# Theory for linearizing behaviour in circuits

#### AF_Maxwell

Joined Dec 12, 2018
34
So suppose you have behaviour in a circuit that depends on a variable capacitance

This capacitance follows an inverse log law such that there is an asymptote and discontinuity in the variable value for capacitance

If you wanted to get rid of this asymptote and linearize the capacitance variation how would you go about it?

My idea is to turn the capacitance into a voltage and then linearize that then turn the result back into a capacitance

The issue is of course the conversion but also the mathematics, what function would you apply to an inverse logarithm 1/ln(x) to make it linear and how would you do this with circuits

I don't want the answer but perhaps a push in the right direction, what area of circuit theory should i be looking at

Analog circuits only btw!

#### Papabravo

Joined Feb 24, 2006
20,604
There are two basic steps:
1. Choose a DC operating point. What will the voltages and currents be when there are no AC sources?
2. Introduce a restricted magnitude AC source, and ignore all 2nd order and higher terms.
Any 1st order approximation to 1/ln(x) will do just fine. Use the Taylor series for ln(1+x) to find the approximation

#### Delta Prime

Joined Nov 15, 2019
1,311
Hello there
If you want to approximate this capacitance into a voltage and then linearize that then turn the result back into a capacitance to first order, it just means that you use up to the term and scrap the rest, meaning that...which is a first-order Taylor series approximation . It's a worse approximation than, say, the 2nd- or 3rd-order approximation, but it's easier to work with if accuracy isn't that important.

#### AF_Maxwell

Joined Dec 12, 2018
34
this feels like a stupid question but how exactly do you do the approximation with physical circuitry

#### Wolframore

Joined Jan 21, 2019
2,593
why does this sound like a homework question?

#### MrChips

Joined Oct 2, 2009
29,848
A capacitor charges at inverse exponential from a constant voltage source.
A capacitor charges linearly from a constant current source.

#### Wolframore

Joined Jan 21, 2019
2,593
I bet it can be done with an opamp and diode.

#### Papabravo

Joined Feb 24, 2006
20,604
this feels like a stupid question but how exactly do you do the approximation with physical circuitry
You replace the exponential function with a power series around the operating point and ignore higher order terms.

#### AF_Maxwell

Joined Dec 12, 2018
34
I've decided to pursue translinear circuits, before I attempt to simulate them I was wondering if anyone knew if LT Spice was capable of such a thing or if I should find a more appropriate simulation tool

#### AF_Maxwell

Joined Dec 12, 2018
34
You replace the exponential function with a power series around the operating point and ignore higher order terms.
okay my attempt with simulating translinear circuits failed and i still can't understand what this means

what kind of circuit do you use to implement this, an op-amp, some bjts? whats the name of the theory that applies this idea so that i can go and read about it (i don't mean the taylor series, although i am confused how you would "ignore the higher order terms" practically)

what would the general block diagram look like with physical implementation?

My question is quite similar to the following question you replied to back in 2006 https://forum.allaboutcircuits.com/threads/analog-linearization-technique.3961/

#### Papabravo

Joined Feb 24, 2006
20,604
okay my attempt with simulating translinear circuits failed and i still can't understand what this means

what kind of circuit do you use to implement this, an op-amp, some bjts? whats the name of the theory that applies this idea so that i can go and read about it (i don't mean the taylor series, although i am confused how you would "ignore the higher order terms" practically)

what would the general block diagram look like with physical implementation?

My question is quite similar to the following question you replied to back in 2006 https://forum.allaboutcircuits.com/threads/analog-linearization-technique.3961/
So to give you some topics to research:
1. Perturbation theory ->> https://en.wikipedia.org/wiki/Perturbation_theory
2. Linear algebra ->> https://www.analyzemath.com/applied_mathematics/electric_circuit_1.html
3. SPICE engines --> https://www.imperial.ac.uk/pls/portallive/docs/1/7292571.PDF and https://en.wikipedia.org/wiki/SPICE (reference list at end of article)
In practice any circuit with linear behavior can operate around an operating point and produce useful results.
In the field of aircraft control the question is often asked "are there ANY good non-linearites?" The consensus answer, the last time I checked, was no. When it comes to aircraft control there don't seem to be any that improve the pilot's ability to properly control the flight characteristics of an air frame. I should note that this discussion is completely separate from the control (fly-by-wire) of an inherently unstable air frame; a flying wing for example.

In electronics, the situation is less clear, since we have aficionados of various stripes that propound the superiority of the sound emanating from an over driven pentode.

#### Tesla23

Joined May 10, 2009
537
If you want to understand how to simulate non-linear circuits, I'd suggest you start with the enormous amount of work that was done with SPICE-like programs. A good book that explains the methods in SPICE is "Fundamentals of Computer-Aided Circuit Simulation" by McCalla. It's relatively short and to the point, maybe you can find it in a library.

There are lots of publications around SPICE, Nagel's thesis is available, but some of the books do a better job with the explanations.