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ARRL Examination Study (Part I)

The ARRL (American Radio Relay League) is an organization for amateur radio enthusiasts. In order to communicate using HAM radio, at least a technician license must be obtained. The following post is meant as a useful information guide for those wishing to obtain a license.

The ARRL provides a complete manual as a study reference for HAMs. The book is divided into nine chapters: Basic info about ARRL, Radio and Signals, Circuit components, propagation and antennas, Amateur radio equipment, HAM communication, License regulation, operating regulation and safety. The questions come directly from each chapter (35 total, 26 to pass).


For Radio and Signal fundamentals, it is important to know basic properties of waves including wavelength, speed of propagation, the relation between wavelength and frequency, identifying frequency bands, the frequency ranges of various bands used by HAMs and so forth. The fundamental equation for propagation of waves is c = fλ. Because radio waves are being transmitted by antennas through air, the speed of propagation is 300 million meters/sec. This is a constant value and therefore if frequency is increased, the wavelength decreases proportionally. This speed value is roughly equivalent to the speed of light in a vacuum. The property of radio waves used to identify different frequency bands is wavelength. HAMs tend to use the frequencies occupied by bands MF through UHF. It is important to know the frequency ranges of these bands.


In this section, it is important to know prefixes for the SI unit system, so conversions between various values can be made. The following table should be committed to memory.


The next section deals with modulation, which is a necessary function to transmit the correct signal to receiver. It is important not to set a transmit frequency to be at the edge of any band to allow for transmitter frequency drift, allow for calibration error, and so that modulation sidebands do not extend beyond the band edge. It is important to know about FM deviation (which is dependent on amplitude of the modulating signal) and that if the deviation is increased, the signal occupies more bandwidth. Setting a microphone gain too high could cause the FM signal to interfere with nearby stations. It is important to know the types of AM modulation (Double Sideband, Single Sideband, etc) and which modulation technique is best for various frequency bands. “Continuous wave” (Morse code-esque) modulation occupies the lowest bandwidth, followed by SSB modulation. The various advantages to certain modulation techniques should be understood. For example, SSB is preferential to FM because it occupies less bandwidth and has longer range. The bandwidth for each modulation technique is shown below.


The final section of Chapter two deals with radio equipment basics. A repeater should be understood to be a station that retransmits a signal onto another channel. The following is an image of a transceiver, which transmits and receives RF signals using a TR switch to switch between each function. A repeater uses a duplexer in place of this switch to transmit and receive simultaneously.


The Human Ear

The Human ear is important to the study of acoustics because it is inborn pressure sensor. It is one of the most sensitive parts of the human body and its job is to sense pressure changes in air and convert these to electrical signals that the brain can process as “sound”. Humans can hear roughly between 20 Hz to 20 kHz but this range decreases with age. The human ear can sense sound intensities from 1 W/sqm to 1 trillionth of a W/sqm. What most people intuitively perceive as music loudness, pitch and timbre roughly corresponds to amplitude (or sound intensity, which is proportional to the square of amplitude), frequency and waveform shape. Of course, these are not one to one relationships because if a tone is too high in frequency (ultrasound) or too low (infrasound) it will effect the perceived loudness because it will not be heard at all, for example.

The human ear consists of three main parts: inner ear, middle ear and outer ear. The outer ear consists of the pinna, auditory canal and eardrum. The pinna (the only visible part of the ear) serves as a guide to guide pressure waves into the ear canal. The ear canal is filled with air which is necessary because sound needs a medium such as air to transmit pressure waves. The waves reach the conically shaped eardrum, which vibrates and sends signals to the brain to process.


The middle ear consists of several dense bones (ossicles) called the hammer, anvil and stirrup. These are elastically connected and serve to transmit and amplify sound from the outer to inner ear. These bones are necessary because the pressure waves are being transferred to a different medium (air to ear fluid called endolymph) and require an impedance matching network to transmit sound effectively. This is not unlike the soundboard of a guitar (for impedance matching to air) or an electrical impedance matching network design for maximum power transfer from a source to a load.

