Circuit Idea/Revealing the Mystery of Negative Impedance
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Circuit idea: Adding the same voltage (current) that appears across (flows through) a corresponding "positive" impedance element.
[edit] Introduction
Negative impedance is a property of special one-port electronic circuits to produce voltage that depends on the current passing through them (qualified as current-driven or S-shaped) or current depending on the voltage applied across them (voltage-driven or N-shaped) in the same manner as in the ordinary passive resistors, capacitors or inductors. So, these circuits act as negative resistors, negative capacitors and negative inductors having properties of negative resistance, negative capacitance and negative inductance. In the general case, they may have a combination of these attributes thus having negative impedance. From energetic viewpoint, negative impedance circuits are actually dynamic electrical sources that inject energy into circuits in the same manner as the according passive elements (resistors, capacitors and inductors) absorb energy from circuits.
The true negative resistor configuration is the most popular negative impedance circuit. It produces voltage/current that is proportional to the current/voltage through/across it according to Ohm's law (there are also negative non-linear resistors, e.g. "negative diodes", that do not obey Ohm's law). On the graphical presentation (see the picture on the right), the IV curve of the absolute negative resistor has a negative slope and passes through the origin of the coordinate system as the voltage and the current have opposite signs for every operating point along the curve. As a result, the ratio R = V/I < 0 (the resistance R is negative).
The amazing feature of negative impedance is to "neutralize" the equivalent "positive" impedance: connecting a current-driven negative impedance element in series witn the same "positive" impedance element gives zero total impedance; connecting a voltage-driven negative impedance element in parallel to the same positive impedance element gives infinite total impedance. Because of these remarkable properties current-driven negative impedance elements are used in circuitry (e.g., in telephony line repeaters) to zero the line resistance, internal voltage source resistance and load resistance. Voltage-driven negative impedance elements are used in such excentric circuits as Howland current source, Deboo integrator, load and stray capacitance cancellers, etc., to increase up to infinity the internal current source resistance and load impedance.
Although the term negative resistance is frequently used to encompass negative differential resistance as well, the two phenomena are quite different. A true negative resistor is not a resistor; it is actually a dynamic electrical source. A negative differential resistor is actually a dynamic positive resistor that cannot be used independently; it may be used in combination with an electrical source to build a true negative resistor.
[edit] Negative versus "positive" impedance
Negative resistance is the most popular negative impedance manifestation. Its nature is revealed below by comparing an ordinary "positive" resistor having a resistance R with a true negative resistor having a resistance -R. For this purpose, two pairs of equivalent electrical circuits are used[2], in which the resistors are connected in series with the loads so that the same current passes through them or in parallel to the loads so that the same voltage is applied across them.
[edit] What a "positive" impedance is
In electrical circuits, passive elements (resistors, capacitors and inductors) impede current by their inherent resistance, capacitance and inductance (impedance is the combination of them). As a result, voltage drops appear across the passive elements that represent the energy losses in them. Passive elements absorb this energy from the exciting electrical source: resistors dissipate energy from them to outside environment[3] while capacitors and inductors accumulate energy into themselves. From the negative impedance viewpoint it is not important if passive elements dissipate or accumulate energy; it is only important that they absorb energy.
[edit] Series-connected "positive" impedance elements
Electrical elements may be connected in series, parallel or mixed. For example, if a resistor R is connected in series with a load (Fig. 1a), a voltage drop VR = R.I that is proportional to the current appears across the resistor. When capacitors or inductors are placed at this place, voltage drops changing through time in the according manner appear across these elements.
A grapho-analytical interpretation (by using superimposed IV curves) of the "positive" resistance phenomenon is shown on Fig. 1b. It is supposed the resistor is supplied by a real current source (its IV curve is green colored). When the input current varies, the crossing operating point slides over an IV curve representing the positive resistance. It is a real, static, immovable IV curve having a positive slope and passing through the origin of the coordinate system.
[edit] Parallel-connected "positive" impedance elements
Conversely, if the resistor R is connected in parallel with the load (Fig. 2a), a current IR = VL/R that is proportional to the voltage flows through the resistor. When capacitors or inductors are placed there, currents changing through time in the according manner flow through them.
A grapho-analytical interpretation (by using superimposed IV curves) of the "positive" resistance phenomenon is shown on Fig. 2b. It is supposed the resistor is supplied by a real voltage source (its IV curve is red colored). When the input voltage varies, the crossing operating point slides over an IV curve representing the positive resistance. It is a real, static, immovable IV curve having a positive slope and passing through the origin of the coordinate system.
[edit] What negative impedance is
Elements with true negative impedance do the opposite - they inject energy into electrical circuits. Whereas passive elements with positive impedance absorb energy from the input source (they are loads), elements with negative impedance add energy to the input source (they are sources[3]). While voltage drops appear across positive elements, negative elements produce voltage; while positive elements sink current, negative elements produce current. However, they are not ordinary, steady electrical sources; they are varying, dynamic, "self-dependent" sources (voltage sources whose voltage across them depends on the current through them or current sources whose current depends on the voltage across them). Besides, the relation between the voltage and the current is as in the corresponding passive resistors, capacitors or inductors.
