## Kleine Geschichte: Das Land ohne Wege

„Der Krieg dauerte so lange, dass er nicht mehr hoffen konnte, das Vaterland wiede zusehen.“

Es war 1943, der kälteste Winter in den letzten vier Jahren. Herr Müller und seine Kollegen–Herr Fischer, Herr Abrecht und Herr Schneider—schliefen bei einer Feuer auf der Wiese in Frankreich.

Gestern kamen sie aus der Schweiz nach Lyon. Der Weg war schlammig, weil es in der Nacht regnete, aber nach Herr Müllers Meinung war es besser als die Wege nach Paris. Alle Wege waren von Zürich nach Paris wegen der Bombardierung seit Juli zerstört geworden. Die Ingenieure wollten innerhalb drei Wochen diese Wege reparieren und sie—die Panzerführer von Panzer-Gebirgs-Division Nr. 31—hatten ein bisschen frei Zeit vor ihrem Urlaub ans Meer des Normandys zu reisen.

Die Feuer war nicht so hell aber sie war warm. Auf der Feuer kocht Herr Müller eine Eisflasche mit wenig grünem Tee. Seine Wache war von zwölf Uhr bis drei Uhr, dann war Herr Fischer der letzte Mann des Wachdiensts.

Er glaubt, dass niemand so dumm sein kann, alleine gegen einen Panzer vorzugehen, aber Herr Fischer glaubt, dass man die verzwitfelte Franzosen unterschätzen nicht konnte. Die Franzosen überfielen deshalb immer in Gruppen und in der Dunkelheit, erklärte Fischer. Deshalb musste man immer die Anderen aufwecken, wenn er etwas hörte und man sollte auf keinen Fall über fünf meter von der Feuer wegging. Das war eine Regel im Krieg, wobei der nichtsahnende Einzelkämpfer tot gewesen wäre. Aber, Herr Müller war achtlos, weil niemand seit Monaten einen einzigen französischen Kämpfer in Lyon finden konnte.

Um ein Uhr hörte er einen Geräusch von kleinen Kieseln, die getreten wurde. Mit seiner Luger in der Hand schrie er in der Dunkelheit. „Wer is da?“ fragt er.

Es gab keine Antwort aber er glaubte, dass er jemanden in einer Fremdsprache hatte sagen hören.

Dann kam ein arisches Bauernmädchen mit graunen Augen und schönen Haaren an, welches bayerishen Akzent hat. „Grüß Gott!“ sie grüßte.

„Hmph! Gute Nacht. Wer sind Sie und woher kommen Sie?“

„Ich heiße Julia, eine Ärztin in der neben Stadt. Hier ist mein Ausweis. Ach! Können Sie bitte die Pistole auf mich nicht richten?“

„Sehr gut. Und was machen Sie hier am Ende der Welt?“

„Am Ende der Welt“, sie gluckste und erklärte: „Ich bin mit meinem Bruder zusammen, er stellte auch in der Stadt auf. Heute besuchte ich einen Patienten in Béligneux, sie war eine Kleinstadt östlich gelegen von hier. Mein Motorrad war kaputt bei einem Unfall. Bitte können Sie mir hilfen?“

„Dort!“ zeigte sie auf der Dunkelheit.

„Wo?“ fragte er nochmals.

Nach der Frage kannte er nicht mehr die ganzen Einzelheiten. Er wurde auf den Kopf eingeschlagen. Wenn er aufwachte, war es am Morgen und er saß in einem fremden Zimmer mit einer einzigen eisen Tür.  Er wurde nicht gebunden oder er hast seine Pistole verloren. Seiner Kopf tat ihn weh und seine Kollegen wäre nirgendwo zu sehen.

Rückblickend sollte er die Regel besser gefolgt haben.

„Verdammt! Wo bin ich hier…?“ stöhnte er und öffnete er den eisen Tür.

Dann sah er einen ganzen Kleinestadt wurde verschandelt in dem Kreig; wenige Gebäude bestehen blieb. Einen Junge mit einem Gewehr, das war höher als der Junge, zielte ihn auf. „Julia“, schrie der Junge und dann auf Französisch sag er etwas mehr, den Herr Müller nicht verstehen könnte.

Julia lief aus anderen Gebäude und nochmals grüßte Herr Müller „Grüß Gott!“ in ihren bayerishen Akzent.

„Verdammt Gott!“ ächzte er. „Wo sind meine Freunde?“ fragt er.

„Mein herzliches Beileid. Ihren Freunde sind tot. Glücklicherweise habe ich Ihren Teeflasche hier.“

Sie gab ihm seinen Flasche, sie mit Blut rot färben wurde.

Er warf den Flasche hin und shrie „Was haben Sie gemacht!?“

„Was macht richtig“ sie antwortete.

Des Mädchens Behauptung zufolge, es gab einen Angriff von Frankreichs-Befreiungskämpfern—die Liberté—nachdem er ohnmächtig geworden war. Alle in seiner Gruppe hatten gefallen im Einsatz und er wurde erbeutet und nach Béligneux gebracht. Sie war keine Kämpferin von Dritte Reich noch Liberté. Sie war eine Ärztin der Menschens. Sie arbeitet in dem „Krankenhaus“, das durch Bombardierung teilweise verstört geworden, und im Moment diese Franzosen wäre ihren Patienten.

Herr Müller, sie sag, war auch einer Patient.

Aber nachdem er den blutroten Flasche gesehen hat, hörte Herr Müller keine Ausrede mehr. Er fiel das Mädchen an und sobald er tat, fund er sich gerungen nieder wurde, des Junges Gewehr auf seinen Kopf.

Fortsetzung folgt…

# Short story: The land without roads

“The war has been raging on for so long that he could not hope to see homeland ever again.”

It was 1943, the coldest winter in the last four years. Mr. Müller and his colleagues—Mr. Fischer, Mr. Abrecht and Mr. Schneider—was sleeping next to a fire, in a plain in France.

They arrived in Lyon from Switzerland yesterday. The road was muddy due to a rain last night, however, Mr. Müller surmised it was still better than the roads to Paris. All the roads from Zürich to Paris had been destroyed in July’s bombing. The engineers wanted at least three weeks to repair these roads and they—the tank crew of the mountain-ranger-tank-division no. 31—had a bit of free time for a vacation on the beach of Normandy.

The fire was not very bright, but it was warm. Mr. Müller cooked a bottle of ice with some green tea on this fire. His watch started from 12AM to 3AM, then it was Mr. Fischer who took the last shift.

He supposed nobody could be dumb enough to attack a tank alone but Mr. Fischer argued that one could not underestimate the desperate Frenchmen. They knew, therefore, they always attacked in group and in the dark, Mr. Fischer explained. That was why one must always wake up the others should he heard something, and he should, under no circumstances, go more than five meters away from the fire. That was the rule of war; the unsuspecting lone soldier was a dead soldier. Though, Mr. Müller remained indifferent since no one had seen any French soldiers in Lyon for many months.

At one o’ clock, he heard the noise of pebbles being stepped on. Armed with a Luger, he shouted in the darkness. “Who’s there?” he asked.

There was no answer but he thought that he heard someone speaking in a foreign language.

Then, an Aryan peasant girl with grey eyes and fair hair appeared, “God bless!”, she greeted in a distinct Bavarian accent.

“Hmph! Good night. Who are you and where did you come from?”

“My name is Julia, a doctor in the nearby city. Here’s my license. Ah! Can you please not point that pistol at me?”

“Very good. And what are you doing here in the middle of nowhere?”

“In the middle of nowhere,” she chuckled and explained: “I was with my brother, who also stationed in the city. Today, I visited a patient in Béligneux. It’s a small town east of here. There was an accident and my motorbike broke down. Can you help me?”

“There,” she gestured into the darkness.

