CZ:Featured article/Current: Difference between revisions

Revision as of 18:36, 29 July 2011

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_M, is given by

C_{M}=C(1-A)\ ,

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.

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.

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

(PD) Image: John R. Brews
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_μ 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 v₁, and at the output end −Av₁. The current through Z_μ according to Ohm's law is given by:

i_{Z}={\frac {v_{1}-(-A)v_{1}}{Z_{\mu }}}={\frac {v_{1}}{Z_{\mu }/(1+A)}}

.

The input current is:

i_{1}=i_{Z}+{\frac {v_{1}}{Z_{11}}}\ .

The impedance of the circuit at node 1 is:

{\frac {1}{Z_{1}}}={\frac {i_{1}}{v_{1}}}={\frac {1+A}{Z_{\mu }}}+{\frac {1}{Z_{11}}}.

This same input impedance is found if the input stage simply is decoupled from the output stage, and the reduced impedance Z_μ / (1+A) is substituted in parallel with Z₁₁. 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_μ reduces input impedance by an amount equivalent to shunting the input with the reduced impedance Z_μ / (1+A) is called the Miller effect.

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-'''[[Gut-brain signalling]]''' describes the interaction between the gastrointestinal tract and the brain, and how secretion of varying hormones from different areas of the body causes appetite-enhancing and satiety signals to be sent to the brain.  The hormones that have been most intensely studied are: ghrelin, obestatin, cholecystokinin (CCK), GLP-1, peptide YY (PYY) and insulin which all play major roles in appetite regulation.  The vagus nerve is also a key mediator of regulation, and all of these inputs are processed by areas in the brain such as the hypothalamus and the nucleus tractus solitarii (NTS).
+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
+:<math>C_{M}=C (1-A)\ ,</math>
+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.
-===Anorexic Signals===
+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'''.
-{{Image|diagram 3.jpg|center|350px|''Gut-Brain signaling Pathways'' Proteins and hormones activate brain pathways in different ways, either by eventual vagal activation or through peripheral circulation. The nucleus tractus solitarii and the arcuate nucleus are then activated. }}
-'''Cholecystokinin''' (CCK) is a peptide hormone synthesised  by L-cells in the mucosal epithelium of the duodenum, and secreted in response to the presence of partly digested lipids and protein]]s. CCK inhibits gastric emptying and stimulates the release of digestive enzymes from the pancreas and bile from the gall bladder by acting at CCK-A receptors (mainly found in the periphery but also found in some areas of the CNS). Because gastric emptying is inhibited, the partly digested lipids and proteins are exposed to the digestive enzymes and bile so are further broken down. As the lipids and proteins are broken down, CCK secretion declines.
-CCK acts as a ‘gatekeeper’ for the response of other gut-brain signalling hormones on the afferent vagal neurons. At low levels (after fasting), CCK stimulates the expression of receptors associated with the stimulation of food intake, including receptors for melanin concentrating hormone (MCH)-1 and cannabinoid CB1 receptors. At high levels (after food consumption), MCH-1 and CB1 receptors are down- regulated. Therefore CCK, at a high or low concentration, can affect how afferent vagal neurons respond to other neurohormones.
+== 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.
-In rats, CCK inhibits food intake in younger individuals more effectively than in older individuals. It also has a greater effect in males than in females.
+== 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:
-'''Glucagon-like peptide-1''' (GLP-1) is a hormone secreted from L-cells in the mucosal epithelium of the duodenum and small intestine. It is derived from the ''pro-glucagon'' gene, and is secreted into the circulation in response to the presence of nutrients. It acts at the pancreas, where it stimulates insulin secretion and suppresses glucagon secretion. It also increases insulin sensitivity. GLP-1 also activates anorexigenic neurons in the arcuate nucleus via the caudal brainstem. Activation of these  neurons induces satiety and decreases food intake/hunger. It also decreases gastric emptying, so adds to the feeling of being ‘full’. At higher concentrations, GLP-1 causes nausea, and can induce conditioned taste aversion, where the brain associates the taste of a certain food with being toxic (usually after an individual consumes a food that had made them sick).
+:<math>i_Z =   \frac{v_1 - (- A)v_1}{Z_\mu} = \frac{v_1}{ Z_\mu / (1+A)}</math>.
-[[Gut-brain signalling|.....]]
+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|....]]

CZ:Featured article/Current: Difference between revisions

Revision as of 18:36, 29 July 2011

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