[edit] Current-driven negative impedance elements
On Fig. 3a a negative resistor NR with a resistance of -R is connected in series with the load so that a voltage VNR = R.I that is proportional to the current appears across the negative resistor. However, while above the "positive" resistor detracts the voltage VR from the input voltage (there VR is a voltage drop), here the negative resistor adds a voltage VNR to the input voltage (here VNR is a voltage). The element named "resistor" is really a resistor while the "negative resistor" here is actually a voltage source, whose voltage is proportional to the current passing through it. If negative capacitors or inductors are placed at this place, they produce voltages changing through time.
A negative resistor can be implemented as a varying (dynamic) voltage source, whose voltage is proportional to the current passing through it; this two-terminal current-controlled voltage source acts as a current-driven negative resistor. Negative impedance elements can be implemented as dynamic voltage sources, whose voltage depends on the current in the same manner as the voltage drop across the according passive elements (resistors, capacitors or inductors) depends on the current passing through them.
A graphoanalytical interpretation of this kind of negative resistance phenomenon is shown on Fig. 3b. It is supposed the negative resistor is driven by a real current source. When the input current varies, the voltage source representing the negative "resistor" changes proportionally its voltage. As a result, its IV curve moves and the crossing operating point slides over a new dynamic IV curve representing the negative resistance. Note it is not a real IV curve; it is an artificial, imaginary IV curve having a negative slope and passing through the origin of the coordinate system.
[edit] Voltage-driven negative impedance elements
Dually, when the negative resistor is connected in parallel to the load (Fig. 4a), a current INR = VL/R that is proportional to the voltage drop across the load flows through the negative resistor. However, while above the positive resistor sinks a current from the input current (diverts it from the load current), here the negative resistor adds the same current to the input current (injects an additional current into the load). If negative capacitors or inductors are placed there, they inject currents changing through time.
A negative resistor can be implemented also as a varying (dynamic) current source, whose current is proportional to the voltage across it; this two-terminal voltage-controlled current source acts as a voltage-driven negative resistor. Negative impedance elements can be implemented as dynamic current sources, whose current depends on the voltage in the same manner as the current passing through the according passive elements (resistors, capacitors or inductors) depends on the voltage drop across them.
The graphoanalytical interpretation of this kind of negative resistance phenomenon is shown on Fig. 4b. Now it is supposed the negative resistor is driven by a real voltage source. When the input voltage varies, the current source representing the negative "resistor" changes proportionally its current. As a result, its IV curve moves and the crossing operating point slides over a new dynamic IV curve representing the negative resistance. Note it is not a real IV curve; it is an artificial, imaginary IV curve having a negative slope and passing through the origin of the coordinate system.
True negative resistors act as supplemental sources that "help" the basic input sources: current-driven negative resistors are voltage sources "helping" the input voltage source to pass a current through the load; voltage-driven negative resistors are current sources "helping" the input current source to create voltage across the load. The input source has the illusion:) that only it determines the current/voltage through/across the load; but actually, both the sources determine these atrtributes (the output quantity depends on two input quantities).
[edit] Negative impedance implementations
Electrical elements with negative impedance are extremely useful in circuitry as they can compensate the losses in passive elements having the equivalent positive impedance. Unfortunately, true negative impedance elements do not exist in nature; there are only ordinary passive elements with positive impedance (resistors, capacitors and inductors). It is a powerful idea to use them as "shaping" elements for creating the according "mirror" elements with negative impedance. Negative impedance converters exploit this idea by converting the positive impedance into a negative one with the same value.
[edit] Current-driven negative impedance elements
[edit] The problem
In real life, we may observe situations when we solve some problem but a disturbance has stood in our way. Similar problems exist in electricity and electronics - see again for example the elementary voltage-supplied electrical circuit shown on Fig. 1a. It contains two elements with positive impedance (in this case, resistance) connected in series: the first element (the load) is useful; the second element (the line resistance, the internal source resistance, etc.) is undesired. Well, what do we do to eliminate the disturbance?
[edit] The solution
We may find a remedy in many situations of our human routine when we remove a disturbance by an equivalent "antidisturbance". In all these cases we add energy (as much as it losses in the disturbance) by an additional power source to "help" the main source and, as a result, to compensate the energy losses. According to this powerful "neutralization" idea (also mentioned in the introduction), the positive impedance of the undesired element can be eliminated (can be made zero) by including in series an additional current-driven element with the same negative impedance (Fig. 5). How does it happen?
A voltage drop appears across the undesired positive impedance element that depends on the current passing through it in a some definite way (linear, non-linear or time-dependent). In order to eliminate this disturbing voltage drop, the same voltage (depending in the same way on the current) has to appear across the compensating negative impedance element.
