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+ v i Power flow (a) Conceptual representation
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(c) Physical representation
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Figure 210 Various representations of an electrical system
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In the analysis of electrical circuits, we choose to represent the physical reality of Figure 210(c) by means of the approximation provided by ideal circuit elements, as depicted in Figure 210(b) Ideal Current Sources An ideal current source is a device that can generate a prescribed current independent of the circuit it is connected to To do so, it must be able to generate an arbitrary voltage across its terminals Figure 211 depicts the symbol used to represent ideal current sources By analogy with the de nition of the ideal voltage source stated in the previous section, we write:
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iS, IS
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An ideal current source provides a prescribed current to any circuit connected to it The voltage generated by the source is determined by the circuit connected to it
Figure 211 Symbol for ideal current source
The same uppercase and lowercase convention used for voltage sources will be employed in denoting current sources Dependent (Controlled) Sources The sources described so far have the capability of generating a prescribed voltage or current independent of any other element within the circuit Thus, they are termed independent sources There exists another category of sources, however, whose output (current or voltage) is a function of some other voltage or current in a circuit These are called dependent (or controlled) sources A different symbol, in the shape of a diamond, is used to represent dependent sources and to distinguish them from independent sources The symbols typically used to represent dependent sources are depicted in Figure 212; the table illustrates the relationship between the source voltage or current and the voltage or current it depends on vx or ix , respectively which can be any voltage or current in the circuit
Source type Voltage controlled voltage source (VCVS) vS
Relationship vS = Avx vS = Aix iS = Avx iS = Aix
Current controlled voltage source (CCVS) Voltage controlled current source (VCCS) Current controlled current source (CCCS)
Figure 212 Symbols for dependent sources
Dependent sources are very useful in describing certain types of electronic circuits You will encounter dependent sources again in s 9, 10, and 12, when electronic ampli ers are discussed
2
Fundamentals of Electric Circuits
ELECTRIC POWER AND SIGN CONVENTION
The de nition of voltage as work per unit charge lends itself very conveniently to the introduction of power Recall that power is de ned as the work done per unit time Thus, the power, P , either generated or dissipated by a circuit element can be represented by the following relationship: Power = Thus, Work Work Charge = = Voltage Current Time Charge Time (29)
The electrical power generated by an active element, or that dissipated or stored by a passive element, is equal to the product of the voltage across the element and the current owing through it
P = VI
(210)
i + v
Source
Power dissipated = = v ( i) = ( v)i = vi Power generated = vi i + Load v Power dissipated = vi Power generated = = v ( i) = ( v)i = vi
Figure 213 The passive sign convention
It is easy to verify that the units of voltage (joules/coulomb) times current (coulombs/second) are indeed those of power (joules/second, or watts) It is important to realize that, just like voltage, power is a signed quantity, and that it is necessary to make a distinction between positive and negative power This distinction can be understood with reference to Figure 213, in which a source and a load are shown side by side The polarity of the voltage across the source and the direction of the current through it indicate that the voltage source is doing work in moving charge from a lower potential to a higher potential On the other hand, the load is dissipating energy, because the direction of the current indicates that charge is being displaced from a higher potential to a lower potential To avoid confusion with regard to the sign of power, the electrical engineering community uniformly adopts the passive sign convention, which simply states that the power dissipated by a load is a positive quantity (or, conversely, that the power generated by a source is a positive quantity) Another way of phrasing the same concept is to state that if current ows from a higher to a lower voltage (+ to ), the power is dissipated and will be a positive quantity It is important to note also that the actual numerical values of voltages and currents do not matter: once the proper reference directions have been established and the passive sign convention has been applied consistently, the answer will be correct regardless of the reference direction chosen The following examples illustrate this point
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