Someone who shows in his actions that he does not want you should be easy to walk away from. But instead of this being an instant turn off, it sometimes merely attracts you all the more. Why is that so? One answer to the question comes from the object relations theory of W. R. D. Fairbairn (1889–1964).
I wasn’t sure whether there is any use to psychoanalysis at all. I had read some psychoanalysis in the past, but it all seemed like the barely coherent writings of someone who has never made a positive difference in any of his patients’ lives, preferring to focus on a systematic method for explaining why it’s actually the patient’s fault that his “cure”, never actually articulated in any concrete way, didn’t work.
Fairbairn, in a cursory reading, turned out to be different: his object relations theory made for an uncomfortably precise description of some situations I had experienced myself.
In the object relations theory, the psyche has a fundamental need to form “object relations”, or relations with an “object”, which is some person or entity from which it gets love, support, safety, etc.
Sometimes it happens that someone pursues a “bad” object, i.e., an object that is unable to provide these essential needs, or does so only intermittently. This is often the case with less than perfectly available parents, or in Fairbairn’s emphasis, specifically the mother. Because of a real or imagined dependence on this person, it is not possible, or it appears not possible, to sever ties with the bad object. Instead, the soul rephrases the unavailability as a consequence of its own intrinsic badness, transferring the blame from the object to itself. This results in a “split self”, a civil war of the soul, where one blames themselves for having simple human needs, and comes to a set of odd beliefs where having these needs is the cause of other people not fulfilling them.
In Fairbairn’s own words:
it also becomes a dangerous procedure for the child to express his libidinal need, i.e. his nascent love, of his mother in face of rejection at her hands; for it is equivalent to discharging his libido into an emotional vacuum. Such a discharge is accompanied by an affective experience which is singularly devastating. [...] the experience is one of shame over the display of needs which are disregarded or belittled.[1]
How to apply this developmental theory to adult relationships? Here’s a personal example. Consider someone who had been my “object”, namely the object of my need for attachment, love, belonging, togetherness, etc. However, he was unable to act the role of even a good enough object; the aforementioned needs did not see satisfaction in my interactions with him. Had my personality been secure (in the sense of not having experienced the aforementioned rejections of “libidinal needs”), I would have very early on been able to drop him and gone on my way looking for a more appropriate object.
However, my past has left me with a risk: rather than leaving a “bad object”, my instinct is to employ the so-called moral defense to exonerate it at the cost of my own self-esteem for fear of being left “objectless”: casting myself as “bad” for having the needs and feelings, for having tried to express them, for not being satisfied, the object remains blameless and thus my relationship to him has been preserved. This is how the bad object gets embedded deeper while at the same time my need for a good object, and therefore for him, is deepened.
Fairbairn’s own image for the moral defense is striking:
it is better to be a sinner in a world ruled by God than to live in a world ruled by the Devil. A sinner in a world ruled by God may be bad; but there is always a certain sense of security to be derived from the fact that the world around is good [...]; and in any case there is always a hope of redemption. In a world ruled by the Devil the individual may escape the badness of being a sinner; but he is bad because the world around him is bad. Further, he can have no sense of security and no hope of redemption.[2]
What is the nature of the fear that keeps one bound to a bad object and which creates the need for the moral defense? In the course of a normal development, one would not have a need to internalize the object; it’s a defensive move that renders it more amenable to control. One would also have been left with an impression that good objects are in fact abundant in one’s environment and would have no trouble recognizing them and rejecting those that do not meet the minimum standards for what it takes to be a good object.
Instead, if one has internalized a bad object which provides little by way of emotional support, then the temptation is to whitewash it to make it appear good. An internalized bad object, like its external counterpart, does a bad job of satisfying any needs, leaving one liable to behavior that is either overly dependent, or else painfully aware of an internal sense of emptiness. One may then also pick up that objects, of any kind, are very scarce, and thus that it’s essential to hold onto the one at hand as tightly as possible.
Keeping a bad object takes more than blaming oneself. Both the object and the self get divided.