Afterward, he could not recall the whole experience. He was struck in the head and when he came to, it was morning and he found himself sitting in a strange room with only one iron door. He was not bound but he had lost his pistol. His head hurt. His colleagues were nowhere to be seen.

In retrospect, he should have followed the rule better.

“Dammit! Where am I…?” he groaned and opened the iron door.

Then, he saw the entirety of a small town devastated by the war; very few buildings were left standing. A youngster aimed a rifle which was taller than the youngster himself at him. “Julia!” the youngster shouted and then spoke something Mr. Müller could not understand in French.

Julia ran out of another building and once again greeted Mr. Müller “God bless” in her Bavarian accent.

“Damn God!” he grunted, “Where are my friends,” he asked.

She gave him his bottle, which was stained red in blood.

He knocked away the bottle and shouted “What have you done!?”

According to the girl, there was an ambush by the France’s liberation fighters—the Liberté—took place after he lost consciousness.  Everyone in his group was killed in action and he was captured and brought to Béligneux. She herself was no soldier for neither the Third Reich or the Liberté. She was a doctor of the people. She worked in the “hospital”, a building half destroyed in bombing, and at the time these French were her patients.

Mr. Müller, she said, was also a patient.

Nevertheless, having seen the blood-red bottle, Mr. Müller could not hear any more excuses. He attacked the girl and, as soon as he did, found himself wrestled to the ground, the young man’s rifle at his head.

To be continued…

## Memo: AC displacement sensors

Resistive sensors, in general, can operate both DC and AC excitation. The frequency characteristics, the output encoding and bridge’s components can vary but the operating principles remain the same. Other sensors that take advantage of capacitive, inductive and magnetic phenomena, however, only accept AC excitation.

##### Capacitive displacement sensor

Capacitative displacement sensors are passive sensors operating solely on AC power supply. They tend to have higher precision than resistive and inductive sensors, though, they don’t work as well in a dirty environment. They are capable of converting physical changes to capacitance difference.

Capacitance is an electrical property created by applying electrical charge (AC current) to two conductive objects (plates) with a gap between them. The magnitude of capacitance can be measured using an AC bridge called Wien bridge as described in the previous memo. This capacitance depends on the area of the plates, the dielectric constant of the gap and the distance between the plates.

$C = \frac{\epsilon_0 K A}{d}$

Where

C is the capacitance

ε0 is the permittivity of free space constant

K is the dielectric constant of the material in the gap

A is the area of the plates

d is the distance between the plates.

Capacitance displacement sensor ties the physical displacement to one of the three factors in the formula: K, A, and d. Using two plates, there are three possible configurations as follow:

Using three plates, three more configurations can be formed:

Configuration C’s are useful as a mean of measuring the thickness of a non-conductive material in a nondestructive manner. Configuration A’s, on the other hand, can measure displacement without physical contact.

Configuration B’s, in particular, are widely used in touchscreens. See the video below for more information:

##### Inductive displacement sensor / Eddy-current sensor

Inductance is the capacitance’s magnetic equivalence in AC systems. It stores energy a magnetic field instead of an electric field, and thus, on its own, has more noise tolerance than capacitance does. However, inductance suffers a multitude of problems when facing magnetic fields from other sources due to mutual inductance effect (as mentioned in the previous memo on AC circuits).

Mutual inductance, on the other hand, is the core principle behind transformers (differential inductive devices). Transformers are not only useful in power engineering but they also see applications in sensor technologies; in particular, as Linear variable differential transformers (LVDT).

The majority of inductive displacement sensors, however, operate on the Eddy current induced by a lone magnetic field. These inexpensive and cruddy Eddy-current sensors are what the electronics shops usually called “inductive displacement sensors”.

Eddy currents are loops of electrical current induced within conductors by a changing magnetic field nearby. These currents produce an opposing magnetic field (via Faraday’s law of induction, with the direction dictated by Lenz’s law) that resists the change in the external magnetic field, similar to how Peltier effect resists the change in thermal gradient caused by Seebeck effect.

The working principle of an Eddy-current sensor is described by Lion Precision Co. as follow:

“Rather than electric fields, eddy-current sensors use magnetic fields to sense the distance to the target. Sensing begins by passing an alternating current through the sensing coil. This creates an alternating magnetic field around the coil. When this alternating magnetic field interacts with the conductive target, it induces a current in the target material called an eddy current. This eddy current produces its own magnetic field which opposes the sensing coil’s field.

As the eddy currents in the target oppose the sensing field, the impedance of the sensing coil will change. The amount of impedance change is dependent on the distance between the target and the sensing coil in the probe. Current flow in the sensing coil, which is impedance dependent, is processed to create the output voltage which is an indication of the position of the target relative to the probe.”

Compared to conductive sensors, Eddy-current sensors are inexpensive, contactless and extremely durable devices. At the core, they still rely on a secondary impedance sensor to interpret the change in position so they lack the precision conductive sensors have. On top of that, since they use a magnetic field, they are vulnerable to interferences from other magnetic sources.

Finally, the involvement of magnetic field, electric field, and AC impedance brings the lesser known proximity effect into play in the secondary impedance sensor’s circuit. As a result of the proximity effect and Joule heating, the mathematical model of these “simple” sensors are not so easy to formulate. Experimentation and regression techniques are often required to calibrate these devices.

##### Linear variable differential transformer (LVDT)

LVDT is one kind of inductive displacement sensor that makes use of mutual inductance for measuring. Its physical configuration is identical to differential capacitive transducer’s configuration A described above.

The linear relationship between the voltage across the primary coil A and that across the secondary coils B involves complex mathematics. It is often easier to derive the proportional coefficient K from the number of loops in the coils or from experimentation than from mathematical formula.

$U_B = U_A\cdot K \cdot x$

Like capacitive displacement sensors, LVDTs are absolute measurement devices. This means unlike photodiode-based encoder wheels when LVDTs are restarted they will show the same measurement as before without any information losses. Without any physical contact, the LVDT produces no wear and tear and this also allows it to be completely sealed against the environment.

Another advantage of LVDT is its single axis sensitivity. Essentially, LVDTs are immune to positional shift in any direction other than the one it is measuring. In addition, the full range output is typically a large signal (a volt or more), allowing it to forgo amplifications, while its infinitesimally small resolution (being an analog device and all) allows it to measure minute changes accurately.

The primary drawback of LVDT compared to its capacitive counterparts is response time. Being a transformer circuit, the response time of LVDT is tied to the frequency of its AC source. Since normal AC sources operate in the utility frequency range of 50-60Hz, this fact limits LVDT to low-frequency applications. This flaw can be rectified using a frequency synthesizer (based on PLL circuit, as described in a previous memo) that can generate high-frequency AC sources.

https://en.wikipedia.org/wiki/Capacitive_displacement_sensor

http://www.lionprecision.com/capacitive-sensors/

https://en.wikipedia.org/wiki/Inductance

http://www.lionprecision.com/eddy-current-sensors/index.html

https://en.wikipedia.org/wiki/Eddy_current

https://en.wikipedia.org/wiki/Lenz%27s_law

http://www.lionprecision.com/tech-library/technotes/article-0011-cve.html

https://en.wikipedia.org/wiki/Proximity_effect_(electromagnetism)

https://en.wikipedia.org/wiki/Linear_variable_differential_transformer

https://www.quora.com/Does-LVDT-work-on-a-DC-power-supply-and-what-is-the-use-of-IC-555-here-What-is-its-function

http://www.te.com/usa-en/industries/sensor-solutions/insights/lvdt-tutorial.html

##### For general sensor selection guidance, see

http://www.machinedesign.com/sensors/finding-right-sensor-linear-displacement

https://www.embedded.com/print/4384454

## End-of-year blog update

Dear frequent patrons,

As you might have noticed, there have been some changes to the blog’s content in recent time and I would like to make a sort of report and keep you up to date on what I’m up to lately.

