Essential concepts that took me years to understand
Let's master some important electrical engineering ideas here. Powerpoint of some ideas.
As a signal travels down a cable the path is ideally a smooth one with minimal bumps. But when it gets to the load there's a chance that some of the energy will be reflected back down the path. This can happen if there is an abrupt change in the impedance. Changes in conductuctivity, capacitive, and inductive reactance can all contribute to a frequency dependent reflection of signal energy. This is desirable in some cases and not so much in others. See Max Power Transfer below for details.
In an electric circuit one can measure the impedence at any part. Between major components exist "transmission lines". If the impedance (relative to ground) on both conductors is the same all the way down the line, it is considered balanced. The benefit is that external noise is induced equally in both lines so common mode rejection is possible.
Signals are often transmitted over balanced connections using the differential mode, meaning the wires carry signals of opposite polarity to each other (for instance, in an XLR connector, pin 2 carries the signal with normal polarity, and pin 3 carries an inverted version of the same signal). Despite popular belief, this arrangement is not necessary for noise rejection. As long as the impedances are balanced, noise will couple equally into the two wires (and be rejected by a differential amplifier), regardless of the signal that is present on them. A simple method of driving a balanced line is to inject the signal into the "hot" wire through a known source impedance, and connect the "cold" wire to the signal's local ground reference through an identical impedance. Due to common misconceptions about differential signalling, this is often referred to as a quasi-balanced or impedance-balanced output, though it is, in fact, fully balanced and will reject common-mode interference. Most explanations of balanced lines assume symmetric (antiphase) signals but this is an unfortunate confusion - signal symmetry and balanced lines are quite independent of each other. Essential in a balanced line is matched impedances in the driver, line and receiver. These conditions ensure that external noise affects each leg of the differential line equally and thus appears as a common mode signal that is removed by the receiver. There are balanced drive circuits that have excellent common-mode impedance matching between "legs" but do not provide symmetric signals. Symmetric differential signals exist to prevent interference to other circuits - the electromagnetic fields are canceled out by the equal and opposite currents. But they are not necessary for interference rejection from other circuits. Lines carrying symmetric signals (those with equal but opposite voltages to ground on each leg) are often incorrectly referred to as "balanced", but this is actually differential signaling. Balanced lines and differential signaling are often used together, but they are not the same thing. Differential signalling does not make a line balanced, nor does noise rejection in balanced cables require differential signalling. Circuits driving balanced lines must themselves be balanced to maintain the benefits of balance. This may be achieved by transformer coupling or by merely balancing the impedance in each conductor. This advantage is not directly due to differential signaling itself, but to the common practice of transmitting differential signals on balanced lines. Single-ended signals are still resistant to interference if the lines are balanced and terminated by a differential amplifier.
In a transmission line there exist at least two conductors carrying a signal from the source to the load. When the signals are perfectly out-of-phase then the effective radiation at a distance is zero. This helps prevent the line from being the source of noise to other parts of the circuit. Single-ended signals by definition are not symmetric and can radiate!
Consider a complete circuit with a source and a load. We define this as a "network" and use network analyzers to measure important properties like impedence and bandwidth. A network analyzer typically has two ports S_1 and S_2. Sometimes you only need one port. For testing antenna bandwidth you would hook the antenna up to just S_1 and the analyzer tells you what frequencies are reflected across a band. Voltage standing waves...hmmmm I need to learn more about VSW ratios.
Max Power Transfer
For ideal load/source impedance ratios it depends on the application. Audio circuits contain wavelengths in the conductors that are huge compared to the circuit itself. This and the fact that modern amplifiers have super low output impedances makes efficiency an option. For audio it is usually desired that one "bridges" the source to the load. This puts a greater voltage drop across the load so more power is used by the load relative to the source. This of course means tons of energy reflected back to the source by the load but that's ok because it does not change the efficiency of the operation. RF circuits are different. RF wavelengths are about the size of the circuit and this makes it possible for tons of power to be wasted in reflection and standing waves. For radio circuits one ideally matches the source impedance to the load impedance such that no energy is reflected back. This is needed when noise is an issue.
In-Phase and Quadrature (analytic) signals
Sometimes it is important to know the instantaneous phase of a signal in real-time. Coherent systems by definition have well known phases along the transmission line. I-Q modulation is one way of doing this. With this modulation there are two signals, one "In-Phase" carries the raw signal of interest. The second carries the instantaneous phase of the raw signal. This is essentially two signals that are 90 degrees out of phase with eachother, hence quadrature because 360/4 = 90. Many times coherency through out the system is not needed untul post-processing. In this case a Hilbert transform takes any raw signal and generates a complex signal where the imaginary part is the instantaneous phase.
