Lecture 1 Review

Transmission Lines

⇶ Transmission Line Theory

In an electronic system, power delivery requires connecting two wires between source and load.

⇆ General Considerations

A TL is a two-port network connecting generator to load.

Sending End
Generator
$V_g$, $R_g$
Receiving End
Load
$R_L$

Used to transmit electrical energy/signals from source to load. Types include wires, coaxial cables, optical fibers, microstrip, etc.

⚙ Types of Transmission Lines
  1. 1 Two-Wire LineTEM
  2. 2 Coaxial CableTEM
  3. 3 WaveguideHigher Order
    ↳ Rectangular ↳ Circular
  4. 4 Planar Transmission Lines
    ↳ Strip Line TEM ↳ Microstrip Line TEM ↳ Slot Line Higher Order ↳ Fin Line Higher Order ↳ Coplanar WG Higher Order ↳ Coplanar Slot Line Higher Order
≡ Low vs High Frequency
$$ \lambda = \frac{c}{f} $$$c$ = speed of light, $f$ = frequency (Hz)

Impact of TL on V/I depends on $l$ and $f$. At high freq the impact is very significant.

☰ Propagation Modes
Propagation modes | ┌──────────┴──────────┐ ▼ ▼ TEM (Transverse EM) Higher Order (TE/TM) ────────────────── ──────────────────── E & H entirely At least one field transverse to component in the propagation dir propagation dir
TEM field config (coaxial cross-section): ↑ ↖ ↑ ↗ ← — ◎ — → ↑ ~ ↓ E: radial (↑) ↗ ↓ ↖ H: concentric (~) ↓ inner (◎) ↔ outer

TEM: Electric field E radiates outward from the inner conductor, magnetic field H forms concentric loops around it. Both are perpendicular to the propagation direction $z$. No cut-off frequency → works from DC to GHz.

Higher Order (TE/TM): At least one field component points along $z$. TE = $E_z=0$, TM = $H_z=0$. Each mode has a cut-off frequency → only propagates above it. Dominant: TE$_{10}$ in rectangular WG.

≣ Lumped-Element Model

A TL is represented by a parallel-wire configuration regardless of its shape (coax, two-wire, any TEM line). The line is split into differential sections $\Delta z$, each modeled as:

┌──────────────────────────────────────────┐ $i(z,t)$ ───$R'\Delta z$──$L'\Delta z$─── $i(z+\Delta z,t)$ │ │ │ │ │ $G'\Delta z$ $C'\Delta z$ │ │ │ │ │ $v(z,t)$ ────────────────────────────────── $v(z+\Delta z,t)$ └──────────────────────────────────────────┘ $z$ ──────────────────────────────────> $z+\Delta z$
Series
$R'\Delta z$   $L'\Delta z$
resistance + inductance
Shunt
$G'\Delta z$   $C'\Delta z$
conductance + capacitance
ℙ Primary Constants (per unit length)
$R'$Ω/m · conductor resistance
$L'$H/m · conductor inductance
$G'$S/m · dielectric conductance
$C'$F/m · dielectric capacitance
Fundamental relation for TEM lines:
$$ L'C' = \mu\varepsilon \qquad \frac{G'}{C'} = \frac{\sigma}{\varepsilon} $$
How we got $L'C' = \mu\varepsilon$
For a TEM line, the fields are transverse. Magnetic energy: $W_m = \frac12 L'I^2 = \frac12 \mu\iint|H|^2 dS$
Electric energy: $W_e = \frac12 C'V^2 = \frac12 \varepsilon\iint|E|^2 dS$
For TEM: $E = \eta H$ with $\eta = \sqrt{\mu/\varepsilon}$. Using $V = \int E\cdot dl$, $I = \oint H\cdot dl$ gives $L'C' = \mu\varepsilon$.
Similarly: $G'/C' = \sigma/\varepsilon$.
√ Surface Resistance $R_s$

At high frequencies, current concentrates at the conductor surface (skin effect).