Internally, the bad object gets embedded as two entities, the exciting and the rejecting object, where the promising and the unavailable (or even punishing) characteristics become allocated to two almost entirely separate beings. Seen as the exciting object, the person holds the promise of love; seen as the rejecting object, they are entirely bad.
Together with the splitting of the object, the self or the ego gets split into two as well, the libidinal and antilibidinal selves. The latter becomes the locus of all hate, rage, anger, etc., against the object—feelings that are not safe for one to possess, since a child angry at their parents may fear losing the parents entirely if the anger were to surface. Instead, the anger may get turned inwards and contained in the antilibidinal ego. Conversely, the libidinal ego contains all the positive feelings of excitement, attraction, love, etc., towards the exciting object.
Altogether, these are Fairbairn’s 4 unconscious entities: antilibidinal ego, rejecting object, libidinal ego, exciting object.[3] Let’s give them names: Al and Lib.
Antilibidinal ego, “Al”:
Forms to deal with rejecting/frustrating parts of attachment figures. Consists of the impotent anger, hatred, vengeance towards the bad object—perfectly justified feelings.
The negative feelings are too dangerous to feel consciously or even express—there’s no alternative to a bad parent. That’s why the feelings get repressed into Al.[4]
Libidinal ego, “Lib”:
Develops in relation to the exciting object, which consists of memories/fantasies of the few times love was given or promised. It contains the hope/promise of love that keeps BPD individuals from collapsing into abandonment depression.[5]
Al hates the bad object, so why doesn’t he leave it? Why hold on to a bad object at all, instead of looking for a better one? The answer has to do with emptiness.
a very strong hostile attachment exists between the antilibidinal ego and its associated rejecting object. This attachment comes from the characterological patient’s desperate need for objects of any type in the inner world to counteract inner emptiness.[6]
This suggests a simple view of BPD: a maladaptive response to an inner sense of emptiness. DBT is effective as a cure to the extent that it provides effective means of dispelling the inner emptiness. But what is this emptiness? It’s a lack of objects, objects that should have been there. Object relation is a basic human need; in absence of good objects just about anything will do, even a frustrating bad object. That’s why the antilibidinal ego doesn’t drop the bad object.
Celani puts it this way:
These children are caught in a Kafkaesque reality where they are empty and abandoned. They are emotionally so young that they are only able to focus on one object whom they absolutely believe is the only possible source of nurturance. It is a cruel fate, as the most severely deprived individuals are bound more tightly to a worse object both externally, and more importantly, internally than those who are moderately deprived.[7]
If Al clings to the rejecting object out of emptiness, Lib clings to the exciting object out of hope.
Of course, the exciting object is far stickier, and the libidinal ego that clings to it “creates the most obvious havoc in the lives of many patients.” Lib’s unrealistic hope “leads the person to make decisions that have absolutely no relation to reality and invariably are detrimental to the individual’s own best interests.”[8] The relationship to the exciting object should be compared to a drug addiction: remembering the few times it felt amazing, but seemingly unaware that most of the time it feels mediocre or even awful, plus the punishment of withdrawal.
As Celani puts it:
it is not rejection that is devastating to the borderline individual but the loss of hope—the loss of the exciting aspect of the bad object[.][9]
Celani continues, quoting Armstrong-Perlman:
The loss of the relationship, or rather the hope of a relationship, cannot be borne.... The need is compulsive and the fantasy of loss is experienced as potentially catastrophic, either leading to the disintegration of the self, or a fear of a reclusive emptiness to which any state of connectedness, no matter how infused with suffering is preferable.[10]
The loss is experienced as final and irretrievable. There is now no hope the object will come back of its own accord. One becomes afraid of being alone forever, of never finding anyone like the lost object. It’s an experience of immense sadness: it has been a very special relationship. Lib is frightened that the Emptiness will engulf it, ending in death in unthinkable pain and suffering.
Lib will always want to leave the door open for the object to come back, to redeem itself. This is perhaps its way of softening the “abandonment crisis”.[11] It cannot give up the exciting object, even if it’s only a fantasy of hope. It can’t afford to lose it. Not until the central ego can win over the emptiness.
