Misplaced Pages

#494505

82-436: Where − A v {\displaystyle -A_{v}} is the voltage gain of the inverting amplifier ( A v {\displaystyle A_{v}} positive) and C {\displaystyle C} is the feedback capacitance. Although the term Miller effect normally refers to capacitance, any impedance connected between the input and another node exhibiting gain can modify

164-466: A Darlington configuration (commonly called as a Darlington pair ) is a circuit consisting of two bipolar transistors with the emitter of one transistor connected to the base of the other, such that the current amplified by the first transistor is amplified further by the second one. The collectors of both transistors are connected together. This configuration has a much higher current gain than each transistor taken separately. It acts like and

246-410: A shared collector. Integrated Darlington pairs come packaged singly in transistor-like packages or as an array of devices (usually eight) in an integrated circuit . A third transistor can be added to a Darlington pair to give even higher current gain, making a Darlington triplet. The emitter of the second transistor in the pair is connected to the base of the third, as the emitter of first transistor

328-509: A capacitor C M o {\displaystyle C_{Mo}} equal to: C M o = ( 1 + 1 A v ) C C . {\displaystyle C_{Mo}=(1+{\frac {1}{A_{v}}})C_{C}.} In order for the Miller capacitance to draw the same current in Figure 2B as the coupling capacitor in Figure 2A, the Miller transformation

410-425: A circuit of an ideal inverting voltage amplifier of gain − A v {\displaystyle -A_{v}} with an impedance Z {\displaystyle Z} connected between its input and output nodes. The output voltage is therefore V o = − A v V i {\displaystyle V_{o}=-A_{v}V_{i}} . Assuming that

492-415: A differential input. The input voltage source has to have internal impedance Z i n t > 0 {\displaystyle Z_{int}>0} or it has to be connected through another impedance element to the input. Under these conditions, the input voltage V i {\displaystyle V_{i}} of the circuit changes its polarity as the output voltage exceeds

574-399: A faster transistor turn-off. The Darlington pair has more phase shift at high frequencies than a single transistor and hence can more easily become unstable with negative feedback (i.e., systems that use this configuration can have poor performance due to the extra transistor delay). Darlington pairs are available as integrated packages or can be made from two discrete transistors; Q 1 ,

656-488: A feedback. Depending on the kind of amplifier (non-inverting, inverting or differential), the feedback can be positive, negative or mixed. The Miller amplifier arrangement has two aspects: The introduction of an impedance that connects amplifier input and output ports adds a great deal of complexity in the analysis process. Miller theorem helps reduce the complexity in some circuits particularly with feedback by converting them to simpler equivalent circuits. But Miller theorem

738-891: A floating impedance element, supplied by two voltage sources connected in series, may be split into two grounded elements with corresponding impedances. There is also a dual Miller theorem with regards to impedance supplied by two current sources connected in parallel. The two versions are based on the two Kirchhoff's circuit laws . Miller theorems are not only pure mathematical expressions. These arrangements explain important circuit phenomena about modifying impedance ( Miller effect , virtual ground , bootstrapping , negative impedance , etc.) and help in designing and understanding various commonplace circuits (feedback amplifiers, resistive and time-dependent converters, negative impedance converters, etc.). The theorems are useful in 'circuit analysis' especially for analyzing circuits with feedback and certain transistor amplifiers at high frequencies. There

820-694: A node, where two currents I 1 {\displaystyle I_{1}} and I 2 {\displaystyle I_{2}} converge to ground, we can replace this branch by two conducting the referred currents, with impedances respectively equal to ( 1 + α ) Z {\displaystyle (1+\alpha )Z} and ( 1 + α ) Z α {\displaystyle {\frac {(1+\alpha )Z}{\alpha }}} , where α = I 2 I 1 {\displaystyle \alpha ={\frac {I_{2}}{I_{1}}}} . The dual theorem may be proved by replacing

902-421: A positive resistor R {\displaystyle R} , a load (the capacitor C {\displaystyle C} acting as impedance Z {\displaystyle Z} ) and a negative impedance converter INIC ( R 1 = R 2 = R 3 = R {\displaystyle R_{1}=R_{2}=R_{3}=R} and the op-amp). The input voltage source and

