# Doppler Effect

RF/Photonics Lab
November 2019
Michael Benker

Doppler Effect

The Doppler Effect is an important principle in communications, optics, RADAR systems and other systems that deal with the propagation of signals through space. The Doppler Effect can be summarized as the resultant change to a signal’s propagation due to movement either by the source or receiving end of the signal. As the distance between two objects changes, so does the frequency. If, for instance, a signal is being propagated towards an object that is moving towards the source, the returning signal will be of a higher frequency. The Doppler Effect is also applied to rotation of an object in optics and RADAR backscatter scenarios. A rotating target of a radar or optical system will return a set of frequencies which reflect the distances of each point on the target. If one side of the target is moving closer while the other side is moving away, there will be both a higher and lower frequency component to the return signal. # Skin Effect

RF/Photonics Lab
Jared Alves
November 2019

Skin Effect

The skin effect is an important characteristic of alternating current within conductors. With direct current, charges are distributed evenly when flowing through a conductor. However, due to the Skin Effect as the frequency of the conduction current is increased, the charges distribute in greater quantities towards the surface of the conductor. In other words, the current density (J) decreases with greater depth in the conductor.   As shown, the current density is per area. The skin depth of the conductor is the length from the surface of the conductor inward in which the majority of the charge is contained at frequencies higher than DC. As shown in the equation above, skin depth is inversely proportional to frequency so at higher frequency values, the effective resistance of the conductor increases which reduces the cross-sectional area, as shown below. The figure demonstrates that the conductor becomes more “hollow” at higher frequencies as the electric charges avoids traveling through the center. This is because the back EMF is strongest towards the center of the conductor. Maxwell’s equations explain that magnetic field strength is proportional to current and therefore as current intensity changes, so does magnetic field strength. The changing magnetic field creates an electric field opposing this change in intensity which causes the counter EMF effect. This creates an almost “Faraday cage” effect with the electrons at the center of conductor as the electric field cannot penetrate as deep into the conductor with increasing frequency. The skin depth is technically defined as the length from the surface to the inside of a conductor in which J (current density) decays to 1/e of Js (current density at the surface). The imaginary part of the above equation shows that for each skin depth of penetration, the current density phase is delayed by 1 radian.

# Interferometry – Introduction

RF/Photonics Lab
Jared Alves
November 2019

Interferometry – Introduction

Interferometry is a family of techniques in which waves are superimposed for measurement purposes. These waves tend to be radio, sound or optical waves. Various measurements can be obtained using interferometry that portray characteristics of the medium through which the waves propagate or properties of the waves themselves. In terms of optics, two light beams can be split to create an interference pattern when the waves combine (superimpose). This superposition can lead to a diminished wave, an increased wave or a wave completely reduced in amplitude. In an easily realizable physical sense, tossing a stone into a pond creates concentric waves that radiate away from where the stone was tossed. If two stones are thrown near each other, their waves would interfere with each other creating the same effect described previously. Constructive interference is the superposition of waves that results in a larger amplitude whereas destructive interference diminishes the resultant amplitude. Normally, the interference is either partially constructive or partially destructive, unless the waves are perfectly out of phase. The following image displays total constructive and destructive interference. A simple way to explain the operation of an interferometer is that it converts a phase difference to an intensity. When two waves of the same frequency are added together, the result depends only on the phase difference between them, as explained previously. The image above shows a Michelson interferometer which uses two beams of light to measure small displacements, refractive index changes and surface irregularities.  The beams are split using a mirror that is not completely reflective and angled so that one beam is reflected, and one is not. The two beams travel in separate paths which combine to produce interference. Whether the waves combine destructively or constructively depends on distancing between the mirrors. Because the device shows the difference in path lengths, it is a differential device. Generally, one leg length is kept constant for control purposes.

# IP3 Distortion and Linearity

RF/Photonics Lab
November 2019
Michael Benker

IP3 Distortion & Linearity

Linearity is the measure of a system’s performance as an output signal being proportional to the input signal level as characterized by Ohm’s Law, V = I*R. Not every system can be expected to perform ideally and thus linearly. Devices such as diodes and transistors are examples of non-linear systems. The intercept point of the third order, IP3 is a measure of the linearity of a system. IP3 is the third order of a Taylor series expansion of the input signal’s presence in the frequency domain. Being third order, this term in a Taylor series expansion is understood as distortion since it is different from the sought output signal. In contrast to the second order harmonics, which fall outside of the frequency band of the first order signal, the third order is found in the same frequency band as the original or first order signal. Similarly, consecutive even orders (4, 6, 8, etc) are found outside of the frequency band of the first order signal. Consecutive odd orders beyond the third order such as IP5 and IP7 also cause distortion but are not of primary focus since the amplitude of these order signals are weaker after consequent exponentiation.