[edit] A voltage follower acting as a simple negative impedance element
According to this idea, a voltage follower with differential input can "copy" the voltage drop VPE2 across the positive impedance element PE2 and add an equivalent voltage VNE = VPE2 in series to the excitation input voltage V (Fig. 6). In this arrangement, the voltage drop across the "positive" element PE2 is an "original" and the voltage across the negative element NE is its "mirror copy". In this way, the voltage follower (including its power supply) acts as the simplest negative impedance element.
Is there any negative feedback in this configuration? No, it is not. Instead, a slight positive feedback can exsist if the Element PE2 is a kind of a resistor. Look at Fig. 6 again to convince yourself of this assertion. When, for example, the input voltage increases the current I, the voltage drop VPE2, the follower output voltage VOUT and the total current-creating voltage V + VOUT increase as well. This increases additionally the current I; we name this phenomenon positive feedback.
In this arrangement, the existing positive impedance element PE2 acts as a functional current-to-voltage converter; it is necessary but it does not belong to the negative impedance element (compare with the 2-terminal negative impedance element below).
[edit] An op-amp behaving as a simple negative impedance element
In circuitry, we implement voltage followers exceptionally by applying a negative feedback. First, we may use a classic op-amp voltage follower (with a local negative feedback between its output and inverting input) as a negative resistor (Fig. 7). For this purpose, we have to supply the op-amp with flying voltage source VS and to reverse its output terminals, in order to add its output voltage to the input one. In this way, the local op-amp's ground serves as an output while the very op-amp's output is connected to the common circuit ground. Although we have guessed for ourselves at this trick, it is proposed as far back as in early 1960s[4].
Note although we have applied a local negative feedback between the op-amp's output and its inverting input the op-amp does not "observe" the virtual ground in this arrangement. In addition, the op-amp has to have a differential input.
The most reliable arrangement (Fig. 8) is if we make the op-amp compare its output voltage VOUT (by subtracting) with the voltage drop VPE2 across the Element PE2 and change it so that to keep a zero difference between them. That means we have applied a global negative feedback. Note in this arrangement, we have made the op-amp "observe" the virtual ground. As a result, the op-amp output voltage is a "mirror" copy of the voltage drop across the positive element PE2. The op-amp may have a bare single-ended input as its input voltage (the difference VPE2 - VNE) is measured regarding to the ground.
In this arrangement, the negative impedance is precisely equal to the positive one as the op-amp "monitors" the difference between them (the presence of the virtual ground). However, for this purpose, the op-amp needs a third wire (a "sense") to measure the voltage drop across the positive impedance element; sometimes this is impossible or inconvenient. The true two-terminal current-driven negative impedance element solves smartly this problem (see below).
All the op-amp inverting circuits with negative feedback are based on this arrangement[5]. In these circuits, the combination of the op-amp and the power supply acts as a simple negative impedance element. It produces a voltage that is a "mirror" copy of the voltage drop across the positive impedance element connected between the op-amp's output and inverting input. The two elements (the "positive" and the "negative" one) are connected in series so that the total voltage and impedance of this combination is zero; as a result, a virtual ground appears at the top of this couple. Actually, it serves as a perfect functional current-to-voltage converter without internal losses.
[edit] Popular op-amp applications
For example, in the popular circuits of transimpedance amplifier, inverting amplifier, diode antilogarithmic converter, capacitive differentiator and inductive integrator (Fig. 9) a "positive" resistor R is connected between the op-amp's output and the inverting input. The combination of the op-amp and the power supply acts as a negative resistor with negative resistance -R that "neutralizes" the positive resistance R (the negative resistor -R adds as much voltage as it loses across the positive resistor R). Similarly, in the capacitive op-amp inverting integrator (Fig. 10), the combination of the op-amp and the power supply acts as a negative capacitor with negative capacitive reactance that "neutralizes" the positive capacitive reactance (the negative capacitor adds as much voltage as it accumulates in the positive capacitor).
Further, in the circuit of the op-amp inverting logarithmic converter (Fig. 11), the op-amp acts as a negative diode; it adds a forward voltage VF to the input voltage instead to subtract a forward voltage drop VF from the input voltage. In the circuit of the op-amp inverting inductive differentiator (Fig. 12) the op-amp acts as a negative inductor, etc.
[edit] A true current-driven negative impedance element
[edit] The problem
The simple current-driven element with negative impedance is just an op-amp that "copies" the voltage drop across the already existing element with positive impedance (see again Fig. 7). For this purpose, it uses an additional third wire (a "sense") to "feel" by its inverting input the difference between the voltage drop across the positive impedance element and its output voltage. However, sometimes this is inconvenient or, unfortunately, it is just impossible. In order to clarify the situation, a well-known problem from circuitry is solved on Fig. 13 - creating a perfect voltage source with zero internal resistance[6].