Unfortunately, the theory itself says little about how to actually go about releasing the bad object, apart from recognizing on an intellectual level what the dynamic has been. However, I know from past experience that merely understanding a pattern or dynamic is very different from disrupting it and replacing it with a better one, just like knowing how chess is played does not give you the power to win a game.
Celani makes the same point:
Fairbairn also assumed, somewhat naively, that the “libidinal cathexis” in the internalized bad object, which is equivalent to the relational concept of “attachment,” would somehow magically dissolve when the contents of the bad object memory was derepressed. Clinical experience strongly suggests that this is not the case. The “release” of a bad object memory is simply the first step in a long process of detoxification of its impact and importance.[12]
Patients often come into therapy with theories about themselves that only serve to protect the bad objects; that may be why they are stuck in life. So how do I know that what I’m saying is true, and not just an instance of my libidinal ego misleading me?
What does the object relations theory say one should do? If being stuck in an obviously inadequate relationship is due to an internalized bad object, then one should begin by dislodging the bad object. It has probably been internalized for a long time, so getting rid of it may be challenging, but it is the only way of reaching any meaningful improvement. One would likely also need to internalize a good object, and come to see them as abundant. Thus, one must
dissolve the cathexis to the exciting object and accept that the wish for the loving acceptance of the object is hopeless. The patient also has to feel that this no longer means that there is no hope for the self. He has to feel that he can survive the trauma of loss with sadness, and can mourn for that which he did not receive.[13]
The goal of treatment is to integrate these formerly intolerable split-off parts back into the conscious central ego and the good object. Strengthening the central ego reduces the illusory power of bad objects. Mature grief on realizing bad objects did not provide the love they should have becomes a core part of the central ego. The method is the creation of “narrative truth”, interpreting one’s life story in terms of Fairbairn’s metapsychology.[14]
But how is the cathexis to be dissolved, if understanding alone does not do it? Fairbairn makes the therapeutic relationship the decisive factor, on which interpretation and everything else the therapist does depends. Already in 1943 he suspected that good therapeutic results come about, in part at least, because
in the transference situation the patient is provided in reality with an unwontedly good object, and is thereby placed in a position to risk a release of his internalized bad objects from the unconscious and so to provide conditions for the libidinal cathexis of these objects to be dissolved[.][15]
In his late paper on treatment, he made the relationship itself the decisive factor:
In my own opinion, the really decisive factor is the relationship of the patient to the analyst, and it is upon this relationship that the other factors [...] depend not only for their effectiveness, but for their very existence, since in the absence of a therapeutic relationship with the analyst, they simply do not occur.[16]
And he did not mean the transference alone:
what I understand by “the relationship between the patient and the analyst” is not just the relationship involved in the transference, but the total relationship existing between the patient and the analyst as persons. After all, it is on the basis of the relationships existing between the individual and his parents in childhood that his personality develops and assumes its particular form; and it seems logical to infer that any subsequent change in his personality that may be effected by psycho-analytical treatment (or any other form of psychotherapy) must be effected primarily on the basis of a personal relationship.[17]
In CBT, by contrast, a good therapeutic relationship is usually considered necessary but not sufficient: it is the techniques that are expected to do the work.
Celani draws the practical conclusion for the exciting object:
the only way patients can finally let go of the fantasy that their exciting object contains love is for them to develop a powerful central ego–ideal object relationship with the therapist or analyst.[18]
Once the cathexis to the bad object has been dissolved, what follows need not be the feared abyss. In my experience, the pain of letting go is real, but it diminishes with time rather than increasing, as the fear had predicted; in its place comes a sense of fullness and freedom, and the ability to pursue relationships that are mutually sensible and meaningful, without feeling stuck in any of them.
Here’s some notes on how to use the 1980s classic spreadsheet program sc, and
why. We’ll start with the usage basics, then discuss the limitations and comment
on the user experience.
Install it from your favorite package manager (sudo apt install sc) or build
the latest version yourself:
git clone https://github.com/n-t-roff/sc.git
cd sc
./configure
make
sudo make installLaunch sc and you’ll have a spreadsheet in milliseconds—likely faster than
your finger can leave the Enter key.
