Acoustics and Sound: The Vocal Apparatus

The study of modulation of signals for wireless transmission can, to some extent, be applied to the human body, In the RF wireless world, a “carrier” signal of a high frequency has a “message” encoded on it (message signal) in some form or fashion. This is then transmitted through a medium (generally air) as a radio frequency electromagnetic wave.

In a similar way, the vocal apparatus of the human body performs a similar function. The lungs forcibly expel air in a steady stream comparable to a carrier wave.  This steady stream gets encoded with information by periodically varying its velocity and pressure into two forms of sound: voiced and unvoiced. Voiced sounds produce vowels and are modulated by the larynx and vocal cords. The vocal chords are bands which have a narrow slit in between them which are flexed in certain ways to produce sounds. The tightening of the cords produces a higher pitch and loosening or relaxing produces a lower pitch. In general, thicker vocal cords will produce deeper voices. The relaxation oscillation produced by this effect converts a steady air flow into a periodic pressure wave. Unvoiced sounds do not use the vocal chords.

The tightness of the vocal cords produces a fundamental frequency which characterizes the tone of voice. In addition, resonating cavities above and below the larynx have certain resonant frequencies which also contribute to the tone of voice through inharmonic frequencies, as these are not necessarily spaced evenly.

Although the lowest frequency is the fundamental and most recognizable tone within the human voice, higher frequencies tend to be of a greater amplitude. Different sounds produced will of course have different spectrum characteristics. This is demonstrated in the subsequent image.


The “oo” sound appears to contain a prominent 3rd harmonic, for example. In none of these sounds is the fundamental of highest amplitude. The image also shows how varying the position of the tongue as well as the constriction or release of the larynx contributes to the spectrum.

It is interesting to note the difference between male and female voices: male voices contain more harmonic content. This is because lower multiples of the fundamentals are more represented in the male voice and are spaced closed to one another in the frequency domain.


The Cavity Magnetron

The operation of a cavity magnetron is comparable to a vacuum tube: a nonlinear device that was mostly replaced by the transistor. The vacuum tube operated using thermionic emission, when a material with a high melting point is heated and expels electrons. When the work function of a material is overcome through thermal energy transfer to electrons, these particles can escape the material.

Magnetrons are comprised of two main elements: the cathode and anode. The cathode is at the center and contains the filament which is heated to create the thermionic emission effect. The outside part of the anode acts as a one-turn inductor to provide a magnetic field to bend the movement of the electrons in a circular manner. If not for the magnetic field, the electrons would simple be expelled outward. The magnetic field sweeps the electrons around, exciting the resonant cavities of the anode block.

The resonant cavities behave much like a passive LC filter circuit which resonate a certain frequency. In fact, the tipped end of each resonant cavity looks much like a capacitor storing charge between two plates, and the back wall acts an inductor. It is well known that a parallel resonant circuit has a high voltage output at one particular frequency (the resonant frequency) depending on the reactance of the capacitor and inductor. This can be contrasted with a series resonant circuit, which has a current peak at the resonant frequency where the two devices act as a low impedance short circuit. The resonant cavities in question are parallel resonant.

Just like the soundhole of a guitar, the resonant cavity of the magnetron’s resonance frequency is determined by the size of the cavity. Therefore, the magnetron should be designed to have a resonant frequency that makes sense for the application. For a microwaves oven, the frequency should be around 2.4GHz for optimum cooking. For an X-band RADAR, this should be closer to 10GHz or around this level. An interesting aspect of the magnetron is when a cavity is excited, another sequential cavity is also excited out of phase by 180 degrees.

The magnetron generally produces wavelength around several centimeters (roughly 10 cm in a microwave oven). It is known as a “crossed field” device, because the electrons are under the influence of both electric and magnetic fields, which are in orthogonal directions. An antenna is attached to the dipole for the radiation to be expelled. In a microwaves oven, the microwaves are guided using a metallic waveguide into the cooking chamber.



Quality Factor

Quality factor is an extremely important fundamental concept in electrical and mechanical engineering. An oscillator (active) or resonator (passive) can be described by its Q-factor, which is inversely proportional to bandwidth. For these devices, the Q factor describes the damping of the system. In some instances, it is better to have either a lower or higher quality factor. For instance, with a guitar you would want to have a lower quality factor. The reason is because a high Q guitar would not amplify frequencies very evenly. To lower the quality factor, complex or strange shapes are introduced for the instrument body. However, the soundhole of a guitar (a Helmholtz resonator) has a very high quality factors to increase its frequency selectivity.

