Once answered what is "passive" and what "active" in electronics,
https://www.researchgate.net/post/What_is_passive_and_what_active_in_electricity_and_electronics_Can_we_make_passive_elements_become_less_passive_neutral_or_even_active?
we should easily answer what is "linear" and what - "nonlinear"... and even see if there are some relations between these four fundamental categories. Some "genius" speculations-:) are already proposed by Gregg and Lutz in the question above.
Actually, the main question here is, "What is a nonlinear resistor?" My deep inner conviction is that only to show the nonlinear IV curve is not enough to answer this question; we have to show how this odd curve is obtained. I have been thinking about this problem since the late 80's when I began marveling at the weird negative resistance phenomenon. Thus I have finally created my own "philosophy" about the nonlinear phenomenon. But I have not yet got a satisfactory confirmation of my speculations from some "scientific luminary" just because famous persons, blinded by fame, do not have time to answer (or are confused by) such unsolicited questions...
My notion of a "linear resistor" (the simplest case) is that this is an element with steady, constant, static, invariable (toward the input quantity variations) resistance; i.e., "linear" means "static" for me. When we change the input quantity (e.g., the voltage across the resistor), the moving operating point draws the real IV curve of this humble element (ohmic resistor) - a line beginning from the coordinates origin and graphically representing the Ohm's law:
http://www.circuit-fantasia.com/my-students/ske2004/classes/class1/v-to-i-old-page6-3.html
This linear resistor can be a "variable linear resistor" (a rheostat) with the reservation that we will not change its resistance together with the input quantity changes. Its IV curve remains a line beginning from the origin but it only changes its slope according to the instant linear resistance.
But if, at some moment and for some reasons, we break our promise to keep the resistance steady while the input quantity wiggles, the operating point changes its trajectory. Now it begins drawing a new imaginary, artificial, virtual, "nonlinear" IV curve with some new (vertical, horizontal, negative, whatever...) slope.
So, my notion of a "nonlinear resistor" is that this is an element with a kind of "self-variable" (toward the input quantity variations) resistance. Thus I present the more complex nonlinear element as a modified simple linear element; i.e., "nonlinear" means "dynamic linear" for me. To show graphically this "dynamization", I superimpose two IV curves (lines) on the same coordinate system: the first - of the instant ohmic resistance; the second - of the input source. In contrast to the linear graphical representation, where only the input source IV curve moves, here both the curves simultaneously move and, as a result, the operating point draws the new nonlinear IV curve. I have developed this idea to the utmost in the Wikibooks story about the negative differential resistance:
http://en.wikibooks.org/wiki/Circuit_Idea/Negative_Differential_Resistance
My questions are: Is it reasonable to present the nonlinear resistor as a dynamic linear resistor? Can we illustrate its operation in this graphical way? Is this the real situation or only a model?
The quantity R is simple the ratio of voltage to current. If the ratio does not change when voltage or current vary, then the resistor is linear. If it does, then it is nonlinear. Many resistors of fixed value will have a very small voltage coefficient of resistance. In this case they are essentially linear. Resistors that are deliberately nonlinear can have large swings in resistance. In this instance two resistance definitions apply. Static resistance is the slope of a line from the origin to the operating point. Dynamic resistance is the slope of a tangent curve at the operating point, or dV/dI. This can be positive or negative depending on how the resistance is created. It can be treated as if it were linear if the change of slope is slight compared to the signal swing. It is possible, for instance, to use a large DC signal to set the apparent resistance seen across the nonlinear resistor by a small AC signal.
There is also the concept of a time variable resistor, for instance a potentiometer that is being adjusted. At any given instant the wiper to end resistance is linear but the value varies from instant to instant. Linearity or nonlinearity are independent concepts from time variability. If it were arranged to dynamically vary the potentiometer according to instantaneous voltage or current, that would emulate a nonlinear resistor, but the insertion of a control system to do so is an artificial means.
Dear Cyril; You and I wrote -
I wonder if we can accept without reservation your explicit formulation - "a passive element is any circuit element that cannot provide signal gain"...
You made me ask yourself why transistors have to be nonlinear to amplify and why, "if they were truly linear we would have no need for the small-signal model". IMO only the 2-terminal elements (diodes) should be nonlinear (to have a negative differential resistance) to amplify...
