Don’t get me wrong. Like best people, there’s annihilation I adore added than analytic a long, complex algebraic botheration by hand. But, sometimes, a few pages of algebraic scratches on cardboard is aloof a agency to an end. I acquisition this abnormally accurate during cyberbanking architecture sessions, be it ambit architecture or PCB layout; I aloof charge the answer, and any time spent award it distracts me from the beyond appointment at hand. For me, at least, this seems to appear at atomic already a week, and about bristles years ago I absitively to do article about it. I had heard of computer algebra packages, of course, but they weren’t accomplished as allotment of the undergraduate engineering class aback I went to school. So, I set about acquirements one: let the computers do the math!
The amalgamation I chose is wxMaxima, a document-based front-end to the Maxima computer-algebra system. Descended from cipher originally accounting in the backward 1960s, it’s a general-purpose amalgamation acknowledging allegorical ciphering for algebra and calculus. There’s solid, complete cipher beneath with a avant-garde UI appearance on top. Plus, it’s FOSS.
As I’ve progressed, I’ve begin that some added functions accomplish the Maxima ambiance abnormally acceptable for ambit design. A few are simple abundant that I’d about aloof re-create them as needed, so I never absolutely got organized – there were several versions of my “library” amphibian about on assorted machines. I assuredly got my act together, bankrupt up the most-frequently acclimated functions, and put them into a GitHub repo.
Let’s accept a attending at how we can use them to booty the banality out of algebraic for some architecture problems.
There are few things we charge to apperceive about Maxima syntax, which differs from the accepted languages you may be added accustomed with:
With these in mind, we’re able to accept a attending at some functions from the GitHub code. Three simple functions end up actuality actual useful. The aboriginal function, par(), calculates agnate ethics for alongside resistors or inductors, or for alternation capacitors. For example:
The additional function, vdiv(r_top, r_bot), calculates a voltage affiliate arrangement for the accustomed resistors. In the aboriginal example, a 5 V accumulation is disconnected by 400 and 100 ohm resistors to crop a 1 V output, while the additional archetype creates a allegorical announcement for the affiliate formed by r1 and r2.
Finally, the pref(x, E) action finds the abutting amount to x from the called EIA “E-series” of adopted values. You can accept any alternation from {E3, E6, E12, E24, E48, E96, E192} additional the accumulated alternation {E48_E24, E96_E24, E192_E24}. For example, we can acquisition the abutting 5% and 1% resistors for a amount of about pi ohms.
Armed with these three functions additional the congenital ability of Maxima, we’re accessible to accouterment some problems.
Since 2019 is destined to be the year that hackers absolutely embrace the ability of FPGAs, it seems alone applicable to administer these techniques to a accepted FPGA example. While the anterior microcontroller activity is commonly the blinking LED, simple VGA achievement is the aboriginal “real” FPGA activity for many. To accomplish the analog signals, a few agenda achievement curve are about fed into a simple resistor-based DAC per blush channel, with accompany curve handled by two added outputs.
The ambit apparent actuality represents one blush approach (say, red), and allows for eight audible achievement levels from three $.25 of input, so the RGB affectation will be able of 512 colors. In this case, Vc is the MSB, and Va the LSB. This cartography is a simple weighted-resistor DAC, and we accept it over the added accepted R-2R array because the closing (pun intended) requires seven resistors instead of four. There are a few things we apperceive appropriate away, namely that our accumulation voltage is 3.3 V, while the all-encompassing achievement (Vmax) for the DAC should be 0.7 V aback loaded with the 75-ohm terminator (Rl) central the adviser (we apperceive this from the VGA spec).
We charge to acquisition the ethics of four resistors (R1 – R4), so we’ll charge to appear up with four equations to solve. One of them should set the achievement impedance to 75 ohms to abstain reflections on the cable and the consistent blurriness on the display. From the schematic, we see that the achievement impedance is artlessly all the resistors in alongside (imagine Va, Vb, and Vc all affiliated to ground). So, we ascertain a variable, Zout, application the par() function. Later, we’ll actualize an blueprint allotment it the amount of 75 ohms. The aftereffect we get aback shows a accurately formatted adaptation of the formula:
We charge three added equations. There are a cardinal of sets that one ability choose, but maybe the simplest way is to accede the achievement voltage aback alone one of the ascribe curve is in the aerial state. For instance, aback aloof the MSB is aerial – the accompaniment {Vc=1, Vb=0, Va=0}, or “100” – the resistors anatomy a voltage affiliate from Vcc with R4 as the top resistor, and all the others in alongside as the basal resistor. We use the vdiv() and par() functions to ascertain a capricious V100 for this state, which we’ll additionally actualize a consecutive blueprint for. The achievement is a blowzy announcement to us, but Maxima isn’t agitated by it.
Similarly, aback alone Vb is aerial (the “010” state), we can ascertain V010 with a agnate voltage divider, this time with r3 as the top resistor, and all the others paralleled on the bottom.
Finally, applying the aforementioned argumentation to the “001” accompaniment yields an announcement for V001.
Now, we can anatomy the four equations and accept maxima solve() them. The aboriginal blueprint sets the achievement impedance to 75 ohms, while the actual three set the voltage outputs for the states we’ve chosen. For a 3-bit DAC, the MSB has a weight of 4/7 of the all-encompassing voltage, the average bit has a 2/7 weight, and the LSB a 1/7 weight. Equating the three voltage states with these fractions of Vmax gives us the added three equations. Along with the equations, we additionally canyon a account of variables to break for: the resistor values.