The inner ear contains the cochlea and the semicircular canals. The cochlea contains thousands of tiny hair cells that are stimulated by the vibrations of sound. The semicircular canals contribute to our sense of balance, but not the sensation of hearing. The inner ear fluid causes the hairs in the cochlea to bend, which are converted to electrical pulses and sent to the brain. These are sent to the auditory nerve and are interpreted as sound.

The following diagram depicts the human ear as a passive electrical circuit using the “impedance analogy”. The eardrum middle ear section is shown to be a transformer to match the outer ear to the middle ear. There could also be another transformer between the middle ear and the cochlea, as stated before. Without going into excruciating detail, it is important to show that the human ear is not all different from an electrical circuit in the sense that it impedance matches and transforms/transduces different forms of energy.



Quarter Wave Transformer Matching – Using Theory of Multiple Reflections

There are two ways to derive an impedance value for a quarter wave transformer line. The transformer is an excellent tool to match a characteristic impedance to a purely resistive load where a large bandwidth is not required. It is much easier to find this relationship by examining it from an impedance viewpoint, however the theory of multiple reflections is an excellent topic because it illustrates the contribution of multiple impedance lines to the overall reflection coefficient.

The following circuit with the matching transformer is shown below.


The addition of the matching transformer introduces discontinuity at the first port. Ideally, the addition of the transformer will match the load resistance to Zo, minimizing all reflection, as will be shown. the bottom figure provides a “step by step” analysis of each trip of the wave as it travels. When the wave first hits the Zo and Z1 junction, it sees Z1 as a “load” and does not yet see the actual load resistance. Depending on the impedance match, some of the wave will be reflected and some will be transmitted. The transmitted part of the wave then travels to the load and a portion is again reflected with reflection coefficient 3. As that portion of the wave travels back to the Z1 and Zo junction, the process repeats. This process continues infinitely and results in the following equation.


Using the definition of a geometric series and writing the reflection coefficient in terms of impedance, the equation reduces to


The reflection is seen to reduce to zero when Z1 (the impedance of the quarter wave section of transmission line) is set to q.PNG

Distributed Antenna Systems

Distributed antenna systems (DAS) provide a convenient, power efficient way to move RF signals within buildings (iDAS or indoor DAS) or in outdoor places such as stadiums or venues (oDAS or outdoor DAS). A DAS consists of two main parts: a signal source and a distribution system. The signal source can be an outdoor antenna or a local base station.


The principle idea is this: to replace a single, high power antenna with several power efficient antennas without losing any area coverage. There are many types of distribution systems. The main idea for each system is to propagate the signals in such as way that maximum power and signal coverage is utilized. In one architecture, a master unit is connected to the base station using RF coaxial cable. The frequency of this energy is then increased to the optical range and carried using fiber optic cables to each remote unit on each floor of the building. The energy is then reduced down to the RF frequency and passive splitters are used to distribute the signals to each antenna. Architectures can be purely passive, purely active or a hybrid combination of both. DAS is advantageous for increased coverage but can result in higher costs due to increased infrastructure required.

Sound – ADSR Envelope

In terms of sound and music, the ADSR envelope describes how sound changes over time. In terms of physics of wave, a general envelope outlines the extreme points (max and mins) of a wave through a smooth curve.

It is obvious to the human ear that when a musical instrument is played, its volume (amplitude) changes over time. For example when a guitar string is plucked, the string vibrates and the initial amplitude is high. After a brief period, the sound amplitude decays. Different musical instruments will have different ADSR envelopes to describe their sound characteristics.

The “Attack” phase of the sound refers to how quickly a sound reaches its maximum amplitude. This is the initial phase of the sound. For most instruments, this period is extremely short (almost instantaneous). The next phase is the “Decay” phase, or the time the note takes to drop to the sustain level. The “sustain” level is generally the longest portion of time, and this is when the amplitude envelope stays relatively constant. The “release” is the period of time the sound takes to go from the sustain level to zero amplitude and is generally short.


Random Variables, PDFs and CDFs

The concept of probability is very important in the field of electrical engineering, where outcomes can be nondeterministic. In a nondeterministic outcome, an experiment can be repeated multiple times and have different outcomes. For example, in a communication link messages may not be delivered the same way each time. Another example would be the failure of manufactured parts even within a scheduled lifetime.