Good question. To rephrase, how about - why would nonlinearity be a necessary condition for amplification?
nonlinear dynamics is a huge field, so briefly, a nonlinear system has many defining properties. One important difference is the question of uniqueness. For a linear element there will always be one, and only one, unique solution for a given set of boundary conditions. This is called homogeneity. This is not always the case in nonlinear circuits. For instance, a linear resistor with a fixed voltage applied to it has only one solution for the current through it. On the other hand, a tunnel diode may have 3 solutions for the current for a given voltage. That is, a particular solution for the current through the diode is not unique; there may be others which are equally valid. In some cases there may not be a solution at all: thus the question of existence of solutions must be considered as well.
Another important consideration is the question of stability. A particular solution may exist, but it may not be stable, rapidly departing from that point at the slightest stimulation. in this case we can create stable oscillations, so long as we have a device that is capable of at least second order behavior. Such a system, when analyzed in it's phase domain, will possess fixed limit cycles - orbits (attractors). It is interesting though that It can be shown that any device absolutely stable for all conditions must have one, and only one, solution for each set of conditions. This is the so called fixed point, or one dimensional NL system. Thus a nonlinear element operating in a perfectly stable region is able to take on the characteristics of a linear element. This is exactly what we see whenever we bias a TD or transistor to some desired operating point in it's transfer function and limit the input signal such that we never stray far from that point of operation.
So to answer your question about whether it is valid to model the TD as a variable resistor, I would say yes based on the picture you drew. By limiting the bias range of Vne, the device is forced into linear operation in the sense that the I-V relationship is 1-1. The other two solutions are ignored.
Well - in order to stimulate the discussion I like to summarize my views concerning linear/non-linear etc:
1.) Strictly spoken, there are no elements/parts/circuits which really are linear. Linearity is a mathematical term that cannot be applied to real electronics.
2.) However: It is good engineering practice to treat some elements as linear as long as the deviations between linear and non-linear are acceptable and are below the application specific requirements.
3.) Thus, using the term "linear" one should know that the real meaning is "quasi-linear" (or "linear enough"). Otherwise, one always would have to mention the "degree of non-linearity". (As another example: In the field of electronics there is no formula that is correct by 100% - but we use, nevertheless, the equality symbol "=").
4.) In this sense: A linear part (example: resistor) has a static resistance (ohmic resistance), which is identical to the differential (dynamic) resistance of the device.
5.) A part having a non-linear characteristic (example: pn-diode) has two different resistive properties:
(a) R(static)=DC voltage-to-DC current ratio
(b) r(dyn)=ac voltage-to-ac current ratio (around a fixed dc bias point).
Comment: Both resistors depend on the selected DC values (bias point)
6.) Explanation to 5(b): This is a small-signal value only - identical to the slope of the non-linear V-I characteristic (differential quotient).
7.) Finally (with respect to earlier discussions): The terms linear/non-linear cannot serve as a criterion for the classification active/passive.
Lutz vW
The quantity R is simple the ratio of voltage to current. If the ratio does not change when voltage or current vary, then the resistor is linear. If it does, then it is nonlinear. Many resistors of fixed value will have a very small voltage coefficient of resistance. In this case they are essentially linear. Resistors that are deliberately nonlinear can have large swings in resistance. In this instance two resistance definitions apply. Static resistance is the slope of a line from the origin to the operating point. Dynamic resistance is the slope of a tangent curve at the operating point, or dV/dI. This can be positive or negative depending on how the resistance is created. It can be treated as if it were linear if the change of slope is slight compared to the signal swing. It is possible, for instance, to use a large DC signal to set the apparent resistance seen across the nonlinear resistor by a small AC signal.
There is also the concept of a time variable resistor, for instance a potentiometer that is being adjusted. At any given instant the wiper to end resistance is linear but the value varies from instant to instant. Linearity or nonlinearity are independent concepts from time variability. If it were arranged to dynamically vary the potentiometer according to instantaneous voltage or current, that would emulate a nonlinear resistor, but the insertion of a control system to do so is an artificial means.