##### Introducing the Memos

The most striking new addition to this blog is “the Memos” series, featuring lecture notes for my current post-grad study in sensor systems technology. Prior to the memos, I had published a few technical documents concerning programming and engineering problems I tackled. These early DIY memos have been of great help to me in more than one instance and I hope some of you will find them useful too. The memos and all previous documents are available under the new “Engineering” category:

https://fujihita.wordpress.com/category/engineering/

I still have a number of displacement sensor topics I would like to cover and a number more on control theory topics but it will take a while to write those. Researches have to be done and all to ensure the most accurate information possible. When it comes to technical researches, it takes time, lots of time.

##### Reviewing past stories

Pertaining my writing endeavors, I have given up the fourth revision of White Destiny novel and shifted my focus elsewhere. I read the fourth revision with a fresh perspective after five months and I found myself cringing at the latter half. It went downhill from the scene in the Alchemist’s quarter onward. There was no direction, everything was reactive and arbitrary; in short, I don’t like it at all.

At long last, I figure I need a change of pace, I need to start small; walking first before running. Early short story experiments revealed multiple issues with my storytelling in general. I’ll just write here the things I learned from all these short stories.

Short story: “Cat”, “Pumpkin”, “Headphone” was a test story using abrupt scene transition. This “cinematic cut” transition style enjoy considerable success in fanfictions for whatever reasons I’m not aware of. I wrote it also as a test for Mystery / Crime Fiction genre, though, it became clear halfway through that I have next to zero talent in Mystery genre. Here, I learned to never use abrupt transitions again.

In my second piece, Short story: “Swimming pool”, “Cicada”, “Airplane”, I played with onomatopoeia, Survival and (subtle) Supernatural genres. I particularly like this one, all the way to the climax. However, the resolution was cheesy and a bit over the top for my taste. It was 4AM in the morning when I wrote it and I was under time pressure. I will definitely rewrite the ending sometimes. It’ll be done, definitely!

Short story: “Rain”, “Bottle”, “Bookstore” focused on creating the atmosphere at the start and delved into Realism / Slice of Life genres. It went well at first but then it broke down in the middle, with the whole tragic backstory as the cringey cheese trap I somehow have not learned to avoid.

After the epic fail third story, I spent a good long break and picked up a few classic books from the store. They served me well in writing Short story: “Lawyer”, “A.I.”, “Wikipedia”. I personally feel this piece marked the end of random, dramatic and illogical cliche shenanigans that have been plaguing my writing for so long. Here, I learned to focus more on the characters’ portrayals.

This focus on characters’ portrayals continued in Short story: “Death”, “Hat”, “Gene” where the cast was composed of only two characters. I wrote this part as a prologue to the book “Sasaki” I am writing so the story might be a bit hard to follow with lots of loose ends and all. Overall, as far as the goal of practicing writing character depth goes, it was a success.

And the latest Short story: Halloween special was just an attempt on Lovecraftian horror with the wordiness from “The life and adventures of Robinson Crusoe”. No doubt it is hard to write and hard to read but the number of details can be packed in that story is astonishing. Perhaps, I’ll need to find a balance between my normal easy-to-read style and the classical literature style in my future works.

##### Let’s write German!

My next short story, “Das Land ohne Wege”, which will be published (hopefully) on Saturday will be written in German.

Yes, you read it right, in German, auf Deutsch!

In the plot, it will be about a World War II German tank driver who was captured by the French resistance and discovered the Allies’ scheme to land in Normandy. In function, it will be a cross between “the memo” and “short stories”. What I mean is that…since I’m still a novice in the German language so these “Kleine Geschichte” will no doubt be nonsensical to the native speakers, whom I know some of you are, and I’m telling you this so as you know what’s in store on Saturday.

My only intention is to learn the language via storytelling, as I did with English. Therefore, if any of you spot mistakes (grammar, word choice or whatever) or if any parts sound unnatural to you, please don’t hesitate to drop a comment and let me know.

Danke schön!

## Memo: AC excitation

All passive sensors discussed so far rely on DC excitation. DC has its charm in simplicity, low cost and fluid integration into existing digital technologies. However, when it comes to analog technologies, AC excitation remains a cornerstone of high-performance circuits employing inductive or capacitive elements.

For the most parts, the working principles of passive sensors are the same in either cases; their encoding might differ but the theory of operation should be the same. As such, AC sensors can use the same conditioning circuitry as their DC counterparts, plus some extra to deal with the additional information from frequency domain.

##### AC excitation characteristic

High tolerance to common resistive effects like corrosion and wear.

Immunity to electrical noise as information can be stored in frequency domain.

Immunity to thermoelectric effects.

Can use high-performance transformer, capacitive and inductive circuits.

Low power losses to self-heating as only the real part of the AC is responsible for heat dissipation.

Vulnerable to phase shifting and race conditions.

Problems with ground leakage / stray capacitance.

Complex interface with existing digital technologies, requiring rectification and rescaling to match the legacy DC signal.

##### AC bridges

Applying AC excitation to Wheatstone bridge circuits changes the quantity being measured from resistance to impedance. Impedance is a complex quantity containing resistance, inductance and capacitance as well. It wouldn’t hurt to revise the conversion of these DC values to AC complex values as follow:

In a nutshell, the operation of AC Wheatstone is similar to DC Wheatstone; an unknown quantity is weighed against known, adjustable quantities. The condition for a balanced bridge in AC domain is:

In other words, both the resistance and the phase factors must match on either arms for the bridge to be balanced. For measuring inductance and capacitance, a few bridge circuits can be used. The simplest and most straightforward bridge is the symmetrical bridge as seen below:

This symmetrical bridge does the trick illustrating the basic idea of impedance bridges but it is hardly appropriate in practice. In the real world, capacitors and inductors possess internal resistance and this must be properly addressed.

Wien bridge takes into account of the capacitor’s internal resistance, which is in series with the capacitive component, and counters it with a parallel resistor-capacitor pair as seen below:

The bridge is balanced when both conditions below are satisfied:

As Wien bridge’s frequency (or that of the AC source) can be calculated from

it is possible to use Wien bridge for measuring frequency of an unknown signal source.

Another thing about the real world is that, it is incredibly difficult to manufacture variable inductors. However, variable capacitors are readily available and fortunately there exists a bridge called Maxwell bridge that permits measurement of inductance using variable capacitors.

The full condition for balance is as follow:

Maxwell bridge’s balance condition is independent of source frequency. This permits some tolerance to mixed frequency AC voltage source. On top of that, Maxwell bridge uses only one inductor and avoids mutual inductance issue which plagues symmetrical bridge using two inductors. Not that two capacitors don’t cause any mutual capacitance; they do, but shielding against electric field is much easier than against magnetic field so capacitors are preferred.

##### Phase shifter

Phase shifter is an important element of AC conditioning circuit. It allows (manual) phase matching of the input and output signal. Capacitive and inductive elements shift the phase by 90 and -90 degree respectively and they are the foundation of phase shifters.

Due to mutual inductance phenomenon as mentioned in Maxwell bridge’s discussion, RL phase shifters are possible though rarely employed in practice. For RC phase shifters, the phase angle is dependent on the value of R and C, and is calculated as follow:

Using phase shifters naturally introduces additional resistance, which in turn causes the signal to drift (off-set). This off-set should be compensated and hence phase shifters should be used before off-set compensation in the conditioning circuitry.

One important note is that, RC phase shifter is also a passive high pass filter and it will suppress low frequency signals; including the signal itself if it is low enough. This is one of the reasons why Phase-locked loop (PLL), albeit complicated in construction, is preferred in conditioning circuits over simple RC phase shifters for phase matching.