There are two problems to be solved in this arrangement: first, real voltage sources have some internal resistance; second, there is some line resistance between the source and the load. As a result, the voltage across the load droops when the source is loaded as disturbing voltage drop appears across the internal resistance and across the line resistance. That is why, in electronics, there is a need of voltage sources having zero internal resistance and lines having zero line resistance. The classical solution is to buffer the imperfect voltage source by a powerful voltage follower. However, the booster has to stay at the end of the line, near the load; unfortunately, in many cases, this is just impossible. More exotic and sometimes, more useful solution is to compensate the internal resistance by an equivalent negative resistance.
If the line resistance has to be compensated, then it would be possible to connect the op-amp inverting input (the "sense") to the left side of the line "resistor" (Fig. 13) realizing the simple current-driven negative resistor from Fig. 8 (a transimpedance amplifier). In this arrangement, the op-amp compares its output voltage with the voltage drop across the line resistance Ri and change it so that to keep (almost) zero difference between them. As a result, the op-amp produces output voltage that compensates the voltage drop across the line resistor R. Note in this arrangement the op-amp has "flying" power supply, in order the load to be grounded.
[edit] The basic idea of current-driven negative impedance
However, if an internal resistance has to be compensated, it is impossible to connect the op-amp "sense" to the left side of the internal "resistor" as the internal resistance is distributed inside the source. Or, if a distant positive resistance has to be compensated, it is inconvinient to stretch an additional third wire.
Two-terminal current-driven negative resistors use a clever trick to solve these problems (Fig. 14): as the existing positive resistor Ri is not accessible or it is not at a short distance, they use an additional positive resistor R with an equivalent resistance (a duplicate of the unaccessible resistor Ri). Since the current passing through the two series connected resistors is the same and they have equal resistances, the voltage drops across them are also equal. In this way, current-driven negative resistors use the common current to measure the voltage drop across the unaccessible or remote positive resistor (the current loop interface exploits the same idea).
However, the additional voltage drop VR across the duplicate resistor R is disturbing and has to be compensated as well. Therefore, the compensating voltage has to be two times higher; so, the doubling voltage source BH produces a compensating voltage VH= 2Ri.I. A half of this voltage compensates the voltage drop across the internal resistance Ri; the rest half compensates the voltage drop across the duplicate resistor R. As a result, only the load resistance RL remains in the circuit and the load voltage VL stays equal to the input voltage.
You may think of the duplicate resistor R as a current-to-voltage converter that converts the flowing current I into a proportional "copy" voltage drop VR = Ri.I, which drives the compensating voltage source BH (an amplifier with K = 2). From this viewpoint, this kind of negative impedance element is a current-driven voltage source consisting of a current-to-voltage converter and a voltage amplifier.
Note the overall voltage across the composed negative resistor -R has an opposite polarity to the voltage drop across the internal "positive" resistor R as though the circuit has inverted the initial voltage.
Simple negative impedance elements only inject a portion of energy into circuits equal to the losses in the positive elements while true current-driven negative impedance elements, in order to inject one portion of energy into circuits, absorb one portion and inject two portions of energy. For this purpose, they use an additional positive impedance element (simple negative impedance elements use the existing, accessible positive impedance element). One might say that true current-driven negative impedance elements convert a positive impedance element into a negative impedance one by inverting the initial voltage; so, they behave as negative impedance converters with voltage inversion (VNIC).
[edit] The op-amp implementation
In the op-amp implementation (Fig. 15), the duplicate resistor R (below the op-amp) acts as a current-to-voltage converter that converts the flowing current I into a proportional "copy" voltage drop VR = Ri.I. A voltage divider consisting of two equal resistors R (above the op-amp) is connected between the op-amp's output and its non-inverting input. So, the op-amp compares half of its output voltage with the "copy" voltage drop across the duplicate resistor R instead with the "original" voltage drop across the internal resistance Ri and changes it so that to keep (almost) zero difference between them. As a result, the op-amp produces two times higher output voltage than the "copy" voltage drop across the duplicate resistor R = Ri. Half the voltage compensates the voltage drop across the internal resistance Ri; the rest half compensates the voltage drop across the duplicate resistor R. In this exemplary arrangement, the op-amp circuit is "flying"; it has a local, internal ground that is not connected with the common, external ground.
Actually, this op-amp circuit converts the positive resistance R of the duplicate resistor into a negative resistance -R, i.e it acts as a negative impedance converter (NIC). As the voltage across this kind of negative resistance circuit has an opposite polarity to the voltage drop across the initial "positive" resistor, this circuit is named negative impedance converter with voltage inversion (VNIC).
[edit] Real current-driven true negative impedance element
[edit] What do current-driven negative resistors actually do?
A current-driven true negative resistor with resistance -R connected in series with a positive resistor with a total resistance RTOT destroys, eats, neutralizes R-part of the total positive resistance thus converting it to a zero resistance. Only, in order to have a stability (see below|), some part of the positive resistance has to remain.