SECTION 10

#1733093634495

984-488: A result, the effective voltage across and the current through the impedance increase; the input impedance decreases. Decreased impedance is implemented by an inverting amplifier having some moderate gain, usually 10 < A v < 1000 {\displaystyle 10<A_{v}<1000} . It may be observed as an undesired Miller effect in common-emitter , common-source and common-cathode amplifying stages where effective input capacitance

1066-543: A voltage amplifier with gain of A V = K {\displaystyle A_{V}=K} serves as such a linear network, but also other devices can play this role: a person and a potentiometer in a potentiometric null-balance meter , an electromechanical integrator (servomechanisms using potentiometric feedback sensors), etc. In the amplifier implementation, the input voltage V i {\displaystyle V_{i}} serves as V 1 {\displaystyle V_{1}} and

1148-568: Is a close relationship between Miller theorem and Miller effect: the theorem may be considered as a generalization of the effect and the effect may be thought as of a special case of the theorem. The Miller theorem establishes that in a linear circuit, if there exists a branch with impedance Z {\displaystyle Z} , connecting two nodes with nodal voltages V 1 {\displaystyle V_{1}} and V 2 {\displaystyle V_{2}} , we can replace this branch by two branches connecting

1230-428: Is a reduction in switching speed or response, because the first transistor cannot actively inhibit the base current of the second one, making the device slow to switch off. To alleviate this, the second transistor often has a resistor of a few hundred ohms connected between its base and emitter terminals. This resistor provides a low-impedance discharge path for the charge accumulated on the base-emitter junction, allowing

1312-473: Is also available in a variety of single packages. One drawback is an approximate doubling of the base–emitter voltage. Since there are two junctions between the base and emitter of the Darlington transistor, the equivalent base–emitter voltage is the sum of both base–emitter voltages: For silicon-based technology, where each V BEi is about 0.65 V when the device is operating in the active or saturated region,

1394-431: Is an example of such a virtual element where the resistance R L {\displaystyle R_{L}} is modified so that to mimic inductance, capacitance or inversed resistance. There is also a dual version of Miller theorem that is based on Kirchhoff's current law ( Miller theorem for currents ): if there is a branch in a circuit with impedance Z {\displaystyle Z} connecting

1476-416: Is connected to the base of the second, and the collectors of all three transistors are connected together. This gives current gain approximately equal to the product of the gains of the three transistors. However the increased current gain often does not justify the sensitivity and saturation current problems, so this circuit is seldom used. Darlington pairs are often used in the push-pull output stages of

1558-556: Is equal to the sum of its own base–emitter voltage and the collector-emitter voltage of the first transistor, both positive quantities in normal operation, it always exceeds the base-emitter voltage. (In symbols, V C E 2 = V C E 1 + V B E 2 > V B E 2 ⇒ V C 2 > V B 2 {\displaystyle \mathrm {V_{CE2}=V_{CE1}+V_{BE2}>V_{BE2}} \Rightarrow \mathrm {V_{C2}>V_{B2}} } always.) Thus

1640-399: Is increased. Frequency compensation for general purpose operational amplifiers and transistor Miller integrator are examples of useful usage of the Miller effect. Zeroed impedance uses an inverting (usually op-amp) amplifier with enormously high gain A v → ∞ {\displaystyle A_{v}\to \infty } . The output voltage is almost equal to

1722-419: Is intentionally introduced. In these applications, the output voltage V o {\displaystyle V_{o}} is inserted with an opposite polarity in respect to the input voltage V i {\displaystyle V_{i}} travelling along the loop (but in respect to ground, the polarities are the same). As a result, the effective voltage across, and the current through,

SECTION 20

#1733093634495

1804-475: Is moderately increased. Infinite impedance uses a non-inverting amplifier with A v = 1 {\displaystyle A_{v}=1} . The output voltage is equal to the input voltage V i {\displaystyle V_{i}} and completely neutralizes it. Examples are potentiometric null-balance meters and op-amp followers and amplifiers with series negative feedback ( op-amp follower and non-inverting amplifier ) where