##### Lock-in amplifier

Using AC excitation, it is possible to encode information in both amplitude and frequency alteration. Because AC bridges are natural AM modulators, they produce an amplitude-modulated output signal of the carrier signal–the AC excitation–and the input signal–the impedance change due to physical changes. It is, therefore, imperative to demodulate the input signal from the output at a later stage for usage in a (possibly digital) control system.

One device that performs this demodulation (the term phase-sensitive detection is the same thing) is a lock-in amplifier. Essentially, a lock-in amplifier takes a modulated signal and a reference, carrier signal and attempt to extract the amplitude “shape” of the modulated signal at the reference’s frequency, reconstructing the sensor’s signal as a result. This is the description of its operation in time domain though it is a mere corollary of its frequency domain’s operation.

In frequency domain, the lock-in amplifier acts as a very narrow band-pass filter which locks in on a specific frequency (based on the reference) and suppresses all other frequencies. The output is a DC signal comprising of the signal’s amplitude at reference frequency and its phase shift from the reference.

This DC output can be used as a representation of the phase shift and as control signal in self-tuning circuits as seen in a previous memo. If reconstructing the sensor’s signal is of interest, this phase shift component can be rectified via a phase shifter beforehand so that the output is a linear representation of only the sensor’s signal (its amplitude, specifically). In other words:

Uout = 1/2 Vsig

when  = 0

The phase shifter can be an RC phase shifter as described in the previous section or, more commonly, a PLL circuit, which actively keeps the reference signal in-phase with the input signal without further intervention. The reference signal can be taken directly from AC excitation source (at the engineer’s discretion) or from an internal oscillator.

More advanced “dual-phase demodulation” lock-in amplifiers use a fixed 90 degree phase shifter to generate two components X and Y for mathematical calculation of both the phase shift and amplitude quantities of the signal separately.

The in-phase component X from input signal and unchanged reference signal:

The quadrature component Y from input signal and phase shifted reference signal:

The magnitude of the signal at reference frequency:

And the phase shift of the signal from the reference signal:

http://www.sensorsmag.com/components/solving-tough-strain-gauge-problems

http://www.play-hookey.com/ac_theory/randr/wheatstone_ac_apps.html

https://en.wikipedia.org/wiki/Alternating_current

https://en.wikipedia.org/wiki/Maxwell_bridge

https://en.wikipedia.org/wiki/Wien_bridge

https://en.wikipedia.org/wiki/Inductance#Coupled_inductors_and_mutual_inductance

https://encyclopedia2.thefreedictionary.com/Phase-Shift+Circuit

http://www.electronics-tutorials.ws/oscillator/rc_oscillator.html

https://forums.ni.com/t5/Multisim-and-Ultiboard/Offset-problem-in-simulating-current-and-voltage-phase-relation/td-p/1194627

https://www.zhinst.com/applications/principles-of-lock-in-detection

http://www.phys.huji.ac.il/~greenwald/el_lab/labC/LockI.pdf

## Memo: Strain gauge bridge circuits

As mentioned in the previous memo, strain gauges are vulnerable to temperature effects such as thermoresistive, thermoelectric and self-heating. Some special alloys like Constantan are designed to be self-compensating while others aren’t. Those that aren’t will require special conditioning circuits utilizing one or more complimentary strain gauges in bridge topology. Three most common bridge circuits for strain gauges are quarter bridge, half bridge and full bridge.

All these bridge circuits take advantage of the fact that Wheatstone bridge is a differential device and temperature-dependent errors occur uniformly on all sensors in thermal contact with the specimen. This allows common-mode error compensation by positioning one or more identical gauges on the adjacent arm(s) to the active gauge.

Half bridge and full bridge circuits also provide increased sensitivity via measuring the negative strains incurred on the specimen due to Newton’s third law of motion (action and reaction) and Poisson’s effect. This negative strains are measured by the extra gauges and then manipulated to add to the positive strain being measured by the active gauge.

##### Quarter bridge strain gauge circuit

A quarter bridge circuit is simply a Wheatsone bridge hooked up to a stressed strain gauge. The basic quarter bridge naturally does not have any temperature compensation mechanics but this can be rectified with an identical, unstressed (adjacent) strain gauge in thermal contact with the active strain gauge. The unstressed, dummy gauge is kept in thermal contact with the active gauge but not bound to the specimen; hence Poisson-induced strains will have negligible effect on the dummy gauge.

One problem with quarter bridge circuit is that, even though there are now two strain gauges, only one of them is responding to mechanical strain.

##### Half bridge strain gauge circuit

Half bridge circuit also uses two strain gauges, however, the second strain gauge is bound to the specimen so that it may respond to mechanical strain. The first half bridge configuration has exactly the same positioning and circuitry as quarter bridge configuration. However, when the active gauge is stretched, the second gauge is compressed due to Poisson-induced strain and this improves sensitivity by a small amount while providing the same temperature compensation effect.

For bending strain measurement, a different, more sensitive configuration is often preferred. The second half bridge configuration changes the position of the second gauge so that it may measure the negative strain caused by Newton’s third law of motion (source from the fixture–or wall connected–point of the specimen). As this reactionary strain is much more prominent than Poisson-induced strain and it has the same magnitude as the positive strain, the half bridge circuit using this second configuration has twice the sensitivity of comparable quarter bridge.

The second half bridge configuration is unaffected by strains from Poisson’s effect and can only measure bending strain (up/down). When axial strain is applied here, both gauges will be stretched/compressed in the same manner and hence will cancel each other out, yielding zero output.

Both half bridge circuit configurations are temperature-compensated and have increased sensitivity. However, their outputs are only approximated and not linear to the actual strain. As explained by AllAboutCircuits.com’s textbook:

“Linearity, or proportionality, of these bridge circuits is best when the amount of resistance change due to applied force is very small compared to the nominal resistance of the gauge(s). With a full-bridge, however, the output voltage is directly proportional to applied force, with no approximation (provided that the change in resistance caused by the applied force is equal for all four strain gauges!)”

##### Full bridge strain gauge circuit

Full bridge circuit provides a directly proportional output to the actual strain by replacing all passive resistance with active gauges. Four gauges are bound to the specimen similar to half bridge circuit in sets of two. National Instruments’ white paper describes three full bridge configurations arise from sides and alignments combinations of the sensors’ placements on the specimen.

The mapping of gauges in Wheatstone bridge circuit to physical placements on the specimen vary from configuration to configuration and this is potentially a source of confusion. In general, they can be organized by the strain they measure. Take a look at the two diagrams below:

R2 and R4 will be measuring negative strains while R1 and R3 will be measuring positive strains.

The first full bridge configuration doubles down on the half bridge configuration II; two positive gauges on top and two negative gauges under. This doubles the sensitivity and provides true linearity on top of temperature compensation but can only measure bending strain.

The second and third full bridge configurations double down on half bridge configuration I; one set of adjacent positive and negative gauge on each side. Both configurations measure the positive strains and the Poisson-induced negative strains. They have lower sensitivity than full bridge configuration I and higher sensitivity than half bridge, as well as true linearity and temperature compensation of all full bridge circuits.

The alignments of the gauges are based on the kind of strain being measured.

For bending strain, the alignments of positive and negative gauges are swapped when crossing the neutral axis.

For axial strain, the alignments remain the same on both sides.

The only drawback of full bridge circuit is the number of strain gauges and wires required. For all their benefits, full bridge sensors are significantly more expensive than half bridge and quarter bridge sensors while uncompensated quarter bridge (using only one gauge) is the cheapest.

##### Bonus: Strain gauge torque sensor (rotary encoder)

Find below some self-explanatory illustrations for a strain gauge-based torque sensor.