A random variable is defined as a function that maps a real number to an outcome within a sample space (a set that contains all possible outcomes of a random experiment). The “real number” is sometimes called an observation. The range includes all the possible observations and the domain includes all of the possible outcomes of the experiment. A single random variable produces a single observation. Random variables are notated by capital letters generally towards the end of the alphabet (eg. X(a), U(a), etc).


A useful function for describing probability of random variables is the Cumulative Distribution function (CDF). This function is defined as the probability that the random variable is less than a certain value, x.


Setting x = infinity shows that the CDF should equal to one. This is equivalent to the probability of the sure event (which is always one because it is the probability of the entire sample space). Setting x = 0 gives a probability of 0 (probability of the null event is always zero). The value of the CDF must always be between these two values and must never decrease. The CDF can be discontinous (in the case of discrete random variables) as well as continuous (in the case of continuous random variables).

Finding the probability between two values is easily obtained from the CDF.


Taking the derivative of the CDF leads to the PDF (probability density function).


This shows that the PDF and CDF are inverse functions. Alternatively, the PDF can be defined as


This is because the area under the PDF is what determines probability. The total integration of the PDF must equal one.


Electric permittivity is an extremely important concept in electromagnetics. This is a material parameter also known as “distributed capacitance” as evidenced by its units [Farads/meter]. The absolute permittivity (∈) is used in the calculation of the capacitance of a parallel plate capacitor and is inherent to a material. The relative permittivity (∈_r) is the ratio of the absolute permittivity of a material to the permittivity of a vacuum (8.85E-12 F/m). This is also known as the dielectric constant.

A good way to understand permittivity is to consider two conductive plates separated by a distance with an equal and opposite amount of charge applied. As you can probably guess, a static electric field exists between the plates due to the charge, since the charges are separated by nonconducting medium.


As the figure demonstrates, a dielectric material will polarize itself and create a field opposing the external field applied. This is because even though the molecules of a dielectric are mostly stationary due to their lattice structure, they can rearrange a bit due to an externally applied field. In addition, dielectrics can become conducting if a large enough field is applied (dielectric breakdown). The rearranging of these molecules reduces the overall field and increases the “distributed capacitance”.

It is important to know that for normal materials, the electric permittivity is generally a complex number, because permittivity is dependent on the frequency of the field applied. This is because the polarization of a dielectric cannot happen instantaneously due to “causality” (a system’s response depends on past or present inputs, not on future inputs). Permittivity can also be affected by temperature and humidity. The complex permittivity equation can be written as


It is seen that the imaginary part of the equation depends on frequency and accounts for conductivity. The response of the material to a static (DC) field is found be decreasing the frequency to zero. The high frequency limit is found by increasing the frequency.

It is important to distinguish between dielectric constant and dielectric strength. Dielectric strength is the ability of the material to resist dielectric breakdown (units V/mil). A high dielectric breakdown means that a high voltage can be applied before the dielectric conducts appreciable current.

Electroacoustic Transducers

An electroacoustic transducer converts energy from electrical to mechanical. Transducers in general convert energy from various forms. For example a “piezoelectric” (stress electric) transducer converts a mechanical force into a voltage. These transducers can be used to convert speech or music signals into electrical signals for processing or to serve as measuring instruments for acoustic quantities. A transducer can be modeled as a two port network relating electrical and mechanical properties. All of these values are RMS (effective) values.

Various electrical quantities can be transformed into mechanical quantities:




Capacitance<—>Inverse of stiffness

Resistance<—>Mechanical Resistance or Damping

Transducers can be reciprocal or nonreciprocal. Crystal or ceramic electroacoustic transducers are considered reciprocal (Transduction coeffecients for electrical and mechanical are equivalent).

Two major types of electroacoustic transducers are the electromagnetic variety (which use the principle of Faraday’s Law of Induction to stimulate charge flow) or electrostatic transducers which store charge on capacitive plates which then vibrate to create changes in pressure. They can be further classified as “active” (not requiring external power) or “passive”. There are resistive, inductive, capacitive or light dependent transducers. Transducers can be characterized by their frequency response or directivity pattern. For example, a microphone with a circular directive pattern would be seen as “omnidirectional” or equal amplification in all directions. The following is the frequency response of a microphone that appears very flat. This means the microphone is very neutral and does not amplify certain frequencies more than others.


