' /''$+z$ wave''

$\Gamma \left( z \right)=\Gamma \left( \ell \right)\cdot {{e}^{+2\gamma \left( z-\ell \right)}}$ -- $\Gamma{}\left(l\right)={\Gamma{}}_L=\left[Z_L-Z_o\right]/\left[Z_L+Z_o\right]$

$\Gamma{}\left(z\right)=\left[Z\left(z\right)-Z_o\right]/\left[Z\left(z\right)+Z_o\right]$ -- $Z\left(z\right)=Z_o\left[1+\Gamma{}\left(z\right)\right]/\left[1-\Gamma{}\left(z\right)\right]$

Reflections are caused by impedance discontinuities "mismatch -- not equal to $Zo$"

Traveling and Standing Waves

$Z_{L}$=$Z_{o}$, $\Gamma{}$$_{L}$=0$\rightarrow{}$$\Gamma{}(z)$=0$\rightarrow{}Z(z)$= $Z_{o}$: $V^{-}$=0, only "$+z$ wave", pure traveling wave, $\vert{}V$ $(z)\vert{}$= $\vert{}V$ $^{+}$ $\vert{}e$ $^{-\alpha{}z}$

Magnitude constant except for decay due to attenuation (loss)

$Z_{L}=0+jX_{L}$, $\vert{}$$\Gamma{}$$_{L}$$\vert{}$=1$\rightarrow{}$$\vert{}$$\Gamma{}$(z)$\vert{}$=1$\rightarrow{}Z(z)$= 0+jX(z): $\vert{}$V$^{+}$$\vert{}$= $\vert{}$V$^{-}$$\vert{}$, pure standing waves, $\vert{}V(z)$$\vert{}$=2$\vert{}V$$^{+}$$\vert{}$cos($\beta{}z+$$\ \varphi{}$)

Wave does not travel, Magnitude has peaks (=2$\vert{}V$$^{+}$$\vert{}$) and valleys of zeroes (nodes)

$Z_{L}$$\not=Z$$_{o}$, $\Gamma{}$$_{L}$$\not=$0$\rightarrow{}$$\Gamma{}$($z$)$\not=$0$\rightarrow{}$$Z(z)$$\not=$ $Zo$: combination of "+$z$'' and reflected "-$z$'' waves
Mix of traveling {$\vert{}V$$^{+}$$\vert{}$-$\vert{}V$$^{-}$$\vert{}$}, and standing {2$\vert{}V$$^{-}$$\vert{}$} waves

Standing Wave Ratio, SWR = $V_{max}$/$V_{min}$=[1+$\vert{}$$\Gamma{}$$\vert{}$]/[1+$\vert{}$$\Gamma{}$$\vert{}$] -- $\vert{}$$\Gamma{}$$\vert{}$=0$\rightarrow{}$SWR=1, $\vert{}$$\Gamma{}$$\vert{}$=1$\rightarrow{}$SWR=$\infty{}$

Reflections: Power and Signal Delivery

Reflections (standing waves) reduce power delivered to load - also degrade signal quality
Mismatch causes dispersion to communication signals-Reflections limit bandwidth of digital communications
For optimum power & signal delivery, impedance matching at both source and load ends is necessary.