5 ms·

Taylor Series and Accelerometers

67 points by signa11 6y ago

deleted 6y ago
[deleted]
kayson 6y ago
Using differences to avoid non-linearity is a pretty common technique used across EE, especially circuit design. Transistors are very non-linear and circuits like amplifiers have problematic squared terms. However, they're almost always built "differentially": there are both positive and negative input terminals, and positive and negative output terminals. The difference of the output voltages is an amplified version of the difference of the input voltages. Because the squared term at each terminal is the same polarity, subtracting them cancels it out very well. In practice, you're limited by how well the positive and negative paths match, and the mismatch allows some second order term to leak out. Unfortunately, this does not help with the third order terms. See: https://en.wikipedia.org/wiki/Differential_amplifier https://en.wikipedia.org/wiki/Differential_amplifier https://en.wikipedia.org/wiki/Third-order_intercept_point https://en.wikipedia.org/wiki/Third-order_intercept_point
ISL 6y ago
To answer the post's final question: Yes. It is quite common to design instrumentation to cancel systematic uncertainties or nonlinearities to leading order. Fancier arrangements go to higher orders. In the particular case of the differential capacitor, that form of differential measurement is particularly common -- if one non-linear system can paired with a symmetric partner with equal and opposite nonlinearity, the leading-order non-linearities are suppressed (to the extent that the matched pair are actually matched). An easy-to-understand example of the compensation of linear effects is the temperature-compensated pendulum clock: https://en.wikipedia.org/wiki/Gridiron_pendulum https://en.wikipedia.org/wiki/Gridiron_pendulum
seamyb88 6y ago
> Nb., I’m not sure why it doesn’t depend on the material the plate is composed of, but I imagine it’s smooshed into the dielectric constant if necessary It isn't smooshed into this. Capacitance is the Charge per Volt. Hence, capacitance says nothing about the total charge or the total voltage (which depend on plate material), just the ratio of these for given plates.
compumike 6y ago
Differential measurements cancelling out unwanted terms is a common "trick" in electronics, and explains why Ethernet cables come wired as 4 twisted pairs. Along similar lines, the "Translinear Principle" [1] may be another fun mathematical-relationship-turned-practical-electronics-tool to explore: it takes advantage of the logarithmic relationship between voltage and current in a PN junction -- specifically a bipolar junction transistor (BJT). Since log(X) + log(Y) = log(X*Y), you can make very simple analog circuits which compute, for example, a square [2] or a square root [3] using just 4 transistors. These can actually be quite high-performance, low-power-consumption circuits. (JavaScript simulations attached; just click the link and then "Run DC Sweep".) [1] https://en.wikipedia.org/wiki/Translinear_circuit https://en.wikipedia.org/wiki/Translinear_circuit [2] https://www.circuitlab.com/editor/33yksg85qf3s/?mode=simulate https://www.circuitlab.com/editor/33yksg85qf3s/?mode=simulat... [3] https://www.circuitlab.com/editor/rfc7h79pvgtb/?mode=simulate https://www.circuitlab.com/editor/rfc7h79pvgtb/?mode=simulat...
- cosmojg 6y ago
  Why haven't analog circuits taken off yet in mobile computing and other spaces? They seem great for things like machine learning and possibly superior to traditional digital circuits. What are the trade-offs?
  - stefan_ 6y ago
    Huge power consumption, extraordinary costs for precision components.
    - dreamcompiler 6y ago
      Power is a big deal with analog. Power wasted in heating a device is (I^2)R where I is the current through and R is the effective resistance of the device. Power wasted is obviously 0 if R=0. Because I=V/R, power wasted is also 0 if R is very high (because in that case I approaches 0). Power wasted is highest if R is between 0 and "very high." This third case is where analog electronics like to live. The first two cases are where digital electronics lives. And that's why switching power supplies and Class D amplifiers are so efficient: Their effective R values spend most of the time near zero or "high" but not much time in between. This is also why even in digital circuits higher clock speed means more power: More transistor transitions between "fully on" and "fully off" mean more time spent in between in the high power dissipation regime.
random314 6y ago
A cleaner derivation would be 1/(1-d) - 1/(1+d) = 2d/(1-d^2) followed by Taylor series, but the conclusion is obvious even without the Taylor series.