1886-495: Is not only an effective tool for creating equivalent circuits; it is also a powerful tool for designing and understanding circuits based on modifying impedance by additional voltage . Depending on the polarity of the output voltage versus the input voltage and the proportion between their magnitudes, there are six groups of typical situations. In some of them, the Miller phenomenon appears as desired ( bootstrapping ) or undesired ( Miller effect ) unintentional effects; in other cases it

1968-447: Is not zero, Figure 2B shows the large Miller capacitance appears at the input of the circuit. The voltage output of the circuit now becomes and rolls off with frequency once frequency is high enough that ω C M R A ≥ 1. It is a low-pass filter . In analog amplifiers this curtailment of frequency response is a major implication of the Miller effect. In this example, the frequency ω 3dB such that ω 3dB C M R A = 1 marks

2050-411: Is often packaged as a single transistor. It was invented in 1953 by Sidney Darlington . A Darlington pair behaves like a single transistor, meaning it has one base, collector, and emitter. It typically creates a high current gain (approximately the product of the gains of the two transistors, due to the fact that their β values multiply together). A general relation between the compound current gain and

2132-406: Is positive), the effective capacitance at their inputs is increased due to the Miller effect. This can reduce the bandwidth of the amplifier, restricting its range of operation to lower frequencies. The tiny junction and stray capacitances between the base and collector terminals of a Darlington transistor , for example, may be drastically increased by the Miller effects due to its high gain, lowering

2214-414: Is replaced on the input side of the circuit by the Miller capacitance C M {\displaystyle C_{M}} , which draws the same current from the driver as the coupling capacitor in Figure 2A. Therefore, the driver sees exactly the same loading in both circuits. On the output side, the same current from the output as drawn from the coupling capacitor in Figure 2A is instead drawn from

2296-479: Is subtracted from or added to V 1 {\displaystyle V_{1}} ; so, the input current decreases/increases and the input impedance of the circuit seen from the side of the input source accordingly increases/decreases So, the Miller theorem expresses the fact that connecting a second voltage source with proportional voltage V 2 = K V 1 {\displaystyle V_{2}=K{V_{1}}} in series with

2378-423: Is used to realize a non-inverting amplifier and the positive feedback – to modify the impedance). In these applications, the output voltage V o {\displaystyle V_{o}} is inserted with the same polarity in respect to the input voltage V i {\displaystyle V_{i}} travelling along the loop (but in respect to ground, the polarities are opposite). As

2460-571: Is used to relate C M {\displaystyle C_{M}} to C C {\displaystyle C_{C}} . In this example, this transformation is equivalent to setting the currents equal, that is or, rearranging this equation This result is the same as C M {\displaystyle C_{M}} of the Derivation Section . The present example with A v {\displaystyle A_{v}} frequency independent shows

2542-399: The quasi-symmetrical push-pull circuit was used, in which only the two transistors connected to the positive supply rail were an NPN Darlington pair, and the pair from the negative rail were two more NPN transistors connected as common-emitter amplifiers. A Darlington pair can be sensitive enough to respond to the current passed by skin contact even at safe zone voltages. Thus it can form

Miller effect - Misplaced Pages Continue

2624-441: The "saturation" voltage of a Darlington transistor is one V BE (about 0.65 V in silicon) higher than a single transistor saturation voltage, which is typically 0.1 - 0.2 V in silicon. For equal collector currents, this drawback translates to an increase in the dissipated power for the Darlington transistor over a single transistor. The increased low output level can cause troubles when TTL logic circuits are driven. Another problem

2706-521: The IV characteristic of an arbitrary element. The circuit of a diode log converter is an example of a non-linear virtually zeroed resistance where the logarithmic forward IV curve of a diode is transformed to a vertical straight line overlapping the y {\displaystyle y} axis. Not constant coefficient. If the coefficient K {\displaystyle K} varies, some exotic virtual elements can be obtained. A gyrator circuit

2788-461: The Miller effect can, at least in theory, be eliminated entirely. In practice, variations in the capacitance of individual amplifying devices coupled with other stray capacitances, makes it difficult to design a circuit such that total cancellation occurs. Historically, it was not unknown for the neutralising capacitor to be selected on test to match the amplifying device, particularly with early transistors that had very poor bandwidths. The derivation of