$$ R_s = \sqrt{\frac{\pi f \mu}{\sigma_c}} $$surface resistivity in $\Omega/\square$ (ohms per square)
How we got $R_s$
Skin depth: $\delta_s = \sqrt{2/(\omega\mu\sigma_c)} = 1/\sqrt{\pi f\mu\sigma_c}$
Resistance of a slab: $R = l/(\sigma_c w \delta_s)$
Surface resistance (per square): $R_s = R \cdot w/l = 1/(\sigma_c \delta_s) = \sqrt{\pi f\mu/\sigma_c}$
∫ Lumped Model for 3 Line Types

$\mu,\sigma,\varepsilon$ pertain to the insulating material between conductors.

ParamCoaxialTwo WireParallel PlateUnit
$R'$$\frac{R_s}{2\pi}(\frac1a+\frac1b)$$\frac{R_s}{\pi a}$$\frac{2R_s}{w}$Ω/m
$L'$$\frac{\mu}{2\pi}\ln\frac{b}{a}$$\frac{\mu}{\pi}\ln\!\left[D\!+\!\sqrt{D^2\!-\!1}\right]$
$D=d/2a$		$\frac{\mu d}{w}$H/m
$G'$$\frac{2\pi\sigma}{\ln(b/a)}$$\frac{\pi\sigma}{\ln[D+\sqrt{D^2-1}]}$$\frac{\sigma w}{d}$S/m
$C'$$\frac{2\pi\varepsilon}{\ln(b/a)}$$\frac{\pi\varepsilon}{\ln[D+\sqrt{D^2-1}]}$$\frac{\varepsilon w}{d}$F/m
📈 Real-World Parameters
Line$Z_0$$R'$$L'$$C'$
RG-59 Coax75 Ω36 mΩ/m430 nH/m69 pF/m
CAT5 Twisted Pair100 Ω176 mΩ/m490 nH/m49 pF/m
Microstrip50 Ω150 mΩ/m364 nH/m107 pF/m
✍ Example 1
Two-Wire Air Line — Click to expand
Given: $d = 2$ cm, $a = 1$ mm, perfect conductors ($\sigma_c = \infty$)
Find: $R', L', G', C'$
Solution:
$R' = R_s/(\pi a)$ → $R_s = \sqrt{\pi f\mu_0/\infty} = 0$ → $\boxed{R' = 0}$
$G' = \frac{\pi\sigma}{\ln[D+\sqrt{D^2-1}]}$ → $\sigma_0 = 0$ → $\boxed{G' = 0}$
$C' = \frac{\pi\varepsilon}{\ln[D+\sqrt{D^2-1}]}$ where $D = d/(2a) = 0.02/0.002 = 10$
$C' = \frac{\pi \times 8.854 \times 10^{-12}}{\ln[10 + \sqrt{100-1}]} = \frac{\pi \times 8.854 \times 10^{-12}}{\ln(19.95)}$
$\boxed{C' = 9.29\ \text{pF/m}}$
$L'$ from $L'C' = \mu_0\varepsilon_0$ → $L' = \mu_0\varepsilon_0 / C'$
✍ Example 2
Coaxial Air Line at 1 MHz — Click to expand
Given: $f = 1$ MHz, inner diam $= 0.6$ cm, outer diam $= 1.2$ cm, copper ($\sigma_c = 5.8\times 10^7$)
$a = 0.003$ m, $b = 0.006$ m, $\mu_c \approx 1$
Solution:
$R_s$: $R_s = \sqrt{\pi f\mu/\sigma_c} = \sqrt{\pi \times 10^6 \times 4\pi \times 10^{-7} / (5.8\times 10^7)}$
$\boxed{R_s = 2.608 \times 10^{-4}\ \Omega}$
$R'$: $R' = \frac{R_s}{2\pi}(\frac1a+\frac1b) = \frac{2.608\times 10^{-4}}{2\pi}(\frac{1}{0.003}+\frac{1}{0.006})$
$\boxed{R' = 0.0208\ \Omega/\text{m}}$
$L'$: $L' = \frac{\mu}{2\pi}\ln\frac{b}{a} = \frac{4\pi\times 10^{-7}}{2\pi}\ln(2) = 2\times 10^{-7} \times \ln 2$
$\boxed{L' = 0.138\ \mu\text{H/m}}$
$C'$: $C' = \frac{2\pi\varepsilon}{\ln(b/a)} = \frac{2\pi \times 8.854\times 10^{-12}}{\ln(2)}$
$\boxed{C' = 80.26\ \text{pF/m}}$
