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In electronics, the '''[[Miller effect]]''' is the increase in the equivalent input capacitance of an inverting voltage [[amplifier]] due to a capacitance connected between two gain-related nodes, one on the input side of an amplifier and the other the output side. The amplified input capacitance due to the Miller effect, called the '''Miller capacitance''' ''C<sub>M</sub>'', is given by
{{:{{FeaturedArticleTitle}}}}
:<math>C_{M}=C (1-A)\ ,</math>
<small>
where ''A''  is the voltage gain between the two nodes at either end of the coupling capacitance, which is a negative number because the amplifier is ''inverting'', and ''C'' is the coupling capacitance.
==Footnotes==
 
{{reflist|2}}
Although the term ''Miller effect'' normally refers to capacitance, the Miller effect applies to any impedance connected between two nodes exhibiting gain. These properties of the Miller effect are generalized in '''Miller's theorem'''.
</small>
 
=== History ===
The Miller effect is named after John Milton Miller. When Miller published his work in 1920, he was working on vacuum tube triodes, however the same theory applies to more modern devices such as bipolar transistors and MOSFETs.
 
=== Derivation ===
{{Image|Miller effect.PNG|right|350px|These two circuits are equivalent. Arrows indicate current flow. Notice the polarity of the dependent voltage source is flipped, to correspond with an ''inverting'' amplifier.}}
Consider a voltage amplifier of gain −''A'' with an impedance ''Z<sub>&mu;</sub>'' connected between its input and output stages. The input signal is provided by a Thévenin voltage source representing the driving stage. The voltage at the input end (node 1) of the coupling impedance is ''v<sub>1</sub>'', and at the output end  −''Av<sub>1</sub>''.  The current through ''Z<sub>&mu;</sub>'' according to Ohm's law is given by:
 
:<math>i_Z =  \frac{v_1 - (- A)v_1}{Z_\mu} = \frac{v_1}{ Z_\mu / (1+A)}</math>.
 
The input current is:
 
:<math>i_1 = i_Z+\frac{v_1}{Z_{11}} \ . </math>
 
The impedance of the circuit at node 1 is:
 
:<math>\frac {1}{Z_{1}} = \frac {i_1} {v_1} = \frac {1+A}{Z_\mu} +\frac{1}{Z_{11}} .</math>
 
This same input impedance is found if the input stage simply is decoupled from the output stage, and the reduced impedance ''{{nowrap|Z<sub>&mu;</sub> / (1+A)}}'' is substituted in parallel with ''Z<sub>11</sub>''. Of course, if the input stage is decoupled, no current reaches the output stage. To fix that problem, a dependent current source is attached to the second stage to provide the correct current to the output circuit, as shown in the lower figure. This decoupling scenario is the basis for ''Miller's theorem'', which replaces the current source on the output side by addition of a shunt impedance in the output circuit that draws the same current. The striking prediction that a coupling impedance ''Z<sub>&mu;</sub>'' reduces input impedance by an amount equivalent to shunting the input with the reduced impedance ''{{nowrap|Z<sub>&mu;</sub> / (1+A)}}'' is called the ''Miller effect''.
 
[[Miller effect|....]]

Latest revision as of 10:19, 11 September 2020

After decades of failure to slow the rising global consumption of coal, oil and gas,[1] many countries have proceeded as of 2024 to reconsider nuclear power in order to lower the demand for fossil fuels.[2] Wind and solar power alone, without large-scale storage for these intermittent sources, are unlikely to meet the world's needs for reliable energy.[3][4][5] See Figures 1 and 2 on the magnitude of the world energy challenge.

Nuclear power plants that use nuclear reactors to create electricity could provide the abundant, zero-carbon, dispatchable[6] energy needed for a low-carbon future, but not by simply building more of what we already have. New innovative designs for nuclear reactors are needed to avoid the problems of the past.

(CC) Image: Geoff Russell
Fig.1 Electricity consumption may soon double, mostly from coal-fired power plants in the developing world.[7]

Issues Confronting the Nuclear Industry

New reactor designers have sought to address issues that have prevented the acceptance of nuclear power, including safety, waste management, weapons proliferation, and cost. This article will summarize the questions that have been raised and the criteria that have been established for evaluating these designs. Answers to these questions will be provided by the designers of these reactors in the articles on their designs. Further debate will be provided in the Discussion and the Debate Guide pages of those articles.

Footnotes
  1. ↑ Global Energy Growth by Our World In Data
  2. ↑ Public figures who have reconsidered their stance on nuclear power are listed on the External Links tab of this article.
  3. ↑ Pumped storage is currently the most economical way to store electricity, but it requires a large reservoir on a nearby hill or in an abandoned mine. Li-ion battery systems at $500 per KWh are not practical for utility-scale storage. See Energy Storage for a summary of other alternatives.
  4. ↑ Utilities that include wind and solar power in their grid must have non-intermittent generating capacity (typically fossil fuels) to handle maximum demand for several days. They can save on fuel, but the cost of the plant is the same with or without intermittent sources.
  5. ↑ Mark Jacobson believes that long-distance transmission lines can provide an alternative to costly storage. See the bibliography for more on this proposal and the critique by Christopher Clack.
  6. ↑ "Load following" is the term used by utilities, and is important when there is a lot of wind and solar on the grid. Some reactors are not able to do this.
  7. ↑ Fig.1.3 in Devanney "Why Nuclear Power has been a Flop"

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