[edit] Stability (operating mode)
Current-driven true negative resistors are circuits with positive feedback where a part of the output quantity adds to the input quantity. The gain of the feedback loop is proportional to the ratio between the negative resistance RN and positive resistance RP. In order to have a stability (to operate in active mode), we need the positive resistance to dominate over the negative one (RN/RP < 1). For the op-amp INIC from Fig. 15 this means: Ri/(Ri + R) > R/(R + R) = 1/2. Otherwise, the circuit will operate in bi-stable (memory) mode.
[edit] Voltage-driven negative impedance elements
[edit] Voltage-driven negative resistor ("neutralizing" a load resistance)
[edit] The problem
In nature, real sources (motors, beings, etc.) have a limited power. If they are not loaded, they behave perfectly. But if they are loaded (for example, if we try to raise a big weight), they droop.
A similar problem exists in electronics (electricity) when imperfect voltage sources are loaded. A more concrete example is the simplest varying voltage source on Fig. 16 that consists of a steady voltage source V and a potentiometer P (a voltage divider r1-r2). If there is no load connected (Fig. 16a), this real voltage source works well - VOUT = r2/(r1 + r2).
However, when a load RL is connected (Fig. 16b), it "sucks" a current IL and the output voltage VL drops.
We may generalize the problem if we complicate a bit the dual current-supplied electrical circuit shown on Fig. 2a by adding another element with positive impedance PE2 (see Fig. 18 below). Now it contains two elements with positive impedance connected in parallel: the first element PE1 (the load) is useful; the second element PE2 (e.g., a leakage resistance, a stray capacitance, a voltmeter internal resistance, etc.) is undesired. Well, what do we do to remove the disturbance?
[edit] The basic idea of voltage-driven negative impedance
The classic remedy is to connect a voltage follower (a unity-gain amplifier acting as a buffer amplifier) before the load, in order to decrease the current IL (to increase the load resistance RL). Unfortunately, this solution introduces some errors inherent for this circuit [7]. Then let's look for a remedy in our routine.
What can we do in real life when some object (being, machine etc.) supplied by a real power source droops? We can just help it. For this purpose, we usually use an additional power source, which "helps" the main source by compensating the losses caused by the load. For example, if someone has to raise a heavy loaded cage, we can help it by an equivalent "anti-weght". This is the well-known powerful idea of mechanics named anti-weight or anti-load that is widely used in the lift systems, cranes etc. (Fig. 17)
According to this powerful "neutralization" idea (also mentioned in the introduction), the positive impedance of the undesired element can be eliminated (can be made infinite) by connecting in parallel an additional voltage-driven element NE with the same negative impedance. How does it happen?
A current flows through the undesired positive impedance element PE2 (Fig. 18) that depends on the voltage across it in a some definite way (linear, non-linear or time-dependent). In order to eliminate this disturbing current, the same current (depending in the same way on the voltage) has to be produced by the compensating negative impedance element NE. As a result, the disturbing element PE2 will not consume any current from the input source; the compensating negative impedance element will provide all the needed current to PE2.
[edit] The basic electrical circuit
As this idea is so wonderful then let's realize it[8]. How do we create the needed voltage-driven negative resistor? We can use various building "scenarios" to do it, let's begin...
Scenario 1. In order to make a current-driven negative resistor, we have produced voltage that is proportional to the current flowing through it. Now, in order to make a voltage-driven negative resistor, we have to do the opposite - to produce current that is proportional to the voltage across it. For this purpose, we connect in series a "helping" voltage source (the output of an amplifier) and a "positive" resistor R acting as a voltage-to-current converter (Fig. 19). The voltage source has to keep the same voltage VR across the resistor as the voltage VL across the load; that means it to produce two times higher "helping" voltage VH= 2VL.
Scenario 2. If a voltage VL is applied across the "positive" load resistor RL, it will consume a load current IL. Conversely, if we apply the same voltage VL across an identical "positive" resistor R = RL, it has to produce the same current IH = VL/RL = IL. So, we have to "lift" the right end of the resistor with voltage VL toward its left end that is connected to the load. For this purpose, we connect a compensating voltage source BH (a non-inverting amplifier with K = 2) in series with the "copy" positive resistor R having the same resistance as the "original" positive resistor RL (Fig. 19).
The voltage source makes a current IH = (VH - VL)/R = (2VL - VL)/RL = VL/RL = IL that is equal to the load current IL flow through the load. In this way, the whole load current IL is provided only by the "helping" current source IH (the negative resistor -RL) instead by the real input voltage source. The load does not consume any energy from the input source since it is supplied completely by the "helping" source. Figuratively speaking, the load "pulls" the point A down toward the ground while the resistor R "pulls" the point A up toward the voltage VH. As a result of this "stretching", the point A experiences "weightlessness" (as it pulls itself up) and it follows easily the point B. There is no current flowing through the "bridge" connecting the point B and point A since the whole right part of the circuit (RL, R and VH) behaves as a load with infinite internal resistance. This is the well-known phenomenon of bootstrapping and it is put in practice for the first time by Baron Munchhausen (the legend says that he was using his own boot straps to pull himself out of the sea:)
Note the current flowing through the composed voltage-driven negative resistor -R has an opposite direction to the current flowing through the initial "positive" resistor R as though the circuit has inverted the initial current.
