0195136047.pdf

16 CIRCUIT CONCEPTS

Notice thatωtrather thantis chosen as the variable for convenience;ω= 2 πf = 2 π/T;

and integration is performed over three discrete intervals because of the discontinuous current

function. Sincei=0 for 0≤ωt < π/3 andπ≤ωt≤ 2 π,

Irms=

√

√

√

√

√

1

2 π

∫π

π/ 3

102 sin^2 ωt d(ωt)= 4 .49 A

Iav=

1

2 π

∫π

π/ 3

10 sinωt d(ωt)= 2 .39 A

Note that the base is the entire period 2π, even though the current is zero for a substantial part of

the period.

1.2 Lumped-Circuit Elements

Electriccircuitsornetworksare formed by interconnecting various devices, sources, and com-

ponents. Although the effects of each element (such as heating effects, electric-field effects,

or magnetic-field effects) are distributed throughout space, one often lumps them together as

lumped elements. Thepassivecomponents are theresistance Rrepresenting the heating effect,

thecapacitance Crepresenting the electric-field effect, and theinductance Lrepresenting the

magnetic-field effect. Their characteristics will be presented in this section. The capacitor models

the relation between voltage and current due to changes in the accumulation of electric charge, and

the inductor models the relation due to changes in magnetic flux linkages, as will be seen later.

While these phenomena are generally distributed throughout an electric circuit, under certain

conditions they can be considered to be concentrated at certain points and can therefore be

represented by lumped parameters.

Resistance

Anideal resistoris a circuit element with the property that the current through it is linearly

proportional to the potential difference across its terminals,

i=v/R=Gv,orv=iR (1.2.1)

which is known asOhm’s law, published in 1827.Ris known as the resistance of the resistor

with the SI unit ofohms(), andGis the reciprocal of resistance calledconductance, with the SI

unit ofsiemens(S). The circuit symbols of fixed and variable resistors are shown in Figure 1.2.1,

along with an illustration of Ohm’s law. Most resistors used in practice are good approximations

tolinearresistors for large ranges of current, and theiri–vcharacteristic (current versus voltage

plot) is a straight line.

The value of resistance is determined mainly by the physical dimensions and theresistivity

ρof the material of which the resistor is composed. For a bar of resistive material of lengthland

cross-sectional areaAthe resistance is given by

R=

ρl

A

=

l

σA

(1.2.2)

whereρis the resistivity of the material in ohm-meters(·m), andσis theconductivityof the

material in S/m, which is the reciprocal of the resistivity. Metal wires are often considered as ideal