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66

Other answers haven't yet hit upon what makes e special: defining the time constant as the time required for something to drop by a factor of e means that at any moment of time, the rate of change will be such that--if that rate were continued--the time required to decay to nothing would be one time constant. For example, if one has a 1uF cap and a 1M ...


49

It's built into the mathematics of exponential decay associated with first-order systems. If the response starts at unity at t=0, then after one "unit of time", the response is \$e^{-1} = 0.36788\$. When you're looking at a risetime, you subtract this from unity, giving 0.63212 or 63.2%. The "unit of time" is referred to as the "time constant" of the system,...


36

Simple: the average of a sine is zero. Power is proportional to voltage squared: \$ P = \dfrac{V^2}{R} \$ so to get average power you calculate average voltage squared. That's what the RMS refers to: Root Mean Square: take the square root of the average (mean) of the squared voltage. You have to take the square root to get the dimension of a voltage ...


35

A divider maps much less elegantly to typical hardware. Take Lattice ICE40 FPGAs as examples. Let us compare two cases: this 8x8 bit to 16 bit multiplier: module multiply (clk, a, b, result); input clk; input [7:0]a; input [7:0]b; output [15:0]result; always @(posedge clk) result = a * b; endmodule // multiply and this divider that ...


30

Typically high resolution sin(x) functions would be implemented with a CORDIC (COrdiate Rotation DIgital Computer) algorithm, which can be accomplished with a small number of iterations using only shifts and add/subtract and a small lookup table. The original paper The CORDIC Computing Technique by Jack Volder is from 1959. It also works nicely when ...


21

The ordinary common stuff is mostly just basic algebra, like Ohm's law, computing one of frequency, resistance, and capacitance from the other two, etc. The important skill here is not so much math but intuitively understanding the physics behind what you are doing. If you can look at a schematic and feel the voltages pushing and currents flowing and how ...


19

These calculations are absolutely used by professional EEs, for some on a daily basis. However, for many this job has been given to simulation software, such as LTSpice, which is also used on a daily basis. Generally the simulation is much faster to complete, so it is much more productive than doing the calculations by hand. I generally use the formulas ...


18

Only a sketch of a solution. Take all 3 axes into consideration. Acceleration due to gravity, regardless of tilt, will always be 1G, as a vector sum of X,Y,Z, no matter what the tilt. You can picture the acceleration at rest or steady motion as a point on a sphere with radius 1G. (If you are perfectly horizontal, that point will be (0, 0, -1) i.e. directly ...


16

Ohm's law $$ 1: V(t) = I(t)R $$ Instantaneous power dissipation is product of voltage and current $$ 2: P(t) = V(t)I(t)\\ $$ Substitute 1 into 2 to get instantaneous power through a resistor in terms of voltage or current: $$ 3: P(t) = I^2(t)R = \frac{V^2(t)}{R}\\ $$ Average power is definitionally the integral of instantaneous power over a period, ...


14

Now speaking in terms of equations: \$ P_{avg}= avg(P_{inst}) \$ Now \$P_{inst} = v(t) \cdot i(t)\$ where \$v(t)\$ and \$i(t)\$ are instantaneous voltage and current resp. Hence \$ P_{inst} = \dfrac{(v(t))^2}{R} \$ \$ P_{avg} = avg(\dfrac{((v(t))^2}{R}) \$ \$ P_{avg}= \dfrac{V_{rms}^2}{R} \$ As RMS = \$\sqrt{\text{average of squares of inst.}}\$


14

Most computer trig libraries are based on polynomial approximations, which gives the best balance between speed an accuracy. For example, a dozen or so multiplication and add/subtract operations is enough to give full single-precision accuracy for sine and cosine.


13

Assume the system is already precharged and operating in a steady state. The bridge has two discrete states: either the capacitor is charging (a diode pair is forward biased), or the capacitor is discharging. Call the period P, the charge time DP, and the discharge time (1-D)P. During the charge cycle, we can approximate the current entering the capacitor ...


13

I assume for "high speed" you mean a small delay from data collection to the resultant FFT. With a low sample rate, your computational ability isn't the limiting factor, given modern computers. The delay problem lies in having enough data for analysis. If you want your 1Hz bin to be different from DC/0Hz, you have to accumulate enough signal data to capture ...


13

You refer to these basic formulae at first and then find the real world has a lot of non-linear characteristics like XOR phase detectors in a second PLL loop response when you exceed the phase limit or that all Low Pass filters cause Inter-Symbol-Interference (ISI) unless the filter resonates within the binary symbol then you apply "Raised Cosine" Filters ...


12

I haven't done this for double precision FP, but the same principles apply as for single precision, for which I have implemented division (as multiply by reciprocal). What these FPGAs do have, instead of FPUs, is hardwired DSP/multiplier blocks, capable of implementing a 18*18 or (Virtex-5) 18*25 multiplication in a single cycle. And the larger devices ...