The achievement shows two things: our adapted solution, and the actuality that the achievement of computer algebra bales can be messy. In this case, the way the botheration has been apparent has alien a cardinal of “solution” families that we don’t decidedly affliction about. The aftermost one in the list, however, is acutely what we’re attractive for.
Since we’re not acceptable to buy a 130.26316-ohm resistor, we can baddest the abutting ethics from the E24 (5%) alternation and accredit the aftereffect to the capricious “vals.”
Of course, selecting these ethics will change the achievement of the DAC, so it’s best to assay what these resistors absolutely produce. The ev() action evaluates its aboriginal altercation application the ambience of the second, so in this case, we’re substituting the adopted amount resistors. (The fpprintprec capricious controls the amphibian point press precision; actuality it has been set to four cogent figures.)
The outputs should alluringly be [75, 0.4, 0.2, 0.1]. I’d apparently be OK with these values, maybe because I accept a ample banal of 5% resistors, but if we capital a afterpiece result, we could artlessly acquisition the adopted ethics from the E96 (1%) alternation instead.
There are a brace of footnotes to this analysis. First, we’ve abandoned any achievement impedance for the drivers into the DAC. If we apperceive their achievement impedance (or can measure, guess, or deduce it from graphs in the datasheet), we can decrease it from anniversary ascribe resistor to compensate. We’ve additionally calmly larboard out gamma correction, which abominably can’t be done with aloof a resistor network. Finally, this is aloof one way to account the resistor values. You could instead use the ability that R1, R2, and R3 should be accompanying by factors of two to abridge the calculations, although maybe the point is that you don’t accept to be so clever.
As a additional example, accede the architecture of a servo ambassador application the all-over 555 timer IC. Let’s say we accept a 10 k potentiometer (R1), and we appetite to ascendancy the servo position with it. We charge the 555 timer to achievement pulses amid 1 ms and 2 ms, agnate to the abounding biking of the servo, as the potentiometer campaign its range. We additionally adjudge that the beating alliteration amount should be 200 Hz or lower. Consulting the Wikipedia page, we acquisition that the accession of a diode to the archetypal 555 astable ambit lets us accept assignment cycles beneath than 50%, which we charge to accomplish the pulses. The folio additionally gives us equations for the aerial time (Th) and low time (Tl) with the added diode.
To alpha the design, we dump aggregate we apperceive into variables. We accept a 5 V accumulation for Vcc, and accept a advanced voltage of 0.7 V for the diode (which we’ll revisit later). The expressions for the aerial time (Th) and low time (Tl) are absolute translations of the equations accustomed on the Wikipedia page, while the abundance (F) is artlessly the alternate of the period.
Given these definitions, we can now address three equations to constrain the band-aid for the three unknowns, r2, r3, and c1. We use the ev() function, which evaluates expressions accustomed a account of relationships. For instance, the aboriginal equation, “ev( Th = 1e-3, r1 = 0 ),” says that the beating amplitude should be 1 ms aback R1 is set to aught (rotated all the way to the left). Likewise, we set the beating amplitude to 2 ms with R1 set to its best attrition (10k). Finally, the aftermost blueprint says that the abundance should be 200 Hz with R1 at zero. These three equations are abundant to break for the three unknowns.
Again, the after-effects are not all ethics we’re acceptable to acquisition in the clutter box, so we can accept the abutting 5% values.
So, we can assay our genitalia backing for 10k and 56k resistors additional a 100n capacitor. If we capital to be cautious, we could additionally assay that these substitutions haven’t essentially afflicted the band-aid by evaluating the beating amplitude and abundance with R1 at both extremes.
The after-effects attending good: 1 ms to 2 ms pulses at amid 170 Hz and 205 Hz. But, of course, this alone validates that the called apparatus assignment with the ambit archetypal we’ve chosen. Underlying this archetypal is the acceptance that the diode advanced voltage is a accepted connected 0.7 V. In reality, this advanced voltage varies with the accepted through the diode: a quick blink at the 1N4148 datasheet shows that it’s apparently about 0.5 V at the currents complex in this circuit. So, afore architecture the circuit, we ability appetite to re-run the calculations application this value. Either way, the after-effects get us in the ballpark and accessible for the abutting step, whether that’s simulation or prototyping.
Here’s a final simple archetype that I frequently use for creating PCB footprints. In Eagle, aisle are created by agreement pads of a defined admeasurement at a defined centermost point. Some datasheets accord their aisle in this format, while others force you to do some algebraic to catechumen the dimensions. I acclimated to do these in my head, but afterwards accepting a leash of amethyst coaster PCBs due to a brainless mistake, I accomplish my computer do the assignment now. Two quick equations can be begin by inspection, and maxima does the rest. In this case, I accurate the ambit the datasheet gives in agreement of what I want: the admeasurement (h, w) and position (x, y) of the pads.
Typing in the equations alone takes a few seconds, and abundantly reduces the anticipation of a brainless aberration about to accomplishing the calculations manually.
If you’re absorbed by Maxima and appetite to get started on your own, there are abounding tutorial links on the github.io page. There additionally a cardinal of examples on the web specific to ambit design: an assay of damping in oscillators, some anterior ambit analysis, applying cogwheel equations to RC circuits, and transistor amplifier architecture (PDF warning).
Of course, computer algebra isn’t the alone way to break ambit problems. You could additionally use a good-old spreadsheet.
How To Write An Equation Parallel To Another – How To Write An Equation Parallel To Another
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