A very important area of discussion is the Quality Factor of a filter. Higher Q filters have higher peaks in the frequency domain and are more selective. The Quality factor is really only valid for a second order filter, which is based off of a second order equation and contains both an inductor and a capacitor. At a certain frequency, the reactances of both the capacitor and inductor cancel, leading to a strong output of current (lower total impedance). For a tuned circuit, the Q must be very high and is considered a “Figure of Merit”.

In terms of equations, the quality factor can be thought of in many different ways. It can be thought of as the ratio of “reactive” or wasted power to average power. It can also be thought of as the ratio of center frequency to bandwidth (NOTE: This is the FWHM bandwidth in which only frequencies that are equal to or greater than half power are part of the band). Another common equation is 2π multiplied by the ratio of energy stored in a system to energy lost in one cycle. The energy dissipated is due to damping, which again shows that Q factor is inversely related to damping, in addition to bandwidth.

Q can also be expressed as a function of frequency:


The full relationship between Q factor and damping can be expressed as the following:

When Q = 1/2, the system is critically damped (such as with a door damper). The system does not oscillate. This is also when the damping ratio is equal to one. The main difference between critical damping and overdamping is that in critical damping, the system returns to equilibrium in the minimum amount of time.

When Q > 1/2 the system is underdamped and oscillatory. With a small Quality factor underdamped system, the system many only oscillate for a few cycles before dying out. Higher Q factors will oscillate longer.

When Q < 1/2 the system is overdamped. The system does not oscillate but takes longer to reach equilibrium than critical damping.



Bragg Gratings

Bragg gratings are commonly used in optical fibers. Generally, an optical fiber has a relatively constant refractive index throughout. With a FBG (Fiber Bragg Grading) the refractive index is varied periodically within the core of the fiber. This can allow certain wavelengths to be reflected while all others are transmitted.


The typical spectral response is shown above. It is clear that only a specific wavelength is reflected, while all others are transmitted. Bragg Gratings are typically only used in short lengths of the optical fiber to create a sort of optical filter. The only wavelength to be reflected is the one that is in phase with the Bragg grating distribution.

A typical usage of a Bragg Grating is for optical communications as a “notch filter”, which is essentially a band stop filter with a very high Quality factor, giving it a very narrow range of attenuated frequencies. These fibers are generally single mode, which features a very narrow core that can only support one mode as opposed to a wider multimode fiber, which can suffer from greater modal distortion.

The “Bragg Wavelength” can be calculated by the equation:

λ = 2n∧

where n is the refractive index and ∧ is the period of the bragg grating. This wavelength can also be shifted by stretching the fiber or exposing it to varying temperature.

These fibers are typically made by exposing the core to a periodic pattern of intense laser light which permanently increases the refractive index periodically. This phenomenon is known as “self focusing” which is when refractive index can be permanently changed by extreme electromagnetic radiation.


Photodetectors and Dark Current

A photodetector simply is a device that converts light energy to an electrical current. These devices are very much similar to lasers, although they are designed to operate in reverse bias. “Dark current” is a term that originates from this reverse bias condition. When you reverse bias any diode, there is some leakage current which is appropriately named reverse bias leakage current. For photsensitive devices, it is called dark current because there is no light absorption involved. The main cause of this current is random generation of electrons and holes in the depletion region. Ideally, this dark current is minimal (<< 1).


The basic structure of the photodiode is the “PIN” structure, similar to a semiconductor laser diode. An intrinsic (undoped) region occurs between the P-doped and N-doped region.  Although PIN diodes are poor rectifiers, they are much better suited for high speed, high frequency applications due to the high level injection process. The wide intrinsic region provides a lowered capacitance at high frequencies. For photodetectors, the process is photon energy being absorbed into the depletion region, causing an electron hole pair to be created when the electron moves to a higher energy level (from valence to conduction band). This is what causes an electrical current to be created from light.

Photodetectors are “photoconductive”. That is, conductivity changes with applied light. Like amplifiers and other devices, photodetectors have “Figures of Merit” which signify characteristics of the device. These will be briefly examined

Quantum Efficiency

Quantum efficiency refers to the number of carriers generated per photon. It is normally denoted by η. It can also be stated as carrier flux/incident photon flux. Sometimes anti-reflection coatings are applied to photodetectors to increase QE.