[edit] An implementation by...
[edit] ...a fixed gain amplifier...
We need a doubling voltage source; a non-inverting amplifier A having a gain of (only) +2 can act as such a "helping" voltage source (Fig. 20). We have just to connect the amp's input to the point A and the amp's output in place of the "helping" voltage source BH. The amplifier is single-supplied since here the input voltage is only positive.
The amplifier doses the voltage +V of the power supply, in order to produce the voltage needed (VA = 2VR). Actually, the steady voltage source +V and the amplifier A constitute the varying voltage source needed. The combination of this composed voltage source and the resistor R acts as a "helping" current source. It injects a current IH through the resistor R into the point A and raises its voltage; as a result, the point A "pulls itself up". The output voltage affects the input voltage as a part of the output voltage adds to the input voltage. This great phenomenon is referred to as positive feedback.
[edit] ...an op-amp amplifier with negative feedback
In electronics, we realize such amplifiers with fixed gain (in this case we need G = 2) by operational amplifiers. There are perfect op-amps having extremely large but unstable voltage gain (typically 200000). By applying a negative feedback we can make an op-amp amplify exactly two times needed. How do we do this magic?
Negative feedback systems have a nice feature to reverse the causality in electronic circuits. For example, if we put a passive circuit (an integrator, differentiator, attenuator, etc.) into the feedback loop, we will obtain the opposite active circuit (a differentiator, integrator, amplifier, etc.) According to this idea, let's build a voltage divider having a ratio 0.5 by connecting in series two equal resistors R1 and R2; then, let's connect it between the op-amp's output and the inverting input. As a result, we obtain an op-amp non-inverting amplifier having the stable gain of 2 needed.
Actually, this op-amp circuit converts the positive resistance R of the duplicate resistor into a negative resistance -R, i.e it acts as a negative impedance converter (NIC). As the current flowing through this kind of negative resistance circuit has an opposite direction to the current flowing through the initial "positive" resistor, this circuit is named negative impedance converter with current inversion (INIC).
[edit] Real voltage-driven true negative impedance element
When the op-amp output voltage approaches supply rails, it stops changing as the op-amp saturates and begins acting as an ordinary constant voltge source. The "magic" of negative resistance ceases.
[edit] What do voltage-driven negative resistors actually do?
A voltage-driven true negative resistor with resistance -R connected in parallel to a positive resistor with a total resistance RTOT destroys, eats, neutralizes R-part of the total positive resistance thus converting it to infinite resistance. Only, in order to have a stability (see below), some portion of negative resistance has to remain.
[edit] Stability (operating mode)
Voltage-driven true negative resistors are also circuits with positive feedback where a part of the output quantity adds to the input quantity. Here, the gain of the feedback loop is proportional to the ratio between the positive resistance RP and negative resistance RN. So, in order to have a stability (to operate in active mode), now we need the negative resistance to dominate over the positive one ( RP/RN < 1). For the op-amp INIC from Fig. 21 this means: RL/(R + RL) < R2/(R1 + R2). Otherwise, the circuit will operate in bi-stable (memory) mode.
[edit] Voltage-driven negative capacitor ("neutralizing" a stray capacitance)
So far we have been using linear ohmic resistors as initial, passive elements with "positive" resistance to make dual active elements with negative resistance (current- and voltage-controlled ones). But with the same success we can transmute every non-linear "positive" resistor into a negative one (e.g., a diode into a negative diode). Finally, by using the same technique, we can create various time-dependent elements with negative impedance, e.g. a negative capacitor.
The concept of negative capacitance is abstract enough; so, let's consider a typical application - "neutralizing" a stray capacitance by a negative capacitance. Although this brilliant idea is proposed as far back as in early 60s[4] maybe we might find its origin in Armstrong's radio times. It seems paradoxical but there are not still clear, simple and intuitive explanations of the capacitive "neutralization" idea. So, it is worth unveiling the mystery of this clever trick.
Negative capacitors are AC circuits driven by sine wave input voltage. In order to really understand how they exactly operate, we will show what the voltages are and where the currents flow in the circuits at one given (arbitrary chosen) moment of the sine wave. So, think of the pictures of voltage bars and current loops superimposed on the figures below as kinds of snapshots.
[edit] The problem caused by the stray capacitance
Imagine a sine wave generator with output resistance RIN drives a load with infinite input resistance (Fig. 22a). As no current flows through the resistance there is no voltage drop across it and, as a result, the output voltage is equal to the input one (VOUT = VIN).