Displacement Current

One of Maxwell’s equations, Ampere’s circuit law, tells us that there are two sources of magnetic fields: conduction currents and displacement currents. Conduction current is very familiar to most people: it is flow of electrons through a conductor due to an applied electric field. The electrons hop from atom to atom within the conductor and rate at which this happens is termed displacement current.

The differential form of the equation also shows another source of magnetic field: displacement current. For a static field (not time varying/DC), there is no displacement current and the Ampere equation is


However for time varying fields, the right hand side contains an extra term, which is displacement current density.


“J” is the conduction current density which is equivalent to conductivity multiplied to E field (also known as Ohm’s Law in point form). Taking the surface integral of the second term on the right hand side yields displacement current.

The important distinction here is that displacement current is not due to the flow of electrons directly, but rather a time varying electric field. A common example is that of a capacitor with an AC voltage source applied to the device. While there is no conduction current flowing through the dielectric which separates the plates, there is still a current through the capacitor (displacement current) .


The following image shows two surfaces about a capacitor with an AC voltage applied. If Ampere’s law is applied to surface one, the right hand side is equal to the conduction current flowing in the wire. However, if the law is applied to surface two it demonstrates that no conduction flows through the capacitor. The same closed path is used in the integration (L) and therefore the right hand side cannot be zero. This means a new term for displacement current must be inserted to satisfy the equation. This is demonstrated in the equation below the figure.



Reflection and Transmission of Sound

Similar to electromagnetics, sound waves that are incident upon a medium with different properties will experience reflection, transmission/refraction or absorption depending on multiple factors. The analysis of the transmission and reflection of sound is greatly simplified when the boundary between media and the incident wave are planar. The amount of transmission and reflection depends on each material’s acoustic or characteristic impedance (r = p*c) and the angle that the incident wave makes with the boundary.

Much like in electromagnetics, transmission and reflection and reflection coefficients can be defined as shown.


The figure shows that the transmission coefficient is the ratio of the transmitted pressure wave to the incident pressure wave. The reflection coefficient is the ratio of the reflected pressure wave to the incident pressure wave. This is similar to the voltage reflection coefficient from transmission line theory.


The angle of the sound wave with the boundary can either be “normal” (at a 90 degree angle) or “oblique”. For normal incidence, the problems are greatly simplified, as there is no refracted wave, only a reflected component and a transmitted component. These are described by plane wave equations:




The propagation vector (k) is dependent on the material. The transmitted wave has a different propagation vector value because it has surpassed the boundary into the second medium. This is due to the fact that a material’s characteristic impedance is dependent on the speed of sound.


In order for boundary conditions to be satisfied, both the normal component of velocity and pressure must be continuous. This means the acoustic pressure on both sides of the boundary must be equal, leaving no net force on the planar boundary separating the fluids. The fluids must also remain in contact, meaning the normal component of the velocity must be continuous.


These equations can be used to derive the reflection and transmission coefficients.

It is important to note that the reflection coefficient is always real. When the second medium (the medium the transmitted wave propagates into) has a greater acoustic impedance than the first, the reflection coefficient is positive. This makes sense because when a sound pressure wave comes in contact with a rigid boundary, sound echoing occurs. When r2 is much greater than r1, this defines the rigid boundary condition. When r1 is much greater than r2, the boundary is termed “pressure release” and there is an 180 degree phase shift between incident and reflected wave.


Having an incident angle other than 90 degrees complicates the solution process a bit. This is termed “oblique incidence”. The pressure equations become a bit more complex.


The angles made by an obliquely incident wave are shown below.


Applying the same continuity of pressure boundary condition from before leads to Snell’s law.


The critical angle can be defined as


This angle, as well as the comparison of speeds of the materials determines the bending of the refracted wave.


