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3.17 Dependent sources

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Dependent sources.

A dependent source is either a voltage or current source whose value is proportional to someother voltage or current in the circuit. Thus, there are four different kinds of dependent sources; to describe an op-amp, weneed a voltage-dependent voltage source. However, the standard circuit-theoretical model for a transistor contains a current-dependent current source. Dependent sources do not serve as inputs to a circuitlike independent sources. They are used to model active circuits : those containing electronic elements. The RLC circuits we have been considering so far areknown as passive circuits .

Dependent sources

Of the four possible dependent sources, depicted is a voltage-dependent voltage source in the context of a genericcircuit.

[link] shows the circuit symbol for the op-amp and its equivalentcircuit in terms of a voltage-dependent voltage source.

Op-amp

The op-amp has four terminals to which connections can bemade. Inputs attach to nodes a and b , and the output is node c . As the circuit model on the right shows, the op-amp serves as an amplifier for the difference ofthe input node voltages.

Here, the output voltage equals an amplified version of thedifference of node voltages appearing across its inputs. The dependent source model portrays how the op-amp works quitewell. As in most active circuit schematics, the power supply is not shown, but must be present for the circuit model to beaccurate. Most operational amplifiers require both positive and negative supply voltages for proper operation.

Because dependent sources cannot be described as impedances, and because the dependent variable cannot "disappear" when you applyparallel/series combining rules, circuit simplifications such as current and voltage divider should not be applied in mostcases. Analysis of circuits containing dependent sources essentially requires use of formal methods, like the node method . Using the node method for such circuits is notdifficult, with node voltages defined across the source treated as if they were known (as with independent sources). Considerthe circuit shown on the top in [link] .

Feedback op-amp

The top circuit depicts an op-amp in a feedback amplifier configuration. On the bottom is the equivalent circuit, andintegrates the op-amp circuit model into the circuit.

Note that the op-amp is placed in the circuit "upside-down," with its inverting input at the top and serving as the onlyinput. As we explore op-amps in more detail in the next section, this configuration will appear again and again and its usefulness demonstrated. To determine how the output voltage is relatedto the input voltage, we apply the node method. Only two node voltages— v and v out —need be defined; the remaining nodes are across sources or serve as the reference. The node equations are

v v in R v R in v v out R F 0
v out G v R out v out v R F v out R L 0
Note that no special considerations were used in applying the node method to this dependent-source circuit. Solving these tolearn how v out relates to v in yields
R F R out R out G R F 1 R out 1 R in 1 R L 1 R 1 R in 1 R F 1 R F v out 1 R v in
This expression represents the general input-output relation for this circuit, known as the standard feedback configuration . Once we learn more about op-amps , in particular what its typical element values are, the expression will simplifygreatly. Do note that the units check, and that the parameter G of the dependent source is a dimensionless gain.

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When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
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Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
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Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
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Answer
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suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
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Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
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