Author Archives: jalves61

The Superheterodyne Receiver

“Heterodyning” is a commonly used term in the design of RF wireless communication systems. It the process of using a local oscillator of a frequency close to an input signal in order to produce a lower frequency signal on the output which is the difference in the two frequencies. It is contrasted with “homodyning” which uses the same frequency for the local oscillator and the input. In a superhet receiver, the RF input and the local oscillator are easily tunable whereas the ouput IF (intermediate frequency) is fixed.

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After the antenna, the front end of the receiver comprises of a band select filter and a LNA (low noise amplifier). This is needed because the electrical output of the antenna is often as small as a few microvolts and needs to be amplified, but not in a way that leads to a higher Noise Figure. The typical superhet NF should be around 8-10 dB. Then the signal is frequency multiplied or heterodyned with the local oscillator. In the frequency domain, this corresponds to a shift in frequency. The next filter is the channel select filter which has a higher Quality factor than the band select filter for enhanced selectivity.

For the filtering, the local oscillator can either be fixed or variable for downconversion to the baseband IF. If it is variable, a variable capacitor or a tuning diode is used. The local oscillator can be higher or lower in frequency than the desired frequency resulting from the heterodyning (high side or low side injection).

A common issue in the superhet receiver is image frequency, which needs to be suppressed by the initial filter to prevent interference. Often multiple mixer stages are used (called multiple conversion) to overcome the image issue. The image frequencies are given below.

image

Higher IF frequencies tend to be better at suppressing image as demonstrated in the term 2f_IF. The level of attenuation (in dB) of a receiver to image is given in the Image Rejection Ratio (the ratio of the output of the receiver from a signal at the received frequency, to its output for an equal strength signal at the image frequency.

RADAR Range Resolution

Before delving into the topic of pulse compression, it is necessary to briefly discuss the advantages of pulse RADAR over CW RADAR. The main difference between the two is with duty cycle (time high vs total time). For CW RADARs this is 100% and pulse RADARs are typically much lower. The efficiency of this comes with the fact that the scattered signal can be observed when the signal is low, making it much more clear. With CW RADARs (which are much less common then pulse RADARs), since the transmitter is constantly transmitting, the return signal must be read over the transmitted signal. In all cases, the return signal is weaker than the transmitter signals due to absorption by the target. This leads to difficulties with continuous wave RADAR.  Pulse RADARs can also provide high peak power without increasing average power, leading to greater efficiency.

“Pulse Compression” is a signal processing technique that tries to take the advantages of pulse RADAR and mitigate its disadvantages. The major dilemma is that accuracy of RADAR is dependent on pulse width. For instance, if you send out a short pulse you can illuminate the target with a small amount of energy. However the range resolution is increased. The digital processing of pulse compression grants the best of both worlds: having a high range resolution and also illuminate the target with greater energy. This is done using Linear Frequency Modulation or “Chirp modulation”, illustrated below.

290px-Linear-chirp.svg

As shown above, the frequency gradually increases with time (x axis).

A “matched filter” is a processing technique to optimize the SNR, which outputs a compressed pulse.

Range resolution can be calculated as follows:

Resolution = (C*T)/2

Where T is the pulse time or width.

With greater range resolution, a RADAR can detect two objects that are very close. As shown this is easier to do with a longer pulse, unless pulse compression is achieved.

It can also be demonstrated that range resolution is proportional to bandwidth:

Resolution = c/2B

So this means that RADARs with higher frequencies (which tend to have higher bandwidth), greater resolution can also be achieved.

 

 

Mathematical Formulation for Antennas: Radiation Integrals and Auxiliary Potentials

This short paper will attempt to clarify some useful mathematical tools for antenna analysis that seem overly “mathematical” but can aid in understanding antenna theory. A solid background in Maxwell’s equations and vector calculus would be helpful.

Two sources will be introduced: The Electric and Magnetic sources (E and M respectively). These will be integrated to obtain either an electric and magnetic field directly or integrated to obtain a Vector potential, which is then differentiated to obtain the E and H fields. We will use A for magnetic vector potential and F for electric vector potential.

Using Gauss’ laws (first two equations) for a source free region:

cfr

And also the identity:

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It can be shown that:

2

In the case of the magnetic field in response to the magnetic vector potential (A). This is done by equating the divergence of B with the divergence of the curl of A, which both equal zero. The same can be done from Gauss Law of electricity (1st equation) and the divergence of the curl of F.

Using Maxwell’s equations (not necessary to know how) the following can be derived:

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For total fields, the two auxiliary potentials can be summed. In the case of the Electric field this leads to:

4

The following integrals can be used to solve for the vector potentials, if the current densities are known:

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For some cases, the volume integral is reduced to a surface or line integral.

An important note: most antenna calculations and also the above integrals are independent of distance, and therefore are done in the far field (region greater than 2D^2/λ, where D is the largest dimension of the antenna).

The familiar duality theorem from Fourier Transform properties can be applied in a similar way to Maxwell’s equations, as shown.

mxw

In the chart, Faraday’s Law, Ampere’s Law, Helmholtz equations and the above mentioned integrals are shown. To be perfectly honest, I think the top right equation is wrong. I believe is should have permittivity rather than permeability.

Another important antenna property is reciprocity… that is the receive and transmit radiation patterns are the same , given that the medium of propagation is linear and isotropic. This can be compared to the reciprocity theorem of circuits, meaning that a volt meter and source can be interchanged if a constant current or voltage source is used and the circuit components are linear, bilateral and discrete elements.

 

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.

fuck

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.

unnamed

 

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:

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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.

spec

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.