Electromagnetics, Transmission Lines, Waveguides and Antennas — ESE E&T

Weightage: Electromagnetics and its applications (lines, waveguides and antennas) make up a regular, formula-led block in the E&T papers. It is also the foundation for microwave and radar topics, so the same results return in several papers.

1. Maxwell's equations

In differential form, in a source-free medium:

Maxwell's addition of the displacement current term $\partial \mathbf{D}/\partial t$ makes the equations predict electromagnetic waves travelling at $c=1/\sqrt{{\mu }_0{\epsilon }_0}=3\times 10^8$ m/s.

Boundary conditions at an interface: the tangential component of $\mathbf{E}$ and the normal component of $\mathbf{B}$ are continuous. The tangential $\mathbf{H}$ is continuous if no surface current flows, and the normal $\mathbf{D}$ is continuous if no surface charge is present. Inside a perfect conductor, fields are zero.

2. Plane waves

In a lossless medium with permittivity $\epsilon$ and permeability $\mu$:

$v=\frac{1}{\sqrt{\mu \epsilon }},\eta =\sqrt{\frac{\mu }{\epsilon }},\lambda =\frac{v}{f}$

In free space ${\eta }_0=120\pi \approx 377\Omega$. A plane wave is transverse: $\mathbf{E}$, $\mathbf{H}$ and the direction of travel are mutually perpendicular, and $|\mathbf{E}|/|\mathbf{H}|=\eta$. The Poynting vector $\mathbf{S}=\mathbf{E}\times \mathbf{H}$ gives the power per unit area.

In a dielectric of relative permittivity ${\epsilon }_r$ the wave is slowed by $\sqrt{{\epsilon }_r}$.

Polarisation describes the path of the $\mathbf{E}$ vector: linear, circular or elliptical.

In a conducting medium the wave attenuates. The skin depth is:

$\delta =\frac{1}{\sqrt{\pi f\mu \sigma }}$

the depth at which the amplitude falls to $1/e$ of its surface value. Copper at 1 MHz has $\delta \approx 66\mu$m, so high-frequency currents flow in a thin skin, and RF conductors are often silver-plated.

3. Reflection and refraction

At normal incidence from medium 1 to medium 2:

$\Gamma =\frac{{\eta }_2-{\eta }_1}{{\eta }_2+{\eta }_1},\tau =1+\Gamma$

For a perfect conductor $\Gamma =-1$, giving a standing wave with a node at the surface. Snell's law is $n_1\sin {\theta }_1=n_2\sin {\theta }_2$. At the Brewster angle, parallel-polarised light is fully transmitted, with $\tan {\theta }_B=n_2/n_1$, and at the critical angle $\sin {\theta }_c=n_2/n_1$ (for $n_1>n_2$) total internal reflection begins, which is the principle of optical fibre.

4. Transmission lines

A transmission line has distributed resistance, inductance, conductance and capacitance. For a lossless line the characteristic impedance and velocity are:

$Z_0=\sqrt{\frac{L}{C}},v=\frac{1}{\sqrt{LC}}$

A load $Z_L$ causes a reflection:

$\Gamma =\frac{Z_L-Z_0}{Z_L+Z_0},VSWR=\frac{1+|\Gamma |}{1-|\Gamma |}$

A matched load has $\Gamma =0$ and $VSWR=1$. A short or open circuit gives $|\Gamma |=1$ and infinite VSWR.

Worked example. A $50\Omega$ line ends in a $100\Omega$ load. Then $\Gamma =50/150=1/3$ and $VSWR=(4/3)/(2/3)=2$. The reflected power fraction is ${\Gamma }^2=1/9$.

Input impedance of a line of length $l$:

$Z_{in}=Z_0\frac{Z_L+j{Z}_0\tan \beta l}{Z_0+j{Z}_L\tan \beta l}$

• A quarter-wave line inverts the load: $Z_{in}=Z_0^2/Z_L$, so it can match two impedances using a section with $Z_t=\sqrt{Z_{in}{Z}_L}$.
• A half-wave line repeats the load: $Z_{in}=Z_L$.
• A short-circuited quarter-wave line looks open, and an open-circuited one looks like a short.