Responsivity is closely related to the QE (quantum efficiency). The units are amperes/watt. It can also be known as “input-out gain” of any photosensitive or detective device. For amplifiers this is known as “gain”. Responsivity can be increased by maximizing the quantum efficiency.

Response Time

This is the time required for the photodiode to increase its output from 10% to 90% of final output level.

Noise Equivalent power

This value corresponds to units of Watts/sqrt(Hz). It is another measure of sensitivity of the device in terms of power that gives a signal to noise ratio of one hertz per output bandwidth, Small NEP is due to increased sensitivity of the device.

Carrier Recombination

Carrier recombination is an effect in which electrons and holes (carriers) interract with each other in a way in which both particles are eliminated. The energy given off in this process is related to the difference between the energy of the initial and final state of the electron that is moved during this process. Recombination can be stimulated by temperature changes, exposure to light or electric fields. Radiative recombination occurs when a photon is emitted in the process. Non-radiative recombination occurs when a phonon (quanta of lattice vibrations) is given off rather than a photon. A special case known as “Auger recombination” causes kinetic energy to be transferred to another electron.


Band to band recombination occurs when an electron moves from one band to another. In thermal equilibrium, the carrier generation rate is equal to the recombination rate. This type of recombination is dependent on carrier density. In a direct bandgap material, this will radiate a photon.

An atom of a different type of defect in the material can form “traps” which can contain one electron when the particle falls into it. Essentially, trap assisted recombination is a two step transitional process as opposed to the one step band to band transition. This is sometimes known as R-G center recombination. A two step recombination is known as “Shockley Read Hall” recombination. This is typically indirect recombinaton, which emits lattice vibrations rather than light.

The final type is Auger Recombination caused by collisions. These collisions between carriers transfer motional energy to another particle. One of the main reasons why this is distinct from the other two types is that this transfer of energy also causes a change in the recombination rate. Like the previous type, this tends to be non radiative.

A distinction should be made for band-to-band recombination between stimulated and spontaneous emission. Spontaneous emission is not started by a photon, but rather due to temperature or some other means (sometimes called luminescence). As stated in a previous post, stimulated emission is what emits coherent light in lasers, however spontaneous emission is responsible for most light emission in general.

Rayleigh Scattering

Rayleigh scattering is an effect of the scattering of light or electromagnetic radiation by particles much smaller in size than the wavelength. For example, when sunlight emits photons which enter the earth’s atmosphere, scattering occurs. The average wavelength for sunlight is around 500nm, which is in the visible light spectrum. However, it is known that the sunlight also emits Infrared waves and of course, ultraviolet radition. Interestingly enough, Rayleigh scattering influences the color of the sky due to diffuse sky radiation.

The reason why a huge wavelength (compare 400 nm with nitrogen and oxygen molecules which are only hundreds of picometers) can scatter on a small particle is because of electromagnetic interractions. When the nitrogen/oxygen molecules vibrate at a certain frequency, the photons interract and vibrate at the same frequency. The molecule essential absorbs and reradiates the energy, scattering it. Because the horizontal direction is the primary direction of vibration, the air scatters the sunlight. The polarization is dependent on the direction of the incoming sunlight. The intensity is proportional to the inverse of the wavelength to the fourth power. The shorter the wavelength, the more scattering. This can explain why the sky is blue because blue is more likely scattered by Raleigh scattering due to higher frequency (smaller wavelength). It is not dark blue because other wavelengths are also scattered, but much less so.


Rayleigh Scattering is quite important in optical fibers. Because the silica glass have microscopic differences in the refractive index within the material, Rayleigh scattering occurs which leads to losses. The following coefficient determines the scattering.


The equation shows that the scattering coefficient is proportional to isothermal compressibility (β), photoelastic coeffecient, the refractive index  as well as fictive Temperatue and is inversely proportional to the wavelength.

Rayleigh scattering accounts for 96% of attenuation in optical fibers. In a perfectly pure fiber, this would not occur. The scattering centers are typically atoms or molecules, so in comparison to the wavelength they are quite small. The Rayleigh scattering sets the lower limit for propagation loss. In low loss fibers, the attenuation is close to the Rayleigh scattering level, such as in Silica Fibers optimized for long distance propagation.

