ELEC3115 Electromagnetic

ELEC3115 Electromagnatic Engineering UNSW

Part A

Potential

Assuming gaussian sphere and charge Q

$$ V = \frac{W}{q} = -\int_{a}^{b} \vec{E} \cdot d \vec{l }= {q \over 4 \pi \epsilon_0 R } = {1 \over 4 \pi \epsilon_0 } \int \frac{\rho_v}{R} \, dv = {1 \over 4 \pi \epsilon_0 } \int \frac{\rho_s}{R} \, ds $$

Capacitance

$$ C = \frac{Q}{V} $$

Parallel Plate $$ C = {\epsilon S \over d } $$
Coaxial Cable $$ C = {{2\pi\epsilon} \over {\ln{b \over a }}} $$
Parallel wires $$ C = {{\pi\epsilon} \over {\ln{ b-a \over a }}} $$

Electron velocity
$$ u = \mu E \text{ (m/sec) where } \mu \text{ is electron mobility} $$

Magnetic

Flux Linkage in linear media
$$L = {\lambda \over I } = \frac{N^2}{\mathfrak{R}}$$

Energy
$$W_m = \frac{1}{2}L I^2= \frac{1}{2}\lambda I $$

Flux in current carrying conductor

for internal flux

$$ B_{int} = \frac{\mu_0 r }{2 \pi b^2 } I $$

for external flux

$$ B_{ext} = \frac{\mu_0 I }{ 2 \pi r} $$

Inductance of current carrying conductor

Turn, N, equivalent

$$ N = \frac{2r \,dr}{a^2} $$

Flux linkage

$$ d\lambda = \frac{2 r \,dr}{a^2} ( \frac{\mu_0 I}{2 \pi a^2} (a^2 – r^2) + \frac{\mu_0 I}{2 \pi} \ln \frac{b}{a}) $$

Inductance for coaxial wire

$$ L = \frac{\mu_0}{8 \pi} + \frac{\mu_0}{2\pi}\ln \frac{b}{a } $$

Inductance for parallel wire

$$ L = \frac{\mu_0}{4\pi} + \frac{\mu_0}{\pi}\ln \frac{b}{a } $$

Force

$$ F = BIL \sin \phi $$

$$ V = Blu \sin \phi $$

Tricks

In spherical coordinate, if \(\theta\) and \(\phi\) are not restricted

$$dv = R^2 \sin{\theta} \, d\theta d\phi dR = 4\pi R^2 dR $$


Part B

$$ Z_0= \sqrt{L \over C} $$

Phase Velocity

$$V_p = {\omega \over \beta} = {1 \over \sqrt{\mu\epsilon}} = {1 \over \sqrt{LC}}$$

Refection Coefficient

$$\Gamma = {Z_L – Z_o\over Z_L + Z_o}$$

Power along the line

$$P_{av}(z) = {1\over2}{|V_o^+|^2 \over Z_o}(1 – |\Gamma|^2)$$

Return Loss

$$RL = -20 \log{\Gamma}$$

Standing Wave Ration

$$\text{VSWR} = {|V_\max| \over |V_\min|} = {{1 + |\Gamma|} \over {1 – |\Gamma|}}$$

Cut-off Number and Frequency

$$ k = \frac{2\pi}{\lambda} \qquad \beta = \sqrt{k^2 – k_c^2} = k \sqrt{1 – (\frac{k_c}{k})^2} = k \sqrt{1 – (\frac{f_c}{f})^2}$$

$$f_{cmn} = \frac{k_{cmn}}{2\pi\sqrt{\epsilon\mu}} = \frac{1}{2\pi\sqrt{\epsilon\mu}} \sqrt{ (\frac{m\pi}{a})^2 + ( \frac{n\pi}{b})^2} $$

Wave Impedance

$$Z_o = \sqrt{\mu_0\mu_r \over \epsilon_0\epsilon_r} = Z_{TEM}$$

$$Z_{TE} = \frac{Z_0}{\sqrt{1 – (\frac{\omega_c}{\omega})^2 }}$$
\(Z_{TE}\) always higher than \(Z_0\)

$$Z_{TM} = Z_0\sqrt{1 – (\frac{\omega_c}{\omega})^2}$$
\(Z_{TM}\) always lower than \(Z_0\)

Group Velocity

$$v_g = c\sqrt{1-(\frac{f_c}{f})^2}$$

Evanescent waves

$$\alpha_{bc} = \omega \sqrt{\mu\epsilon} \sqrt{(\frac{\omega_{cmn}}{\omega})^2 -1 } $$