If the load has a significant stray capacitance CSTR, it constitutes (in conjunction with the resistance RIN) an integrating circuit (Fig. 22b). As a result, the output voltage begins lagging and thus differring from the input one.
The problem is that the capacitor draws a current from the input source; it is a passive element that absorbs energy from the exciting electrical source and accumulates the "stolen" energy into itself (see above).
[edit] The basic electrical circuit
Exactly in the same way as above, we may "neutralize" the positive impedance of the stray capacitance CSTR by connecting in parallel an additional voltage-driven negative capacitor with the same but negative impedance. While the ordinary "positive" capacitor consumes energy from the input source (it is a load); the negative capacitor does the opposite - it injects energy into the circuit (it is a source). Speaking more concrete, while a series connected "positive" capacitor detracts a voltage drop from the input voltage, a current-driven negative capacitor adds voltage to the input voltage (it is a voltage source); while a parallel connected "positive" capacitor "sucks" current, a voltage-driven negative capacitor produces current (it is a current source). As above, a current flows through the undesired stray capacitance (Fig. 23) that is a differential of the voltage across it. In order to eliminate this disturbing current, the negative capacitor has to produce the same current (depending in the same way on the voltage through time). As a result, the stray capacitance will not consume any current from the input source; the negative capacitor will provide all the current needed to charge the stray capacitance. It is wonderful but yet... how do we make a negative capacitor?
We may use the same trick as above - to convert a "positive" capacitor into a negative one. For this purpose, we connect a "helping" voltage source VH = 2.VSTR (a non-inverting amplifier with K = 2) in series with a "positive" capacitor C having the same as the stray capacitance CSTR. The voltage source makes a current IH flow that is equal to the current IC flowing through the stray capacitance. In this way, the whole current IC is provided only by the "helping" voltage source BH instead by the real input voltage source. The load does not consume any energy from the input source since it is supplied completely by the "helping" source. The input voltage source works at ideal load conditions; it has the "feeling" that there is not a capacitive load connected and the output voltage VSTR is equal to the input one VIN. The situation is exactly as it is shown on Fig. 22a.
Where do we take the output voltage from?
We may use, as usual, VSTR as an output voltage (OUT1 serves as an output). Only, if the load has some resistance RL, it will constitute a voltage divider with the internal resistance RIN and the output voltage will droop - VOUT = VIN.RL/(RL + RIN (note that the negative capacitor compensates only the stray capacitance; it does not compensate the load resistance). But we have the unique possibility to use the compensating voltage VH = 2VSTR as an output voltage (OUT2 serves as an output)! As a result, the load will consume energy from the helping voltage source instead from the input voltage source and the output voltage will not droop. In addition, it will be amplified two times (whether we whish it or not).
[edit] Op-amp implementation
The op-amp implementation of a negative capacitor (Fig. 24) is similar to the op-amp circuit of a negative resistor (Fig. 21) with only one difference - a capacitor C is connected instead the resistor R. As above, the op-amp and the voltage divider (the resistors R1 and R2) constitute a non-inverting amplifier with gain of 2 that serves as a compensating voltage source.
Finally, let's look at the scanned image on Fig. 25 giving thanks to pioneers. It is an extract from page 8 of the remarkable genuine paper written by Dan Sheingold in the enthusiastic issue The Lightning Empiricist of Philbrick Researches in the distant 60's. As you can see, the basic idea behind this exotic circuit solution is thoroughly hidden there...and it was staying hidden as many as 45 years...and we have finally managed to reveal it relying only on our human intuition and common sense!
[edit] General properties of negative impedance elements
[edit] True negative versus "positive" impedance elements
- True negative impedance elements are sources that inject energy into circuits while the according "positive" impedance elements (resistors, capacitors and inductors) absorb energy from circuits.
- True negative impedance elements add so much energy to the input sources as it loses into "positive" impedance elements having the same impedance.
- True negative impedance elements are electronic circuits while "positive" impedance elements are really elements (components).
[edit] Current-driven versus voltage-driven negative impedance elements
- Both the true negative impedance elements are composed circuits consisting of two connected in series components: an internal "positive" impedance element and (the output of) an amplifier with gain of 2. The amplifier of a current-driven negative impedance element amplifies the voltage drop across the internal "positive" impedance element; the amplifier of a voltage-driven negative impedance element amplifies the voltage drop across the terminals of the very negative impedance element.
- A current-driven negative impedance element is a current-driven voltage source consisting of a current-to-voltage converter that drives a voltage amplifier; a voltage-driven negative impedance element is a voltage-driven current source consisting of a voltage amplifier and a voltage-to-current converter.
- Current-driven negative impedance elements are connected in series while voltage-driven negative impedance elements are connected in parallel to "positive" impedance elements.
- Current-driven negative impedance elements add so much voltage to the input voltage source as it would appear across an equivalent "positive" impedance element; voltage-driven negative impedance elements add so much current to the input current source as it would flow through an equivalent "positive" impedance element.