This is the idea that we tend to measure an unecessary amount of information when analyzing common natural systems. Intelligent systems are usually "sparse" in some domain. That domain unfortunately is a transform away from its natural state. However, upon transformation, this means that one can get a perfectly accurate sample of data with one-tenth the sampling rate for example. The trick is to find the domain in which it's sparse and transform the signal before or during the measurement!
For years Short-Time Fast-Fourier Transforms were the only way we analyzed the spectrum of signals in real-time.
ICA and PCA
To take advantage of a small voltage and control larger voltages both negative and positive one uses the H-bridge
EE Principles used in Project 8
Here are some of the presentations that were made while working at Penn State:
- Antenna Factor and CRES
- Antenna Factor
- Defining EE Concepts
- Colloquium UVU
- Longitudinal Beam Shapes
- PNA Studies
- Narrowband DBF
- Normalization in HFSSfull
- Phase II Hardware Change part3 Results
- Phase III AntennaeDec2017
- Phase III Call_1_15_09
- Phase III Design Topics Case Western
- Phase III DesignDec2017
- Phase III DesignJan2018
- Project 8 Phase IIIwendler
- Scattering Studies
- Sim WG and Phase III Call_12_19_18
- Simulation and Design of the next phase
T. Wendler Final Report (semi-Technical)During my postdoc appointment at Penn State I was tasked with developing the Phase III detector for Project 8. A major change from previous phases of the experiment was that we moved into the far-field of the CRES source to cover a larger volume. Having no structures near the source means we have to rely purely on the radiative power transfer and can not exploit any reactive “modal” coupling (i.e. TE01 in a waveguide) as in the previous phases. A wide variety of antenna topics have been covered while studying the large volume demonstrator. This folder contains most of my presentations while at Penn State. A few helpful references for antennas and array design are here. Below is a list the general research developments in this effort followed by an synopsis of each: CRES Synthesis Power Combining Transmit vs. Receive mode Far/Near-field properties Maximum Power Transfer Theorem From Frequency to Time-domain with Impulse Response (HFSS to Locust)
CRES SynthesisA Locust CRES point source based on a numerical routine for the Lienard-Wiechert potentials has been rigorously proven to be accurate in the frequency and time-domains (Nick B. and M.Jones Reference). The general properties of the CRES point source are that it has an instantaneous gain (from the relativistic lobing) of 6dBi and a time-avg gain of 1.4dBi. The CRES source is very close to an isotropic radiator and thus naturally introduces the “hairy-ball” theorem. The polarization is similar to a magnetic dipole, but has some dynamic polarization effects equivalent to the classic turnstile antenna in order to achieve this isotropy. Therefore, point source antennas have been developed in frequency domain simulation software with the capability of synthesizing radiation from a 90 degree CRES event in both polarization and power. It is essentially an electrically small, K-band, turnstile antenna, or equivalently a crossed Hertzian dipole. The CRES synth antenna includes the spatial quadrature we see as a spiral in the analytical cyclotron field which includes the LHCP/RHCP in the +/- z-dir. The polarization in the φ direction is proving difficult to create in practice and we are forced to use advanced fabrication techniques to realize the synth CRES antenna in the lab. This is expected since θ pol from the electric dipole moment is naturally much stronger and as Griffiths points out, making a magnetic dipole radiator basically comes down to just trying to minimize the electric dipole moment. A. Ziegler has continued the work into a new level of precision, A. Ziegler Email: email@example.com for any questions. Intellectual Property Evolution: This idea had started with Hamish and Brent as a concept with no real design before I arrived at P8, then Nick (at 2018 PNNL collab meeting) and I talked about the idea as an actual testing device for P8 to more detail (reproducing different CRES features). I prototyped a few ideas with PNNL’s help: mag-dipole, octo-micro-helix array (which made the full field), etc...then Erik van Bronkhorst suggested Turnstile, Helical variations, and a turnstile array. Kirk MacDonald's hertzian fields matched well to time-avg CRES so we sim prototyped Ideas, did a “lobed” proof of concept in the lab, then Andrew Ziegler took from there and added a slot-guide approach, among other phased structures needed to condition the signal to be in spatial quadrature.