Signal conditioning for load and torque sensors:

Excitation to power the Wheatstone bridge circuitry

Remote sensing to compensate for errors in excitation voltage from long lead wires

Amplification to increase measurement resolution and improve signal-to-noise ratio

Filtering to remove external, high-frequency noise

Offset nulling to balance the bridge to output 0 V when no strain is applied

Shunt calibration to verify the output of the bridge to a known, expected value

If DC excitation of Wheatstone bridge is bipolar (as it should), the direction of rotation is determined by the sign of the output.

If AC excitation is provided instead, the direction of rotation can be encoded in the peak-to-peak amplitude of the output; for example, 10VAC output range can have stationary output rated at 5VAC, fastest clockwise output at 10VAC and fastest counter-clockwise output at 0VAC. In this encoding, DC output can be obtained from AC output after passing the signal through a rectifier and Schmitt trigger circuit.

http://www.ni.com/white-paper/3642/en/

https://en.wikipedia.org/wiki/Strain_gauge#Variations_in_temperature

https://www.omega.co.uk/literature/transactions/volume3/strain2.html

http://www.continuummechanics.org/beambending.html

https://measurementsensors.honeywell.com/techresources/appnotes/Pages/Ways_to_Measure_the_Force_Acting_on_a_Rotating_Shaft.aspx

https://datum-electronics.co.uk/news/experts-in-torque-measurement/

http://www.machineryspaces.com/Torsionmeters.html

https://appmeas.co.uk/products/torque-sensors/

http://sine.ni.com/np/app/main/p/ap/daq/lang/en/pg/1/sn/n17:daq,n21:17557/fmid/6673/

## Memo: Resistive displacement sensors

Displacement sensors (aka position transducers) convert spatial information; linear distance, rotary position and deformation; into electrical quantities. One family of such sensors are called “encoders” ergo, devices that encode position into signal. Based on their coordinate system, encoders can be roughly classified into linear encoder (Cartesian coordinate) or rotary encoder (Polar coordinate) classes.

The most basic displacement sensors are resistive. They are measured and conditioned in the same way as resistive thermometers (with a Wheatstone bridge, instrumentation amplifier, low-pass filter, phase compensation, three-wire sensing and Kelvin’s sensing, etc.) but they have additional circuits to cope with temperature-induced errors.

##### Potentiometer

The simplest encoders are contact-based, and among them the most straightforward is potentiometer. Potentiometer is a voltage divider device used for converting spatial information to resistance, and in turn voltage. It’s a passive sensor requiring a  power supply to work. Its construction according to Wikipedia is as follow:

“Potentiometers consist of a resistive element, a sliding contact (wiper) that moves along the element, making good electrical contact with one part of it, electrical terminals at each end of the element, a mechanism that moves the wiper from one end to the other, and a housing containing the element and wiper.”

The wiper can be a rotating shaft or a linear slider. Depending on the construction, a potentiometer can be a rotary or linear encoder.

The relationship between position and resistance is, known as “taper”, is controlled by the manufacturer. There are two common tapers: linear and logarithmic. As the name suggested, linear taper potentiometers have a linear position-resistance relationship with the center position is usually 1/2 total R value. These find applications as encoders while logarithmic taper potentiometers are commonly used for audio amplifiers as human hearing is logarithmic.

Potentiometer can be used as it is, with the supply going into two outer terminals and the signal coming out of the middle terminal; or it can be used as a variable resistor (aka rheostat) in Wheatstone bridge implementation (though this configuration is rarely used as applications using potentiometer tend to value simplicity more than precision), using only one outer terminal and the middle terminal.

Assuming the potentiometer above is linear taper, the relation of position x of the wiper to the resistance is:

R2 = R * x

R1 = R * (1-x)

And the unloaded relation (RL >> R) of input and output voltage is given by

VL = R2/(R1 + R2) * VS

As such that it is simplified into:

VL = R*x/R * VS

VL = x * VS

For output of a loaded circuit with impedance RL, the formula is:

VL = x * VS / (1 + (x – x^2) * R/RL)

In reality, a voltage buffer is often used as conditioning circuit for potentiometer, in which cases the simpler, unloaded formula will be used.

Potentiometers, like most contact-based encoders, are slow, have small measurement surface, short measuring distance, varying degrees of accuracy (multi-turn potentiometers can be very accurate, single-turn ones…not so much) and low resistance to dust, oil and water. In addition, they are affected by friction, which can reduce the sensor’s lifespan among other annoyances (noise, heat, abrasion).

Trimmers are potentiometers that are rated for fewer adjustments over their lifetime. They are meant to be set once on installation by the technician and never to be seen or used by the user.

##### Strain gauge

Strain, ergo the displacement between particles in the body relative to a reference length (or deformation), can be measured by strain gauge. Strain gauge relies on piezoresistive effect and geometric deformation for its sensing. In a homogeneous metal, the resistance is given by its geometry and material as follow:

where

is the conductor’s length [m]

A is the cross-sectional area of the current flow [m²]

p is the specific resistance or resistivity of the material

The change in strain gauge’s resistance can then be expressed as:

dR/R = dp/p + dL/L – dA/A

When the strain gauge is tensed or compressed, piezoresistive effect changes the resistivity p of the material, causing the resistance to rise.

As in the case of thermoresistive effect, piezoresistive effect applies, in varying degrees, to both metals and semiconductors. The term “strain gauge effect” refers to unwanted piezoresistive behavior in non-strain gauge devices like thermometers; such devices are often designed in such a way that minimizes geometric changes during operation.

Likewise, strain gauges are designed to minimize thermoresistive effect and self-heating (Joule heating). The thin zigzag pattern improves heat dissipation (compared to a single thick trace) and the material (i.e. constantan alloy) is chosen such that the temperature effects on the resistance of the strain gauge itself cancel out the resistance change due to the thermal expansion. Ones made of such materials are called self-temperature-compensated strain gauges.

Those that are not self-compensating will need to be compensated during conditioning stage. Several strategies for temperature compensation are available and will be discussed in the next memo.

In constantan, a popular self-temperature-compensating material, piezoresistive effect contributes up to 20% of the resistance change. The rest of the resistance change is on account of geometric deformation, ergo, the change of the conductor’s length and cross-sectional area.

This geometric deformation follows a number of mechanical physics properties, namely: Poisson effect.

“Poisson effect is the phenomenon in which a material tends to expand in directions perpendicular to the direction of compression. Conversely, if the material is stretched rather than compressed, it usually tends to contract in the directions transverse to the direction of stretching”

The Poisson’s ratio v measures the magnitude of Poisson effect.

εT = ε * v

where

ε is the strain

εT is the trasverse strain, the strain caused by Poisson effect in perpendicular direction due to the original strain.

v is the Poisson’s ratio (normally in the range of 0 to 0.5)

The contribution of geometric deformation into Gauge factor, the ratio of relative change in electrical resistance R to mechanical strain ε, is given by:

and the full gauge factor with both geometric deformation and piezoresistive effect is:

where p is the resistivity.

For the exam, the key part of the above formula is:

GF = dR/(R * ε)

GF and R will be given by the datasheet. dR can be measured. With these values, the strain ε can be derived from the formula above. Inversely, dR can be calculated given the strain ε, R and GF. In these sort of problems, the strain ε might be given indirectly as displacement dL from the formula below:

ε = dL/L

When strain gauge is used to measure force instead of strain, the construct is called a load cell. A load cell usually consists of four strain gauges in a Wheatstone bridge configuration.

The change in force is linearly proportional to the change in strain in accordance to Hooke’s law and Young’s modulus. The formula for this physical property is given as follow:

where

E is Young’s modulus (material dependent) [Pa] or [N/m2] or [kg·m−1·s−2]

is the tensile stress

ε is the strain

At this point, it is also important to know the difference between stress and strain. Here’s a quick run down from an answer by Erik Carton at ResearchGate.net

Stress and strain do not have the same units at all. Stress is a force (N) per unit of area (m^2), hence it has the unit (N/m^2) = Pascal (Pa).