- A current-driven true negative resistor with resistance -R connected in series with a positive resistor with a total resistance RTOT destroys, "eats", neutralizes R-part of the total positive resistance; the result of this neutralization is zero resistance. A voltage-driven true negative resistor with resistance -R connected in parallel to a positive resistor with a total resistance RTOT neutralizes R-part of the total positive resistance; the result of this neutralization is infinite resistance.
- The voltage across a current-driven negative impedance element has an opposite polarity to the voltage drop across the initial "positive" resistor; so, it behaves as a negative impedance converter with voltage inversion (VNIC). The current flowing through a voltage-driven negative impedance element has an opposite direction to the current flowing through the initial "positive" resistor; so, it behaves as a negative impedance converter with current inversion (INIC).
- To operate in active mode, the positive resistance has to dominate over the negative resistance in circuits with current-controlled negative resistors while the negative resistance has to dominate over the "positive" resistance in circuits with voltage-controlled negative resistors. Otherwise, these circuits will operate in bi-stable mode (acting as Schmitt trigger).
[edit] True negative resistors versus differential negative resistors
- True negative resistors are electronic circuits while negative differential resistors are elements (components).
- Both the negative resistors are dynamic electronic elements (circuits).
- True negative resistors are dynamic electrical sources while negative differential resistors are just dynamic resistors that cannot be used independently; they may be used in combination with electrical sources to build true negative resistors.
[edit] General rules for using negative impedance elements
- Current-driven negative impedance elements:
- Transform a "positive" impedance element into a current-driven negative one by connecting in series an amplifier with gain of 2 that amplifies the voltage drop across the "positive" impedance element.
- Connect current-driven negative impedance elements in series with "positive" impedance elements to decrease their impedance.
- Make the negative impedance equal to some part of the positive one to destroy it and to give zero impedance.
- Reserve some positive resistance to operate in an active mode.
- Voltage-driven negative impedance elements:
- Transform a "positive" impedance element into a voltage-driven negative one by connecting in series an amplifier with gain of 2 that amplifies the voltage drop across the terminals of the very negative impedance element.
- Connect voltage-driven negative impedance elements in parallel to positive impedance elements to decrease their impedance.
- Make the negative impedance equal to some part of the positive one to annihilate it and to give infinite impedance.
- Reserve some negative resistance to operate in an active mode.
[edit] References
- ↑ In this area of electronics, negative impedance has the meaning of anti-impedance as it symbolizes the ability of special active electronic circuits (NICs) to produce energy in the same manner as the according passive electrical elements with "positive" impedance consume energy. In contrast to this "with regard to energy" viewpoint, the classical AC electricity uses a "with regard to time" viewpoint at the same terms. There, negative impedance and positive impedance terms symbolize the two opposite kinds of behavior (reactance) that reactive elements show through time - increasing voltage opposition (capacitors) or decreasing voltage opposition (inductors). As a result, in negative impedance circuitry both capacitors and inductors have positive impedance while in AC electricity capacitors have negative impedance and inductors have positive impedance.
- ↑ Mechkov C. (2006) A heuristic approach to teaching negative resistance phenomenon. COMPUTER SCIENCE' 2006, Istanbul, Turkey.
- ↑ a b Negative resistance process versus positive resistance process (an expressive picture)
- ↑ a b Impedance and admittance transformations using operational amplifiers is a genuine source from Philbrick Reserches (written by D. H. Sheingold).
- ↑ Voltage compensation (talk) reveals the secret of op-amp inverting circuits with negative feedback.
- ↑ How to compensate resistive losses by series connected negative resistors
- ↑ Negative-resistance load canceller helps drive heavy loads shows a typical voltage-driven negative resistance application.
- ↑ How to compensate resistive losses by parallel connected negative resistors
[edit] See also
Investigating the linear mode of negative impedance converters with voltage inversion (uncompleted)
Negative resistance is the main Wikipedia article dedicated to the phenomenon.
Negative impedance converter considers NIC with current inversion (INIC).
[edit] External links
Understanding negative impedance converters (VNIC) - reveals in three consecutive steps the basic idea behind negative impedance converters with voltage inversion (VNIC).
Negative Resistance Revived - condensed version of article originally published in Amateur Radio, November 1995.
Negative-resistance circuits - nice material from Answers.com.
Handbook of operational amplifier active RC networks - a formal but well-written electronic book.
How I revealed the secret of parallel negative feedback circuits reveals the philosophy of circuits with parallel negative feedback.
How do we create a virtual ground? reveals the secret of the great circuit phenomenon.
Reinventing transimpedance amplifier is a story about the op-amp current-to-voltage converter.
Op-amp inverting summer is an animated Flash tutorial about the famous op-amp summing circuit.
Keeping a constant current by adding an additional voltage
How do we build an op-amp RC integrator?
Analog electronics 2004, Class 2: Elementary passive converters with voltage output