2870-484: The Miller effect in its extreme manifestation. In all these op-amp inverting circuits with parallel negative feedback , the input current is increased to its maximum. It is determined only by the input voltage and the input impedance according to Ohm's law ; it does not depend on the impedance Z {\displaystyle Z} . Negative impedance with voltage inversion is implemented by applying both negative and positive feedback to an op-amp amplifier with

2952-522: The advantage of this technique as the bandwidth of tubes improved. Later tubes had to employ very small grids (the frame grid) to reduce the capacitance to allow the device to operate at frequencies that were impossible with the screen grid. Figure 2A shows an example of Figure 1 where the impedance coupling the input to the output is the coupling capacitor C C {\displaystyle C_{C}} . Thévenin voltage source V A {\displaystyle V_{A}} drives

3034-480: The amplifier contained implicitly in A v . Such frequency dependence of A v also makes the Miller capacitance frequency dependent, so interpretation of C M as a capacitance becomes more difficult. However, ordinarily any frequency dependence of A v arises only at frequencies much higher than the roll-off with frequency caused by the Miller effect, so for frequencies up to the Miller-effect roll-off of

3116-460: The amplifier has a high impedance output, such as if a gain stage is also the output stage, then this RC can have a significant impact on the performance of the amplifier. This is when pole splitting techniques are used. The Miller effect may also be exploited to synthesize larger capacitors from smaller ones. One such example is in the stabilization of feedback amplifiers , where the required capacitance may be too large to practically include in

3198-556: The amplifier input draws no current, all of the input current flows through Z {\displaystyle Z} , and is therefore given by The input impedance of the circuit is In the Laplace domain (where s {\displaystyle s} represents complex frequency), if Z {\displaystyle Z} consists of just a capacitor forming a complex impedance Z = 1 s C {\displaystyle Z={\frac {1}{sC}}} , then

3280-580: The amplifier input impedance via this effect. These properties of the Miller effect are generalized in the Miller theorem . The Miller capacitance due to undesired parasitic capacitance between the output and input of active devices like transistors and vacuum tubes is a major factor limiting their gain at high frequencies. When Miller published his work in 1919, he was working on vacuum tube triodes . The same analysis applies to modern devices such as bipolar junction and field-effect transistors . Consider

3362-402: The circuit input impedance is enormously increased. This technique is referred to as bootstrapping and is intentionally used in biasing circuits, input guarding circuits, etc. Negative impedance obtained by current inversion is implemented by a non-inverting amplifier with A v > 1 {\displaystyle A_{v}>1} . The current changes its direction, as

Miller effect - Misplaced Pages Continue

3444-423: The circuit with Thévenin resistance R A {\displaystyle R_{A}} . The output impedance of the amplifier is considered low enough that the relationship V o = − A v V i {\displaystyle V_{o}=-A_{v}V_{i}} is presumed to hold. At the output, Z L {\displaystyle Z_{L}} serves as

3526-548: The circuit's resulting input impedance will be equivalent to that of a larger capacitance C M {\displaystyle C_{M}} : This Miller capacitance C M {\displaystyle C_{M}} is the physical capacitance C {\displaystyle C} multiplied by the factor ( 1 + A v ) {\displaystyle (1+A_{v})} . As most amplifiers are inverting ( A v {\displaystyle A_{v}} as defined above

3608-457: The circuit. This may be particularly important in the design of integrated circuits , where capacitors can consume significant area, increasing costs. The Miller effect may be undesired in many cases, and approaches may be sought to lower its impact. Several such techniques are used in the design of amplifiers. A current buffer stage may be added at the output to lower the gain A v {\displaystyle A_{v}} between

3690-411: The coefficient K {\displaystyle K} is replaced by 1 K {\displaystyle {\frac {1}{K}}} Most frequently, the Miller theorem may be observed in, and implemented by, an arrangement consisting of an element with impedance Z {\displaystyle Z} connected between the two terminals of a grounded general linear network. Usually,

3772-421: The common ground. In practice, one of them acts as a main (independent) voltage source with voltage V 1 {\displaystyle V_{1}} and the other – as an additional (linearly dependent) voltage source with voltage V 2 = K V 1 {\displaystyle V_{2}=K{V_{1}}} . The idea of the Miller theorem (modifying circuit impedances seen from