The Electronic Oscillator

The semiconductor laser is a device that can be compared to an electronic oscillator. An oscillator can be thought of as a resonator (a circuit that resonates or produces a strong output at a specific frequency) with gain. Resonators naturally decay over time by some factor, so adding in gain (so long as the gain is greater than or equal to the loss) can allow the resonator to become an oscillator that does not decay or dampen.

The stimulation of the oscillations of an oscillator is caused by electronic noise. A block diagram can demonstrate an oscillator in an abstract, easier to understand way.


The oscillator is built using an amplifier (transistor that is biased into active/saturation region) or op amp with positive and negative feedback. Noise in the circuit begins the oscillation, and this output is fed back into the input and is filtered along the way. This becomes an oscillation at a single frequency.

Oscillators can be built from RC circuits, LC circuits or can be crystal oscillators. RC circuit oscillators tend to be lower frequency oscillators in the audio range. The LC oscillator is often compared to the laser in terms of functionality. The negative reactance of the capacitor and positive inductive reactance cancel at a specific frequency, leaving the circuit with only resistance and a strong current is achieved. LC oscillators are much more important for RF/microwave purposes. A crystal oscillator produces its frequency through mechanical vibrations and has a much higher Q factor than the other resonator types, which provides greater temperature and frequency stability.

Two very important oscillator types for RF/microwave/mmWave circuits are dielectric resonators and SAW (surface acoustic wave) resonators. Dielectric resonators are mainly used as mmWave oscillators to drive antennas. They are generally made of a “puck” of ceramic which oscillates at a certain frequency dependent on its dimensions. Waves are confined inside the material due to an abrupt change in the permittivity. When the waves inside interfere and produce a standing wave, this increase of amplitude creates the resonance effect. SAW resonators are often used in cell phones and have distinct advantages over the LC oscillator or other types due to cost and size.

In a semiconductor laser (laser diode), the source of oscillations is the noise generated by spontaneous emission. Spontaneous emission is the result of recombination of electron and hole pairs within the material which produces photons. This spontaneous emission is how lasers begin their operation, and this is continued by stimulated emission. Stimulated emission is electron hole recombination due to photon energy which also produces a photon. The light emitted by this type of emission is coherent, a characteristic of a laser.

Pseudomorphic HEMT

The Pseudomorphic HEMT makes up the majority of High Electron Mobility Transistors, so it is important to discuss this typology. The pHEMT differentiates itself in many ways including its increased mobility and distinct Quantum well shape. The basic idea is to create a lattice mismatch in the heterostructure.

A standard HEMT is a field effect transistor formed through a heterostructure rather than PN junctions. This means that the HEMT is made up of compound semiconductors instead of traditional silicon FETs (MOSFET). The heterojunction is formed when two different materials with different band gaps between valence and conduction bands are combined to form a heterojunction. GaAs (with a band gap of 1.42eV) and AlGaAs (with a band gap of 1.42 to 2.16eV) is a common combination. One advantage that this typology has is that the lattice constant is almost independent of the material composition (fractions of each element represented in the material). An important distinction between the MESFET and the HEMT is that for the HEMT, a triangular potential well is formed which reduces Coloumb Scattering effects. Also, the MESFET modulates the thickness of the inversion layer while keeping the density of charge carriers constant. With the HEMT, the opposite is true. Ideally, the two compound semiconductors grown together have the same or almost similar lattice constants to mitigate the effects of discontinuities. The lattice constant refers to the spacing between the atoms of the material.

However, the pseudomorphic HEMT purposely violates this rule by using an extremely thin layer of one material which stretches over the other. For example, InGaAs can be combined with AlGaAs to form a pseudomorphic HEMT. A huge advantage of the pseudomorphic typology is that there is much greater flexibility when choosing materials. This provides double the maximum density of the 2D electron gas (2DEG). As previously mentioned, the field mobility also increases. The image below illustrates the band diagram of this pHEMT. As shown, the discontinuity between the bandgaps of InGaAs and AlGaAs is greater than between AlGaAs and GaAs. This is what leads to the higher carrier density as well as increased output conductance. This provides the device with higher gain and high current for more power when compared to traditional HEMT.


The 2DEG is confined in the InGaAs channel, shown below. Pulse doping is generally utilized in place of uniform doping to reduce the effects of parasitic current. To increase the discontinuity Ec, higher Indium concentrations can be used which requires that the layer be thinner. The Indium content tends to be around 15-25% to increase the density of the 2DEG.