Strain is the elongation of a (stressed) material divided by the original length, hence m/m, which is unit-less and can be expressed in percentage (%).

The Young’s modulus provides the ratio between stress and strain (in the elastic regime): Pa/(m/m) = Pa  and therefore has the same unit as stress (or pressure).

So from stress, it is possible to determine the force given the area upon which the force was exerted. In practice, though, the force-to-voltage ratio is rarely modeled mathematically due to numerous complications (starting with the method to precisely measure the area upon which the force was exerted). The ratio is usually given by the datasheet or discovered via experimentation as a black box system.

One important note when using load cells: Hooke’s law is only valid when within the material’s elastic limit (reversible deformation). Beyond that limit and the response will no longer linear.

##### Bonus: Another way to calculate strain gauge’s resistance

For very small values, Poisson’s ratio can be approximated directly from length L and width (or normal of length) L’ as follow:

Using this, the change in resistance (the second formula in strain gauge section) can be simplified in term of Poisson’s ratio:

dR/R = (1 + 2v) dL/L + dp/p

This is one way to calculate strain gauge’s resistance from Poisson’s ratio and resistivity change but as resistivity change is hardly detectable, the formula above exists mainly in theoretical realm and is included here for the sake of completeness.

https://en.wikipedia.org/wiki/Linear_encoder

https://en.wikipedia.org/wiki/Rotary_encoder

https://www.keyence.com/ss/products/measure/measurement_library/basic/products_info/

https://en.wikipedia.org/wiki/Potentiometer

https://en.wikipedia.org/wiki/Trimmer_(electronics)

https://en.wikipedia.org/wiki/Strain_gauge

https://en.wikipedia.org/wiki/Deformation_(mechanics)#Strain

https://en.wikipedia.org/wiki/Piezoresistive_effect

https://en.wikipedia.org/wiki/Gauge_factor

https://en.wikipedia.org/wiki/Poisson%27s_ratio

https://en.wikipedia.org/wiki/Hooke%27s_law

https://en.wikipedia.org/wiki/Young%27s_modulus

https://www.engineeringtoolbox.com/stress-strain-d_950.html

https://www.researchgate.net/post/what_is_the_difference_between_pressure_and_strain

https://www.omega.co.uk/literature/transactions/volume3/strain2.html

## Short story: Halloween special

They ventured into the crypt: five militias, two knights, two priests, and one dog; torches in hands, provisions on backs, and a primal fright at heart. Leading the pack, the younger of the knights, Sir Rathamul brandished a sword and a spiked lantern shield. The oil lantern, a part of his shield’s gauntlet, emanated a dim orange light the radius of two meter–about two third the reach of his sword–accompanied by a sweet incense whose cost per pouch far surpassed the blue fur cap on his shoulders, and whose sole purpose was to veer away the stench of bat droppings.

Father Germini claimed the incense warded evil spirits and kept one sane by the grace of God; how much of this was a priest’s belief and how much was an apothecary’s consideration for the laymen, Sir Rathamul knew not. The priest knew his craft; that was certain. In this party, he was the only person who could speak Gondrash—the language of the dragons—and his healing arts could bring a man back from the brink of death. His young age was of no relevance to his wisdom, or so Sister Forse, who on the same day resorted to young age as an excuse for her inability to read, spoke about the man in white cassock, clutching a golden pectoral cross in his hands.

When it came to wisdom and age, however, no one in his homeland or that marble city of Azeth to the west could match Sir Winfried of Ursland in the south. The city of Ironheart had always produced the finest knights in the kingdom and Winfried was this generation’s most skilled red-helm guard. With his blessed tower shield, which was being strapped to his back for ease of travel, he could deflect arrows back to their shooters from fifty meter away.

In the end, only Sister Forse and three fifteen-year-old and two sixteen-year-old militias were childish enough to argue who, Germini or Winfried, was the wisest of them all. The children had grown attached to her since the day the chieftain commanded them to assist Sir Rathamul on his first quest as a blue-shield crusader. That was a mere fortnight ago, and yet their homeland—his homeland—the fishing village of Merlock, already seemed a distant memory.

Sister Forse, in a standard Azethan church’s black habit, would turn twelve this year. She was at the age where she could be courted by and married to nobles if not for her clergy duties. Sir Winfried opposed her coming with them; he saw her as a dead weight, a fine companion at the dining table but a dead weight in battle nevertheless. The one-meter-and-eight giant watched her slender back from the end of their human column, keeping a short half meter distance from the militia before him so that his torch would not drip oil on the boy’s leather helm. At times, he would nag her and her boys to keep quiet else they would wake the dead.

One last member of the party was Sister Forse’s dog: Dust. It was a large brown dog; large enough to carry Sister Forse on its back when she was younger, Father Germini said, but he could no longer. His mistress had outgrown him to the point he could be picked up in her arms and cuddled. This, Sir Rathamul later learned, was no exaggeration. Her work in Azethan church’s eatery apparently gave her the strength to pick up sacks of potatoes with one hand and the dexterity to outrun a chicken.

Inscribed along the crypt’s walls were oval glyphs indecipherable by either Father Germini or Sir Winfried. These black, obsidian glyphs gave off an apprehensive sensation when gazed upon and strange warmth when touched—in contrast to the cold haze permeating from the blizzard outside through the rocks and into this ancient sanctuary. Everyone was as tense as a harp’s string when Sir Winfried ordered one of the boys to inspect the glyphs. Reluctantly, the militia obliged and poked his spear at the wall, breathing a sigh of relief when nothing happened.

They split up into two groups. The first group consisted of Father Germini, Sir Rathamul and three militias traveled eastward along the wall. The second group; Sister Forse, Sir Winfried, two militias and the dog; went in the opposite direction, following the western wall. The knights with their shields and swords guarded their own respective group from frontal attacks, the militias with bronze spears shortened for indoor fighting covered the flanks, and the priests with torches remained protected at the center of the formation. The goal was to circle the crypt, estimate its layout and gather fuel for a camp fire.

Every step he made, Sir Rathamul counted it in his mind. Every hundred steps he made, he counted it aloud for Father Germini and the boys. Meanwhile, the priests and the militias kept their eyes peeled for dry branches, hay, linen or anything of interest in the premise, or anything that lurked in the shadows. As the step count reached a staggering thousand, Father Germini grew anxious. Even the Common’s Hall—the parliament building in Azeth city—was not as vast as a thousand steps in diameter, he said, and something was out of the ordinary, he concluded.

Sir Rathamul had also noticed the peculiarity of their situation. His senses told him they had been walking downhill for a while; but he knew not how steep the slope had been without a natural reference such as the sky, whereby the hypnotic, swirly patterns on the ceiling did nothing to help as they induced the same apprehension as the glyphs on the walls, then seemed to be coiling like a giant, living, stone serpent in his eyes. His feet, however, told a different story. The unarmored priest and light armored militias might not have felt this, but Sir Rathamul, cladded in steel from head to toe, had an acute awareness of this so terrible a weariness for a leisure stride downhill.

This surreal discrepancy struck him as dreadful for they could have walked themselves to exhaustion and been swallowed up by the eternal darkness, had they taken the trustworthiness of their senses for granted. Having understood the situation—that the crypt might be so vast, they would not be able to circle it with their depleting oil reserve and waning stamina—Father Germini urged them to withdraw if they ever hoped to see the sun again. He commanded Sir Rathamul to burn a bit of the expensive incense they were saving for the whole party after they rendezvoused.

The way the priest gave the commands got on the young knight’s nerves and the knight became infuriated. Sir Winfried was the party’s leader and Sir Rathamul was the second in command, this made the knight the group’s bona fide leader when the party split. Young as he might be, Sir Rathamul was still four years older than the priest and in his village, age and wisdom came in hand.

