Transmission Lines
A transmission line guides EM waves between two parallel conductors.
It is used to transmit microwave signals from transmitter to antenna.
Two common types: two-wire line and coaxial cable.
Transmission line parameters (per unit length)Quantity | Two-wire (sep. \(d\), radius \(a\)) | Coaxial (inner \(a\), outer \(b\)) |
Inductance \(L\) | \(\dfrac{\muz}{\pi}\ln\!\dfrac{d}{a}\) | \(\dfrac{\muz}{2\pi}\ln\!\dfrac{b}{a}\) |
[8pt]
Capacitance \(C\) | \(\dfrac{\pi\eps}{\ln(d/a)}\) | \(\dfrac{2\pi\eps}{\ln(b/a)}\) |
[8pt]
\(LC\) | \(\muz\eps\) | \(\muz\eps\) |
Key transmission line quantities\begin{align}
v &= \frac{1}{\sqrt{LC}} = \frac{c}{\sqrt{\varepsilon_r\mu_r}}
\quad\text{(signal/phase velocity)} \\[6pt]
Z_0 &= \sqrt{\frac{L}{C}}
\quad\text{(characteristic impedance)} \\[6pt]
\Gamma &= \frac{Z_L-Z_0}{Z_L+Z_0}
\quad\text{(voltage reflection coefficient)} \\[6pt]
\text{VSWR} &= \frac{1+|\Gamma|}{1-|\Gamma|}
\quad\text{(voltage standing wave ratio)}
\end{align}
Limiting cases[noitemsep]
Lossless (\(R=0\), \(G=0\)):
\(\gamma_\text{prop}=i\omega\sqrt{LC}\),
\(v=1/\sqrt{LC}\), \(Z_0=\sqrt{L/C}\)
Distortionless (\(R/L=G/C\)):
no dispersion — all frequencies travel at same speed
Low frequency (\(\omega\to0\)):
\(\gamma\approx\sqrt{RG}\), \(Z_0\approx\sqrt{R/G}\)
High frequency (\(\omega\to\infty\)):
\(\gamma\approx i\omega\sqrt{LC}\), \(Z_0\approx\sqrt{L/C}\)
Matched load (\(Z_L=Z_0\)): \(\Gamma=0\), no reflection
Example 5.1 Finding \(L\) and \(C\) from line parametersA lossless microwave transmission line has phase velocity \(v=2\times10^7\) m/s
and characteristic impedance \(Z_0=70\,\Omega\). Find \(L\) and \(C\) per unit length.
\(v=1/\sqrt{LC}\) and \(Z_0=\sqrt{L/C}\).
Solving: \(L=Z_0/v = 70/(2\times10^7) = 3.5\,\mu\)H/m;
\(C=1/(Z_0 v)=1/(70\times2\times10^7)=714\) pF/m.
\[\boxed{L=3.5\,\mu\text{H/m},\quad C=714\,\text{pF/m}}\]
Example 5.2 Reflection coefficient and VSWRTransmission line \(Z_0=50\,\Omega\) terminated with \(Z_L=100\,\Omega\).