Navigation. To move around, use the arrow cells or the hjkl keys familiar
from Vim and similar programs. To quit, press q. To save, press P.
Text Entry. Free-form text can be inserted in the current cell using <` for
left-aligned text, `> for right aligned, and \ for centered. To change
alignment subsequently, navigate to the cell and press { or } to left or
right align, and | to center.
Annotations. Instead of adding hidden notes to cells, in sc one adds links
to other cells. (I tend to place these notes in the rightmost column, so they do
not clutter the main data.) To add a link, press *a, navigate to the target
cell, and press Enter twice. Press ** to jump from the original cell to the
linked target cell. Simple!
Numerical Values. Press = and enter a number or a formula. Special
functions start with @, for example entering @now will enter the current
Unix time. To enter a specific date, use @dts(2026,11,30).
Cell Format. Press F to set the format string for the current cell. The
manual details the format, but in practice I use this most commonly to make one
cell (or a few cells) format a date/time value. I use the format string
^D%m/%d/%y, where ^D is the literal Ctrl-D character (ASCII 0x04, “end of
transmission”).
Column Format. Press f and use the arrow keys or hjkl: left/right (or
h/l) to make the column narrower or wider; up/down (or k/j) to increase
or decrease the number of decimal places.
For more flexibility, press f followed by a digit: f0 is the default, f1
formats numbers in scientific notation like 1.00E+04, f2 is the same in
engineering notation (10.00e+03 such that the exponential is a multiple of
three), f3 and f4 format dates as 5 Nov 26 and 5 Nov 2026, resp. To
define a custom numerical format, open spreadsheet in the text editor and enter
a line like
format 5 = "^D%m/%d/%y"Careful: if you want the date formatting to work correctly, the ^D must be the
literal end-of-transmission character, not a caret followed by uppercase D.
Rows/Cols. To insert new rows or columns, type ir or ic respectively. To
delete rows or columns, type dr and dc. As with most commands, you can
precede it with a number; for example, 5ir inserts five rows. Careful when
deleting multiple rows, there is no undo! Save the sheet first, before making
large deletions.
Ranges. To operate on several cells at once, use the range
commands that start with r. There are several range commands but I only use
four:
Delete a range: type rx then use motion commands (such as the arrow keys) to
select the range, followed by Enter.
Convert formulas to values: rv + motion + Enter and the formulas are
replaced with their numeric values.
Copy a range: navigate to the destination cell and type rc followed by
spacebar. Then use motion commands to navigate to the first cell in the source
range. Press . (period) to begin range selection. Press Enter to execute the
copy.
Move a range: starting on the destination cell, type rm. Then go to the
source, press . (period), select the range you moved moved, and press Enter
to execute the move.
Copy/Cut/Paste. Besides the range copy and move commands explained just above, it is also possible to operate on a single cell, or one or more rows or columns, as follows:
| Object | Copy | Delete | Paste |
|---|---|---|---|
| Cell | yy | x | pp |
| Row | yr | dr | pr |
| Column | yc | dc | pc |
Marks and Jumping. When working on two parts of the spreadsheet that are far
apart, for example when copying and pasting, or moving things around, it helps
to give cells a temporary “mark”. To mark a cell, press m followed by any
other lowercase letter, for example ma. To jump to a cell previously marked in
this way, press backtick followed by the same lowercase letter.
To copy the marked cell, mark it, navigate somewhere else, and press c
followed by the letter. Even though one can copy and paste a single cell using
yy and pp as explained above, I got used to copying with ma and ca
instead (Y is far away from fingers on QWERTY keyboards).
Type Ctrl-A to jump to cell A0.
Scrolling. To scroll the sheet down, press Ctrl-e followed by Ctrl-e
(again). To scroll up, press Ctrl-y. To scroll such that the current cell
becomes the top row, press z followed by Enter. To center the current cell,
press zc.
Searching. To search for a particular string (case sensitive), type g"
followed by the desired string. To repeat the search without re-typing the
string, press n. To repeat the search in the backwards direction, type N.
