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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
'''[[Félix d'Hérelle]]''' (April 25, 1873 – February 22, 1949) was a French-Canadian scientist, who, with only a high-school education, became one of the most accomplished bacteriologists of his day. d'Herelle is credited with discovering bacteriophages  and inventing phage therapy and modern biological pest control. Subsequently, bacteriophages became the model organisms for the studies that spawned much of our knowledge of molecular genetics.<BR><BR>[[Image:Young d'Herelle.jpg|thumb|200 px|Felix d'Herelle]]
:<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.


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'''.
=== Early Years ===


=== History ===
D'Herelle was born in Montreal, Quebec, the son of French emigrants. His father, 30 years older than his wife, died when Félix was 6 years old. Following his father's death, Félix, his mother and his younger brother Daniel, moved to Paris. From 7 to 17 years of age, d'Herelle attended school in Paris, including the lycee. In the fall of 1891, d'Herelle traveled to Bonn where he attended lectures at the University of Bonn "for several months." Thus, d'Herelle only obtained a high school education and was self-taught in the sciences.  
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 ===
Between 16 and 24, d'Herelle traveled extensively via money given by his mother. When 16 years old, he started to travel through western Europe on bike. When 17, after finishing school, he traveled through South America. Afterwards, he continued his travels through Europe, including Turkey, where he, at 20 years of age, met his wife, Marie Caire.
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>.
At age 24, now father of a daughter, d'Herelle and his family moved to Canada. He built a home laboratory and studied microbiology from books and his own experiments. Through the influence of a friend of his late father, he earned a commission from the Canadian government to study the fermentation and distillation of maple syrup to schnapps. His father's friend shrewdly pointed out that Pasteur "made a good beginning by studying fermentations, so it might be interesting to you, too." He also worked as a medic for a geological expedition to Labrador, even though he had no medical degree or real experience. Together with his brother, he invested almost all his money in a chocolate factory, which soon went bankrupt.  


The input current is:
During this period, d'Herelle published his first scientific paper, "De la formation du Carbone par les vegetaux" in the May 1901 issue of ''Le Naturaliste Canadien''. The paper is noteworthy for two reasons: it shows an exceptional level of scientific development for a self-taught scientist and reveals a broad level of interest, namely the global balance of carbon in nature. However, the claims of the paper were in error, as d'Herelle contended that the results of his experiments indicated that carbon was a compound, not an element.


:<math>i_1 = i_Z+\frac{v_1}{Z_{11}} \ . </math>
=== Guatemala and Mexico ===


The impedance of the circuit at node 1 is:
With his money almost gone and his second daughter born, d'Herelle took a contract with the government of Guatemala as a bacteriologist at the General Hospital in Guatemala City. Some of his work included organizing defenses against the dread diseases of the time: malaria and yellow fever. He also studied a local fungal infection of coffee plants, and discovered that acidifying the soil could serve as an effective treatment. As a side job, he was asked to find a way to make whiskey from bananas. Life in the rough and dangerous environment of the country was hard on his family, but d'Herelle, always adventurer at heart, rather enjoyed working close to "real life", compared to the sterile environments of a "civilized" clinic. He later stated that his scientific path began on this occasion.


:<math>\frac {1}{Z_{1}} = \frac {i_1} {v_1} = \frac {1+A}{Z_\mu} +\frac{1}{Z_{11}} .</math>
In 1907, he accepted an offer from the Mexican government to continue his studies on fermentation. He and his family moved to a sisal plantation near Mérida, Yucatán. Disease struck at him and his family, but in 1909.... ''[[Félix d'Hérelle|(read more)]]''
 
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''.
{{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.}}
 
[[Miller effect|....]]

Revision as of 17:29, 7 August 2011

Félix d'Hérelle (April 25, 1873 – February 22, 1949) was a French-Canadian scientist, who, with only a high-school education, became one of the most accomplished bacteriologists of his day. d'Herelle is credited with discovering bacteriophages and inventing phage therapy and modern biological pest control. Subsequently, bacteriophages became the model organisms for the studies that spawned much of our knowledge of molecular genetics.

Felix d'Herelle

Early Years

D'Herelle was born in Montreal, Quebec, the son of French emigrants. His father, 30 years older than his wife, died when Félix was 6 years old. Following his father's death, Félix, his mother and his younger brother Daniel, moved to Paris. From 7 to 17 years of age, d'Herelle attended school in Paris, including the lycee. In the fall of 1891, d'Herelle traveled to Bonn where he attended lectures at the University of Bonn "for several months." Thus, d'Herelle only obtained a high school education and was self-taught in the sciences.

Between 16 and 24, d'Herelle traveled extensively via money given by his mother. When 16 years old, he started to travel through western Europe on bike. When 17, after finishing school, he traveled through South America. Afterwards, he continued his travels through Europe, including Turkey, where he, at 20 years of age, met his wife, Marie Caire.

At age 24, now father of a daughter, d'Herelle and his family moved to Canada. He built a home laboratory and studied microbiology from books and his own experiments. Through the influence of a friend of his late father, he earned a commission from the Canadian government to study the fermentation and distillation of maple syrup to schnapps. His father's friend shrewdly pointed out that Pasteur "made a good beginning by studying fermentations, so it might be interesting to you, too." He also worked as a medic for a geological expedition to Labrador, even though he had no medical degree or real experience. Together with his brother, he invested almost all his money in a chocolate factory, which soon went bankrupt.

During this period, d'Herelle published his first scientific paper, "De la formation du Carbone par les vegetaux" in the May 1901 issue of Le Naturaliste Canadien. The paper is noteworthy for two reasons: it shows an exceptional level of scientific development for a self-taught scientist and reveals a broad level of interest, namely the global balance of carbon in nature. However, the claims of the paper were in error, as d'Herelle contended that the results of his experiments indicated that carbon was a compound, not an element.

Guatemala and Mexico

With his money almost gone and his second daughter born, d'Herelle took a contract with the government of Guatemala as a bacteriologist at the General Hospital in Guatemala City. Some of his work included organizing defenses against the dread diseases of the time: malaria and yellow fever. He also studied a local fungal infection of coffee plants, and discovered that acidifying the soil could serve as an effective treatment. As a side job, he was asked to find a way to make whiskey from bananas. Life in the rough and dangerous environment of the country was hard on his family, but d'Herelle, always adventurer at heart, rather enjoyed working close to "real life", compared to the sterile environments of a "civilized" clinic. He later stated that his scientific path began on this occasion.

In 1907, he accepted an offer from the Mexican government to continue his studies on fermentation. He and his family moved to a sisal plantation near Mérida, Yucatán. Disease struck at him and his family, but in 1909.... (read more)