To match a load, use the quarter-wave transformer, a single stub at a determined distance, or a double stub. The Smith chart maps the reflection coefficient and normalised impedance on one diagram, with a full turn corresponding to a half wavelength of line.

Worked example. To match a $100\Omega$ load to a $50\Omega$ line, the quarter-wave transformer needs $Z_t=\sqrt{50\times 100}=70.7\Omega$.

5. Waveguides

A hollow metal rectangular waveguide of broad dimension $a$ supports only TE and TM modes, not TEM. The dominant mode is TE${}_{10}$, with cutoff frequency:

$f_c=\frac{c}{2a}$

Below $f_c$ the wave is evanescent. Above it, the guide wavelength is longer than the free-space wavelength:

${\lambda }_g=\frac{\lambda }{\sqrt{1-(f_c/f{)}^2}}$

The phase velocity exceeds $c$ and the group velocity (the speed of energy) is less than $c$, with $v_p{v}_g=c^2$.

Worked example. For $a=2.29$ cm, $f_c=3\times 10^8/(2\times 0.0229)=6.55$ GHz. At 10 GHz, $\lambda =3$ cm and ${\lambda }_g=3/\sqrt{1-0.429}=3.97$ cm.

A coaxial line carries TEM and has no cutoff, while a waveguide has lower loss at microwave power levels. An optical fibre guides light by total internal reflection in a high-index core.

6. Antennas

An antenna converts guided waves to radiated waves, and the reverse. Its main parameters are:

• Radiation pattern: the angular distribution of radiated power.
• Directivity $D$: peak radiation intensity relative to an isotropic source. Gain $G={\eta }_{rad}D$.
• Beamwidth: the angular width between half-power points.
• Radiation resistance, which links radiated power to the feed current.
• Effective aperture $A_e=G{\lambda }^2/(4\pi )$.
• Bandwidth and polarisation.

A half-wave dipole has a gain of 1.64 (2.15 dBi) and a radiation resistance of about 73 ohms. A short dipole has a gain of 1.5. Arrays raise directivity by combining several elements with controlled phase, and the Yagi-Uda antenna uses a driven element with parasitic reflector and directors. A parabolic reflector provides high gain at microwave frequencies.

7. Link calculations

The Friis transmission formula gives the received power in free space:

$P_r=P_t{G}_t{G}_r{\left(\frac{\lambda }{4\pi R}\right)}^2$

The free-space path loss rises by 6 dB for each doubling of distance or frequency. The radar equation for a target of cross-section $\sigma$ gives $P_r\propto 1/R^4$, so doubling the range cuts the echo to one-sixteenth. In radio propagation, ground waves serve low frequencies, sky waves reflect from the ionosphere in the HF band, and line-of-sight links serve VHF and above.

Common traps

• Treating TEM propagation as possible in a hollow waveguide.
• Using $c$ for velocity in a dielectric without the $\sqrt{{\epsilon }_r}$ factor.
• Confusing group and phase velocity. Energy travels at the group velocity.
• Reading VSWR as a power ratio directly. It is a voltage ratio.
• Forgetting the fourth-power dependence on range in radar.

Memory aids

• "377 ohms": free-space impedance.
• "Quarter-wave inverts, half-wave repeats": line sections.
• "Vp times Vg equals c squared": waveguide.

Summary

Maxwell's equations lead to plane waves with intrinsic impedance and skin-depth loss in conductors, and boundary conditions give reflection and refraction. Transmission lines add reflection coefficient, VSWR and matching with quarter-wave sections or stubs.

Waveguides have a cutoff and dispersive velocities, antennas are described by gain, beamwidth and aperture, and the Friis and radar equations link everything into a link budget.

Exam protocol

• Write the impedance or reflection formula before the numbers.
• Check the operating frequency against cutoff.
• Convert dBi and dB to ratios before using the Friis formula.
• Remember the $1/R^4$ law for radar echoes.