To build electronic circuits, first of all we need the natural electrical elements resistors, capacitors and inductors. However, in many cases we are not satisfied with the performance of these passive components and try to improve them in an artificial way. For this purpose, in electronics we have been inventing a variety of clever and sophisticated techniques to create artificial (synthetic, virtual) elements. The question is, "How do we make virtual elements?"
Like magicians, in electronics we convert the imperfect passive elements into perfect active "elements" (by applying the virtual ground configuration)... or we transform some element (a capacitor) into its dual (an inductor) by swapping the voltage across and the current through it (gyrators)... or we transmute the passive circuits into their opposite mirror doubles (negative impedance)... or we even create completely new electrical elements (memristors)... Thus, for some reasons, we frequently replace the natural electrical elements by their circuit equivalents - a gyrator, multiplier, memristor, negative resistor (capacitor, inductor...)
It is important to note that all these virtual "elements" (electronic circuits) emulate only particular properties (usually, the time behavior) of the genuine elements... they are not real, they are just an illusion...
Genuine elements. The general property of passive electrical elements is taking (consuming) energy from the input source; resistors dissipate this energy while capacitors and inductors store (accumulate, "steal") it. But how do they do it?
Let's assume the considered passive element is connected in series to the exciting voltage source. What does it do in this case? It subtracts a portion of voltage from the whole input voltage: the resistor "creates" an opposing voltage drop across itself while the capacitor and inductor "create" an opposing voltage (a kind of emf). Resistors do this by throwing out (dissipating) energy while capacitors and inductors do it by taking energy from the excitation source, accumulating it into itself and setting it against the input source. In the first case there is a voltage drop while in the second case there is a voltage (emf).
So, we can emulate these passive elements by replacing them with some other elements producing the same opposing voltage (having an opposite to the input voltage polarity when travelling along the loop). Then, we can modify or even create mirror active (negative) "copies" of these passive elements by replacing them with sources producing the same but now "helping" voltage (having the same as the input voltage polarity when travelling along the loop). This is the main idea of the substitution and inverse substitution theorem perfectly considered by Prof. Lutz von Wangenheim in his work:
https://www.researchgate.net/file.PostFileLoader.html?id=540dd638d039b1ee348b458f&key=d62e286c-fe94-4019-b22b-4b550b6c8bfa
* Emulating by (varying) voltage. First, we may replace the original elements by varying voltage sources and this is the most natural way of making emulated capacitors and inductors (as they behave as varying through time voltage sources). Op-amp gyrator, multiplying, memcapacitive and meminductive circuits do it in this way. In these circuits, the op-amp output voltage represents the voltage across the according capacitor or inductor.
In the case of the true negative resistor, the ordinary ohmic resistor is replaced again with a voltage source (exactly as in the case of gyrators and multipliers) but it has the same polarity as the input voltage source so that it adds an additional voltage to the input voltage. For example, the negative impedance converter with voltage inversion (VNIC) is a dynamic voltage source emulating a negative resistor by adding a voltage that is equal to the voltage drop across a real ohmic resistor.
It is interesting that we can change the properties of the ordinary constant voltage source by properly varying its voltage (as in the attached picture below).
The emulation by including an additional voltage source is the basis of the Miller theorem (see https://en.wikipedia.org/wiki/Miller_theorem#Applications).
* Emulating by (varying) resistance. But a memristor can do the same by replacing the voltage by an equivalent voltage drop across a dynamic time-dependent resistor. Transistor gyrator and multiplying circuits do it in a similar way.
It is interesting that we can change the properties of the ordinary ohmic resistor by properly varying its resistance. A good example of this technique is the creation of the negative differential resistor:
https://en.wikibooks.org/wiki/Circuit_Idea/Negative_Differential_Resistance#General_considerations_about_creating_the_NDR
So, to emulate passive elements (to create virtual elements), we may replace the elements behaving as resistors with (properly varying) resistors, and elements behaving as sources - with (properly varying) sources. But it seems we can do it by swapping these correspondences - replacing the elements behaving as sources with resistors, and elements behaving as resistors with sources... Am I right? Please, discuss.