The knight was being unreasonable; Father Germini contested. But surmised he had authority, what suppose his plan was? Was it not the same? The priest challenged. To this point, Sir Rathamul had no retort and as soon as the incense was lit and they began marching to the crypt’s entrance, he felt petty and angry at himself for crying attention to someone younger than he was. Perhaps having seen through the knight’s frustration, Father Germini offered the first apology, saying, he had been a big brother to Sister Forse for so long that he had forgotten his place in the larger group. Afterward, words of confession poured out of his mouth so easily that Sir Rathamul had a vision: that this priest had brought out a church from the cross on his chest and placed them all within its sanctuary.

When they backtracked to the entrance, it took them only eight hundred steps and the uphill climb felt much worse than the other way—and yet his mind was exhilarated, thrilled by the normalcy, so filled with euphoria that the toll on his body eluded him until they reached the exit and finally got to settle down on the dusty floor. They welcomed the sound, the sensation, and the vision of a sensible and cruel world like an old friend they had not seen for many years. The roar of winds, the chill of snows, and the orange glow of lantern reflected off the dull rock face beyond the crypt’s entrance had become so beloved in their eyes. They had no sense of time in that strange place, but given the blizzard had yet to subside, it could not have been more than a few hours.

They sensed an imposing desire to flee the shelter and run into the raging storm. Indeed had they not been so weary to the point of collapse, the militias would have done so, and there would have been nothing Sir Rathamul or Father Germini could do to stop them. At the border of the blizzard and the crypt, they were assaulted by both the cold and the darkness. While the blizzard soon subsided, the chill lingered and the snow kept pelting down till dawn of the next day. But, dawn was seen by none of them for at midnight they heard a series of loud barks in the crypt.

Father Germini was the first to notice and he cried the dog’s name and brought a torch to the dark entrance. The soft galloping of a dog answered, drawing closer to their camp until the dog jumped out from the void like a javelin. All of them froze at the terrible sight in front of them: the dog with blank white eyes growled and bared his canine teeth at them, his mouth foaming crimson saliva, dark stains splattered his brown fur, and a nauseating stench distinctive of blood radiated from the creature. Sir Rathamul drew his weapons and the militias followed suit. Hearing the sound of sheathing metal, the dog flinched and darted back into the crypt. Father Germini, with only a torch in hand chased after the dog; all shouting from the knights and the militias went into deaf ear.

Sir Rathamul and his fellow warriors, who could not bear the guilt of abandoning their comrades, held their breaths, gathered their courage and once again plunged into the crypt. This time, they were much less prepared than before; much of their provisions were left by the camp fire and they only had on them the bare minimum oilskin for half an hour of fire. As they ran, the knight shed his plate armor pieces by pieces, with each piece making a resounding clang when it hit the stone floor, until there were only greaves and gauntlets on him. When he did so, he felt his body wrapped in both a physical and mental chill. Without the iron clad he would have nothing but the lantern shield and the short sword to fend off the mortal dangers implied by the sorry state of the dog. However, his mind rejoiced as his legs became nimbler and he could sprint faster than ever.

They covered a great distance before they caught up with Father Germini. The priest lost sight of the dog and was desperately shouting its name. They were at the center of the vast chamber, darkness stretching in all directions for as far as the eyes could see and the torchlight could reach. Sir Rathamul feared the worst had come to pass to Sir Winfried’s party, and should it have been the case, he feared there would be no going back from this adventure for he was nothing but a mere shadow of the great red-helm knight. He would rather fight bandits and wolves than fumbling in the dark, looking for a danger he could not perceive. Even swamp giants stank less than this crypt, said one of the militias, they would be better off in the blizzard than here, the other two militias added in unison.

This was the foolish priest’s fault, Sir Rathamul accused, he just had to run after the dog and drag them all into this situation; now without food and water, they would starve faster, if whatever was lurking in the shadow did not get them first. Father Germini said nothing. He let them vent their frustrations on him until they exhausted, then sighed and knelt down on the floor. Holding the golden cross in his hand, he prayed God for their deliverance and requested them to join him in prayer, which they eventually did. They all closed their eyes and chanted prayers after Father Germini. After a while, they could recite different prayers from memory and this, for a time, brought their souls closer to God, closer to salvation.

As it came to pass, he told them the method to find the exit. They would travel in zigzag pattern until they caught a breeze, which could only come from the outside world, and following the wind would lead them to safety. Sir Rathamul found the knowledge assuring, he removed the gauntlet of his sword hand—then felt like a shackle—and licked one side of his index finger, raising it up high to sense the cold embrace of the mountain wind and began to lead the party to and fro. It took no time for them to encounter a welcoming headwind, which they followed in the previous formation, having regained a sense of unity, and all the while reciting biblical passages after Father Germini. The sweet incense, the priest explained, was often used in cleansing rituals at the church and when combined with prayers from three clergymen formed the essences of God and called upon his divine power to vanquish the evil one. They did not have much of the incense left in reserve but he surmised it should last long enough till the exit.

At length, Sir Rathamul noticed the fragrant smoke from the incense was flowing against the wind, toward the direction they were traveling instead of trailing behind them. When he spoke his discovery to Father Germini, the priest dismissed it with a hint of ire in his tone, saying this was a miracle, proof that God was leading them to deliverance, and that they must be as silent as a lamb and have faith in the Lord’s guidance. Thus, Sir Rathamul, having been proven to be the foolish one for the second time, kept his doubts to himself and spoke his own foolishness no more. But the fragrant led them not unto their deliverance but into temptation, and into the hand of the evil one, as creeping into the foremost corner of their vision at this very moment was the remnant of Sir Winfried’s party, which they thought to have left behind in the void and ought to remain there for eternity.

There laid the corpses of two boys on the floor: one headless and the other had one arm impaled by a bronze spear, and a fatal stabbing wound at his side. The blood of their bodies had ceased to flow, the limbs were stiff, cold and blue, and when the party approached, the men caught glimpse of a few rats feasting on the boys’ flesh. A distance away, they found Sir Winfried in a frightening state; the man was groveling, hands holding his bleeding eyes, his scarlet helm thrown away, his large rectangular tower shield and his sword, covered in blood, laid flat on the ground nearby. Upon hearing Sir Rathamul and his aides coming, Sir Winfried picked up his weapons and charged at them, cursing witches and shouting repentance.

Father Germini, who had been squeamish and shaking, and the militias, who had cried their hearts out to the loss of their spear brothers, shouted at Sir Winfried to stop but it was in vain. Their leader saw them as witches, as heretics to be vanquished and his ears were shut as his mind was clouded. Get up and run! Screamed Sir Rathamul as he picked up a spear on the ground and flung it at the red-helm knight’s exposed head. This halted Sir Winfried for a moment, he was forced to bring up his tower shield to defend, angling the shield at an angle to deflect the projectile upward and bashing the spear back at Sir Rathamul when it fell back down. This was Sir Winfried of Ursland’s famous counter skill and it was only because Sir Rathamul had anticipated this coming that he was able to parry it.

Sir Rathamul knew he would not last against such a mighty foe on equal footings, let alone at a disadvantage; he had no armor on him while Sir Winfried was almost in full iron clad. But he had to buy time for the others to get away. He lunged his sword to intercept Sir Winfried’s charge and raised his shield to protect his chest. His strike landed squarely on the steel cuirass and did nothing to Sir Winfried, who knew his advantage so well that he did not bother dodging or deflecting the attack, while his arm received a blow from the tower shield so tremendous that his joints crackled and his forearm twisted as though dislocated. Sir Rathamul staggered and fell on his back to dodge the horizontal slash at his abdomen, shoving a circular opening on his shield, from which the lantern light shone through and with which he could blind his opponent, into Sir Winfried’s face. In light and the intense fragrant took Sir Winfried’s off guard long enough for Sir Rathamul to escape. As he was without armor, he was nimbler than Sir Winfried and was able to put some distance between them with ease, eventually catching up with the priest’s group before the oil in his lantern ran out.