3854-434: The corresponding nodes to ground by impedances respectively Z 1 − K {\displaystyle {\frac {Z}{1-K}}} and K Z K − 1 {\displaystyle {\frac {KZ}{K-1}}} , where K = V 2 V 1 {\displaystyle K={\frac {V_{2}}{V_{1}}}} . The Miller theorem may be proved by using

3936-402: The driver and the amplifier, which reduces the apparent driver impedance seen by the amplifier. The output voltage of this simple circuit is always A v v i . However, real amplifiers have output resistance. If the amplifier output resistance is included in the analysis, the output voltage exhibits a more complex frequency response and the impact of the frequency-dependent current source on

4018-530: The dual version is a powerful tool for designing and understanding circuits based on modifying impedance by additional current. Typical applications are some exotic circuits with negative impedance as load cancellers, capacitance neutralizers, Howland current source and its derivative Deboo integrator. In the last example (see Fig. 1 there), the Howland current source consists of an input voltage source V i n {\displaystyle V_{in}} ,

4100-423: The end of the low-frequency response region and sets the bandwidth or cutoff frequency of the amplifier. The effect of C M upon the amplifier bandwidth is greatly reduced for low impedance drivers ( C M R A is small if R A is small). Consequently, one way to minimize the Miller effect upon bandwidth is to use a low-impedance driver, for example, by interposing a voltage follower stage between

4182-435: The equivalent two-port network technique to replace the two-port to its equivalent and by applying the source absorption theorem. This version of the Miller theorem is based on Kirchhoff's voltage law ; for that reason, it is named also Miller theorem for voltages . The Miller theorem implies that an impedance element is supplied by two arbitrary (not necessarily dependent) voltage sources that are connected in series through

SECTION 50

#1733093634495

4264-436: The gain, A v is accurately approximated by its low-frequency value. Determination of C M using A v at low frequencies is the so-called Miller approximation . With the Miller approximation, C M becomes frequency independent, and its interpretation as a capacitance at low frequencies is secure. Miller theorem The Miller theorem refers to the process of creating equivalent circuits . It asserts that

4346-582: The high frequency response of the device. It is also important to note that the Miller capacitance is the capacitance seen looking into the input. If looking for all of the RC time constants (poles) it is important to include as well the capacitance seen by the output. The capacitance on the output is often neglected since it sees C ( 1 + 1 A v ) {\displaystyle {C}({1+{\tfrac {1}{A_{v}}}})} and amplifier outputs are typically low impedance. However if

4428-516: The impedance Z {\displaystyle Z} thus decreasing/increasing the current. The proportion between the voltages determines the value of the obtained impedance (see the tables below) and gives in total six groups of typical applications . The circuit impedance, seen from the side of the output source, may be defined similarly, if the voltages V 1 {\displaystyle V_{1}} and V 2 {\displaystyle V_{2}} are swapped and

4510-576: The impedance decrease; the input impedance increases. Increased impedance is implemented by a non-inverting amplifier with gain of 0 < A v < 1 {\displaystyle 0<A_{v}<1} . The (magnitude of) output voltage is less than the input voltage V i {\displaystyle V_{i}} and partially neutralizes it. Examples are imperfect voltage followers ( emitter , source , cathode follower, etc.) and amplifiers with series negative feedback ( emitter degeneration ), whose input impedance

4592-552: The implications of the Miller effect, and therefore of C C {\displaystyle C_{C}} , upon the frequency response of this circuit, and is typical of the impact of the Miller effect (see, for example, common source ). If C C {\displaystyle C_{C}} is 0, the output voltage of the circuit is simply A v v A {\displaystyle A_{v}v_{A}} , independent of frequency. However, when C C {\displaystyle C_{C}}

4674-580: The individual gains is given by: If β 1 and β 2 are high enough (hundreds), this relation can be approximated with: A typical Darlington transistor has a current gain of 1000 or more, so that only a small base current is needed to make the pair switch on much higher switched currents. Another advantage involves providing a very high input impedance for the circuit which also translates into an equal decrease in output impedance. The ease of creating this circuit also provides an advantage. It can be simply made with two separate NPN (or PNP) transistors, and