Power CombiningAfter the incident field on an antenna has been converted to a travelling-wave voltage signal on a transmission line, the voltage wave propagates to the amplifier. Since the PIII design involves passive arrays of antennas, it is often the case that multiple antennas are coupled to one amplifier through a feed network. The purpose of power combining in Locust is to sum and propagate these signals accurately through the network to the amplifier. The signal propagation is done semi-analytically in Locust: the amplitude of the signal is derived either from HFSS or an equivalent circuit model, and the phase is calculated analytically assuming mod(2pi) electrical spacing between elements. Some of the details are described in presentations here including “power combining” (or dividing) in the title such as this patch array study and this slot array study. Many of the Locust results are shown alongside analogous computations in HFSS by Arina T. and Penny S. presentations on Basecamp. This Locust/HFSS interface has served as a general cross-check for the coding of the power combiner, although simple analytic impedance calculations have been used as well. One fundamental discovery made by the antenna team was that the ideal junction S matrix in a passively combined linear array would need to include a high directivity for the LNA-side output, a single port of a 3 port junction. The following S-matrix represents the ideal junction for a phased array with maximal axial coverage and ideal power combining losses through evading reradiation: S=[1,1,1:0,0,0:0,0,0]. In this matrix one can see that port 1 output is highly favored over the others. Unfortunately, EM laws prevent us from creating a simultaneously matched, reciprocal, and lossless network (Pozar p.318). Circulators and Isolators are devices that come close as they use ferrous materials to enforce non-reciprocal behavior. The Wilkinson divider also has a similar property to that which we are looking for, isolation between two ports. Moreover, the junction efficiencies highly depend on what mode they are in, even-mode (100%) or odd (0%) or something in between (asymmetric amplitude and/or phase between the inputs). For example, in the Wilkinson divider odd-mode produces total destructive interference while even-mode produces the complete opposite. Perhaps the most important feature we had to address in power combining was the possibility of the incident signals on each junction not only being unequal in phase but also amplitude. This was mostly a near-field effect. We do indeed anticipate this near-field phenomena where the signal’s incident on each of the junctions are not only asymmetric in phase but also in amplitude in the FSCD. A last but most important part is that these are specifically called Power Dividers and NOT Voltage Dividers, therefore, the outputs are NOT simply a superposition of voltages of the inputs! One way to reconcile this with the irrefutable idea of EM being a linear theory is by considering the transformation of impedances of the junction in question.
Transmit Mode vs Receive ModeBefore discussing the details of the antenna elements for a given array it is important to establish the differences between transmit mode and receive mode analysis. The Friis formula and antenna patterns are all generated under the far-field condition. The detector designed here is designed to be sensitive to near-field phenomena and therefore many of the traditional antenna engineering methods break down. One particular instance of that breakdown begins in the differences between receive and transmit modes for electromagnetically large arrays. The first place receive mode and transmit mode diverge is in the current distributions on the elements. The current distribution on a transmitting antenna follows that of a pure standing wave. The current distribution on that same antenna in receive mode depends both on the excitation condition and the loading conditions and differs with that on the transmitting antenna. In transmit mode, the field at infinity is the interference of all the differential elements on the antenna with certain amplitude distribution, uniform electric phase (in the case of a half-wavelength dipole), and progressive space phase difference caused by the path difference in space. In receive mode the excitation is a plane wave with uniform amplitude and progressive phase difference caused by the path difference between the wave-front in space and the antenna (note the taper is evident in an array in transmit mode and not in receive mode). Therefore, the electric phase is different at the different points of the receive antenna. The induced current through the load is the superposition of the differential current produced by the differential elements with certain amplitude and electric phase distributions. Although the meaning of superposition is different for transmitting and receiving modes, the final integral expression is the same, resulting in the same pattern. For the most part, simple array code using electrically small dipole-like elements (even superposition of scalar isotropic sources![“Phase is King” M. Jones]) can be used for analysis of the detector’s coverage. However, a nuance to the detector is that reciprocity breaks down, especially in the near-field with a non-uniform-copolar source. The CRES source has an elliptical polarization up/down stream which affects the up/down stream fields with a -3dB hit of the phi polarization, this changes the reciprocity enough to see significant differences in pure transmit mode and Friis maps using a CRES source in some cases.