$G'$: Air dielectric $\sigma = 0$ → $\boxed{G' = 0\ \text{S/m}}$
? MCQ Bank
1. TL defined as in microwave region?
A Wire delivering power via current
B Structure guiding EM waves place to place
C Two-terminal lumped network
D Element with zero resistance
2. Low freq elements are lumped because:
A $\lambda \ll$ circuit dimensions
B V/I affect entire circuit simultaneously
C Freq too high
D $\gamma = 0$
3. Which is NOT a type of TL?
A Coaxial cable
B Waveguide
C Optical fiber
D Antenna array
4. $\lambda = c/f$ explains at high freq:
A Elements remain lumped
B Dimensions $\sim \lambda$
C $R' = 0$
D TL = short circuit
5. Most common planar TL in microwave PCBs?
A Slot line
B Fin line
C Microstrip line
D Coplanar slot line
6. TEM lines have:
A Field component in propagation dir
B E and H entirely transverse
C Only H perpendicular
D Must be hollow metal
7. Rectangular waveguide is classified as:
A TEM line
B Higher-order line
C Microstrip
D Two-wire
8. TL is modeled electrically as:
A Single-port network
B Two-port network
C Three-port network
D Four-port network
9. $R'$ represents:
A Insulation resistance
B Combined conductor resistance ($\Omega$/m)
C Dielectric conductance
D Capacitance per length
10. $G'$ represents:
A Inductance per length
B Capacitance per length
C Insulation conductance (S/m)
D Resistance per length
11. Two-wire $\sigma_c=\infty$, $R'$ = ?
A $\infty$
B 1 $\Omega$/m
C 0 $\Omega$/m
D Freq dependent
12. Coaxial air line $G'$ = ?
A $\infty$
B Zero
C Equal to $C'$
D Equal to $L'$
13. $R_s$ formula:
A $\sqrt{\sigma_c/\pi f\mu}$
B $\sqrt{\pi f\mu/\sigma_c}$
C $\pi f\mu\sigma_c$
D $\sigma_c/\pi f\mu$
14. Coaxial $L'$:
A $\frac{\mu}{2\pi}\ln(b/a)$
B $\frac{2\pi\varepsilon}{\ln(b/a)}$
C $\frac{\mu}{\pi}\ln(b/a)$
D $\frac{2\pi\sigma}{\ln(b/a)}$
15. $L'C' = \mu\varepsilon$ holds for:
A TEM lines
B Only coax
C Only waveguides
D Lossless lines only
16. Secondary constants depend on:
A Line length only
B $R',L',G',C'$ and $\omega$
C Conductor material only
D Dielectric only
17. $\gamma$ defined as:
A $(R'+j\omega L')+(G'+j\omega C')$
B $\sqrt{(R'+j\omega L')(G'+j\omega C')}$
C $(R'+j\omega L')/(G'+j\omega C')$
D $\sqrt{(R'+j\omega L')+(G'+j\omega C')}$
18. $\alpha$ indicates:
A Phase lag (rad/m)
B Frequency
C Wave magnitude reduction (Np/m)
D $Z_0$
19. $\beta$ units:
A Np/m
B $\Omega$
C rad/m
D S/m
20. $Z_0$ formula:
A $\sqrt{(G'+j\omega C')/(R'+j\omega L')}$
B $(R'+j\omega L')(G'+j\omega C')$
C $\sqrt{(R'+j\omega L')/(G'+j\omega C')}$
D $(R'+j\omega L')+(G'+j\omega C')$
made by @ar0xh