67 points by signa11 6y ago

deleted 6y ago

[deleted]

Using differences to avoid non-linearity is a pretty common technique used across EE, especially circuit design. Transistors are very non-linear and circuits like amplifiers have problematic squared terms. However, they're almost always built "differentially": there are both positive and negative input terminals, and positive and negative output terminals. The difference of the output voltages is an amplified version of the difference of the input voltages. Because the squared term at each terminal is the same polarity, subtracting them cancels it out very well. In practice, you're limited by how well the positive and negative paths match, and the mismatch allows some second order term to leak out. Unfortunately, this does not help with the third order terms. See: https://en.wikipedia.org/wiki/Differential_amplifier https://en.wikipedia.org/wiki/Differential_amplifier https://en.wikipedia.org/wiki/Third-order_intercept_point https://en.wikipedia.org/wiki/Third-order_intercept_point

ISL 6y ago

To answer the post's final question: Yes. It is quite common to design instrumentation to cancel systematic uncertainties or nonlinearities to leading order. Fancier arrangements go to higher orders. In the particular case of the differential capacitor, that form of differential measurement is particularly common -- if one non-linear system can paired with a symmetric partner with equal and opposite nonlinearity, the leading-order non-linearities are suppressed (to the extent that the matched pair are actually matched). An easy-to-understand example of the compensation of linear effects is the temperature-compensated pendulum clock: https://en.wikipedia.org/wiki/Gridiron_pendulum https://en.wikipedia.org/wiki/Gridiron_pendulum

seamyb88 6y ago

> Nb., I’m not sure why it doesn’t depend on the material the plate is composed of, but I imagine it’s smooshed into the dielectric constant if necessary It isn't smooshed into this. Capacitance is the Charge per Volt. Hence, capacitance says nothing about the total charge or the total voltage (which depend on plate material), just the ratio of these for given plates.

compumike 6y ago

Differential measurements cancelling out unwanted terms is a common "trick" in electronics, and explains why Ethernet cables come wired as 4 twisted pairs. Along similar lines, the "Translinear Principle" [1] may be another fun mathematical-relationship-turned-practical-electronics-tool to explore: it takes advantage of the logarithmic relationship between voltage and current in a PN junction -- specifically a bipolar junction transistor (BJT). Since log(X) + log(Y) = log(X*Y), you can make very simple analog circuits which compute, for example, a square [2] or a square root [3] using just 4 transistors. These can actually be quite high-performance, low-power-consumption circuits. (JavaScript simulations attached; just click the link and then "Run DC Sweep".) [1] https://en.wikipedia.org/wiki/Translinear_circuit https://en.wikipedia.org/wiki/Translinear_circuit [2] https://www.circuitlab.com/editor/33yksg85qf3s/?mode=simulate https://www.circuitlab.com/editor/33yksg85qf3s/?mode=simulat... [3] https://www.circuitlab.com/editor/rfc7h79pvgtb/?mode=simulate https://www.circuitlab.com/editor/rfc7h79pvgtb/?mode=simulat...

cosmojg 6y ago
Why haven't analog circuits taken off yet in mobile computing and other spaces? They seem great for things like machine learning and possibly superior to traditional digital circuits. What are the trade-offs?
- stefan_ 6y ago
  Huge power consumption, extraordinary costs for precision components.
  - dreamcompiler 6y ago
    Power is a big deal with analog. Power wasted in heating a device is (I^2)R where I is the current through and R is the effective resistance of the device. Power wasted is obviously 0 if R=0. Because I=V/R, power wasted is also 0 if R is very high (because in that case I approaches 0). Power wasted is highest if R is between 0 and "very high." This third case is where analog electronics like to live. The first two cases are where digital electronics lives. And that's why switching power supplies and Class D amplifiers are so efficient: Their effective R values spend most of the time near zero or "high" but not much time in between. This is also why even in digital circuits higher clock speed means more power: More transistor transitions between "fully on" and "fully off" mean more time spent in between in the high power dissipation regime.

random314 6y ago

A cleaner derivation would be 1/(1-d) - 1/(1+d) = 2d/(1-d^2) followed by Taylor series, but the conclusion is obvious even without the Taylor series.