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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>
 
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|center|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

Nuclear weapons proliferation is one of the four big issues that have held back worldwide deployment of peaceful nuclear power. This article will address the proliferation questions raised in Nuclear power reconsidered.

As of 2022, countries with nuclear weapons have followed one or both of two paths in producing fissile materials for nuclear weapons: enrichment of uranium to very high fractions of U-235, or extraction of fissile plutonium (Pu-239) from irradiated uranium nuclear reactor fuel. The US forged the way on both paths during its World War II Manhattan Project. The fundamental aspects of both paths are well understood, but both are technically challenging. Even relatively poor countries can be successful if they have sufficient motivation, financial investment, and, in some cases, direct or illicit assistance from more technologically advanced countries.

The International Non-proliferation Regime

The International Atomic Energy Agency (IAEA) has a vigorous program to prevent additional countries from acquiring nuclear weapons. The Treaty on the Non-Proliferation of Nuclear Weapons (NPT) is the cornerstone arrangement under which strategic rivals can trust, by independent international verification, that their rivals are not developing a nuclear weapons threat. The large expense of weapons programs makes it very unlikely that a country would start its own nuclear weapons program, if it knows that its rivals are not so engaged. With some notable and worrying exceptions, this program has been largely successful.

Paths to the Bomb

It is frequently claimed that building a civil nuclear power program adds to the weapons proliferation risk. There is an overlap in the two distinct technologies, after all. To build a bomb, one needs Highly Enriched Uranium (HEU) or weapons-grade plutonium (Pu-239). Existing reactors running on Low Enriched Uranium (LEU, under 5% U-235) or advanced reactors running on High Assay LEU (HALEU,up to 20% U-235) use the same technology that can enrich uranium to very high levels, but configured differently. Enrichment levels and centrifuge configurations can be monitored using remote cameras, on-site inspections, and installed instrumentation -- hence the value of international inspections by the IAEA. Using commercial power reactors as a weapons plutonium source is an extremely ineffective, slow, expensive, and easily detectable way to produce Pu. Besides the nuclear physics issues, refueling pressurized water reactors is both time-consuming and obvious to outside observers. That is why the US and other countries developed specialized Pu production reactors and/or uranium enrichment to produce fissile cores for nuclear weapons.

Future Threats and Barriers

Minimizing the risk of future proliferation in states that want to buy nuclear reactors or fuel might require one or more barriers:
1) Insisting on full transparency for all nuclear activities in buyer states, including monitoring and inspections by the International Atomic Energy Agency (IAEA).
2) Limiting fuel processing to just a few supplier states that already have weapons or are approved by the IAEA.
3) Ensuring that fuel at any stage after initial fabrication has an isotopic composition unsuitable for weapons. "Spiking" the initial fuel with non-fissile isotopes, if necessary.
4) Limiting the types of reactors deployed to buyer states. In general, breeders are less secure than burners. Sealed reactor modules are more secure than reactors with on-site fuel processing.
5) Providing incentives and assurances for buyer states to go along with all of the above.
6) Application of diplomatic pressure, sanctions, and other economic measures to non-compliant states.
7) Agreement that any reactor declared rogue by the IAEA will be "fair game" for any state feeling threatened.

Footnotes