Colors. There are a number of commands to set the color of cells. I do not remember what these are. Instead, I open the spreadsheet file in a text editor, and simply add the following self-explanatory lines:
set color
color 1 = @white;@black
color 2 = @black;@green
color A1:C4 2Frames. Set a frame if you want one part of the spreadsheet to scroll independently of another. I use this to have the table header with column names always visible, even when scrolling through many pages of data rows. There are many commands for manipulating frames and as with colors, I do not remember any of them. Instead, open the spreadsheet in a text editor and add a line like
frame A0:N1000 B3:N1000This sets the first three lines to be the fixed header, while the rest scrolls freely.
Function keys. To use shortcuts, the function keys (F1 through F12) can be given custom actions, such as motion commands or calling macros. For example:
fkey 4 = "endright"
fkey 11 = "merge \"|~/projects/config/scmacros/copy_above.sh\""Macros. To convert a complex sequence of keystroke into a single keypress,
define macros. These are simple shell scripts such as the
copy_above.sh
that I have mapped to F11: it moves one cell up and one cell right, copies the
cell, then moves one cell down to paste it. To create your own, take a look at
my macros, or, better
yet, point your favorite AI at the cloned sc repo and tell it to write it for
you.
Technically sc distinguishes “simple” macros, which are interpreted by sc
itself, and advanced macros, which are shell filters. However, the advanced
macros can do anything the simple ones do, so I never used the simple ones.
Config file. Common settings that you want applied by default to all
spreadsheets can be placed in the ~/.scrc file. Rather than reading this file
in addition to the spreadsheet file, sc will place the content of scrc into
all newly created spreadsheets. In my
scrc, you will find only
the format 5 definition and a couple function key shortcuts.
A key part of the sc experience lies in the features that are not present.
Of course, you could add any of these with AI, or look at projects like
sc-im but in this instance I choose
to “embrace the constraint” instead, to see what I can learn from it.
No Sync. Instead of every change being reflected on the cloud, I sync the sheets I want to sync by adding them to a git repo. Works great since the file format is text-based and very simple.
No Multiple Sheets. I was used to grouping several related sheets in one
file, but after sc-ing for a while, this makes no more sense to me than
bundling several source code files in one. The file system works great for
organizing files—why does it seem like people are suddenly afraid of using
files? Why do Apple and Google go to such lengths to hide it from users?
Limited Conditional Formatting. I love making sheets automatically get colored based on data values but in retrospect it’s a huge distraction. Why not make the data speak for itself instead? (There are some commands that can change cell color, but I do not find it useful enough to learn.)
UI is not Discoverable. You cannot figure out how to do things, or even if something is possible, except by reading the manual and/or source code. The learning code feels steep but for a program I use every day, it totally does not matter. I have it all in finger memory and I’m glad there’s not “beginner friendly” menus, popups, AI chats slowing me down.
Limited External Copy/Paste. It’s nice to copy a table from a web page seamlessly into a spreadsheet but here you have to import it with a text editor, or Python script, or Claude. No CSV in/out either, without external tools. In other words, use the right tool for the right job. Does one program really have to do everything? For me, the key insight of Unix is the partitioning of tasks into separate small programs, even if sometimes at the cost of slight inconvenience.
No Undo/Redo. Instead, I keep history with git, and make sure to save
before and after making edits. If I make a mistake, just quit and reopen the
program: qn, Up, Enter. Four keystrokes and it’s undone. (Usually Ctrl-Z
in Google Sheets takes longer to execute, amazingly.)
No Unicode. I live in ASCII.
In light of these limitations, why bother with sc? Isn’t old software
obsolete, by definition of “old”? I disagree. If a program worked well on
ancient computers a long time ago, then on modern machines it will work even
better. The sc user experience is just so fast it blew me away. There is
simply no waiting; everything happens instantaneously: opening/closing the
program, entering data, making edits.
Conversely, if a piece of software—indeed any technical project–is worth implementing, it’s worth doing it right so it can stand the test of time. Sure, some tech is necessarily disposable, if it interfaces with some frequently changing protocols, for example. But most problem domains allow decomposition into small tools, where the tool remains essentially unchanged over decades.
