How can I realize this nonlinear resistor?

Question

I want to know if it's possible to realize a circuit described by the equation

$$I = \sin(w V)$$

where \$I\$ is the current, \$V\$ the potential and \$w\$ a variable which characterizes the single circuit.

Answer 1

A common way to implement this is to use a piecewise-linear approximation to a sine curve, built using multiple resistors and diodes. One such implementation is described here, regarding the HP3311A function generator.

Answer 2

A bit like a SQUID, which has a (ideally) sinusoidal response.

If your function is monotonic you could use some simple nonlinear circuit arrangement.

If your allowable values of wV cause the current function to be multivalued you could use a voltage-driven LUT (like an arbitrary function generator) and current source, but of course the resulting current would be dependent on the voltage history and initial conditions.

Answer 3

If what you truly want is a resistor, consider using a motor with a lever arm. The motor position (which can be servo-controlled to be a number of clockwise turns equal to V *W/(2*pi) ) will then put the end of the lever arm at an excursion which can be linkage-connected so that it drives a variable resistor. The negative resistance cases will be a bit of a problem, but a current driven positive or negative is a relatively easy thing to arrange (the resistor can span +1 to -1 volts, and a voltage-controlled current source driven from it).

A related probem, building a sinusoidal voltage-controlled oscillator, requires only proportioning a frequency to the input voltage, not holding any absolute phase relationship.

Answer 4

In theory you can look at the taylor series expansion of \$\sin x\$ which is $$ \sin x = \sum_{n = 0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \pm \dots$$

Using a set of transistors / diodes all in their nonlinear region you somehow can hack together something that fits this curve quite tightly, at least in the interval \$[0;\pi/2]\$.

You asked for a circuit that essentially emulates a passive element with the V/I characteristic of $$I(V) = \sin(\omega V)$$

Where you get problems, is, when the sine expression is negative. In this case your "resistor" has to somehow generate power, which requires some sort of active circuit.

Maybe someone will suggest a nice analog way to do this. However, I spent the last minutes in spice and couldn't get a decent result.

So I will go with another approach. Essentially we will build an electrically controlled load that emulates a nonlinear load with your specified characteristic. It can (easily, depends on your know-how) be build with an of the shelf microcontroller and an ADC/DAC which some controllers already contain.

The opamp, transistor and the shunt resistor form a constant current source. It is controlled by the DAC connected to your controller. Using an ADC the controller measures the current voltage at the input terminal and calculates the desired output current and sets it using the DAC.

This is just a quick sketch. But this is essentially an easy way, that doesn't involve a lot of nonlinear circuits and finetuning. The curve can be cahnged anytime in the microcontrollers source code.

A downside of this circuit is: It will not handle negative input voltages or negative output currents. It is only usable for input voltages between 0 and PI.

Depending on your Opamp, ADC, DAC and controller, there is a maximum bandwidth this whole control loop can handle. For low frequency input voltages it will be just fine. But if you want to use a 100 MHz sine input, this circuit is definitely not what you want.