A resistor is a voltage source controlled by the current imposed on the source i.e. the voltage is a function of the current. If the ratio between the voltage and the current is constant in a certain time interval the resistor is linear in this time interval.
Hi Erik,
I am happy to hear from you again.
However, I am a bit confused about the "time interval".
What about the voltage-to-current ratio for a pn diode (for example 0.65V/1mA), which is constant over - let`s say - 1 minute. Is the I-V characteristic of the diode linear?
I think, this violates the definition of linearity, does it not?
Lutz
Electrical resistance is used as a means to model the heat generated in any material when subjected to EM fields. Whilst in many materials as Orin notes resistance is found to be a linear material property even when it becomes heated it is used as the basis to define the non-linear process of heat generation within the material itself. The source of heat is the electrical power (I^2*R) which when dissipated within the material may itself be non-linear. Furthermore, if the material is unable to dissipate such heat its material properties and hence resistive properties will change and become non-linear in the process. Just explore the dynamics of any fuse material. So in the end the complexity of the model depends on the process trying to be modelled. However despite this basic laws such as KVL and KCL will not be violated.
On a lighter side are you aware of this new and much simpler electrical theory which suggests that all electrical devices and equipment work on smoke. There is conclusive evidence, even amongst lay people, that once this smoke is seen to escape, electrical equipment no longer works. I have not yet been able to disprove this theory.
Erik, I hope you are not saying "a resistor is a voltage source" seriously... Really, it can be simulated by a "self-current-controlled voltage source" that is contrary connected to the main voltage source, and this will be a kind of a virtual resistor... Cyril
Hi Cyril:
In our circuit simulation programs we model the lumped elements as controlled sources. We may control with signal or time derivative of signal. An inductor is a voltage source controlled by the time derivative of the imposed current. A capacitor is a current source controlled by the time derivative of the imposed voltage. Our equations are the extended node equations with node-voltages and impedance branch currents as variables.
Hi Lutz,
Nice to hear from you too. A linear element (two terminal) is a component for which the ratio of voltage and current is constant for all times e.g V/I = R and I/V=G.
A diode is a current source controlled by the imposed voltage. If the voltage is constant over a certain time interval you observe a constant current in this time interval i.e. the diode is a linear conductance in this time interval. For a linear capacitor you have charge Q = C times V ;-)
Please note that electrical and electronic circuits are fractal patterns of coupled oscillators if you take all parasitic into account e.g. a resistor may be modeled as a linear damped oscillator. Steady state oscillators are nonlinear circuits. Chaotic behavior is more natural than limit cycle behavior.
Quote Erik: "If the voltage is constant over a certain time interval you observe a constant current in this time interval i.e. the diode is a linear conductance in this time interval."
Hi Erik, I have some problems to follow this definition of linearity.
Doesn`t the classical definition of linearity include Delta-values (small voltage and current variations)?
Hi Lutz,
The diode is of-course a nonlinear component but the behavior may be observed as linear in a time interval. If you by Delta-values mean step and pulse variations to me they are nonlinear. What is a small variation ? Noise is nonlinear to me !
do I see a cat amongst the pigeons!
Erik what you are stating is fairly "standard" practise in simulation and one may readily use such representations at small time steps as you have noted. However basic conservation laws like KVL and KCL are not violated.
Hello Erik,
Here comes the definition of linearity - as can be found in "Signals and Systems" (Oppenheim/Willsky):
When y1(t) and y2(t) are the responses to x1(t) resp. x2(t), the system is linear if:
* the response to x1(t)+x2(t) is y1(t)+y2(t)
* the response to a*x1(t) is a*y1(t) where a is a constant.
_______________
The first condition involves the well known superposition theorem and the second one a change of the input variable (voltage variation in case of a diode).
Thus, if I observe that the voltage-to-current ratio is constant over a certain time interval, I have no information about linearity (assuming a constant voltage, as you have mentioned). Or am I wrong?
Hi Lutz,
You are of course right concerning the basic definition of linearity. Sorry for my mistake !!! :-(
Erik - don`t mention it.
Such things happen from time to time - some time ago I have confused transconductance with transresistance (and there are other examples which are even worse: Yesterday I have confused the diffusion capacitance of a diode with the diffusion voltage).
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