High Speed Electronics

The electric field propagates radially from a charge (Couloumb's law). Radiation is a disturbance in the electric field. The disturbance's power propagates with \(1/r^2\) rather than changing the electric field with \(1/r^3\). The radial field depends on acceleration and decreases less sharply than the couloumb field.

Maxwell's equations, particularly the variation on ampere's law, presents an electric field that alters itself. The electric field becomes radially dependent, and has a magnetic field in the gap region capable of storing energy. Hence the ideal parallel plate capacitor has an inherent inductance and thus a self resonant frequency. As the frequency changes, the capacitor behaves more like an inductor. A real capacitor also has a resistance.

A field applied to media causes polarisation to the atoms and molecules. In the media the displacement flux becomes: \[\vec{D}=\epsilon_0\vec{E}+\vec{P}_e\] In a linear medium: \[\vec{P}_e=\epsilon_0\chi_e\vec{E}\] Where \(\chi_e\) can be complex and is the electric susceptibility. \[\vec{D}=\epsilon_0\vec{E}+\vec{P}_e=\epsilon_0(1+\chi_e)\vec{E}=\epsilon\vec{E}\] Where \(\epsilon=\epsilon'-j\epsilon''=\epsilon_0(1+\chi_e)\). The imaginary part accounts for the loss in the medium. In a material with conductivity \(\sigma\), the conductive current density is given by the electromagnetic form of Ohm's law: \[\vec{J}=\sigma\vec{E}\] This would allow us to write ampere's law as: \[\nabla\times H=j\omega\vec{D}+\vec{J}=j\omega\left(\epsilon'-j\epsilon''-j\frac{\sigma}{\omega}\right)\vec{E}\] Material data sheets define a quality loss tangent: \[\tan\delta=\frac{\omega\epsilon''+\sigma}{\omega\epsilon'}\] A material will have a relative permittivity quoted as: \[\epsilon|_\omega=\epsilon_0\epsilon_r(1-j\tan\delta)\]

A material may have permittivity that may vary depending on direction.

At a surface boundary, the difference in electric flux normal to the boundary is equal to the charge on the surface. There is no change in magnetic flux as there is no monopole. The difference in magnetic fields is due to current at the boundary. The electric fields on the boundary are continuous. On a perfect conductor in a field, charge redistributes at the surface and current flows to cancel the electric and magnetic fields.

The wavelength of a wave is dependent on the material it is travelling through, as the material's permittivity increases, the wavelength decreases. The wave impedance relates the magnitude of the magnetic field to the electric field. \[\eta=\omega\mu/k=\sqrt{\mu/\epsilon}\] The above equation is for a lossless medium. The wave impedance of free space is \(377\Omega\). In a general lossy medium, there is a propagation constant: \[\gamma=\alpha+j\beta=j\omega\sqrt{\mu\epsilon}\sqrt{1-j\frac{\sigma}{\omega\epsilon}}\] For a dielectric \(\sigma=0\) and \(\gamma=j\omega\sqrt{\mu\epsilon}=jk=j\omega\sqrt{mu\epsilon'(1-j\tan\delta)}\), the wave impedance is given my: \[\eta=\frac{j\omega\mu}{\gamma}\] The complex wave impedance is the loss in the medium. In a good conductor, the propagation constant becomes: \[\gamma=(1+j)\sqrt{\frac{\omega\mu\sigma}{2}}\] We define the skin depth to be the travelling distance before the wave decays by \(e^{-1}\). \[\delta_s=\frac{1}{\alpha}=\sqrt{\frac{2}{\omega\mu\sigma}}\]

Antennas are structures that are capable of converting power into radiated energy.

• Isotropic antennas have equal hypothetical losses in all directions
• Directional antennas is more effective at radiating in some directions
• Omnidirectional antennas have essentially nondirectional pattern in a given plane and a directional pattern in an orthogonal plane.

Directional antennas will have a main power lobe and multiple smaller side lobes. The side lobes can cause interference, affecting the signal.

An antenna's field regions are:

Reactive near field

Within \(R_1=0.62\sqrt{D^3/\lambda}\)

Radiating near field (Fresnel zone)

Within \(R_2=2D^2/\lambda\)

Far field (Fraunhofer)

Here \(D\) is the largest antenna dimension. Objects closer to the antenna affect its radiative pattern, detuning the antenna and changing its frequency.

