Lesson 3 Ideal Elements: Resistor，Inductor，and Capacitor

Our purpose in this text is to introduce ideal network elements that represent energy dissipation，magnetic field energy storage，and electric field energy storage. Before presenting the details，we must discuss several general features of this procedure. The network description of an electrical element is a statement about the voltage（s）and current（s）of that element.The task is therefore one of determining the proper voltage-current relationship that corresponds to the effect or behavior that the ideal element is supposed to represent.Each ideal element is a definition in which only essential effects are approximated.However，definitions are not unique;the merits of each must be judged in terms of the degree to which the behavior of ideal element corresponds to that of the real device. Moreover，since all network analysis and design takes place in a mathematical framework，an essential feature of a useful definition is mathematical simplicity.

Our goal at this point is to present a logical development of ideal elements，each one having a behavior that corresponds to the effects to be approximated and involving a mathematically simple voltage-current relation.The existence of practical devices whose response is well approximated by these ideal elements can obviously be verified only by students from experiences with real devices and systems as encountered，for example，in laboratories.Approaching our task from the viewpoint of energy allows us to introduce resistors，inductors，and capacitors in a unified manner without reference to the detailed properties of the electric and magnetic fields that actually govern the operation of real devices.

Ideal elements that represent the dissipation of energy is called resistors，symbolized in Fig.3.1. In defining these elements，we must satisfy the key condition implied by energy dissipation，that is，the one-way transfer of energy from the system to which resistors are connected. Mathematically，this means that p≥0，where p is the power flow into the resistor;We define the resistor by giving the relation for calculating the power in terms of the current，that is，p=f（i）.Note that since p is never negative in a resistor，the formula for p must not depend on the algebraic sign of i，that is，the reference direction for i.Otherwise，a different choice of reference direction would change the sign of p and violate the condition p≥0.These considerations lead to a simple definition for received power in a resistor which satisfies these requirements as

The proportionality constant R is called resistance;its SI unit is ohm.

It is clear from Fig.3.1 that power may also be expressed as p =vi. This fact in conjunction with Eq.（3.1）allows us to write the following:

From this it follows that

This last equation is the familiar Ohm's law，a relation that many conducting materials follow quite closely.Note that Eq.（3.2）is valid only for the choice of current reference in the direction of the voltage drop（Fig.3.1）.If the current reference is in the voltage-rise direction，then v=?Ri.

The ideal inductive element is called the inductor，symbolized in Fig.3.2.The purpose of the inductor is to represent the storage of magnetic field energy.It is defined by giving the relation for stored energy in terms of the current，Wm=f（i）.In making this definition，we are guided by the earlier knowledge concerning the dependence of magnetic field energy on current.In particular，the condition Wm＞0 must be satisfied.Accordingly，the inductor is defined through the following relation for stored magnetic field energy.

In this formula L is called inductance;its SI unit is henry.An important result may be derived from Eq.（3.4）by noting that the power flow into inductor equals the time rate of the increase of stored magnetic energy W0.

However，since p=vi，it follows from Eq.（3.5）that

It is interesting to observe that since the energy in magnetic field can change only when the corresponding current changes，the power flow to inductor will be zero whenever di/dt=0.Note that Eq.（3.6）is valid only for the choice of reference direction and polarity in Fig.3.2 and that a minus sign must be introduced if either reference is reversed.The symbol for inductor（Fig.3.2）reflects the fact that a coil of wire tends to behave like an inductor and has a voltage-current relation that is often closely approximated by Eq.（3.6）.

To represent electric field stored energy，we define an ideal element called capacitor.The symbol for the capacitor is that of Fig. 3.3. Since the electric field strength，and therefore the electric field stored energy，is related to the voltage，we expect the stored energy in a capacitor to be a function of the voltage.However，the stored energy must not depend on the polarity of the voltage.Accordingly，we define the ideal capacitor in terms of

where We，is the electric field stored energy and the constant C is called capacitance，having an SI unit known as farad.The relation between voltage and current in a capacitor is obtained as follows:

Equation（3.8）yields the following equation for the current in terms of the voltage:

which applies for the choice of references in Fig.3.3.Note that a comparison of Eq.（3.4）with Eq.（3.7）and Eq.（3.6）with Eq.（3.9）suggests strong similarities between inductors and capacitors. Indeed，if v in Eq.（3.9）is replaced by i，i is replaced by v，and C is replaced by L，the result becomes Eq.（3.6）.When two elements have a symmetrical relation of this type，they are said to be the duals of one another.