Basic Energy Band Theory

Band theory is essential in the study of solid state physics. The basic idea tends to center around two bands: the conduction and valence band (for reasons discussed later on). Between the two bands is a forbidden energy level (Energy gap) which depends on the resistivity or conductance of the material. In order to fully understand solid state devices such as transistors or solar cells, this must be discussed.

For a single atom, electrons occupy discrete energy levels called bands. When two atoms join together to form a diatomic element (such as Hydrogen), their orbitals overlap. The Pauli Exclusion Principle states that no two electrons can have the same quantum numbers. Now keep in mind that there are four types of quantum numbers. This means that when these two atoms combine the atomic orbitals must split to compensate so that no two electrons have the same energy. However for a macroscopic piece of a solid, the number of atoms is quite high (on the power of 10^22) and therefore the number of energy levels is also high. For this reason, adjacent energy levels are almost continuous, forming an energy band. The main bands under consideration are the valence (outermost band involved in chemical bonding) and conduction because the inner electron bands are so narrow. Band gaps or “forbidden zones” are leftover energy levels that are not covered by a band.

In order to apply band theory to a solid, the medium must be homogeneous or evenly distributed. The size of material must be considerable as well, which is not unreasonable considering the number of atoms in an appreciable piece of a solid. The assumption also must include that electrons do not interract with phonons or photons.

The “density of states” is a function that describes the number of states per unit volume, per unit energy. It is represented by a Probability Density function.

A Fermi-Dirac distribution function demonstrates the probability of a state of energy being filled with an electron. The probability is given below.


The μ is generally expressed as EF which is the Fermi energy level or total chemical potential. kT is the familiar thermal energy which is the product of the Boltzmann constant and the temperature. From this equation it is clear that absolute zero temperature, the exponential term increases to infinity, causing the entire term to trend to zero. This leads to the conclusion that semiconductors behave as insulators at 0K.

The density of electrons can be calculated by multiplying this value with the density of states function and integrating over all energy.

Band-gap engineering is the process of changing a material’s band gap. This is usually done to semiconductors by changing the composition of alloys in the material.

Object Oriented Programming and C#: Dictionaries/Hash Tables

A “dictionary” in C# is a ADT (Abstract data type) that maps “keys” to “values”. Normally with an array, the values within this collection of data are accessed using indexing. For the dictionary, instead of indexes there are keys. Another name for a dictionary is a “hash table”, although the distinction can be made in the sense that the hash table is a non-generic type and the dictionary is of generic type. The namespace required for dictionaries is the “System.Collections.Generics” namespace.

The dictionary is initialized much like a list (dynamic array), however the dictionary take two parameters (“TKey”,”TValue”). The first is the data type of the key and the second the data type of the value. Similarly to dynamic arrays, values can be added to the dictionary using a “Add(key,value)” command. Similarly, a value can be deleted using the “Delete(key)” command. However it is important to note that keys do not have to be integers, unlike an index. They can be of any data type imaginable. However, a dictionary cannot contain duplicate keys.

The functionality of a dictionary in C# is similar to a physical dictionary. A dictionary contains words and their definitions and analogously, a programming dictionary maps a key (word) to a value (definition).

The following program illustrates adding values to a dictionary. The key is of type integer and the value of type string. The values “one”, “two” and “three” are added with corresponding integer keys.


Much like with arrays, a “foreach” statement can be used to iterate over all the values of a dictionary.


It is important to note for a hash table, the relationship between the key and its value as that this must be one to one. When different keys have the same hash value, a “collision” occurs. In order to resolve the collision, a link list must be created in order to chain elements to a single location.

An important concept with hash table: speed of processing does not depend on size. For arrays, in order to find a specific value a linear search must be performed. This takes a long time to complete if the array is very long. With a hash table, size does not matter because the hashing function is a constant time. The “ContainsKey()” method can be used to find a specific key without the need for a linear search.

When would you use a dictionary/hash table over a list? Dictionaries can be helpful in instances where indexes have special meaning. A particular use of a dictionary could be to count the words in a text using the “String.Split()” method and adding each word to the dictionary. In this instance, the “foreach” statement could easily be used to iterate over every value and find the number of words. In short, the dictionary maps meaningful keys to values whereas the list simply maps indexes to values.