The antenna gain is the amount of power in a given area compared to all the power radiated if it were radiated isotropically. The radiation power density is given by the Poynting vector. \[\vec{W}=\vec{E}+\vec{H}\] Where \(\vec{W}\) is the instantaneous Poynting vector, \(\vec{E}\) is the electric field intensity and \(\vec{H}\) is the magnetic field intensity. The total power radiated is: \[P=\oiint_s\vec{W}\dot ds=\oiint_s\vec{W}\dot\hat{n}da \] The power is \(P\), \(\hat{n}\) is the unit vector normal to the surface and \(da\) is the infinitesimal area of the closed surface. The reactive power component contributes to creating the electric and magnetic fields, whereas the real power carries the signal.

A quick approximate method of estimating directivity is estimating the half beam angle from the radiation pattern. \[D_0=\frac{4\pi}{\Omega_A}\approx\frac{4\pi}{\Theta_{1r}\Theta_{2r}}\] Where \(\Theta\) is the angle in the x and y directions. This produces linear units, but antenna directivity is in dB commonly.

The antenna gain is reduced due to metal and dielectric losses, but doesn't account for reflected power from impedance mismatches. Antenna efficiency does account for the reflected power.

Polarisation of a radiated wave is the time varying magnitude and relative magnitude of the electric field vector. Different antennas produce radiation with different polarisation. Polarisation can be vertical, horizontal, circular right, circular left and others. The power is reduced if the antenna is cross polarised to the signal, reducing to 0 received power in the worst case.

The antenna's input impedance is given by: \[Z_A=R_A+jX_A\] The resistive component is given by the radiation resistance (\(R_r\)) and loss resistance (\(R_L\)). \[R_A=R_r+R_L\] The radiation resistance gives the amount of power that is radiated, so a larger resistance means more power is radiated. In order to deliver maximum power, the impedances must be conjugately matched.

Transmission lines are waveguides for energy. Usually a significant portion of the wavelength. The impedance along the line is complex and is dependence on its resistance, capacitance, inductance and shunt conductance. The capacitance is dependent on the geometry of the line, whereas its resistance is on the material used. There is a relationship between the forward and reverse travelling voltage wave. \[\frac{V_+}{I_+}=Z_0=-\frac{V_-}{I_-}\] We can characterise the fields in the conductor into lumped circuit elements.

The voltage and current at a point along the wave can be written as: \[V(z)=V_0^+e^{-j\beta z}+V_0^-e^{j\beta z}\] \[I(z)=\frac{V_0^+}{Z_0}e^{-j\beta z}+\frac{V_0^-}{Z_0}e^{j\beta z}\] At the load: \[Z_L=...\] And we can make reflection coefficient: \[\Gamma=\frac{Z_n-1}{Z_n+1}\] Which lets us rewrite the equation as: \[V(z)=...\]

Return loss is the amount of power that is reflected away from the load, and is lost power sent back to the generator. For an open and short circuit, the reflection coefficient is magnitude 1, and the return loss is 0 dB (all power returned).

The voltage wave standing ratio is: \[SWR=\frac{V_{max}}{V_{min}}=\frac{1+|\Gamma|}{1-|\Gamma|}\] The SWR is bounded between 1 and infinity. The min and max voltages are \(|V_0^|(1-|\Gamma|)\) and \(|V_0^+|(1-|\Gamma|)\) respectively.

The transmission coefficient is: \[T=1+\Gamma=\frac{2Z_1}{Z_1+Z_0}\] The insertion loss is the power lost (in dB) on the load. \[IL=-20\log|T|dB\]

For a general wave travelling along a lossless transmission line, the voltage and current are: \[V(Z,t)=V^+\left(t-\frac{Z}{V_P}\right)+V^-\left(t+\frac{Z}{V_P}\right)\] \[I(Z,t)=\frac{1}{Z_0}\left[V^+\left(t-\frac{Z}{V_P}\right)+V^-\left(t+\frac{Z}{V_P}\right)\right]\] The first term represents the forward travelling wave and the second is the reverse.