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I was inspired to ask this question mainly by the numerous discussions between me and Prof. Lutz von Wangenheim mostly in the questions below...
https://www.researchgate.net/post/Are_electrical_sources_elements_with_static_negative_impedance_If_so_is_there_any_benefit_from_this_concept
https://www.researchgate.net/post/Does_the_op-amp_in_all_the_inverting_circuits_with_negative_feedback_behave_as_a_negative_impedance_element_negative_resistor_capacitor_etc
https://www.researchgate.net/post/Does_the_amplifier_in_negative_feedback_systems_possess_negative_impedance?
...and especially, by his idea about the inverse substitution theorem (IST) proposed by him in the question below:
https://www.researchgate.net/post/Can_we_formulate_Kirchhoffs_laws_for_resistances_KRL_and_conductances_KGL_based_on_KVL_and_KCL/2
So, to emulate passive elements (to create virtual elements), we may replace the elements behaving as resistors with (properly varying) resistors, and elements behaving as sources - with (properly varying) sources. But it seems we can do it by swapping these correspondences - replacing the elements behaving as sources with resistors, and elements behaving as resistors with sources... Am I right? Please, discuss.
Cyril - I am not sure if I really understand the main question you have asked. Do you refer with the above cited text to the substitution theorem (and its inverse) only? Or is it something else?
Dear Lutz, I have just added a reference to the theorem:
"This is the main idea of the substitution and inverse substitution theorem perfectly considered by Prof. Lutz von Wangenheim in his work:
https://www.researchgate.net/file.PostFileLoader.html?id=540dd638d039b1ee348b458f&key=d62e286c-fe94-4019-b22b-4b550b6c8bfa"
I mentioned the theorem but I do not think the reference to it is absolutely necessary. To replace a resistor (capacitor, inductor ...) with battery and vice versa is so simple, clear and intuitive idea that it does not require special theoretical evidence...
"To replace a resistor (capacitor, inductor ...) with battery and vice versa...."
The substitution theorem applies not only to parts but also to complete branches containing more than one element only. And - in case of frequency-dependent impedances - we need voltage/current sources with phase shift capabilities (instead of batteries).
Maybe, the idea of virtual elements seems a bit strange to traditionally minded people in electronics... and it would be well if we illustrate it by some clear examples...
I have had many times the opportunity to convince myself that the truth about circuit phenomena is hidden sooner in the circuit evolution than in the final perfect circuit solutions. So, I suggest to you to trace the metamorphosis of а typical virtual element - the true voltage-inverted (S-shaped) negative resistor from the initial natural ohmic resistor with high positive resistance to the final virtual negative resistor with negative resistance... to see the evolution of the bare ohmic resistance in the succession high ohmic > decreased > zeroed > S-negative... This would be a fancy story about how to decrease virtually the resistance...
The best way to see how such weird electronic devices operate is to place ourselves in their place and begin performing their functions... to emulate their behavior by an empathy. Then how do we emulate a virtually decreased resistor?
Following the powerful Miller idea (modifying the impedance by inserting an additional voltage), we have just to connect a variable voltage source VH in series to a constant resistor R to obtain the emulating setup shown in the attached figure. And as I frequently suggest, let's do it as a funny game: you will control the input current source, and I will control the "helping" voltage source... you are the source, I am the virtual resistor:) Thus I, the voltage source and the resistor combine to form a "man-controlled" virtual resistor and you drive it with current...
https://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance#A_setup_for_emulating_an_S-shaped_negative_resistor
We can present graphically the circuit operation (the circuit KVL equation VA = VH - IA.R) of the simple electrical circuit above if we consider that the voltage across the two elements (the input current source and the virtual resistor) is the same and the current flowing through them is the same as well.
