Category: Electromagnetics

Point Charge

Electric Field of a Point Charge

The electric field of a point charge is given by:

$$E = \frac{q}{4 \pi \epsilon _0 r^2} \hspace{5 mm}V/m$$



Charge (nC)


Distance (m)


Electric Field (V/m)

Wave Impedance

Any material has a wave impedance for propagating electromagnetic waves.
This impedance relates the electric and magnetic field magnitudes.

$$\eta = \frac{E}{H}$$

Notice the similarity to Ohm’s law:
$$Z = \frac{V}{I}$$

For a general material:
$$\eta = \sqrt{\frac{j\omega\mu}{\sigma+j\omega\epsilon}} \hspace{5 mm}\Omega$$

Special Cases

The impedance formula simplifies for several special cases described below.

Free space

Free space is primarily air or vacuum. Any lossless dielectric with a dielectric close to 1 can be considered free space.
$$
\epsilon = \epsilon_0=8.854\times 10^{-12} \hspace{5 mm}F/m\\
\mu = \mu_0=4\pi\times 10^{-7} \hspace{5 mm}H/m\\
\sigma = 0\\
\eta = \sqrt{\frac{\mu_0}{\epsilon_0}} = 377 \hspace{5 mm}\Omega
$$

Lossless Dielectric

Any dielectric with low loss, such as teflon, glass, etc.
$$
\epsilon = \epsilon_r\epsilon_0\\
\mu = \mu_r\mu_0\\
\sigma = 0\\
\eta = \sqrt{\frac{\mu_r\mu_0}{\epsilon_r\epsilon_0}} = 377\sqrt{\frac{\mu_r}{\epsilon_r}} \hspace{5 mm}\Omega
$$

Conductor

Any good conductor, primarily metallic materials.
$$
\mu = \mu_r\mu_0\\
\sigma \gg \omega\epsilon\\
\eta = \sqrt{\frac{\omega\mu_r\mu_0}{\sigma}}\angle{-45^{\circ}} \hspace{5 mm}\Omega
$$

Inductance of Loops

These equations are from Inductance: Loop and Partial — Clayton R Paul.

Note that the inductance is fairly similar for either square or circular loops for a given length of wire (circumference/perimeter).

Inductance of a circular loop:

\(L_{circular} = \mu_0a(\ln{\frac{8a}{r_w}}-2)\)

with:
a = radius of loop
r_w = radius of wire
(assumed a >> r_w)

For a 22 AWG wire:
1 cm = 45 nH (wire length 6.3 cm)
10 cm = 740 nH (wire length 63 cm)
100 cm = 10 uH (wire length 630 cm)

Inductance of a square loop:

\(L_{square} = 2\frac{\mu_0}{\pi}l(\ln{\frac{l}{r_w}}-0.774)\)

with:
l = length of side
r_w = radius of wire
(assumed l >> r_w)

For a 22 AWG wire:
1 cm = 22 nH (wire length 4 cm)
10 cm = 400 nH (wire length 40 cm)
100 cm = 6 uH (wire length 400 cm)

Inductance of a rectangular loop:

\(L_{rectangle} = \frac{\mu_0}{\pi}[-l\ln(1+\sqrt{1+(\frac{w}{l})^2})
-w\ln(1+\sqrt{1+(\frac{l}{w})^2})
+l\ln\frac{2w}{r_w}+w\ln\frac{2l}{2w}
+2\sqrt{l^2+w^2} -2w -2l]\)

Transmission Line Introduction


In reality nothing happens instantaneously, no matter how fast it appears to be.
When it comes to electronics, all signals are electromagnetic waves and they travel at the speed of light.
The wiring, circuit board material, or whatever the medium it is, affects the speed of light.
Wire or conductor structures that guide these waves are called transmission lines.
Continue reading “Transmission Line Introduction”

Radiation

Radiation refers to the transmission of electromagnetic energy through propagating waves in free space.

There are two components to an EM wave – electric and magnetic.
A radiation source will inevitably be dominant in either electric or magnetic fields.

Magnetic Sources

A magnetic field is generated by current, and loop geometries emphasize magnetic fields.
A coil geometry strengthens magnetic fields by the square of the number of turns.
These structures are associated with inductance and low impedance.

Example magnetic sources:
-multi-turn coils
-loops
-apertures (openings) in metal structure

Electric Sources

Electric field is generated by voltage between conductive elements. This is the most common antenna type.

Larger surface areas tend to increase electric field.
These structures are associated with capacitance and high impedance.

Example electric sources:
-monopoles and dipoles
-parallel plates
-horn antennas
-wire antennas

Capacitance

Capacitance is physically defined as the ratio of stored charge on two electrodes to the voltage between them.

\(C = Q/V\)

This is a very broad definition and it can apply to any two objects though typically it is reserved for conductive ones.

All objects have a capacitance to all other objects in the vicinity. This makes capacitance very difficult to measure or control in any general way.

An electronic component designed to maximize its capacitance is called a capacitor, however all devices have intrinsic capacitance.