The meaning of an intercept point of an nth order (IPn) on a dBm-dBm axis is the point at which the first-order and nth-order powers would be equal for a given input power. In the case of IP3, this indicates the power level needed for a third-order power to potentially drown out the first-order signal with distortion. The 1 dB compression point defines the range of linear operation for a system. # Scattering Parameters

RF/Photonics Lab
Jared Alves
November 2019

Scattering Parameters

After the mid-1900s, high frequency networks became increasingly prevalent. When analyzing low frequency circuits parameters such as voltages and currents are easily realized. From these signals, Y and Z (admittance and impedance) parameters can be used to describe a network. For the Radio Frequency and Microwave range, S parameters are much more applicable when studying a network of a single port or multiple ports. Each S parameter can be placed in an NxN square matrix where N is the number of ports. For a single port network, only the parameter S11 (also known as ᴦ (gamma or voltage reflection coefficient)) can be realized. The S parameters are unitless because they are ratios of voltages. The parameters can be viewed as both reflection and transmission coefficients for multi-port networks. S parameters with subscripts of the same number are reflection coefficients, as they describe the ratio of voltage waves at a single port (reflected to incident).

For a two-port network, only parameters S11, S12, S21, S22 exist. For a simple network like this, S11 represents return loss or reflection at port 1. S22 is the output reflection coefficient.  S12 and S21 are transmission coefficients where the first subscript is the responding port and the second the incident port. For example, S21 would be the “forward gain” at port 2 incident from port 1. The following diagram shows an abstracted view of a two-port network, where each “a” and “b” are normalized by the system’s characteristic impedance. Each S parameter can be calculated by terminating a port with a matched load equal to the characteristic impedance. For example, when calculated return loss for a two-port network, port 2 should be terminated by a matched load reducing a2 to zero.  For calculating S1,2 or S2,2 port 1 would be terminated with a matching load to reduce a1 to zero. Each “a” is an incident wave and “b” a reflected wave. Having a matched load at a port results none of the incident wave being reflected due to impedance mismatching. This leads to the following voltage ratios: An amplitude with a negative superscript indicates a reflected wave, and an amplitude with a positive superscript indicates a forward propagating wave.

SParameters

# Smith Chart

RF/Photonics Lab
Jared Alves
November 2019

Smith Chart The Smith Chart, named after laboratories engineer Phillip Smith, is a graphical tool for solving RF transmission line problems. There are many specific uses for a Smith Chart, but it is most commonly used to visually represent impedance matching problems. Although paper Smith Charts are outdated, RF equipment such as Network Analyzers display information using the chart as well.

The Smith Chart is a unit circle (radius of one) plotted on the complex plane of the voltage reflection coefficient (ᴦ – gamma). As with any complex plane, the vertical axis is the imaginary and the horizontal axis the real. The Smith Chart can be used as an admittance or impedance chart or both. For a load impedance to be plotted on the chart, it must be normalized (divided by) the characteristic impedance of the system (Zo) which is the center of the chart. With this information in mind, it is apparent that a matched load condition would result in traveling to the center of the chart (where ZL=Zo). Along the circumference of the chart, there are two scales: wavelength and degrees. The degrees scale can be used to find the angle of the complex reflection coefficient. Since the plot is the polar representation of the reflection coefficient, if a line is drawn from the load impedance point to the center of the chart this would be considered the magnitude of the reflection coefficient. By extending the line to the circumference of the circle, the angle (in degrees) can be found. The wavelength scale shows distance across a transmission line in meters. A clockwise rotation represents moving towards the generator whereas a counter-clockwise rotation represents moving towards the load side.

It is important to note that a Smith Chart can only be used at one specific frequency and one moment in time. This is because waves are functions of both space and time as shown by the equations: VF is the forward propagating voltage wave and VR is the reverse propagating voltage wave. If a transmission line system is not impedance matched, a reflected wave will exist on the line which will cause partial or fully standing waves to occur on the line (the reflected wave will add to the incident wave). For the matched condition the reflected wave is zero. Because the Smith Chart can only be used at a specific instant in time and at one frequency the first exponential term in each equation drops out. Because the reflection coefficient is the ratio of the reflected wave to the forward propagating wave, the reflection coefficient becomes: Where C is the ratio of the amplitudes of both waves. For a passive load, the reflection coefficient must be equal to one or less because the reflected wave cannot be greater in amplitude than the incident wave.

Many transmission lines can be approximated as lossless and therefore have zero attenuation. This leads to: The propagation constant is a complex number that describes how a wave changes as it propagates down a transmission line. The real part is attenuation constant (Nepers/meter) and the imaginary part is the phase constant or wave number (radians/meter). For the lossless condition the attenuation is zero, as stated previously.

On the Smith Chart, the wavelength λ = 720. This is because the reflected wave must travel the roundtrip distance moved (it must propagate forward and then back again). Using the piece of information, a half wavelength distance is one complete revolution on the chart. This leads to the conclusion that a transmission line that is a half wavelength long does not transform impedance.

The following image shows common points on the Smith Chart. The left-hand side of the chart (lying on the real axis) represents a short circuit load. This makes intuitive sense because the reflection coefficient must be real and negative for a short circuit. This is because short circuits have a voltage drop of zero across them which would require a same-amplitude wave with a 180-degree phase shift to cancel the forward propagating wave. The right-hand part of the real axis represents the open circuit load, where the reflection coefficient is purely real but has no phase shift. For an open circuit, the current wave would have to be phase shifted by 180-degrees, but since the reflection coefficient is a voltage reflection coefficient it is not necessary for it to be phase shifted. As shown in the image, the upper half plane is inductive (positive reactance) and the lower half is capacitive (negative reactance).

Smith Chart