The Half Wave Dipole Antenna

The dipole is a type of linear antenna which commonly features two monopole antennas of a quarter wavelength in size bent at 90 degree angles to each other. Another common size for the dipole is 1.25λ. These sizes will be discussed later.

It is important for beginning the study of the dipole antenna to discuss the infinitesimal dipole. This is the dipole which is smaller than 1/50 of the wavelength and is also known as a Hertzian dipole. This is an idealized component which does not exist, although it can serve as an approximation to large antennas which can be broken into smaller segments. The mathematics behind this can be found in “Antenna theory:Analysis and Design” by Constantine Balanis.

More importantly, three regions of radiation can be defined: the far field (where the radiation pattern is constant – this is where the radiation pattern is calculated), the reactive near field and the radiative near field.


As shown in the image, the reactive near field is when the range is less than the wavelength divided by 2π or when the range is less than 1/6 of the wavelength. The electric and magnetic fields in this region are 90 degrees out of phase and do not radiate. It is known that the E and H fields must be in phase to propagate. The radiating near field is where the range is between 1/6 of the wavelength and the value 2D^2 divided by the wavelength. This is also known as the Fresnel zone. Although the radiation pattern is not fully formed, propagating waves exist in this region. For the far field, r must be much, much greater than λ/2π.

The radiating patterns of the dipole antenna is pictured below, with both the E and H planes. The E plane (elevation angle pattern) is pictured on the bottom right and the H plane (Azimuthal angle) beside it on the left. The plots are given in dB scale. The radiation patterns can be understood by considering a pen. While facing the pen you can see the full length of the pen, but if you look down on the pen you can only see the tip or end. This is analogous to the dipole antenna where maximum radiation is broadside to the antenna and minimum radiation on the ends, leading to the figure 8 radiation pattern. When this radiation pattern in extended to three dimensions, the top left image is derived.



Object Oriented and C#: Quadratic Roots Program

The following program is designed to accept three doubles as inputs and prints the roots of a quadratic, whether complex or real. If a non-double is inputted into the program, the program should display “Bad Input”. The program contains two files: a “program” file to run several of the main methods and a “complex” file which creates the class for handling complex numbers and overrides the built in “ToString” method.

The first goal is to initialize the part of the program that handles real roots. The easiest portion is to create a method that reads doubles. It is important that the method contains a nullable type because the method should return null if a non-double such as a string is put into the method. This provides an easy way to use a conditional statement upon using the “Tryparse” method. The “Tryparse” method returns a boolean value of true or false. The “if” statement checks if the return is true and if so, returns the result. If not, null is returned.


Next, the “getquadraticstring” method is implemented to format the printed result in the form “AX^2+BX+C”. This is also done within the “program” file. Format specifiers are put within the placeholders to set the printed values to two decimal places if neccessary.


The “getrealroots” method produces the roots of the quadratic given that they are purely real. First the discriminant (the part in the quadratic formula under the square root symbol) is calculated. Several if statements are provided to check how many real roots there are and returns that quantity as an integer. For example, if the discriminant is negative, there will be no real roots returned. This means both of the “out” variables should be set to null and the function should return a 0. For a discriminant = 0, the quadratic formula reduces to -B/2A and the second root should be null. The return value is again the number of roots (1). It is important to note that an “if-else” statement must only end in “else” rather than “else if”. The “else” statement must cover all other possibilities.


Within the “main” function, three numbers are taken from the console using the getDouble method. An integer value is obtained from the getRealroots method which states the number of roots. This will be used for the conditional statements. For ease of reading code, a string variable is created to store the return from the “getQuadraticString” method.

Next, an “if” statement is used to print a bad input if any of the a, b, c variables are null. A return statement is included within the “if” statement so that an else does not have to be provided. This will exit the statement after it has completed.


Now the logic for the imaginary numbers must be implemented. The default constructor is shown with default inputs of zero. It doesn’t need any code within it because it inherits the Complex constructor. The “ToString()” method must be overridden because the formatting must be changed to adhere to complex numbers.


In addition, logic must be implemented for the “getImaginaryRoots()” method. The discriminant is calculated the same way as before, however the absolute value is taken. The real part must be calculated separately and the denominator is split for this reason. For clarification, this is the real part of a complex root. The two roots are the same, but complex conjugates.