For this purpose, we have to superimpose their IV curves on the same coordinate system: the IV curve of the virtual resistor consisting of the "helping" voltage source and the ohmic resistor is an inclined line with a slope depending on the ohmic resistance; the IV curve of the input current source is a horisontal line vertically shifted from the X-axis. The intersection point A (alias operating point) represents the instant magnitudes of the current IA and the voltage VA.
When the input current increases (changes from the most negative to the most positive value), the IV curve of the input current source moves up; the operating point moves up from point 0 to point 7 along the S-shaped IV curve below and gradually draws the curve. To help the understanding of the operation, we will draw step-by-step the particular segments of the IV curve and will explain how they are obtained. A light grey colored guide line will direct us during our "excursion" showing the trajectory of the operating point...
1. High positive resistance (0-1). In the beginning, I set the maximum positive voltage VH and keep it constant while you begin increasing (in the sense that you make it more positive continuously the input current IIN. As a result, a voltage drop VR appears across the resistor R. According to Ohm's law, it is proportional to the current passing through the resistor - VR = R.IIN. The difference VH - VR (negative) appears across the whole element (the future virtual resistor). Note this voltage depends only on the input current and the Ohm's law equation is a function of one variable.
On the graphical representation (the attached picture below), when you vary the current IIN of the input current source, its (your) IV curve moves vertically remaining parallel to itself (i.e., it translates). As a result, the operating point A slides over the IV curve of the ohmic resistor R from point 0 to point 1 that is a straight line. The slope of the R IV curve represents graphically the value of the ohmic resistance R... and this is a real, static, ohmic, "positive" resistance...
https://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance#High_positive_resistance
2. Virtually decreased resistance (1-2). Now lets decrease virtually the resistance R in the exotic Miller way. Imagine when you reach the point 1 (in the attached picture below), I decide to help you along the whole section 1-2 as follows...
When you increase the current IIN of the input source from point 1 to point 2, its IV curve translates upward. But, at the same time, I begin moderately decreasing the voltage VH (and its magnitude) thus helping you to increase the current (to decrease its magnitude). The composed VH-R IV curve moves to the left remaining parallel to itself (translates). As a result, the operating point A slides along a new more vertical IV curve, which represents the new virtual resistance dR1 < R.
Actually, the voltage across this new virtual resistor depends both on the current IIN and the voltage VH, and the Ohm's law equation becomes a function of two variables - VOUT = f(IIN, VH). You have the illusion that the resistance R has decreased and you see new, lower dynamic resistance dR1 < R; thus the initial ohmic resistance R is converted into a smaller virtual resistance dR1.
Note the segment 1-2 is a straight line and it resembles ordinary ohmic resistance. Looking only at this part of the curve, you may think that you investigate an ohmic resistor... but this is just an illusion...
We can observe such virtually decreased resistance in circuits with imperfect parallel negative feedback implemented by an imperfect inverting amplifier with finite gain (usually realized by discrete transistors): transimpedance amplifier, inverting amplifier, etc.
https://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance#Virtually_decreased_resistance
3. Virtually zeroed resistance (2-3). As this Miller idea is so wonderful, let's apply it to zero virtually the resistance R in such a marvellous manner...
Now imagine when you reach the point 2 (in the attached picture below), I decide to help exactly you along the whole section 2-3. You continue increasing (make more positive) the current IIN of the input source from point 2 to point 3 so that its IV curve continues translating up. But now I begin vigorously decreasing the voltage VH so the composed VH-R IV curve quckly moves (translates) to the left. As a result, the operating point A slides along a new vertical IV curve, which represents the new zero virtual resistance dR2 = 0 and you have the illusion the resistance R has become zero... we have created a virtual zero resistor...
This great virtual ground idea may be observed in all kinds inverting op-amp circuits with perfect parallel negative feedback: transimpedance amplifier, active ammeter, inverting amplifier, etc. In all these circuits, the op-amp acts as the compensating voltage source VH - it adds so much voltage to the input voltage as it loses across the resistor connected between the op-amp output and the inverting input. The compensating voltage serves as output voltage supplying the load...
https://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance#Virtually_zeroed_resistance
4. Virtually voltage-inverted (negative) resistance (3-4). Finally, we can go too far enormously strengthening the great Miller idea...