4756-427: The input and output terminals of the amplifier (though not necessarily the overall gain). For example, a common base may be used as a current buffer at the output of a common emitter stage, forming a cascode . This will typically reduce the Miller effect and increase the bandwidth of the amplifier. Alternatively, a voltage buffer may be used before the amplifier input, reducing the effective source impedance seen by

4838-418: The input terminals. This lowers the R C {\displaystyle RC} time constant of the circuit and typically increases the bandwidth. The Miller capacitance can be mitigated by employing neutralisation . This can be achieved by feeding back an additional signal that is in phase opposition to that which is present at the stage output. By feeding back such a signal via a suitable capacitor,

4920-407: The input voltage source changes the effective voltage, the current and respectively, the circuit impedance seen from the side of the input source . Depending on the polarity, V 2 {\displaystyle V_{2}} acts as a supplemental voltage source helping or opposing the main voltage source to pass the current through the impedance. Besides by presenting the combination of

5002-433: The left-hand transistor in the diagram, can be a low power type, but normally Q 2 (on the right) will need to be high power. The maximum collector current I C (max) of the pair is that of Q 2 . A typical integrated power device is the 2N6282, which includes a switch-off resistor and has a current gain of 2400 at I C =10 A. Integrated devices can take less space than two individual transistors because they can use

SECTION 60

#1733093634495

5084-484: The load is constant and the circuit impedance seen by the input source is increased. As a comparison, in a load canceller , the INIC passes all the required current through the load; the circuit impedance seen from the side of the input source (the load impedance) is almost infinite. Below is a list of circuit solutions, phenomena and techniques based on the two Miller theorems. Darlington transistor In electronics ,

5166-461: The load. (The load is irrelevant to this discussion: it just provides a path for the current to leave the circuit.) In Figure 2A, the coupling capacitor delivers a current j ω C C ( V i − V o ) {\textstyle j\omega C_{C}(V_{i}-V_{o})} to the output node. Figure 2B shows a circuit electrically identical to Figure 2A using Miller's theorem. The coupling capacitor

5248-400: The main and the auxiliary one. As in the case of the main Miller theorem, the second voltage is usually produced by a voltage amplifier. Depending on the kind of the amplifier (inverting, non-inverting or differential) and the gain, the circuit input impedance may be virtually increased, infinite, decreased, zero or negative. As the main Miller theorem, besides helping circuit analysis process,

5330-439: The necessary base–emitter voltage of the pair is 1.3 V. Another drawback of the Darlington pair is its increased "saturation" voltage. The output transistor is not allowed to saturate (i.e. its base–collector junction must remain reverse-biased) because the first transistor, when saturated, establishes full (100%) parallel negative feedback between the collector and the base of the second transistor. Since collector–emitter voltage

5412-401: The nodes. It is supposed also a constant coefficient K {\displaystyle K} ; then the expressions above are valid. But modifying properties of Miller theorem exist even when these requirements are violated and this arrangement can be generalized further by dynamizing the impedance and the coefficient. Non-linear element. Besides impedance, Miller arrangement can modify

5494-406: The output side must be taken into account. Ordinarily these effects show up only at frequencies much higher than the roll-off due to the Miller capacitance, so the analysis presented here is adequate to determine the useful frequency range of an amplifier dominated by the Miller effect. This example also assumes A v is frequency independent, but more generally there is frequency dependence of

5576-421: The output voltage V o {\displaystyle V_{o}} as V 2 {\displaystyle V_{2}} . In many cases, the input voltage source has some internal impedance Z i n t {\displaystyle Z_{int}} or an additional input impedance is connected that, in combination with Z {\displaystyle Z} , introduces

5658-428: The output voltage is higher than the input voltage. If the input voltage source has some internal impedance Z i n t {\displaystyle Z_{int}} or if it is connected through another impedance element, a positive feedback appears. A typical application is the negative impedance converter with current inversion (INIC) that uses both negative and positive feedback (the negative feedback