Introduction to Acoustic Waveguides/Cavities

Waveguides are of great importance to both electromagnetics (for example, guiding microwaves into the cooking chamber of a microwave oven to cook food) and the world of acoustics. Waveguides are hollow tubes that guide waves by reflecting them. Without waveguides, waves propagate spherically and decay with range. The waveguides restricts the propagation of the wave to one dimension. These devices can be rectangular or circular in shape. A major area of importance when studying waveguides is boundary conditions. For acoustic waveguides, the boundary conditions are governed by the linearized force equation:


This equation shows that the spacial derivative (gradient) of the pressure (a scalar field) is proportional to density and acceleration. The equation can be used to show that at a rigid boundary where the pressure is maximum, the velocity must be minimum. Calculus tells us that taking the derivative of a function and setting it to zero (and hence the velocity in this case) will yield maximum values. The opposite is also true for a “pressure release” boundary.

We will first consider the case of a rectangular boundaried cavity. A cavity is similar to a waveguide, however the dimensions of a cavity are comparable to each other whereas a waveguide will have one direction that is much longer than the others (to propagate the waves). Applying boundary conditions, it is apparent that if all boundaries are rigid, only standing waves can be contained within the cavity. A pressure equation can be derived from the boundary conditions through substitution into the wave equation. The wave equation will always be satisfied for any kind of wave, including pressure waves. The following equation is substituted into the wave equation:


This results in an equation of three sinusoids in each direction. In this case, the boundary conditions result in cosine (which is maximal at zero (x=0,y=0,z=0)).


Then, from the boundary conditions it must be true that these cosine functions be equal to zero at the boundaries x =Lx, y = Ly, z = Lz.


This leads to solving for the cutoff frequency of the cavity. The wave number (k) is defined as w/c. Solving for w leads to


For waveguides, it is best for the frequency of propagation to be much higher than the cutoff for a decrease in waveguide dispersion. Frequencies below cutoff produce evanescent waves, or waves that die off without propagating.

Simple Harmonic Oscillator

Any spring that obeys Hooke’s Law is appropriately named a “simple harmonic oscillator”. This law of physics describes the behavior of a mass-spring system disturbed from its equilibrium by pulling or pushing on the mass in such a way that the mass experiences a “restoring force” described by Hooke’s Law,  F = -sx where “F” is the restoring force, “s” is the spring constant (units – Newtons/meter) and “x” is the displacement of the mass. If x is positive, this refers to stretching the spring, and if x is negative this refers to compression. The spring constant determines how easily the spring is deformed (also known as stiffness). The negative sign refers to the fact that the force opposes the force applied (it is a restoring force), which aligns with Newton’s Third Law of Motion.

Another important Newtonian equation is F = ma which states that force is proportional to both an object’s mass and acceleration. If these two equations are brought together, using the relation that acceleration is the second derivative of position, a second order linear differential equation can be formed.


It is seen that the solution to this undamped mechanical system is sinusoidal in nature. This makes intuitive sense, because sinusoids are proportional to their second derivative, meaning if a sine or cosine is plugged into the differential equation, it can be shown to be a solution. The system’s “natural frequency” is obtained by equating the potential and kinetic energy of the system. This is intuitively satisfying because the kinetic energy dominates the system at frequencies below the resonance/undamped natural frequency and above this frequency, potential energy dominates.

For damped system, the equation becomes slightly more complex as a new force must be considered (the product of mechanical resistance and acceleration). This is a much more realistic approach because in a real mass spring system, air creates a frictional force effect on the spring as it oscillates, causing the oscillations to die out as mechanical/motional energy is lost to heat. The solution to this new differential equation contains a decaying exponential term.


Three cases of damping are shown above: critical damping, overdamping and underdamping. These cases depend upon whether the resonance/undamped natural frequency of the system is equal to the real part of gamma (called temporal absorption coefficient) in which case the system would be critically damped. An underdamped system would occur when the absorption coefficient is lower than the resonance frequency and the opposite would be considered an overdamped system. As shown above, critically damped systems have no oscillations because the imaginary part of gamma is zero (no reactance). Underdamped systems experiences oscillations that decay to zero over time. Overdamped systems decay to equilibrium without oscillating, but not as quickly as critically damped systems.