Near/Far field AnalysisWe have discovered that a major detection sensitivity challenge comes from the CRES source being in the near-field and in some cases the extreme near-field of the receiving array. In the FSCD the general idea is that the fields incident across the array aperture, and therefore the currents induced on the respective elements, are not uniform if the driving source is within the near field of the array. This near-field non-uniformity is exaggerated for a CRES source as the Huygens surface is a changing wave-front of linear to circularly polarized waves on top of that. On top of this the source is also moving relativistically within the near-field of the arrays, in and out of coverage regions producing both amplitude and frequency modulation in the received wave front. It is for these reasons that we employ sanity checks in the Receive mode and do not study the detector focal coverage only in transmit mode as commonly done in HFSS. In the extreme near-field of a microwave/radio source the fields are reactive. This reactivity can be seen in most easily in the frequency-domain (E, H, V, and I in steady state) in three ways: 1) Non-orthogonality of E and H vector phasors 2) E_u/H_v not equal to 377 3) Phase between the E and H vector phasors 4) Non-zero Imaginary part of V/I (in a circuit t-line/port/termination/load) Bullet points 2 and 3 are connected when you consider instantaneous vs time-avg E/H ratios. Bullet point 1 is related to axions which are coupled to E dot H. 1 is also the reactance one sees in the extreme near-field of a CRES source. The above 3 ways for a field to be reactive are mostly only seen in the extreme near-field, or, reactive near-field. Once you reach the Fresnel distance the fields are mostly all radiative and the only thing that keeps it from being a “far-field” zone is the fact that the pattern is still a function of distance. Bullet 4 is only a concern after the signal has been coupled to a transmission line when matching circuits such as those between the radiating elements in the passive combiner network, or between the antenna output port and the LNA, but it’s still a key reactivity in the full Rx chain nonetheless. At this point one can see the classic ELI the ICE man rule can be used for the free-field reactivity, where inductance “L” is likened to the energy stored in the magnetic field and capacitance “C” to that in the electric field.
Maximum Power Transfer Theorem and the Losses in FSCDBeing as how the Project 8 experiment is challenged by the extremely low power levels of the source, losses by any mechanism should be well understood by the engineer/physicist. Some general electromagnetic losses are shown in the Fig  below. One fundamental EE principle that is of constant concern in matching the antenna to the free-field is the Maximum Power Transfer Theorem. It is more typically found in a circuit context, however, for the free-field power propagating to the antenna t-line output it is stated as: MPTT: Under the condition for which a maximum amount of power is delivered to the antenna terminal impedance, an equal amount of power is reradiated. This is also true for any impedance transformation along transmission lines in the receiving chain such as LNA input, mixer RF input etc, but the “equal amount of power” being not in the form of reradiation, but rather power dissipated by the “load”. It is not quite intuitively understood as to whether or not this implies a -3dB loss for our already tiny 1fW in the free-field we are working with, as the theorem is in a circuit context. For our application the physical circuit can be drawn to include a source voltage drop across free-space (377 Ohms) as well as the load radiation resistance of the antenna for a full loop. One thing to consider is the subsequent collection of this equally reradiated power, which indeed is theoretically possible. The conclusion is that there is no law of physics that directly claims we cannot collect more than half of the incident power asymptotically through any arbitrary number scatters. It can be seen with Friis far-field theory that an antenna with a physical aperture of 4pi solid angle (surrounding the transmitter) and matching polarization can capture ~100% of the transmitted power from a point source. Figure 1: Various losses when considering a patch antenna array for Phase III. They’re ordered in inevitability vs. Significance. Note the MPTT is there but may not be an existential threat for a full 4pi solid angle surrounding antenna as it is not placed at the top of ‘inevitably’ scale. For a single ring of antennas it will be inevitable as the scattering up/down stream will not be regathered, but rather lost.
From Frequency to Time-domain with Impulse Response
Traditional antenna engineering and design is dominated by frequency-domain analysis (Balanis, Krauss). Unfortunately for us, the FSCD of Project 8 has a relativistically moving source in the near-field. This includes time-domain effects that cannot be simply imported into a frequency domain software like HFSS. When modeling the antennas in Locust we needed a seamless way to accurately convert the incident E fields into antenna voltages. Since the antennas are Linear Time-Invariant systems we exploited what is called the Impulse Response. A Finite Impulse Response filter can be derived from HFSS data with an IFFT of the Transfer Function of the system being modeled. This is one of the more important contributions I made to Project 8 and the FSCD modeling in Locust and was rigorously validated by J. Tedeschi at PNNL. It is crucial that we accurately model the antenna’s phase response above all since that is most likely how we will localize events through model based processing. Especially for sensitive or resonant structure where the phase response is highly frequency dependent. We have developed a robust connection between HFSS and Locust so we can exploit the best of both worlds, commercial FEM packages and homemade software. Here is a set of tutorials focused on HFSS specific to Project 8 needs