The user experience deserves particular focus in the earliest stages of design, to ensure that it does not irritate or slow down the advanced user. If a tool is supposed to be used frequently by most of its users, then most of its users will become “advanced users” and will not tolerate convoluted keystrokes or, worse, time-consuming mouse or touch navigation. And certainly there is no excuse for slow user interfaces—that ship sailed back when processor clock speeds exceeded a megahertz.
Android is a particularly nasty counterexample: in a misguided attempt to
appease the “beginner”, everything is hidden behind seven layers of tiny
hamburger menu icons. Essentially all web apps take several orders of magnitude
too many clock cycles for any actions. The problem has become so bad and so
pervasive that no one even notices anymore—everything is slow, slow slow. But
programs such as sc can wake us up to realize: not everything has to
suck.
I’m no longer convinced anybody knows anything about user interface. It’s clear some things are easier to use and some things are harder to use. That’s okay. But it’s also clear that people learn a lot.[1]
When the rules of a game are simple, don’t keep inventing new rules to complicate things. In particular, at all levels an engineering project needs to be as simple as possible while still functioning as expected. Beyond that, leave all rules behind.
Indeed, there is only one rule: the thing has to work. Besides, the less time and effort spent in the implementation, the more there is left for interesting applications. In sum, always only make the minimum working example.
These are some notes I’ve taken when creating a circuit that takes a 10 MHz sine-wave input and outputs several copies of the same, with a minimum of added phase noise or distortion. First, select a topology and simulate it. Then, build the circuit and measure the key specs: noise, crosstalk, distortion.
The goal is simple, 10 MHz in, 10 MHz out. Many approaches have been explored:
Take a differential 10 MHz input signal, pass it through a difference amplifier, output a single ended signal.
Start with 20 MHz, pass it through a flip-flop to divide by 2, and filter the resulting square wave to kill the harmonics and obtain a sine wave.
Design a differential amplifier using discrete transistors to drive a signal transformer to lower the output impedance.
Emitter follower, with AC coupled output (series LC).
Inverting op-amp plus some filtering and the impedance transformer at the output.
Non-inverting op-amp output circuit.
Of these, I have chosen the version that I find easiest to explain to myself: an inverting buffer drives several non-inverting output drivers. This is shown in the image below.
A straightforward transient LTspice simulation (.tran 0 1.5u 0.5u 25p) shows
that the output amplitude is 1Vrms. In other words, the circuit should work as
shown.
In the simulation I used the spice model for the LMH6609 op-amp provided by
Texas Instruments (original model by National Semiconductor). It works just fine
in LTspice except for commenting out a comment. Apparently PSPICE supports the
/ symbol as comment in addition to the * used by LTspice.
A key spec on any multi-channel instrument is crosstalk: if a signal is injected in one output, does it show up in the other output?
Notice the V4 signal source in the LTspice schematic shown above. If we
disable the main signal source V3 to AC 0 and enable V4 as AC 1, then an
AC simulation (.ac dec 200 1k 1G) shows about 10μV of crosstalk at out1.
Including a rough estimate for layout-dependent capacitive coupling between two
outputs directly (e.g. 10fF at C7), this rises to 17μV, or about -95 dB.
Assuming the op-amp model is accurate, we can use this result to estimate the phase pulling between the two outputs. Assume one of the outputs is connected to an unterminated cable such that the full signal strength—all 1Vrms or +13dBm of it—comes back in quadrature. The peak amplitude is 2.82 × 17 μV = 48μV.
Adding a small phasor a at angle φ to a carrier A gives a resultant phase
shift Δφ ≈ (a/A)·sin φ. The angle φ is set by the cable’s round-trip delay
plus the coupling path’s fixed phase—and since the user can present any
length, the worst case is φ = 90° (quadrature), where the interferer is pure
phase: Δφ = 48 µV / 1.38 V = 35 µrad.
At 10MHz one period is 100ns spanning 2π rad, so Δt = Δφ/(2πf) = 34e-6 /
(6.28e7) ≈ 0.55 ps. The reflection on Ch-2 will cause Ch-1 to move in phase by
less than a picosecond. Pretty good!