Now, when you reach the point 3 (in the attached picture below), I begin overhelping you along the whole section 3-4. As usual, you are continuously increasing (make more positive) the current IIN of the input source from point 3 to point 4 so that its IV curve is translating up remaining parallel to itself. Following the Miller idea, I am extremely vigorously decreasing the voltage VH so the composed VH-R IV curve extremely quckly translates to the left. As a result, the operating point A slides up along the new IV curve of the inverted positive resistance that is inclined (folded up) to the left and has a negative slope. You have the illusion the resistance R has become a new kind of virtual resistance - true negative resistance dR3 < 0.
This idea is directly implemented in voltage-inversion negative impedance converters (VNIC)...
https://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance#.22Inverted_positive.22_.28S-shaped_negative.29_resistance
The ingeniously simple Miller's idea (inserting a "helping" voltage source in series to an electrical element) has allowed us to create virtual "elements" with artificially decreased, zeroed and voltage-inverted (negative) resistance. But if we reverse the voltage source (make it "opposing"), we can create more virtual "elements" with artificially increased, infinite and current-inverted (N-shaped negative) resistance.
As above, I suggest to you again to trace the metamorphosis of а typical virtual element - the true N-shaped negative resistor from the initial natural ohmic resistor with low positive resistance to the final virtual negative resistor with negative resistance... to see the evolution of the bare ohmic resistance in the succession low ohmic > increased > infinite > N-negative... This would be another fancy story about how to increase virtually the resistance...
We can emulate this type of virtual resistor in a similar way as above - by connecting a variable voltage source in series to the constant resistor R and contrary to the input voltage source. Well, let's do it again as a funny game: you will control the input voltage source; I will control the additional contrary voltage source. Thus I, the contrary voltage source and the resistor combine to form a "man-controlled" virtual resistor and you drive it with voltage (to operate in a linear mode).
https://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance#A_setup_for_emulating_an_N-shaped_negative_resistor
We can present graphically the circuit operation (the circuit KVL equation VA = VO - IA.R) if we consider as above that the voltage across the two elements is the same and the current flowing through them is the same as well. This allows to superimpose their IV curves on the same coordinate system: the IV curve of the combination "voltage source + resistor" is an inclined line (in green color) with a slope depending on the resistance; the IV curve of the input voltage source is a vertical line (in red) horizontally shifted from the Y-axis. The intersection (operating) point A represents the instant magnitudes of the current IA and the voltage VA.
When the input voltage increases (changes from the most negative to the most positive value), the IV curve of the input voltage source moves to right (in contrast to the S-type negative resistor, here the two IV curves move in the same direction); the operating point moves to right from point 0 to point 7 along the N-shaped IV curve and gradually draws the curve. To help the understanding of the operation as above, we will draw step-by-step the particular segments of the IV curve and will explain how they are obtained; the light-grey line will guide us during our "excursion" showing the trajectory of the operating point...
Now connect a voltmeter across the input current source and an ammeter in series with it to monitor the voltage and the current and start the "game"!
5. Low positive resistance (0-1). In the beginning, I set the maximum negaitive voltage VO and keep it constant while you begin increasing (in the sense that you make it more positive) continuously the input voltage VIN. As VIN is more negative than VO, the current flows through the resistor R from the left to the right. According to Ohm's law, it is proportional to the difference between the two voltages. Note the current depends only on the input voltage and the Ohm's law equation is a function of one variable.
In the graphical representation (the attached picture below), when you vary the voltage VIN of the input voltage source, its (your) IV curve moves horizontally remaining parallel to itself (i.e., it translates). As a result, the operating point A slides over the IV curve of the ohmic resistor R from point 0 to point 1 that is a straight line. The slope of the R IV curve represents graphically the value of the ohmic resistance R. This is a real, static, ohmic, "positive" resistance...
https://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance#Low_positive_resistance
6. Virtually increased resistance (1-2). Now Imagine when you reach the point 1 (the attached picture below), I decide to oppose you along the whole section 1-2. When you increase the voltage VIN of the input source from point 1 to point 2, its IV curve translates to the right. But, at the same time, I begin moderately increasing the voltage VO (and its magnitude) thus opposing you to increase the current (to decrease its magnitude). The composed VO-R IV curve translates to the right. As a result, the operating point A slides along a new more inclined IV curve, which represents the new virtual resistance dR1 > R.