5740-402: The output voltage, although it is applied to the inverting op-amp input; the input source has an opposite polarity to both the circuit input and output voltages. The original Miller effect is implemented by capacitive impedance connected between the two nodes. Miller theorem generalizes Miller effect as it implies arbitrary impedance Z {\displaystyle Z} connected between

5822-414: The phase inverted signal usually requires an inductive component such as a choke or an inter-stage transformer. In vacuum tubes , an extra grid (the screen grid) could be inserted between the control grid and the anode. This had the effect of screening the anode from the grid and substantially reducing the capacitance between them. While the technique was initially successful other factors limited

5904-514: The place of the impedance Z {\displaystyle Z} ). The rest of them have additional impedance connected in series to the input: voltage-to-current converter (transconductance amplifier), inverting amplifier , summing amplifier , inductive integrator, capacitive differentiator, resistive-capacitive integrator , capacitive-resistive differentiator , inductive-resistive differentiator, etc. The inverting integrators from this list are examples of useful and desired applications of

5986-491: The power audio amplifiers that drive most sound systems. In a fully symmetrical push-pull circuit two Darlington pairs are connected as emitter followers driving the output from the positive and negative supply: an NPN Darlington pair connected to the positive rail providing current for positive excursions of the output, and a PNP Darlington pair connected to the negative rail providing current for negative excursions. Before good quality PNP power transistors were available,

6068-430: The resistor R {\displaystyle R} constitute an imperfect current source passing current I R {\displaystyle I_{R}} through the load (see Fig. 3 in the source). The INIC acts as a second current source passing "helping" current I − R {\displaystyle I_{-R}} through the load. As a result, the total current flowing through

6150-421: The second current source may be thought as of a new virtual element with dynamically modified impedance. Dual Miller theorem is usually implemented by an arrangement consisting of two voltage sources supplying the grounded impedance Z {\displaystyle Z} through floating impedances (see Fig. 3 ). The combinations of the voltage sources and belonging impedances form the two current sources –

6232-417: The sides of the input and output sources) is revealed below by comparing the two situations – without and with connecting an additional voltage source V 2 {\displaystyle V_{2}} . If V 2 {\displaystyle V_{2}} were zero (there was not a second voltage source or the right end of the element with impedance Z {\displaystyle Z}

6314-448: The two voltage sources as a new composed voltage source, the theorem may be explained by combining the actual element and the second voltage source into a new virtual element with dynamically modified impedance . From this viewpoint, V 2 {\displaystyle V_{2}} is an additional voltage that artificially increases/decreases the voltage drop V z {\displaystyle V_{z}} across

6396-410: The two-port network by its equivalent and by applying the source absorption theorem. Dual Miller theorem actually expresses the fact that connecting a second current source producing proportional current I 2 = K I 1 {\displaystyle I_{2}=KI_{1}} in parallel with the main input source and the impedance element changes the current flowing through it,

6478-408: The voltage and accordingly, the circuit impedance seen from the side of the input source. Depending on the direction, I 2 {\displaystyle I_{2}} acts as a supplemental current source helping or opposing the main current source I 1 {\displaystyle I_{1}} to create voltage across the impedance. The combination of the actual element and

6560-414: The voltage drop V z {\displaystyle V_{z}} across the impedance ( V i = V z − V o < 0 {\displaystyle V_{i}=V_{z}-V_{o}<0} ). A typical application is a negative impedance converter with voltage inversion (VNIC). It is interesting that the circuit input voltage has the same polarity as

6642-604: The voltage drop V z {\displaystyle V_{z}} across the impedance and completely neutralizes it. The circuit behaves as a short connection and a virtual ground appears at the input; so, it should not be driven by a constant voltage source. For this purpose, some circuits are driven by a constant current source or by a real voltage source with internal impedance: current-to-voltage converter (transimpedance amplifier), capacitive integrator (named also current integrator or charge amplifier ), resistance-to-voltage converter (a resistive sensor connected in

6724-403: Was just grounded), the input current flowing through the element would be determined, according to Ohm's law, only by V 1 {\displaystyle V_{1}} and the input impedance of the circuit would be As a second voltage source is included, the input current depends on both the voltages. According to its polarity, V 2 {\displaystyle V_{2}}

#494505