12

In order to how much cap to use where, you need to know a fair bit about capacitors in general: The different types (electrolytic, film, ceramic, tantalum, OS-CON, metalized film, etc.) Their characteristics (impedance, ESR, ESL, polarity, temperature rise, dielectric, etc.) Their failure modes (aging, over voltage, reverse voltage, thermal runaway, etc.) ...


11

The Fourier series: \$ V_t = \dfrac{a_0}{2} + \displaystyle \sum_{i=1}^{\infty}[a_i sin(i \omega_0 t) + b_i cos(i \omega_0 t) ] \$ The term \$\dfrac{a_0}{2}\$ is a constant, that's the DC level. It could also have been written without dividing by two, but this is the convention. The terms of the infinite sum are the sum of a weighted sine and a ...


11

I find I use mostly simple Algebra day to day. Calculating power consumption, currents, resistor values, and thermal issues. For everyday practical circuit design like you're talking about it's more about creative problem solving than math. I'd take a guy who was a good debugger over a good mathematician any day ;) That being said there are days when it ...


11

If you can only take measurement at discrete times, then summing up and dividing by the time between measurements is the only way possible – the integral $$E_\text{total}=\int\limits_{T_\text{start}}^{T_\text{end}} P(t) dt$$ really collapses to a sum, it \$P(t)\$ is only known for set of points. For example, assume the power value is constant for amount ...


11

The decay of an RC parallel circuit with capacitor charged to Vo v(t) = \$Vo(1-e^{-t/\tau})\$ , where \$\tau\$ is the time constant R\$\cdot\$C. So v(\$\tau\$)/Vo is approximately 0.63212055882855767840447622983854 In other words, the time constant is defined by the RC product (or L/R ratio), and the seemingly arbitrary voltage is a result of that ...


11

The standard C library is providing the optimized solutions for many problems with considerations based on the architecture, compiler in use and others. The abs() function defined in stdlib.h is one of these, and it is used for your purpose exactly. To emphasize the point, here is ARM compiler result when using abs vs a version of a homebrew abs: https://arm....


10

I'm not sure that will work well; you probably won't see a factor 5 difference between 5 ml and 25 ml. I would suggest something capacitive. The relative permittivity of water is about 80 times that of air, so you should be able measure a difference in capacitance between two plates with water between them compared to air. So the capacitance will be a ...


10

There are some topics you should pay attention concerning the cable thickness selection: the voltage dropout, cable heating, electro-magnetic interference, impedance matching. The maximum allowed voltage dropout is a good parameter to calculate the cable thickness. In most situations this is the limitation factor and may define alone (without the need to ...


10

The transfer function makes sense only for linear systems (in summary, systems for which \$f(a+b)=f(a)+f(b)\$ and \$a\cdot f(x) = f(a \cdot x)\$). Yours is not a linear system, because it contains nonlinear elements (the diodes, due to the exponential relationship between voltage and current). You can linearize the model at the neighborhood of a certain ...


10

The Laplace variable \$s\$ relates to Fourier's \$j\omega\$ as follows: $$ s = \sigma + j\omega $$ Fourier transform can be seen as a Laplace transform when \$\sigma=0\$. The \$\sigma\$ allows the Laplace integral transformation to converge for signals that Fourier transform does not, e.g. a unitary step (Heaviside function). If you are working with real ...


10

One usually needs to acquire multiple samples per waveform period to get good results from an FFT. The Nyquist limit of 2 samples per period is a lower bound but usually 10 samples per period or more is what is practically used. So to analyze a 64Hz signal you probably want to acquire samples at a rate of 640Hz or more. Also (up to a point) you will get ...


9

To derive this properly from basic principles, you have to work with the amplifier's open loop gain. You seem to be confusing open loop and closed loop gain, or perhaps misunderstanding how op-amps work. The output voltage is not the difference between + and -. It is the difference between + and - multiplied by the open loop gain. In amplifying ...


9

Slow division is inherently iterative so it tends to take longer. There are somewhat faster slow division algorithms than the simple ones, using lookup tables. The SRT algorithm produces two bits per cycle. An error in such a table was the cause of the infamous Pentium FDIV bug (ca. 1994). Then there are so-called fast division algorithms. Of course, in ...


9

We can have multiple layers of logic per clock cycle but there is a limit, exactly how many layers of logic we can have an how complex those layers can be will depend on our clock speed and our semiconductor process. However, there are many different multiplication algorithms, and I don't have a clue which one may be used by microcontrollers Afaict most ...


9

Your main mistake is in not treating acceleration as a single vector. When the car is at rest, that vector will always be 1 g upwards. Don't look at just the X component of the raw accelerometer data. Do the real vector math. But my problem is that when the device is on tilt (0g when no tilt) the acceleration is between (downward) 0g->-1g or between (...


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