The trick is that actually the current through the virtual resistor depends both on the input voltage VIN and the opposing voltage VO, and the Ohm's law equation becomes a function of two variables - IOUT = f(VIN, VO). You have the illusion that the resistance R has increased and you see new, higher dynamic resistance dR1 > R; as though, the initial ohmic resistance R is converted into a higher virtual resistance dR1. Note the segment 1-2 is a straight line and it resembles ordinary ohmic resistance. Looking only at this part of the curve, you may think that you investigate an ohmic resistor... but this is just an illusion...
We can observe virtually increased resistance in imperfect voltage followers made by low gain amplifiers (usually realized by discrete transistors) - emitter, source and cathode followers.
https://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance#Virtually_increased_resistance
7. Virtual infinite resistance (2-3). Let's then apply the modified Miller's idea to make virtual infinite resistance in such a marvellous manner as we made virtual zero resistance above.
So imagine when you reach the point 2 (the attached picture below), I decide to oppose "exactly" you along the whole section 2-3. You continue increasing (make more positive) the voltage VIN of the input voltage source from point 2 to point 3 so that its IV curve continues translating to the right. But now I begin vigorously increasing the voltage VO so the composed VO-R IV curve quckly moves (translates) to the right as well. As a result, the operating point A slides along a new horizontal IV curve, which represents the new virtual infinite resistance dR2 = ∞ and you have the illusion the resistance R has become infinite...
We can see this clever trick in the Ohm's equation - I = (VO - VIN)/R. In the numerator, the two voltages change with the same rate and direction; so, their difference and respectively the current, stays constant.
This great idea is known as bootstrapping. It may be naturally observed in perfect op-amp voltage followers. It is frequently used in bootstrapped amplifiers, constant current sources and load cancelers to increase extremely their input impedance.
https://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance#Virtual_infinite_resistance
8. Virtually current-inverted (negative) resistance (3-4). Finally, we can go too far enormously strengthening the modified Miller's idea...
Now, when you reach the point 3 (in the attached picture below), I begin "overopposing" you along the whole section 3-4. As usual, you are continuously increasing (make more positive) the voltage VIN of the input voltage source from point 3 to point 4 so that its IV curve continues translating to the right. But now I am extremely vigorously increasing the voltage VO so the composed VO-R IV curve extremely quckly moves (translates) to the right as well. As a result, the operating point A slides down along the new IV curve of the inverted positive resistance that is inclined (folded up) to the right and has a negative slope. You have the illusion the resistance R has become true negative resistance dR3 < 0.
This idea is directly implemented in current-inversion negative impedance converters (INIC).
https://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance#.22Inverted_positive.22_.28N-shaped_negative.29_resistance
Yes, Vasile... there is only circuits acting as elements... and we can name them figuratively "virtual elements"... just like we name NIC "negative resistor"...
However, this brings up the question: How do we define an "element"? Are "elements" restricted to one-ports (two-poles)? What about transistors? Is the opamp an element?
I think that depending on the objectives we have set, we look at circuits from different perspectives. When a circuit is very complicated, we might brake it into separate functional blocks that represent its constituents (components). Then we do not mind what these units contain rather than their behavior. These "elements" can be 2-, 3- or 4-terminal devices.
From this perspective, a NIC can be viewed as a 2-terminal "element" - a "resistor", a GIC - as an "inductor", etc.
Vasile, I also do not use simulation... I prefer to "simulate" circuits in my mind relying on my intuition and imagination. The simulation is only a tool like math... it shows the final result but it does not (cannot) show how it is obtained...
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