There are many mechanisms that contribute to the phase noise at our outputs. Let us immediately ignore the noise of the 10 MHz source and instead focus on what adds to the source’s phase noise.
At large offsets away from carrier, the noise floor is set by op-amp voltage
noise, resistor Johnson noise, and current noise. The same LTspice model from
above (remember to disable V4 and re-enable V3!) can be used to calculate
the voltage noise density at the output, by means of a .noise V(out1) V3 lin
401 9.9Meg 10.1Meg simulation directive. At 10 MHz, it reads 9.7 nV/rtHz.
Relative to the 1 Vrms output amplitude, this is a phase noise floor of
L = 20 log10(9.7e-9 / 1.00) − 3.01 = −163 dBc/Hz.Note the 3 dB subtracted, due to the fact that the noise decomposes into components in phase and in quadrature with the carrier phasor. The in-phase part wiggles the amplitude; the quadrature part wiggles the phase. For additive noise these are equal in power, so exactly half the noise power becomes phase modulation.
Any air moving across the board, whether forced or natural convection, can cause the temperature of PCB components to fluctuate. Ferrite inductors, such as the 1.5 μH seen in our tank circuit, might have as much as 500 ppm/K of temperature coefficient. We can step the inductance value in the LTspice simulation to find about 0.45° of phase change due to 1% change in inductance.
Putting the numbers together, the phase sensitivity is 3.9e-4 rad/K, or, in time units, about 6.3 ps/K. If we estimate the inductors temperature noise at 1 Hz to be about 1.5 mK/rtHz, then the phase noise would be
L = 20 log10(3.9e-4 * 1.5e-3) - 3.01 = -127 dBc @ 1 Hz offset.That’s a significant amount of phase noise, but easily fixed by reducing the air
currents or increasing thermal insulation. The inductor temperature changes are
heavily low-pass filtered by means of the part’s thermal inertia. In other
words, above the thermal pole the response falls as f^{-2}, so lowering the
corner buys 20 dB per decade of shift.
Any sound or mechanical vibration modulates the electrical characteristics of the circuit. Non-NP0 capacitors are particularly affected, so it’s important to verify there’s none in the critical signal path.
Inductors are also prone to microphonics, via magnetostrictive modulation of inductance. The datasheet will most likely not specify it, so instead we take a wild guess, say 1.5e-4 inductance change per “g” of acceleration.
Reusing the 0.45° per 1% coefficient from the previous section, the phase sensitivity is 1.2e-4 rad/g, or about 1.9 ps/g. A fan blade-pass tone at 120 Hz depositing 0.01 g peak gives a peak phase deviation of 1.2 μrad, and a sinusoidal modulation of peak deviation Δφ produces sidebands at
20 log10(1.2e-6 / 2) = -124 dBc.
Disclaimer: I’m not a theologian, proceed at your own risk.
An astute reader of the Notes on Cassian noticed a wild claim of mine, that “being dead to the wold” means afterlife, when examining Cassian’s thoughts on divine contemplation and “seeing God”, in Conf. 1, Ch. XV. Let’s take a closer look.
I don’t think you can “see God”, no matter how “pure” or “dead to the world” you are. To be fair, Cassian only makes the converse claim: if, any carnal affections survive in you, then you cannot lastingly contemplate God. Am I going too far by assuming Cassian implied the reverse claim: you could lastingly contemplate God if only no carnal affections survive in you?
A parallel set of claims arises from his scriptural quote. Exodus states that if you see God, then you cannot live. Cassian elaborates: you cannot live to “this world and to earthly affection” (mundo huic affectibusque terrenis). The implied reverse statement in this case is, if you’re “dead to the world”, then you may be able to see God. Again, did Cassian intend this implication or am I reading into it too far?
There’s only two options, either he did mean it or he did not. If he did not mean it, then we could discuss whether carnal affections (carnalis affectus) prevent you from contemplating God “lastingly” (perpetuo), or from seeing God. Since I’m not at this point interested in studying perpetuity, i.e., eternity or afterlife, so let’s pass on to the other option: Cassian did mean the “reversed claim”: purify your heart and you’ll be able to see God now and forever.