They found an extra hour of oil from the corpses of the militias and a torch on the ground; the torch that was supposed to be in the hand of Sister Forse. She was nowhere to be seen but given the torch, the militias, the dog and Sir Winfried, they came to a hard conclusion that she must have been killed and eaten by creatures of the dark. Father Germini did not take the news well, he was quiet and falling behind, at times, stopping and staring into the darkness as though hoping to catch a fleeting glimpse of the twelve-year-old nun. Perhaps, it had been for the best that they did not find her half-eaten body else the man might have lost the will to live.

Sir Rathamul asked the militias if they were okay, one of them had been sniffling for a while and the other two dragged their feet in silence, grim looks on their faces. They were okay, they said, as men at arms, they knew what they had signed up for and they were thankful they were still alive. But Sister Forse, she did not deserve this, the sobbing boy uttered, Sir Winfried was right: they should have left her in Azeth. At his words, the other two started weeping as well, swearing to get stronger, stronger than Sir Winfried, stronger than the darkness that begat this tragedy and swearing to avenge their losses this day.

These too were what Sir Rathamul would have wished for, had he not faced the wild blank eyes of Sister Forse’s dog, rummaged the pale and rotten corpses of young boys for provisions, and then stared down the killer end of Sir Winfried’s bloody sword, but he had and the young knight could not imagine escaping from the shadow of such primal, paralyzing and overwhelming fear for the rest of his mortal days. Thus, he wished for a life less strange, less otherworldly than this one; perhaps he should quit his adventuring ambition and settle down as a city guard in Azeth.

At last, the little incense they had ran dry and the smell of sweats mixed in the smog of burning oil were all that left for them. Up on the ceiling, the faint purple patterns continued swirling about; it was strange, when he thought about how they could see these phantasms when it was so dark, they could not see their own hands without a lantern; and then there was the wind, and the glyphs, and the witches. He realized his consciousness was fading and salvation drifting away. It felt almost as if…

God had forsaken them, Sir Rathamul spoke aloud.

This was God’s trial, Father Germini said—his voice seemed hollow and devoid of emotions—and only the most devoted of souls shall receive his divine guidance. He asked them to pause, get on their knees, put down their weapons, and pray the most sincere prayers to God, may his almighty show their sinful hearts mercy. That, they did. However, unlike before, when Sir Rathamul closed his eyes this time, he could not see the sanctuary of a church, nor could he feel his soul overflowed with hopes. He felt powerless, wretched and hopeless. He said he would fight and prevail against any enemy he could perceive and there he was, running away from Sir Winfried like a coward he was. He said he would lead them to safety as the rightful leader and there they were, fumbling in the dark, praying to a heavenly power for deliverance. His face grimaced, twisted in agony, as he struggled to cite the Lord’s prayers amid a torrent of self-pity and regrets.

Then, abruptly, a spear punched through Sir Rathamul’s chest, his eyes flung wide open and, behold, Father Germini’s bloodshot eyes were staring down at him from the other end of the bronze spear. It was so sudden that the knight could not utter a word as blood gushed forth from his chest like a fountain, and he slumped to the ground, making a dull thud. All three militias heard the sound and opened their eyes. It took them precious seconds to realize what was happening and when they did, the priest had already knocked their weapons into the darkness. Without their weapons, the militias were helpless children before this spear-wielding man, who then also armed himself with the fallen knight’s lantern shield.

God wills!

Father Germini cried and slew two of them: his spear skewed one boy’s throat and bashed the other’s head, splitting it in two. The two boys who moved their fingers toward their weapons were mercilessly put down on the spot, in manners so brutal that the last boy who was too stunned to react had no choice but to grovel on the ground, his face so low he could taste the blood of his brothers on his lips, and pledged allegiance to the priest in the name of his ancestors to save his own skin. This, in Merlock’s tradition, was the highest oath he could make.

At the boy’s pleading, Father Germini halted, lowered his weapon and slowly looked at the scene he had caused. His breaths were heavy, his movements rigid and his limbs were drained of strength. As his eyes drifted from the groveling boy, still too horrified to face reality, he saw it: the bloody spear in his hand, red droplets splattered on his white cassock and ponds of viscos liquid crept toward him, converging into a single mass centered at his feet, formed by the blood of three humans, the boy’s tears and his dreadful sin. The bloody spear hit the ground with a cold, metallic clang, which made the boy shriveled and started begging the priest not to kill him again.

But, Father Germini had stumbled backward, barely remained on his feet, both hands grabbed hold of his own head. Forgive me, Lord! Forgive me, Lord! Forgive me, Lord! So the priest moaned as the boy continued begging for his life. The lantern shield flickered, became dimmer and dimmer until finally it went out, and darkness swallowed them both. Then, the boy felt something sharp, oozy and furry wrapped around his arm and something else sweaty, soft and powerful wrapped around the other arm. These invisible creatures tugged at him, pulling him away from the priest and his brothers. He let out a high-pitched shriek which was muffled almost instantaneously. Nothing else could be heard but the sound of the boy’s feet being dragged away.

Soon, the unseen forces of the void turned their vicious jaws at the priest. Father Germini picked up an eerie sensation of things shuffling around him. He swung his lantern shield in the dark, crouching down and using his sense of touch to search for Sir Rathamul’s corpse, in the process he stumbled upon an oozy, mushy substance on one of the boys’ heads. His hand jerked away from it, his imagination spun wild and he became panicked for he did not recognize what this unknown substance was. Gulping down his saliva, he pressed on and his hand came into contact with a human’s hand so warm that almost felt like it could still be moving. Each artifact he touched fueled his paranoia further; by the time he found the oilskin on the knight’s body and refueled the lantern shield, he was so paranoid that he was convinced Sir Rathamul and the militias were alive and stabbed their dead bodies repeatedly with the spike of the lantern shield.

The fifteen year-old boy, the last survivor of the spear brothers, stiffened in fear by his plight and found it hard to breath. These unknown things had dragged him so far away from the priest that when the lantern light was lit, it appeared to him as a tiny orange dot at the distance. He knew neither what these creatures wanted nor what fate awaited him in the void. He could not cry, he was scared, too scared, in fact, that he could not remember his own name when a familiar voice asked him. His mind blanked out and when he came to be once more, he was already at the crypt’s entrance, next to a smoldering camp fire, next to Sister Forse, who was pouring a waterskin on her dog and washing his fur clean of blood. When asked, she told him that the dog—Dust—could navigate in the dark with his keen noses and ears, and while he had become both blinded and deaf after receiving a shield bash from Sir Winfried, as long as she was there, she could be his eyes, ears and mind.

Her escape in itself was as absurd as the boy’s own. These two children and a dog knocked on the doors of every church, every garrison in the kingdom, hoping someone would believe their story and help them free Father Germini and Sir Winfried from the clutch of that accursed crypt, recover the remains of the fallen ones, and vanquish the evil forces that lurked within. When at long last, the queen of Ironheart agreed to spare them a band of elite knights, who had been close friends of Sir Winfried, and they traveled to the snowy mountain north east of Silver Gallop harbor in order to challenge the accursed crypt once more, the landscape had changed so much that they had to question if their memories were real or dream. After two weeks scouring every crook and cranny of the mountain in search of the crypt without any luck, the party went back to Ironheart, disappointed and doubtful.

While years later the boy had given up the spear and became an apprentice of a blacksmith in Ironheart, he heard Sister Forse had never given up her quest. Last they spoke in Ironheart, she was chasing after the legend of the witches Sir Winfried mentioned in his fit of rage to the forest west of the city, and later her name, too, vanished into obscurity…