The “main” function must be updated to reflect the imaginary roots.


The “getQuadraticString()” method is updated as shown. Three pieces of string must be created with several conditions imposed. They begin as empty strings and are filled in. Separating them into parts lets the logic be implemented for when each coefficient is 1 or -1. When C is zero, an empty string will be printed.



E-K Diagrams

As previously concluded, solids can be characterized based on energy band diagrams. A conductor has a valence and conduction bands that are very close or overlap. In addition a conductor will have a completely filled valence band and an almost full conduction band. The “forbidden region of the conductor is very small and little energy is required for an electron to move from conduction to valence band. In the presence of an external field, it is very easy for electrons to move from the valence band to the conduction band.

For semiconductors, at absolute zero the valence band is also completely full and the bandgap is typically about 1eV to 3eV, however even a bandgap of .1eV could be considered a semiconductor. Therefore, a semiconductor at 0K is an insulator. Semiconductors are very temperature sensitive. The subsequent figure illustrates the temperature dependence. The resistivity is very high at absolute zero, making the semiconductor behave like an insulator. However at higher temperatures the semiconductor can become quite conductive. At room temperature (300k), the semiconductor behaves more like a conductor.


With band diagrams, not much information is given therefore it is necessary to also analyze an E-K (Energy momentum) diagram. E is the energy require for an electron to traverse the bandgap. For example in Silicon with a bandgap of 1.1eV, it would take an energy level of 1.1eV for an electron to move from conduction to valence band. Energy is given as E = kT where T is a given temperature.

For intrinsic semiconductors like Silicon, the structure is crystalline and periodic. The wavefunction (which describes probability of finding an electron) should therefore be of periodic nature (sinusoidal). From the Schrodinger equation, it can be found that the Energy is periodic with k as well. For the diagrams, E is plotted against k.


The borders of the first Brillouin zone are from -π/a to π/a. These are cells of the crystalline lattice. Since the wavefunction is periodic, we only care about one of the zones. The above figure can be considered the “reduced zone” figure. Sometimes the x axis is given as the moment or wavenumber, since these only differ by a factor of Planck’s constant. From this diagram: the bandgap energy is shown, the effective mass of electrons and holes are shown as well as the density of states. The effective mass is shown by the curvature of the bands. For example, a heavy hole band could be found by observing the diagram that is less curved. From the above diagram, it is also noticeable that the material is direction bandgap (such as GaAs). The basic energy gap diagram compares to the E-k diagram in that the maximums and minimums correspond. However, the original band gap diagram does not give any other characteristics. It is for this reason the E-k diagram is so useful.

The Radar Range Equation

To derive the RADAR range equation, it is first necessary to define the power density at a distance from an isotropic radiator. An isotropic radiator is a fictional antenna that radiates equally in all directions (azimuthal and elevation angle accounted for). The power density (in watts/sq meter) is given as:


However, of course RADARs are not going to be isotropic, but rather directional. The power density for this can be taken directly from the isotropic radiator with an additional scaling factor (antenna gain). This simply means that the power is concentrated into a smaller surface area of the sphere. To review, gain is directivity scaled by antenna efficiency. This means that gain accounts for attenuation and loss as it travels through the input port of the antenna to where it is radiated into the atmosphere.


To determine the received power to a target, this value can be scaled by another value known as RCS (RADAR Cross section) which has units of square meters. The RCS of a target is dependent on three main parameters: interception, reflection and directivity. The RCS is a function of target viewing angle and therefore is not a constant. So in short, the RCS is a unit that describes how much from the target is reflected from the target, how much is intercepted by the target as well as how much as directed back towards the receiver. An invisible stealth target would have an RCS that is zero. So in order to determined received power, the incident power density is scaled by the RCS:


The power density back at the receiver can then be calculated from the received power, resulting in the range being to the fourth power. This means that if the range of the radar to target is doubled, the received power is reduced by 12 dB (a factor of 16). When this number is scaled by Antenna effective area, the power received at the radar can be found. However it is customary to replace this effective area (which is less than actual area due to losses) with a receive gain term:




The symbol η represents antenna, and is coefficient between 0 and 1. It is important to note that the RCS value (σ) is an average RCS value, since as discussed RCS is not a constant. For a monostatic radar, the two gain terms can be replaced by a G^2 term because the receive and transmitted gain tends to be the same, especially for mechanically scanned array antennas.